0.3 Performance comparsion with other fir design methods  (Page 2/2)

The meaning of the design parameter α

More insight into the meaning of the design parameter $\alpha$ can be gained by examining all three aforementioned design methods in terms of the inversediscrete Fourier transform. Suppose that our objective, as it is, is to synthesize an N-point FIR filter. Suppose further that weuse the approach of specifying the frequency response we desire with equallyspaced samples in the frequency domain and then use the inverse discrete Fourier transform (DFT) to transform the frequency specification into atime-domain impulse response. This approach is shown in graphical form in [link] .

Analytically there is a one-to-one relationship between the N points of an FIR impulse response and the frequency response of the filter measuredat N equally-spaced frequencies between 0 and $f_s$ Hertz. Specifically it is straight-forward to show that the impulse response $h\left(k\right)$ and the complex gains ${\widehat{h}}_n$ , for $0\le n\le N-1$ , are invertibly related, where the filter's frequency response is given by

Thus choosing the complex gains ${\widehat{h}}_n$ is equivalent to choosing the impulse response $h\left(k\right),0\le k\le N-1$ , and, through [link] , to the filter frequency response at all values of $f$ between 0 and $f_s$ Hertz. By examining [link] 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 typenow used on home stereos. Each lever sets the gain, denoted here as ${\widehat{h}}_n$ , of a filter given by

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

The analogy of the graphic equalizer can be followed somewhat further. [link] suggests that the FIR design problem can be thought in the terms of the structure shown in [link] . The input signal is applied to all $N$ of what we'll the basis filters, where the frequency response of the $n$ -th filter is given by [link] . As noted earlier these basis filters, so called because they form the linearly independent set of filters used toconstruct $H\left(f\right)$ , are frequency-shifted versions of the same fairly sloppy bandpass filter. These filter outputs are then scaled by thecomplex coefficients ${\widehat{h}}_n$ and then added together to produce the observable filter output. Thus the basis filters are fixed and the ${\widehat{h}}_n$ control the frequency and hence impulse response of the digital filter. It should be noted that the filter is not usually actuallyconstructed Frequency-domain filters are of course the counterexample. as shown in [link] but it is a very convenient analogy when trying to understand the relationships betweenthe various filter synthesis methods.

Now we shall use the model. In our quest for the true meaning of $\alpha$ , consider first the design of a simple lowpass filter. We desire the cutoff frequency $f_c$ and the stopband edge $f_{st}$ to be as low as possible and allow the peak stopband ripple to be quite large. Using thegraphic equalizer model just discussed yields the design shown in [link] . 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 peakstopband 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 powerlevel (measured at DC).

What is $\Delta f$ in this case? Graphically it can be seen to be somewhat less than than the frequency interval between DC and the first transmissionzero of $H_n\left(f\right)$ which occurs at $f=\frac{f_s}{N}$ . Suppose that we now rewrite equation 2 from the module titled "Filter Sizing" as

Thus we see that in the simple filter designed in [link] that associated value of $\alpha$ 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 stopbandripple. To do this we leave ${\widehat{h}}_0$ set to unity and increase the values of ${\widehat{h}}_1$ and ${\widehat{h}}_2$ slightly so that their positive mainlobe values cancel thenegative-going first sidelobe of ${\widehat{h}}_0$ . All other filter gain levels willremain set to zero. The effects of this strategy are seen in [link] .

The first objective, that of reducing the peak stopband ripple, is achieved. By choosing ${\widehat{h}}_1$ and ${\widehat{h}}_2$ just right, the first sidelobe of ${\widehat{h}}_0$ can be effectively cancelled, leaving the other sidelobes to compete for thepeak value. The second effect is less desirable, however. From graphical inspection it is clear that $\Delta f$ , the frequency interval between $f_c$ and $f_{st}$ , has grown. It now exceeds $\frac{f_s}{N}$ , thus making $\alpha$ greater than unity.

These trends continue as more and more filter gains ${\widehat{h}}_n$ 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 transitionband stretches out as more filters are used to try to constrain the stopband frequency response to the stopband ripple goals. The design parameter $\alpha$ is just a measure of the number of filters, or, equivalently, the number of equalizer levers, needed to transit from one gainlevel (e.g., the passband) to another (e.g., the stopband) while achieving the desired passband and stopband ripple performance. Since $\frac{f_s}{N}$ is the spacing between the bins of an N-point DFT, the term $\alpha$ can also be thought of as the number of DFT bins needed to make a gain transition. Thisinterpretation is explored next.