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308 Chapter 7 FIR FILTER DESIGN
Symmetric Impulse Response: M odd
h(n)
0
0 (M – 1)/2 (M – 1)
n
where α is a constant phase delay. Then we know from Chapter 6 that
h(n) must be symmetric, that is,
h (n) =h(M − 1 − n), 0 ≤ n ≤ (M − 1) with α = M − 1
2
(7.3)
Hence h(n) issymmetric about α, which is the index of symmetry. There
are two possible types of symmetry:
• M odd: In this case α =(M − 1)/2 isaninteger. The impulse response
is as shown below.
• M even: In this case α =(M − 1)/2 isnot an integer. The impulse
response is as shown here.
Symmetric Impulse Response: M even
h(n)
0
0 M/2 + 1 M/2 M – 1
n
We also have a second type of “linear-phase” FIR filter if we require
that the phase response ̸ H(e jω ) satisfy the condition
̸ H(e jω )=β − αω
which is a straight line but not through the origin. In this case α is not a
constant phase delay, but
d̸ H(e jω )
dω
= −α
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