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The module "Performance Comparison with other FIR Design Methods" alluded to the fact that three basic methods have traditionally been used for the design of FIR digital filters. Figure 1 in the module titled "Performance Comparison with other FIR Design Methods" in fact compares their relative performance in terms of the value of $\alpha $ (which was shown to be proportional to the filter's required run-timecomputation rate). Given the background of the previous subsection it is now possible to understand each of the methodsand to gain some insight into the differences between their performance.
As described earlier, one of the first class of FIR filters is that based on the use of a “smoothing window". This window, constructed tohave 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 theeffect of smearing the desired frequency response.
The stopband ripple specification is obtained by using a window capable of suppressing all sidelobes to the desired degree. This can be seen in [link] . The windowed filter basis function has substantially lower sidelobes than the original $\frac{sin\phantom{\rule{3.33333pt}{0ex}}Nq}{sin\phantom{\rule{3.33333pt}{0ex}}q}$ filter basis function, in trade for substantial widening of the main lobe. This widening means growth in the equivalent design parameter $\alpha $ 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. Thussome of the filter's degrees of freedom are given up in perhaps overdesigning the passband response rather than focusing them on the stopbandperformance.
In the simplest DFT-based FIR filter design method, the desired frequency response is sampled at frequency intervals of $\frac{{f}_{s}}{N}$ Hertz and the filter gains ${\widehat{h}}_{n}$ 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 itssimplicity and the fact that any desired response, no matter how complicated, can be approximated. The big disadvantage is its uncontrolledripple performance in both the stopband and passband. The traditional cures for this are the use of a window function to suppress the ripple and theexpansion of the filter order N to compensate for the window's smearing of the desired response. Increasing $N$ , 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 frequencyresponse could be attained by allowing some of the basis filter gains ${\widehat{h}}_{n}$ to vary slightly from the exact sampled values (e.g., 1 and 0 for a lowpass filter). This idea is shown in [link] . A simple lowpass filter is the desired response. Solid dots show the frequency samples of this desired responsetaken every $\frac{{f}_{s}}{N}$ Hertz. These samples have values of 1 and 0 for ${\widehat{h}}_{n}$ in the passband and stopband respectively. Now suppose that the values of ${\widehat{h}}_{n}$ for $n$ in the vicinity of the cutoff frequency ${f}_{c}$ are allowed to be modified slightly with the goal of minimizing the peak stopband ripple. These values of $n$ are denoted with small circles instead of solid dots in [link] . Rabiner and his coworkers [4] showed in1970 that it was possible to use the linear programming optimization technique to manipulate two or three of the filter gains to obtaingreat improvement in stopband ripple performance. The computational complexity of the linear programming method, however, limited thenumber of the ${\widehat{h}}_{n}$ which could be so chosen.
It was generally known in 1971 that equal-ripple passband and stopband behavior would lead to the best filter performance, where “best" meansthe smallest transition band (and hence $\alpha $ ) for a given set of peak passband and stopband ripple specifications. In fact a great deal wasknown 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, butprincipally Parks, McClellan (Parks' graduate student), and Rabiner showed that four different variants of FIR linear phase filters couldall be represented by the same set of equations The variants are odd and even filter order and symmetric and antisymmetric impulse responses. and could therefore be solved the same way. The second part was Parks' suggestion of using the the Remez exchange algorithm for doingthe actual optimization. The Remez exchange algorithm effectively allows all degrees of freedom in the filter impulse response to be adjustedsimultaneously while the linear programming technique allows the adjustment of only one at a time. For high order filters this distinction makes atremendous difference in the number of computations needed to iteratively optimize a design. Refering again to [link] , the Remez algorithm allows all of the frequency samples to be modified,even for filter orders as high as $N=1000$ or more, thus permitting the best possible filter performance to be achieved. McClellan also proved that the linearphase FIR filter design problem satisfied the conditions needed to guarantee convergence of the Remez algorithm.
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