20 asptvsslms

Purpose

Sample per sample filtering and coefficient update using the Variable Step Size LMS (VSSLMS) algorithm.

Syntax

[w,g,mu,y,e]= asptvsslms(x,w,g,d,mu,roh)
[w,g,mu,y,e]= asptvsslms(x,w,g,d,mu,roh,mu_min,mu_max)

Description

asptvsslms() implements the Variable Step Size LMS adaptive algorithm used to update transversal adaptive filters. VSSLMS does not only adjust the filter coefficients but also adjusts the step size $mu$ to obtain fast convergence rate as well as small final misadjustment, a combination impossible to achieve with constant step size. Referring to the general adaptive filter shown in Fig. 2.6, asptvsslms() takes an input samples delay line $x(n)$, a desired sample $d(n)$, the vector of the adaptive filter coefficients from previous iteration $w(n-1)$, the previous vector of step sizes $mu(n-1)$, the previous gradient vector $g(n-1)$ (used to update $mu$), and returns the filter output $y(n)$, the error sample $e(n)$, the updated gradient vector $g(n)$, the updated step size vector $mu(n)$, and the updated vector of filter coefficients $w(n)$. If the mu_min and mu_max optional input arguments are given, each element of the step size vector is constrained to those limits. The update equation of asptvsslms() is given by

$$\vw (n) = \vw (n) + \vmu (n) e(n) \vx (n).$$ (42)

The input and output parameters of asptvsslms() for an FIR adaptive filter of $L$ coefficients are summarized below.

Input Parameters [Size] :: 
   x      : input samples delay line [L x 1]
   d      : desired response [1 x 1]
   w      : filter coef. vector w(n-1) [L x 1]
   g      : gradient vector g(n-1) [L x 1]
   mu     : vector of step sizes mu(n-1) [L x 1]
   roh    : gradient vector step size [1 x 1]
   mu_min : lower bound for mu [1 x 1]
   mu_max : higher bound for mu [1 x 1]

Output parameters::
   w      : updated filter coefficients w(n)
   y      : filter output y(n)
   g      : updated gradient vector g(n)
   mu     : updated vector of step sizes mu(n)
   e      : error sample, e(n)=d(n)-y(n)

Example

Figure 4.23: The adaptive filter coefficients after convergence, the learning curve, and the evolution of the mean value of the step size for the complex FIR system identification problem using the VSSLMS algorithm.

% VSSLMS used in a system identification application.
% By the end of this script the adaptive filter w should
% have the same coefficients as the unknown filter h. 
iter = 5000;                  % Number of samples to process
% Complex unknown impulse response
h    = [.9 + i*.4; 0.7+ i*.2; .5; .3+i*.1; .1];     
xn   = 2*(rand(iter,1)-0.5);  % Input signal, zero mean random.
% although xn is real, dn will be complex since h is complex
dn   = osfilter(h,xn);        % Unknown filter output 
en   = zeros(iter,1);         % vector to collect the error
mu0  = 0.05*ones(10,1);       % initial step size
muv  = zeros(iter,1);         % evolution of mu with time
% Initialize the VSSLMS algorithm with a filter of 10 coef.
[w,x,d,y,e,g,mu] 	= init_vsslms(10,[],[],[],mu0); 
%% Processing Loop
for (m=1:iter)
   % update the input delay line
   x = [xn(m,:); x(1:end-1,:)];  
   d = dn(m,:) + 1e-3*rand;      % additive noise of var = 1e-6   
   % call VSSLMS to calculate the filter output, estimation error
   % and update the coefficients and step sizes. 
   [w,g,mu,y,e] = asptvsslms(x,w,g,d,mu,1e-3,1e-6,.99);
   % save the last error sample to plot later
   en(m,:) = e;   muv(m) = mean(mu);
end;
% display the results
% display the results
subplot(3,3,1);stem([real(w) imag(conj(w))]); grid;
eb = filter(.1, [1 -.9], en .* conj(en));
subplot(3,3,2);plot(10*log10(eb  ));grid
subplot(3,3,3);plot(muv); grid;

Running the above script will produce the graph shown in Fig. 4.23. The left-side graph of the figure shows the adaptive filter coefficients after convergence which are almost identical to the unknown filter h. The middle graph shows the mean square error in dB versus time during the adaptation process, which is usually called the learning curve. The lower limit of the error signal power in the learning curve is defined here by the additive white noise added at the filter output (-60 dB). The right-side graph shows the evolution of the mean value of the step size vector with time.

Remarks

Like the LMS, the VSSLMS is also a stochastic implementation of the steepest-descent algorithm where the mean value of the filter coefficients converge towards their optimal solution. Therefore, the filter coefficients will fluctuate about their optimum values given by the Wiener solution. The amplitude of the fluctuations is controlled by the step size. The smaller the step size, the smaller the fluctuations (less final misadjustment) but also the slower the adaptive coefficients converge to their optimal values. The improvement the VSSLMS introduces is that a separate step size is used for each filter coefficient, and the algorithm adapts those step sizes. When a coefficient is far from its optimal value, its corresponding step size is increased to converge faster. Conversely, when a coefficient is near its optimal value, the step size is decreased to decrease the final misadjustment. Similar to the LMS, the following points also apply to the VSSLMS.

• The VSSLMS algorithm shows stable convergence behavior only when all elements of the step size vector $mu(n)$ take values between zero and an upper limit, at all time indexes n, defined by the statistics of the filter's input signal. The fastest convergence will be achieved for a white noise input sequence with zero mean and unit variance. Such white input signal has all its eigenvalues equal to unity and therefore has a diagonal autocorrelation matrix with diagonal values equal to unity.
• The more colored the spectrum of the input signal, the slower the convergence will be. This is due to the large eigenvalue spread for such colored signals. This makes the convergence composed of several modes, each associated with one of the eigenvalues.
• asptvsslms() supports both real and complex data and filters. The adaptive filter for the complex VSSLMS algorithm converges to the complex conjugate of the optimum solution.
• asptvsslms() does not update the input delay line for $x(n)$, this has been chosen to provide more flexibility, so that the same function can be used with transversal as well as linear combiner structures. Delay line update, by inserting the newest sample at the beginning of the buffer and shifting the rest of the samples to the right, has to be done before calling asptvsslms() as in the example above.

Resources

The resources required to implement the VSSLMS algorithm for a transversal adaptive FIR filter of $L$ coefficients in real time is given in the table below. The computations given are those required to process one sample.

MEMORY	$4L + 6$
MULTIPLY	$4L$
ADD	$3L$
DIVIDE	0

See Also

INIT_ VSSLMS, MODEL_ VSSLMS, ASPTNLMS, ASPTLMS, ASPTLCLMS.

Reference

[11] for extensive analysis of the LMS and the steepest-descent search method.