Digital Signal Processing Reference
In-Depth Information
Consider the FIR direct-form structure shown in Fig.
4.13
(a), which assumes
that every coefficient multiplier output is quantized before the addition operation is
performed. According to the linear noise model shown in Fig.
4.13
(b), the total
noise will be equal to the superposition of all the noise components. Accordingly,
the output noise variance is given by
r
t
¼
Mr
q
;
ð
4
:
36
Þ
where r
2
q
is the quantizer noise variance. Recall that under the assumption of
round-off based arithmetic, r
2
q
= 2
-
2b/12. For a symmetric linear-phase FIR
filter, the output noise variance would be approximately half of that value, as the
number of required multipliers is halved.
As the input to each multiplier branch in the FIR structure is just a delayed
version of the original input signal x(n), the coefficients could be scaled if an
overflow is expected. Otherwise, only the output node needs to be scaled. Cascade
form implementation of FIR filters is reduced to ordering of zero-pair sections
(since FIR filters have no poles), rather than pairing and ordering, as was the case
for IIR filters.
x
(
n
)
z
-
z
-
z
-
z
-
1
1
1
1
h
(
0
)
h
(
h
(
2
h
(
M
1
h
(
M
2
)
Q
Q
Q
Q
Q
+
+
+
+
+
y
(
n
)
(a)
x
(
n
)
z
-
z
-
z
-
z
-
1
1
1
1
h
(
0
)
h
(
h
(
2
h
(
M
1
h
(
M
2
)
+
+
+
+
+
e
0
n
)
e
1
n
)
e
M
1
n
)
e
2
n
)
e
M
2
n
)
-
-
+
+
+
+
+
y
(
n
)
(b)
Fig. 4.13 Direct-form implementation of FIR filter, a the effect of arithmetic quantization, b the
linear noise model
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