Design and complexity analysis of an improved adaptive filtering algorithm for non-sparse impulse response
© Sun et al.; licensee Springer. 2014
Received: 21 October 2013
Accepted: 13 January 2014
Published: 22 January 2014
An improved adaptive filtering (IMPNLMS+) algorithm has been proposed for non-sparse impulse responses by incorporating an adaptive parameter μ of the μ-law compression into the improved μ-law PNLMS algorithm (IMPNLMS) algorithm. It not only achieves optimal step-size control factors but also overcomes that the convergence of classical μ-law PNLMS (MPNLMS) is even slower than conventional NLMS algorithm for dispersive channels. In this paper, we propose IMPNLMS++ algorithm, where normalized algorithm is analyzed to reduce computational complexity of proposed improved μ-law PNLMS (IMPNLMS+) algorithm. The validity has been proved by the simulation results.
In echo cancelation systems, the adaptive filters are used to identify the impulse responses for echo paths. However, the impulse responses are usually sparse in nature [1–3]. For these systems, the classical normalized least-mean-square (NLMS) algorithm which assigns the same step-size to all filter coefficients converges slowly. Some adaptive algorithms exploiting the sparse nature of the impulse response have been proposed to resolve this problem.
The proportionate NLMS (PNLMS) algorithm  presented by Duttweiler converges slowly dramatically after the initial fast period. The μ-law PNLMS (MPNLMS) algorithm  was proposed to solve this problem. Although it achieves optimal proportionate step size, it converges even slower than the classical NLMS algorithm in dispersive channels. The improved MPNLMS (IMPNLMS) algorithm  for non-sparse impulse responses was proposed to improve the performance of MPNLMS with an automatic adjustable parameter. If the convergence speed of IMPNLMS algorithm can be further improved, this algorithm will get a good application in time-varying environment. In , we have presented the improvement of the IMPNLMS algorithm using a variable parameter μ instead of the constant value. As a consequence, the convergence features of the MPNLMS algorithm are improved significantly.
In this paper, the improved IMPNLMS algorithm, referred to as IMPNLMS + algorithm all over the paper, is reviewed. The computational complexity is then compared with other adaptive filter algorithms. Non-normalized algorithm is analyzed to bring down the computational complexity of the IMPNLMS+; the feasibility is confirmed by numerical situations.
This paper is organized as follows. In Section 2, the IMPNLMS + algorithm is reviewed. In Section 3, in order to lower computational complexity, the non-normalized technology is analyzed. In Section 4, numerical simulations confirm the computational complexity of IMPNLMS + is lower without normalization. Finally, Section 5 presents conclusions.
2. Review of the improved IMPNLMS algorithm
For non-sparse response, we have proposed an IMPNLMS + algorithm  by applying time variable parameter μ, which does not perform worse than NLMS even for dispersive channels. It can be applied in time-varying environment as well. In this section, the IMPNLMS + algorithm is recalled.
In IMPNLMS + algorithm, thanks to the adaptation of μ (k), the algorithm is more flexible to minimize the MSE related to the time-varying μ (k). The IMPNLMS + algorithm can achieve better convergence even in time-varying environment where the echo path changes obviously. However, the computation of step-size control matrix with μ-law in IMPNLMS + algorithm is expensive. In the next section, we analyze the computational complexity based on IMPNLMS + algorithm. Non-normalized technology is discussed to reduce the computational complexity of IMPNLMS+, which can be also applied to all proportionate NLMS algorithms.
3. The analytical solution of computational complexity
The computational complexity of each filter coefficient update equation of NLMS
2 L + 1
2 L - 1
4 L + 1
4 L + 1
2 L - 1
4 L + 2
5 L + 2
5 L + 5
5 L + 3
5 L + 8
In , the denominator of coefficient update equation in PNLMS is x T (k) x (k), Duttweiler normalized the step-size control factors g l (k) to avoid direct influence of target impulse response amplitude (i.e., |h m | ) on efficient step-size parameter β g l (k). Thus, the value of final efficient step-size assigned to filters is only proportional to parameter β. However, when the denominator is x T (k)G(k + 1)x(k), normalization can be skipped. The analysis can be described as follows.
Where δ is a small positive number only to prevent the algorithm from stalling when denominator equals zero. Hence, in general, x T (k)[cG(k + 1)]x(k) ≫ cδ, the operation (11) which is just the same as normalization does not affect the coefficient update item dramatically. Therefore, the absence of normalization has little effect on adaptive algorithms, but also reduces L divisions. This thesis is applicable to MPNLMS and IMPNLMS as well.
4. Numerical simulations
In this section, the proportionate algorithm IMPNLMS + is simulated to confirm the little influence on algorithms without normalization. The input of the simulation system, which is similar as , is described as follows.
The coefficients of the unknown system and the adaptive filter is 100 (L = 100). The signal-to-noise ratio (SNR) is 40 dB, and the disturbance z (n) is a Gaussian signal. The constant step size β is 0.5. The initial value of ξ is 0.96. The forgetting factor ρ used to estimate channel sparseness is 0.1. In IMPNLMS + algorithm, η = 0.99, and ν = 1,000.
In Figures 1, 2, and 3, with the increasing of iterations, IMPNLMS+ and IMPNLMS++ converges gradually in 1,000 iterations. Furthermore, the lower the sparseness degree is, the faster IMPNLMS+ and IMPNLMS++ converge. As illustrated in figures above, IMPNLMS++ performs nearly the same as the normalized IMPNLMS+. However, the computational complexity of IMPNLMS++ is L divisions less than normalized IMPNLMS+ .
In this paper, we first recall the improved IMPNLMS algorithm for non-sparse impulse response. The complexity of each filter coefficient update equation is then compared. To reduce computational complexity of the IMPNLMS + algorithm, the possibility of employing the non-normalization technique is verified through theoretical derivations. Simulation results have proven the effectiveness and feasibility of the non-normalized IMPNLMS + algorithm.
This work is supported in part by NSFC 61143008, National High Technology Research and Development Program of China (no. 2011AA01A204), the Fundamental Research Funds for the Central Universities.
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