CN108848043B - Low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing - Google Patents

Low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing Download PDF

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CN108848043B
CN108848043B CN201810574993.1A CN201810574993A CN108848043B CN 108848043 B CN108848043 B CN 108848043B CN 201810574993 A CN201810574993 A CN 201810574993A CN 108848043 B CN108848043 B CN 108848043B
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马璐
宋庆军
乔钢
万磊
刘凇佐
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Harbin Engineering University
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Abstract

The invention provides a low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing. Pre-calculating an alternative path characteristic Hermitian inner product matrix; then, the joint estimation of the time delay and the Doppler factor is carried out in an iterative mode. The method avoids the repeated calculation of the matrix inner product in iteration by the conventional orthogonal matching pursuit algorithm through a mode of pre-calculating the Hermitian inner product matrix of the alternative path characteristics, greatly reduces the calculation complexity, verifies the effectiveness of the method under the underwater sound time-varying channel through performance simulation, and verifies that the method can realize the same channel estimation precision under the condition that the calculation complexity is far lower than that of the conventional orthogonal matching pursuit algorithm and can provide the estimation precision higher than that of the orthogonal matching pursuit algorithm under the condition of the same calculation complexity, thereby having practical application value.

Description

Low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing
Technical Field
The invention relates to an underwater acoustic communication method, in particular to a low-complexity underwater acoustic sparse time-varying channel estimation method.
Background
With the increasing of ocean development activities of people, more and more ocean information monitoring and collecting devices are applied underwater, and an underwater acoustic communication technology for bearing underwater information transmission tasks becomes a key point of attention. In recent years, Orthogonal Frequency Division Multiplexing (OFDM) technology has been widely used in underwater communication systems due to its high spectral efficiency and good resistance to channel frequency selective fading. However, the underwater acoustic channel is one of the most complex wireless channels due to its fast time, large spreading delay, and severe doppler shift. Therefore, for the OFDM system, accurate underwater acoustic channel estimation is an important link for ensuring communication performance. The underwater acoustic channel is generally modeled into a sparse multi-path channel, each path of the sparse multi-path channel is determined by path gain, path delay and path Doppler factor, and channel estimation can be performed by adopting a compressed sensing algorithm under the channel model, wherein the channel estimation comprises an orthogonal matching tracking algorithm, a basis tracking algorithm and the like. However, when such an algorithm is adopted, an observation matrix with a larger dimension needs to be constructed for high-precision channel estimation, which causes the high computational complexity of the algorithm and is not beneficial to practical application.
Disclosure of Invention
The invention aims to provide a low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing, which can realize the same channel estimation precision under the condition of complexity and has practical application value.
The purpose of the invention is realized as follows: pre-calculating an alternative path characteristic Hermitian inner product matrix; then, the joint estimation of the time delay and the Doppler factor is carried out in an iterative mode.
The joint estimation of the delay and the doppler factor in an iterative manner specifically includes:
(1) inputting channel estimation parameters including a received symbol vector, an observation matrix, an alternative path characteristic Hermitian inner product matrix, sparsity and a termination threshold;
(2) initializing, including residual error initialization, time delay set, Doppler factor set and null matrix initialization, iteration count initialization and time delay index set initialization;
(3) initializing a Hermitian inner product matrix;
(4) searching parameters of the maximum amplitude of the Hermitian inner product matrix;
(5) updating the matrix of the set and the extracted atoms;
(6) determining a path complex gain;
(7) updating a path characteristic Hermitian inner product matrix, and setting the appointed row to be zero;
(8) judging iteration termination conditions, and if the conditions are met, terminating the iteration; if not, returning to the step (4);
(9) and outputting channel estimation parameters comprising a time delay estimation set, a Doppler factor estimation set and a path complex gain estimation set.
The invention provides a low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing through formula derivation aiming at the problem that the calculation complexity of the existing underwater sound time-varying channel estimation method based on an orthogonal matching pursuit algorithm is too high. The method avoids the repeated calculation of the matrix inner product in iteration by the conventional orthogonal matching pursuit algorithm through a mode of pre-calculating the Hermitian inner product matrix of the alternative path characteristics, greatly reduces the calculation complexity, verifies the effectiveness of the method under the underwater sound time-varying channel through performance simulation, and verifies that the method can realize the same channel estimation precision under the condition that the calculation complexity is far lower than that of the conventional orthogonal matching pursuit algorithm and can provide the estimation precision higher than that of the orthogonal matching pursuit algorithm under the condition of the same calculation complexity, thereby having practical application value.
The method has the advantages that the Hermitian inner product matrix of the alternative path characteristics is calculated in advance, repeated calculation of the inner product of the matrix of the orthogonal matching pursuit algorithm is avoided in iteration of channel estimation, the calculation complexity is greatly reduced, the estimation precision identical to that of the orthogonal matching pursuit algorithm can be provided under the identical parameters, and the method has practical application value in an actual underwater acoustic OFDM communication system.
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Fig. 1 is a flowchart of an underwater acoustic sparse time-varying channel estimation method based on an orthogonal matching pursuit algorithm.
FIG. 2 is a flow chart of a low-complexity underwater acoustic sparse time-varying channel estimation method based on compressed sensing designed by the present invention.
Fig. 3 is a graph comparing the signal-to-noise ratio-bit error rate performance of the method of the present invention and the OMP channel estimation method.
Fig. 4 is a table of the comparison of computational complexity for the inventive method and the OMP channel estimation method.
Detailed Description
The method mainly comprises the following two parts: and (3) calculating a Hermitian inner product matrix of the alternative path characteristics in advance, and based on the iterative sparse time-varying channel estimation method. Based on a compressed sensing model, a matrix inner product matrix in traditional OMP iteration is avoided from being repeatedly calculated by using a pre-calculated alternative path characteristic Hermitian inner product matrix, and then the time delay and Doppler factor are jointly estimated in an iteration mode, so that the complexity of a traditional OMP algorithm is reduced while the channel estimation performance is ensured.
The invention is described in more detail below by way of example.
The method aims at the characteristics of the underwater acoustic channel, formula derivation is carried out on the basis of the compressive sensing OMP algorithm, the problem of repeated calculation of the inner product matrix of the traditional orthogonal matching tracking algorithm is solved by pre-calculating the alternative path characteristic Hermitian inner product matrix, and the same estimation precision as the OMP algorithm is provided while the calculation complexity is reduced by the same parameters.
The following is detailed in the following according to four parts of a basic underwater sound OFDM communication system model, a channel estimation method based on an OMP algorithm, an underwater sound sparse time-varying channel estimation method based on compressed sensing and low complexity, and simulation performance analysis:
1. basic underwater sound OFDM communication system model
The invention considers a CP-OFDM system, assuming that an OFDM block has K subcarriers in total, then the symbol transmitted at the K subcarrier is s [ K ]]. Defining one OFDM block period as T and cyclic prefix length as Tcp. Will f iscDefined as the center frequency, the k-th subcarrier frequency is
fk＝fc+kΔf,k＝-K/2,…,K/2-1. (0.1)
The transmitted OFDM signal can be written as
Figure BDA0001685826590000031
For the underwater acoustic sparse time-varying channel model, assuming there are L paths, the channel impulse response can be represented as
Figure BDA0001685826590000032
Wherein A isl，τlAnd alRespectively, the amplitude, delay and doppler factor of the ith path. The channel parameters are assumed to be constant within one CP-OFDM block.
OFDM signals received via a channel can be written as
Figure BDA0001685826590000033
Wherein
Figure BDA0001685826590000034
Is additive noise. Doppler spread factor from coarse estimate before demodulation
Figure BDA0001685826590000035
And residual average Doppler estimate frequency offset ε, then defining a new residual Doppler factor, path delay and path complex gain of
Figure BDA0001685826590000036
After Doppler compensation, the m sub-carrier symbol of OFDM demodulation is
Figure BDA0001685826590000037
Wherein wmIs additive noise, m is ∈ [ -K/2, K/2-1]And is and
Figure BDA0001685826590000038
Figure BDA0001685826590000039
according to (0.6), can be obtained
z＝Hs+w， (0.9)
Where z is a K × 1-dimensional received symbol vector, s is a K × 1-dimensional transmitted symbol vector, and w is a K × 1-dimensional noise vector. While the K x K dimensional hybrid channel matrix H is expressed as
Figure BDA0001685826590000041
Wherein, K x K dimension matrix gammalThe (m, k) -th element in (b) is expressed as
Figure BDA0001685826590000042
Simultaneous lambdalIs a diagonal matrix of K x K dimensions, and the diagonal elements satisfy
Figure BDA0001685826590000043
Considering that the complexity of the algorithm is not too high, only 2D +1 diagonal lines in H are generally reserved, and D is a small integer.
2. Existing channel estimation method based on OMP algorithm
Can be rewritten as (0.9)
Figure BDA0001685826590000044
Equation (2.1) satisfies the compressed sensing algorithm model. Where Φ is a K × L-dimensional observation matrix, and ξ is an L × 1-dimensional sparse vector to be estimated. The solution to the sparse vector ξ can be estimated using the OMP algorithm. Suppose the search has the following delay and Doppler factor ranges
Figure BDA0001685826590000045
Wherein, the baseband sampling interval is T/K, I is an oversampling factor, and considering that all paths are in the cyclic prefix interval, the time delay has N in totalτ＝ITcpK/T possible values; suppose the maximum Doppler factor is bmaxThe search range satisfies [ -b ]max,bmax]And the search interval is Δ b, then the Doppler factor is N in totalb＝2bmax/(. DELTA.b) +1 possible values. Search delay and doppler factor based on assumptionsIn the range, we can construct K × NτNbDimension observation matrix
Figure BDA0001685826590000051
Wherein
Figure BDA0001685826590000052
Representing the Doppler factor selection biAll paths possible in time K NτAnd (5) dimension observation matrix.
The method comprises the following specific steps:
(1) inputting: receiving a symbol vector z, observing a matrix phi, sparsity L and a termination threshold e
(2) Initialization: initial residual r0Z, set of delays
Figure BDA0001685826590000053
Doppler factor set
Figure BDA0001685826590000054
One empty matrix Ψ0The iteration count l is 1.
(3) Searching parameters corresponding to the maximum amplitude of the residual error and the observation matrix Hermitian inner product:
Figure BDA0001685826590000055
wherein
Figure BDA0001685826590000056
Denotes xi(i)Q column of<·,·>And | represents the absolute value of the Hermitian inner product.
(4) Updating the matrix of sets and abstracted atoms:
Figure BDA0001685826590000057
(5) updating the path complex gain vector:
Figure BDA0001685826590000058
(6) and updating a residual vector:
Figure BDA0001685826590000059
(7) and (3) iteration termination judgment: repeating step (3-6) if L < L; or if rl||2And e is less than or equal to e, the iteration is terminated.
(8) And (3) outputting: path delay estimation set
Figure BDA00016858265900000510
Path Doppler factor estimation set
Figure BDA00016858265900000511
Path complex gain estimation vector
Figure BDA00016858265900000512
And finishing the channel estimation based on the orthogonal matching tracking algorithm through the steps. However, referring back to step (6), the residual of the ith iteration can be represented as
Figure BDA00016858265900000513
Then step (3) of the (l + 1) th iteration, the residual and the observation matrix Hermitian inner product is represented as
Figure BDA00016858265900000514
It can be found that for Hermitian the inner product consists of two parts: one part is the Hermitian inner product of the initial residual signal and the observation matrix; the other part is based on the Hermitian inner product of the previously estimated reconstructed signal with the observation matrix. For the iterative process of the whole algorithm, the former part is repeatedly calculated and brings great calculation complexity, and the patent provides a low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing aiming at the problem.
3. Low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing
First of all, c ═ is defined<Ξ(i),r>Is the Hermitian inner product matrix of the residual signal and the observation matrix. For q e [1, Nτ]And i ∈ [1, N ]b]The (q, i) element in the Hermitian inner product matrix c is represented as
Figure BDA0001685826590000061
When c is an Nτ×NbA matrix of dimensions. If the noise is not considered, the initial residual signal without noise is
Figure BDA0001685826590000062
Initial Hermitian inner product matrix c without noise at this time0The (q, i) element in (1) is represented by
Figure BDA0001685826590000063
Wherein, G islDefined as the ith path characteristics Hermitian inner product matrix. GlIs a number Nτ×NbDimension matrix, in which (q, i) elements are represented as
Figure BDA0001685826590000064
Wherein (·)*It is meant a conjugate operation of the two,
Figure BDA0001685826590000065
representing a dot product operation of corresponding elements of two vectors, (.)HRepresenting a conjugate transpose operation. Gamma rayqThe q-th column representing γ, where γ is part of the DFT matrixWhen m is ∈ [ -K/2, K/2-1],n∈[1,Nτ]The (m, n) element in γ is represented as
Figure BDA0001685826590000066
Derived by equation (3.2), the initial Hermitian inner product matrix c without noise0Is the sum of the complex gain of each path and the product of the Hermitian inner product matrix of the corresponding path characteristics. From the preset search range of equation (2.2), the signal Λ Γ s can be reconstructed for all possible combinations of path delays and doppler factors. Defining an alternative path characteristic Hermitian inner product matrix at the moment
Figure BDA0001685826590000071
At the same time can obtain
Figure BDA0001685826590000072
Wherein
Figure BDA0001685826590000073
Is one (2N)τ-1)×NbDimension matrix corresponding to a Doppler factor of buAnd (3) a Hermitian inner product matrix of alternative path characteristics including all path time delays, wherein u is equal to [1, Nb]。G(u)(q-v + N) of (1)τI) the element is represented as
Figure BDA0001685826590000074
Wherein
Figure BDA0001685826590000075
q,v∈[1,Nτ]，i∈[1,Nb]And when the search range of (2.2), G(u)All parameters are known and can be pre-calculated. Then according to Hermiti in the l iterationDetermining the path time delay and Doppler factor searched at this time by the maximum amplitude of an inner product matrix
Figure BDA0001685826590000076
And buThen, selecting a corresponding path characteristic Hermitian inner product matrix from the alternative path characteristic Hermitian inner product matrix to order
Figure BDA0001685826590000077
Wherein
Figure BDA0001685826590000078
Represents G(u)(1-v + N) of the matrixτ) Go to (2N)τV) all columns of rows. Each iteration is completed by means of iterative table look-up.
The method comprises the following specific steps:
(1) inputting: receiving a symbol vector z, an observation matrix phi, an alternative Hermitian inner product matrix G, sparsity L and a termination threshold e.
(2) Initialization: initial residual r0Z, set of delays
Figure BDA0001685826590000079
Doppler factor set
Figure BDA00016858265900000710
One empty matrix Ψ0Iteration count l is 1, delay index set V0。
(3) Initializing a Hermitian inner product matrix:
Figure BDA00016858265900000711
wherein
Figure BDA00016858265900000712
Denotes xi(u)V.e [1, N)τ]，u∈[1,Nb]。
(4) Searching parameters of the maximum amplitude of the Hermitian inner product matrix:
Figure BDA00016858265900000713
(5) updating the matrix of sets and abstracted atoms:
Figure BDA0001685826590000081
(6) determining the complex gain of the ith path:
Figure BDA0001685826590000082
(7) updating the ith path characteristic Hermitian inner product matrix:
Figure BDA0001685826590000083
(8) updating a Hermitian inner product matrix:
Figure BDA0001685826590000084
and c islV inlThe row is set to zero.
(9) And (3) iteration termination judgment: repeating step (4-8) if L < L; or if
Figure BDA0001685826590000085
The iteration is terminated.
(10) And (3) outputting: estimated set of path delays
Figure BDA0001685826590000086
Estimated path doppler factor set
Figure BDA0001685826590000087
Estimated path complex gain vector
Figure BDA0001685826590000088
4. Simulation performance analysis
In order to verify the performance of the channel estimation method, an underwater sound OFDM system is built, wherein the underwater sound OFDM system comprises 1024 sub-carriers, the bandwidth B is 6kHz, and the center frequency fc9kHz, sample rate fs48kHz, signal length T171 ms, cyclic prefix TcpQPSK modulation, LDPC coding is used for 20 ms. The underwater sound sparse time-varying channel model adopts 8 randomly generated paths, the time delay interval obeys exponential distribution of a mean value of 1ms, and the path amplitude obeys Rayleigh distribution along with the path time delay. Assuming a maximum Doppler factor bmax＝2×10-4Doppler search interval Δ b is 1 × 10-4(ii) a Path delay search range 0, Tcp) The search interval T/KI, the sparsity L is 8, the threshold e is 0.1, and D is 3.
In the simulation, an LS channel estimation method, an OMP algorithm-based channel estimation method and the channel estimation method are respectively adopted for comparison, and a simulation result under the known channel complete information is given as a lower performance bound.
Fig. 3 is a graph comparing the signal-to-noise ratio-bit error rate performance of the method of the present invention and the OMP channel estimation method. From simulation, it can be seen that the LS channel estimation method is poor under the time-varying channel, and the receiving end can hardly decode correctly. When the oversampling factor I is 1,2 or 8, the performance of the OMP channel estimation method and the channel estimation method of the present invention are gradually improved, and the performance is very close to each other under the same oversampling factor. Therefore, the channel estimation method can effectively carry out underwater sound sparse time-varying channel estimation.
Fig. 4 is a table of the comparison of computational complexity for the inventive method and the OMP channel estimation method. When the oversampling factors I are the same, it can be seen that the computation complexity of the method of the present invention is much smaller than that of the OMP channel estimation method, and in the case of obtaining the same delay and doppler estimation accuracy by combining with fig. 3, the computation complexity of the method of the present invention is significantly smaller than that of the OMP channel estimation method, and it can also be illustrated that the method of the present invention can provide higher estimation accuracy under the same computation complexity, which embodies the advantages of the method of the present invention.

Claims (1)

1. A low-complexity underwater sound sparse time-varying channel estimation method based on compressed sensing is characterized by comprising the following steps:
definition c ═<Ξ(i),r>Is the Hermitian inner product matrix of the residual signal and the observation matrix, for q ∈ [1, Nτ]And i ∈ [1, N ]b]The (q, i) element in the Hermitian inner product matrix c is represented as
Figure FDA0002791747560000011
c is an Nτ×NbA matrix of dimensions, the initial residual signal without noise, regardless of noise, being
Figure FDA0002791747560000012
Noiseless initial Hermitian inner product matrix c0The (q, i) element in (1) is represented by
Figure FDA0002791747560000013
Wherein G islFor the l-th path characteristics Hermitian inner product matrix, GlIs a number Nτ×NbDimension matrix, in which (q, i) elements are represented as
Figure FDA0002791747560000014
Wherein (·)*It is meant a conjugate operation of the two,
Figure FDA0002791747560000019
representing a dot product operation of corresponding elements of two vectors, (.)HDenotes a conjugate transpose operation, YqQ-th column representing Y, Y being part of a DFT matrix, K/2-1 when m ∈ [ -K/2],n∈[1,Nτ]When, the (m, n) element in Y is represented by
Figure FDA0002791747560000015
Alternative path characteristic Hermitian inner product matrix
Figure FDA0002791747560000016
Figure FDA0002791747560000017
Wherein
Figure FDA0002791747560000018
Is one (2N)τ-1)×NbDimension matrix corresponding to a Doppler factor of buAnd (3) a Hermitian inner product matrix of alternative path characteristics including all path time delays, wherein u is equal to [1, Nb]，G(u)(q-v + N) of (1)τI) the element is represented as
Figure FDA0002791747560000021
Wherein
Figure FDA0002791747560000022
And when determining
Figure FDA0002791747560000023
Search range of G(u)All the parameters are known, and then the searched path delay and Doppler factor are determined according to the maximum amplitude of the Hermitian inner product matrix in the first iteration
Figure FDA0002791747560000024
And buThen, selecting a corresponding path characteristic Hermitian inner product matrix from the alternative path characteristic Hermitian inner product matrix to order
Figure FDA0002791747560000025
Wherein
Figure FDA0002791747560000026
Represents G(u)(1-v + N) of the matrixτ) Go to (2N)τV) all columns of rows;
each iteration is completed in an iterative table look-up mode, and the specific steps are as follows:
(1) inputting: receiving a symbol vector z, an observation matrix phi, an alternative Hermitian inner product matrix G, sparsity L and a termination threshold e;
(2) initialization: initial residual r0Z, set of delays
Figure FDA0002791747560000027
Doppler factor set
Figure FDA0002791747560000028
One empty matrix Ψ0Iteration count l is 1, delay index set V0；
(3) Initializing a Hermitian inner product matrix:
Figure FDA0002791747560000029
wherein
Figure FDA00027917475600000210
Denotes xi(u)V.e [1, N)τ]，u∈[1,Nb]；
(4) Searching parameters of the maximum amplitude of the Hermitian inner product matrix:
Figure FDA00027917475600000211
(5) updating the matrix of sets and abstracted atoms:
Figure FDA00027917475600000212
(6) determining the complex gain of the ith path:
Figure FDA00027917475600000213
(7) updating the ith path characteristic Hermitian inner product matrix:
Figure FDA0002791747560000031
(8) updating a Hermitian inner product matrix:
Figure FDA0002791747560000032
and c islV inlSetting lines to zero;
(9) and (3) iteration termination judgment: repeating step (4-8) if L < L; or if
Figure FDA0002791747560000033
The iteration is terminated;
(10) and (3) outputting: estimated set of path delays
Figure FDA0002791747560000034
Estimated path doppler factor set
Figure FDA0002791747560000035
Estimated path complex gain vector
Figure FDA0002791747560000036
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