Improved proportionate FONLMS algorithm based direct adaptive Turbo equalization for MIMO underwater acoustic communications

In this paper, a novel normalized least mean squares (NLMS) algorithm that jointly updates the efficient of the linear equalizer and soft interference canceller (SIC) in an adaptive turbo equalizer for multiple-input multiple-output (MIMO) underwater acoustic (UWA) communications. To exploit the sparsity of MIMO UWA channels and enhance the convergence speed of adaptive equalization, improved proportionate fast self-optimized NLMS algorithm (IPFONLMS), is proposed to well adapt to sparse channel with the similar complexity as improve proportionate NLMS (IPNLMS) algorithm. Then we extend the proposed algorithm to the adaptive turbo equalization for MIMO UWA communications. The performance of the proposed adaptive algorithm is evaluated by numerical results. Simulation results show that the improved data efficiency and bit error ratio (BER) performance of the proposed receiver is achieved over adaptive turbo equalizer based on the IPNLMS algorithm.


Introduction
In recent years, due to the competition for marine resources, every country has strengthened the research on UWA communications. However, owing to the characteristic of UWA channels, the UWA communications still facing significant challenges [1][2][3].
UWA communications using MIMO technique shown effective in exploiting the time and spatial diversity characteristics of UWA channel. However, MIMO systems have intersymbol interference (ISI) caused by the multipath phenomenon, which affects communication systems performance [2]. The equalization technique is used to remove ISI. Thus, the turbo equalization (TEQ) techniques for UWA communication is commonly used in large applications [2,3].
In [4], P. Bragard and G. Jourdain demonstrated a linear equalizer that directly adapted its weights via LMS algorithm. However, in the direct adaptation decisionfeedback equalizer (DA-DFE) receiver, the conventional adaptive equalizer algorithms can't obtain a good performance for UWA communications. To improve the performance of UWA communications, a lot of modified LMS algorithms have been investigated [5][6][7]. In [5], the proportionate normalized least-mean-squares (PNLMS) algorithm is proposed; it converges faster than the normalized NLMS algorithm used in echo cancelers. However, the PNLMS algorithm converges much slower than NLMS algorithm when the impulse response is dispersive. To resolve this problem, a new and simple rule was derived in [7]. In [6], a fast self-optimized LMS (FOLMS) algorithm has been presented. The FOLMS update the step-size with the gradient-search procedure. However, the existing algorithms do not strike a balance between utilize the sparsity of UWA channels and adaptive varying step-size. In order to solve the above issues, in this paper, we present a new adaptive algorithm IPFONLMS to update the coefficients of TEQ for MIMO UWA communications.
The rest of the paper is organized as follows. In section II, the MIMO system model is described and DA-TEQ techniques are briefly described. In section III, the MIMO UWA receiver architecture based on the IPFONLMS algorithm is presented. In section IV presents the numerical results, and in section V, some conclusions are discussed.
Notation: In this paper, we describe matrices and vectors by bold-face italic letters in capital cases and small cases, respectively. The superscripts ( ) *   At the time k , the received signal by M hydrophones can be expressed as is the l -th tap of the length-L equivalent channel between the n -th transducer element and the m -th hydrophone element. m η is the zero-mean valued Gaussian random noise with variance 2 η σ .

DA-TEQ for MIMO systems
The structure of DA-TEQ is depicted in Fig.2. The DA-TEQ is made up of feedforward filter, feedback filter and SIC filter. The length of channel is where f L and b L denote the length of precursor and postcursor response, respectively.
We consider an observation window containing The ( ) n x k obtained from the priori log likelihood ratio (LLR) of the decoder [2].

Proposed IPFONLMS algorithm for MIMO systems
For the IPFONLMS algorithm, we apply the idea of update step size to the IPNLMS algorithm. Without loss of generality, we take the updating of feedforward filter coefficient for example. The new cost function can be rewritten as

Convergence and Steady-state Analysis of Algorithms
In this section, we compare the performance of LMStype algorithms in sparse channel. The channel impulse response length is 64. The step size is shown in Fig.3(a), and ( ) 0 µ is set to 0.6 for IPFONLMS algorithm.  Fig.3(a) shows the performance of three algorithms in time invariant sparse channel. And the channel sparsity level is 10. It can be seen from Fig.3(a) that proposed IPFONLMS algorithm reaches the steady-state faster than other algorithms. The variable step size algorithm will improve performance as the number of iteration increases. We draw a conclusion that the adaptive step size can greatly improve the performance. Under the condition of sudden change sparse channel in Fig.3(b), the proposed algorithm shows better performance than others at sudden change position 2500 n = .

Results of System Performance
For MIMO transmission, the encoding scheme was shown in Fig

Conclusions
In this paper, we have proposed IPFONLMS algorithm to update the coefficients of the filters for MIMO UWA communications. By providing variable step-size, the proposed algorithm improves the tracking ability of the adaptive filter in UWA channels. Numerical simulations results show that it can provide a faster convergence rate and a lower MSE than alternative algorithms. And results show that the DA-TEQ based on the proposed algorithm has a superior performance for the MIMO UWA communication system. Therefore, the DA-TEQ sheme based on the IPFONLMS algorithm is a good candidate for practical applications.