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 of an optimal technique buy over some other conventional methods. This question is conveniently answered using Figure 1, 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 αα needed to attain a specific degree of stopband suppression in lowpass filters. Since the filter order NN and therefore the amount of computation1R=NfsR=Nfs are directly proportional to αα, it serves as an excellent indicator for comparisons.

Figure 1: Comparisons among Windowed, Frequency Sampling, and Optimal Lowpass Filters (drawn from [1])

Curves for three design methods are shown, windowing techniques, so-called “frequency sampling" techniques, and the optimal, equal-ripple design produced by the Parks-McClellan program. In each case there are some variations depending on the choice of design parameters other than stopband ripple. For example, the optimal technique shows a band of results indexed by the amount of passband ripple (hence δ1δ1) specified. The figure shows that, for modest degrees of stopband suppression, all of the methods work about equally well. For high degrees of suppression, however, the optimal technique allows values of αα to be attained which are on the order of half of those attainable with the windowing methods and about 60-70% of the frequency sampling method. Since computation is directly proportional to αα, 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 is that 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. For example, 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 allowed the magnitude of the passband ripple to be almost 29 times larger than the stopband ripple. Since the 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 δ1δ1 as stopband ripple δ2δ2. Effectively the optimal design methods allow the degrees of freedom in the impulse response to be focused on the most stressing parts of the frequency response design while the window method treats all parts equally. The frequency-sampling method falls in between.

More insight into the meaning of the design parameter αα can be gained by examining all three aforementioned design methods in terms of the inverse discrete Fourier transform. Suppose that our objective, as it is, is to synthesize an N-point FIR filter. Suppose further that we use the approach of specifying the frequency response we desire with equally spaced samples in the frequency domain and then use the inverse discrete Fourier transform (DFT) to transform the frequency specification into a time-domain impulse response. This approach is shown in graphical form in Figure 2.

Figure 2: Using the Discrete Fourier Transform (DFT) as the Basis of FIR Filter Design

Analytically there is a one-to-one relationship between the N points of an FIR impulse response and the frequency response of the filter measured at N equally-spaced frequencies between 0 and fsfs Hertz. Specifically it is straight-forward to show that the impulse response h(k)h(k) and the complex gains h^nh^n, for 0≤n≤N-10≤n≤N-1, are invertibly related, where the filter's frequency response is given by

Thus choosing the complex gains h^nh^n is equivalent to choosing the impulse response h(k),0≤k≤N-1h(k),0≤k≤N-1, and, through Equation 1, to the filter frequency response at all values of ff between 0 and fsfs Hertz. By examining Figure 2 it can be seen that choosing a frequency response (and hence an impulse response) can be intuitively viewed as adjusting the gain levers on a graphic equalizer of the type now used on home stereos. Each lever sets the gain, denoted here as h^nh^n, of a filter given by

By setting these N gain values optimally the best possible frequency response is attained.

The analogy of the graphic equalizer can be followed somewhat further. Figure 2 suggests that the FIR design problem can be thought in the terms of the structure shown in Figure 3. The input signal is applied to all NN of what we'll the basis filters, where the frequency response of the nn-th filter is given by Equation 2. As noted earlier these basis filters, so called because they form the linearly independent set of filters used to construct H(f)H(f), are frequency-shifted versions of the same fairly sloppy bandpass filter. These filter outputs are then scaled by the complex coefficients h^nh^n and then added together to produce the observable filter output. Thus the basis filters are fixed and the h^nh^n control the frequency and hence impulse response of the digital filter. It should be noted that the filter is not usually actually constructed2 as shown in Figure 3 but it is a very convenient analogy when trying to understand the relationships between the various filter synthesis methods.

Figure 3: The FIR Filter Design Problem Models as a Bank of Bandpass Filters

Now we shall use the model. In our quest for the true meaning of αα, consider first the design of a simple lowpass filter. We desire the cutoff frequency fcfc and the stopband edge fstfst to be as low as possible and allow the peak stopband ripple to be quite large. Using the graphic equalizer model just discussed yields the design shown in Figure 4. Only one filter, the one centered at DC, is used. Its gain is set to unity and that of all others is set to zero. The peak stopband ripple is determined by the first sidelobe of the only active filter. It can be computed to be about 13 dB below the maximum passband power level (measured at DC).

Figure 4: A Simple Lowpass Filter Designed Using the Graphic Equalizer Analogy

What is ΔfΔf in this case? Graphically it can be seen to be somewhat less than than the frequency interval between DC and the first transmission zero of Hn(f)Hn(f) which occurs at f=fsNf=fsN. Suppose that we now rewrite equation 2 from the module titled "Filter Sizing" as

Thus we see that in the simple filter designed in Figure 4 that associated value of αα is slightly less than one.

Now suppose that we attempt to design a better filter, again using the graphic equalizer method. Our first objective is to reduce the size of the stopband ripple. To do this we leave h^0h^0 set to unity and increase the values of h^1h^1 and h^2h^2 slightly so that their positive mainlobe values cancel the negative-going first sidelobe of h^0h^0. All other filter gain levels will remain set to zero. The effects of this strategy are seen in Figure 5.

Figure 5: Lowpass Filter Obtained Using the Second and Third DFT Basis Functions

The first objective, that of reducing the peak stopband ripple, is achieved. By choosing h^1h^1 and h^2h^2 just right, the first sidelobe of h^0h^0 can be effectively cancelled, leaving the other sidelobes to compete for the peak value. The second effect is less desirable, however. From graphical inspection it is clear that ΔfΔf, the frequency interval between fcfc and fstfst, has grown. It now exceeds fsNfsN, thus making αα greater than unity.

These trends continue as more and more filter gains h^nh^n are allowed to become non-zero in the quest of further reducing the peak stopband ripple. The peak is reduced, the ripple structure begins to approach the Chebyshev equal-ripple firm seen in Figure 1 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem", and the transition band stretches out as more filters are used to try to constrain the stopband frequency response to the stopband ripple goals. The design parameter αα is just a measure of the number of filters, or, equivalently, the number of equalizer levers, needed to transit from one gain level (e.g., the passband) to another (e.g., the stopband) while achieving the desired passband and stopband ripple performance. Since fsNfsN is the spacing between the bins of an N-point DFT, the term αα can also be thought of as the number of DFT bins needed to make a gain transition. This interpretation is explored next.