are the lower and upper cut-off frequencies, respectively. The
bandwidth is the difference between the two cut-off frequencies for a bandpass filter.
The 3-dB bandwidth commonly used in practice is defined as
Another way of describing a resonator (or notch) filter is the quality factor defined as
There are many applications that require high Q filters.
When a signal passes through a filter, it is modified both in amplitude and phase. The
phase response is an important filter characteristic because it affects time delay of the
different frequency components passing through the filter. If we consider a signal that
consists of several frequency components, the phase delay of the filter is the average
time delay the composite signal suffers at each frequency. The group delay function is
where f! is the phase response of the filter.
A filter is said to have a linear phase if its phase response satisfies
À p ! p
f! b À a!,
À p ! p
These equations show that for a filter with a linear phase, the group delay T
! given in
(5.1.13) is a constant a for all frequencies. This filter avoids phase distortion because all
sinusoidal components in the input are delayed by the same amount. A filter with a
nonlinear phase will cause a phase distortion in the signal that passes through it. This is
because the frequency components in the signal will each be delayed by a different
amount, thereby altering their harmonic relationships. Linear phase is important in data
communications, audio, and other applications where the temporal relationships
between different frequency components are critical.
The specifications on the magnitude and phase (or group delay) of H! are based on
the steady-state response of the filter. Therefore they are called the steady-state speci-
fications. The speed of the response concerns the rate at which the filter reaches the
steady-state response. The transient performance is defined for the response right after
DESIGN AND IMPLEMENTATION OF FIR FILTERS