2.2 Least squared error design of fir filters  (Page 4/13)

Although developed here for the linear-phase filter, [link] is a very general design approach for the FIR filter that allows arbitraryphase, as well as uneven frequency sampling and a weighting function in the error definition. For the arbitrary phase case, a complex $\mathbf{F}$ is obtained from sampling Equation 28 from FIR Digital Filters and the full $h\left(n\right)$ is used. For the special case of the equally-spaced frequency samples and linear- phasefilter with unity weighting, the solution of [link] or [link] is the same as given by the frequency sampling design formulas.

One of the important uses of the unequally spaced frequency samples is to create a transition band between the pass and stopbands where there are nosamples. This “don't care" band does not contribute to the error measure q and allows better approximation to occur over the pass and stopbands.

Of the many ways to solve [link] or [link] , one of the easiest and most reliable is the use of Matlab, which has a special command to solve this least-mean-squared errorproblem. Equation [link] should not be solved directly. For large $L$ , it is ill-conditioned and a direct solution will probably have large errors.Matlab uses special algorithms to minimize these numerical errors.

This approach was applied to the same problems that were solved by frequency sampling in the previous section. For $N=L$ , the same results are obtained, thus verifying the theoretical prediction. As $L$ becomes larger compared to $N$ , more control is exerted over the behavior between the original sample points. As $L$ becomes large compared to $N$ , the solution approaches the same results as obtained where theerror is defined as a continuous function of frequency and the integral of the squared error is minimized.Although the solution of the normal equations is a powerful and flexible technique, it can be slow, have numerical problems, and require largeamounts of computer memory.

Examples of discrete least squared error filter design

Here we will give examples of several least squared error designs of FIR filters.

As for the frequency sampling design, we see a good lowpass filter frequency response with the actual amplitude interpolating the desiredvalues at different points from the frequency sampling example in [link] even though the length and band edge are the same. Notice there is less over shoot but more ripple near $f=0$ . The Gibbs phenomenon is the same as for the Fourier series.

If a transition band is introduces in the ideal amplitude response between $f=0.4$ and $f=0.6$ with a straight line, the overshoot is reduced significantly but with a slightly slower transition from the pass to stop band. This is illustrated in [link] .

Continuous frequency definition of error

Because the energy of a signal is the integral of the sum of the squares of the Fourier transform magnitude and because specifications are usuallygiven in the frequency domain, a very reasonable error measure to minimize is the integral squared error given by