Incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method

Abstract

The invention discloses an incremental Markov chain Monte Carlo (MCMC) underwater acoustic channel blind equalization method which is used for solving the problem that traditional MCMC blind equalization cannot be converged due to excessive unknown parameters in a long pulse response underwater acoustic channel. The method is divided into two stages: in the first stage, MCMC blind equalization is started with a smaller initial channel length, and the length is gradually increased until a normalized reconstruction error is lower than a threshold; and in the second stage, the result of the previous stage is inherited, the channel length is increased by the step length delta, and MCMC blind equalization is initialized and executed based on the result of the previous stage until the reconstruction error is not reduced any more. Through an incremental strategy and intermediate result multiplexing, the calculation complexity is significantly reduced, and the convergence and accuracy of long channel estimation are improved. The method is suitable for an underwater communication system, the blind equalization of the long pulse response underwater acoustic channel becomes possible through an incremental strategy, the high calculation complexity is avoided by reasonably utilizing the blind equalization intermediate result, and the method has high practical application value.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of underwater acoustic channel blind equalization, and particularly relates to an incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method. BACKGROUND

[0002] The blind equalization technology can recover the sending symbols without pilot signals by directly using the statistical characteristics or modulation structure of the signal itself, and is widely applied in mobile communication, underwater acoustic communication and other fields.

[0003] At present, the blind equalization algorithms for underwater acoustic channels are mainly divided into four categories: firstly, the method based on high-order statistics, such as the fourth-order cumulants matching method. This kind of method mainly recovers the phase and amplitude information destroyed by the channel through high-order statistics. This kind of method has high computational complexity and is sensitive to noise. Secondly, the method based on signal structure characteristics, such as the constant modulus algorithm. This kind of method uses the physical characteristics of the modulation signal (such as constant modulus, limited value, cyclostationarity) to design the cost function, and recovers the sending symbols by optimizing the cost function. This kind of method is strongly dependent on the structure characteristics of the signal, is easy to converge to a local optimal solution or produce error propagation, and has high steady-state error. Thirdly, the method based on machine learning, such as the deep learning equalizer. This kind of method uses neural network to estimate the sending symbols from data without explicit input-output relationship. This kind of method depends on a large amount of training data, and the generalization ability is unstable. Finally, the method based on Bayesian inference, which establishes a joint probability model of channel and symbol, recovers the symbol through Bayesian inference, has the advantages of complex channel adaptability and low signal-to-noise ratio robustness. Under the condition of complex underwater acoustic channel, the blind equalization based on Bayesian inference has advantages among the above four kinds of methods, and is widely used. The representative Bayesian inference blind equalization is based on Markov chain Monte Carlo algorithm (MCMC). MCMC is a classical algorithm for Bayesian inference, which can sample the joint distribution of multi-dimensional variables through Gibbs sampling. By updating the conditional distribution of each variable in turn, a Markov chain is constructed, and a sample chain is generated through iteration. Finally, the sample converges to the target distribution.

[0004] However, when the channel is too long and the unknown parameters are too many, the constraint ability of the likelihood function on the parameters is weakened, leading to the posterior distribution being mainly dominated by the prior distribution, and the flatness or multi-peakness, which further leads to low sampling efficiency. The existing MCMC blind equalization estimation result cannot converge, and the technical bottleneck needs to be broken through. Summary of the Invention

[0005] The present invention proposes an incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method to solve the problem that the existing Markov chain Monte Carlo blind equalization method cannot converge when the channel is too long and there are too many unknown parameters.

[0006] In order to solve the above technical problems, the specific technical solutions of the present invention are as follows:

[0007] An incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method comprises the following steps:

[0008] Step 1: Establish the received signal model y = Xh + n and set the reconstruction error threshold S threshold and the initial channel length L0; N is the baseband received signal length, h is the channel vector, and X is the transmitted symbol x=[x1,…,x N ] T The Toeplitz matrix is ​​constructed, and the transmitted symbols are taken from the set A={a1,a2,…,a |A|}, |A| is the number of elements in set A; n is a noise vector, each element has a mean of 0 and a variance of σ 2 Gaussian distribution;

[0009] Step 2: Take the current channel length L c Perform MCMC blind equalization:

[0010] Generate initial estimates of channel, noise, and symbols based on prior distributions Perform n Gibbs samplings and output the results

[0011] Step 3: Use the output from step 2 and Calculate the reconstructed signal and normalized error S; if S≤S threshold , record the current channel length as L1 and go to step 4; otherwise, gradually increase the channel length with a step size of 1 and repeat steps 2 and 3 until S≤S threshold ;

[0012] Step 4: Based on the channel length L1 obtained in step 3 and the corresponding blind equalization result Perform incremental channel length extension:

[0013] Set the current channel length L according to the number of growth times q q+1 =L1+q×δ, where δ is the growth step size, q≥1 and δ>0;

[0014] Inherit L q Results under length Initialize the current length L q+1 of the MCMC process;

[0015] Perform MCMC blind equalization, and record the channel length L Q if the reconstruction error S no longer decreases, output the final estimate

[0016] Preferably, the generating of the initial estimate in step two comprises:

[0017] According to the prior complex Gaussian distribution of the channel, i.e. sampling to obtain the initial estimate of the channel is the prior mean of the channel, is the prior variance;

[0018] According to the prior inverse chi-square distribution of the noise, i.e. 2 p(σ -2 ) ~ χ k (v0, λ0), sampling to obtain the initial estimate of the noise variance v0 is the prior degree of freedom of the noise variance, and λ0 is the prior scale parameter;

[0019] According to the prior uniform distribution of the transmitted symbol, i.e. sampling to obtain the initial estimate of the symbol x k is the kth transmitted symbol, and a j is the jth element in the set A.

[0020] Preferably, the Gibbs sampling of step two comprises the following sub-steps which are sequentially performed:

[0021] (1) Calculate the posterior probability of the channel as follows

[0022]

[0023] where

[0024]

[0025] where, (·) H is the conjugate transpose,

[0026] Sample according to the posterior probability of the channel to obtain the channel sample value

[0027] (2) Calculate the posterior probability of the noise as follows

[0028]

[0029] where, the expression of

[0030]

[0031] Sampling according to the noise posterior probability to obtain a noise variance sampling value

[0032] (3) The kth symbol x k The posterior probability of the element a j in the symbol set is as follows

[0033]

[0034] Wherein,

[0035] For the kth symbol, by calculating the posterior probability of each value in the symbol set A, the value with the largest probability is selected to obtain the symbol estimation value Iterate k = 1 to N to perform the process to obtain the complete symbol vector estimation value

[0036] Preferably, in step three, the reconstruction error is calculated as:

[0037]

[0038] Where ||·|| represents the L2 norm of the vector.

[0039] Preferably, the initialization of step four specifically includes:

[0040] According to the channel estimation value of the L q length channel The prior mean value of the extended channel is:

[0041] The prior covariance matrix is constructed as: Where ε is the uncertainty;

[0042] According to the channel prior information, that is, Sampling to obtain the initial estimation value of the channel under L q+1 length Inherit the noise estimation value and symbol estimation value obtained by blind equalization under the L q length channel And As the initial value of the noise variance and the symbol under the L q+1 channel length;

[0043] Based on the initial value MCMC blind equalization under the L q+1 length channel.

[0044] Beneficial effects: the application is an incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method, compared with the prior art, the method has the following advantages:

[0045] 1) The application adopts incremental channel length setting to realize MCMC blind equalization, so that reliable channel and symbol estimation results can be obtained when the channel is too long and the unknown parameters are too many, and the application situation of MCMC blind equalization is significantly improved.

[0046] Specifically, the application divides the blind equalization into two stages: the initial stage: starting MCMC with a smaller channel length L0, gradually increasing the length until the reconstruction error S≤S threshold ; the incremental stage: based on the results of the previous stage, the channel length is expanded by a fixed step δ, the historical estimation value is reused to initialize the MCMC process of the new length, and the error is no longer reduced.

[0047] The traditional MCMC cannot converge due to the flattening of the posterior distribution caused by too many parameters in a long channel. The application gradually increases the complexity to avoid one-time sampling in a high-dimensional parameter space, so that blind equalization of long impulse response underwater acoustic channels is possible.

[0048] 2) The application initializes the next blind equalization by reasonably inheriting the results of the previous blind equalization, reduces the number of Gibbs sampling iterations and the calculation complexity, and improves the convergence speed and performance.

[0049] Specifically, the application retains the converged channel structure to avoid re-exploring low-probability areas, that is, the final channel estimation of the previous stage is used as the core part of the prior mean value under the new length, and δ new taps are initialized to a zero vector; a block diagonal matrix is constructed to distinguish the confidence of the historical and new taps, wherein the unit matrix I locks the high confidence, forces the algorithm to maintain the optimized results, and the diagonal matrix εI gives high uncertainty to encourage exploration of the new parameter space; the random initialization is skipped, and the initial values close to the true distribution, that is, the noise variance initial value and the symbol vector initial value, are directly injected. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 is a time-domain channel measured according to a detection signal;

[0051] Figure 2 is a workflow diagram of the incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method;

[0052] Figure 3 is a relationship diagram of the initial channel length L0 and the corresponding received signal reconstruction error;

[0053] Figure 4 is a diagram of the reconstruction error of the incremental MCMC blind equalization changing with the number of channel growth (q=0 corresponds to the final blind equalization reconstruction error of the first stage);

[0054] Figure 5 Comparison of channel estimation results of incremental MCMC blind equalization and MCMC blind equalization

[0055] Figure 6 The comparison chart of bit error rate between incremental MCMC blind equalization and MCMC blind equalization under different channel lengths. DETAILED DESCRIPTION

[0056] The technical solutions provided by the present invention will be described in detail below in conjunction with specific implementation cases. It should be understood that the following specific implementation methods are only used to illustrate the present invention and are not used to limit the scope of the present invention.

[0057] The present invention proposes an incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method, which includes the following steps:

[0058] Step 1: The baseband received signal y=[y1,…,y N ] T It can be expressed as y=Xh+n, where h is the channel vector, whose length L is unknown and obeys the multi-dimensional complex Gaussian distribution. X is the transmitted symbol x=[x1,…,x N ] T The Toeplitz matrix is ​​constructed, and the transmitted symbols are taken from the set A={a1,a2,…,a |A|}, and the probability of each symbol in the set is equal. n is a noise vector, each element is independent, and all obey the mean of 0 and the variance of σ 2 Gaussian distribution, and the variance σ 2 And it obeys the inverse chi-square distribution. Set the reconstruction error threshold S threshold . Set the initial channel length L0;

[0059] Step 2: Set L according to the current channel length c , based on the prior distribution information of step 1, set the channel prior mean to The prior variance is Set the noise variance prior degrees of freedom to v0 and the prior scale parameter to λ0. Generate initial estimates of the channel, noise, and symbols Set the number of iterations n of MCMC blind equalization, based on the initial estimate And the received symbol y, perform n Gibbs sampling to get the blind equalization result

[0060] Step 3: Reconstruct the received signal using the channel and transmitted symbol estimates from step 2 Calculate the normalized received signal reconstruction error S. If S≤S thresholdIf S > S then go to step four, record the channel length as L1; otherwise, increase the channel length by one step and repeat step two and step three until S≤S threshold ;

[0061] Step four: increase the channel length by δ based on the previous blind equalization result, set the iteration number n of MCMC blind equalization, and perform MCMC blind equalization. Stop the channel length growth when the normalized reconstruction error is no longer reduced, record the channel length as L Q at this time, and output the final estimated value

[0062] In step two, set L c based on the prior distribution information in step one, generate initial estimates of the channel, noise, and symbols, and the specific implementation is as follows:

[0063] First, according to the prior complex Gaussian distribution of the channel, i.e. sampling obtains the initial estimate of the channel

[0064] Second, according to the prior inverse chi-square distribution of the noise, i.e. 2 p(σ -2 (v0, θ0), sampling obtains the initial estimate of the noise variance

[0065] Finally, according to the prior uniform distribution of the transmitted symbol, i.e. sampling obtains the initial estimate of the symbol

[0066] In step two, the i-th (1≤i≤n) Gibbs sampling is performed, including the following sub-steps:

[0067] (1) Sampling the channel: calculate the channel posterior probability as follows

[0068]

[0069] where

[0070]

[0071] (·) H is the conjugate transpose, According to the posterior probability given in (1), sampling obtains the channel sampling value

[0072] (2) Sampling the noise variance: calculate the posterior probability of the noise as follows

[0073]

[0074] where The expression is

[0075]

[0076] According to the posterior probability given by (2), sampling is performed to obtain a noise variance sampling value

[0077] (3) Estimation of symbols: under the prior equal probability condition, the kth (1≤k≤N) bit symbol x k is equal to the posterior probability of the element a j in the symbol set

[0078] wherein Here, the symbol sampling can be performed according to the probability in (3), but considering that the symbol value is in a limited value in the symbol set, the present application performs hard decision by comparing the posterior probabilities of different values in the symbol set to obtain the estimation value of the kth bit symbol The above estimation is performed on all N symbols to obtain the symbol vector estimation value

[0079] In step three, the reconstructed received signal is obtained using the channel in step two and the symbol estimation value The normalized received signal reconstruction error S is calculated. The specific implementation is as follows:

[0080] First, the channel estimation value and the symbol estimation value in step two are used to calculate the reconstructed received signal wherein the kth (1≤k≤N) bit reconstructed received signal is calculated as

[0081] Then, the normalized received signal reconstruction error is calculated, wherein ||·|| represents the L2 norm of the vector.

[0082] In step four, the channel length is increased by δ based on the blind equalization result in the previous stage, the iteration number n of the MCMC blind equalization is set, and the MCMC blind equalization is performed. The specific implementation is as follows:

[0083] First, the current channel length L q+1 is set according to the growth number q, wherein L q+1 =L 1+q×δ, and δ is the growth step, and it is required that q≥1 and δ>0. In particular, the previous blind equalization result when q=1 is the final blind equalization result in step two The channel estimation value q of the L q length channel is set The channel mean prior information is set, that is, wherein ε is the uncertainty;

[0084] Secondly, according to the channel prior information, that is, Sampling to obtain L q+1 Initial channel estimate at length Inherit L q Noise estimation value and symbol estimation value obtained by blind equalization under length channel and As L q+1 The noise variance and initial value of the symbol under the channel length;

[0085] Finally, based on the initial value Carry out L q+1 MCMC blind equalization under long channels.

[0086] The present invention will be further explained below with reference to the embodiments.

[0087] The proposed method was validated using experimental data from a single-transmitter, single-receiver, single-carrier underwater acoustic communication tank. The experimental tank was 15 meters long, 12 meters wide, and 7 meters deep. The transmitting transducer and receiving array elements were both located at a depth of 2 meters underwater, resulting in a communication range of 3.2 meters. The carrier frequency was 6 kHz, and the symbol period was 0.25 ms. The channel coding employed a rate-1 / 2 convolutional code, and the symbol modulation scheme was BPSK, with the symbol set A = {+1, -1}. A transmission block contained M = 512 BPSK symbols. Figure 1 The time domain channel measured based on the detection signal is shown, with a length of L = 20. This channel is used as the reference true value in the following text. The workflow of the method of the present invention is as follows Figure 2 shown.

[0088] Set the uncertainty ε = 1000. According to the initial channel length L0, the channel is set to obey the prior mean The prior variance is the variance Multidimensional complex Gaussian distribution; set the noise variance σ 2 Obey the inverse chi-square distribution with degree of freedom v0=1 and scale parameter λ0=1; each transmitted symbol has equal probability of taking two values ​​in the BPSK set A. Set the normalized reconstruction error threshold S threshold = 0.1. Based on the above prior information, the initial estimated values ​​of the channel, noise and symbol under the initial channel length are obtained. Set the number of MCMC blind equalization iterations n = 200, and then perform the first stage of blind equalization: estimate the estimated value corresponding to the initial channel Figure 3 The reconstruction errors corresponding to different initial channel lengths L0 are shown. It can be seen that when L0=4, the reconstruction error is lower than the threshold value for the first time. At this time, the bit error rate BER=1.76%, and the equalization result is acceptable. Therefore, the final channel length L1=4 of the first stage is obtained, and the second stage blind equalization is performed on this basis.

[0089] The second stage channel growth step is set to be 4, and the channel length is sequentially increased for blind equalization until the reconstruction error is no longer reduced. Since the current blind equalization in this stage is initialized by the result of the previous blind equalization, the iteration number of the MCMC blind equalization is reduced to n=60. Figure 4 The change of the incremental MCMC blind equalization reconstruction error with the channel growth number is shown, and it can be seen that the reconstruction error is not obviously reduced when q=3, 4, and the reconstruction error is slightly increased when q=5, so the blind equalization result corresponding to q=4 is the final blind equalization result.

[0090] Figure 5 For comparison of the channel estimation results of the incremental MCMC blind equalization and the MCMC blind equalization, it can be seen that the incremental MCMC blind equalization can estimate the channel with a length of 20, while the MCMC blind equalization cannot estimate the channel.

[0091] Figure 6 For comparison of the bit error rates of the incremental MCMC blind equalization and the MCMC blind equalization under different channel lengths, L2=8, L3=12, L4=16, and L5=20. It can be seen that when the channel length is greater than L3=12, the performance of the MCMC blind equalization continuously decreases, while the bit error rate of the incremental MCMC blind equalization continuously decreases with the increase of the channel length.

[0092] The above only describes the preferred embodiments of the present application, and it should be noted that for those skilled in the art, some improvements and refinements can be made without departing from the principles of the present application, and these improvements and refinements should also be considered as the protection scope of the present application.

Claims

1. An incremental Markov chain Monte Carlo underwater acoustic channel blind equalization method, characterized in that: The steps include: Step 1: Establish the received signal model y = Xh + n and set the reconstruction error threshold S threshold and the initial channel length L0; N is the baseband received signal length, h is the channel vector, and X is the transmitted symbol x=[x1,…,x N ] T The Toeplitz matrix is ​​constructed, and the transmitted symbols are taken from the set A={a1,a2,…,a |A| }, |A| is the number of elements in set A; n is a noise vector, each element has a mean of 0 and a variance of σ 2 Gaussian distribution; Step 2: Take the current channel length L c Perform MCMC blind equalization: Generate initial estimates of channel, noise, and symbols based on prior distributions Perform n Gibbs samplings and output the results Step 3: Use the output from step 2 and Calculate the reconstructed signal and normalized error S; if S≤S threshold , record the current channel length as L1 and go to step 4; otherwise, gradually increase the channel length with a step size of 1 and repeat steps 2 and 3 until S≤S threshold ; Step 4: Based on the channel length L1 obtained in step 3 and the corresponding blind equalization result Perform incremental channel length extension: Set the current channel length L according to the number of growth times q q+1 =L1+q×δ, where δ is the growth step size, q≥1 and δ>0; Inherit L q Results under length Initialize the current length L q+1 MCMC process; Perform MCMC blind equalization. If the reconstruction error S no longer decreases, the channel length at this time is L. Q , output the final estimate 2. The method according to claim 1, characterized in that The generation of the initial estimated value in step 2 includes: According to the prior complex Gaussian distribution of the channel, Sampling to obtain an initial estimate of the channel is the channel prior mean, is the prior variance; According to the prior inverse chi-square distribution of noise, that is, p(σ 2 )~χ -2 (v0,λ0), sample to get the initial estimate of the noise variance v0 is the prior degrees of freedom of the noise variance, λ0 is the prior scale parameter; According to the prior uniform distribution of the transmitted symbols, Sampling to obtain an initial estimate of the symbol x k is the kth transmitted symbol, a j is the jth element in set A.

3. The method according to claim 1, characterized in that The Gibbs sampling of step 2 includes sub-steps performed sequentially: (1) Calculate the channel posterior probability as follows in in,(·) H is the conjugate transpose, Sampling is performed according to the channel posterior probability to obtain the channel sampling value (2) Calculate the posterior probability of noise as follows in, The expression is Sampling is performed according to the posterior probability of the noise to obtain the noise variance sampling value (3) Under the prior equal probability condition, the kth symbol x k Equal to element a in the symbol set j The posterior probability is as follows in, For the k-th symbol, by calculating the posterior probability of each value in the symbol set A, the value with the largest probability is selected for hard decision to obtain the symbol estimate Traverse k = 1 to N and perform this process to obtain the complete symbol vector estimate 4. The method according to claim 1, wherein In step 3, the reconstruction error is calculated as: Where ‖·‖ represents the L2 norm of the vector.

5. The method according to claim 1, wherein The initialization of step 4 specifically includes: According to L q Channel estimate for a channel of length Extended channel prior mean: Construct the prior covariance matrix: Where ε is the uncertainty; According to the channel prior information, Sampling to obtain L q+1 Initial channel estimate at length Inherit L q Noise estimation value and symbol estimation value obtained by blind equalization under length channel and As L q+1 The noise variance and initial value of the symbol under the channel length; Based on the initial value Carry out L q+1 MCMC blind equalization under long channels.

Images