maxstep

Maximum step size for LMS adaptive filter convergence

Syntax

mumax = maxstep(lmsFilt,x)

[mumax,mumaxmse] = maxstep(lmsFilt,x)

Description

mumax = maxstep(lmsFilt,x) predicts a bound on the step size to provide convergence of the mean values of the coefficients of the adaptive filter, lmsFilt.

[mumax,mumaxmse] = maxstep(lmsFilt,x) predicts a bound, in mean squared sense, on the adaptive filter step size to provide convergence of the adaptive filter coefficients.

Examples

Compute Maximum Step of LMS Adaptive Filter

The maxstep function computes the maximum step size of the adaptive filter. This step size keeps the filter stable at the maximum possible speed of convergence. Create the primary input signal, x, by passing a signed random signal to an IIR filter. Signal x contains 50 frames of 2000 samples each frame. Create an LMS filter with 32 taps and a step size of 0.1.

x = zeros(2000,50);
IIRFilter = dsp.IIRFilter('Numerator',sqrt(0.75),...
    'Denominator',[1 -0.5]);
for k = 1:size(x,2)   
  x(:,k) = IIRFilter(sign(randn(size(x,1),1))); 
end
mu = 0.1;     
LMSFilter = dsp.LMSFilter('Length',32,...
    'StepSize',mu);

Compute the maximum adaptation step size and the maximum step size in mean-squared sense using the maxstep function.

[mumax,mumaxmse] = maxstep(LMSFilter,x)

mumax = 
0.0625

mumaxmse = 
0.0536

Input Arguments

lmsFilt — LMS adaptive filter System object™

dsp.LMSFilter | dsp.BlockLMSFilter

LMS adaptive filter, specified as either a dsp.LMSFilter System object or a dsp.BlockLMSFilter System object.

x — Input signal

scalar | column vector | matrix

Columns of the matrix x contain individual input signal sequences. The signal set is assumed to have zero mean or close to zero mean.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical
Complex Number Support: Yes

Output Arguments

mumax — Maximum step size

scalar

Maximum step size value, returned as a scalar. This is the step size you can specify for the adaptive filter without causing the filter to become unstable. For details on how this parameter is calculated, see Algorithms.

Data Types: double

mumaxmse — Maximum step size in mean squared sense

scalar

Maximum adaptive filter step size to provide convergence of the LMS adaptive filter coefficients in the mean squared sense, returned as a scalar. For details on how this parameter is calculated, see Algorithms.

Data Types: double

Algorithms

The step size of the adaptive filter must satisfy the following equation in order for the adaptive filter to be stable:

$$0<\mu <{\mu }_{\max }$$

where, μ_max is the maximum step size.

The value of μ_max depends on the LMS filter System object and the adaptive filter algorithm the object uses.

dsp.LMSFilter

LMS

When the Method property of the dsp.LMSFilter object is set to 'LMS', maximum step size μ_max is calculated using the following equation:

$${\mu }_{\max }=\frac{2}{\text{mean}(xt\circ xt)L}$$

where,

• xt –– Concatenated columns of the input matrix, x(:).

• xt◦xt –– Hadamard or entrywise product of the two vectors.

• L –– Length of the filter coefficients.

The maximum step size in mean-square sense, μ_maxMSE is computed using the following equation:

$${\mu }_{\text{maxMSE}}=\frac{2}{{\lambda }_{\text{max}}(Kurt+2)+\text{sum}(\lambda )}$$

where,

• sum(λ) –– Sum of the eigenvalues of the input auto correlation matrix.

• λmax –– Maximum eigenvalue of the input auto correlation matrix.

• Kurt –– Average kurtosis value of eigenvector-filtered signals.

Normalized LMS

When the Method property of the dsp.LMSFilter System object is set to 'Normalized LMS':

• Maximum step size, μ_max = 2.

• Maximum step size in mean-square sense, μ_maxMSE = 2.

For Other Methods

For all other methods such as Sign-Data LMS, Sign-Error LMS, and Sign-Sign LMS:

• μ_max = ∞.

• μ_maxMSE = ∞.

dsp.BlockLMSFilter

The maximum step size for dsp.BlockLMSFilter is computed using the following equation:

$${\mu }_{\max }=\frac{2}{\text{mean}(xt\circ xt)L}$$

where,

• xt –– Concatenated columns of the input matrix, x(:).

• xt◦xt –– Hadamard or entrywise product of the two vectors.

• L –– Length of the filter coefficients.

The maximum step size in mean-square sense, μ_maxMSE is computed using the following equation:

$${\mu }_{\text{maxMSE}}=\frac{{\mu }_{\max }}{3}$$

References

[1] Hayes, M.H. Statistical Digital Signal Processing and Modeling. New York: John Wiley & Sons, 1996.

Version History

Introduced in R2012a