Digital filters - Continuous and Discrete Time Signals and Systems - page 627

infinite number of impulse functions, even if the amplitudes of the constituent
impulse functions are finite, the summation

h [ k ] in an IIR filter may not be
finite. In other words, it is not guaranteed that an IIR filter will always be stable.
Therefore, care should be taken when designing IIR filters so that the filter is
stable.
The implementation cost, typically measured by the number of delay ele-
ments used, is another important criterion in the design of filters. IIR filters are
implemented using a feedback loop, in which the number of delay elements is
determined by the order of the IIR filter. The number of delay elements used in
FIR filters depends on its length, and so the implementation cost of such filters
increases with the number of filter taps. An FIR filter with a large number of
taps may therefore be computationally infeasible.

14.3 Phase of a dig ital filter

In Section 14.1, we introduced ideal frequency-selective filters as having rect-
angular magnitude response with sharp transitions between the pass band and
stop band. The phase of ideal filters is assumed to be zero at all frequencies. An
ideal filter is physically unrealizable because of the sharp transitions between
the pass bands and stop bands and also because of the zero phase. In this sec-
tion, we illustrate the effect of the phase on the performance of digital filters.
In particular, we show that distortionless transmission within the pass band can
be achieved by using a filter having a linear phase within the pass band.
Consider the following sinusoidal sequence:

x [ k ] = A 1 cos( Ω 1 k ) + A 2 cos( Ω 2 k ) + A 3 cos( Ω 3 k ) ,

consisting of three tone frequencies Ω 1 < Ω 2 < Ω 3 applied at the input of a
physically realizable lowpass filter with the frequency response H ( Ω ) illustrated
in Fig. 14.4. The magnitude spectrum H ( Ω ) of the filter is shown by a solid
line, while the phase spectrum < H ( Ω ) is shown by a dashed line. The filter
has a cut-off frequency Ω c , such that Ω 2 < Ω c < Ω 3 , and the cut-off frequency
lies within the transition band. Based on the frequency response H ( Ω ) shown

H ( W )

stop
band

stop
band

pass band

w

− p

− W 3

0

− W 2 − W 1 W 1 W 2 W 3 p

Fig. 14.4. Physically realizable
lowpass filter with transition
bands and non-zero phase.

transition
band

transition
band