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# A new joint channel equalization and estimation algorithm for underwater acoustic channels

- Bo Li
^{1, 2}, - Hongjuan Yang
^{1}, - Gongliang Liu
^{1}Email author and - Xiyuan Peng
^{1, 3}

**2017**:169

https://doi.org/10.1186/s13638-017-0955-7

© The Author(s). 2017

**Received:**27 June 2017**Accepted:**5 October 2017**Published:**23 October 2017

## Abstract

Underwater acoustic channel (UAC) is one of the most challenging communication channels in the world, owing to its complex multi-path and absorption as well as variable ambient noise. Although adaptive equalization could effectively eliminate the inter-symbol interference (ISI) with the help of training sequences, the convergence rate of equalization in sparse UAC decreased remarkably. Besides, channel estimation algorithms could roughly figure out channel impulse response and other channel parameters through several specific mathematical criterions. In this paper, a typical channel estimation method, least square (LS) algorithm, is applied in adaptive equalization to obtain the initial tap weights of least mean square (LMS) algorithm. Simulation results show that the proposed method significantly enhances the convergence rate of the LMS algorithm.

## Keywords

- Equalization
- ISI
- Sparse underwater acoustic channel
- LMS
- Channel estimation

## 1 Introduction

With further exploration of ocean resources, underwater communication is playing a more critical role in both military and civilian aspects. Owing to the fact that electromagnetic wave attenuates severely in underwater channels, sound wave becomes the only effective communication mode. But compared with electromagnetic wave, sound velocity is extremely slow which would cause a severe propagation delay. When transmitting signals, sound wave would continually reflect between sea surface and bottom owing to restrained underwater channel. As a result, transmitted signals in underwater acoustic channel (UAC) have more severe inter-symbol interference (ISI) due to complex multi-path propagation in contrast to other kinds of communication channel. Besides, underwater channels have variable and unknown impulsive ambient noises which are often related to wind, rainfall, tide, vessels, and so on [1].

Adaptive equalizers are often utilized to effectively mitigate the inter-symbol interference (ISI), but they have a considerably low convergence rate in UAC. Therefore, information frame needs to carry longer training sequences to guarantee that iterations could reach the steady state of convergence during training mode. But it would occupy more bandwidth and reduce communication effectiveness, and this would be a deadly drawback for the fact that underwater acoustic channel is badly band-limited due to low-frequency ship noise and absorption of high-frequency energy [2]. It can be concluded that enhancing convergence rate of equalizer is a better option than enlarging the training sequences in underwater acoustic communication.

In general, the convergence rate of standard least mean square (LMS) adaptive equalizer mainly depends on the step size of each iteration. Therefore, a series of variable step-size least mean square (VSSLMS) algorithms [3–5] were proposed, which adjusted the variable step-size by minimizing the error at each iteration. Tong et al. [6] proposed a data reuse least mean square (DR-LMS) algorithm, which reuse the known training sequences to achieve a better equalization performance. Cui et al. [7] combined LMS with recursive least square (RLS) algorithms to realize a faster convergence rate and simplify complexity of implementation at the same time. However, the initial coefficients of equalizer tap weights are always neglected among improved equalization methods, which are also critical to the whole iterations and convergence rate.

Channel estimation is another way to impede and compensate channel fading, which obtains an approximate channel response through a series of mathematical analysis and calculations. But those estimation algorithms would get poor performances in sparse underwater acoustic channels. This paper aims to exploit the typical channel estimation algorithm—least square (LS) [8] to obtain the initial coefficients of equalizer tap weights. Simulation results under UAC reveal that our proposed algorithm improves the convergence rate and BER compared with the traditional LMS adaptive equalizer, especially in a low SNR region.

## 2 UAC communication model

### 2.1 Sound velocity

*c*is the corresponding sound velocity,

*T*stands for temperature,

*S*is salinity (‰), and

*P*stands for pressure (atm). However, those environmental factors are slow time-varying during the communication, and sound velocity is usually considered as a constant, which is 1500 m/s.

### 2.2 Ray model

### 2.3 System mode for simulation

Note that error correction coding and orthogonal frequency division multiplex (OFDM) could improve system performance. However, it would impede the understanding of how efficiently the receiver equalizer mitigates the ISI. For this reason, channel coding is omitted and OFDM is replaced by binary ASK modulation in this paper. For more details, the modulation frequency is 8 kHz and each transmitting frame consists of 400 training symbols and 1000 data symbols.

## 3 Related works: channel equalization and estimation

*N*delay units and

*N*tap coefficients [11]. If we just define the input signal as

*x*(

*n*) and the corresponding tap weights as

*w*

_{ i }(

*n*), then the output signal

*y*(

*n*) can be defined as:

While channel estimation generally use complex probability theory and information theory to approximately deduce channel response with the aid of training sequences or pilot signal, those algorithms vary in time domain and frequency domain, and least square is the most conventional principle. The specific equations would be illustrated in the next section.

## 4 The proposed algorithm

Our new algorithm effectively combines channel equalization with estimation methods, which make a better use of training sequences.

*N*th order finite impulse response (FIR) filter with an impulse response which is

*h*= [

*h*

_{0},

*h*

_{1}, ⋯,

*h*

_{N ‐ 1}]

^{ T }. In addition, the corresponding input vector regression of the adaptive equalization filter is assumed as

*x*(

*n*) = [

*x*(

*n*),

*x*(

*n*− 1), ⋯,

*x*(

*n*−

*M*+ 1)]

^{ T }and the tap weight vector is

*w*(

*k*) = [

*w*

_{0}(

*k*),

*w*

_{1}(

*k*), ⋯,

*w*

_{ M − 1}(

*k*)]

^{ T }, where

*n*is the time index and

*M*is the length of the equalizer taps, given that

*s*(

*n*) is the initial training sequences and

*v*(

*n*), which is to substitute the ambient noise of real underwater environment, is the independent white Gaussian noise with zero mean and variance \( {\delta}_n^2 \). Besides, the inner structure of training sequences is [0, 1, 0, 1, ⋯, 0, 1], because changeable sequences could better track channels. Finally, the desired output sequences

*d*(

*n*) can be defined as

*d*is the outcome of the training sequences

*s*influenced by channel response

*h*and

*v*(

*n*).

Prior to starting with iterations and updating the tap weights, a significant step needs to be done, which is roughly estimating the tap weights with the aid of LS channel estimation algorithm. Firstly, we need to cut out *s* and *d* so that they are in the same length of equalizer taps, then we obtain \( \overline{s}=\left[s(0),s(1),\cdots, s\left(M-1\right)\right] \) and \( \overline{d}=\left[d(0),d(1),\cdots, d\left(M-1\right)\right] \). Next, a discrete fast Fourier transform is conducted on both of them as

*k*= 0, 1, ⋯,

*M*− 1.

*w*

_{0}which is as follows:

*w*(

*n*) are obtained, we can utilize the general LMS algorithm to recursively update them as follows:

*μ*is the step size of updating tap weights and

*e*(

*n*) is the error calculation output.

## 5 Simulation results and discussions

*μ*(

*n*) is set to a constant, 0.005, and the length of the equalizer taps is 70. The final outcome shows that the proposed algorithm has a faster convergence rate with less than 2000 iterations to reach the steady state. Since the ambient noises are neglected during this simulation, the MSE in steady state approximately reaches − 600 dB.

## 6 Conclusions

In this paper, a novel equalization algorithm is proposed which utilize channel estimation to define the initial values of receiver equalizer taps. Simulations show that our new method has better performances both in convergence rate and BER compared with the original LMS algorithm. In addition, the proposed method could lessen the transmission of training sequences and save energy for underwater communication devices.

In future work, MIMO channel equalization will gain more attention. And the relevant simulations would take more practical factors into account. Furthermore, we would attempt to figure out the optimal inner structure of training sequences by virtue of mathematical derivation and computing experiments.

## Declarations

### Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (No. 61401118, No. 61371100, and No. 61671184), the Natural Science Foundation of Shandong Province (No. ZR2014FP016), the Foundation of Science and Technology on Communication Networks Key Laboratory, and the Fundamental Research Funds for the Central Universities (No. HIT.NSRIF.2016100 and 201720).

### Funding

The National Natural Science Foundation of China (Grant Nos. 61401118, 61371100, and 61671184) are supporting the data acquisition devices and materials; the Natural Science Foundation of Shandong Province (Grant No. ZR2014FP016), the Foundation of Science and Technology on Communication Networks Key Laboratory are supporting the simulations, and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2016100 and 201720) are supporting the data analyses.

### Authors’ contributions

BL and GL conceived and designed the experiments; HY performed the experiments; XP contributed the simulation tools; and BL wrote the paper. All authors have read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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