Page 87 - Analog and Digital Filter Design

84 Analog and Digital Filter Design

Frequency and Time Domain Relationship

In the frequency domain there are three parameters to consider: frequency,
amplitude, and phase. The amplitude and phase versus frequency give the trans-
fer function of a network in the frequency domain. The transfer function can
be measured if a pure sine wave is applied at the network’s input, and the ampli-
tude and phase of the output signal is recorded for each frequency. Through
analysis of the network, an equation for the transfer function can also be found.
Transfer functions are normally calculated, then used for comparison with the
actual circuit implementation.

The time domain parameters include delay, rise time, overshoot, and ringing.
Delay can be measured by applying a step-input voltage to a network, where the
time for the output to reach 50% of the final value is measured. Rise time uses
the same step input and the time difference between the output reaching 10%
and 90% of the final output level. Overshoot and ringing are related, and they
also use a step input. Overshoot is where the output rises above the steady state
final value; the maximum output is recorded in terms of percent above the
nominal output. Ringing is where there is insufficient damping and the output
has an exponentially decaying sinusoidal waveform superimposed upon it.

Frequency and time domain transfer functions were described in Chapter 2. It
was made clear that a relationship between the frequency response and the time
domain response exists. If a filter’s frequency responsp has a gentle transition
between the passband and the stopband, it also has constant group delay
(the Bessel response). If the frequency response has a steep roll-off outside the
passband, it’s group delay (in the time domain) peaks where the change in
attenuation is greatest.

The time domain response can be converted into the frequency domain using
the Fourier Transform. Unfortunately, this transform can only be applied to
continuous periodic signals, so a variant of this, the Laplace Transform, is used
instead. The Laplace Transform is used to analyze transient signals; that is,
signals that appear at time, t = 0. When the Laplace Transform is applied to a
signal that is a function of time, At), it produces a response as a function
of complex frequency, F(s). The frequency response F(s) is complex, where
S = okjw. This leads us nicely to the S-plane.

The S-Plane

The S-plane can be used to describe both time and frequency domain responses.
It is just a graphical representation of mathematical ideas. However, these visual
aids are very powerful in helping us to understand filters and signals.