Page 383 - Analog and Digital Filter Design

380 Analog and Digital Filter Design

Frequency versus Time-Domain Responses

The following subsections provide coefficient values for lowpass, highpass,
bandpass, and bandstop filters. In each case the central coefficient value h[O] is
given separately and is derived using L'Hopital's rule (see Appendix). The coef-
ficients h[n] apply to all nonzero values of n.

Denormalized Lowpass Response Coefficients
As discussed previously, in Chapter 15, the lowpass frequency domain response
becomes a sinc(x) function in the time domain. Denormalization to give a par-
ticular lowpass response is quite simple. The normalized response has a sam-
pling rate of 1 Hz (2n radians per second), so the cutoff frequency is relative to
this (cutoff at oc) and the value of o, is given by the equation:

The relationship between sampling frequency and the filter cutoff frequency for
lowpass filters is shown in Figure 16.4.

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I I 1

For example, let 4. = 3.4kHz and Fs = 16 kHz. The value of a,= 3.4/16 = 0.2125.

The value of the central coefficient is given by:

The values of the other coefficients are given by:

sin(w,n)
k[n] = ~
zn

Using the example value of w,. = 0.2125 in the above equations gives a set
of coefficient values, which are: h[O] = 0.06764, h[l] = O.O67133,h[2] = 0.065623,
. . . , and so on.