Digital Signal Processing Reference
In-Depth Information
1
2
(
b
)
(
a
)
Input time series
Filter output
0.8
1.5
Input series
0.6
1
0.4
0.5
0.2
0
0
-0.2
-0.5
-0.4
-1
-0.6
-1.5
Filter output
-0.8
Start of Filter output at
L
/2 = 49
-2
-1
0
100
200
300
400
500
0
30
60
90
120
150
Time
Time
Time
Time
5
2
(
d
)
(
c
)
4
1.6
3
1.2
(0.02, 6.7x10
-6
)
2
0.8
1
0.4
0
0
-1
-0.4
Effective
Bandwidth for
precision work
-2
-0.8
-3
-1.2
-4
-1.6
-5
-2
0
100
200
300
400
0
0.05
0.1
0.15
0.2
0.25
Normalized Frequency
Normalized Frequency
Time
Time
Figure 2.14
Filtering in the frequency domain. (a) Both frequency components passed. (b) Low-pass
component passed with rejection of higher component. (c) Error between predicted and filtered output.
(d) Pass-band error of filter showing effective bandwidth for precision work.
band-reject (band-stop), and other filters will be built up from the impulse
response of the low-`pass filter.
2.6.1 The All-Pass Filter
The all-pass filter is derived directly from the low-pass filter. It has a cutoff at the
Nyquist frequency, which means that it exhibits a flat characteristic in the pass
band of magnitude 1 (0 dB). The phase characteristics are linear, as for all FIR
filters, and cover the entire Nyquist half-space, 0
1. The discrete impulse
response for the all-pass filter
h
AP
is given in terms of the low-pass impulse
response by
F
