Digital Signal Processing Reference
In-Depth Information
Fig. 2.20 Efficient imple-
mentation of a linear-phase
FIR digital filter with sym-
metric impulse response
h(n) a with odd length
N = 2M ? 1, b with even
length N = 2M
x
(
n
)
z
−1
z
−1
z
−1
z
−1
z
−1
z
−1
h
M−1
h
o
h
1
h
2
h
M
y
(
n
)
(a)
x
(
n
)
z
−1
z
−1
z
−1
z
−1
z
−1
z
−1
z
−1
h
M−2
h
M−1
h
o
h
1
h
2
y
(
n
)
(b)
Since there are non-zero values of h(n) for for n \ 0, the ideal LPF is non-
causal, and is hence physically unrealizable. It cannot therefore be used to process
real-time signals. In addition,
P
j
h
L
ð
n
Þj!1;
and so the filter is also unstable.
Since -?\ n \?, the ideal LPF is an IIR filter. A practical 2M ? 1 sample
FIR approximation to the ideal LPF can be obtained by first shifting h
L
(n) right by
M samples to get H
1
(n) = h
L
(n - M). Note that this time-shift causes only a phase
shift in the frequency domain (see Tables, Fourier Transform Properties), and
hence |H
1
(f)| = |H
L
(f)|. Second, h
1
(n) needs to be truncated (symmetrically around
n = M) by putting h
1
(n) = 0 for n \ 0orn [ 2M (total length is N = 2M ? 1).
This process yields a finite impulse response h
Lt
(n), which is symmetric about
n = M, as shown in Fig. (
2.22
) for the scenario where M = 5, N = 11, f
c
= 1.25,
and f
s
= 5 Hz.
As would be expected intuitively, larger values of M tend to give rise to
better approximations of the ideal impulse response. It is important to note that
when the original infinite length time domain impulse response is truncated, the
Search WWH ::
Custom Search