A novel normalized subband adaptive filter algorithm based on the joint-optimization scheme

Herein, we propose a normalized subband adaptive filter (NSAF) algorithm that adjusts both the step size and regularization parameter. Based on the random-walk model, the proposed algorithm is derived by minimizing the mean-square deviation of the NSAF at each iteration to calculate the optimal parameters. We also propose a method for estimating the uncertainty in an unknown system. Consequently, the proposed algorithm improves performance in terms of tracking speed and misalignment. Simulation results show that the proposed NSAF outperforms existing algorithms in system identification scenarios.

INDEX TERMS Adaptive filter, normalized subband adaptive filter, variable step size, variable regularization parameter, mean-square deviation.

I. INTRODUCTION
A DAPTIVE filter algorithms have been used in a wide range of signal processing applications, such as acoustic echo cancellation, system identification, and channel equalization [1]- [5]. The normalized least-meansquare(NLMS) algorithm is one of the most widely used adaptive filter algorithms owing to its low computational complexity and ease of implementation [6], [7]. However, it exhibits substantial performance degradation in terms of the convergence rates for highly correlated input signals. To address this problem, affine projection (AP) algorithm and normalized subband adaptive filter (NSAF) algorithm were introduced [3], [8], [9]. The AP algorithm achieves fast convergence speeds using multiple input vectors; thus, its computational complexity increases for higher-order systems, such as acoustic echo cancellation. However, despite the lower computational complexity than the AP algorithm, the NSAF algorithms have better performance in terms of tracking speed owing to their self-whitening property. Moreover, some variants of the NSAF, such as sign subband adaptive filter (SSAF) and its variants [10]- [12], M-estimate NSAF [13], [14], sparsity-aware SSAF and NSAF [15], [16], and bias-compensated NSAF [17]- [19], have been proposed to improve performance.
The NSAF algorithms have two important parameters that affect the performance in terms of convergence rates and steady-state errors, i.e., step size and regularization parameter. The fixed step size and the regularization parameter lead to a trade-off between the convergence rate and misalignment in the NSAF. To control these parameters, variable stepsize (VSS) algorithms and variable regularization-parameter (VR) algorithms have been proposed for the NSAF [20]- [28]. Unlike other variable parameter NSAF algorithms, the joint-optimization step size and regularization parameter NSAF(JOSR-NSAF) algorithm controls both the step size and regularization parameter [28]. The JOSR-NSAF algorithm is derived from joint-optimization (JO) scheme [29] based on minimizing the mean-square deviation (MSD). The result of the MSD analysis from the JOSR-NSAF algorithm is similar to that of the VR-NSAF algorithm developed by Jeong et al. [27] under stationary environments. However, this analysis method does not accurately reflect the change in the regularization parameters for colored input signals well [26].
In this study, we propose an NSAF algorithm that controls both the step size and regularization parameter owing to improvements in terms of tracking speed and misalignment. The contribution of this study is threefold. First, an MSD analysis of the NSAF algorithm is provided for both the step size and regularization parameter based on the randomwalk model. The proposed method is derived from a method similar to that of the adaptive regularization NSAF algorithm [26]; however, we propose a method that considers the step size and non-stationary systems to obtain both the optimal step size and regularization parameter at each iteration. Second, an optimized NSAF algorithm is derived from the MSD analysis, which control both the step size and regularization parameter to minimize the MSD of the NSAF algorithm at each iteration. Third, a method for estimating the uncertainty of an unknown system is proposed. Simulations tested in the system identification scenario and the performance of the proposed algorithm were compared with those of existing algorithms.
The remainder of this paper is organized as follows. A brief review of the NSAF algorithm is presented in Section II. The MSD performance analysis of the NSAF algorithm is developed in Section III. Section IV introduces an optimal NSAF algorithm. Practical issues are presented for the proposed algorithm in Section V. Finally, in Section VI, the performance of the proposed NSAF algorithm is verified through various computer simulations.

II. REVIEW OF CONVENTIONAL NSAF ALGORITHM
We consider a system identification problem such as acoustic echo cancellation. The weight coefficient vector of the unknown system and input signal at the discrete-time index ,n, are denoted by: where M is filter lengths. The desired signal derived from an unknown system is where v(n) represents measurement noise with zero mean and variance σ 2 v . Figure 1 shows the structure of the NSAF. In this figure, the desired signal, d i (n), and input signal, u i , are derived by the analysis filter, H i (z), which is defined as where N represents the number of subbands, L is the length of the analysis filter, and h i (j) is the jth coefficient of the analysis filter for the ith subband. d i,D and y i D are obtained by critically sub-sampling d i and y i , respectively. Therefore, the downsampled output signal of the ith subband is defined as y i,D (k) = u T i (k) w(k), where u i (k) = [u i (kN ), u i (kN − 1), · · · , u i (kN − M + 1)] T , w(k) = [ŵ 0 (k),ŵ 1 (k), · · · ,ŵ M −1 (k)] T indicates the estimation of w o at index k, and k is used to index the decimated signal. We define the decimated output error signal of the ith subband as follows: where is the ith subband noise with zero-mean white Gaussian noise with variance σ 2 vi . Therefore, the update equation of the conventional NSAF algorithm is: where µ i and β i are the step size and regularization parameter of the ith subband, respectively.

III. MSD ANALYSIS OF NSAF WITH THE RANDOM-WALK MODEL A. RANDOM-WALK MODEL
We assume that an unknown weight vector w o has the following random-walk model [29], [30] where q(k) is a white Gaussian noise vector with zero mean and variance mσ 2 q = E q T (k)q(k) and covariance matrix E q(k)q T (k) = σ 2 q I M with I M being an M × M identity matrix. We have an unknown weight vector at the kth iteration as follows:

B. PRELIMINARIES
Let us consider the following four assumptions as proposed in [26], [31]- [34] for analysis of the MSD performance of the NSAF with step-size and regularization parameters. Assumption 1: The input signal vector u(n) has zero mean and an independent and identically distributed (i.i.d.) and covariance matrix: where Λ = diag(λ 1 λ 2 · · · λ M ) are the eigenvalues of covariance matrix R, and V = [ν 1 ν 2 · · · ν M ] are the corresponding orthonormal eigenvectors(V T V = I).
Here, r i ∼ ∥u i (k)∥ indicates that r i has the same distribution as the norm of u i (k). Note that m j=1 p j = 1.

C. MSD ANALYSIS OF NSAF
The NSAF update equation (6) can be rewritten in terms of w using (9) as follows: wherew(k) ≜ w o (k) − w(k). By subtracting (5) from (9), the decimated output error signals become Considering (13), we rearrange (12) as Applying Assumption 2 and 3, the covariance matrix of w(k) can be obtained as follows: where by multiplying ν T j and ν j on both side of (15), we obtain: where α i ≜ µ i /(r 2 i + β i ). From [2, eq.(18) and (20)], the recursion of π j (k) for the ith subband can be expressed as follows: when M ≫ 1.

VOLUME 4, 2016
To conduct the MSD analysis of the NSAF, the MSD is defined as MSD(k) ≜ E( w T (k) w(k)) = T r(P(k)). Finally, we obtain where

IV. OPTIMIZED NSAF ALGORITHM BASED ON THE JOINT-OPTIMIZATION SCHEME
To rapidly minimize the MSD of the NSAF at each iteration, a joint-optimization strategy can be derived by ∂MSD(k)/∂µ i (k) = ∂MSD(k)/∂β i (k) = 0. Therefore, we can obtain a novel update equation of the NSAF with optimal step size µ i (k) and regularization parameter β i (k) as follows: where Substituting (21) and (22) into (23), we obtain the quadratic equation for the steady-state value of f i (k) as follows: where f i,ss ≜ lim k→∞ f i (k). Therefore, the steady-state value of MSD can be obtained as where The performance limit of the proposed algorithm is related to the system parameters (N, M ), measurement noise, (σ 2 v ), and non-stationary environments, (σ 2 q ).

V. PRACTICAL CONSIDERATIONS
The proposed NSAF algorithm depends on the parameters r i , σ 2 vi , and σ 2 q . Therefore, practical considerations are necessary.
The first parameter, r i , is related to the subband input variance σ 2 ui . Therefore, we can use the subband input variance instead of r 2 i as follows: Moreover, the input variance is easily estimated as follows [28], [29]: The second parameter is the subband noise variance, σ 2 vi , which can be easily estimated [31]- [34]. We assume that the subband noise variance, σ 2 vi , is already known, because it is out of the scope of this study.
The third parameter, σ 2 q , denotes the uncertainties in unknown system w o (n). The performance of the proposed NSAF algorithm is affected by the value of σ 2 q . Therefore, it is important to estimate the value of σ 2 q for implementing the proposed NSAF algorithm. A method for estimating σ 2 q using the random-walk model (9) and replacing w o (k) with w(k) is as follows: We define the estimated parameter, σ 2 q,1 , which is estimated using the above method as When a system undergoes an abrupt change, the parameter σ 2 q,1 (k) has large values, which provides a fast convergence speed. However, an error in the estimation of the parameter, σ 2 q,1 (k) causes an error in deriving the optimal parameter, α * i (k), which leads to significant misalignment. Thus, we propose a method to estimate σ 2 q using multiband meansquared error (MSE), which is defined as follows: Applying Assumption 2, the multiband MSE can be rewritten as Initialization.

VI. SIMULATION RESULTS
Computer simulations were performed using the system identification model to verify the MSD performance of the proposed algorithm. The unknown system coefficient, w o was randomly generated with unit variance for the MSD analysis comparison and the acoustic impulse response of a room for performance comparison. It was assumed that the adaptive filter and the unknown system have the same number of filter length, which was set M = 512 for computer simulations. The colored input signals were generated by filtering the white Gaussian noise through which are referred to as AR 1 , and AR 2 . The signal-to-noise ratio (SNR) was set to 20dB or 30dB to add the measurement noise at the output signal y i where the SNR is defined as In addition, we assumed that the noise variance was known.
The measure of performance is the normalized mean squared deviation (NMSD), which is defined as: The simulation results were obtained by ensemble averaging over 10 trials. For the proposed algorithm, we set γ = 1 − κN/M , where κ = 2.

A. SELF WHITENING EFFECT DEPENDING ON THE NUMBER OF SUBBANDS
The convergence performance of the subband adaptive filter is almost same for white Gaussian input signal; the proposed NSAF algorithm is shown in Figure 2(a). However, the convergence rate of the subband adaptive filter is further improved with an increased number of subbands owing to the self-input whitening effect for colored input signals. Figure  2(b) and 2(c) show that the NMSD learning curves of the proposed NSAFs according to the number of subbands in AR 1 and AR 2 input environments, respectively. As can be seen, the convergence performance of the proposed NSAF is saturated at N = 8, implying that the colored input signals are sufficiently whited at N = 8. Figure 3 and 4 show the NMSD learning curves of the experimental results and theoretical analysis results for white Gaussian and colored input signals. As shown in Figure 3(a) and 4(a), the analysis methods of [28] and proposed are match well with the experimental results for white Gaussian input signals. However, Figure 3(b), 4(b), 3(c), and 4(c) show a discrepancy between the theoretical analysis results that are obtained by [28] and the experimental results for both convergence and steady-state behaviors. The proposed analysis can properly estimate the NMSD learning curves of the NSAFs that have various step sizes and regularization   NLMS [29], and JOSR-NSAF [28] with white Gaussian, AR 1 , and AR 2 input signals. All algorithms needed to tune several parameters that were set according to the recommendations provided in [21], [26], [28], [29]. The NSAF-VSS algorithm controls only step-size values with zero regularization parameters and the AR-NSAF algorithm controls only regularization parameters with a unit step size. However, the  proposed algorithm and JO algorithms control both parameters. As can be seen in figure 5 and 6, the JO algorithms have low convergence rates and high misalignments than the existing VSS and VR algorithms, especially when the input signal is white Gaussian. The proposed algorithm has steady-state errors performance similar to VSS and VR algorithms, and provides faster convergence rates than existing  algorithms when the unknown system changes abruptly. In Figure 7, the proposed algorithm outperforms the existing VSS, VR, and JO algorithms in terms of the steady-state error and the convergence speed when the input signal is a speech sequence.

VII. CONCLUSION
In this paper, we presented an NSAF algorithm that controls both the step size and regularization parameter by minimizing the MSD performance of the NSAF at each iteration. To deal with optimal parameters, the MSD analysis method for step size and regularization parameter was proposed based on the random-walk model. We then proposed an estimation algorithm for calculating the uncertainty of the unknown system to improve the performance of the proposed NSAF algorithm in terms of misalignments and convergence rates. Simulation results verified that the proposed NSAF with adjusting parameters had a fast convergence rate and low misalignment as compared with the existing algorithms in the system identification scenario.