Designing Adaptive Filters (Filter Design Toolbox)

Filter Design Toolbox

adaptlms Example -- System Identification

To use the adaptive filter functions in the toolbox you need to provide three things:

An unknown system or process to adapt to. In this example, the filter designed by gremez is the unknown system.
Appropriate input data to exercise the adaptation process. In terms of the generic LMS model, these are the desired signal d(k) and the input signal x(k).
Both the adaptive LMS function to use and the matching initialization function to set up the adapting algorithm. Here we use adaptlms and initlms.

Start by defining an input signal x.

```
x = 0.1*randn(1,500);
```

The input is broadband noise. For the unknown system filter, use gremez to create a twelfth-order lowpass filter:

[b,err,res] = gremez(12,[0 0.4 0.5 1], [1 1 0 0], [1 0.2],...
{'w' 'c'});

Although you do not need them here, include the err and res output arguments.

Now filter the signal through the unknown system to get the desired signal.

```
d = filter(b,1,x);
```

With the unknown filter designed and the desired signal in place you can apply the adaptive LMS filter to identify the unknown.

Preparing the adaptive filter algorithm requires that you provide starting values for estimates of the filter coefficients and the LMS step size in a single structure s. We use initlms to populate the structure. You could start with estimated coefficients of some set of nonzero values; this example uses zeros for the 12 initial filter weights. For the step size, 0.8 is a reasonable value -- a good compromise between being large enough to converge well within the 500 iterations (500 input sample points) and small enough to create an accurate copy of the unknown filter.

w0 = zeros(1,13);
mu = 0.8;
s = initlms(w0,mu);

Structure s now comprises the following fields.

Structure Element
Element Contents
initlms Element

s.coeffs
LMS FIR filter coefficients. Should be initialized with the initial coefficients for the FIR filter prior to adapting. You need (adapting filter order + 1) entries in s.coeffs. Updated filter coefficients are returned in s.coeffs when you use s as an output argument.
wo

s.step
Sets the LMS algorithm step size. Determines both how quickly and how closely the adative filter adapts to the filter solution.
mu

s.states
Returns the states of the FIR filter after adaptation. This is an optional element. If omitted, it defaults to a zero vector of length equal to the filter order. When you use adaptlms in a loop structure, use this element to specify the initial filter states for the adapting FIR filter.
zi

s.leakage
Specifies the LMS leakage parameter. Allows you to implement a leaky LMS algorithm. Including a leakage factor can improve the results of the algorithm by forcing the LMS algorithm to continue to adapt even after it reaches a minimum value. Ranges between 0 and 1. This is an optional field. Defaults to one if omitted (specifying no leakage) or set to empty, [ ].
lf

s.iter
Total number of iterations in the adaptive filter run. Although you can set this in s, you should not. Consider it a read-only value.

Structure Element	Element Contents	initlms Element
`s.coeffs`	LMS FIR filter coefficients. Should be initialized with the initial coefficients for the FIR filter prior to adapting. You need (adapting filter order + 1) entries in `s.coeffs`. Updated filter coefficients are returned in `s.coeffs` when you use `s` as an output argument.	`wo`
`s.step`	Sets the LMS algorithm step size. Determines both how quickly and how closely the adative filter adapts to the filter solution.	`mu`
`s.states`	Returns the states of the FIR filter after adaptation. This is an optional element. If omitted, it defaults to a zero vector of length equal to the filter order. When you use `adaptlms` in a loop structure, use this element to specify the initial filter states for the adapting FIR filter.	`zi`
`s.leakage`	Specifies the LMS leakage parameter. Allows you to implement a leaky LMS algorithm. Including a leakage factor can improve the results of the algorithm by forcing the LMS algorithm to continue to adapt even after it reaches a minimum value. Ranges between 0 and 1. This is an optional field. Defaults to one if omitted (specifying no leakage) or set to empty, [ ].	`lf`
`s.iter`	Total number of iterations in the adaptive filter run. Although you can set this in s, you should not. Consider it a read-only value.

Finally, using the desired signal, d, the input to the filter, x, and the structure that contains the algorithm initialization settings, s, we run the adaptive filter to determine the unknown system and plot the results, comparing the actual coefficients from gremez to the coefficients found by adaptlms.

[y,e,s] = adaptlms(x,d,s);
stem([b.' s.coeffs.'])

In the stem plot the actual and estimated filter weights are the same. As an experiment, try changing the step size to 0.2. Repeating the example with mu = 0.2 results in the following stem plot. The estimated weights fail to approximate the actual weights closely.

Since this may be because we did not iterate over the LMS algorithm enough times, try using 1000 samples. With 1000 samples, the stem plot, shown in the next figure, looks much better, albeit at the expense of much more computation. Clearly you should take care to select the step size with both the computation required and the fidelity of the estimated filter in mind.

Examples of Adaptive Filters That Use LMS Algorithms adaptnlms Example -- System Identification