Page 48 - Analog and Digital Filter Design

45
Time and Frequency Response

output signal that is not proportional to the frequency. The rate of change
in phase with frequency is known as the group delay. The group delay increases
with the number of filter stages, so a fourth-order filter will produce a greater
delay than a third-order filter.

The group delay is the delay seen by all signal frequencies as they pass through
the filter. For example, a signal of 1 kHz may see a phase shift of 36", which is
a delay of 0.1 ms (the period of a 1 kHz signal is I ms and 36" is 0.1 cycle). If
the phase change is proportional to frequency, a 2 kHz signal will see a phase
shift of 72", which is also a delay of 0.1 ms (the period of a 2 kHz signal is
0.5ms and 72" is 0.2 cycle). This represents a constant group delay because
both signals are delayed by the same amount.

An example of a nonconstant group delay filter would be one where the I kHz
signal is delayed by 0.1 ms, as before. Now suppose that the 2 kHz signal sees a
phase shift of 90", which is a delay of 0.125ms. The timing relationship between
the two signals has changed because the 2 kHz signal is delayed by 0.025 ms more
than the 1 kHz signal.

The consequence of a nonconstant group delay can be seen when pulses are
applied to the filter input. Pulses contain signal harmonics several times the fun-
damental frequency of the pulse. As these harmonics propagate through the
filter, they each experience different delays. Summing the delayed fundamental
and harmonic signals results in slowly rising and falling pulse edges and causes
ripple on top of the pulse. This distortion can produce errors when subsequent
circuits process the pulse.

Butterworth, Chebyshev, Inverse Chebyshev. and Cauer filters have a group
delay that increases with frequency and reaches a peak value close to the cutoff
point. Beyond the cutoff point, the group delay gradually reduces to a constant
value. An example of group delay for a third-order Butterworth filter is given
in Figure 2.3. The delay shown is for a filter having a 1 rad/s cutoff frequency,
which is about 2 seconds at low frequencies. The delay is inversely proportional
to the cutoff frequency, so a filter having a 1 kHz (6283 rad/s) cutoff frequency
will have a delay of 2/6283, or 3 I8 ps, at low frequencies.

Further information on group delay is provided in Chapter 9, which describes
all-pass filters. All-pass filters have a phase response that can be used to correct
group delay variations of band-shaping filters. All-pass filters can also be used
to create Hilbert transform filters. These have two filter branches whose outputs
have the same signal amplitude but with a 90" phase difference. Hilbert filters
are used to create single-sideband modulation of a radio carrier signal.