Estimated mean squared error for adaptive filters

The mean squared error (MSE) measures the average of the squares of the errors between the desired signal and the primary signal input to the adaptive filter. Reducing this error converges the primary input to the desired signal. Determine the predicted value of MSE and the simulated value of MSE at each time instant using the `msepred`

and `msesim`

functions. Compare these MSE values with each other and with respect to the minimum MSE and steady-state MSE values. In addition, compute the sum of the squares of the coefficient errors given by the trace of the coefficient covariance matrix.

**Note**: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, `obj(x)`

becomes `step(obj,x)`

.

**Initialization**

Create a `dsp.FIRFilter`

System object™ that represents the unknown system. Pass the signal, `x`

*,* to the FIR filter. The output of the unknown system is the desired signal, `d`

, which is the sum of the output of the unknown system (FIR filter) and an additive noise signal, `n`

.

num = fir1(31,0.5); fir = dsp.FIRFilter('Numerator',num); iir = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iir(sign(randn(2000,25))); n = 0.1*randn(size(x)); d = fir(x) + n;

**LMS Filter**

Create a `dsp.LMSFilter`

System object to create a filter that adapts to output the desired signal. Set the length of the adaptive filter to 32 taps, step size to 0.008, and the decimation factor for analysis and simulation to 5. The variable `simmse`

represents the simulated MSE between the output of the unknown system, `d`

, and the output of the adaptive filter. The variable `mse`

gives the corresponding predicted value.

l = 32; mu = 0.008; m = 5; lms = dsp.LMSFilter('Length',l,'StepSize',mu); [mmse,emse,meanW,mse,traceK] = msepred(lms,x,d,m); [simmse,meanWsim,Wsim,traceKsim] = msesim(lms,x,d,m);

**Plot the MSE Results**

Compare the values of simulated MSE, predicted MSE, minimum MSE, and the final MSE. The final MSE value is given by the sum of minimum MSE and excess MSE.

nn = m:m:size(x,1); semilogy(nn,simmse,[0 size(x,1)],[(emse+mmse)... (emse+mmse)],nn,mse,[0 size(x,1)],[mmse mmse]) title('Mean Squared Error Performance') axis([0 size(x,1) 0.001 10]) legend('MSE (Sim.)','Final MSE','MSE','Min. MSE') xlabel('Time Index') ylabel('Squared Error Value')

The predicted MSE follows the same trajectory as the simulated MSE. Both these trajectories converge with the steady-state (final) MSE.

**Plot the Coefficient Trajectories**

`meanWsim`

is the mean value of the simulated coefficients given by `msesim`

. `meanW`

is the mean value of the predicted coefficients given by `msepred`

.

Compare the simulated and predicted mean values of LMS filter coefficients 12,13,14, and 15.

plot(nn,meanWsim(:,12),'b',nn,meanW(:,12),'r',nn,... meanWsim(:,13:15),'b',nn,meanW(:,13:15),'r') PlotTitle ={'Average Coefficient Trajectories for';... 'W(12), W(13), W(14), and W(15)'}

`PlotTitle = `*2x1 cell array*
{'Average Coefficient Trajectories for'}
{'W(12), W(13), W(14), and W(15)' }

title(PlotTitle) legend('Simulation','Theory') xlabel('Time Index') ylabel('Coefficient Value')

In steady state, both the trajectories converge.

**Sum of Squared Coefficient Errors**

Compare the sum of the squared coefficient errors given by `msepred`

and `msesim`

. These values are given by the trace of the coefficient covariance matrix.

semilogy(nn,traceKsim,nn,traceK,'r') title('Sum-of-Squared Coefficient Errors') axis([0 size(x,1) 0.0001 1]) legend('Simulation','Theory') xlabel('Time Index') ylabel('Squared Error Value')

Identify an unknown system by performing active noise control using a filtered-x LMS algorithm. The objective of the adaptive filter is to minimize the error signal between the output of the adaptive filter and the output of the unknown system (or the system to be identified). Once the error signal is minimal, the unknown system converges to the adaptive filter.

**Note**: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, `obj(x)`

becomes `step(obj,x)`

.

**Initialization**

Create a `dsp.FIRFilter`

System object that represents the system to be identified. Pass the signal, `x`

, to the FIR filter. The output of the unknown system is the desired signal, `d`

, which is the sum of the output of the unknown system (FIR filter) and an additive noise signal, `n`

.

num = fir1(31,0.5); fir = dsp.FIRFilter('Numerator',num); iir = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iir(sign(randn(2000,25))); n = 0.1*randn(size(x)); d = fir(x) + n;

**Adaptive Filter**

Create a `dsp.FilteredXLMSFilter`

System object to create an adaptive filter that uses the filtered-x LMS algorithm. Set the length of the adaptive filter to 32 taps, step size to 0.008, and the decimation factor for analysis and simulation to 5. The variable `simmse`

represents the error between the output of the unknown system, `d`

, and the output of the adaptive filter.

l = 32; mu = 0.008; m = 5; fxlms = dsp.FilteredXLMSFilter(l,'StepSize',mu); [simmse,meanWsim,Wsim,traceKsim] = msesim(fxlms,x,d,m); plot(m*(1:length(simmse)),10*log10(simmse)) xlabel('Iteration') ylabel('MSE (dB)') title('Learning curve for Filtered-x LMS filter used in system identification')

With each iteration of adaptation, the value of `simmse`

decreases to a minimal value, indicating that the unknown system has converged to the adaptive filter.

**Note**: If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step`

syntax. For example, `obj(x)`

becomes `step(obj,x)`

.

ha = fir1(31,0.5); fir = dsp.FIRFilter('Numerator',ha); % FIR system to be identified iir = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iir(sign(randn(2000,25))); n = 0.1*randn(size(x)); % Observation noise signal d = fir(x)+n; % Desired signal l = 32; % Filter length m = 5; % Decimation factor for analysis % and simulation results ha = dsp.AdaptiveLatticeFilter(l); [simmse,meanWsim,Wsim,traceKsim] = msesim(ha,x,d,m); plot(m*(1:length(simmse)),10*log10(simmse)); xlabel('Iteration'); ylabel('MSE (dB)'); title('Learning curve for Adaptive Lattice filter used in system identification')

**Note**: If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step`

syntax. For example, `obj(x)`

becomes `step(obj,x)`

.

fir = fir1(31,0.5); firFilter = dsp.FIRFilter('Numerator',fir); % FIR system to be identified iirFilter = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iirFilter(sign(randn(2000,25))); n = 0.1*randn(size(x)); % Observation noise signal d = firFilter(x)+n; % Desired signal l = 32; % Filter length mu = 0.008; % Block LMS Step size. m = 32; % Decimation factor for analysis % and simulation results fir = dsp.BlockLMSFilter(l,'StepSize',mu); [simmse,meanWsim,Wsim,traceKsim] = msesim(fir,x,d,m); plot(m*(1:length(simmse)),10*log10(simmse)); xlabel('Iteration'); ylabel('MSE (dB)'); title('Learning curve for block LMS filter used in system identification')

**Note**: If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step`

syntax. For example, `obj(x)`

becomes `step(obj,x)`

.

ha = fir1(31,0.5); fir = dsp.FIRFilter('Numerator',ha); % FIR system to be identified iir = dsp.IIRFilter('Numerator',sqrt(0.75),... 'Denominator',[1 -0.5]); x = iir(sign(randn(2000,25))); n = 0.1*randn(size(x)); % Observation noise signal d = fir(x)+n; % Desired signal l = 32; % Filter length mu = 0.008; % Affine Projection filter Step size. m = 5; % Decimation factor for analysis % and simulation results apf = dsp.AffineProjectionFilter(l,'StepSize',mu); [simmse,meanWsim,Wsim,traceKsim] = msesim(apf,x,d,m); plot(m*(1:length(simmse)),10*log10(simmse)); xlabel('Iteration'); ylabel('MSE (dB)'); title('Learning curve for Affine Projection filter used in system identification')

`adaptFilt`

— Adaptive filter System object™adaptive filter System object

Adaptive filter, specified as one of the following System objects:

`x`

— Input signalscalar | column vector | matrix

Input signal, specified as a scalar, column vector, or matrix. Columns of
the matrix `x`

contain individual input signal sequences.
The input, `x`

, and the desired signal,
`d`

, must have the same size and data type.

If `adaptFilt`

is a `dsp.BlockLMSFilter`

object, the
input signal frame size must be greater than or equal to the value you
specify in the `BlockSize`

property of the object.

**Data Types: **`single`

| `double`

`d`

— Desired signalscalar | column vector | matrix

Desired response signal, specified as a scalar, column vector, or matrix.
Columns of the matrix `d`

contain individual desired
signal sequences. The input, `x`

, and the desired signal,
`d`

, must have the same size and the data
type.

If `adaptFilt`

is a `dsp.BlockLMSFilter`

System
object, the desired signal frame size must be greater than or equal
to the value you specify in the `BlockSize`

property of
the object.

**Data Types: **`single`

| `double`

`m`

— Decimation factor`1`

(default) | positive scalarDecimation factor, specified as a positive scalar. Every
`m`

th value of the estimated sequences is saved into
the corresponding output arguments, `mse`

,
`meanw`

, `w`

, and
`tracek`

. If `m`

equals 1, every
value of these sequences is saved.

If `adaptFilt`

is a `dsp.BlockLMSFilter`

System
object, the decimation factor must be a multiple of the value you
specify in the `BlockSize`

property of the object.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `logical`

`mse`

— Sequence of mean squared errorscolumn vector

Estimates of the mean squared error of the adaptive filter at each time instant, returned as a column vector.

If the adaptive filter is `dsp.BlockLMSFilter`

and the
decimation factor `m`

is specified, length of
`mse`

equals floor(*M*/m).
*M* is the frame size (number of rows) of the input
signal, `x`

. If `m`

is not specified,
the length of `mse`

equals floor(*M/B*),
where *B* is the value you specify in the
`BlockSize`

property of the object. The input signal
frame size must be greater than or equal to the value you specify in the
`BlockSize`

property of the object. The decimation
factor, if specified, must be a multiple of the
`BlockSize`

property.

For the other adaptive filters, if the decimation factor,
`m`

= 1, the length of `mse`

equals the frame size of the input signal. If `m`

> 1,
the length of `mse`

equals
floor(*M*/m).

**Data Types: **`double`

`meanw`

— Sequence of coefficient vector meansmatrix

Sequence of coefficient vector means of the adaptive filter at each time instant, estimated as a matrix. The columns of this matrix contain estimates of the mean values of the adaptive filter coefficients at each time instant.

If the adaptive filter is `dsp.BlockLMSFilter`

and the
decimation factor `m`

is specified, the dimensions of
`meanw`

is
floor(*M*/m)-by-*N*.
*M* is the frame size (number of rows) of the input
signal, `x`

. *N* is the length of the
filter weights vector, specified by the `Length`

property
of the adaptive filter. If `m`

is not specified, the
dimensions of `meanw`

is
floor(*M/B*)-by-*N*, where
*B* is the value you specify in the
`BlockSize`

property of the object. The input signal
frame size must be greater than or equal to the value you specify in the
`BlockSize`

property of the object. The decimation
factor, if specified, must be a multiple of the
`BlockSize`

property.

For the other adaptive filters, If the decimation factor,
`m`

= 1, the dimensions of `meanw`

is *M*-by-*N*. If `m`

>
1, the dimensions of `meanw`

is
floor(*M*/m)-by-*N*.

**Data Types: **`double`

`w`

— Final values of adaptive filter coefficientsrow vector

Final values of the adaptive filter coefficients for the algorithm
corresponding to `adaptFilt`

, returned as a row vector. The
length of the row vector equals the value you specify in the
`Length`

property of the object.

**Data Types: **`single`

| `double`

`tracek`

— Sequence of total coefficient error powerscolumn vector

Sequence of total coefficient error powers, estimated as a column vector. This column vector contains estimates of the total coefficient error power of the adaptive filter at each time instant.

If the adaptive filter is `dsp.BlockLMSFilter`

and the
decimation factor `m`

is specified, length of
`tracek`

equals floor(*M*/m).
*M* is the frame size (number of rows) of the input
signal, `x`

. If `m`

is not specified,
the length of `tracek`

equals
floor(*M/B*), where *B* is the value
you specify in the `BlockSize`

property of the object.
The input signal frame size must be greater than or equal to the value you
specify in the `BlockSize`

property of the object. The
decimation factor, if specified, must be a multiple of the
`BlockSize`

property.

For the other adaptive filters, if the decimation factor,
`m`

= 1, the length of `tracek`

equals the frame size of the input signal. If `m`

> 1,
the length of `tracek`

equals
floor(*M*/m).

**Data Types: **`double`

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

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