 Research
 Open Access
A family of variable stepsize affine projection adaptive filter algorithms using statistics of channel impulse response
 Mohammad Shams Esfand Abadi^{1}Email author and
 Seyed Ali Asghar AbbasZadeh Arani^{1}
https://doi.org/10.1186/16876180201197
© Esfand Abadi and AbbasZadeh Arani; licensee Springer. 2011
 Received: 2 December 2010
 Accepted: 8 November 2011
 Published: 8 November 2011
Abstract
This paper extends the recently introduced variable stepsize (VSS) approach to the family of adaptive filter algorithms. This method uses prior knowledge of the channel impulse response statistic. Accordingly, optimal stepsize vector is obtained by minimizing the meansquare deviation (MSD). The presented algorithms are the VSS affine projection algorithm (VSSAPA), the VSS selective partial update NLMS (VSSSPUNLMS), the VSSSPUAPA, and the VSS selective regressor APA (VSSSRAPA). In VSSSPU adaptive algorithms the filter coefficients are partially updated which reduce the computational complexity. In VSSSRAPA, the optimal selection of input regressors is performed during the adaptation. The presented algorithms have good convergence speed, low steady state mean square error (MSE), and low computational complexity features. We demonstrate the good performance of the proposed algorithms through several simulations in system identification scenario.
Keywords
 Adaptive filter
 Normalized Least Mean Square
 Affine projection
 Selective partial update
 Selective regressor
 Variable stepsize
1. Introduction
Adaptive filtering has been, and still is, an area of active research that plays an active role in an ever increasing number of applications, such as noise cancellation, channel estimation, channel equalization and acoustic echo cancellation [1, 2]. The least mean squares (LMS) and its normalized version (NLMS) are the workhorses of adaptive filtering. In the presence of colored input signals, the LMS and NLMS algorithms have extremely slow convergence rates. To solve this problem, a number of adaptive filtering structures, based on affine subspace projections [3, 4], data reusing adaptive algorithms [5, 6], block adaptive filters [2] and multi rate techniques [7, 8] have been proposed in the literatures. In all these algorithms, the selected fixed stepsize can change the convergence and the steadystate mean square error (MSE). It is well known that the steadystate MSE decreases when the stepsize decreases, while the convergence speed increases when the stepsize increases. By optimally selecting the stepsize during the adaptation, we can obtain both fast convergence rates and low steadystate MSE. These selections are based on various criteria. In [9], squared instantaneous errors were used. To improve noise immunity under Gaussian noise, the squared autocorrelation of errors at adjacent times was used in [10], and in [11], the fourth order cumulant of instantaneous error was adopted.
In [12], two adaptive stepsize gradient adaptive filters were presented. In these algorithms, the step sizes were changed using a gradient descent algorithm designed to minimize the squared estimation error. This algorithm had fast convergence, low steadystate MSE, and good performance in nonstationary environment. The blind adaptive gradient (BAG) algorithm for codeaided suppression of multipleaccess interference (MAI) and narrowband interference (NBI) in directsequence/codedivision multipleaccess (DS/CDMA) systems was presented in [13]. The BAG algorithm was based on the concept of accelerating the convergence of a stochastic gradient algorithm by averaging. The authors shown that the BAG had identical convergence and tracking properties to recursive least squares (RLS) but had a computational cost similar to the LMS algorithm. In [14], two low complexity variable step size mechanisms were proposed for constrained minimum variance (CMV) receivers that operate with stochastic gradient algorithms and are also incorporated in the channel estimation algorithms. Also, the low lowcomplexity variable step size mechanism for blind codeconstrained constant modulus algorithm (CCM) receivers was proposed in [15]. This approach was very useful for nonstationary wireless environment.
In [16], a generalized normalized gradient descent (GNGD) algorithm for linear finiteimpulse response (FIR) adaptive filters was introduced that represents an extension of the NLMS algorithm by means of an additional gradient adaptive term in the denominator of the learning rate of NLMS. The simulation results show that the GNGD algorithm is robust to significant variations of initial values of its parameters.
Important examples of two new variable stepsize (VSS) versions of the NLMS and the affine projection algorithm (APA) can be found in [17]. In [17], the stepsize is obtained by minimizing the meansquare deviation (MSD). This introduced algorithms show good performance in convergence rate and steadystate MSE. This approach was successfully extended to selective partial update (SPU) adaptive filter algorithm in [18].
To improve the performance of adaptive filter algorithms, the adaptive filter algorithm was proposed based on channel impulse response statistics [19, 20]. In [21] a new variablestepsize control was proposed for the NLMS algorithm. In this algorithm, the stepsize vector with different values for each filter coefficient was used. In this approach, based on prior knowledge of the channel impulse response statistics, the optimal stepsize vector is obtained by minimizing the meansquare deviation (MSD) between the optimal and estimated filter coefficients.
Another feature which is important in adaptive filter algorithms is computational complexity. Several adaptive filters with fixed stepsize, such as the adaptive filter algorithms with selective partial updates have been proposed to reduce the computational complexity. These algorithms update only a subset of the filter coefficients in each time iteration. The MaxNLMS [22], the variants of the SPUNLMS [23], and SPUAPA [24, 25] are important examples of this family of adaptive filter algorithms. Recently an affine projection adaptive filtering algorithm with selective regressors (SR) was also proposed to reduce the computational complexity of APA [26–28]. In this algorithm, the recent regressors are optimally selected during the adaptation.
In this paper, we extend the approach in [21] to the APA, SPUNLMS, SPUAPA, and SRAPA and four novel VSS adaptive filter algorithms are established. These algorithms are computationally efficient. We demonstrate the good performance of the presented algorithms through several simulation results in system identification scenario. The comparison of the proposed algorithms with other algorithms is also presented.
What we propose in this paper can be summarized as follows:

The establishment of the VSSAPA.

The establishment of the VSSSPUNLMS.

The establishment of the VSSSPUAPA.

The establishment of the VSSSRAPA.

Demonstrating of the proposed algorithms in system identification scenario.

Demonstrating the tracking ability of the proposed algorithms.
We have organized our paper as follows. In section 2, the NLMS and SPUNLMS algorithms will be briefly reviewed. Then, the family of APA, SRAPA and SPUAPA are presented and the family of variable stepsize adaptive filters is established. In the following, the computational complexity of the VSS adaptive filters is discussed. Finally, before concluding the paper, we demonstrate the usefulness of these algorithms by presenting several experimental results.
Throughout the paper, the following notations are adopted:
. norm of a scalar
.^{2} squared Euclidean norm of a vector
(.) ^{ T } transpose of a vector or a matrix
tr(.) trace of a matrix
E[.] expectation operator
2 Background on family of NLMS algorithm
21 Background on NLMS
22 Selective Partial Update NLMS
which is the maxNLMS algorithm [22]. For M = B and L = B, the SPUNLMS algorithm becomes identical to NLMS algorithm in (3).
3. Background on APA, SRAPA and SPUAPA
31. Affine projection algorithm (APA)
32 Selective Regressor APA (SRAPA)
 1.Compute the following values for 0 ≤ i ≤ K  1:$\frac{{e}^{2}\left(ni\right)}{{\parallel X\left(n+i\right)\parallel}^{2}}$(17)
 2.
The indices of G correspond to P largest values of (17).
33. Selective Partial Update APA (SPUAPA)
 1.Compute the following values for 1 ≤ i ≤ B:$Tr\left({X}_{i}^{T}\left(n\right){X}_{i}\left(n\right)\right)$(22)
 2.
The indices of F correspond to S largest values of relation (22).
4. VSSNLMS and the proposed VSS Adaptive Filter Algorithms
41. Variable StepSize NLMS algorithm using statistics of channel response
where h = [h_{0}, h_{1}, ..., h_{M1}] ^{ T } is the true unknown system with memory length M, X(n) = [x(n), ..., x(nM+1)] ^{ T } is the system input vector, and v(n) is the additive noise.
where the stepsize matrix is defined as U(n) = diag[μ_{0}(n), ..., μ_{M1}(n)].
where 0 < λ < 1 is the forgetting factor. Also, the initial value for $E\left[{\stackrel{\u0303}{w}}_{i}^{2}\left(0\right)\right]$ is given by the secondorder statistics of the channel impulse response, i.e. $E\left[{h}_{i}^{2}\right]$.
42. Variable StepSize Selective Partial Update NLMS algorithm using statistics of channel response
and I_{ SL } is the SL × SL identity matrix.
Also, E(e^{2}(n)) is obtained according to (30).
43. Variable StepSize APA using statistics of channel response
and I_{ M } is the M × M identity matrix.
44. Variable StepSize Selective Regressor AP algorithm using statistics of channel response
45. Variable StepSize Selective Partial Update AP algorithm using statistics of channel response
Also E[e(n)^{2}] is obtained according to (59).
5. Computational complexity
The computational complexity of NLMS, SPUNLMS, VSSNLMS, and VSSSPUNLMS algorithms.
Algorithm  multiplication  Additional division  Additional multiplication  Comparisons 

NLMS  3M+1       
SPUNLMS  3SL+1    1  B log_{2}S+O(B) 
VSSNLMS  4M  M  3M+4   
VSSSPUNLMS  4SL  SL  3SL+5  B log_{2}S+O(B) 
The computational complexity of APA, SPUAPA, SRAPA, VSSAPA, VSSSPUAPA, and VSSSRAPA.
Algorithm  multiplication  Additional division  Additional multiplication  Comparisons 

APA  (K^{2} + 2K)M+K^{3}+K^{2}       
SPUAPA  (K^{2} + 2K)SL+K^{3}+K^{2}    1  B log_{2}S+O(B) 
SRAPA  (P^{2} + 2P)M+P^{3}+P^{2}  K  (KP)M + K+1  K log_{2}P+O(K) 
VSSAPA  (K^{2} + 2K)M+K^{3}+K^{2}+KM^{2}  M  3M+K+1   
VSSSPUAPA  (K^{2} + 2K)SL+K^{3}+K^{2}+KS^{2}L^{2}  SL  3SL + K + 2  B log_{2}S+O(B) 
VSSSRAPA  (P^{2} + 2P)M+P^{3}+P^{2}+PM^{2}  M + K  (KP+3)M+K+P+2  K log_{2}P+O(K) 
The SPUAPA needs (K^{2}+2K)SL+K^{3}+K^{2} multiplications. This algorithm needs 1 additional multiplication and B log_{2}S+O(B) comparisons. The SRAPA needs (P^{2}+2P)M+P^{3}+P^{2} multiplications and K divisions. This algorithm needs (KP)M+K+1 additional multiplications and K log_{2}P+O(K) comparisons. Comparing the updated equation for APA and VSSAPA shows that the VSSAPA needs KM^{2}+3M+K+1 additional multiplications and M divisions due to variable stepsize. In VSSSPUAPA, the additional multiplication is KS^{2}+L^{2}+3SL+K+1 and additional division is SL. Also, this algorithm needs B log_{2}S+O(B) comparisons. It is obvious that the computational complexity of VSSSPUAPA is lower than VSSAPA. The number of reductions in multiplication and division for VSSSPUAPA is respectively (K^{2}+2K+4)(MSL)+K(M^{2}S^{2}L^{2}) and MSL. In VSSSRAPA, the additional multiplication is 3M+P+1 and additional division is M compared with SRAPA. Also this algorithm needs K log_{2}P+O(K) comparisons. It is obvious that the computational complexity of VSSSRAPA is lower than VSSAPA.
6. Experimental results
The white zeromean Gaussian noise was added to the filter output such that the SNR = 15dB.
In all simulations, the MSD curves are obtained by ensemble averaging over 200 independent trials.
7. Conclusions
In this paper, we presented the novel VSS adaptive filter algorithms such as VSSSPUNLMS, VSSAPA, VSSSRAPA and VSSSPUAPA based on prior knowledge of the channel impulse response statistic. These algorithms exhibit fast convergence while reducing the steadystate MSD as compared to the ordinary SPUNLMS, APA, SRAPA and SPUAPA algorithms. The presented algorithms were also computationally efficient. We demonstrated the good performance of the presented VSS adaptive algorithms in system identification scenario.
Declarations
Authors’ Affiliations
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