Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 659

why this is an efficient approach consider the following issues that derive from the application of the
bilinear transformation to the Butterworth design:

n Since the discrete magnitude specifications are not changed by the bilinear transformation,
we only need to change the analog frequency term in the formulas obtained before for the
Butterworth low-pass analog filter.
n It is important to recognize that when finding the minimal orderNand the half-power relation
the value ofKis not used. This constant is only important in the final step where the analog filter
is transformed into the discrete filter using the bilinear transformation.
n When considering thatK= 2 /Tsdepends onTs, one might think that a small value forTs
improves the design, but that is not the case. Given that the analog frequency is related to the
discrete frequency as

=

2

Ts

tan

(ω

2

)

(11.30)

for a given value ofωif we choose a small value ofTsthe specified analog frequency would
increase, and if we choose a large value ofTsthe analog frequency would decrease. In fact, in
the above equation we can only choose eitherorTs. To avoid this ambiguity, we ignore the
connection ofKwithTsand concentrate onK.
n An appropriate value forKfor the Butterworth design is obtained by connecting the normalized
half-power frequency

′

hp=1 in the analog domain with the corresponding frequencyωhpin

the discrete-domain. This allows us to go from the discrete-domain specificationsdirectlyto the

analog normalized frequency specifications. Thus, we map the normalized half-power frequency

′hp=1 into the discrete half-power frequencyωhp, by means ofKb.

n Once the analog filterHN(s)is obtained, using the bilinear transformation with theKbwe
transformHN(s)into a discrete filter

HN(z)=HN(s)

∣

∣

∣s=Kbz− 1

z+ 1

n The filter parameters(N,ωhp)can also be obtained directly from the discrete loss function

α(ejω)=10 log

[

1 +(tan(0.5ω)/tan(0.5ωhp))^2 N

]

(11.31)

and the loss specifications

0 ≤α(ejω)≤αmax 0 ≤ω≤ωp

α(ejω)≥αmin ω≥ωst

just as we did in the continuous case. The results coincide with those where we replace the warping

frequency relation.

nExample 11.6

The analog signal

x(t)=cos( 40 πt)+cos( 500 πt)