Digital Signal Processing Reference
In-Depth Information
thoroughly analyzed in the next section. We can already say, however, that
while the minimum square error is an integral criterion, the minimax is a
pointwise criterion; or, mathematically, that the MSE and the minimax are
respectively
L
2
[
−
π
,
π
]
-and
L
∞
[
−
π
,
π
]
-norm minimizations for the er-
ror function
E
l
g
r
,
y
i
d
.
,
©
,
L
s
(
ω
)=
H
e
j
ω
)
−
e
j
ω
)
. Figure 7.11 illustrates the typical result
of applying both criteria to the ideal lowpass problem. As can be seen, the
minimum square and minimax solutions are very different.
(
H
(
1
1
0
0
0.4
π
0.4
π
0
π/
2
0
π/
2
Figure 7.11
Error shapes in passband for MSE and minimax optimization meth-
ods.
7.2.2 Minimax FIR Filter Design
As we saw in the opening example, FIR filter design by windowing mini-
mizes the overall mean square error between the desired frequency response
and the actual response of the filter. Since this might lead to a very large er-
ror at frequencies near the transition band, we now consider a different ap-
proach, namely the design by minimax optimization. This technique mini-
mizes the maximum allowable error in the filter's magnitude response, both
in the passband and in the stopband. Optimality in the minimax sense re-
quires therefore the explicit stating of a set of
tolerances
in the prototypi-
cal frequency response, in the form of design specifications as seen in Sec-
tion 7.1.2. Before tackling the design procedure itself, we will need a series
of intermediate results.
Generalized Linear Phase.
In Section 5.4.3, we introduced the concept
of linear phase; a filter with linear phase response is particularly desirable
since the phase response translates to just a time delay (possibly fractional)
and we can concentrate on the magnitude response only. We also intro-
duced the notion of group delay and showed that linear phase corresponds
to constant group delay. Clearly, the converse is not true: a frequency re-
sponse of the type
e
j
ω
)=
H
e
j
ω
)
e
−j
ω
d
+
j
α
H
(
(
