WO2011041834A1

WO2011041834A1 - Reconstruction of a recorded sound field

Info

Publication number: WO2011041834A1
Application number: PCT/AU2010/001312
Authority: WO
Inventors: Craig Jin; Andre Van Schaik; Nicolas Epain
Original assignee: The University Of Sydney
Priority date: 2009-10-07
Filing date: 2010-10-06
Publication date: 2011-04-14
Also published as: AU2010305313B2; EP2486561A1; JP5773540B2; EP2486561B1; US20120259442A1; AU2010305313A1; EP2486561A4; US9113281B2; JP2013507796A

Abstract

Equipment (10) for reconstructing a recorded sound field includes a sensing arrangement (12) for measuring the sound field to obtain recorded data. A signal processing module (14) is in communication with the sensing arrangement (12) and processes the recorded data for the purposes of at least one of (a) estimating the sparsity of the recorded sound field and (b) obtaining plane-wave signals to enable the recorded sound field to be reconstructed.

Description

"Reconstruction of a recorded sound field"

Cross-Reference to Related Applications

The present application claims priority from Australian Provisional Patent Application No. 2009904871 filed on 7 October 2009, the contents of which are incorporated herein by reference in their entirety.

Field

The present disclosure relates, generally, to reconstruction of a recorded sound field and, more particularly, to equipment for, and a method of, recording and then reconstructing a sound field using techniques related to at least one of compressive sensing and independent component analysis.

Background

Various means exist for recording and then reproducing a sound field using microphones and loudspeakers (or headphones). The focus of this disclosure is accurate sound field reconstruction and/or reproduction compared with artistic sound field reproduction where creative modifications are allowed. Currently, there are two primary and state-of-the-art techniques used for accurately recording and reproducing a sound field: higher order ambisonics (HOA) and wave-field synthesis (WFS). The WFS technique generally requires a spot microphone for each sound source. In addition, the location of each sound source must be determined and recorded. The recording from each spot microphone is then rendered using the mathematical machinery of WFS. Sometimes spot microphones are not available for each sound source or spot microphones may not be convenient to use. In such cases, one generally uses a more compact microphone array such as a linear, circular, or spherical array. Currently, the best available technique for reconstructing a sound field from a compact microphone array is HOA. However, HOA suffers from two major problems: (1) a small sweet spot and (2) degradation in the reconstruction when the mathematical system is under-constrained (for example, when too many loudspeakers are used). The small sweet spot phenomenon refers to the fact that the sound field is only accurate for a small region of space.

Several terms relating to this disclosure are defined below.

"Reconstructing a sound field" refers, in addition to reproducing a recorded sound field, to using a set of analysis plane- wave directions to determine a set of plane- wave source signals and their associated source directions. Typically, analysis is done in association with a dense set of plane-wave source directions to obtain a vector, g, of plane-wave source signals in which each entry of g is clearly matched to an associated source direction.

"Head-related transfer functions" (HRTFs) or "Head-related impulse responses" (HRIRs) refer to transfer functions that mathematically specify the directional acoustic properties of the human auditory periphery including the outer ear, head, shoulders, and torso as a linear system. HRTFs express the transfer functions in the frequency domain and HRIRs express the transfer functions in the time domain.

"HOA-domain" and "HOA-domain Fourier Expansion" refer to any mathematical basis set that may be used for analysis and synthesis for Higher Order Ambisonics such as the Fourier-Bessel system, circular harmonics, and so forth. Signals can be expressed in terms of their components based on their expansion in the HOA-domain mathematical basis set. When signals are expressed in terms of these components, it is said that the signals are expressed in the "HOA-domain". Signals in the HOA-domain can be represented in both the frequency and time domain in a manner similar to other signals.

"HOA" refers to Higher Order Ambisonics which is a general term encompassing sound field representation and manipulation in the HOA-domain.

"Compressive Sampling" or "Compressed Sensing" or "Compressive Sensing" all refer to a set of techniques that analyse signals in a sparse domain (defined below).

"Sparsity Domain" or "Sparse Domain" is a compressive sampling term that refers to the fact that a vector of sampled observations y can be written as a matrix- vector product, e.g., as:

where Ψ is a basis of elementary functions and nearly all coefficient in x are null. If S coefficients in x are non-null, we say the observed phenomenon is S-sparse in the sparsity domain Ψ .

The function "pinv" refers to a pseudo-inverse, a regularised pseudo-inverse or a

Moore-Penrose inverse of a matrix.

The LI -norm of a vector x is denoted and is given by

ί

The L2-norm of a vector x is denoted by ||x||₂ and is given by ||

The L1-L2 norm of a matrix A is denoted by | and is given by:

wher ] is the z^'-th element of u , an

j] is the element in

the z^'-th row and y^'-th column of A .

"ICA" is Independent Component Analysis which is a mathematical method that provides, for example, a means to estimate a mixing matrix and an unmixing matrix for a given set of mixed signals. It also provides a set of separated source signals for the set of mixed signals.

The "sparsity" of a recorded sound field provides a measure of the extent to which a small number of sources dominate the sound field.

"Dominant components" of a vector or matrix refer to components of the vector or matrix that are much larger in relative value than some of the other components. For example, for a vector x , we can measure the relative value of component x_i compared by computing the ratio or the logarithm of the ratio, If the ratio or

log-ratio exceeds some particular threshold value, say θ_th , x_i may be considered a dominant component compared to x_j ,

"Cleaning a vector or matrix" refers to searching for dominant components (as defined above) in the vector or matrix and then modifying the vector or matrix by removing or setting to zero some of its components which are not dominant components.

"Reducing a matrix M" refers to an operation that may remove columns of M that contain all zeros and/or an operation that may remove columns that do not have a Dominant Component. Instead, "Reducing a matrix M" may refer to removing columns of the matrix M depending on some vector x. In this case, the columns of the matrix M that do not correspond to Dominant Components of the vector x are removed. Still further, "Reducing a matrix M" may refer to removing columns of the matrix M depending on some other matrix N. In this case, the columns of the matrix M must correspond somehow to the columns or rows of the matrix N. When there is this correspondence, "Reducing the matrix M" refers to removing the columns of the matrix M that correspond to columns or rows of the matrix N which do not have a Dominant Component.

"Expanding a matrix M" refers to an operation that may insert into the matrix M a set of columns that contains all zeros. An example of when such an operation may be required is when the columns of matrix M correspond to a smaller set of basis functions and it is required to express the matrix M in a manner that is suited to a larger set of basis functions. "Expanding a vector of time signals x(t) " refers to an operation that may insert into the vector of time signals x(t) , signals that contain all zeros. An example of when such an operation may be required is when the entries of x(t) correspond to time signals that match a smaller set of basis functions and it is required to express the vector of time signals x(t) in a manner that is suited to a larger set of basis functions.

"FFT" means a Fast Fourier Transform.

"IFFT" means an Inverse Fast Fourier Transform.

A "baffled spherical microphone array" refers to a spherical array of microphones which are mounted on a rigid baffle, such as a solid sphere. This is in contrast to an open spherical array of microphones which does not have a baffle.

Several notations related to this disclosure are described below:

Time domain and frequency domain vectors are sometimes expressed using the following notation: A vector of time domain signals is written as x(t) . In the frequency domain, this vector is written as x . In other words, x is the FFT of x(t) . To avoid confusion with this notation, all vectors of time signals are explicitly written out as x(i) .

Matrices and vectors are expressed using bold-type. Matrices are expressed using capital letters in bold-type and vectors are expressed using lower-case letters in bold-type.

A matrix of filters is expressed using a capital letter with bold-type and with an explicit time component such as M(t) when expressed in the time domain or with an explicit frequency component such as M(eo) when expressed in the frequency domain.

For the remainder of this definition we assume that the matrix of filters is expressed in the time domain. Each entry of the matrix is then itself a finite impulse response filter. The column index of the matrix M(t) is an index that corresponds to the index of some vector of time signals that is to be filtered by the matrix. The row index of the matrix M(t) corresponds to the index of the group of output signals. As a matrix of filters operates on a vector of time signals, the "multiplication operator" is the convolution operator described in more detail below.

"® " is a mathematical operator which denotes convolution. It may be used to express convolution of a matrix of filters (represented as a general matrix) with a vector of time signals. For example, represents the convolution of the

matrix of filters M(t) with the corresponding vector of time signals in x(t) . Each entry of M(t) is a filter and the entries running along each column of M (t) correspond to the time signals contained in the vector of time signals x(t) . The filters running along each row of M (?) correspond to the different time signals in the vector of output signals y (t) . As a concrete example, x(t) may correspond to a set of microphone signals, while y (t) may correspond to a set of HOA-domain time signals. In this case, the equation indicates that the microphone signals are

filtered with a set of filters given by each row of M(t) and then added together to give a time signal corresponding to one of the HOA-domain component signals in y (t) .

Flow charts of signal processing operations are expressed using numbers to indicate a particular step number and letters to indicate one of several different operational paths. Thus, for example, Step 1.A.2.B.1 indicates that in the first step, there is an alternative operational path A, which has a second step, which has an alternative operational path B, which has a first step.

Summary

In a first aspect there is provided equipment for reconstructing a recorded sound field, the equipment including

a sensing arrangement for measuring the sound field to obtain recorded data; and

a signal processing module in communication with the sensing arrangement and which processes the recorded data for the purposes of at least one of (a) estimating the sparsity of the recorded sound field and (b) obtaining plane-wave signals and their associated source directions to enable the recorded sound field to be reconstructed.

The sensing arrangement may comprise a microphone array. The microphone array may be one of a baffled array and an open spherical microphone array.

The signal processing module may be configured to estimate the sparsity of the recorded data according to the method of one of aspects three and four below.

Further, the signal processing module may be configured to analyse the recorded sound field, using the methods of aspects five to seven below, to obtain a set of plane- wave signals that separate the sources in the sound field and identify the source locations and allow the sound field to be reconstructed.

The signal processing module may be configured to modify the set of plane- wave signals to reduce unwanted artifacts such as reverberations and/or unwanted sound sources. To reduce reverberations, the signal processing module may reduce the signal values of some of the signals in the plane- wave signals. To separate sound sources in the sound field reconstruction so that the unwanted sound sources can be reduced, the signal processing module may be operative to set to zero some of the signals in the set of plane-wave signals.

The equipment may include a playback device for playing back the reconstructed sound field. The playback device may be one of a loudspeaker array and headphones. The signal processing module may be operative to modify the recorded data depending on which playback device is to be used for playing back the reconstructed sound field.

In a second aspect, there is provided, a method of reconstructing a recorded sound field, the method including

analysing recorded data in a sparse domain using one of a time domain technique and a frequency domain technique; and

obtaining plane-wave signals and their associated source directions generated from the selected technique to enable the recorded sound field to be reconstructed.

The method may include recording a time frame of audio of the sound field to obtain the recorded data in the form of a set of signals, s_mic (t) , using an acoustic sensing arrangement. Preferably, the acoustic sensing arrangement comprises a microphone array. The microphone array may be a baffled or open spherical microphone array.

In a third aspect, the method may include estimating the sparsity of the recorded sound field by applying ICA in an HOA-domain to calculate the sparsity of the recorded sound field.

The method may include analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_HOA (t) , and computing from b_H0A (t) a mixing matrix, M_ICA , using signal processing techniques. The method may include using instantaneous Independent Component Analysis as the signal processing technique.

The method may include projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing S ^{is me} transpose (Hermitian conjugate) of the real-

value (complex- valued) HOA direction matrix associated with the plane-wave basis directions and the hat-operator onYj_w._HOA indicates it has been truncated to an HOA- order M.

The method may include estimating the sparsity, S, of the recorded data by first determining the number, N_source , of dominant plane-wave directions represented by v. source and then computing where _w is the number of analysis plane-

wave basis directions.

In a fourth aspect, the method may include estimating the sparsity of the recorded sound field by analysing recorded data using compressed sensing or convex optimization techniques to calculate the sparsity of the recorded sound field.

The method may include analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_H0A (t) , and sampling the vector of

HOA-domain time signals over a given time frame, L, to obtain a collection of time samples at time instances t₁ to t_N to obtain a set of HOA-domain vectors at each time instant: ) expressed as a matrix, B_H0A by:

The method may include applying singular value decomposition to B_HOAto obtain a matrix decomposition:

The method may include forming a matrix S_reduced by keeping only the first m columns of S , where m is the number of rows of B_H0A and forming a matrix, Ω , given by

The method may include solving the following convex programming problem for a matrix Γ :

where Y_pIwis the matrix (truncated to a high spherical harmonic order) whose columns are the values of the spherical harmonic functions for the set of directions corresponding to some set of analysis plane waves,

and

ε_λ is a non-negative real number.

The method may include obtaining G

from Γ using:

G

where V^T is obtained from the matrix decomposition of B_H0A .

The method may include obtaining an unmixing matrix, , for the Z-th time frame, by calculating:

where;

is an unmixing matrix for the L-l time frame,

or is a forgetting factor such that 0 < a < 1 .

The method may include obtaining G,,,^.^ using:

The method may include obtaining the vector of plane-wave signals,

from the collection of plane-wave time samples, G_plw._smooth , using standard overlap-add techniques. Instead when obtaining the vector of plane-wave signals , the

method may include obtaining, g_pIw._cs (t) , from the collection of plane-wave time samples, G_plw , without smoothing using standard overlap-add techniques.

The method may include estimating the sparsity of the recorded data by first computing the number, N_comp , of dominant components of and then

computing is the number of analysis plane-wave basis

directions.

In a fifth aspect the method may include reconstructing the recorded sound field, using frequency-domain techniques to analyse the recorded data in the sparse domain; and obtaining the plane-wave signals from the frequency-domain techniques to enable the recorded sound field to be reconstructed.

The method may include transforming the set of signals,

to the frequency domain using an FFT to obtain

The method may include analysing the recorded sound field in the frequency domain using plane-wave analysis to produce a vector of plane-wave amplitudes,

In a first embodiment of the fifth aspect, the method may include conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes,

where:

Tpiw/mic is a transfer matrix between plane-waves and the microphones, s_mic is the set of signals recorded by the microphone array, and

ε₁ is a non-negative real number. In a second embodiment of the fifth aspect, the method may include conducting the plane- wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes,

_p

minimise

subject to

IPmic II2

where:

Tpiw/mio is ^a transfer matrix between the plane-waves and the microphones, s_mic is the set of signals recorded by the microphone array, and

ε_χ is a non-negative real number,

TpiwHOA i^{s a} transfer matrix between the plane-waves and the HOA-domain Fourier expansion,

b_H0A is a set of HOA-domain Fourier coefficients given by b_H0A = T_mic/H0As_mic where T_mjc/H0A is a transfer matrix between the microphones and the HOA-domain

Fourier expansion, and

ε₂ is a non-negative real number.

In a third embodiment of the fifth aspect, the method may include conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes, g_pIw._cs :

minimise

where:

T_plw/mi0 is a transfer matrix between plane-waves and the microphones,

T_mj_C/HOA i^{s a} transfer matrix between the microphones and the HOA-domain Fourier expansion,

b_H0A is a set of HOA-domain Fourier coefficients given by b_H0A = T_mic/H0As_mic , ε_ι is a non-negative real number.

In a fourth embodiment of the fifth aspect, the method may include conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes, g_plw._os :

where:

Tpiw/mic ^{1S a} transfer matrix between plane-waves and the microphones,

£·, is a non-negative real number,

T_plw/H0A is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

b_H0A is a set of HOA-domain Fourier coefficients given by b_H0A = T_mjc/H0As_mjc where T_mic/H0A is a transfer matrix between the microphones and the HOA-domain

Fourier expansion, and

ε₂ is a non-negative real number.

The method may include setting ε_λ based on the resolution of the spatial division of a set of directions corresponding to the set of analysis plane-waves and setting the value of ε₂ based on the computed sparsity of the sound field. Further, the method may include transforming g_pi_w._cs back to the time-domain using an inverse FFT to obtain g_p|W._cs (^) . The method may include identifying source directions with each entry of g_plw._cs or g_plw-cs (t) .

In a sixth aspect, the method may include using a time domain technique to analyse recorded data in the sparse domain and obtaining parameters generated from the selected time domain technique to enable the recorded sound field to be reconstructed.

The method may include analysing the recorded sound field in the time domain using plane-wave analysis according to a set of basis plane- waves to produce a set of plane-wave signals, g_plw._cs (t) . The method may include analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_H0A (t) , and sampling the vector of HOA-domain time signals over a given time frame, L, to obtain a collection of time samples at time instances t, to t_N to obtain a set of HOA-domain vectors at each time instant: b

expressed as a matrix, B

The method may include computing a correlation vector,

where b_omni is an omni-directional HOA-component of

In a first embodiment of the sixth aspect, the method may include solving the following convex programming problem for a vector of plane- wave gains,

minimise

subject to

where:

TTpiwHOA is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

ε_ι is a non-negative real number.

In a second embodiment, of the sixth aspect, the method may include solving the following convex programming problem for a vector of plane-wave gains, p_plw._os :

minimise subject to

where:

γ = Β HOA^omni '

^plw/HOA is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

ε_χ is a non-negative real number,

ε₂ is a non-negative real number.

The method may include setting ε_λ based on the resolution of the spatial division of a set of directions corresponding to the set of analysis plane-waves and setting the value of ε₂ based on the computed sparsity of the sound field. The method may include thresholding and cleaning p_plw._cs to set some of its small components to zero.

The method may include forming a matrix,

according to the plane- wave basis and then reducing Y to by keeping only the columns

corresponding to the non-zero components in where Y_P1W._HOA is an HOA

direction matrix for the plane-wave basis and the hat-operator on _A indicates it

has been truncated to some HO A-order M.

The method may include computing

(0 ^as

(0 ( ) (0 - ^Further' ™e method may include expanding

( ) to obtain g_plw._cs (t) by inserting rows of time signals of zeros so that ( ) matches the plane-wave basis.

In a third embodiment of the sixth aspect, the method may include solving the following convex programming problem for a matrix

where Y_plw is a matrix (truncated to a high spherical harmonic order) whose columns are the values of the spherical harmonic functions for the set of directions corresponding to some set of analysis plane waves, and

ε_λ is a non-negative real number.

The method may include obtaining an unmixing matrix, H_L , for the L-th time frame, by calculating:

where

n , , refers to the unmixing matrix for the L-l time frame and

a is a forgetting factor such that 0≤ a≤ 1 .

In a fourth embodiment of the fifth aspect, the method may include applying singular value decomposition to B_HOAto obtain a matrix decomposition:

m

s

where ε_χ and Y_pIw are as defined above.

The method may include obtaining G_plw from Γ using:

where V^T is obtained from the matrix decomposition of

The method may include obtaining an unmixing matrix, Π, , for the I-th time frame, by calculating:

where;

Π^_, is an unmixing matrix for the L-\ time frame,

or is a forgetting factor such that 0 < a < 1 .

The method may include obtaining G_plw._sraooth using:

The method may include obtaining the vector of plane-wave signals, ,

from the collection of plane-wave time samples, G_plw._sm00th , using standard overlap-add techniques. Instead when obtaining the vector of plane-wave signals the

method may include obtaining,

from the collection of plane-wave time samples, without smoothing using standard overlap-add techniques. The method

may include identifying source directions with each entry of

The method may include modifying g_p]w.cs (t) to reduce unwanted artifacts such as reverberations and/or unwanted sound sources. Further, the method may include, to reduce reverberations, reducing the signal values of some of the signals in the signal vector, The method may include, to separate sound sources in the sound

field reconstruction so that the unwanted sound sources can be reduced, setting to zero some of the signals in the signal vector,

Further, the method may include modifying dependent on the means

of playback of the reconstructed sound field. When the reconstructed sound field is to be played back over loudspeakers, in one embodiment, the method may include modifying as follows:

where:

i^{s a} loudspeaker panning matrix.

In another embodiment, when the reconstructed sound field is to be played back over loudspeakers, the method may include converting back to the HOA-

domain by computing:

where ) is a high-resolution HOA-domain representation of g_pIw._cs (i)

capable of expansion to arbitrary HOA-domain order, where

_P *^{s an} HOA direction matrix for a plane- wave basis and the hat-operator on

_{p A} indicates it has been truncated to some HOA-order M. The method may include decoding

O using HOA decoding techniques.

When the reconstructed sound field is to be played back over headphones, the method may include modifying ) to determine headphone gains as follows:

where:

0 is a head-related impulse response matrix of filters corresponding to

the set of plane wave directions.

In a seventh aspect, the method may include using time-domain techniques of Independent Component Analysis (ICA) in the HOA-domain to analyse recorded data in a sparse domain, and obtaining parameters from the selected time domain technique to enable the recorded sound field to be reconstructed.

The method may include analysing the recorded sound field in the HOA-domain to obtain a vector of HOA-domain time signals The method may include

analysing the HOA-domain time signals using ICA signal processing to produce a set of plane-wave source signals,

In a first embodiment of the seventh aspect, the method may include computing from b_H0A (t) a mixing matrix,

using signal processing techniques. The method may include using instantaneous Independent Component Analysis as the signal processing technique. The method may include projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing ^{is the} transpose (Hermitian conjugate)

of the real- value (complex- valued) HOA direction matrix associated with the plane- wave basis and the hat-operator on indicates it has been truncated to some

HOA-order M.

The method may include using thresholding techniques to identify the columns of V_source that indicate a dominant source direction. These columns may be identified on the basis of having a single dominant component.

The method may include reducing the matrix Y

to obtain a matrix, bY removing the plane-wave direction vectors in Y

_pI that do not

correspond to dominant source directions associated with matrix V

The method may include estimating gpi_w-ica._reduoed (0 ^as gpi_w._ica._reduoed (0 ⁼ P^inv(Ypiw-HOA-reduoed ) ^BHOA(0- Instead, the method may include estimating g_pi_w-ica._reduced (t) by working in the frequency domain and computing s_mic as the FFT of s_mi0 (t) .

The method may include, for each frequency, reducing a transfer matrix, between the plane-waves and the microphones, T_plw/mio , to obtain a matrix, T_plw/mic._reduced , by removing the columns in T_plw/mic that do not correspond to dominant source directions associated with matrix V_source .

The method may include estimating _d by computing:

( ) ^Md transforming g_plw,_ca,_educei back to the time-

domain using an inverse FFT to obtain g_pIw-ica._reduced (t) · The method may include expanding g_plw._ica._reduoed (t) to obtain g_plw._ica (t) by inserting rows of time signals of zeros so that g_plw._ica (t) matches the plane-wave basis.

In a second embodiment of the seventh aspect, the method may include computing from b_H0A a mixing matrix, M_ICA , and a set of separated source signals, g_ica (t) using signal processing techniques. The method may include using instantaneous Independent Component Analysis as the signal processing technique. The method may include projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing ¾ ^where is the transpose (Hermitian conjugate) of the real-

value (complex-valued) HOA direction matrix associated with the plane-wave basis and the hat-operator on Y indicates it has been truncated to some HOA-order M.

The method may include using thresholding techniques to identify from V_source the dominant plane- wave directions. Further, the method may include cleaning g_ioa (t) to obtain g which retains the signals corresponding to the dominant plane-wave

directions in and sets the other signals to zero.

The method may include modifying g_plw-ioa (t) to reduce unwanted artifacts such as reverberations and/or unwanted sound sources. The method may include, to reduce reverberations, reducing the signal values of some of the signals in the signal vector, g_pi_w._ica (t) · Further, the method may include, to separate sound sources in the sound field reconstruction so that the unwanted sound sources can be reduced, setting to zero some of the signals in the signal vector,

Still further, the method may include modifying g_pUv._ica (t) dependent on the means of playback of the reconstructed sound field. When the reconstructed sound field is to be played back over loudspeakers, in one embodiment the method may include modifying g_plw._ica (t) as follows:

where:

Ppiw/spk is a loudspeaker panning matrix.

In another embodiment, when the reconstructed sound field is to be played back over loudspeakers, the method may include converting g_plw._ioa (t) back to the HOA- domain by computing:

where:

bHOA-hi_ghres (0 is a high-resolution HOA-domain representation of

capable of expansion to arbitrary HOA-domain order,

Ypiw-HOA is an HOA direction matrix for a plane- wave basis and the hat-operator on Ypi_w-HOA indicates it has been truncated to some HOA-order M.

The method may include decoding using HOA

decoding techniques.

When the reconstructed sound field is to be played back over headphones, the method may include modifying g_plw-cs (t) to determine headphone gains as follows:

where:

Ppiw/hph (t) is a head-related impulse response matrix of filters corresponding to the set of plane wave directions.

The disclosure extends to a computer when programmed to perform the method as described above.

The disclosure also extends to a computer readable medium to enable a computer to perform the method as described above. Brief Description of Drawings

Fig. 1 shows a block diagram of an embodiment of equipment for reconstructing a recorded sound field and also estimating the sparsity of the recorded sound field;

Figs. 2-5 show flow charts of the steps involved in estimating the sparsity of a recorded sound field using the equipment of Fig. 1; Figs. 6-23 show flow charts of embodiments of reconstructing a recorded sound field using the equipment of Fig. 1 ;

Figs. 24A-24C show a first example of, respectively, a photographic representation of an HOA solution to reconstructing a recorded sound field, the original sound field and the solution offered by the present disclosure; and

Figs. 25A-25C show a second example of, respectively, a photographic representation of an HOA solution to reconstructing a recorded sound field, the original sound field and the solution offered by the present disclosure. Detailed Description of Exemplary Embodiments

In Fig. 1 of the drawings, reference numeral 10 generally designates an embodiment of equipment for reconstructing a recorded sound field and/or estimating the sparsity of the sound field. The equipment 10 includes a sensing arrangement 12 for measuring the sound field to obtain recorded data. The sensing arrangement 12 is connected to a signal processing module 14, such as a microprocessor, which processes the recorded data to obtain plane-wave signals enabling the recorded sound field to be reconstructed and/or processes the recorded data to obtain the sparsity of the sound field. The sparsity of the sound field, the separated plane-wave sources and their associated source directions are provided via an output port 24. The signal processing module 14 is referred to below, for the sake of conciseness, as the SPM 14.

A data accessing module 16 is connected to the SPM 14. In one embodiment the data accessing module 16 is a memory module in which data are stored. The SPM 14 accesses the memory module to retrieve the required data from the memory module as and when required. In another embodiment, the data accessing module 16 is a connection module, such as, for example, a modem or the like, to enable the SPM 14 to retrieve the data from a remote location.

The equipment 10 includes a playback module 18 for playing back the reconstructed sound field. The playback module 18 comprises a loudspeaker array 20 and/or one or more headphones 22.

The sensing arrangement 12 is a baffled spherical microphone array for recording the sound field to produce recorded data in the form of a set of signals, s_mic (T ).

The SPM 14 analyses the recorded data relating to the sound field using plane- wave analysis to produce a vector of plane-wave signals, g_plw (t) . Producing the vector of plane- wave signals, g_plw (t) , is to be understood as also obtaining the associated set of pale-wave source directions. Depending on the particular method used to produce the vector of plane wave amplitudes, g_pIw (i) is referred to more specifically as §piw-os (0 if Compressed Sensing techniques are used or _ica ( ) if IC A techniques are

used. As will be described in greater detail below, the SPM 14 is also used to modify gpiw (ή > i^f desired.

Once the SPM 14 has performed its analysis, it produces output data for the output port 24 which may include the sparsity of the sound field, the separated plane- wave source signals and the associated source directions of the plane-wave source signals. In addition, once the SPM 14 has performed its analysis, it generates signals, s_out (t) , for rendering the determined g_plw (t) as audio to be replayed over the loudspeaker array 20 and/or the one or more headphones 22.

The SPM 14 performs a series of operations on the set of signals, s_mic (t) , after the signals have been recorded by the microphone array 12, to enable the signals to be reconstructed into a sound field closely approximating the recorded sound field.

In order to describe the signal processing operations concisely, a set of matrices that characterise the microphone array 12 are defined. These matrices may be computed as needed by the SPM 14 or may be retrieved as needed from data storage using the data accessing module 16. When one of these matrices is referred to, it will be described as "one of the defined matrices".

The following is a list of Defined Matrices that may be computed or retrieved as required:

is a transfer matrix between the spherical harmonic domain and the

microphone signals, the matrix T_sph/mic being truncated to order M, as: where:

is the transpose of the matrix whose columns are the values of the spherical harmonic functions, Y where ( are the spherical coordinates for the 1-

th microphone and the hat-operator on indicates it has been truncated to some

order M; and

is the diagonal matrix whose coefficients are defined by

^ where R is the radius of the sphere of the

microphone array, is the spherical Hankel function of the second kind of order m, j_m is the spherical Bessel function of order m, j'_m and

are the derivatives of j_m and ¾²⁾ , respectively. Once again, the hat-operator on W_mic indicates that it has been truncated to some order M.

T_sph/mi0 is similar to T_sph/mio except it has been truncated to a much higher order

M' with (M' > M) .

Y_pIw is the matrix (truncated to the higher order M' ) whose columns are the values of the spherical harmonic functions for the set of directions corresponding to some set of analysis plane waves.

Y_plw is similar to Y_plw except it has been truncated to the lower order

M with (M < M') .

Tpiw/HOA *^{s a} transfer matrix between the analysis plane waves and the HOA- estimated spherical harmonic expansion (derived from the microphone array 12) as:

Tpiw/mic is a transfer matrix between the analysis plane waves and the microphone array 12 as:

where:

^Tsph/mic ^{is as} defined above.

E_mic/HOA (7) is a matrix of filters that implements, via a convolution operation, that transformation between the time signals of the microphone array 12 and the HOA- domain time signals and is defined as:

where:

each frequency component of E_raic/H0A (ω) is given by E_mic/H0A = pinv(T_sph/mic ) . The operations performed on the set of signals, s_mio (t) , are now described with reference to the flow charts illustrated in Figs. 2-16 of the drawings. The flow chart shown in Fig. 2 provides an overview of the flow of operations to estimate the sparsity, S, of a recorded sound field. This flow chart is broken down into higher levels of detail in Figs. 3-5. The flow chart shown in Fig. 6 provides an overview of the flow of operations to reconstruct a recorded sound field. The flow chart of Fig. 6 is broken down into higher levels of detail in Figs. 7-16.

The operations performed on the set of signals, s_mjc (t) , by the SPM 14 to determine the sparsity, S, of the sound field is now described with reference to the flow charts of Figs.2-5. In Fig. 2, at Step 1, the microphone array 12 is used to record a set of signals, s_mic (t) . At Step 2, the SPM 14 estimates the sparsity of the sound field. The flow chart shown in Fig. 3 describes the details of the calculations for Step 2. At Step 2.1, the SPM 14 calculates a vector of HOA-domain time signals b_H0A (t) as:

At Step 2.2, there are two different options available: Step 2.2.A and Step 2.2.B.

At Step 2.2.A, the SPM 14 estimates the sparsity of the sound field by applying ICA in the HOA-domain. Instead, at Step 2.2.B the SPM 14 estimates the sparsity of the sound field using a Compressed Sampling technique.

The flow chart of Fig. 4 describes the details of Step 2.2. A. At Step 2.2.A.1, the SPM 14 determines a mixing matrix, M_ICA , using Independent Component Analysis techniques.

At Step 2.2.A.2, the SPM 14 projects the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions. This projection is obtained by computing V where is the transpose of the Defined

Matrix Y_plw .

At Step 2.2.A.3, the SPM 14 applies thresholding techniques to clean V_source in order to obtain V_source._olean . The operation of cleaning V_source occurs as follows. Firstly, the ideal format for V_source is defined. V_source is a matrix which is ideally composed of columns which either have all components as zero or contain a single dominant component corresponding to a specific plane wave direction with the rest of the column's components being zero. Thresholding techniques are applied to ensure that V_souree takes its ideal format. That is to say, columns of V_source which contain a dominant value compared to the rest of the column's components are thresholded so that all components less than the dominant component are set to zero. Also, columns of V_source which do not have a dominant component have all of its components set to zero.

Applying the above thresholding operations to V_source gives V_source._clean .

At Step 2.2. A.4, the SPM 14 computes the sparsity of the sound field. It does this by calculating the number, N_source , of dominant plane wave directions in V_source._olean .

The SPM 14 then computes the sparsity, S, of the sound field as where

N _lw is the number of analysis plane-wave basis directions.

The flow chart of Fig. 5 describes the details of Step 2.2.B in Fig. 3, step 2.2.B being an alternative to Step 2.2. A. At Step 2.2.B.1, the SPM 14 calculates the matrix B_H0A from the vector of HOA signals b_H0A (t) by setting each signal in b_H0A (t) to run along the rows of B_H0A so that time runs along the rows of the matrix B_H0A and the various HOA orders run along the columns of the matrix B_H0A . More specifically, the SPM 14 samples b_H0A (t) over a given time frame, labelled by L, to obtain a collection of time samples at the time instances to t_N . The SPM 14 thus obtains a set of HOA- domain vectors at each time instant: . The SPM 14

then forms the matrix, B_H0A by:

At Step 2.2.B.2, the SPM 14 calculates a correlation vector, γ , as

where b_omni is the omni-directional HOA-component of expressed as a

column vector.

At Step 2.2.B.3, the SPM 14 solves the following convex programming problem to obtain the vector of plane-wave gains, P_plw._cs :

where T_plw/H0A is one of the defined matrices and ε_χ is a non-negative real number.

At Step 2.2.B.4, the SPM 14 estimates the sparsity of the sound field. It does this by applying a thresholding technique to P_plw.cs in order to estimate the number,

N_comp , of its Dominant Components. The SPM 14 then computes the sparsity, S, of the sound field as where N is the number of analysis plane-wave basis

directions.

The operations performed on the set of signals,

by the SPM 14 to reconstruct the sound field is now described and is illustrated using the flow charts of Figs.6-23.

In Fig. 6, Step 1 and Step 2 are the same as in the flow chart of Fig. 2 which has been described above. However, in the operational flow of Fig. 6, Step 2 is optional and is therefore represented by a dashed box.

At Step 3, the SPM 14 estimates the parameters, in the form of plane-wave signals g_plw (t) , that allow the sound field to be reconstructed. The plane-wave signals are expressed either as depending on the method of

derivation. At Step 4 there is an optional step (represented by a dashed box) in which the estimated parameters are modified by the SPM 14 to reduce reverberation and/or separate unwanted sounds. At Step 5, the SPM 14 estimates the plane-wave signals,

( ),(possibly modified) that are used to reconstruct and play back the sound field.

The operations of Step 1 and Step 2 having been previously described, the flow of operations contained in Step 3 are now described.

The flow chart of Fig. 7 provides an overview of the operations required for Step 3 of the flow chart shown in Fig. 6. It shows that there are four different paths available: Step 3. A, Step 3.B, Step 3.C and Step 3.D.

At Step 3. A, the SPM 14 estimates the plane- wave signals using a Compressive

Sampling technique in the time-domain. At Step 3.B, the SPM 14 estimates the plane- wave signals using a Compressive Sampling technique in the frequency-domain. At Step 3.C, the SPM 14 estimates the plane- wave signals using ICA in the HOA-domain. At Step 3.D, the SPM 14 estimates the plane-wave signals using Compressive Sampling in the time domain using a multiple measurement vector technique.

The flow chart shown in Fig. 8 describes the details of Step 3. A. At Step 3.A.1 b_HOA (t) and B_H0A are determined by the SPM 14 as described above for Step 2.1 and

Step 2.2.B.1, respectively.

At Step 3.A.2 the correlation vector, γ , is determined by the SPM 14 as described above for Step 2.2.B.2.

At Step 3.A.3 there are two options: Step 3.A.3.A and Step 3.A.3.B. At Step

3.A.3.A, the SPM 14 solves a convex programming problem to determine plane-wave direction gains, p_plw-cs . This convex programming problem does not include a sparsity constraint. More specifically, the following convex programming problem is solved:

where:

γ is as defined above and is one of the Defined Matrices,

ε_λ is a non-negative real number.

At Step 3.A.3.B, the SPM 14 solves a convex programming problem to determine plane-wave direction gains, , only this time a sparsity constraint is

included in the convex programming problem. More specifically, the following convex programming problem is solved to determine

where:

γ , ε₁ are as defined above ,

Tpiw_/HOA i^{s one} °f ^me Defined Matrices, and

ε₂ is a non-negative real number.

For the convex programming problems at Step 3.A.3, ε_χ may be set by the SPM

14 based on the resolution of the spatial division of a set of directions corresponding to the set of analysis plane waves. Further, the value of ε₂ may be set by the SPM 14 based on the computed sparsity of the sound field (optional Step 2).

At Step 3.A.4, the SPM 14 applies thresholding techniques to clean p__lw._os so that some of its small components are set to zero.

At Step 3.A.5, the SPM 14 forms a matrix, Υ_Ρ1ΝΝ._{Η0Α 5} according to the plane- wave basis and then reduces Yp_lw-H0A to Y_pi_w-reduce_ by keeping only the columns corresponding to the non-zero components in P_plw._cs , where Y_P1W._HOA ^^{s an} HOA direction matrix for the plane-wave basis and the hat-operator on Y_PIW._H0A indicates it has been truncated to some HO A-order M.

At Step 3.A.6, the SPM 14 calculates g_pIw-0,_reduced (i) as:

where Y_PLW._REDUCED an

At Step 3.A.7, the SPM 14 expands g_plw._c,_reduced (t) to obtain g_pIw,_s (t) by inserting rows of time signals of zeros to match the plane-wave basis that has been used for the analyses.

As indicated, above, an alternative to Step 3. A is Step 3.B. The flow chart of Fig. 9 details Step 3.B. At Step 3.B.1, the SPM 14 computes b_H0A (t) as b_H0A (t) = E_MIC/H0A (t) ® s_mi0 (t) . Further, at Step 3 JB.1 , the SPM 14 calculates a FFT, smio . ^{of s}mic (0 and/or a FFT, b_HOA , of b_H0A (t) .

At Step 3.B.2, the SPM 14 solves one of four optional convex programming problems. The convex programming problem shown at Step 3.B.2.A operates on s_mjo and does not use a sparsity constraint. More precisely, the SPM 14 solves the following convex programming problem to determine

where:

T_plw/mic is one of the Defined Matrices,

s_mic is as defined above, and

ε_λ is a non-negative real number.

The convex programming problem shown at Step 3.B.2.B operates on s_mjo and includes a sparsity constraint. More precisely, the SPM 14 solves the following convex programming problem to determine g_plw-cs :

where:

^Tpiw/mio ' ^Tpiw/HOA are each one of the Defined Matrices,

smic ' ^HOA ^Ε\ ^{3X6 as} defined above, and

ε₂ is a non-negative real number.

The convex programming problem shown at Step 3.B.2.C operates on b and does not use a sparsity constraint. More precisely, the SPM 14 solves the following convex programming problem to determine

where:

are each one of tne Defined Matrices, and

b_H0A , and ε₁ are as defined above. The convex programming problem shown at Step 3.B.2.D operates on b_H0A and includes a sparsity constraint. More precisely, the SPM 14 solves the following convex programming problem to determine g_plw._cs :

where:

are each one of the Defined Matrices, and

b_H0A , ε₁ , and ε₂ are as defined above.

At Step 3.B.3, the SPM 14 computes an inverse FFT of g_plw._cs to obtain gpiw-cs (0 · When operating on multiple time frames, overlap-and-add procedures are followed.

A further option to Step 3.A or Step 3.B is Step 3.C. The flow chart of Fig. 10 provides an overview of Step 3.C. At Step 3.C.1, the SPM 14 computes b_H0A (t) as

At Step 3.C.2 there are two options, Step 3.C.2.A and Step 3.C.2.B. At Step 3.C.2.A, the SPM 14 uses ICA in the HOA-domain to estimate a mixing matrix which is then used to obtain g_plw-ica (t) . Instead, at Step 3.C.2.B, the SPM 14 uses ICA in the

HOA-domain to estimate a mixing matrix and also a set of separated source signals.

Both the mixing matrix and the separated source signals are then used by the SPM 14 to obtain g_plw-ica (t ) .

The flow chart of Fig. 11 describes the details of Step 3.C.2.A. At Step

3.C.2.A.1, the SPM 14 applies ICA to the vector of signals b_H0A (t) to obtain the mixing matrix, M_ICA .

At Step 3.C.2.A.2, the SPM 14 projects the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions as described at Step 2.2.A.2. That is to say, the projection is obtained by computing V_souroe = Yj_wM_ICA , where Y_plw is the transpose of the defined matrix Y_plw .

At Step 3.C.2.A.3, the SPM 14 applies thresholding techniques to V_source to identify the dominant plane-wave directions in V_souroe . This is achieved similarly to the operation described above with reference to Step 2.2.A.3. At Step 3.C.2.A.4, there are two options, Step 3.C.2.A.4.A and Step 3.C.2.A.4.B. At Step 3.C.2.A.4.A, the SPM 14 uses the HOA domain matrix, ¾_w , to compute ) Instead, at Step 3.C.2.A.4.B, the SPM 14 uses the

microphone signals s_mic (t) and the matrix T_plWmioto compute

)

The flow chart of Fig. 12 describes the details of Step 3.C.2.A.4.A. At Step

3.C.2.A.4.A.1, the SPM 14 reduces the matrix Yj_w to obtain the matrix, Yj_w._reduced , by removing the plane-wave direction vectors in Y_plw that do not correspond to dominant source directions associated with matrix V_source .

At Step 3.C.2.A.4.A.2, the SPM 14 calculates g_plw-ica._reduced (f ) as:

where Y_plw.reduced and b_H0A (t) are as defined above.

An alternative to Step 3.C.2.A.4.A, is Step 3.C.2.A.4.B. The flow chart of Fig.

13 details Step 3.C.2.A.4.B.

At Step 3.C.2.A.4.B.1, the SPM 14 calculates a FFT, s At Step

3.C.2.A.4.B.2, the SPM 14 reduces the matrix T_plw/mio to obtain the matrix, T

by removing the plane- wave directions in T_plw/miothat do not correspond to dominant source directions associated with matrix V_source .

At Step 3.C.2.A.4.B.3, the SPM 14 calculates g_plw,_ca,_educed as:

)

where T_{plw/mic.reduced} and s_mio are as defined above.

At Step 3.C.2.A.4.B.4, the SPM 14 calculates g_{plw-ioa-reduced} (f) as the IFFT of

Splw-ica-reduced '

Reverting to Fig. 11, at Step 3.C.2.A.5, the SPM 14 expands

to obtain g_plw.ica (t) by inserting rows of time signals of zeros to match the plane-wave basis that has been used for the analyses.

An alternative to Step 3.C.2.A is Step 3.C.2.B. The flow chart of Fig. 14 describes the details of Step 3.C.2.B.

At Step 3.C.2.B.1, the SPM 14 applies ICA to the vector of signals b_H0A (t) to obtain the mixing matrix, M_ICA , and a set of separated source signals g_ica (t) .

At Step 3.C.2.B.2, the SPM 14 projects the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions as described for Step 2.2. A.2, i.e the projection is obtained by computing

_> where

is the transpose of the defined matrix

At Step 3.C.2.B.3, the SPM 14 applies thresholding techniques to V_source to identify the dominant plane-wave directions in V_source . This is achieved similarly to the operation described above for Step 2.2.A.3. Once the dominant plane-wave directions in V_source have been identified, the SPM 14 cleans g_ioa (t) to obtain g_plw-ica (t) which retains the signals corresponding to the dominant plane- wave directions V_source and sets the other signals to zero.

As described above, a further option to Steps 3. A, 3.B and 3.C, is Step 3.D. The flow chart of Figure 15 provides an overview of Step 3.D.

At Step 3.D.1, the SPM 14 computes b_H0A (t) as

T^{he SPM 14} then calculates the matrix, B_H0A , from

the vector of HOA signals b_H0A (t) by setting each signal in b_H0A (t) to run along the rows of B_H0A so that time runs along the rows of the matrix B_H0A and the various HOA orders run along the columns of the matrix B_H0A . More specifically, the SPM 14 samples b_H0A (t) over a given time frame, L, to obtain a collection of time samples at the time instances t₁ to t_N . The SPM 14 thus obtains a set of HOA-domain vectors at each time instant: b_H0A (t_j) , b_H0A (t₂) , . . ., b_H0A (t_N) . The SPM 14 forms the matrix, B_HOA by:

At Step 3.D.2 there are two options, Step 3.D.2.A and Step 3.D.2.B. At Step 3.D.2.A, the SPM 14 computes g_p!w-cs using a multiple measurement vector technique applied directly on B_H0A . Instead at Step 3.D.2.B, the SPM 14 computes g_plw._os using a multiple measurement vector technique based on the singular value decomposition of

The flow chart of Fig. 16 describes the details of Step 3.D.2.A. At Step 3.D.2.A.1, the SPM 14 solves the following convex programming problem to determine G_PLW :

where:

Y_PLW is one of the Defined Matrices,

B_H0A is as defined above, and

ε₁ is a non-negative real number.

At Step 3.D.2.A.2, there are two options, i.e. Step 3.D.2.A.2.A and Step 3.D.2.A.2.B. At Step 3.D.2.A.2.A, the SPM 14 computes g_plw._cs(t) directly from G_plw using an overlap-add technique. Instead at Step 3.D.2.A.2.B, the SPM 14 computes g_plw._os(t) using a smoothed version of G_plw and an overlap-add technique.

The flow chart of Fig. 17 describes Step 3.D.2.A.2.B in greater detail.

At Step 3.D.2.A.2.B.1, the SPM 14 calculates an unmixing matrix, Tl_L , for the Z-th time frame, by calculating:

where Π_Ι_₁ refers to the unmixing matrix for the L-l time frame and « is a forgetting factor such that 0 < a≤ 1 , and B_H0A is as defined above.

At Step 3.D.2.A.2.B.2, the SPM 14 calculates G_plw._sra00th as:

where H_L and B_H0A are as defined above.

At Step 3.D.2.A.2.B.3, the SPM 14 calculates g_plw-os(0 from G_plw._smooth using an overlap-add technique.

An alternative to Step 3.D.2.A is Step 3.D.2.B. The flow chart of Fig. 18 describes the details of Step 3.D.2.B.

At Step 3.D.2.B.1, the SPM 14 computes the singular value decomposition of B_HOA to obtain the matrix decomposition:

At Step 3.D.2.B.2, the SPM 14 calculates the matrix, S_reduced , by keeping only the first m columns of S , where m is the number of rows of B_H0A .

At Step 3.D.2.B.3, the SPM 14 calculates matrix Ω as:

^ ⁼ US_reduced .

At Step 3.D.2.B.4, the SPM 14 solves the following convex programming problem for matrix Γ :

minimize

subject to

where:

Y_pIw is one of the defined matrices,

Ω is as defined above, and

ε_χ is a non-negative real number.

At Step 3.D.2.B.5, there are two options, Step 3.D.2.B.5.A and Step 3.D.2.B.5.B. At Step 3.D.2.B.5.A, the SPM 14 calculates G _lw from Γ using:

where V^T is obtained from the matrix decomposition of B_H0A as described above. The SPM 14 then computes ) directly from _w using an overlap-add technique.

Instead, at Step 3.D.2.B.5.B, the SPM 14 calculates g_plw._cs (0 using a smoothed version of G_p]w and an overlap-add technique.

The flow chart of Fig. 19 shows the details of Step 3.D.2.B.5.B.

At Step 3.D.2.B.5.B.1, the SPM 14 calculates at unmixing matrix, n_L, for the i-th time frame, by calculating:

where Π_L_₁ refers to the unmixing matrix for the L-l time frame and a is a forgetting factor such that 0 < a < 1 , and Γ and Ω are as defined above.

At Step 3.D.2.B.5.B.2, the SPM 14 calculates G_plw._smooth as:

where and _A are as defined above.

At Step 3.D.2.B.2.B.3, the SPM 14 calculates g_plw._cs (0 from G_plw._sm00th using an overlap-add technique.

As described above, an optional step of reducing unwanted artifacts is shown at

Step 4 of the flow chart of Fig. 6 The SPM 14 controls the amount of reverberation present in the sound field reconstruction by reducing the signal values of some of the signals in the signal vector Instead, or in addition, the SPM 14 removes

undesired sound sources in the sound field reconstruction by setting to zero some of the signals in the signal vector g_plw (t) .

In Step 5 of the flow chart of Fig. 6, the parameters g_plw (t) are used to play back the sound field. The flow chart of Fig. 20 shows three optional paths for play back of the sound field: Step 5. A, Step 5.B, and Step 5.C. The flow chart of Fig. 21 describes the details of Step 5. A.

At Step 5.A.1, the SPM 14 computes or retrieves from data storage the loudspeaker panning matrix, P_plw/spk , in order to enable loudspeaker playback of the reconstructed sound field over the loudspeaker array 20. The panning matrix, P_pi_w/spk , can be derived using any of the various panning techniques such as, for example, Vector Based Amplitude Panning (VBAP). At Step 5.A.2, the SPM 14 calculates the loudspeaker signals

( ) ( ) ( )

Another option is shown in the flow chart of Fig. 22 which describes the details of Step 5.B. At Step 5.B.1, the SPM 14 computes b_H0A._highres (t) in order to enable loudspeaker playback of the reconstructed sound field over the loudspeaker array 20. ^HOA-highres (0 *^{s a} high-resolution HOA-domain representation of g_plw (^) that is capable of expansion to an arbitrary HOA-domain order. The SPM 14 calculates k_H0A._hj_ghres (0 ^as

where Y_plw is one of the Defined Matrices and the hat-operator on Y_plw indicates it has been truncated to some HOA-order M.

At Step 5.B.2, the SPM 14 decodes ) using HOA decoding techniques .

An alternative to loudspeaker play back is headphone play back. The operations for headphone play back are shown at Step 5.C of the flow chart of Fig. 20. The flow chart of Fig. 23 describes the details of Step 5.C.

At Step 5.C.1, the SPM 14 computes or retrieves from data storage the head- related impulse response matrix of filters, P_pi_w/hph (0 _> corresponding to the set of analysis plane wave directions in order to enable headphone playback of the reconstructed sound field over one or more of the headphones 22. The head-related impulse response (HRIR) matrix of filters, P_plw/hph (t) , is derived from HRTF measurements.

At Step 5.C.2, the SPM 14 calculates the headphone signals as

using a fllter convolution operation.

It will be appreciated by those skilled in the art that the basic HOA decoding for loudspeakers is given (in the frequency domain) by:

where:

N_spk is the number of loudspeakers,

Y_spk is the transpose of the matrix whose columns are the values of the spherical harmonic functions, Y where are the spherical coordinates for the

k-th loudspeaker and the hat-operator on Y_spk indicates it has been truncated to order M, and

b_H0A is the play back signals represented in the HOA-domain.

The basic HOA decoding in three dimensions is a spherical-harmonic-based method that possesses a number of advantages which include the ability to reconstruct the sound field easily using various and arbitrary loudspeaker configurations. However, it will be appreciated by those skilled in the art that it also suffers from limitations related to both the encoding and decoding process. Firstly, as a finite number of sensors is used to observe the sound field, the encoding suffers from spatial aliasing at high frequencies (see N. Epain and J. Daniel, "Improving spherical microphone arrays," in the Proceedings of the AES 124^th Convention, May 2008). Secondly, when the number of loudspeakers that are used for playback is larger than the number of spherical harmonic components used in the sound field description, one generally finds deterioration in the fidelity of the constructed sound field (see A. Solvang, "Spectral impairment of two dimensional higher-order ambisonics," in the Journal of the Audio Engineering Society, volume 56, April 2008, pp. 267-279).

In both cases, the limitations are related to the fact that an under-determined problem is solved using the pseudo-inverse method. In the case of the present disclosure, these limitations are circumvented in some instances using general principles of compressive sampling or ICA. With regard to compressive sampling, the applicants have found that using a plane- wave basis as a sparsity domain for the sound field and then solving one of the several convex programming problems defined above leads to a surprisingly accurate reconstruction of a recorded sound field. The plane wave description is contained in the defined matrix

The distance between the standard HOA solution and the compressive sampling solution may be controlled using, for example, the constraint When ε₂ is zero, the compressive sampling solution

is the same as the standard HOA solution. The SPM 14 may dynamically set the value of ε₂ according to the computed sparsity of the sound field.

With regard to applying ICA in the HO A-domain, the applicants have found that the application of statistical independence benefits greatly from the fact that the HOA- domain provides an instantaneous mixture of the recorded signals. Further, the application of statistical independence seems similar to compressive sampling in that it also appears to impose a sparsity on the solution.

As described above, it is possible to estimate the sparsity of the sound field using techniques of compressive sampling or techniques of ICA in the HO A-domain.

In Figs. 24 and 25 simulation results are shown that demonstrate the power of sound field reconstruction using the present disclosure. In the simulations, the microphone array 12 is a 4 cm radius rigid sphere with thirty two omnidirectional microphones evenly distributed on the surface of the sphere. The sound fields are reconstructed using a ring of forty eight loudspeakers with a radius of 1 m.

In the HOA case, the microphone gains are HOA-encoded up to order 4. The compressive sampling plane-wave analysis is performed using a frequency-domain technique which includes a sparsity constraint and using a basis of 360 plane waves evenly distributed in the horizontal plane. The values of ε₁ and ε₂ have been fixed to

10^~3 and 2, respectively. In every case, the directions of the sound sources that define the sound field have been randomly chosen in the horizontal plane.

Example 1

Referring to Fig. 24, in this simulation four sound sources at 2 kHz were used. The HOA solution is shown in Fig. 24A; the original sound field is shown in Fig. 24B; and the solution using the technique of the present disclosure is shown in Fig. 24C. Clearly, the method as described performs better than a standard HOA method.

Example 2

Referring to Fig. 25, in this simulation twelve sound sources at 16kHz were used. As before, the HOA solution is shown in Fig. 25A; the original sound field is shown in Fig. 25B; and the solution using the technique of the present disclosure is shown in Fig. 25C. It will be appreciated by those skilled in the art, that the results for Figure 25 are obtained outside of the Shannon-

Nyquist spatial aliasing limit of the microphone array but still provide an accurate reconstruction of the sound field.

It is an advantage of the described embodiments that an improved and more robust reconstruction of a sound field is provided so that the sweet spot is larger; there is little, if any, degradation in the quality of the reconstruction when parameters defining the system are under-constrained; and the accuracy of the reconstruction improves as the number of the loudspeakers increases.

It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the disclosure as shown in the specific embodiments without departing from the scope of the disclosure as broadly described.

The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Claims

CLAIMS:

1. Equipment for reconstructing a recorded sound field, the equipment including a sensing arrangement for measuring the sound field to obtain recorded data; and

2. The equipment of claim 1 in which the sensing arrangement comprises a microphone array.

3. The equipment of claim 2 in which the microphone array is one of a baffled array and an open spherical microphone array.

4. The equipment of any one of the preceding claims in which the signal processing module is configured to estimate the sparsity of the recorded data.

5. The equipment of any one of the preceding claims in which the signal processing module is configured to analyse the recorded sound field to obtain a set of plane-wave signals that separate the sources in the sound field and identify the source directions and allow the sound field to be reconstructed.

6. The equipment of claim 5 in which the signal processing module is configured to modify the set of plane-wave signals to reduce unwanted artifacts.

7. The equipment of any one of the preceding claims which includes a playback device for playing back the reconstructed sound field.

8. The equipment of claim 7 in which the signal processing module is operative to modify the recorded data depending on which playback device is to be used for playing back the reconstructed sound field.

9. A method of reconstructing a recorded sound field, the method including

analysing recorded data in a sparse domain using one of a time domain technique and a frequency domain technique; and obtaining plane-wave signals and their associated source directions generated from the selected technique to enable the recorded sound field to be reconstructed.

10. The method of claim 9 which includes recording a time frame of audio of the sound field to obtain the recorded data in the form of a set of signals, s_mic (t) , using an acoustic sensing arrangement.

11. The method of claim 9 or claim 10 which includes estimating the sparsity of the recorded sound field by applying ICA in an HOA-domain to calculate the sparsity of the recorded sound field.

12. The method of claim 11 which includes analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_H0A (t) , and computing from b_H0A (t) a mixing matrix, M_ICA , using signal processing techniques.

13. The method of claim 12 which includes projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing ^{is the} transpose (Hermitian conjugate)

of the real-value (complex-valued) HOA direction matrix associated with the plane- wave basis directions and the hat-operator on Y indicates it has been truncated to

an HOA-order M.

14. The method of claim 13 which includes estimating the sparsity, S, of the recorded data by first determining the number, N_source , of dominant plane-wave directions represented by V_source and then computing where N_plw is the

number of analysis plane- wave basis directions.

15. The method of claim 9 or claim 10 which includes estimating the sparsity of the recorded sound field by analysing recorded data using compressed sensing or convex optimization techniques to calculate the sparsity of the recorded sound field.

16. The method of claim 15 which includes analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_H0A (t) , and sampling the vector of HOA-domain time signals over a given time frame, L, to obtain a collection of time samples at time instances t_x to t_N to obtain a set of HOA-domain vectors at each time instant _w) expressed as a matrix,

^BHOA by:

17. The method of claim 16 which includes applying singular value decomposition to B_H0A to obtain a matrix decomposition:

18. The method of claim 17 which includes forming a matrix S_reduced by keeping only the first m columns of S , where m is the number of rows of B_H0A and forming a matrix, Ω , given by

19. The method of claim 17 which includes solving the following convex programming problem for a matrix Γ :

where

Y_plwis the matrix (truncated to a high spherical harmonic order) whose columns are the values of the spherical harmonic functions for the set of directions corresponding to some set of analysis plane waves,

and

£_j is a non-negative real number.

20. The method of claim 19 which includes obtaining G_plw from Γ using:

where V^T is obtained from the matrix decomposition of B

21. The method of claim 20 which includes obtaining an unmixing matrix,

for the X-th time frame, by calculating:

where;

is an unmixing matrix for the L-l time frame,

a is a forgetting factor such that 0≤ 1≤ 1 .

22. The method of claim 21 which includes obtaining using:

23. The method of claim 22 which includes obtaining the vector of plane-wave signals, , from the collection of plane- wave time samples, using

standard overlap-add techniques.

24. The method of claim 22 which includes obtaining, , from the collection

of plane- wave time samples, without smoothing using standard overlap-add

techniques.

25. The method of claim 24 which includes estimating the sparsity of the recorded data by first computing the number, of dominant components of g_plw._os (t) and

then computing where is the number of analysis plane-wave basis

directions.

26. The method of any one of claims 9 to 25 which includes reconstructing the recorded sound field, using frequency-domain techniques to analyse the recorded data in the sparse domain; and obtaining the plane-wave signals from the frequency-domain techniques to enable the recorded sound field to be reconstructed.

27. The method of claim 26 which includes transforming the set of signals, s_mic (t) , to the frequency domain using an FFT to obtain s_mio .

28. The method of claim 27 which includes analysing the recorded sound field in the frequency domain using plane-wave analysis to produce a vector of plane-wave amplitudes, g_plw._cs .

29. The method of claim 28 which includes conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes,

where:

T _lw/mjo is a transfer matrix between plane-waves and the microphones, s_mic is the set of signals recorded by the microphone array, and

ε_χ is a non-negative real number.

30. The method of claim 29 which includes conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes, :

minimise

subject to

where:

T_piwmio is a transfer matrix between the plane- waves and the microphones, s_mi0 is the set of signals recorded by the microphone array, and

ε_λ is a non-negative real number,

lplw/HOA is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

b_H0A is a set of HOA-domain Fourier coefficients given by b_H0A = T_mic/H0As_mic where T_mic/H0A is a transfer matrix between the microphones and the HOA-domain

Fourier expansion, and

ε₂ is a non-negative real number.

31. The method of claim 28 which includes conducting the plane- wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes, g_plw._os :

where:

plw/mic is a transfer matrix between plane-waves and the microphones, lmic/HOA is a transfer matrix between the microphones and the HOA-domain

Fourier expansion, b_H0A is a set of HOA-domain Fourier coefficients given by

£·, is a non-negative real number.

32. The method of claim 28 which includes conducting the plane-wave analysis of the recorded sound field by solving the following convex programming problem for the vector of plane-wave amplitudes,

where:

T_piw/mic i^{s a} transfer matrix between plane- waves and the microphones, ε₁ is a non-negative real number,

lplw/HOA is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

b_H0A is a set of HOA-domain Fourier coefficients given by

where ^Tmic/HOA is a transfer matrix between the microphones and the HOA-domain Fourier expansion, and

ε₂ is a non-negative real number

33. The method of claim 32 which includes setting ε_λ based on the resolution of the spatial division of a set of directions corresponding to the set of analysis plane-waves and setting the value of ε₂ based on the computed sparsity of the sound field.

34. The method of any one of claims 28 to 33 which includes transforming

back to the time-domain using an inverse FFT to obtain g_plw-cs (t) .

35. The method of any one of claims 9 to 25 which includes using a time domain technique to analyse recorded data in the sparse domain and obtaining parameters generated from the selected time domain technique to enable the recorded sound field to be reconstructed.

36. The method of claim 35 which includes analysing the recorded sound field in the time domain using plane- wave analysis according to a set of basis plane- waves to produce a set of plane-wave signals, g_plw-os (t) .

37. The method of claim 36 which includes analysing the recorded sound field in the HOA domain to obtain a vector of HOA-domain time signals, b_H0A (t) , and sampling the vector of HOA-domain time signals over a given time frame, L, to obtain a collection of time samples at time instances t_x to t_N to obtain a set of HOA-domain vectors at each time instant: expressed as a matrix, B_H0A by:

38. The method of claim 37 which includes computing a correlation vector, γ , as

w^nere

*^{s an} omni-directional HOA-component of

39. The method of claim 38 which includes solving the following convex programming problem for a vector of plane-wave gains, P_plw.cs :

minimise

|| ||

where:

Tpiw/HOA is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

ε_χ is a non-negative real number.

40. The method of claim 38 which includes solving the following convex programming problem for a vector of plane-wave gains,

minimise

subject to

)

to

where:

is a transfer matrix between the plane-waves and the HOA-domain

Fourier expansion,

ε_χ is a non-negative real number,

ε₂ is a non-negative real number.

41. The method of claim 40 which includes setting ε_χ based on the resolution of the spatial division of a set of directions corresponding to the set of analysis plane-waves and setting the value of ε₂ based on the computed sparsity of the sound field.

42. The method of claim 40 or claim 41 which includes thresholding and cleaning Ppiw-cs to set some of its small components to zero.

43. The method of claim 42 which includes forming a matrix, Y_plw._H0A , according to the plane-wave basis and then reducing

_p to Y

by keeping only the columns corresponding to the non-zero components in

where is an

HOA direction matrix for the plane-wave basis and the hat-operator on _A

indicates it has been truncated to some HOA-order M.

44. The method of claim 43 which includes computing

as

45. The method of claim 44 which includes expanding

_p to obtain Spiw-cs (0 by inserting rows of time signals of zeros so that

matches the plane- wave basis.

46. The method of claim 38 which includes solving the following convex programming problem for a matrix

where Y_plw is a matrix (truncated to a high spherical harmonic order) whose columns are the values of the spherical harmonic functions for the set of directions corresponding to some set of analysis plane waves, and ε_χ is a non-negative real number.

47. The method of claim 46 which includes obtaining an unmixing matrix, Tl_L , for the ί,-th time frame, by calculating:

5

where

II^_j refers to the unmixing matrix for the L-\ time frame and

a is a forgetting factor such that 0 < a≤ 1 . 0 48. The method of claim 47 which includes applying singular value decomposition to B_HOAto obtain a matrix decomposition:

49. The method of claim 48 which includes forming a matrix S_reduced by keeping only 5 the first m columns of S , where m is the number of rows of B_H0A and forming a matrix, Ω , given by

50. The method of claim 49 which includes solving the following convex 0 programming problem for a matrix Γ :

i i i |

where ε_λ and Y _lw are as defined above.

51. The method of claim 50 which includes obtaining G_plw from Γ using:

where V^T is obtained from the matrix decomposition of B_H0A .

52. The method of claim 51 which includes obtaining an unmixing matrix, TL_L , for the Z-th time frame, by calculating:

30

where;

n_L_, is an unmixing matrix for the L-l time frame,

a is a forgetting factor such that 0 < a≤ 1 .

35 53. The method of claim 52 which includes obtaining G_plw._smooth using:

54. The method of claim 53 which includes obtaining the vector of plane-wave signals, ) from the collection of plane-wave time samples, using

standard overlap-add techniques.

55. The method of claim 52 or claim 53 which includes obtaining,

from the collection of plane-wave time samples, G_plw , without smoothing using standard overlap-add techniques.

56. The method of any one of claims 34, 36-45 or 54-55 which includes modifying t^o reduce unwanted artifacts such as reverberations and/or unwanted sound

sources.

57. The method of claim 56 which includes, to reduce reverberations, reducing the signal values of some of the signals in the signal vector,

58. The method of claim 56 or claim 57 which includes, to separate sound sources in the sound field reconstruction so that the unwanted sound sources can be reduced, setting to zero some of the signals in the signal vector, g_plw._cs (t) .

59. The method of any one of claims 56 to 58 which includes modifying

dependent on the means of playback of the reconstructed sound field.

60. The method of claim 59 which includes modifying

as follows:

where:

Ppiw/spk is a loudspeaker panning matrix.

61. The method of claim 59 which includes converting back to the HOA-

domain by computing:

where b_H0A._highres (t) is a high-resolution HOA-domain representation of g_plw._cs (t) capable of expansion to arbitrary HOA-domain order, where Y_plw-H0A is an HOA direction matrix for a plane- wave basis and the hat-operator on indicates it has

been truncated to some HO A-order M.

62. The method of claim 61 which includes decoding

HOA decoding techniques.

63. The method of claim 59 which includes modifying to determine headphone gains as follows:

where:

is a head-related impulse response matrix of filters corresponding to

the set of plane wave directions.

64. The method of any one of claims 9 to 20 which includes using time-domain techniques of Independent Component Analysis (ICA) in the HOA-domain to analyse recorded data in a sparse domain, and obtaining parameters from the selected time domain technique to enable the recorded sound field to be reconstructed.

65. The method of claim 64 which includes analysing the recorded sound field in the HOA-domain to obtain a vector of HOA-domain time signals b_H0A (t) .

66. The method of claim 65 which includes analysing the HOA-domain time signals using ICA signal processing to produce a set of plane-wave source signals,

67. The method of claim 66 or claim 67 which includes computing from b_H0A (t) a mixing matrix, M_ICA , using signal processing techniques.

68. The method of claim 67 which includes projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing where is me transpose (Hermitian conjugate)

of the real-value (complex-valued) HOA direction matrix associated with the plane- wave basis and the hat-operator on indicates it has been truncated to some

HO A-order M.

69. The method of claim 68 which includes using thresholding techniques to identify the columns of V_source that indicate a dominant source direction.

70. The method of claim 69 which includes reducing the matrix

Y_p ^to obtain a matrix ^by removing the plane- wave direction vectors in _p that do

not correspond to dominant source directions associated with matrix

71. The method of claim 70 which includes estimating as

72. The method of claim 70 which includes estimating g_plw-ica._reduced (t) by working in the frequency domain and computing

73. The method of claim 72 which includes, for each frequency, reducing a transfer matrix, to obtain a matrix, T , by removing the columns in T_plw/mic that

do not correspond to dominant source directions associated with matrix V_source .

74. The method of claim 73 which includes estimating g_plw.ica._reduced by computing:

and transforming g_p!w,_ca,_eduoed back to the time-

domain using an inverse FFT to obtain g_{plw-ica-reduced} (t) .

75. The method of claim 74 which includes expanding

to obtain gpiw-ica (0 inserting rows of time signals of zeros so that g_plw._ica (t) matches the plane- wave basis.

76. The method of claim 65 or claim 66 which includes computing from b_H0A a mixing matrix, M_ICA , and a set of separated source signals, using signal

processing techniques.

77. The method of claim 76 which includes projecting the mixing matrix, M_ICA , on the HOA direction vectors associated with a set of plane-wave basis directions by computing

where is the transpose (Hermitian conjugate)

HOA-order M

78. The method of claim 77 which includes using thresholding techniques to identify from V the dominant plane-wave directions.

79. The method of claim 78 which includes cleaning g_ioa (t) to obtain

which retains the signals corresponding to the dominant plane-wave directions in V_source and sets the other signals to zero.

80. The method of claim 79 which includes modifying g_plw-ica (i) to reduce unwanted artifacts such as reverberations and/or unwanted sound sources.

81 . The method of claim 80 which includes, to reduce reverberations, reducing the signal values of some of the signals in the signal vector,

82. The method of claim 80 or claim 81 which includes, to separate sound sources in the sound field reconstruction so that the unwanted sound sources can be reduced, setting to zero some of the signals in the signal vector,

83. The method of any one of claims 80 to 82 which includes modifying g_plw._ica (t) dependent on the means of playback of the reconstructed sound field.

84. The method of claim 83 which includes modifying g_p (t)

as follows:

where:

is a loudspeaker panning matrix.

85. The method of claim 83 which includes converting g_plw._ioa (t) back to the HOA- domain by computing:

where:

is a high-resolution HOA-domain representation of

capable of expansion to arbitrary HOA-domain order, Ypiw-HOA i^{s m} HOA direction matrix for a plane-wave basis and the hat-operator on Y_plw._H0A indicates it has been truncated to some HOA-order M.

86. The method of claim 85 which includes decoding

HOA decoding techniques.

87. The method of claim 83 which includes modifying to determine

headphone gains as follows:

where:

is a head-related impulse response matrix of filters corresponding to the set of plane wave directions.

88. A computer when programmed to perform the method of any one of claims 9 to 87.

89. A computer readable medium to enable a computer to perform the method of any one of claims 9 to 87.