Digital Signal Processing Reference
In-Depth Information
and, for band 10 (
i
=
10),
=
(
)(
)
+
(
)(
)
+
(
)(
)
HHLHLHL
HHLHLHL
0
l
0
28
m
0
29
h
0
30
=
(
)(
)
+
(
)(
)
+
(
)(
)
1
l
1
28
m
1
29
h
1
30
M
=
(
)(
)
+
(
)(
)
+
(
)(
)
HHL HL HL
40
l
40
28
m
40
29
h
40
30
For an efficient design with the multirate technique, lower-octave bands are
processed at a lower sampling rate, then interpolated up to a higher sampling rate,
by a factor of 2, to be summed with the next higher octave band filter output, as
shown in Figure 10.60. Each interpolation filter is a 21-coefficient FIR lowpass filter,
with a cutoff frequency of approximately one-fourth of the sampling rate. For each
input, the interpolation filter provides two outputs, or
yxI IxI I
=
+
0
+
+
0
+◊◊◊+
xI
1
0
0
1
1
2
3
10
20
yI
=
0
+
xI
+
0
I
+
I
+◊◊◊+
xI
2
0
0
1
2
1
3
9
19
where
y
1
and
y
2
are the first and second interpolated outputs, respectively,
x
n
are
the filter inputs, and
I
n
are the interpolation filter coefficients. The interpolator is
processed in two sections to provide the data-rate increase by a factor of 2.
For the multirate filter, the approximate number of multiplication operations
(with accumulation) per second is
(
)
(
)
MAC S
=+
41
21 32
++ + + +
64
128
256
512
1 024
,
+
2 048
,
+
4 096
,
+
8 192
,
+
( (
)
41 16 384
1 686
,
.
¥
10
6
The approximate number of multiplications/accumulation per second for an
equivalent single-rate filter is then
(
)
=¥
MAC S
=¥
F
s
41 1
+++++
2
2
2
2
3
...
2
9
687
10
6
which would considerably increase the processing time requirements.
A brief description (recipe) of the main processing follows, for the first time
through (using three buffers
B
1
,
B
2
,
B
3
).
Band 1
1.
Run the bandpass filter and obtain one output sample.
2.
Run the lowpass interpolation filter twice and obtain two outputs. The inter-
polator provides two sample outputs for each input sample.
3.
Store in buffer
B
2
, size 512, at locations 1 and 2 (in memory).
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