WO2021207748A2 - Methods and apparatus for channel reconstruction in intelligent surface aided communications - Google Patents

Abstract

Methods and apparatus are provided for reconstructing a communication channel that includes an intelligent reflecting surface (IRS)-aided reflective channel formed by an IRS and a first and second devices and a direct channel between the first and second devices. In a TDD system, the first device receives a pilot signal from the second device, generates sparsity information or subspace information of the communication channel, and reconstructs the communication channel based on the received pilot signal, the IRS' reflective pattern and the sparsity information or subspace information. In a FDD system, the second device receives a pilot signal from the first device in the communication channel, and sends signal strength information of the received pilot signal to the first device. The first device reconstructs the communication channel based on the signal strength information, the IRS' reflective pattern and subspace information of the communication channel.

Description

Methods and Apparatus for Channel Reconstruction in Intelligent Surface Aided Communications

CROSS-REFERENCE TO RELATED APPLIATIONS

This patent application claims priority to U.S. Provisional Application No. 63/065,257, filed on August 13, 2020 and entitled “Methods and Apparatus for Channel Reconstruction in Intelligent Surface Aided Communications,” which is hereby incorporated by reference herein as if reproduced in its entirety.

TECHNICAL FIELD

The present disclosure relates generally to wireless communications, and in particular embodiments, to techniques and mechanisms for channel reconstruction in intelligent surface aided communications.

BACKGROUND

Intelligent reflecting surface (IRS) has recently emerged as a promising cost-effective technology for enhancing the capacity and energy- efficiency in future wireless communication systems, and has received growing interest from the wireless research community. An IRS includes many low-cost passive antenna elements that can smartly reflect the impinging electromagnetic waves for performance enhancement. However, channel estimation is challenging for the IRS-aided wireless communications due to the lack of sensing capability in the passive IRS elements. It is desirable to develop mechanisms for channel estimation or reconstruction in IRS-aided wireless communications.

SUMMARY OF THE INVENTION

Technical advantages are generally achieved, by embodiments of this disclosure which describe methods and apparatuses for channel reconstruction in intelligent surface aided communications.

According to one aspect of the presort disclosure, a method is provided that includes: receiving, by a first communication device, a first pilot signal sent by a second communication device to the first communication device in a first time duration in a first communication channel of a time division duplex (TDD) system; generating, by the first communication device, sparsity information of the first communication channel, the sparsity information comprising of a set of array beamforming directions, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel including a first IRS channel between an IRS and the second communication device and a second IRS channel betw een the IRS and the first communication device, the first communication channel further comprising a direct channel between the first communication device and the second communication device, the IRS being configured to reflect signals incident to the IRS; performing, by the first communication device, channel reconstruction of the first communication channel based on the received first pilot signal, the sparsity information, a location of tire IRS, a location of the second communication device and a first reflective patter of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating, by the first communication device with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

Optionally, in any of the preceding aspects, the method further includes: receiving, by the first communication device, a second pilot signal sent by tire second communication device to the first communication device in a second time duration in the first communication channel; and wherein performing the channel reconstruction of the first communication channel comprises: reconstructing, by the first communication device, the first communication channel based on the received first pilot signal, tire received second pilot signal, the sparsity information, the location of the IRS, the location of the second communication device, tire first reflective pattern of the IRS for the first time duration, and a second reflective pattern of tire IRS for the second time duration.

Optionally, in any of the preceding aspects, the method further includes: determining, by tire first communication device, a reflective patter of the IRS based on the reconstructed- channel information of first communication channel.

Optionally, in any of the preceding aspects, at least one of the first reflective patter, the second reflective pattern or the reflective pattern of the IRS comprises phase shifts of reflective elements of the IRS.

Optionally, in any of the preceding aspects, the first reflective pattern and the second reflective pattern are different from each other.

Optionally, in any of the preceding aspects, performing the channel reconstruction of the first communication channel comprises: determining, by the first communication device, a line of sight (LOS) steering vector of the IRS aided reflective channel.

Optionally, in any of the preceding aspects, the first communication channel is represented as:

represents the first communication channel, h and g represent the first IRS channel and the second IRS channel of the IRS aided reflective channel, h_dir represents the direct channel, T denotes transpose, Θ denotes Hadamard product, Ĝ represents a dictionary matrix whose columns correspond to a plurality of array beamforming directions, and t represents a combining vector, the channel reconstruction of the first communication channel comprising: estimating, by the first communication device, the combining vector t based on the first received pilot signal.

Optionally, in any of the preceding aspects, the method further includes determining the set of array beamforming directions from the plurality of array beamforming directions. Optionally, in any of the preceding aspects, the first communication device is an access point (AP) and the second communication device is a user equipment (UE), or the second communication device is an AP and the first communication device is a UE.

Optionally, in any of the preceding aspects, the reconstructed-channel information of the first communication channel comprises a channel model of the first communication channel.

According to another aspect of the present disclosure, a method is provided that includes: receiving, by a first communication device, a first pilot signal sent by a second communication device to the first communication device in a first time duration in a first communication channel of a TDD system; determining, by the first communication device, subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical data about channel measurement and reconstruction of the first communication channel accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the first communication device, the first communication channel further comprising a direct channel between the first communication device and the second communication device, and the IRS being configured to reflect signals incident to the IRS; performing, by the first communication device, channel reconstruction of the first communication channel based on the received first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating by the first communication device with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

Optionally, in any of the preceding aspects, the method further includes: receiving, by the first communication device, a second pilot signal sent by the second communication device to the first communication device in a second time duration in the first communication channel; and wherein performing the channel reconstruction of the first communication channel comprises: reconstructing, by the first communication device, the first communication channel based on the received first pilot signal, the received second pilot signal, the subspace information, the first reflective patter of the IRS for the first time duration, and a second reflective pattern of the IRS for tire second time duration. Optionally, in any of the preceding aspects, the method further includes: determining, by the first communication device, a reflective patter of the IRS based on the reconstructed- channel information of the first communication channel.

Optionally, in any of the preceding aspects, at least one of the first reflective pattern, the second reflective patter or the reflective pattern of the IRS comprises phase shifts of reflective elements of the IRS.

Optionally, in any of the preceding aspects, tire first reflective pattern and the second reflective pattern are different from each other.

Optionally, in any of the preceding aspects, the first communication channel is represented as: [(h Θ g)^T, h_dlr]^T = Uζ, wherein[(h Θ g)^T, h_dlr]^T represents the first communication channel, h and g represent the first IRS channel and the second IRS channel of the IRS aided reflective channel, h_dir represents the direct channel, T denotes transpose, Θ denotes Hadamard product, U represents a (N +1)xs matrix whose column space equals the dominant subspace of the covariance matrix of the first communication channel [(h Θ g)^T, h_dlr]^T, N is the number of reflective dements of the IRS, s represents the dimension of the dominant subspaoe, and ζ represents a combining vector; and the channd reconstruction of the first communication channd comprising: estimating, by the first communication device, the combining vector ζ based on the first received pilot signal. Optionally, in any of the preceding aspects, the first communication device is an access point (AP) and the second communication device is a user equipment (UE), or the second communication device is an AP and the first communication device is a UE. Optionally, in any of the preceding aspects, the reconstructed-channel information of the first communication channel comprises a channel model of the first communication channel.

According to another aspect of the present disclosure, a method is provided that includes: sending, by an access point (AP) to a user equipment (UE), a first pilot signal in a first time duration in a first communication channel of a frequency division duplex (FDD) system, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel including a first IRS channel between an IRS and the UE and a second IRS channel between the IRS and the AP, the first communication channel further comprising a direct channel between the AP and the UE, the IRS being configured to reflect signals incident to the IRS; receiving, by the AP from the UE, information of signal strength of a received first pilot signal, the received first pilot signal being the first pilot signal received by the UE through the first communication channel; determining, by the AP, subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical received signal strength measurement data of signals received by the UE through the first communication channel, the historical received signal strength measurement data being accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix; performing, by the AP, channel reconstruction of the first communication channel based on the information of the signal strength of the received first pilot signal, the subspace information, and a first reflective patter of the IRS for the first time duration to generate reconstructed- channel information of the first communication channel; and communicating by the AP with the UE in the first communication channel based on the reconstracted-channel information of the first communication channel. Optionally, in any of the preceding aspects, the subspace information is further determined based on historical channel reconstruction data of the first communication channel.

Optionally, in any of the preceding aspects, the method further includes: sending, by the AP to the UE, a second pilot signal in a second time duration in the first communication channel; receiving, by the AP from the UE, information of signal strength of a received second pilot signal, the received second pilot signal being the second pilot signal received by the UE through the first communication channel; performing the channel reconstruction of the first communication channel comprising: reconstructing, by the AP, the first communication channel based on the signal strength information of the received first pilot signal and the received second pilot signal, the subspace information, the first reflective pattern of the IRS for the first time duration, and a second reflective pattern of the IRS for the second time duration.

Optionally, in any of the preceding aspects, the first reflective patter, the second reflective patter or the reflective pattern of the IRS comprises phase shifts of reflective elements of the IRS.

Optionally, in any of the preceding aspects, the first reflective pattern and the second reflective patter are different from each other.

Optionally, in any of the preceding aspects, the method further includes: determining, by the first communication device, a reflective patter of the IRS based on the reconstructed- channel information of first communication channel.

Optionally, in any of the preceding aspects, the first communication channel is represented as: [(h Θ g)^T, h_dlr]^T = Uζ , wherein [(h Θ g)^T, h_dlr]^T represents the first communication channel, g and h represent the first IRS channel and the second IRS channel of the IRS-aided reflective channel, ha, represents the second direct channel, T denotes transpose, Θ denotes Hadamard product, U represents a (N+1)xs matrix whose column space equals the dominant subspace of the covariance matrix of the first communication channel [(h Θ g)^T, h_dlr]^T _> N is the number of reflective elements of the IRS, s represents the dimension of the dominant subspace, and ζ represents a combining vector; and wherein performing the channel reconstruction of the first communication channel further comprises: estimating, by the AP, the combining factor ζ based on the information of the signal strength of the received first pilot signal.

Optionally, in any of the preceding aspects, the information of the signal strength of the received first pilot signal comprises quantized signal strength of the received first pilot signal.

Optionally, in any of the preceding aspects, the information of the signal strength of the received first pilot signal comprises quantized average received signal strength of the received first pilot signal.

According to another aspect of the present disclosure, an apparatus is provided that includes: a non-transitory memory storage comprising instructions, and one or more processors in communication with the memory storage, wherein the instructions, when executed by the one or more processors, cause the apparatus to perform a method in any of the preceding aspects. According to another aspect of the present disclosure, a non-transitory computer-readable media storing computer instructions that when executed by one or more processors of an apparatus, cause the apparatus to perform a method in any of the preceding aspects.

The above aspects of the present disclosure have advantages of providing channel reconstruction for IRS aided communications with improved channel reconstruction accuracy, reduced channel reconstruction complexity and reduced pilot overheads.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present disclosure, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

Figure 1 is a diagram of an embodiment w ireless communications network;

Figure 2 is a diagram of an example communications system, providing mathematical expressions of signals transmitted in the communications system and a channel model; Figure 3 is a diagram of an embodiment network for IRS aided communications; Figure 4 is a diagram of an embodiment channel model of a netw ork for IRS aided communications;

Figure 5 are graphs showing simulation results of data communication rates varying with hyper-parameters used in channel reconstruction;

Figure 6 are graphs showing simulation results of data communication rates varying with a hyper-parameter used in channel reconstruction and phase resolutions of an IRS;

Figure 7 is a graph showing simulation results of data communication rates varying with tire number of simulations;

Figure 8 is a diagram illustrating an embodiment method for IRS aided communications in a TDD system; Figure 9 is a diagram illustrating an embodiment method for IRS aided communications in a FDD system;

Figure 10 is a diagram of an embodiment method for IRS-aided communications;

Figure 11 is a diagram of another embodiment method for IRS-aided communications; Figure 12 is a diagram of another embodiment method for IRS-aided communications; Figure 13 is a block diagram of an embodiment processing system; and Figure 14 is a block diagram of an embodiment transceiver.

Corresponding numerals and symbols in the different figures generally refer to corresponding parts unless otherwise indicated. The figures are drawn to clearly illustrate the relevant aspects of the embodiments and are not necessarily drawn to scale. DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS The making and using of embodiments of this disclosure are discussed in detail below. It should be appreciated, however, that the concepts disclosed herein can be embodied in a wide variety of specific contexts, and that the specific embodiments discussed herein are merely illustrative and do not serve to limit the scope of the claims. Further, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of this disclosure as defined by the appended claims. Intelligent reflecting surface (IRS) has recently emerged as a promising cost-effective technology for improving wireless communications performance. An IRS is a collection of small antennas that are configured to receive and re-radiate incident signals. The IRS can reflect an incident signal and generate a directional beam in a desired intended direction, thus enhancing the link quality and coverage. Wireless communications utilizing one or more IRSs may be referred to as IRS aided (or assisted) communications. Embodiments of the present disclosure provide channel reconstruction schemes for IRS- aided wireless communications in TDD and FDD systems. Specifically, embodiment methods are provided for reconstructing a communication channel. The communication channel includes an IRS aided reflective channel formed by an IRS, a first device and a second device, and a direct channel between the first and second devices. In some embodiments, in the TDD system, the first device may receive a pilot signal from the second device, generate sparsity information or subspace information of the communication channel, and reconstruct the communication channel based on the received pilot signal, a reflective patter of the IRS, and the sparsity information or subspace information of the communication channel. In some embodiments, in the FDD system, the second device may receive a pilot signal from the first device in the communication channel, and send signal strength information of the received pilot signal to the first device. The first device may reconstruct the communication channel based on the signal strength information, a reflective pattern of the IRS, and subspace information of the communication channel. The embodiments have advantages of providing channel reconstruction for IRS aided communications with improved channel reconstruction accuracy, reduced channel reconstruction complexity and reduced pilot overheads. Further details are provided in the following.

Figure 1 illustrates a network 100 for communicating data. The network 100 comprises a base station 110 having a coverage area 101, a plurality of user equipments (UEs) 120, and a backhaul network 130. As shown, the base station 110 establishes uplink (dashed line) and/or downlink (dotted line) connections with the UEs 120, which serve to carry data from the UEs 120 to the base station 110 and vice-versa. Data carried over the uplink/downlink connections may include data communicated between the UEs 120, as well as data communicated to/from a remote-end (not shown) by way of the backhaul network 130. As used herein, the term “base station” refers to any component (or collection of components) configured to provide wireless access to a network, such as a Node B, an evolved Node B (eNB), a next generation (NG) Node B (gNB), a master eNB (MeNB), a secondary eNB (SeNB), a master gNB (MgNB), a secondary gNB (SgNB), a network controller, a control node, an access node, an access point, a transmission point (TP), a transmission-reception point (TRP), a cell, a carrier, a macro cell, a femtocell, a pico cell, a relay, a customer premises equipment (CPE), a WI-FI access point (AP), or other wirelessly enabled devices. Base stations may provide wireless access in accordance with one or more wireless communication protocols, e.g., long term evolution (LTE), LTE advanced (LTE-A), 5G, 5G LTE, 5G NR, High Speed Packet Access (HSPA), WI- FI 802.11a/b/g/n/ac, etc. As used herein, the term “user equipment” refers to any component (or collection of components) capable of establishing a wireless connection with a base station. UEs may also be commonly referred to as mobile stations, mobile devices, mobiles, terminals, terminal devices, users, subscribers, stations, communication devices, CPEs, relays, Integrated Access and Backhaul (IAB) relays, and the like. It is noted that when relaying is used (based on relays, picos, CPEs, and so on), especially multi-hop relaying, the boundary between a controller and a node controlled by the controller may become blurry, and a dual node (e.g., either the controller or the node controlled by the controller) deployment where a first node that provides configuration or control information to a second node is considered to be the controller. Likewise, the concept of UL and DL transmissions can be extended as well. In some embodiments, the network 100 may comprise various other wireless devices, such as relays, low power nodes, etc. While it is understood that communications systems may employ multiple base stations capable of communicating with a number of UEs, only one base station, and two UEs are illustrated for simplicity. Figure 2 illustrates an example communications system 200, providing mathematical expressions of signals transmitted in the communications system 200 and a channel model. Communications system 200 includes an access point 205 communicating with a UE 210. As shown in Figure 2, access point 205 is using a transmit filter v and UE 210 is using a receive filter w. Both access point 205 and UE 210 use linear precoding or combining. Assuming a channel matrix (or channel model or channel response) H is an N_rx X N_tx matrix of a multiple-input multiple-output (ΜΓΜΟ) system, i.e., there are N_tx transmit antennas and N_rx receive antennas. The transmit filter v of dimension N_tx x Ns enables the transmitter to precode or beamform the transmitted signal, where Ns is the number of layers, ports, streams, symbols, pilots, messages, data, or known sequences transmitted. The receive filter w of multi-antenna systems is of dimension N_rx x Ns and represents the combining matrix, which is usually applied on the received signal y according to w^Hy. The above description is for a transmission from access point 205 to UE 210, i.e., a DL transmission. The transmission may also occur at the reverse direction (an UL transmission), for which the channel matrix becomes H^H in the case of TDD

(where H^H is the Hermitian of channel model H), and w may be seen as the transmit filter and v as the receiver filter. The w for transmission and the w for reception may or may not be the same, and likewise for v.

A DL (or forward) channel 215 between access point 205 and UE 210 has channel model or response H, while an UL (or backward, or reverse) channel 220 between UE 210 and access point 205 has channel model or response H^H. (Another convention is that the UL channel is denoted as H^T, which is the transposition of channel model H). Although Figure 2 depicts only one access point and one UE, communication system 200 is not limited to this case. Multiple UEs may be served by the access point, on different time- frequency resources (such as in frequency division multiplexed-time division multiplexed (FDM-TDM) communication systems, as in typical cellular systems) or on the same time- frequency resources (such as in multi-user MIMO (MU-MIMO) communication systems, wherein multiple UEs are paired together and transmissions to each UE are individually precoded). Among the paired UEs, there is intra-cell interference. Also, multiple access points may exist in the network, some of which may be cooperatively serving UE 210 in a joint transmission fashion (such as in coherent joint transmission, non-coherent joint transmission, coordinated multipoint transmission, etc.), a dynamic point switching fashion, and so on. Some other access points may not serve UE 210 and their transmissions to their own UEs cause inter-cell interference to UE 210. The scenario of multiple access points and multiple UEs, with access point cooperation to serve a UE and with MU-MIMO, is an example scenario considered herein.

Wireless networks have experienced a substantial increase in capacity over the past decade due to several technological advances, including massive multiple-input multiple- output (mMIMO), millimeter wave (mmWave) communications and ultra-dense deployments of small cells. However, implementing these technologies efficiently is a challenging task due to increased hardware cost as well as increased power consumption. To address this need for reliable and energy efficient future cellular networks without incurring prohibitive costs, researchers across the globe are studying different techniques to improve the system performance and have particularly focused on providing control over the propagation environment.

Intelligent reflecting surface (IRS) has recently emerged as a promising cost-effective technology for enhancing the capacity and energy efficiency as well as improving coverage in future wireless communication systems. An IRS is a collection of small antennas that are configured to receive and re-radiate incident signals (e.g., electromagnetic waves) without amplification, but with configurable phase-shifts/time- delays on the signals. An IRS may be a thin two-dimensional metamaterial (i.e., a material that is engineered) that has the ability to control and impact electromagnetic waves to some degree. Note that IRS operates differently than other related technologies such as amplify and forward relaying, backs catter communications, etc. IRS impacts an incident signal passively without additional amplification and thereby avoids energy consumption entailed by the need for amplification. Moreover, the need for RF chains that allow baseband processing of incident signals can also be eschewed. IRS achitectures may employ varactor diodes or other micro electrical mechanical systems (MEMS) technologies, so that in such IRS architectures, electromagnetic (EM) properties of the IRS may be defined by its micro-structure, which can then be programmed, to a certain degree, to vary the phase, amplitude, frequency, and even an orbital angular momentum, of an incident EM wave. Consequently, an IRS can modulate a radio signal without using a mixer and a radio frequency (RF) chain, and real-time reconfigurable propagation environments may be achieved. Furthermore, by smartly adjusting the phase shifts of all the passive elements at the IRS, reflected signals of the IRS can sum up coherently with signals from other paths at a desired receiver to boost the received signal power, or destructively at non-intended receivers to suppress interference as well as enhancing security and privacy'.

From the implementation perspective, IRSs possess appealing advantages such as low profile and conformal geometry. These advantages enable them to be easily attached to or removed from a wall or ceiling, thus providing high flexibility for their practical deployment. For example, by installing IRSs on the walls or ceilings that are in line-of- sight (LoS) with an access point (AP) or base station (BS), signal strength in the vicinity of each IRS may be significantly improved. Moreover, integrating IRSs into the existing networks (such as a cellular or WiFi network) can be made transparent to users without the need of any change in the hardware and software of their devices. These features described above make IRSs a compelling new technology for future wireless networks, particularly in indoor applications with high density of users (such as in stadiums, shopping malls, exhibition centers, airports, etc.). Utilizing a large number of elements whose electromagnetic response (e.g. phase shifts) can be controlled by simple programmable PIN diodes, an IRS can reflect an incident signal and generate a directional beam in a desired intended direction, thus enhancing the link quality and coverage. The phase of each IRS element can be adjusted for instance through the PIN diodes which are controlled by an IRS controller. As an example, an IRS controller may be connected to an access point (AP) via a backhaul link and commanded by that AP to employ an appropriate phase patter across its elements. As another example, the IRS controller itself may be capable to determining an appropriate phase patter.

The above advantages of IRSs make it a very promising technology for helping further improve wireless communications systems. Wireless communications utilizing one or more IRSs may be referred to as IRS aided (or assisted) communications. IRSs have been deployed from various communication perspectives, such as for secrecy' rate maximization, unmanned aerial vehicle UAV/drone communication, energy efficiency optimization, rate (weighted, sum, or minimum) maximization, wireless power transfer and localization, mobile edge computing, tera-hertz (THz) communication, etc.

However, the advantages of using IRS to assist communications, such as throughput, gains or coverage improvements in an IRS assisted communication system, depend on quality of estimates of underlying channels. For example, without availability of reliable estimates of the communication channels, the phase patters of an IRS (or an IRS response) cannot be optimized. However, channel estimation in IRS aided communications is quite challenging in practice in view of the passive nature of IRS elements. In order to reduce implementation costs, the IRS is generally not equipped with any radio frequency (RF) chain and thus lacks requisite baseband processing capability. As a result, it is impossible for the IRS itself to estimate its channels for a transmitter or any of the users. There have been solutions proposed for channel estimation in IRS assisted communications (passive or semi-passive), and for IRS passive reflection pattern optimization (continuous or discrete). Clustering IRS elements, compressive sensing, and deep learning-based methods have been proposed for channel estimation in IRS aided communications. However, in time division duplex (TDD) systems for example, at least N+l overhead resources are necessary to estimate for a single input single output (SISO) system, and in a frequency division duplex (FDD) system, even more overhead is required, where N is the number of IRS elements. A pattern selection dilemma is also observed during a channel estimation training phase. It has been observed that if side information is available, it may be exploited to reduce the overheads.

Embodiments of the present disclosure provide channel reconstruction schemes for intelligent reflecting surface (ERS)-aided wireless communications in TDD and FDD communication systems. An embodiment scheme does not require active elements on an IRS panel. Instead, by exploiting the reciprocity in TDD systems, analog (complex) observations obtained at a receiver in the uplink (e.g., an AP) are utilized to reconstruct a downlink channel. For FDD systems, quantized signal strength measurements fed-back by a receiver to a transmitter (e.g., an AP) or IRS controller are utilized to reconstruct a downlink channel. An embodiment for TDD systems combines matrix completion with beam-domain sparsity or subspace information (also referred to as subspace side information) and leads to implementable alternating direction method of multipliers (ADMM) based algorithms. These algorithms in turn may yield a reconstructed channel model or vector, and an optimized IRS pattern that, subject to further processing (which may be lightweight processing), may be suited for facilitating data transmissions to an intended user. An embodiment for FDD systems exploits subspace side information and is based on either phase-retrieval techniques or certain non-convex quadratically constrained quadratic programming. In each case, effective low-complexity algorithms are provided. Further, embodiment subspace estimation algorithms are proposed. An embodiment ADMM based algorithm for subspace estimation is proposed, which exploits a particular sparsity structure possessed by a downlink channel, and another subspace estimation technique is proposed that establishes and exploits the structure of the covariance of the downlink channel. Moreover, an embodiment is also provided for designing IRS patter vectors that can be gainfully used during a training phase for channel estimation/reconstruction. In addition, simulation results generated using an open source software tool “SimRis” are provided. This tool allows for simulating IRS assisted communications.

The embodiments may be applied for reconstructing a downlink channel or an uplink channel, and may be performed by an AP, a UE or an IRS controller. By use of the sparsity information or the subspace information, the embodiments reduce complexity for channel reconstruction in IRS aided communications, and reduce pilot overheads for the channel reconstruction. Furthermore, use of such sparsity information or subspace information can significantly improve accuracy of channel reconstruction for a given acceptable pilot overhead. Using such information may yield a channel reconstruction that meets a sufficient level of accuracy for much less pilot overhead. Improving channel reconstruction accuracy in turn allows for achieving higher rate and higher reliability in data communications.

Figure 3 is a diagram of an embodiment network 300 for IRS aided communications. The network 300 includes an access point (AP) 302 in communication with a UE 304, and an IRS 306. The IRS 306 is configured to reflect incident signals and generate directional beams in desired directions. The IRS 306 may be located in the proximity of the UE 304, and assist the communications between the AP 302 and the UE 304, especially the downlink transmissions. The IRS may include a number of tunable reflecting elements 308, which may be controlled and adjusted by an IRS controller 310, e.g., for phase patter adjustment. The IRS controller 310 may be connected to the AP 302, which may control the IRS controller 310 to perform the adjustment on the IRS 306.

The AP 302 is configured to transmit signals to the UE 304. Signals received by the UE 304 from the AP 302 may include two portions SI and S2 in this example. SI includes signals received by the UE 304 directly from the AP 302 on an AP-UE link (or channel) 322. The AP-UE link (or channel) 322 may be referred to as a direct channel. S2 includes signals that are sent by the AP 302, incident on the IRS 306 and reflected by the IRS 306, and received by the UE 304 from the IRS 306 on a IRS-UE link (or channel) 324. Signals sent by the AP 302 and incident on the IRS 306, which may be received by the UE 304, may include, in this example, signals incident on an AP-IRS line-of-sight (LoS) link 326, and signals incident on AP-IRS non-LoS (NLoS) links 328, 330. Signals on the AP-IRS NLoS links 328 and 330 may be reflected by respective obstacles 312 and 314 and incident on the IRS 306. The links 326, 328 and 330 may be collectively referred to as an AP-IRS link (channel). The AP-IRS link (channel) and the IRS-UE link may be collectively referred to as an IRS aided reflective link (channel), a reflective channel, or an IRS reflective link (or channel). Those of ordinary skills in the art would recognize that the network 300 may include more than one AP, more than one UE, less or more obstacles than what is illustrated, and/or more than one IRS. Figure 4 is a diagram of an embodiment channel model of a netw ork 400 for IRS aided communications. Similar to the network 300, the network 400 includes a transmitting device (or node) 402 in communication with a receiving device (or node) 404, and an IRS 406 configured to reflect incident signals. As an example, the transmitting device 402 may be an AP, and the receiving device 404 may be a UE. As another example, the transmitting device 402 may be a UE, and the receiving device 404 may be an AP.

Signals transmitted by the transmitting device 402 may arrive at the receiving device 404 on a transmitting device-receiving device channel (or link) h_dir 412 and an IRS -receiving device channel (link) g 414 in this example. The signals may be transmitted by the transmitting device 402 directly to the receiving device 404 on the transmitting device- receiving device channel h_dir 412. The signals transmitted by the transmitting device 402 may also arrive at the IRS 406 on a transmitting device- IRS channel h 416, which are reflected by the IRS 406 onto the receiving device 404 on the IRS-receiving device channel g 414. Thus, a communication channel between the transmitting device 402 and the receiving device 404 may include the transmitting device-receiving device channel h_dir 412, the IRS receiving device channel g 414 and the transmitting device-IRS channel h 416. Note that the transmitting device-IRS channel h 416 may include multiple communications links, such as the links 326, 328, 330 as illustrated in Figure 3. Channels g 414 and h 416 may be collectively referred to as an IRS (aided) reflective channel. Channels g 414 and h 416 may be referred to as two constituent channels or two IRS channels of the IRS (aided) reflective channel. It would be desirable that the receiving device may estimate or reconstruct such a communication channel, so that transmissions from the transmitting device 402 to tiie receiving device 404 may be performed adaptively according to tiie communication channel, and the IRS 406 may be adjusted (e.g., phase shifts of IRS elements may be adjusted) to adapt to the communication channel. Embodiments of the present disclosure provide methods for estimation or reconstruction of the communication channel based on tiie channel model of tiie network 400. Embodiments in the following are provided for reconstructing a downlink channel in IRS aided communications. Those of ordinary skill in the art would readily recognize that the embodiments may also be applied for reconstructing an uplink channel in IRS aided communications, without departing from tiie spirit and principle of tiie present disclosure. Throughout the disclosure, scalar, vector, and matrix are denoted by lowercase, bold lowercase, and bold uppercase, i.e., a, a, and A, respectively. (·)^τ, -(_· ), (·)^† _, and rank(·) denote transpose, conjugate, conjugate transpose, and rank, respectively. ®,Q are used to denote the kronecker product and Hadamard product, respectively, between any two identically sized matrices. We reserve j to denote imaginary unit of a complex number, i.e.,j √- 1. For any vector with m complex-valued elements, a ∈ IC^m, we let diag{a) ∈ IC^mxm denote the associated diagonal matrix, and for any m x m complex- valued matrix A ∈ IC^mxm, we let diag{A} ∈ IC^m denote a vector comprising of diagonal elements of A. I_m denotes the m x m identity matrix.

In some embodiments, the IRS (e.g., the IRS 406) may include N passive elements, and the transmitting device (e.g., an AP) may include one transmit antenna and the receiving device (e.g., a UE) may include one receive antenna. As an example, the IRS is installed indoor, and a UE is in communication with an AP in a single-user single-input-single- output (SISO) system The IRS is a passive surface that applies a pattern Φ ∈ F^N , where the n^th entry of Φ, denoted by Φ _n, models the multiplicative impact of the n^th IRS element on its incident signal. Here F is a finite set of complex scalars with magnitudes no greater than (but not necessarily equal to) unity, which models practical (non-ideal) lossy elements. One proposed choice for T suggests elements of the form

for different uniformly sampled choices of a ∈ [— π, π ), and other parameters may be set as

In an example case, there may be a given set of J pattern vectors

denotes a J x N matrix whose / rows contain these J given pattern vectors, p =

[p₁, — , p_T]^T denotes a vector with unit magnitude entries, where is a pilot symbol

transmitted by a transmitting node in the t^th slot of a training phase spanning T training symbol durations (or T slots).

The training phase may also be referred to a measurement phase, in which measurement, estimation or reconstruction of a communication channel is performed, e.g., during a time interval (referred to as a training or measurement duration). T represents a training or measurement duration with T<J. Information about the estimated or reconstructed communication channel may be used for future communications. In the training phase, training signals, e.g., pilot signals (or referred to as pilots), may be sent by a transmitting node (or referred to as a transmitting device or a transmitter) to a receiving node (or referred to as a receiving device or a receiver), and the receiver may estimate or reconstruct, based on the received pilots, a channel from the transmitter to receiver. For example, during a training phase, an AP sends pilots to a UE, the UE may estimate or reconstruct the downlink channel based on the received pilots. In TDD systems, based on reciprocity, it may also be that the receiver of interest (i.e., a UE) transmits pilots in the uplink during the training phase and the transmitter of interest (i.e., an AP) collects received observations (i.e., received pilots) to reconstruct the downlink channel from the transmitter to the receiver. In FDD systems, the transmitter of interest (i.e., an AP) may transmit pilots, and the receiver of interest (i.e., a UE) may collect and process observations, and report quantized processed observations to the transmitter. The transmitter may use the feedback reports from the receiver to reconstruct the downlink channel from the transmitter to the receiver. In either case, the transmitter may decide on the choice of an IRS phase pattern for a data communication phase based on the estimated/reconstructed channel and convey this choice to an IRS controller, e.g., via a low-rate side-channel (or link). The IRS controller may control to adjust the IRS phase pattern of the IRS assisting communications between the transmitter and the receiver.

The observations at the receiver corresponding to the t^th symbol duration may be expressed as:

(1) where h_dir models the direct channel and denotes the patter employed by the IRS during the t^th training symbol duration, and ~ CN (0, N₀) denotes the additive complex normal noise. In the expression (1), g ∈ IC^N and h ∈ IC^N, which denote the channel vectors modeling the IRS-reveiving node link (e.g., the link 414 as illustrated in Figure 4) and the transmitting node-IRS link (e.g., the link 416 as illustrated in Figure 4), respectively, h and g may be referred to as two constituent channels (or two IRS channels) of the IRS reflective channel. Ω denotes the index set of patter vectors of the IRS that are employed in the training or measurement phase. All patter vectors indexed by Ω are also subsumed as rows of

Ψ denotes the J x (N+ 1) matrix obtained by appending aJ x 1 column of all ones to

. Ψ may be referred to as an IRS patter matrix or a patter matrix for simplicity. All z_t may be saved as non-zero entries in a J x T observation matrix

[(h Θ g)^T, h_dlr]^T may be referred to as a composite product channel. According to the expression (1) above, the communication channel may be reconstructed based on Ψ, 2

and P. In the following, embodiment approaches are provided for estimating or reconstructing the communication channel based on the expression (1). The expression (1) is based on the channel model as illustrated in Figure 4, where the receiving device (e.g., UE) receives pilots sent by the transmitting device (e.g., AP) through the communication channel (observations z_t ). The UE may perform downlink channel reconstruction using the expression (1). In a TDD system, based on the reciprocity between the downlink channel and uplink channel, expression (1) may also apply in a case where the AP receives pilots from the UE (observations of the AP) and performs downlink channel reconstruction based on the AP’s observations using the expression (1). In a FDD system, an AP may send pilots to a UE, the UE may feeds back received pilots (observations) to the AP, and the AP may perform downlink channel reconstruction based on the feedback using the expression (1). Those of ordinary sill in the art w'ould recognize various scenarios that the expression (1) may be applied for channel estimation and reconstruction.

Subspace Aided Approach

In some embodiments, a subspace aided approach may be provided for reconstructing the communication channel in a TDD system, where subspace information of the communication channel model may be used to estimate/reconstruct the channel. We define aJ x T matrix

, whose (i_t, t)^th element is z_t for all 1 ≤ t ≤ T and zero elsewhere. Consider an imaginary genie-aided noiseless system, i.e., a system in which an all knowing genie provides additional information, where in each time-slot, an expanded set of / observations may be obtained (with genie’s assistance), one for each of the / available pattern vectors. These observations can be represented over T slots as Ψ [(h Θ g)^T, h_dlr]^TP^†· Let U denote an (N + 1) x S matrix whose columns span the S —dimensional subspace in which the composite product channel [(h Θ g)^T, h_dlr]^T lies. That is, the column space of the matrix U is the subspace in which, with high probability, most of the energy of the composite product channel [(h Θ g)^T, h_dlr]^T lies. Such a matrix U may be constructed, e.g., by using the first L dominant Eigenvectors of tire (N + 1) x (N + 1) covariance matrix of [(h Θ g)^T, h_dlr]^T· In some embodiments, subspace information of a dominant subspace of a covariance matrix of the communication channel may be determined, e.g., based on historical data about measurement, estimation and reconstruction of the communication channel. For example. the historical data may include historical training signals, channel vector pattern information, channel reconstruction information, channel measurement data, prior/historical subspace information, etc. The historical data may be obtained and stored in memory or in a storage device, e.g., in various data structures, and retrieved from the memory/storage device and used to determine the subspace information.

For example, Eigen decomposition of the covariance matrix may yield a set of Eigenvectors and associated Eigenvalues. Each Eigenvector may represent a beamforming direction in a signal subspace. The beamforming direction may correspond to a combination of a transmit beam of the transmitting device, a receive beam of the receiving device, and a reflective direction of the IRS. The elements of each such vector are complex-valued scalars. In one example, Eigenvectors whose associated Eigenvalues are above a configurable threshold may be deemed dominant. In another example, Eigenvectors (also referred to as Eigen directions) whose associated Eigenvalues exceed a threshold, e.g., a fraction of the total sum of all Eigenvalues, may be deemed dominant. The span of these dominant Eigenvectors represents the subspace information. These Eigenvectors max' be used to construct the matrix U, e.g., by selecting the dominant Eigenvectors to be the columns of the matrix U. The matrix U may be referred to as a subspace matrix, including subspace information of the composite product channel (i.e., tiie communication channel). As a result, the communication channel can be modeled as: [(h Θ g)^T, h_dlr]^T ≃ Uζ (2) where ζ represents a combining vector. In (2) the composite product channel is approximated as a linear combination of the columns of U. Since we know the columns of U, the unknowns in right hand side of equation (2) are the elements of the combining vector ζ. Therefore the problem of reconstructing the composite product channel denoted by the left hand side of equation (2) is simplified to that of determining the combining vector ζ with much fewer unknowns.

In the absence of any subspace side-information, we can set U = I. Alternatively, by setting U = I, we are disregarding any subspace information. By use of equation (2), the aforementioned genie-aided noiseless observations (i.e., the expanded set of / observations, or 2) over T slots may be represented as a matrix ΨUζρ^†. In some cases, only a subset of elements from this matrix may be accessible, which may further be corrupted by noise. Upon defining aJ x T sampling matrix S whose (i_t, t)^th element is 1 for all 1 ≤ t ≤ T and is zero elsewhere, we have available in

noisy versions of the sampled elements in the matrix S Θ Ψυζρ^†. Thus, reconstructing the communication channel [(h Θ g)^T, h_dlr]^T becomes a problem of determining ζ.

A joint problem of optimized pattern selection/determination and channel reconstruction may be provided as:

where ||. II_* denotes the nuclear norm, and τ, y > 0 denote hyper-parameters. It is known that the nuclear norm may ensure a low-rank recovery. The factor τ may bias the retrieved solution towards having a lower rank. The l₂ norm imposed on ζ is as per ridge regression. Note that in many relevant scenarios, S « N, and thus no further sparsity in ζ is expected, and using l_x norm instead would be inappropriate.

The problem expressed in the expression (3) is a convex optimization problem, and it is desirable to have an efficient algorithm to solve tire problem. In some embodiments, additional variables may be introduced to formulate the problem in the expression (3) as:

Adopting tiie framework of alterating direction method of multipliers (ADMM) approach, as an example, we obtain the augmented Lagrangian denoted by £(Y, X, C, ζ, L₁, L₂) as shown in expression (4) below:

where p₁₍ p₂ ≥ 0 are additional hyper-parameters, L₁ L₂ are Lagrange variables, and IRtr(. ) denotes the real part of the output yielded by matrix trace operation. In some embodiments, an ADMM based alterating optimization (ADMM-AO) approach to solve (4) may include the following steps in each of its iteration:

Solve argmin_Y∈1CjxT{L(Y, X, C, ζ, L_1, L₂)}, which is equivalent to which permits a

standard singular value thresholding solution, which is shown in Lemma 2 subsequently provided.

Solve a which is equivalent to an

unconstrained quadratic programming problem.

Finally, the Lagrange matrix variables are updated as:

The inputs for the ADMM-AO may include the subspace matrix U, initial choice values for all matrix variables, and a choice of the hyper-parameter values. By utilizing block coordinate descent (BCD), all the variables may be optimized sequentially in a closed form. Optimized results may be highly dependent on the hyper-parameters, and it is desirable that the hyper-parameters are tuned carefully.

From the output ζ

, we may obtain the reconstruction

which is the reconstructed communication channel, or referred to as a reconstructed channel model. The reconstructed channel model may be used to update or adjust the vector pattern of the IRS. As an example, using the obtained matrix

which may be maintained by the algorithm, a row of this matrix having the largest norm may be determined. This row may be referred to as a maximal normed row. The patter vector in Ψ corresponding to this row may be used as the starting point of a low-complexity enhancement process, which may herein be referred to as “linear pass”. In particular, let

be the pattern vector corresponding to the maximal normed row. The linear pass may include N steps for example. Let

denote the i^th element of the vector , and define

denoting the i^th element of the vector

, and in the i^th step of the linear pass,

may be updated or adjusted as:

The patter vector obtained post linear pass may be declared to be the optimized patter for single-user data communications.

Figure 5 are graphs 500, 530 and 550 showing simulation results of data communication rates varying with hyper-parameters p₁, p₂ and τ, respectively. In the example of Figure 5, a communication channel in a TDD system is reconstructed according to the embodiment subspace aided approach using different hyper-parameters p_1, τ₂ and τ, generating respective reconstructed channel models. IRS patters are adjusted and communications are performed over the communication channel based on the respective reconstructed channel models, and data communication rates are measured. In this example, transmit power is set to be 35dBm, noise power is set to be -lOOdBm, the number N of elements of the IRS is 400, the training pilot durations include 50 slots, and phase resolution is 4 bit (i.e., 16 IRS patters available for choose). Figure 5 shows simulation results utilizing the embodiment approach implemented in various manners, including using ADMM (indicated as “ADMM” in Figure 5), using randomly selected IRS patterns (indicated as “Random”), using highly refined IRS patterns (indicated as “AdmaxO”), using generally refined IRS patterns (indicated as “upNN”). Figure 5 also shows an ideal situation indicated as “Upper Bound”. It can be seen from the simulation results, the results generated in the manners of ADMM and AdmaxO are close to the ideal situation.

Figure 6 are graphs 600 and 620 showing simulation results of data communication rates varying with the hyper-parameter τ and phase resolutions. As used herein, the phase resolution indicates the number of phase patterns that the IRS may have. For example, a 1 bit phase resolution indicates that two (2) phase patterns that the IRS may use. A 4 bit phase resolution indicates that sixteen (16) phase patterns that the IRS may use. The number 2 or 16 may be the value of / described above. The information of the reconstructed channel may be used to determine a patter of the IRS so that the IRS may operate to enhance the transmission to a receiver. The patter may be selected from the available patterns based on the reconstruction information. In the example of Figure 6, a communication channel in a TDD system is reconstructed according to the embodiment subspace aided approach using different τ, generating respective reconstructed channel models. IRS patters are adjusted based on the reconstructed channel models. Figure 6 shows simulation results utilizing the embodiment approach implemented in various manners, including using ADMM (indicated as “ADMM” in Figure 5), using randomly selected IRS patterns (indicated as “Random”), using different levels of refined IRS patters (indicated as “Admax”, “AdmaxO”, “Admax02”, “upN”, and “upNN”, respectively). Figure 6 also shows an ideal situation indicated as “Upper Bound”. Graph 600 shows the simulation results in the case of 1 bit phase resolution. Graph 620 shows the simulation results in the case of 4 bit phase resolution. The above described embodiment may determine subspace information based on historical knowledge. The subspace information may also be estimated using ADMM or covariance projection methods that are described subsequently. For a given subspace, an optimized set of probing patterns, or a set of IRS patterns that are used during training, may be obtained using a Generalized Lloyd algorithm which is also proposed subsequently.

Sparsity Aided Approach

In some embodiments, a sparsity (e.g., beam domain sparsity) aided approach may be provided for reconstructing the communication channel in a TDD system. This approach may be used when the subspace information of the communication channel is not available. In this approach, we may consider that one of the two constituent channels g and h in the composite product channel is LoS-only (Line-of-Sight only) and that the associated LoS steering vector is known. By LoS-only we mean that the constituent channel does not have any contribution from any obstacle in the propagation environment it models. Taking Figure 3 as an example, the channel 324 is LoS-only, and AP-IRS channel is not LoS-only. The LoS steering vector is a function of the known IRS array geometry (including number and arrangement of IRS antenna elements) and an LoS direction. The LoS direction may be determined based on location information of the IRS, of the transmitter transmitting the pilots, and/or of the receiver. As an example, suppose, without loss of generality, that g is LoS only and is known up-to a complex scaling factor, and is known. Let G_ft be an N x D_h dictionary matrix for h such that h permits a

sparse representation under this dictionary, i.e., we have that h = G_ht_h for some sparse vector t_/i e IC°^h. In this case, in the absence of any other subspace side information, G_ft may be identical to a matrix whose columns are steering vectors (defined by the IRS array geometry) uniformly sampled on an angular grid. For a commonly used uniform planar array (UFA) layout, the elements of each column of this matrix may be given by:

where 1 < a_1, a₂ ≤ √ N, λ denotes the wavelength, and Φ, θ are the azimuth and elevation angles measured with respect to the IRS boresight, respectively. Beam domain sparsity analysis of the communication channel (composite product channel) may be performed to determine sparsity information, which may be used to determine its sparse representation. The sparsity information may include a set of array beamforming directions. Each array beamforming direction corresponds to a combination (e.g., elementwise product) of a pair of steering vectors, where one steering vector in that pair is a steering vector determined for the channel between the IRS and the receiver (e.g., 324 in FIG. 3), and the other steering vector in that pair is determined for the channel between the IRS and the transmitter (e.g., 328 or 326 in FIG. 3). For instance, such a set of array beamforming directions may include all directions that are likely to contribute to (or be present in) the communication channel. In other words, the communication channel is expected to have some fraction of its energy along those directions. As an example, based on the orientation of the IRS array, orientation of the transmitter and the receiver, some of the directions may be ruled out or precluded from having any contribution in the communication channel. A sparsity analysis may generate array beamforming directions along with weights. The channel may be decomposed or expressed as the weighted summation of several components. The weights may be complex scalars (each including a magnitude and a phase). A sparsity analysis output may provide all likely components (each such component being an array beamforming direction) as well as their associated complex scalar weights.

Let , The composite product channel can be expressed in this

special case as:

for some sparse vector t having a length of

represents a dictionary matrix whose columns may be constructed from a plurality of array beamforming directions, and t represents a combining vector that is expected to be sparse, i.e., most of the elements of t are expected to be zero. The columns of

correspond to the plurality of array beamforming directions. Since as well as each column of G_h is a steering vector

(corresponding to different azimuth and elevation angle pairs of IRS elements), each column of the matrix represents a steering vector (corresponding to some

azimuth and elevation angle pair) as well. In this case, the expression in (3) may thus be expressed as:

where W_h ≥ 0 denotes a given diagonal matrix of weights. II Y II_* represents a nuclear norm || W_ht ||₁ represents a weighted l₁ norm.

fits sampled matrix to observations in

² fits to the analytical formula using predefined Ψ, dictionary and pilots. As an example, we can set a low weight (relative to others) for steering vectors (corresponding to columns of Ĝ), which may be known a-priori to be more likely to be presort. For instance, if tire LoS component direction in h is known, the corresponding weight may be set to be much smaller relative to other directions. An augmented Lagrangian may be obtained as given by (8) below, where auxiliary matrix variables X,C ∈ lC^JxT are introduced.

f

(8)

Based on expression (8), an embodiment ADMM based algorithm, referred to as ADMM- Ll, may be provided following the same approach as discribed above with respect to the subspace aided approach. The details are not described herein for brevity, but note that a main difference is the sub-problem to update t, which is

which can be solved using the corresponding result summarized in Lemma 2.

Subspace Aided Approach With Quantized Observations

In some embodiments, subspace aided approach using quantized observations may be provided for channel reconstruction in an FDD system In the embodiments, quantized signal strength (or magnitude) of observations (received signals) may be available along with the subspace information of the communication channel. For example, a UE may receive pilots sent by an AP, and the UE may said quantized information of the received pilots (e.g., quantized signal strength or magnitude of the received pilots) to the AP. The AP may reconstruct the downlink channel from the AP to the UE based on the received quantized information. Due to the coarseness of available observations, channel reconstruction may be performed based on the quantized observations without performing patter optimization. The patter vector for data communications may be obtained or optimized using the reconstructed channel, e.g., in a second step of optimization. Two embodiment approaches are provided in the following for the channel reconstruction with quantized observations available, i.e., a phase retrieval based approach and a non-convex quadratically constrained quadratic programming approach. Phase Retrieval Based Approach

This approach may be applied in an FDD scenario in which only Q(|z_t|) are available in the expression (1 ) for 1 ≤ t ≤ T, in addition to U, where Q(.) denotes a scalar quantizer. The composite product channel may be expressed as the expression (2) using subspace information. Upon tailoring the phase-retrieval techniques to the problem at hand, we obtain the formulation given in (9),

where D_p = diag{p₁, ···, p_T), and λ ≥ 0 is any given hyper-parameter. Ψ_Ω denotes rows of Ψ formed by vectors [Φ_ίt, 1], 1 ≤ t ≤ T. ∈ ∈ IC^S is used to model a quantization error vector, where each component of e may be any complex scalar that has a magnitude within a bound, and the bound in turn is determined by a given quantization resolution.

In some embodiments, the problem of (9) may be solved via alternating optimization. As an example, ζ may firstly be optimized by keeping the phases and the error

vector term fixed (step 1). Then, the phases may be optimized by keeping

ζ, e fixed (step 2). Finally, the additive error vector e may be optimized by keeping other two sets

of variables fixed (step 3). This process (steps 1-3) may be repeated till convergence. Note that this approach is a block coordinate descent, since in each step the objective value is minimized over a block of variables (while keeping the other blocks fixed). Since each step (step 1, 2, 3) can be optimally solved, convergence is guaranteed since the objective value decreases after each step.

Non-convex Quadratically Constrained Quadratic Programming Approach This approach may be applied in an FDD scenario in which a user (UE) of interest determines average received signal strength (associated with each choice of IRS pattern vector) and feeds it back to a controller (e.g., an AP) after quantization. The composite product channel may be expressed as the expression (2) using subspace information. In cases where, for each t: 1 ≤ t ≤ T, the controller has access to Q(|[Φ_ίt, 1] [(h Θ g)^T, h_dlr]^T |²), the problem to be solved is shown below: min{|| ζ II²},

where Y_t ^b',Y_t'^b' are upper and lower bounds corresponding to the boundaries of the quantizer’s bin in which |[Φ_ί(, 1]ϋζ|² lies. The problem in (10) is a non-convex quadratically constrained quadratic programming. In one example, it may be solved in an efficient albeit sub-optimal manner using the K-best methodology. Alternatively, a proximal distance based algorithm may be employed, which is provided below. Note that the formulation in (10) appears intractable since it has T constraints each describing a non-convex constraint set. A feasible set of (10) may be the intersection of T feasible sets, each of which is described by one associated constraint. Moreover, this feasible set may be guaranteed to be non-empty since it is based on actual user feedback. An observation is that the following problem (which can be regarded as projection of any vector in ζ e I C^s onto a feasible set described by any one constraint) is a rare non-trivial example of a non-convex problem that may be efficiently and optimally solved according to the following problem: min {II ζ — x II²},

XEIC⁵

Y_t ^b' and Y_t ^b· are bounds of the quantizer’s bin in which < | [Φ_ί(, l]Ux|² lies. Let S_t c I C^s denote the set of all x E IC^S for which the t^th constraint is satisfied, i.e., (x: Yt^b- <

I [Φ^, l]Ux|² < Yt'^b }. Similarly, let D_t(Q, V ζ e IC^S V t denote the objective value obtained by solving (11) for the given ζ input. In other words, D_t(Q is the distance of ζ to the feasible set of the t^th constraint. Then, it can be verified that for all p greater than a large enough penalty, i.e., V p > p > 0, (11) is equivalent to:

Based on the above observations, an embodiment proximal distance-based algorithm may be used to solve the problem. Specifically, this algorithm is an iterative one, and in each iteration it successively solves two sub-problems as shown below:

In a first step, the following sub-problem is sovled for a given choice of

which is solvable in a closed form to obtain ξ ^xt) ·

In a second step, each x_t is updated by by solving (11) using the obtained ξ as input

It can be readily seen that this procedure converges since the objective in (13) is monotonic non-increasing across iterations and is bounded below. Indeed, for p > p and any given choice of x_t e S_t V t = 1, — , T, the optimal value of (13) is an upper bound to the true optimal value of (12) (or (10)). The performance of the algorithm is sensitive to the choice of p as well as the initial x_t e S_t V t = 1, — , T. To address the first dependency, we may progressively increase the choice of p over batches, each batch including several iterations, starting from a small value p «- pi_nu. and amplifying it from one batch to the next as p <- θρ for some amplification factor Θ > 1. Regarding the initial choice of x_t e S_t V t = 1, — , T, we may choose each x_t to be the solution to a relaxation of (10) when only the t^th constraint is retained, which may be retrieved via an efficient and optimal algorithm.

Figure 7 is a graph 700 showing simulation results of data communication rates varying with the number of simulations. In this example, a communication channel is reconstructed in different numbers of simulations (number of simulation runs) utilizing the subspace aided approach with quantized observations, and the data communication rates are measured for communications performed based on information of the corresponding reconstructed channels. Figure 7 shows simulation results obtained with the approach implemented using different manners, including using the phase retrieval- based approach (which is also referred to here as a block coordinate descent (BCD) approach (indicated as “BCD” in Figure 7), using the K-best methodology (indicated as “K-best”), using random IRS patterns (indicated as “Random”). Figure 7 also shows an ideal situation indicated as “Upper”. Patter Vectors for Training

In some embodiments, a problem is considered for obtaining a suitable set of T training vectors that exploit the subspace side information provided as the column-span of some given matrix U e IC^(N+1)xS. This problem is made particularly challenging due to the finite alphabet constraint that is imposed on the entries of all pattern vectors. In one embodiment, a tailored Generalized Lloyd algorithm may be used for this purpose. In an embodiment, a set of all feasible pattern vectors may be defined:

One objective is to design a set of T feasible patterns (for training), all of which lie in T, via the following formulation: max (E \ max | [Φ_£, llr |²1) , (15) (♦_te£lLi ^{l Ll≤t≤T JJ} where the expectation is over the composite product channel, denoted here as r, conditioned upon the given subspace side information. The challenges are two-fold. The first one is that we do not have knowledge of the conditional distribution of r and need to rely on the subspace side information. The other one is the finite alphabet constraint on the pattern vectors. In an embodiment, the first challenge may be addressed by generating a large set of realizations as r_e = U¾ for £ e {1, — , L'} for some large V » 1, where {ζ*} are realizations of an isotropically distributed vector in IC^S. This represents a worst case assumption in that no other prior information is assumed apart from the subspace side information. The finite alphabet restriction challenge may be addressed via an iterative proximal distance based approach. Let

denote a candidate codebook of pattern vectors at the current iteration and let its feasible twin be {$_t e QLI. let p > 0 denote a given penalty factor. Each iteration may include of the following two steps:

• Step 1: partitioning the set of sample vectors {r*} into T regions Jf_t, where H_t = {£: |[φ_ί, l]r_f|² > |[φ_¾, l]i>|² Vk ≠ t}. Then the following sub-problem is solved to update {φ_ί}?=ι^:

(4»_teic^lxW} lstsT

The problem in (16) decouples across 1 < t < T and is an unconstrained quadratic program that can be optimally solved in closed form. Update {4>t}?=i to be the obtained solution.

• Step 2: projecting each φ_£ onto i.e., updating each <j>_t as $t <- arflrmm^rtll Φ — φ_£ ||²}. This problem is also readily solvable by elementwise quantizing φ_£ to its closest point in T.

These above tw o steps may be repeated until convergence which may be guaranteed since the objective of (16) is monotonically decreasing across iterations. The performance of the above algorithm is sensitive to the choice of p as well as the initial {φ_£, φ_£}; V t = 1, — ,T. In an example, multiple random initializations have been used with each such initialization including min{5, T] columns of the matrix U in {φ_£} and their projections onto T in {$_£}. To address the other dependency, as an example, the choice of p may be progressively increased over batches, each batch including several iterations, starting from a small value p «- and amplifying it from one batch to the next as p «- θρ for some amplification factor Θ > 1. Upon convergence, ($_£}_£=i may be selected as the pattern vectors for training. The above procedure may then be repeated to find additional J — T pattern vectors in T_ in order to determine the matrix Ψ used in the matrix completion approach. The iterative procedure may be used for designing the J pattern vectors, and T of them may always be fixed to the T training vectors obtained before. Subspace Estimation

In some embodiments, methods for subspace estimation (and subspace information) are provided. Let G_h be the /V x D_h dictionary matrix for h and let G_g be the N x D_g dictionary matrix for g. Note here that both h and g are assumed to permit a sparse representation under their respective dictionaries, i.e., we suppose that h = G_nt_h & g = G_flt_fl for some sparse t_ft e IC°^h and some sparse t_g e IC ¾, respectively. Here, in the absence of any other side information, we can assume both these dictionaries are identical to tiie matrix whose columns are steering vectors (defined by the IRS array geometry) sampled on the angular grid. Indeed, recall that for a commonly used uniform planar array (UP A) layout, elements of each column of the latter matrix may be given by (6). The following results may be readily obtained after some algebra.

Lemma 1 The product channel h Qg permits a sparse representation under the N x D_hD_g expanded dictionary matrix, i.e..

where » denotes the face-splitting product (or row-wise Kronecker product).

Note that from Lemma 1 we can deduce that the product channel h Q g exhibits rowwise product sparsity. In particular, if {tr_ft>£) and {t_gj} denote the non-zero rows of t„ and t _g, respectively, then {t_fti£t_{5 ;}} will be the non-zero rows (coefficients) in t_ft ® t_fl. We may then augment 6 to form the (JV + 1) x (D_hD_fl + 1) matrix G = [G, 0_N; 0%, 1] using whidi the composite product channel [(h © g)^T, h_dlr ]^T may be expressed as a vector in the column span ofG. Suppose that observations are collected over L frames and these collected observations may form the T x L matrix Z =

where the vector

≤ L, denotes the T observations obtained over the -6^th frame, z^ =

[zip, — , Ζ_γ\ where each ∑P permits the channel model as shown in (1), albeit with unknown composite product channel given by

We are assuming that these underlying channels can change across (but not within) frames. Across all L frames, we suppose that {h^} exhibit common sparsity in that all these channel vectors are obtained as a linear combination of a common set of steering vectors from G_h. An analogous comment may apply to (g^) over the dictionary G_fl. Further, let us collect all the used pattern vectors as rows of the Ψ_Ω as before. Finally, defining the matrix G = Ψ_Ω6, we can pose the problem of interest as: f|GW-ZB² min h T ll W IU+ χΣρ^ II n^W II +y∑ ¾ II ¾W ||}, (18) weic^(Dfc^Dtf⁺¹⁾*ⁱ <7=1 where T, y > 0 are hyper-parameters. picks row's (p — 1 )D_g + 1 through pD_g of W for all 1 < p < D_h, whereas IlJW picks rows q,q + D_g, — ,q + (D_h — 1 )D_g of W for all l ≤ q ≤ D_g. Some comments on the formulation in (18) may be on order. Note that the term y

enforces row-wise product sparsity assured by

Lemma 1. On the other hand, II W II. enforces low-rank property, which is important when the underlying channels do not change sufficiently across frames. An embodiment ADMM based algorithm is provided to solve (18) which is referred to as ADMM-SS. To formulate the ADMM-SS algorithm, (18) may be re-written, by introducing auxiliary matrix variables along with additional constraints, as in (19): j|GW-Z|² min T II U IU+ (19) w,u,v,xeic^(Dh°fl^+1)xL x=w,v=w,u=w

Then, the corresponding augmented Lagrangian may be formulated as in (20):

f II W - U ||²+ f || W - V ||²+ | || W - X ||²+ Ifltr(L*(W - U)) + Ifitr(L½(W - V)) + Ifltr(L½(W - X)))}. (20)

The ADMM-SS algorithm may include the following steps in each of its iteration: Solve ar£rrmn_UeIC(£>_hD_fl+i)xi.{£'(W, U, V, X, L_lf L₂, L₃)} to obtain 0, which is equivalent to:

which permits a standard singular value thresholding solution summarized in Lemma 2.

Solve arpmtn_Ve|C(o_hDfl+i)x_L{£^,(W, U, V, X, L_lt L₂, L₃)} to obtain 9, which is equivalent to:

which can be solved using the corresponding result summarized in Lemma 2. Solving arymi?i_XeIC(DhDfl+i₎x_L{_>C^,(W, U,V,X, L₁, L₂, L3)} to obtain £ proceeds in an exactly analogous manner.

Solve arymtn_Welc(D_hD_a+i)xL{X^,(W, 0, V, £, L_!, L₂, L₃)} to obtain W is equivalent to an unconstrained least-squares problem.

Update the Lagrange matrix variables as

L₂ + p(W- V)& L₃ = L₃ +p(W-X).

Note that the inputs required for the ADMM-SS are the expanded dictionary matrix G, the observations matrix Z, initial choice values for all matrix variables, and a choice of hyper- parameter values. Then, using the output W of this algorithm, the span of the columns of G corresponding to rows of W having respective norms above any given threshold, may be declared to be the desired subspace.

Covariance Projection based Methods

In some emodiments, a covariance projection based method is provided for channel reconstruction. Assume that there is no direct path between the transmitter and the receiver for convenience purposes.. We also suppose that the LoS components in both h, g are known (upto arbitrary phase terms), i.e., we suppose thah = h +

where h^los, g^los are known (based on location information), and exp(/0_h), exp (/¾) are arbitrarily chosen unknown phase terms. Moreover, following conventional modeling, we may suppose that the non-LoS (nLoS) component vectors can be modeled as h = Ggtg & g = Ggtg. In this case, tg, tg need not be sparse vectors, and are instead complex proper normal and mutually independent. In addition, each one of them has a diagonal covariance matrix respectively, i.e., Dg =

£[tgt~] & = £[tgtg] for some diagonal matrices Dg ≥ 0, D_a > 0. Two diagonal matrices may then be formed: = diag{h^los) and D^⁰⁵ = diag{g^los). Let X = ™ —

(h^los Q g^los)(h^los Q g^los)^† denote the sample covariance matrix. In an embodiment, the following results (Preposition 1 and Corollary 1) are provided based on which subsequent subspace estimation techniques may be derived.

Proposition 1 Lei Δ g = E[/i/i^†] & Ag = £[gf9^†] Then, the covariance, Δ of the element-wise product channel h Q g permits the expansion

Δ = Θ Δ, + Djr4e(Dln^† + D£¾*(DjP)t. (21)

Moreover,

^ΔΗ o = (Gg 0 Gg)(Dg 0 ¾)(Gg » G_d)^†. (22)

Corollary 1 For a uniform planar array RIS, both dg and Ag are block Toeplitz so that from array response (6) and Proposition 1, we can deduce that covariance of the product channel h Q g is also block Toeplitz.

Based on above, an embodiment subspace estimation method is provided in a case where the LoS components are zero vectors. The techniques of the embodiment may also be applied to other cases with departing from the principle and spirit of the embodiments. Note first that using the Schur product theorem, it may be deduced that Δ¾ Q Δ¾ > 0 and rank(Ag Q Ag) < rank(Ag) rank(Ag). Then, let dg = diag{Dg), dg = diag(Dg) with D = Dg 0 Dg and G = Gg o Gg. the following problem may be considered:

Further, letting d = vec(D) and x = vec(X), (23) may be re-formulated as:

After further manipulations, we obtain )(dg <g> dg) ||²}, a (25)

where G □ G denotes the Khatri-Rao product or the column-wise kronecker product of G and G. (25) may be sub-optimally solved using alternating non-negative least squares minimization, where dg, dg may be optimized in an alternating manner subject to nonnegativity constraints. On the other hand, if we drop the non-negativity constraints on dg, dg in (25), a relaxed formulation may be obtained as: )(dg ® dg) ||²}. (26)

The problem (26) remains anon-convex problem, and it can be efficiently solved via variable projection methods. In an embodiment, we can employ alternating least squares minimization with the advantage that each step may be solved in closed form. Let dg, dg be tiie optimized solutions so obtained. Then, 5 may be re-constructed using dg, d_g as 6 = diag(dg ® dg) and the desired subspace is the dominant Eigenspace of GOG. A useful observation is that the power angle spectrum can be assumed to be invariant across relatively widely separated frequency bands. To obtain the desired subspace for a different frequency band, we may only need to determine the corresponding matrix G, e.g., G' (by simply computing the underlying array response vectors (cf. 6) using the right carrier frequency), upon which the desired subspace is the dominant Eigenspace of

G'BG'. The following provides Lemma 2.

Lemma 2 For any X E IC^mxn, τ > 0, let X = USV^* denote its economy-sized SVD so

where [.]+ sets all negative diagonal elements of its diagonal matrix argument to zero and retains the non-negative ones. For any X E IC mxn , T > 0, we have: a^r¾Mf^|IY-^{xl|2 +IIY|}} (|X|-1/T)₊

IXI X, where (x)₊ = max{x, 0} V x e IR. For any x e IC^m,r > 0 and any diagonal matrix D = diag{d} > 0, let X = 77 , then we have:

|X| +11 Dy II,}

where now |x| denotes element-wise magnitude operation on the input vector x, whereas [x]₊ element-wise performs the (.)+ operation.

Figure 8 is a diagram illustrating an embodiment method 800 for IRS aided communications in a TDD system. The method 800 performs reconstruction of a communication channel based on the expression (1). An AP 802 in the TDD system receives pilots from a UE 804 (step 820). The TDD system also includes an IRS 808 for assisting communications between the AP and the UE, and an IRS controller 806 for controlling the IRS. The AP 802 receives the pilots from the UE 804 (820), and performs channel construction of the downlink channel between the AP 802 and the UE 804 based on the received pilots (step 824). Optionally, the UE 804 may send another set of pilots to tiie AP 802 (step 822). The pilots sent by the UE 804 in steps 820 and 822 may be salt at different time, where the IRS 808 has different patterns. The AP 802 may perform the channel construction based on both the pilots sent in steps 820 and 822. The channel to be reconstructed may be modeled as the expression (2) based on subspace information, or as the expression (7) based on sparsity information. From thereon, a joint optimization problem may be formulated and solved to obtain the reconstructed channel, as discussed above. The AP 802 may determine an IRS patter based on the reconstructed channel (826). The AP 802 sends the determined IRS pattern to the IRS controller 806 (step 828), instructing the IRS controller 806 to adjust the pattern of the IRS 808 according to the determined pattern. The IRS controller 806 adjusts the pattern of the IRS 808 (step 830). The AP 802 may perform transmissions to the UE 804 based on the reconstructed channel (step 832).

Figure 9 is a diagram illustrating an embodiment method 900 for wireless communications in a FDD system. The method 900 performs reconstruction of a communication channel based on the expression (1) using the subspace aided approach with quantized observations. An AP 902 in the FDD system may send pilots to a UE 904 in the FDD system (step 920). The FDD system also includes an IRS 908 for assisting communications between the AP 902 and the UE 904, and an IRS controller 906 for controlling the IRS 908. The UE 904 receives the pilots from the AP 902 (920). The UE 904 may generate quantized observations of the received pilots and send the quantized observations to the AP 902 (step 922). The AP 902 receives the quantized observations, and performs channel construction of the downlink channel between the AP 902 and the UE 904 based on the quantized observations (step 924). As discussed above, an optimization problem may be formulated based on the subspace information and solved to obtain the reconstructed channel. The optimization problem may be formulated using a phase retrieval based approach or a non-convex quadratically constrained quadratic programming approach, as shown in the expressions (9) and (10), respectively. The AP 902 may determine an IRS pattern based on the reconstructed channel (926). For example, the AP 902 may select an IRS pattern from available patterns based on the reconstructed channel. The AP 902 sends the determined IRS patter to the IRS controller 906 (step 928), instructing the IRS controller 906 to adjust the patter of the IRS 908 according to the determined pattern. The IRS controller 906 adjusts the pattern of the IRS 908 (step 930). The AP 902 may perform transmissions to the UE 904 based on the reconstructed channel (step 932). Figure 10 is a diagram of an embodiment method 1000 for IRS-aided communications. The method 1000 may be indicative operations performed by a first communication device, such as an access point (AP) or a user equipment (UE), in a time division duplex (TDD) system. As shown, at step 1002, the first communication device receives a pilot signal sent by a second communication device to the first communication device in a first communication channel. The first communication channel includes an intelligent reflecting surface (IRS) aided reflective channel, and the IRS aided reflective channel includes a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the first communication device. The first communication channel further includes a second direct channel between the first communication device and the second communication device. At step 1004, the first communication device generates sparsity information of the first communication channel, optionally by performing beam domain sparsity analysis of the first communication channel. At step 1006, the first communication device performs channel reconstruction of the first communication channel based on the received pilot signal, the sparsity information and a location of the IRS, to generate reconstructed-channel information of the first communication channel. The channel reconstruction may further be performed based on a location of the second communication device and a first reflective pattern of the IRS. At step 1008, the first communication device communicates with the second communication device in the first communication channel based on the reconstructed- channel information of the first communication channel.

Figure 11 is a diagram of another embodiment method 1100 for IRS-aided communications. The method 1100 may be indicative operations performed by a first communication device, such as an access point (AP) or a user equipment (UE), in a time division duplex (TDD) system. As shown, at step 1102, the first communication device receives a pilot signal sent by a second communication device to the first communication device in a first communication channel. The first communication channel includes an intelligent reflecting surface (IRS) aided reflective channel, and the IRS aided reflective channel includes a first IRS channel between an IRS and the second communication device and a second IRS channel betw een the IRS and the first communication device. The first communication channel further includes a second direct channel between the first communication device and the second communication device. At step 1104, the first communication device determines subspace information of the first communication channel based on historical data about channel measurement and reconstruction of the first communication channel. At step 1106, the first communication device performs channel reconstruction of the first communication channel based on the received pilot signal, the subspace information and a reflective pattern of the IRS, to generate reconstructed-channel information of the first communication channel. At step 1108, the first communication device communicates with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

Figure 12 is a diagram of another embodiment method 1200 for IRS-aided communications. The method 1200 may be indicative operations performed by an access point (AP) in a frequency division duplex (TDD) system. As shown, at step 1202, the AP sends a pilot signal to a UE in a first communication channel of the TDD system. The first communication channel includes an intelligent reflecting surface (IRS) aided reflective channel, and the IRS aided reflective channel includes a first IRS channel between an IRS and the UE and a second IRS channel between the IRS and the AP. The first communication channel further includes a second direct channel between the AP and the UE. At step 1204, the AP receives, from the UE, information of signal strength of a received pilot signal, where the received pilot signal is the pilot signal received by die UE in the first communication channel. At step 1206, the AP determines subspace information of the first communication channel based on historical received signal strength measurement data of signals received by the UE in the first communication channel. At step 1208, die AP performs channel reconstruction of the first communication channel based on the information of the signal strength of the received pilot signal, the subspace information, and a first reflective patter of the IRS, to generate reconstructed- channel information of the first communication channel. At step 1210, the AP communicates with the UE in the first communication channel based on the reconstructed-channel information of the first communication channel.

Figure 13 is a block diagram of an embodiment processing system 1300 for performing methods described herein, which may be installed in a host device. As shown, the processing system 1300 includes a processor 1302, a memory 1304, and interfaces 1306- 1310, which may (or may not) be arranged as shown in FIG. 13. The processor 1302 may be any component or collection of components adapted to perform computations and/or other processing related tasks, and the memory 1304 may be any component or collection of components adapted to store programming and/or instructions for execution by the processor 1302. In an embodiment, the memory 1304 includes anon-transitory computer readable medium. The interfaces 1306, 1308, 1310 may be any component or collection of components that allow the processing system 1300 to communicate with other devices/components and/or a user. For example, one or more of the interfaces 1306, 1308, 1310 may be adapted to communicate data, control, or management messages from the processor 1302 to applications installed on the host device and/or a remote device. As another example, one or more of the interfaces 1306, 1308, 1310 may be adapted to allow a user or user device (e.g., personal computer (PC), etc.) to interact/communicate with the processing system 1300. The processing system 1300 may include additional components not depicted in FIG. 13, such as long term storage (e.g., non-volatile memory, etc.).

In some embodiments, the processing system 1300 is included in a network device that is accessing, or part otherwise of, a telecommunications network. In one example, the processing system 1300 is in a network-side device in a wireless or wireline telecommunications network, such as a base station, a relay station, a scheduler, a controller, a gateway, a router, an applications server, or any other device in the telecommunications network. In other embodiments, the processing system 1300 is in a user-side device accessing a wireless or wireline telecommunications network, such as a mobile station, a user equipment (UE), a personal computer (PC), a tablet, a wearable communications device (e.g., a smartwatch, etc.), or any other device adapted to access a telecommunications network. In some embodiments, one or more of the interfaces 1306, 1308, 1310 connects the processing system 1300 to a transceiver adapted to transmit and receive signaling over the telecommunications network. Figure 14 is a block diagram of an embodiment transceiver 1400 adapted to transmit and receive signaling over a telecommunications netw ork. The transceiver 1400 may be installed in a host device. As shown, the transceiver 1400 comprises a network-side interface 1402, a coupler 1404, a transmitter 1406, a receiver 1408, a signal processor 1410, and a device-side interface 1412. The network-side interface 1402 may include any component or collection of components adapted to transmit or receive signaling over a wireless or wireline telecommunications network.

The coupler 1404 may include any component or collection of components adapted to facilitate bi-directional communication over the network-side interface 1402. The transmitter 1406 may include any component or collection of components (e.g., up- converter, power amplifier, etc.) adapted to convert a baseband signal into a modulated carrier signal suitable for transmission over the network-side interface 1402. The receiver 1408 may include any component or collection of components (e.g., down-converter, low noise amplifier, etc.) adapted to convert a carrier signal received over the network-side interface 1402 into a baseband signal. The signal processor 1410 may include any component or collection of components adapted to convert a baseband signal into a data signal suitable for communication over the device-side interfaced) 1412, or vice-versa. The device-side interfaced) 1412 may include any component or collection of components adapted to communicate data-signals between the signal processor 1410 and components within the host device (e.g., the processing system 1300, local area network (LAN) ports, etc.).

The transceiver 1400 may transmit and receive signaling over any type of communications medium. In some embodiments, the transceiver 1400 transmits and receives signaling over a wireless medium. For example, the transceiver 1400 may be a wireless transceiver adapted to communicate in accordance with a wireless telecommunications protocol, such as a cellular protocol (e.g., long-term evolution (LTE), etc. ), a wireless local area network (WLAN) protocol (e.g. , Wi-Fi, etc.), or any other type of wireless protocol (e.g., Bluetooth, near field communication (NFC), etc.). In such embodiments, the network-side interface 1402 comprises one or more antenna/radiating elements. For example, the network-side interface 1402 may include a single antenna, multiple separate antennas, or a multi-antenna array configured for multi-layer communication, e.g., single input multiple output (SIMO), multiple input single output (MISO), multiple input multiple output (MIMO), etc. In other embodiments, the transceiver 1400 transmits and receives signaling over a wireline medium, e.g., twistedpair cable, coaxial cable, optical fiber, etc. Specific processing systems and/or transceivers may utilize all of the components shown, or only a subset of the components, and levels of integration may vary from device to device. While the present application is described, at least in part, in terms of methods, a person of ordinary skill in the art will understand that the present disclosure is also directed to the various components for performing at least some of the aspects and features of the described methods, be it by way of hardware components, software or any combination of the two. Accordingly, the technical solution described in the present disclosure may be embodied in the form of a software product. A suitable software product may be stored in a pre-recorded storage device or other similar non-volatile or non-transitory computer readable medium, including DVDs, CD-ROMs, USB flash disk, a removable hard disk, or other storage media, for example. The software product includes instructions tangibly stored thereon that enable a processing device (e.g., a personal computer, a server, or a network device) to execute embodiments of the methods disclosed herein.

It should be appreciated that one or more steps of the embodiment methods provided herein may be performed by corresponding units or modules. For example, a signal may be transmitted by a transmitting unit or a transmitting module. A signal may be received by a receiving unit or a receiving module. A signal may be processed by a processing unit or a processing module. Other steps may be performed by a determining unit/module, a channel reconstructing unit/module, an IRS adjusting unit/module, an IRS pattern selecting unit/module, a channel estimating unit/module, a subspace estimation unit/module, a sparsity analysis unit/module, and/or an observation quantization unit/module. The respective units/modules may be hardware, software, or a combination thereof. For instance, one or more of the units/modules may be an integrated circuit, such as field programmable gate arrays (FPGAs) or application-specific integrated circuits (ASICs).

Please also refer to an Appendix to the specification for further description of the present disclosure. Although the description has been described in detail, it should be understood that various changes, substitutions and alterations can be made without departing from the spirit and scope of this disclosure as defined by the appended claims. Moreover, the scope of the disclosure is not intended to be limited to the particular embodiments described herein, as one of ordinary skill in the art will readily appreciate from this disclosure that processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, may perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.

APPENDIX

Methods and Apparatus for Channel Reconstruction in Intelligent Surface Aided Communications bstract — InteUgent reflecting surface (IRS) has recently waves to some degree. Note that IRS operates differently rged as a promising cost-effective technology for enhancing than other related technologies such as amplify and forward capacity and energy efficiency In fotnre wireless communicarelaying, backscatter communications etc. Indeed IRS impacts systems. An IRS comprises of many low-cost passive antenna ments that can smartly reflect the impinging electromagnetic the incident signal passively without additional amplification es for performance enhancement However, channel estimaand thereby avoids energy consumption entailed by need Is challenging for the IRS- aided wireless communications for amplification. Moreover, need for RF chains that allow to the lack of sensing capability in the passive IRS elements. baseband processing of incident signals can also be eschewed his work, we propose novel channel reconstruction schemes [4], Indeed, proposed IRS achitectures employ varactor diodes ch do not require any active elements on the IRS panel. ead, by exploiting the reciprocity In TDD systems, we rely or other micro electrical mechanical systems (MEMS) techhe analog (complex) observations obtained at the transmitter, nology, so that in such architectures electromagnetic (EM) reas in-case of FDD we rely on quantized signal strength properties of the IRS are defined by its micro-structure, which surements fedback by the receiver to the transmitter (or IRS can then be programmed, to a certain degree, to vary the phase, roller). Our novel formulations for TDD systems combine amplitude, frequency and even orbital angular momentum of rix completion with beam-domain sparsity or suhspace side rmation and lead to !mp!ementable ADMM based algorithms. an incident EM wave. Consequently, an IRS can modulate a se algorithms In turn yield the reconstructed channel vector, radio signal without using a mixer and radio frequency (RF) as a byproduct an optimized IRS pattern that, subject chain. Furthermore, by smartly adjusting the phase shifts of urther ightweight processing, Is we· suited for facilitating all the passive elements at IRS, its reflected signals can sum- transmission to the intended user. Our novel formulations FDD systems also exploit subspace side-information and up: coherently with the signals from other paths at a desired based on either phase-retrieval techniques or on certain receiver to boost the received signal power, or destructively convex quadratiraly constrained quadratic programming. at non-intended receivers to suppress interference as well as ach case, we propose effective low-complexity algorithms. enhancing security and privacy. thermore, we propose candidate algorithms for subspace Moreover, from the implementation perspective. IRSs posmation and also provide a rec^e for designing IRS pattern ors that can be gainfully used during the training phase. sess appealing advantages such as low profile and conformal ally, we provide simulation results generated using the ShnRIs geometry. These advantages enable them to be easily attached to or removed from a wall or ceiling, thus providing high ndex Term § — Intelligent reflecting surface, channel reconflexibility for their practical deployment [5], For example, ction, ADMM, non-con vex QCQP, Proximal distance based by installing IRSs on the walls or ceilings which are in rithms. line-of-sight (LoS) with an access point (AP) or base-station (BS), signal strength in the vicinity of each IRS can be significantly improved. In addition, integrating IRSs into the

I. INTRODUCTION existing networks (such as cellular or WiFi) can be made

Wireless networks have experienced a substantial increase transparent to the users without the need of any change in capacity over the past decade due to several technothe hardware and software of their devices. All the above cal advances, including massive multiple-input multiple- features make IRS a compelling new technology for future put (mMIMO), millimeter wave (mmWave) communicawireless networks, particularly in indoor applications with high s and ultra-dense deployments of small cells. However, density of users (such as stadium, shopping mall, exhibition lementing these technologies efficiently is a challenging center, airport, etc.). Utilizing a large number of elements due to increased hardware cost as well as increased whose electromagnetic response (e.g. phase shifts) can be er consumption [1] — [31- To address this need for reliable controlled by simple programmable PIN diodes [6], an IRS energy efficient future cellular networks without incurring can reflect the incident signal and generate a directional beam hibitive costs, researchers across the globe are studying difin a desired intended direction, and thus enhancing the link nt techniques to improve the system performance and have quality and coverage. The phase of each IRS clement can icularly focused on providing control over the propagation be adjusted through the PIN diodes which are controlled by ronment an RIS-controller. In one architecture such a controller is ntelligent reflecting surface (IRS) has recently emerged as a connected to an access point (AP) via a backhaul link and mising cost-effective technology for enhancing the capacity can be commanded by that AP to employ an appropriate energy efficiency as well as improving coverage in future phase pattern across its elements. In another architecture the eless communication systems. IRS can be a thin two- controller itself can be capable to deciding such an appropriate ensional metamaterial (i.e., a material that is engineered) phase patter. ch has the ability to control and impact electromagnetic All the aforementioned advantages such as throughput

estmat on t at explo ts t e part cular spars ty structure

«∞«««∞ Direct link Indkeet/llefleet link possessed by the effective channel. In addition, we also propose another subspace estimation technique that estab1. An illustration of IRS assisted network. lishes and then exploits the structure of the covariance of the effective channel.

• We provide a recipe for designing IRS pattern vectors s/coverage improvements in an IRS-assisted communicathat can be effectively used during the training phase. systems critically depends on quality of the estimates • Finally, we demonstrate the performance of our algohe underlying channels. Indeed, without availability of rithms over an indoor scenario emulated using the open- able estimates the phase patterns (or IRS response) cannot source SimRis simulator [9] ptimized. However, channel estimation in IRS aided com¬nications is quite challenging in practice. This is because, rder to reduce implementation costs, the IRS generally is C. Notations equipped with any radio frequency (RF) chain and thus Throughout this paper, scalar, vector, and matrix are denoted s requisite baseband processing capability. As a result, it by lowercase, bold lowercase, and bold uppercase, i.e., a,mpossible for the IRS itself to estimate its channels to the a, and A respectively. (·)^τ, (.), (·) _\ and rank(·) denote smitter or any of the users. transpose, conjugate, conjugate transpose, and rank, respectively. ®, Θ are used to denote the kronecker product and Prior Arts Hadamard product, respectively, between any two identically n [3], authors studied channel estimation over an IRS aided sized matrices. We reserve j to denote imaginary unit of a ti-user MIMO system by exploiting compressive sensing. complex number, i.e., j — >/—!· Following Matlab notation, on sparsity among all the users for any a G (C^m. we let ding fa) G ID roxm y further utilize the comm denote the associated diagonal matrix while for mxm arises due to the channel between BS and IRS being any A G ID we mon for all users. let diag{A} G (D^m denote a vector comprising of diagonal n [7], authors proposed a joint beamforming and IRS elements of A. Finally, I_m denotes the m x m identity matrix. raction matrix optimization in an IRS assisted downlink work, wherein the IRS elements can be assigned discrete II. SYSTEM MODEL se shifts Consider a network comprising of an IRS with N elements n [8], authors proposed a wireless virtual reality prototype and a transmitting and receiving node with one transmit and mmWave link by introducing an mmWave mirror that one receive antenna, respectively. The IRS is a passive surface reconfigure itself. Their prototype named MoVR ensures which applies a pattern φ G F^N such that the n^th entry ofh data rate in the presence of mobility by overcoming the φ, denoted by φ„, models the multiplicative impact of the kage problem of mmWave links but requires an active n^th IRS element on its incident signal. Here J⁷ is a finite Wave mirror that can also amplify signals. set of complex scalars with magnitudes no greater than (but not necessarily equal) to unity, which models practical (non- Contributions ideal) lossy elements. Indeed, one possible choice for F is ur contributions in this work can be summarized as the model from [10] which suggests elements of the form ows: exp tfa) (fl_mi_n +· (1 - /¾nin)(sin(a - a) -·- 1)/2)"') for differWe propose novel channel reconstruction schemes which ent uniformly sampled choices of a G [— π, π) and where other do not require any active elements on die IRS array. parameters as set as ά = 1.6, ft_mm = 0.2 & ·γ = 1.6. Then, Instead, by invoking reciprocity in TOD systems we rely suppose there is any given set of J pattern vectors, {ø(}/__!, on the analog observations obtained in the uplink to and let Φ denote a J x N matrix whose J rows contain these reconstruct the effective downlink channel. On the other J given pattern vectors. Further, let p = [p_t , · · · , p-r]^T denote hand, in-case of FDD we rely on quantized signal strength a vector with unit magnitude entries such that pi is the pilot measurements fedback by the receiver to the transmitter. symbol transmitted by the transmitting node in the t^th slot of a Our novel formulations for TOD systems combine matrix training phase spanning T training symbol durations. In TDD completion with beam-domain sparsity or subspace side- systems relying on reciprocity we suppose that the receiver of information and lead to implementable ADMM based alinterest transmits pilots in the uplink during the training phase gorithms. These algorithms in turn yield the reconstructed and the transmitter of interest collects received observations econstruct the downlink channel. In FDD systems the We pose the joint problem of optimized pattern recovery smitter of interest transmits these pilots and the receiver and channel reconstruction as nterest can collect observations, process them and report ntized processed observations to the transmitter. The latter min I ^T!IY|U -t- TjrlK!l² + ^||s® Y - z|| eedback reports to reconstruct the channel. Yec^{Jzr 2} + then use the f ,ce3B^s ither case the transmitter can then decide the choice of ±!|γ ~ Φυζ_Ρηι²} , (3) phase pattern for the data communication phase based on reconstructed channel and convey this choice to the IRS where ||.||, denotes the nuclear norm and T, 7 > 0 denote roller via a low-rate side-channel (or link). hyper-parameters. Recall that the nuclear norm ensures a low- he observation at the receiving node corresponding to the rank recovery and indeed the factor τ will bias the retrieved symbol duration can be expressed as solution towards having a lower rank. The (2 norm imposed

Z_f = (g^Tdiag{0,.}h 4- /i_dirJP_f 4- m on ζ is as per ridge regression. We note here that in many relevant scenarios S « N so no further sparsity in ζ is

= [<Ai,, l][(hG g)^T, h_di_r.J^Tpt + %, expected and using (_\ norm instead would be inappropriate. it e Ω & 1 < t < T, (1) Although, (3) is a convex optimization problem it still requires an efficient algorithm. Towards this end, we introduce re A_dir. models the direct channel and ø<, denotes the additional variables in order to formulate the problem of ern employed by the IRS during the t^th training symbol interest as ration and r/t ~ C.V(0, N₀) denotes the additive complex mal noise. In (1) we have let g and h €⁽C^N denote mm {T||Y||. + channel vectors modeling the IRS to reveiving node link C,Y,x,c JllCII² + ^!icil² + i||s Θ x - z||²} , the IRS to transmitting node link, respectively. We let Ω s.t. X = Y, C = X - *UCp^f. ote the index set of pattern vectors that are employed in raining or measurement phase. All pattern vectors indexed Then adopting the framework of ADMM we obtain the augΩ are also subsumed as rows of Φ. We also let Φ denote mented Lagrangian denoted by £(Y, X, C, ζ, L_lt L^) as in J x ( N ÷ 1) matrix obtained by appending a J x 1 column (4). where ≥ 0 are additional hyper-parameters while ll ones to Φ. Li , L₂ are Lagrange variables and BZtr(.) denotes the real part of the output yielded by matrix trace operation. Our ADMM based alternating optimization (ADMM-AO) approach to solve SUBSPACE AIDED JOINT CHANNEL RECONSTRUCTION (4) comprises of the following steps in each of its iteration²: AND PATTERN OPTIMIZATION

• Solve argminYgi;jxr{£(Y,X,C, (.Li.Lz)} which is

We first focus on a TDD system. We define a J x T matrix equivalent to whose (it, l)^th element is ¾ for all 1 < t < T and which zeros elsewhere. Consider a genie-aided noiseless system Ϋ “ re in each time-slot we could (with genie’s assistance)

in an expanded set of J observations, one for each of which permits a standard singular value thresholding J available pattern vectors. These observations can be solution which we have recalled in Lemma 2. esented over T slots as Φ[(1ι 0 g)^T, hair.]¹ P^†- Now, • Solve argmin_Xgi_:jxT _CgI.j » {£( Ϋ , X, C, ζ, Li , L₂)} U denote an (IV + 1) x S matrix whose columns span which is equivalent to an unconstrained quadratic S -dimensional subspace in which the composite product programming problem. nnel [(h 0 g)^T, h_dir.]^T lies.¹ Such a matrix for instance • Finally, the Lagrange matrix variables are simply updated be constructed by the first L dominant Eigenvectors of as Li — Li 4- pi(Y — X) & Γ1₂ — 1»₂ 4- /¾{C - X (N 4- 1) x (IV 4- 1) covariance matrix of [(h®g)^T, Adir.]⁷"· a result we can express The inputs required for the ADMM-AO are subspace matrix U, initial choice values for all matrix variables, and a choice

[(h 0 g)^T, /i_dir.]^r « 11ζ. (2) of hyper-parameter values. arly in the absence of any subspace side-information, we From the output ζ we obtain the reconstruction U(. In simply set U = I. Thus, the aforementioned genie-aided addition, using the obtained *U(p^f, which we note is eless observations can also be represented as the matrix maintained by the algorithm, we determine the row of this Ιζρ*. Clearly, we have access to only a subset of elements matrix having the largest norm. We use the pattern vector inm this matrix which are further corrupted by noise. Upon Φ corresponding to this row as the starting point of a low- ning a J x T sampling matrix S whose (it, t)^ih element complexity enhancement which we refer to as linear pass. for all 1 < f < T and which has zeros elsewhere we In particular, let [φ, 1] be the pattern vector corresponding that we have available in Z noisy versions of the sampled to the aforementioned maximal normed row. The linear passments in the matrix 8Φυζρ⁷. comprises of N steps. Let ώ denote the i^th element of the vector \φ, 1] and define f = U£ with f< denoting the latterMore precisely the subspace in which must of the enetgy of the composite uct channel is concentrated with high probability. ² Each iterations takes outputs of previous iteration as its starting point.

tor’s i^th component. Then, letting M — {!, · ■ · , N}, in the directions. We can now proceed to obtain an augmented step of the said linear pass, we update φ_ί as Lagrangian given by (8). where we have introduced auxiliary

2 matrix variables X, C e <C^JxT. Based on (8), we can devise an ADMM based algorithm, referred to as ADMM-L1. following *>i <- arg max < f_N+i + V' r_ki>_k ÷ frf (¾ the same approach adopted in Section Hi For brevity, we skip { KWH the details but note that the main difference is the sub-problem pattern vector obtained post linear pass is declared to be to update t which is optimized patter for single-user data communications. t

V. BEAM DOMAIN SPARSITY AIDED JOINT CHANNEL which can be solved using the corresponding result summaRECONSTRUCTION AND PATTERN OPTIMIZATION rized in Lemma 2. n this section we assume no subspace information is V. OPTIMIZING PATTERN VECTORS FOR TRAINING lable. We first consider the case where one of the two stituent channels in the product channel is LOS-only and In this section we consider the problem of obtaining a her that the associated LOS steering vector is known. suitable set of T training vectors that exploit the subspace side- latter can be determined based on location information. information provided as the cohimn-span of some given matrix pose, without loss of generality, that g is known up-to a U e(C^,/v'+1)^xS. This problem is made particularly challenging plex scaling factor and so that g = η¾|· is known. Then, let due to the finite alphabet constraint that is imposed on the be the Λ^Γ x D_k dictionary matrix for n such that h permits entries of all patter vectors. We will tailor a Generalized arse representation under this dictionary, i.e„ we have that Lloyd algorithm for this purpose. Let us first define a set of — Gftth for some sparse vector t_¾ e (C⁰¹¹. Here, in the all feasible pattern vectors ence of any other side information we can assume G_¾ to {Φ eC^lxiV N} (9) dentical to the matrix whose columns are steering vectors fined by the IRS array geometry) uniformly sampled on Our objective is to design a set of T feasible patters (for angular grid. Indeed, for a commonly used uniform planar training), all of which lie in £ via the following formulation y (UPA) layout the elements of each column of the latter max max l<f< |[0t. l]r|²] I (10) rix are given by {*.esr„ M T where the expectation is over the composite product chanp (6)

nel, denoted here as r, conditioned upon the given side- information. The challenges are two-fold. The first one is that re 1 < αι,α₂ < '/N, A denotes the wavelength and φ,θ we do not have knowledge of the conditional distribution of the azimuth and elevation angles measured with respect to r and must rely on only subspace side-information. The other IRS boresight, respectively. is the finite alphabet constraint on the patter vectors. We ext, let G — [diag(g)G_/,, 0jv; 0^,, lJ and note that the address the first issue by generating a large set of realizations posite product channel can be expressed in this special as r t = U(r for t 6 {1, · · · , L'} for some large L' » 1 as and where {(r) are realizations of an isotropically distributed

[(h Θ g)^T, /idir.]^T = Gt (7) vector ini . This represents a worst case assumption in that no other prior information is assumed apart from the subspace some sparse J¾ 4- 1 length vector t. We can tailor the side-information. We tackle the finite alphabet restriction viamulation in (3) to this scenario as an iterative proximal distance based approach. Let {4>t £ φΐ^χΝ- }ιτ/₌₁ denote a candidate codebook of pattern vectors at min {pY||. + 7l|w

Y.t _/1ti|₁ + ±||s eY --- z||²-l·- the current iteration and let its feasible twin be {ø* e £}£. _lt and let p > 0 denote a given penalty factor Each iteration , comprises of the following two steps:

e In the first step we partition the set of sample vectors re W_¾ t 0 denotes a given diagonal matrix of weights. {r_f} into T regions H_t where H_t = {t : |[<p_t, l]r₍|² > articular, we can set a low weight (relative to others) for |[0_¾, 1]Γ₍|² Vfc ≠ t). Then we solve the following subring vectors (corresponding to columns of G) which we problem to update riori know to be more likely to be present For instance, T we know the LOS component direction in h we can set max ding weight to be much smaller relative to other {♦ l_t<* Et<T {1∑ 1= li∑ |[0ti l]r<|² — ||0t - 0t!| (11) correspon ¹-*¹ "l eH_t 1=1 1

problem in (11) decouples across 1 < t < T and is an arc optimized keeping ζ, e fixed. Finally, the additive error onstrained quadratic program that can be optimally solved vector c is optimized keeping other two sets of variables fixed. losed form. Update {iiYt-ι to be the obtained solution. This process is repeated till convergence. Note that since each n second step we project each it onto £, i.e., we update step can be optimally solved convergence is guaranteed since h it as it t- argrnin^_e^{||<^— ||²}. This latter problem the objective value decreases after each step. lso readily solvable by elementwise quantizing it to its est point in T. se two steps are repeated until convergence which we note B. Non-convex Quadratically constrained Quadratic programuaranteed since the objective of (II) is monotonically ming approach reasing across iterations. The performance of the above The second approach targets an FDD scenario in which the rithm is sensitive to the choice of p as well as the initial user of interest determines average received signal strength , it};V t = 1, · · · , T. We have tried multiple random ini(associated with each choice of pattern vector) and feeds zations with each such initialization including min{S, T} it back to the controller after quantization. Specifically, we mns of the matrix U in {<£_#} and their projections onto suppose that for each f : 1 < t < T the controller has access to n {<¾}. To address the other dependency we progressively C(|[0i.· Ij[(h0g)^r, h_dir.H²) Using this we propose another ease the choice of p over batches, each batch comprising formulation introduced below: everal iterations, starting from a small value p <- amplifying it from one batch to the next as p <— Op “•MUCH²} - some amplification factor Θ > 1. Upon convergence we ose {i_t}T-i ^as the pattern vectors for training. We can

v i < t < T. (13) repeat the above procedure to find additional J—T pattern ors in £ in order to determine the matrix Ψ used in our where 7t'^b', 7“'^h' are upper and lower bounds corresponding rix completion approach. Indeed, we again use the iterative to the boundaries of the quantizer’s bin in which |[0,₍, 1]U£|² cedure for designing J pattern vectors but where T of them lies. The problem in (13) is a non-convex quadratically conalways fixed to the T training vectors obtained before. strained quadratic programming. It can be solved in an efficient albeit sub-optimal manner using the K-best methodology proposed in [11]. Alternatively, we can employ a proximal SUBSPACE AIDED CHANNEL RECONSTRUCTION USING distance based algorithm that is detailed below. Note that SIGNAL STRENGTH OBSERVATIONS the formulation in (13) appears intractable since it has T n this section we consider the scenario in which quantized constraints each describing a non-convex constraint seL The al strength (or magnitude) observations arc available along feasible set of (13) is the intersection of T feasible sets each h subspace information. Due to the coarseness of available described by one associated such constraint Moreover, this ervations we focus on channel reconstruction only. The feasible set is guaranteed to be non-empty since it is based ern vector for data communications can be obtained in a on actual user feedback. ³ However, a key observation is that ond step optimization using the reconstructed channel. We the following problem (which can be regarded as projection pose two methods for channel reconstruction. of any vector in ζ e(E^s onto a feasible set described by any one constraint) is a rare non-trivial example of a non-convex Phase retrieval based approach problem that can be efficiently and optimally solved: he first approach targets an FDD scenario in which only z,|) in (1) for 1 < f < T are available in addition to U, re Q(.) denotes a scalar quantizer. Upon tailoring phase- s.t. 7^bb· < l]Ux|² < 7_t ^{u b} , (14) eval techniques to the problem at hand, we obtain the mulation given in (12), where D_p = diag{pi, · · · ,pr}, Let St C <C^,S denote the set of all x £ <C^S for which the Λ > 0 is any given hyper-parameter, and where we have t‘^h constraint is satisfied, i.e„ {x : 7|-^b· < l]Ux|² < d *n to denote rows of Ψ formed by vectors [φ,, , 1 ]. 1 < 7“^-b·}. Similarly, let 1¼(ζ). V ζ effi⁶’ V f denote the objective T. in addition, we have used e e (D^s to model the value obtained by solving (14) for the given ζ input. In other ntization error vector such that each component of e can words I >t(C) is the distance of ζ to the feasible set of the t^th ny complex scalar that has its magnitude within a bound, constraint Then, it can be verified that for all p greater than ch in mm is determined by given quantization resolution. a large enough penalty, i.e.. V p > p > 0, (14) is equivalent propose to solve (12) via alternating optimization. In icular, firstly ζ is optimized keeping the phases and error ³Wc ate ignoring errors in the feedback channel and have also assumedm {expOSt)}^, 6 fixed. Then, the phases {exp(j^'e_f)}^L₁ enough processing at the user to suppress noise.

where we have let o denote die face-splitting product (or rowwise Kronecker product).

(15) Note that from Lemma 1 we can deduce that the product

channel h0g exhibits row-wise product sparsity. In particular se observations prompt our proximal distance based algoif and {t_gJ} denote the non-zero rows of t#, and m. Specifically this algorithm is an iterative one which in t_s, respectively, then {t_h,it_gj} will be the non-zero rows h iteration successively solves two sub-problems: (coefficients) in t_/,St_s. Next, we augment G to form the (N+ In the first step we solve the following sub-problem for 1) x ( D^D_g + 1) matrix 6 — [G, O_jy; 0 J_f, lj using which we a given choice of X| e St V t — 1, · · · , T, can express the composite product channel [(h® g)^T, h_dir.]^T as a vector in the column span of 6. We next suppose that

(16) observations are collected over L frames and collect these

observations in the T x L matrix Z — [z*¹*, · · · where which is solvable in closed form to obtain ζ — the vector z·*' , 1 < t < L denotes the T observations obtained

over the I^th frame. Indeed, z^(<* = [zjf®, · · · , a£P] where In step 2 we update each x₍ by solving (14) using ζ as each z}*® permits the model as in (1) albeit with unknown input composite product channel given by [(h^® ® gW)^T, hffl ]^T. an be readily seen that this procedure converges since the We are assuming that these underlying channels can change ctive in (16) is monotonic non-increasing across iterations across (but not within) frames. Across all L frames we suppose is bounded below. Indeed for p > p and any given choice that {h(^<;} exhibit common sparsity in that all these channel X( e S_t V t — 1, · · · , T the optimal value of (16) is an vectors are obtained as a linear combination of a common set er bound to the true optimal value of (15) (or (13)). The of steering vectors from G_/,. An analogous comment applies ormance of the algorithm is sensitive to the choice of p as to {g(*®} over the dictionary G_s. Further, let us collect all as the initial x_t e St V f = 1, · · · , T. To address the first the used pattern vectors as rows of the *n ^as before. Finally, endency we progressively increase the choice of p over defining the matrix G — *n&, we can pose the problem of hes, each batch comprising of several iterations, starting interest as

be retrieved via an efficient and optimal algorithm. where r, 7 > 0 are hyper-parameters. Il£W picks rows

VII. SUBSPACE ESTIMATION (p — 1)D_S 4 1 through pD_g of W for all 1 < p < We let G_h be the NxD^ dictionary matrix for h and let G whereas Π* W picks rows q,q + D

9 _g, · · · ,94 (¾ — l)D_g of he N x D_g dictionary matrix for g. Note here that both h W for all 1 < q < D_g. Some comments on the formulation g are assumed to permit a sparse representation under their in (18) are on order. Note that the term

ective dictionaries, i.e., we suppose that h — G_/,t_/, & g — 7 ∑9=_i ||n¾W|| enforces row- wise product sparsity assured t_e for some sparse tj, e<C^D,‘ and some sparse t_g eC°“, by Lemma 1. On the other hand, ||W||„ enforces low-rank ectively. Here, in the absence of any other side information property, which is important when underlying channels do can assume both these dictionaries to be identical to the not change sufficiently across frames. We propose an ADMM rix whose columns are steering vectors (defined by the IRS based algorithm to solve (18) which we henceforth refer to y geometry) sampled on the angular grid. Indeed, recall as ADMM-SS. Towards formulating our algorithm, we first for a commonly used uniform planar array (UFA) layout re-write (18), by introducing auxiliary matrix variables along ments of each column of the latter matrix are given by with additional constraints, as in (19).

|||W - U||² + |||W - V||² + |||W - X||² + IRtr(L_j(W - U)) + ifltr(Lt(W - V)) + l/<tr(L’(W - X)))} , (20) owing steps in each of its iteration⁴: matrices Dj“ = diag{h‘“} and D¾“ = diag{g^loe}. Rnally, olve argmin_ugl[,(D_fcD,₊i)_xi.{£^,(W, U, V, X, Li, L₂, L₃)} we let X — - (h^loe Θ g^loe)(h^loe 0 g^lo")^t denote the btain LT, which is equivalent to sample covariance matrix. Next, we introduce the following useful results based on which subsequent subspace estimation argminu

techniques are derived. ch permits a standard singular value thresholding solution Proposition 1. Let Δ¾ — £'[hh^f] it Δ5 — £[ggt] Then, marized in Lemma 2. the covariance, A of the element-wise product channel h0g olve argmin_vgI.(D_fcD,₊i)xr. {£'( W, U, V, X, Li, L₂, L₃)} permits the expansion btain V, which is equivalent to Δ = Δ_¾ Θ Δ₅ + Djf Δ,ΡΪ·)¹ + I¾-A_t(I¾^»)t (21) rgmm_v |₇∑; iin?v|| + |i| W - V - L₂/pj|² Moreover,

} ^Δ¾ © ^δ5 = (^G¾ ⁰ Gg)(D_s ® Dg)(G¾ o Gg)^† (22) ch can be solved using the correspond- Corollary 1. For a uniform planar array R1S, both Δ¾ and result summarized in Lemma 2. Solving Δ$ are block Toeplitz so that from array response (6) and minygnco,, D„-H) XI, •_! L, ,L_2! to Proposition 1, we can deduce that covariance of the product in X proceeds in an exactly analogous manner. channel hOg is also block Toeplitz. arginin_w¾.₍o_¾D,+i₎»_/.{£^,(W,ir, V,X,LI ,L2,L₃)} to We next illustrate subspace estimation for the important in W is equivalent to an unconstrained least-squares special case where the LoS components are zero vectors. blem. This case illustrates all the key techniques that can then be Finally, the Lagrange matrix variables are simply updated applied to other cases. Note fust that using the Schur product L, = L, + p(W - U), L₂ = L₂ + p(W - V)& L₃ = theorem we deduce that Δ^ Θ A_§ >z 0 while rank(A^ 0 + p(W - X). A_g) < rank(A¾)rank(A_g). Then, let d¾ — diag{D¾},d_j — e that the inputs required for the ADMM-SS are diagfD_g} with D = D¾®D_g and G = G¾oG_g. We consider anded dictionary matrix G, observations matrix Z. initial the following formulation: ce values for all matrix variables, and a choice of er-parameter values. Then, using the output W of this rithm, the span of the columns of G corresponding to (23) s of W having respective norms are above any given

shold, is declared to be the desired subspace. Further, letting d = vec(D) and x = vec(X), we can reformulate (23) as

n this section, for convenience we assume that there d---.v_*cf_<li&g{d_£ }) no direct path between transmitter and receiver. We After further manipulations, we obtain suppose that the LoS components in both h,g are wn (upto arbitrary phase terms), i.e., we suppose that _dk±a!_d}to {7-KH²||d₆||² + ||x - (G∞)(d_¾ β d₈)||²} (25) = h i exp(j0_¾)h¹“ & g = g + exptjd_gjg¹” where ,gi°^a are known (based on location information) while where GDG denotes the Khatri-Rao product or the column(j <?_/,), exp (j 0_S ) are arbitrarily chosen unknown phase wise kronecker product of G and G. (25) can be sub- ms. Moreover, following conventional modeling we furoptimally solved using alternating non-negative least squares minimization where d¾,d₈ are optimized in an alternating suppose that the non-LoS (nLoS) component vectors manner subject to non-negativity constraints. On the other be modeled as h = G¾t¾ it g = G_gt_g but now t_g need not be sparse vectors but instead are complex hand, if we drop the non-negativity constraints on d¾, d₈ in per normal and mutually independent. In addition, each (25) we obtain the relaxed formulation of them has a diagonal covariance matrix respectively, min {r||d_A||²||d_g||² + ||x - (GDG)(d_s ® d_d)||²} (26) D¾ = £[t¾ti] it D_j = £[t_jtt] for some diagonal ^dA'^dV 0,D_g t. 0. Also, we form two diagonal While (26) remains a non-convex problem it can be efiiciendy

solved via variable projection methods. Instead, we can emach iterations takes outputs of previous iteration as its starting point ploy alterating least squares minimization with the advantage now each step can be solved in closed form. Let d¾, d_§ be [7] Q. Wu and R. Zhang, “Beamforming optimization ft* wireless network optimized solutions so obtained. Then, we can re-construct aided by intelligent reflecting surface with discrete phase shifts," IEEE Trans. Common., vol. 68, no. 3, pp. 1838-1851, 2020. using

the desired [8] O. Abari, D. Bharadia, A. Duffield, and D. Katabi, “Enabling high- space is the dominant Eigenspace of GDG. Finally, we quality untetbercd virtual reality," in 14th USEN1X Symposium on e a useful observation that the power angle spectrum can Networked Systems Design and Implementation ( NSDl 17), (Boston. ΜΛ). pp. 531-544. USENDC Association, Mar. 2017. assumed to be invariant across relatively widely separated [9] B. Baser, I. Ytldrim. and I. F. Akyildiz, “Simris channel simulator for uency bands. Then to obtain the desired subspace for «configurable intelligent surface-empowered communication systems," fferent frequency band we only need to determine the arXiv ., May. 2020.

[10] S. Abcywickrama, R. Zhang, Q. Wu, and C. Yuen, “Intelligent reflecting esponding matrix G say G' (by simply computing the surface; Practical phase shift model and beamformiug optimization" erlying array response vectors (cf. 6) using the right carrier arXiv., Feb. 2020. uency) upon which the desired subspace is die dominant [11] N. Prasad, Q. Zhang, and X. Qi, “Channel reconstruction via quadratic programming in massive mimo networks,” WiOpt, Jun. 2019. enspace of G'DG'.

VIII. CONCLUSIONS AND FUTURE WORK APPENDIX mma 2. For any X G® ffixn ,r > Q, let X = USV* denote conomy-sized SVD so that S is a square diagonal matrix ingular values. Then, we have arg rSUi^i|Y-^{Xr + m}-}

REFERENCES Q. Wu and R. Zhang, ‘Intelligent reflecting surface enhanced wireless network via joint active and passive beamfomting," IEEE Trans. Wireless Common., vol. 18, no. 11, pp. 5394-5409, 2019. C. Huang, A. Zappone, G. C. AJexandropoulos. M. Debbah. and C. Yuen, “Rcconfigurable intelligent surfaces for energy efficiency in wireless communication” IEEE Dans. Wireless Common., vol. 18, no. 8, pp. 4157-4170, 2019. J. Cben, Y.-C. Liang, R V. Cheng, and W. Υϊι, “Channel Estimation for Rcconfigurable Intelligent Surface Aided Multi-Use MIMO Systems,” arXiv e-prints, p. arXiv:1912.03619, Dec. 2019. Q. Wu and R. Zhang, “Towards smart and «configurable environment: Intelligent reflecting surface aided wireless network,” IEEE Common. Mag , vol. 58, no. I, pp. 106-112, 2020. L. Subtt and P. Rechac. “Intelligent walls as autonomous parts of smart indoor environments,” ΙΕΓ Communications, vol. 6, no. 8, pp. 1004- 1010, 2012. T. J. Cui, D. Smith, and R. Liu, Metamaterials : Theory, Design, and Applications. Springer Publishing Company, Incorporated, 1st ed.. 2009.

Claims

What is Claimed: l. A method comprising: receiving, by a first communication device, a first pilot signal sent by a second communication device to the first communication device in a first time duration in a first communication channel of a time division duplex (TDD) system; generating, by the first communication device, sparsity information of the first communication channel, the sparsity information comprising of a set of array beamforming directions, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel including a first IRS channel between an IRS and the second communication device, and a second IRS channel between the IRS and the first communication device, the first communication channel further comprising a direct channel between the first communication device and the second communication device, the IRS being configured to reflect signals incident to the IRS; performing, by the first communication device, channel reconstruction of the first communication channel based on the received first pilot signal, the sparsity information, a location of the IRS, a location of the second communication device and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating, by the first communication device with the second communication device in the first communication channel, based on the reconstructed- channel information of the first communication channel.

2. The method of claim 1, further comprising: receiving, by the first communication device, a second pilot signal sent by the second communication device to the first communication device in a second time duration in the first communication channel; and wherein performing the channel reconstruction of the first communication channel comprises: reconstructing, by the first communication device, the first communication channel based on the received first pilot signal, the received second pilot signal, the sparsity information, the location of the IRS, the location of the second communication device, the first reflective pattern of the IRS for the first time duration, and a second reflective pattern of the IRS for the second time duration. 3. The method of any one of claims 1-2, further comprising: determining, by the first communication device, a reflective pattern of the IRS based on the reconstructed-channel information of first communication channel.

4. The method of any one of daims 2-3, wherein at least one of the first reflective pattern, the second reflective pattern or the reflective pattern of the IRS comprises phase shifts of reflective elements of the IRS.

5. The method of any one of daims 2-4, wherein the first reflective pattern and the second reflective pattern are different from each other.

6. The method of any one of claims 1-5, wherein performing the channel reconstruction of the first communication channel comprises: determining, by the first communication device, a line of sight (LOS) steering vector of the IRS-aided reflective channel.

7- The method of daim 6, wherein the first communication channd is represented as:

[(h Q g)^T,/i_diry = et, wherein [(h Q g)^T, h_iir]^T represents the first communication channel, h and g represent the first IRS channel and the second IRS channel of the IRS-aided reflective channel, h_di_r represents the direct channel, T denotes transpose, Θ denotes Hadamard product, 6 represents a dictionaiy matrix whose columns correspond to a plurality of array beamforming directions, and t represents a combining vector, the channel reconstruction of the first communication channel comprising: estimating, by the first communication device, the combining vector t based on the first received pilot signal.

8. The method of daim 7, further comprising: determining the set of array beamforming directions from the plurality of array beamforming directions. 9. The method of any one of claims 1-8, wherein the first communication device is an access point (AP) and the second communication device is a user equipment (UE), or the second communication device is an AP and the first communication device is a UE.

10. The method of any one of daims 1-9, wherein the reconstructed-channd information of the first communication channd comprises a channd model of the first communication channel. li. A method comprising: receiving, by a first communication device, a first pilot signal sent by a second communication device to the first communication device in a first time duration in a first communication channel of a TDD system; determining, by the first communication device, subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical data about channel measurement and reconstruction of the first communication channel accessed from memory, the subspaoe information comprising a set of Eigenvectors of the covariance matrix, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the first communication device, the first communication channel further comprising a direct channel between the first communication device and the second communication device, and the IRS being configured to reflect signals incident to the IRS; performing, by the first communication device, channel reconstruction of the first communication channel based on the received first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating by the first communication device with the second communication device in the first communication channel based on the reconstructed- channel information of the first communication channel.

12. The method of daim n, further comprising: receiving, by the first communication device, a second pilot signal sent by the second communication device to the first communication device in a second time duration in the first communication channel; and wherein performing the channel reconstruction of the first communication channel comprises: reconstructing, by the first communication device, the first communication channel based on the received first pilot signal, the received second pilot signal, the subspace information, the first reflective pattern of the IRS for the first time duration, and a second reflective pattern of the IRS for the second time duration. 13. The method of any one of claims 11-12, further comprising: determining, by the first communication device, a reflective pattern of the IRS based on the reconstructed-channel information of the first communication channel.

14. The method of any one of daims 12-13, wherein at least one of the first reflective pattern, the second reflective pattern or the reflective pattern of the IRS comprises phase shifts of reflective elements of the IRS.

15. The method of any one of daims 12-14, wherein the first reflective pattern and the second reflective pattern are different from each other. i6. The method of any one of claims 11-15, wherein the first communication channel is represented as:

[(h Θ g)^T, h_dir.]^T = ϋζ, wherein[(h O g)^T, /i_dtr.]^T represents the first communication channel, h and g represent the first IRS channel and the second IRS channel of the IRS-aided reflective channel, hai_r represents the direct channel, T denotes transpose, Θ denotes Hadamard product, U represents a(/V+i) xs matrix whose column space equals the dominant subspace of the covariance matrix of the first communication channel [(h Q g)^T, h<ur.]^T _> N is the number of reflective elements of the IRS, s represents the dimension of the dominant subspace, and ζ represents a combining vector, the channel reconstruction of the first communication channel comprising: estimating, by the first communication device, the combining vector ζ based on the first received pilot signal.

17. The method of any one of claims 11-16, wherein the first communication device is an access point (AP) and the second communication device is a user equipment (UE), or the second communication device is an AP and the first communication device is a UK

18. The method of any one of daims 11-17, wherein the reconstructed-channel information of the first communication channd comprises a channel modd of the first communication channd.

19. A method comprising: sending, by an access point (AP) to a user equipment (UE), a first pilot signal in a first time duration in a first communication channel of a frequency division duplex (FDD) system, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel including a first IRS channel between an IRS and the UE and a second IRS channel between the IRS and the AP, the first communication channel further comprising a direct channel between the AP and the UE, the IRS being configured to reflect signals incident to the IRS; receiving, by the AP from the UE, information of signal strength of a received first pilot signal, the received first pilot signal being the first pilot signal received by the UE through the first communication channel; determining, by the AP, subspace information of a dominant subspaoe of a covariance matrix of the first communication channel based on historical received signal strength measurement data of signals received by the UE through the first communication channel, the historical received signal strength measurement data being accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix; performing, by the AP, channel reconstruction of the first communication channel based on the information of the signal strength of the received first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration to generate reconstructed-channel information of the first communication channel; and communicating by the AP with the UE in the first communication channel based on the reconstructed-channel information of the first communication channel. 20. The method of claim 19, the subspace information being further determined based on historical channel reconstruction data of the first communication channel.

21. The method of daim 19, further comprising: sending, by the AP to the UE, a second pilot signal in a second time duration in the first communication channel; and receiving, by the AP from the UE, information of signal strength of a received second pilot signal, the received second pilot signal being the second pilot signal received by the UE through the first communication channel; performing the channel reconstruction of the first communication channel comprising: reconstructing, by the AP, the first communication channel based on the signal strength information of the received first pilot signal and the received second pilot signal, the subspace information, the first reflective pattern of the IRS for the first time duration, and a second reflective pattern of the IRS for the second time duration.

22. The method of claim 21, wherein the first reflective pattern and the second reflective pattern are different from each other.

23. The method of any one of daims 19-22, further comprising: determining, by the first communication device, a reflective pattern of the IRS based on the reconst ructed-channel information of first communication channel.

24. The method of daim 23, wherein the first reflective pattern, the second reflective pattern or the reflective pattern of the IRS comprises phase shifts of reflective dements of the IRS. 25- The method of any one of claims 19-24, wherein the first communication channel is represented as:

[(h Q g)^T,h_diry = ϋζ, wherein [(h © g)^T, h_dir ]^Trepresents the first communication channel, g and h represent the first IRS channel and the second IRS channel of the IRS-aided reflective channel, h_di_r represents the direct channel, T denotes transpose, Θ denotes Hadamard product, U represents a (/V+i)xs matrix whose column space equals the dominant subspace of the covariance matrix of the first communication channel [(h © g)^T, /i_dir ]^T, N is the number of reflective elements of the IRS, s represents the dimension of the dominant subspace, and ζ represents a combining vector; wherein performing the channel reconstruction of the first communication channel further comprises: estimating, by the AP, the combining factor ζ based on the information of the signal strength of the received first pilot signal. 26. The method of any one of claims 19-25, wherein the information of the signal strength of the received first pilot signal comprises quantized signal strength of the received first pilot signal.

27. The method of any one of daims 19-25, wherein the information of the signal strength of the received first pilot signal comprises quantized average received signal strength of the received first pilot signal.

28. An apparatus comprising: a non-transitory memory storage comprising instructions; and one or more processors in communication with the memory storage, wherein the instructions, when executed by the one or more processors, cause the apparatus to perform: receiving a first pilot signal sent by a second communication device to the apparatus in a first time duration in a first communication channel of a time division duplex (TDD) system; generating sparsity information of the first communication channel, the sparsity information comprising of a set of array beamforming directions, wherein the first communication channel comprises an intelligent reflecting surface (IRS)-aided reflective channel that comprises a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the second communication device, the IRS being configured to reflect signals incident to the IRS; performing channel reconstruction of the first communication channel based on the received first pilot signal, the sparsity' information, a location of the IRS, a location of the second communication device and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

29. An apparatus comprising: a non-transitory memory storage comprising instructions; and one or more processors in communication with the memory' storage, wherein the instructions, when executed by the one or more processors, cause the apparatus to perform: receiving a first pilot signal sent by a second communication device to the apparatus in a first time duration in a first communication channel of a TDD system; determining subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical data about channel measurement and reconstruction of the first communication channel_, accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the second communication device, the IRS being configured to reflect signals incident to the IRS; performing channel reconstruction of the first communication channel based on the received first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

30. An apparatus comprising: a non-transitory memory storage comprising instructions; and one or more processors in communication with the memory' storage, wherein the instructions, when executed by the one or more processors, cause the apparatus to perform: sending, to a user equipment (UE), a first pilot signal in a first time duration in a first communication channel of a frequency' division duplex (FDD) system, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the UE and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the UE, the IRS configured to reflect signals incident to the IRS; receiving, from the UE, information of signal strength of a received first pilot signal, the received first pilot signal being the first pilot signal received by the UE through the first communication channel; determining subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical received signal strength measurement data of signals received by the UE through the first communication channel, the historical received signal strength measurement data being accessed from memory', the subspace information comprising a set of Eigenvectors of the covariance matrix; performing channel reconstruction of the first communication channel based on the information of the signal strength of the received first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration to generate reconstructed-channel information of the first communication channel; and communicating with the UE in the first communication channel based on the reconstructed-channel information of the first communication channel.

31. A non-transitory computer-readable media storing computer instructions that when executed by one or more processors of an apparatus, cause the apparatus to perform: receiving a first pilot signal sent by a second communication device to the apparatus in a first time duration in a first communication channel of a time division duplex (TDD) system; generating sparsity information of the first communication channel, the sparsity information comprising of a set of array beamforming directions, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the second communication device, and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the second communication device, the IRS being configured to reflect signals incident to the IRS; performing channel reconstruction of the first communication channel based on the received first pilot signal, the sparsity information, a location of the IRS, a location of the second communication device and a first reflective pattern of the IRS for the first time duration, to generate reconstructed-channel information of the first communication channel; and communicating with the second communication device in the first communication channel based on the reconstructed-channel information of the first communication channel.

32. A non-transitoiy computer-readable media storing computer instructions that when executed by one or more processors of an apparatus, cause the apparatus to perform: receiving a first pilot signal sent by a second communication device to the apparatus in a first time duration in a first communication channel of a TDD system; determining subspace information of a dominant subspace of a covariance matrix of the first communication channel based on historical data about channel measurement and reconstruction of the first communication channel accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that includes a first IRS channel between an IRS and the second communication device and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the second communication device, the IRS being configured to reflect signals inddent to the IRS; generating reconstructed-channd information of the first communication channd based on the recdved first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration; and communicating with the second communication device in the first communication channd based on the reconstructed-channd information of the first communication channd. 33· A non-transitoiy computer-readable media storing computer instructions that when executed by one or more processors of an apparatus, cause the apparatus to perform: sending, to a user equipment (UE), a first pilot signal in a first time duration in a first communication channel of a frequency division duplex (FDD) system, the first communication channel comprising an intelligent reflecting surface (IRS)-aided reflective channel that indudes a first IRS channel between an IRS and the UE and a second IRS channel between the IRS and the apparatus, the first communication channel further comprising a direct channel between the apparatus and the UE, the IRS bdng configured to reflect signals inddent to the IRS; receiving, from the UE, information of signal strength of a received first pilot signal, the received first pilot signal bdng the first pilot signal recdved by the UE through the first communication channel; determining subspace information of a dominant subspace of a covariance matrix of the first communication channd based on historical received signal strength measurement data of signals received by the UE through the first communication channd, the historical received signal strength measurement data bdng accessed from memory, the subspace information comprising a set of Eigenvectors of the covariance matrix; performing channel reconstruction of the first communication channd based on the information of the signal strength of the recdved first pilot signal, the subspace information, and a first reflective pattern of the IRS for the first time duration to generate reconstructed-channel information of the first communication channd; and communicating with the UE in the first communication channd based on the reconstructed-channd information of the first communication channd.