Digital Signal Processing Reference
In-Depth Information
The variance of the error filter frequency response r
2
can then be calculated as
2
4
3
5
x
M
3
2
1
þ
4
X
M
1
2
r
Q
¼Ef
Q
ð
x
Þ
Q
H
ð
x
Þg ¼
r
q
cos
2
m
m
¼
0
ð
4
:
24
Þ
M
1
þ
sin
ð
Mx
Þ
sin
ð
x
Þ
¼
r
q
where r
2
q
is the variance of the error as a result of coefficients quantization. It can
be shown that (
4.24
) reduces to
r
Q
r
q
ð
2M
1
Þ;
ð
4
:
25
Þ
(See Ref. [
2
])
Equation
4.25
gives us a bounding formula for the error in the frequency
response. Assume now that each coefficient is quantized to b bits via rounding, and
that there is a uniformly distributed error in each of the frequency domain coef-
ficients given by r
q
¼
D
2
=
12, where D
¼
Full-scale
=
2
b
is the quantization step.
With these assumptions, the error bounding relation becomes
r
Q
ð
2M
1
Þ
D
2
ð
4
:
26
Þ
12
The bound given in (
4.26
) is useful for estimating accuracies in filter design. It
can be seen that for a certain level of tolerable error in the frequency response, the
larger the FIR filter length M the finer the quantization step D should be. Similarly,
as in the case of high-order IIR filters, improved high-order FIR filter operation
can be achieved by realizing the filter as a cascade of short (second-order) sections
rather than with one large FIR filter. However, as usual, the price paid to obtain the
cascade structure is more multiplication operations compared to that of the direct
form structure. Additionally, the errors induced by quantization are less dangerous
than in the case of IIR filters, because they do not have the potential to send the
filter into unstable modes.
MATLAB: The Matlab object
dfilt
provides a convenient means to realize
and simulate both FIR and IIR filters in a variety of structures. The possible
structures include direct-form, second-order sections, lattices, and state-space
formulations. Readers interested in further detail are referred to Mitra [
1
].
4.5 Quantization Errors in Arithmetic Operations
The key elemental processing operations used in DSP are multiplication and
accumulation. Note that accumulation is effectively addition to an existing sum.
As these mathematical operation are commonly carried out using fixed-point
digital machines, additional errors can be introduced with each operation. As each
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