21 init_ arlmsnewt

Purpose

Creates and initializes the variables required for the efficient implementation of the LMS-Newton algorithm using autoregressive modeling.

Syntax

[k,w,x,b,u,P,d,y,e]=init_arlmsnewt(L,M)
[k,w,x,b,u,P,d,y,e]=init_arlmsnewt(L,M,k0,w0,x0,b0,u0,P0,d0)

Description

The LMS-Newton is a stochastic implementation of the Newton search method which solves the eigenvalue spread problem in adaptive filters with colored input signals. The update equation for the LMS-Newton is given by (see Fig. 2.6)

$$\vw (n+1) = \vw (n) + 2 \mu \; e(n) \mR ^{-1} \; \vx (n),$$ (43)

where $\mR$ is the autocorrelation matrix of the adaptive filter input signal $x(n)$. Direct implementation of the LMS-Newton update equation (4.17) requires estimation and inversion of $\mR$ and matrix vector multiplication $\mR ^{-1} \; \vx (n)$ each sample which is of course very computational demanding. arlmsnewt() implements the LMS-Newton method efficiently by recursively estimating the term $u=\mR ^{-1} \; \vx (n)$ using autoregressive modeling. A lattice predictor of $M$ stages is used for the autoregressive modeling part. When the input signal can be modeled with an autoregressive model of length $M$ much less than the adaptive filter length $L$, a significant computational saving can be achieved. The variables of the ARLMSNEWT are summarized below.

Input Parameters:: 
   L   : number of adaptive filter coefficients
   M   : number of autoregressive model coefficients (M << L)
   k0  : vector of initial lattice predictor coefficients [Mx1]
   w0  : vector of initial filter coefficients [Lx1]
   x0  : vector of initial input samples [Lx1]
   b0  : vector of initial backward prediction errors [Lx1]
   u0  : vector of initial normalized gradients [Lx1]
   P0  : initial power of b [(M+1)x1]
   d0  : initial desired response [1x1]
Output parameters::
   k   : initialized lattice predictor coefficients [zeros]
   w   : initialized linear combiner coefficients [zeros]
   x   : initialized input samples vector [random]
   b   : initialized backward prediction errors [random]
   u   : initialized normalized gradient vector [zeros]
   P   : initialized estimated power of b [b .* b]
   d   : initialized desired response [random]
   y   : initialized filter output [w' * x]
   e   : initialized error signal [e = d - y]

Example

L  = 1024;                 % adaptive filter length
M  = 4;                    % lattice predictor stages 
k0 = zeros(M,1);           % initial PARCOR coef. 
w0 = zeros(L,1);           % initial filter coef.
b0 = rand(L,1);            % initial backward errors
P0 = b0(1:M+1).*b0(1:M+1); % initial power of b
d0 = .22;                  % initial desired sample

% Create and initialize an LMS-Newton filter
[k,w,x,b,u,P,d,y,e]=init_arlmsnewt(L,M,k0,w0,[],b0,[],P0,d0);

Remarks

• Supports both real and complex signals and filters.
• Use input parameters 3 through 9 to initialize the algorithm storage. This is helpful when the adaptation process is required to start from a known operation point calculated off-line or from previous simulations.

See Also

ASPTARLMSNEWT, MODEL_ ARLMSNEWT.