U. Glaeser

TABLE 26.1 Filter Design Comparison
Filter # Zeros # Poles Mag. Error GD. Error RMS GD. Error
Elliptic (Fig. 26.1) 6 6 0.082 13.90 3.57
Equalized elliptic (Fig. 26.3) 31 6 0.079 3.15 0.76
IIR (Fig. 26.4) 31 6 0.015 2.40 0.32
Note: The table shows three features: maximal magnitude error, maximal group delay error, and RMSs of the
group delay error, of the optimal response of the filter in Figs. 26.3 and 26.4 designed under different approaches.
|H |
1
FIGURE 26.4 Filter with optimal general weighted norm. The figures show the magnitude and the group delay of
an IIR frequency selective filter (31 zeros, 6 poles) with flat delay passband. The ideal filter is the same as in Fig. 26.3,
and the filter order is the same as the equalized elliptic filter.
Several optimization techniques are available to solve these general problems. In addition, the error
can be controlled by the selection of an error constraint, an error weight, or a design norm; however,
the optimization of general norms is often a difficult problem, especially in the complex domain. Most
recent research has studied these general norm problems in order to improve the design when the goal
is a simultaneous approximation of the magnitude and phase.
Filter Design as a Norm Problem
Filter design is usually done by minimizing either the worst-case error (Chebyshev norm), or the root mean
squares (RMS) (least-squares norm) of the weighted error. Important norms from classical mathematics
are listed below:
• Chebyshev norm: E ∞ = max
ω E(ω )
• Least-squares norm: =
• p-norm: = for p ∈ [1, ∞]
• Combined norm: E α = { α E ∞ + (1 – α) E 2}
for α ∈ [0,1]
where E = W(I − H) is the weighted complex error. When optimizing the Chebyshev norm, the resulting
optimal filters have the smallest maximal error, while filters with minimal least-squares norm have the
smallest RMS error. Preference for one norm over the other will generally depend on the application. In
many cases, where both norms need to be small, filters should be designed under either the p-norm or
the combined norm. Along with the norm, the numerical optimization can be done under design
constraints, e.g., the most obvious one is a constraint on the magnitude of the error
where ε(ω) is the error constraint.
Figure 26.5 shows the error of the filter with the same specification designed under four different
norms. The RMS and maximal errors are summarized in Table 26.2.
© 2001 by CRC Press LLC
0 ω ω
c1
c2 π
ω
D
g
D
I
π 2 1/2
E 2 {∫−π E(ω ) dω}
π p 1/p
E p {∫−π E(ω ) dω}
2
min E such that E(ω ) ≤
ε(ω )
0 ω ω
c1
c2 π
2
ω