Page 365 - Analog and Digital Filter Design

362 Analog and Digital Filter Design

More sophisticated windows are available. Why are they necessary? Quite
simply, truncation distorts the sin(x)/x envelope, whch limits the maximum
stopband attenuation that can be achieved. The use of a rectangular window,
which is simply truncation, limits the maximum achievable attenuation to a little
over 20 dB. Using the triangular (Bartlett) window is a little better, because the
sin(x)/x envelope is gradually reduced in amplitude on either side of the peak
value. The triangular window limits the maximum achievable attenuation to
25 dB. By comparison we have a few other windows listed below:

Hanning 43 dB
Hamming 54 dB

Blackman 75 dB
Harris flat top 97 dB

Clearly the Harris flat top is the best of these windows, so why bother with the
others? Well, as the stopband attenuation increases, the width of the transition
band between the passband and stopband also increases. So there is clearly a
tradeoff between the transition band (skirt) width and the stopband attenua-
tion limit. Also, the windows that achieve the highest levels of attenuation
generally require more filter taps, which increases both time delay and filter
complexity.

Transforming the Lowpass Response

So far I have described the lowpass filter and its time-domain impulse response.
Before I describe highpass, bandpass, and bandstop responses, it is necessary to
consider how these may be affected by the sampling operation. Sampling was
briefly described earlier with reference to analog-to-digital conversion.

Engineers familiar with radio and analog signal processing techniques will
appreciate that sampling performs the same function as mixing. Mixing, or
amplitude modulation, multiplies one signal with another and produces a spec-
trum containing the sum and difference frequencies. The analog mixing process
considers a signal of frequency F1 with a signal of frequency F2, which will
produce the mixed product F2 + F1 and F2 - F1.

The sampling signal in a digital system has a narrow impulse, which contains
many harmonics. The analysis of the mixing process must consider a signal of
frequency F1 with an impulse having spectrum frequencies at N times F2, where
N = 1, 2, 3, and on to infinity (in theory, for infinitely narrow samples). This
will not only produce the mixed product F2 + F1 and F2 - F1 (amplitude
modulation), it will also produce signals at frequencies of { (N times F2) + F 1 }
and ((N times F2) - Fl}.