The frequency response
H
f
ω
H
f
ω
of an FIR filter can be evaluated at
LL equally spaced frequencies
between 0 and π using the DFT.
Consider a causal FIR filter with an impulse response $h(n)$
of length-NN, with
N≤L
N
L
. Samples of the frequency
response of the filter can be written as
H2πLk=∑n=0N−1hne(−i)2πLnk
H
2
L
k
n
0
N
1
h
n
2
L
n
k
Define the LL-point signal
gn
0≤n≤L−1
g
n
0
n
L
1
as
gn={hn if 0≤n≤N−10 if N≤n≤L−1
g
n
h
n
0
n
N
1
0
N
n
L
1
Then
H2πLk=Gk=
DFT
L
gn
H
2
L
k
G
k
DFT
L
g
n
where
Gk
G
k
is the LL-point DFT of
gn
g
n
.

Suppose the FIR filter
hn
h
n
is either a Type I or a Type II FIR filter. Then
we have from above