Digital Signal Processing Reference
In-Depth Information
20
0
-20
-40
Cascade
Filter L = 64
-60
-80
-100
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalized Frequency,
F
Figure 2.21
Effect of cascading notch filter with itself.
2.7.2 Alternative Cascade Technique
The alternative technique to generate filter cascade coefficients is to apply (2.11)
directly. To do this, we must first zero-pad the shorter of the two filters, say
h
, to
create an initial length of 2
L
+1. The parameter
x
in (2.11) is replaced by
h
to give
1
∑
+
=
L
1
(2.27)
h
=
h
h
m
=
1
2
,
(
L
+
1
m
k
k
+
m
1
K
k
1
This gives identical results to (2.25); however the number of operations associated
with (2.25) is
L
(
L
+1) whereas in (2.27) it is (2
L
+1)(
L
+1). The reduction
associated with (2.25) results from the exclusion of all operations involving the
zero values that are used for zero padding in (2.27).
2.8 DECIMATION
In many applications, there is a requirement to reduce the number of data, or
consequently, the sampling rate by a factor of say,
M
, for example, to a new
sampling rate of
f
s
/
M
. The obvious way to do this is to discard (
M
- 1) samples
from successive groups of
M
samples, a process known as
decimation
or
downsampling
. However, this process could lead to aliasing of higher frequency
signals down into the working frequency band. To counter this effect,
downsampling is usually combined with the filtering operation. Decimation of the
measurements by
M
results in the lowering of the Nyquist frequency to
f
s
/2
M
and
consequently a shift in the position of the normalized frequency components.
Downsampling could also be carried out in either the time or frequency domain.
In this section, a very brief description of the decimation process will be given,
but for further details, refer to reference works such as [3,5].
