As described earlier, one of the first class of FIR filters is that based on the use of a “smoothing window". This window, constructed to have only N non-zero points, is multiplied point-by-point by an impulse response of infinite duration which has the “perfect" frequency response. This multiplication or windowing has the effect of making the filter impulse response finite in duration (hence FIR), but also has the effect of smearing the desired frequency response.

Figure 1: The Effect of a Window Function on the Basis Filter

The stopband ripple specification is obtained by using a window capable of suppressing all sidelobes to the desired degree. This can be seen in Figure 1. The windowed filter basis function has substantially lower sidelobes than the original sinNqsinqsinNqsinq filter basis function, in trade for substantial widening of the main lobe. This widening means growth in the equivalent design parameter αα and is monotonic with the degree of sidelobe suppression attained.

It should also be observed that the sidelobe reduction has the effect of reducing the ripple in the passband as well as in the stopband. Thus some of the filter's degrees of freedom are given up in perhaps overdesigning the passband response rather than focusing them on the stopband performance.

In the simplest DFT-based FIR filter design method, the desired frequency response is sampled at frequency intervals of fsNfsN Hertz and the filter gains h^nh^n are set to those values. This is in essence the method used for the simple lowpass filter shown in Figure 4 from the module titled"Performance Comparison with other FIR Design Methods". The big advantages of this method are its simplicity and the fact that any desired response, no matter how complicated, can be approximated. The big disadvantage is its uncontrolled ripple performance in both the stopband and passband. The traditional cures for this are the use of a window function to suppress the ripple and the expansion of the filter order N to compensate for the window's smearing of the desired response. Increasing NN, of course, increases the filter's run-time computation rate.

Relatively early in the development of FIR design techniques it was discovered that much better adherence to the desired frequency response could be attained by allowing some of the basis filter gains h^nh^n to vary slightly from the exact sampled values (e.g., 1 and 0 for a lowpass filter). This idea is shown in Figure 2. A simple lowpass filter is the desired response. Solid dots show the frequency samples of this desired response taken every fsNfsN Hertz. These samples have values of 1 and 0 for h^nh^n in the passband and stopband respectively. Now suppose that the values of h^nh^n for nn in the vicinity of the cutoff frequency fcfc are allowed to be modified slightly with the goal of minimizing the peak stopband ripple. These values of nn are denoted with small circles instead of solid dots in Figure 2. Rabiner and his coworkers [4] showed in 1970 that it was possible to use the linear programming optimization technique to manipulate two or three of the filter gains to obtain great improvement in stopband ripple performance. The computational complexity of the linear programming method, however, limited the number of the h^nh^n which could be so chosen.

Figure 2: Comparison of “Frequency Sampling" and Equal-ripple Design

It was generally known in 1971 that equal-ripple passband and stopband behavior would lead to the best filter performance, where “best" means the smallest transition band (and hence αα) for a given set of peak passband and stopband ripple specifications. In fact a great deal was known about the properties of such filters. What was lacking was a computationally satisfactory method of designing such optimal filters. As just noted, the linear programming technique provided a big step but still fell short. The breakthrough came in two parts. Several workers, but principally Parks, McClellan (Parks' graduate student), and Rabiner showed that four different variants of FIR linear phase filters could all be represented by the same set of equations1 and could therefore be solved the same way. The second part was Parks' suggestion of using the the Remez exchange algorithm for doing the actual optimization. The Remez exchange algorithm effectively allows all degrees of freedom in the filter impulse response to be adjusted simultaneously while the linear programming technique allows the adjustment of only one at a time. For high order filters this distinction makes a tremendous difference in the number of computations needed to iteratively optimize a design. Refering again to Figure 2, the Remez algorithm allows all of the frequency samples to be modified, even for filter orders as high as N=1000N=1000 or more, thus permitting the best possible filter performance to be achieved. McClellan also proved that the linear phase FIR filter design problem satisfied the conditions needed to guarantee convergence of the Remez algorithm.