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310 Chapter 7 FIR FILTER DESIGN
7.2.2 FREQUENCY RESPONSE H(e jω )
When the cases of symmetry and antisymmetry are combined with odd
and even M, weobtain four types of linear-phase FIR filters. Frequency
response functions for each of these types have some peculiar expressions
and shapes. To study these responses, we write H(e jω )as
H(e jω )=H r (ω)e j(β−αω) ; β = ± π 2 ,α= M − 1
2
(7.5)
where H r (ω) isanamplitude response function and not a magnitude response
function. The amplitude response is a real function, but unlike
the magnitude response, which is always positive, the amplitude response
may be both positive and negative. The phase response associated with
the magnitude response is a discontinuous function, while that associated
with the amplitude response is a continuous linear function. To illustrate
the difference between these two types of responses, consider the following
example.
□ EXAMPLE 7.3 Let the impulse response be h(n) ={1, 1, 1}. Determine and draw frequency
↑
responses.
Solution
The frequency response function is
H(e jω )=
2∑
h(n)e jωn = 1+1e −jω + e −j2ω = { e jω +1+e −jω} e −jω
0
= {1+2cos ω} e −jω
From this the magnitude and the phase responses are
|H(e jω )| = |1+2cosω| , 0