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# Documentation

Adaptive filter that uses discrete Fourier transform

## Syntax

delta,lambda,...coeffs,states)

## Description

delta,lambda,...coeffs,states)
constructs a transform-domain adaptive filter object ha using a discrete Fourier transform.

For information on how to run data through your adaptive filter object, see the Adaptive Filter Syntaxes section of the reference page for filter.

### Input Arguments

Entries in the following table describe the input arguments for adaptfilt.tdafdft.

Input Argument

Description

l

Adaptive filter length (the number of coefficients or taps) and it must be a positive integer. l defaults to 10.

step

Adaptive filter step size. It must be a nonnegative scalar. You can use maxstep to determine a reasonable range of step size values for the signals being processed. step defaults to 0.

leakage

Leakage parameter of the adaptive filter. When you set this argument to a value between zero and one, you are implementing a leaky version of the TDAFDFT algorithm. leakage defaults to 1 — no leakage.

offset

Offset for the normalization terms in the coefficient updates. You can use this argument to avoid dividing by zeros or by very small numbers when any of the FFT input signal powers become very small. offset defaults to zero.

delta

Initial common value of all of the transform domain powers. Its initial value should be positive. delta defaults to 5.

lambda

Averaging factor used to compute the exponentially-windowed estimates of the powers in the transformed signal bins for the coefficient updates. lambda should lie between zero and one. For default filter objects, LAMBDA equals (1 - step).

coeffs

Initial time domain coefficients of the adaptive filter. Set it to be a length l vector. coeffs defaults to a zero vector of length l.

states

Initial conditions of the adaptive filter. states defaults to a zero vector with length equal to (l - 1).

## Properties

Since your adaptfilt.tdafdft filter is an object, it has properties that define its behavior in operation. Note that many of the properties are also input arguments for creating adaptfilt.tdafdft objects. To show you the properties that apply, this table lists and describes each property for the transform domain filter object.

Name

Range

Description

Algorithm

None

AvgFactor

Averaging factor used to compute the exponentially-windowed estimates of the powers in the transformed signal bins for the coefficient updates. AvgFactor should lie between zero and one. For default filter objects, AvgFactor equals (1 - step). lambda is the input argument that represent AvgFactor.

Coefficients

Vector of elements

Vector containing the initial filter coefficients. It must be a length l vector where l is the number of filter coefficients. coeffs defaults to length l vector of zeros when you do not provide the argument for input.

FilterLength

Any positive integer

Reports the length of the filter, the number of coefficients or taps

Leakage

0 to 1

Leakage parameter of the adaptive filter. When you set this argument to a value between zero and one, you are implementing a leaky version of the TDAFDFT algorithm. leakage defaults to 1 — no leakage.

Offset

Offset for the normalization terms in the coefficient updates. You can use this argument to avoid dividing by zeros or by very small numbers when any of the FFT input signal powers become very small. offset defaults to zero.

PersistentMemory

false or true

Determines whether the filter states get restored to their starting values for each filtering operation. The starting values are the values in place when you create the filter. PersistentMemory returns to zero any state that the filter changes during processing. States that the filter does not change are not affected. Defaults to false.

Power

2*l element vector

A vector of 2*l elements, each initialized with the value delta from the input arguments. As you filter data, Power gets updated by the filter process.

States

Vector of elements, data type double

Vector of the adaptive filter states. states defaults to a vector of zeros which has length equal to (l + projectord - 2).

StepSize

0 to 1

Step size. It must be a nonnegative scalar, greater than zero and less than or equal to 1. step defaults to 0.

## Examples

Quadrature Phase Shift Keying (QPSK) adaptive equalization using a 32-coefficient FIR filter (1000 iterations).

```D  = 16;                       % Number of samples of delay
b  = exp(1j*pi/4)*[-0.7 1];     % Numerator coefficients of channel
a  = [1 -0.7];                 % Denominator coefficients of channel
ntr= 1000;                     % Number of iterations
s  = sign(randn(1,ntr+D)) + 1j*sign(randn(1,ntr+D));% Baseband QPSK signal
n  = 0.1*(randn(1,ntr+D) + 1j*randn(1,ntr+D));      % Noise signal
r  = filter(b,a,s)+n;           % Received signal
x  = r(1+D:ntr+D);              % Input signal (received signal)
d  = s(1:ntr);                  % Desired signal (delayed QPSK signal)
L  = 32;                       % filter length
mu = 0.01;                     % Step size
[y,e] = filter(ha,x,d);
subplot(2,2,1); plot(1:ntr,real([d;y;e]));
title('In-Phase Components');
legend('Desired','Output','Error');
xlabel('Time Index'); ylabel('Signal Value');
subplot(2,2,2); plot(1:ntr,imag([d;y;e]));
legend('Desired','Output','Error');
xlabel('Time Index'); ylabel('Signal Value');
subplot(2,2,3); plot(x(ntr-100:ntr),'.');
axis([-3 3 -3 3]); title('Received Signal Scatter Plot');
axis('square');  xlabel('Real[x]'); ylabel('Imag[x]'); grid on;
subplot(2,2,4); plot(y(ntr-100:ntr),'.');
axis([-3 3 -3 3]); title('Equalized Signal Scatter Plot');
axis('square'); xlabel('Real[y]'); ylabel('Imag[y]'); grid on;```

All of the time domain adaptive filter reference pages use this QPSK example. By comparing the results for each variation you get an idea of the differences in the way each one performs.

This figure demonstrates the results of running the example code shown.

## References

Haykin, S.,Adaptive Filter Theory, 3rd Edition, Prentice Hall, N.J., 1996