The severe multipath delay of the underwater acoustic channel, the Doppler shift, the severe time-varying characteristics, and sparsity make it difficult to obtain the channel state information in the channel estimation of the virtual time-reverse mirror OFDM, which makes the virtual time mirror subcarrier orthogonality easy to suffer damage; the focusing effect is not obvious. Therefore, this paper proposes a virtual time-inverse OFDM underwater acoustic channel estimation algorithm based on compressed sensing. The algorithm extracts the detection signal, constructs a sparse signal model of the delay-Doppler shift, and then performs preestimation of the underwater acoustic channel based on the compressed sensing theory. Then, by predicting the timing of the underwater acoustic channel and convolving with the received signal, the algorithm improves the focusing effect better. Experimental simulations show that compared with LS and OMP algorithms, the algorithm can accurately recover channel information from a small number of observations, reduce the bit error rate by 10%, and improve the accuracy of channel estimation and the time-inverse OFDM performance of virtual time.

Underwater acoustic communication is an important means to achieve underwater integrated sensing and information interaction. In the field of commercial strategy such as informatization marine data collection, marine resource development, and marine environmental monitoring, underwater acoustic communication plays an important role. At present, Orthogonal Frequency Division Multiplexing (OFDM) [

The majority of scientific research workers have invested a lot of manpower and material resources in the research of time-inverse OFDM technology in virtual time. In a time-varying underwater acoustic channel with a large multipath delay and severe Doppler frequency shift, the orthogonality between subcarriers is easily destroyed, resulting in a decline in communication performance. In response to this phenomenon, time-inversion technology is introduced into OFDM underwater acoustic communication to improve subcarrier focusing ability. Laigui Xu of Zhejiang University [

The bit error rate of the inverse OFDM technique can be reduced by inserting the known pilot technique. However, the sparse nature of the underwater acoustic channel makes the traditional Nyquist theorem not satisfactorily meet the high-precision channel estimation requirements of the sparse characteristic channel. Different from the new theory of the Nyquist sampling theorem, compressed sensing utilizes sparse signals with sparse characteristics and can accurately reconstruct signals with only a small number of observations. Due to the sparsity of underwater acoustic channels [

In the underwater acoustic OFDM communication system, the compressed sensing algorithm can fully utilize the sparsity of the underwater acoustic channel, and the underwater acoustic channel can be accurately estimated and has a smaller amount of calculation than the conventional channel estimation method. Compressed sensing also has the characteristics of suppressing interference noise. In the underwater acoustic OFDM communication system, the system can further improve the ability of the system to resist noise interference and reduce the bit error rate of the system.

In summary, for the Doppler frequency shift, multipath delay and time-frequency characteristics, and channel sparsity of underwater acoustic channel, this paper reconstructs the delay-Doppler based on the OFDM communication technology based on compressed sensing channel estimation. Frequency-shifted sparse signal model, combined with the virtual time reversal technology, enhances space-time-focusing effects and improves time-varying effects in data blocks.

The block diagram of the time-inverse OFDM system based on the compressed sensing virtual type is shown in Figure

Time-inverse OFDM system diagram based on compressed sensing virtual under baseband: the OFDM symbol can be expressed as follows.

The baseband signal after IFFT is

After the transmitted signal passes through the underwater acoustic channel, the received signal can be expressed as

In order to better eliminate the multipath effects of the underwater acoustic channel, the passive virtual time reversal mirror completes the underwater acoustic channel estimation by means of the received detection signal

It can be seen that

It can be seen that the virtual time-inverse OFDM communication technology preferably suppresses the Doppler shift and multipath effects. Channel estimation is the key to the antifocusing effect in the virtual mode. The difference from the Nyquist sampling method is that the compressed sensing channel estimation method can recover the channel information with higher precision through better sample sampling values. Therefore, this paper proposes a virtual time-inverse OFDM underwater acoustic channel estimation method based on compressed sensing.

The virtual time-inverse OFDM technique has a better focusing effect and effectively suppresses ICI (subcarrier interference) caused by multipath effects and the Doppler shift. Therefore, the virtual time-inverse OFDM technique is applied. The channel estimation method is the key to the orthogonality of the OFDM subcarriers. Therefore, this paper proposes a virtual time-inverse OFDM underwater acoustic channel estimation method based on compressed sensing. Since the signal reconstruction process is an ill-conditioned problem-solving process, it is necessary to make full use of the a priori information of the sparsity in the signal and use a specific sparse reconstruction algorithm to complete the signal reconstruction. The virtual time-inverse OFDM channel estimation method based on compressed sensing is mainly divided into three steps: the delay-Doppler shift sparsity signal representation under time-varying channel; constructing uncorrelated measurement equations; using underwater acoustic channel prior information, reduce the number of iterations, establish a priority set, and thus improve the efficiency of data recovery. The flowchart is shown in Figure

The flowchart of system.

A large amount of research has focused on the problem of signal sparsity in compressed sensing. In addition to sparsity, the underwater acoustic signal is still affected by the characteristics of time-space-frequency variation during transmission, resulting in severe multipath effects and the Doppler shift. Therefore, this paper proposes a compressed sensing sparsity modeling for delay-Doppler signals.

If

Combine with

When

Solving the channel estimate

Step 1. Initialization: for any

Step 2. Randomly select

Step 3. Normalize the

The signal reconstruction algorithm solves the original signal from the under sampled measurement matrix and is one of the key links in the whole channel estimation. Common methods include the convex optimization algorithm, Bayesian compressed sensing algorithm, and distributed compressed sensing algorithm. However, due to the time-space-frequency variation characteristics of the underwater acoustic channel and the sparseness characteristics of the delay-Doppler frequency shift signal, the above method has a large amount of iteration in the iterative process. Therefore, this paper proposes to construct a priori support set, select the atom that best matches the residual as the a priori support set, perform orthogonalization processing, and then project the signal in the space formed by these orthogonal atoms to obtain the signal in each the component and margin on the atom have been selected, and then, the remainder is decomposed in the same way. In each step of decomposition, the selected atoms satisfy certain conditions, so the margin decreases rapidly with the decomposition process. The number of iterations is reduced by iterating the selected prior set optimally. The flowchart is as follows:

Assuming an iteration of

Step1: Initialize

Step2: Calculate the inner product

Step3: Add an index to index set

Step4: Find sparse representation by sparse least squares method

Step5: Update the residual to get

Step6: If the condition

It can be seen that after a finite iteration, the algorithm can converge to the sparse solution of the signal, and the optimal atomic position can be obtained in each iteration. Therefore, the method can recover the signal more accurately using only the sparse signal and better complete the function of underwater acoustic channel estimation.

The validity of the algorithm is verified in a shallow water pool of

Experimental scene.

Parameter of the UAC OFDM system.

Item | Parameter |
---|---|

Sampling rate, | 48 kHz |

Bandwidth, | 6 kHz |

OFDM symbol period | 85.33 ms |

IFFT points, | 4096 |

Number of carriers | 512 |

Protection interval, | 25 ms |

Frequency interval, | 11.72 Hz |

Modulation | QPSK |

Two shallow seawater acoustic channels were simulated using the Bellhop model.

Shallow seawater acoustic channel model 1: the sound velocity is always 1500 m/s, the water depth is 15 m, the seafloor reflection coefficient is 1, the sea surface reflection coefficient is -1, the emission transducer is located 3 m below the water surface, the receiving transducer is located 6 m below the water surface, the horizontal distance of the transceiving transducer is 500 m, and the shock response of the underwater acoustic channel (denoted as h_1), as shown Figure

Shallow seawater acoustic channel model 2: the sound velocity varies with depth, the sound velocity profile is as shown in the figure, the water depth is 100 m, the seafloor reflection coefficient is 1, the sea surface reflection coefficient is -1, both the source and the receiving vibration source are below the water surface, 20 m, the transceiving transducer has a horizontal distance of 400 m, and the resulting channel (denoted as h_2) is shown Figure

The channel impulse response of h_3 is shown in reference [

Channel h_1: ((a) sound speed profile, (b) channel impulse response).

Channel h_2: ((a) sound speed profile, (b) channel impulse response).

Channel impulse response h_3.

As shown in Figure

BER and MSE of OMP with different pilot number in h_1. (a) BER. (b) Mean squared error.

As shown in Figure

BER and MSE of OMP with different pilot number in h_2. (a) Bit error rate. (b) Mean squared error.

As shown in Figure

BER and MSE of OMP with different pilot numbers in h_3. (a) Bit error rate. (b) Mean squared error.

In [

Performance comparisons on BER and MSE versus SNR. (a) Bit error rate. (b) Mean squared error.

The severe delay, multipath, and Doppler effect of the underwater acoustic channel make the time-inverse OFDM channel estimation algorithm a huge challenge. In order to obtain reliable channel state information, this paper constructs a delay-Doppler sparse signal by extracting the probe information and completes the channel preestimation based on the compressed sensing theory. The timing of the channel estimation is reversed, and the influence of the Doppler effect on the focus of the time-inverse OFDM subcarrier is reduced. The experimental results show that compared with LS and OMP algorithms, the algorithm can complete channel estimation with sparse data under different channel conditions, reduce the bit error rate and mean square error, and improve the performance of the Lee channel estimation.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported in part by the following projects: the National Natural Science Foundation of China through the grants 61861014, the Guangxi Nature Science Fund (2015GXNSF AA139298, 2016GXNSFAA380226), Guangxi University high level innovation team and outstanding scholar program, Guangxi Thousand Bones Project, Guangxi Science and Technology Project (AC16380094, AA17204086, and 1598008-29), Guangxi Nature Science Fund Key Project (2016 GXNSFDA380031), and Guangxi University Science Research Project (ZD 2014146).