Signals and Systems - Electrical Engineering

11.3 Filter Specifications 649

FIGURE 11.7
Lossα(dB) for
normalized gain|G|.

10 −^5

10 −^4

10 −^3

10 −^2

10 −^1

100

Loss α (dB)

Magnitude |

G

|

0 10 20 30 40 50 60 70 80 90 100

Defining theloss functionfor a discrete filter as

α(ejω)=−10 log 10 |H(ejω)|^2 =−20 log 10 |H(ejω)| dB (11.12)

equivalent magnitude specifications for a discrete low-pass filter are (Figure 11.6(b)):

Passband: 0 ≤α(ejω)≤αmax 0 ≤ω≤ωp

Stopband: αmin≤α(ejω) <∞ ωst≤ω≤π (11.13)

whereαmax=−20 log 10 δ 1 andαmin=−20 log 10 δ 2 (which are positive since bothδ 1 andδ 2 are positive

and smaller than 1 ).

nExample 11.3

Consider the following specifications for a low-pass filter:

0.9≤|H(ejω)|≤1.0 0 ≤ω≤π/ 2

0 <|H(ejω)|≤0.1 3 π/ 4 ≤ω≤π

Determine the equivalent loss specifications.