0.3 Performance comparsion with other fir design methods

Performance comparsion with other fir design methods

A commonly asked question among filter designers is why should the optimal design methods be used at all, or, equivalently, how much does the use ofan optimal technique buy over some other conventional methods. This question is conveniently answered using [link] , a figure extracted from [1]and modified to use the definitions of variables employed in this technical note. The figure shows the value of the design parameter $\alpha$ needed to attain a specific degree of stopband suppression in lowpass filters. Since the filter order $N$ and therefore the amount of computation The actual amount of computation depends on whether the data is real- or complex-valued, whether the impulse response symmetry isexploited, and whether interpolation or decimation is used. In all cases, however, $R$ is proportional to $f_s$ and $\alpha$ , and therefore [link] provides an accurate indication of the relative computational complexity of the filters resulting from the different designmethods. $R=N{f}_s$ are directly proportional to $\alpha$ , it serves as an excellent indicator for comparisons.

Curves for three design methods are shown, windowing techniques, so-called “frequency sampling" techniques, and the optimal, equal-ripple designproduced by the Parks-McClellan program. In each case there are some variations depending on the choice of design parameters other than stopband ripple. Forexample, the optimal technique shows a band of results indexed by the amount of passband ripple (hence ${\delta }_1$ ) specified. The figure shows that, for modest degrees of stopband suppression, all of the methods work aboutequally well. For high degrees of suppression, however, the optimal technique allows values of $\alpha$ to be attained which are on the order of half of those attainable with the windowing methods and about 60-70% of thefrequency sampling method. Since computation is directly proportional to $\alpha$ , these saving are directly translatable into hardware and/or runtime improvements.

Why, one might ask, is the optimal method significantly better than, say, the window method? A fuller answer is presently shortly, but a simple one isthat the optimal methods allow the designer to avoid overdesigning portions of the frequency response about which he or she needn't exert as much control. Forexample, recall the design example discussed in the section "Conversion of Specifications" from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem" . In that case a set of reasonable specifications was developed which allowedthe magnitude of the passband ripple to be almost 29 times larger than the stopband ripple. Sincethe Parks-McClellan design package allows the design of weighted equal-ripple filters this disparity can be accommodated. Window-designed filters,however, are constrained to have exactly the same passband ripple ${\delta }_1$ as stopband ripple ${\delta }_2$ . Effectively the optimal design methods allow the degrees of freedom in the impulse response to be focused on themost stressing parts of the frequency response design while the window method treats all parts equally. The frequency-sampling method falls in between.