Signals and Systems - Electrical Engineering

11.3 Filter Specifications 651

FIGURE 11.8

Not-normalized loss specifications for a

low-pass filter.

ω

α(ejω)(dB)

α 3

α 2

α (^1) ω π
p ωst
thedcloss ofα 1 is achieved by multiplying a magnitude-normalized filter by a constantK, such that
α(ˆej^0 )=α 1 =−20 log 10 K or K= 10 −α^1 /^20 (11.15)
The normalized specifications are then
0 ≤α(ejω)≤αmax 0 ≤ω≤ωp
αmin≤α(ejω) ωst≤ω≤π
whereαmax=α 2 −α 1 andαmin=α 3 −α 1.
nExample 11.4
Suppose the loss specifications of a low-pass filter are
10 ≤ ˆα(ejω)≤ 11 0 ≤ω≤
π
2
α(ˆ ejω)≥ 50
3 π
4
≤ω≤π
Determine the loss specifications that can be used to design a magnitude-normalized filter, and
the gainKto make it satisfy the above specifications.
Solution
If we let
α(ˆejω)= 10 +α(ejω)
the loss specifications for a normalized filter would be
0 ≤α(ejω)≤ 1 0 ≤ω≤
π
2
α(ejω)≥ 40
3 π
4
≤ω≤π