Digital Signal Processing Reference
In-Depth Information
N
Zero-padded
Zero-padded
Filter
h
h
h
h
h
2
L
L
L
+
1
1
2
+
FFT
0
-20
-40
(
a
)
-60
-80
-100
(
b
)
-120
-140
-160
(
c
)
-180
-200
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalized Frequency,
F
Figure 2.4
FFT of impulse response function
H
k
. (a) No windowing and no truncation. (b) Gaussian
windowed and truncated to order
L
= 98. (c) Gaussian windowed, but no truncation.
response affected when both windowing and truncation are applied to
H
k
? In
curve (b) of Figure 2.4,
H
k
is both Gaussian windowed and truncated to
L
+ 1 =
99, but zero-padded to array length
N
= 1024 for FFT purposes as noted earlier.
Clearly, there is a significant loss of attenuation in comparison to (c) of about 80
dB. Curve (a) in Figure 2.4 is identical to that given in Figure 2.1(c) and is used
for comparison: it represents a Gaussian window of infinite width which
corresponds to an infinitely wide rectangular window. Moreover, curve (
a
) serves
to highlight the trade-off between filter edge transition sharpness and relatively
poor attenuation on one hand, and the softer transition edge with relatively
good attenuation on the other. Thus with standard precision floating-point
arithmetic, Gaussian windowing and truncation give a good compromise between
