Page 247 - Analog and Digital Filter Design

244 Analog and Digital Filter Design

some means of correcting for the nonlinear phase shift of Buttenvorth,
Chebyshev, and Cauer filters is desirable.

”Group delay” is the term used to describe the time delay versus frequency rela-
tionship of the transmitted signal. It is defined as the rate of phase change with
frequency. The term “group delay” is very descriptive, in that it is the delay seen
by a group of frequencies that are being transmitted through a filter. A constant
group delay implies that all frequencies experience the same delay. A frequency-
dependent group delay implies that some frequencies are delayed more than
others.

Bessel filters have a constant group delay, because the phase change of signals
passing through them is proportional to the frequency. Other filter types, such
as the Butterworth, have a group delay that is frequency dependent, and the
rate of phase change generally increases as the filter’s cutoff frequency is
approached. The amount of group delay variation with frequency depends on
the filter type, and generally increases for filters that have a rapid increase in
attenuation outside their passband (a steep skirt response). Group delay varia-
tions can be minimized by the use of phase-equalizing all-pass filters. All-pass
filters can be designed to have a group delay that is virtually complementary to
a lowpass filter, so the two filters connected in series produce an almost con-
stant group delay.

Detailed Analysis
Impulsive signals contain many harmonics, and Fourier analysis can be used to
show that summing all the odd harmonics can produce a square wave. Consider
a square wave of amplitude “A”; each harmonic will have an amplitude of 4A/n
multiplied by the inverse of the harmonic number. The sum of harmonics, up
to the fifth order, is thus: square wave = 4A/n x fundamental + 4A/(3n) x third
harmonic + 4A/(5n) x fifth harmonic.

Some distortion is inevitable if the signal passes through a lowpass filter,
because restricting the bandwidth will reduce the amplitude of the higher har-
monics. Generally the distortion caused by restricting the bandwidth is pulse-
edge rounding and some amplitude ripple in the pulse. This is illustrated in
Figure 9.1.