Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 667

Replacing

′=



p

=Kctan(0.5ω)=

tan(0.5ω)

tan(0.5ωp)

(11.33)

into the magnitude-squared function for the Chebyshev analog filter it yields the magnitude-squared function

of the discrete Chebyshev low-pass filter,

|HN(ejω)|^2 =

1

1 +ε^2 C^2 N(tan(0.5ω)/tan(0.5ωp))

(11.34)

whereC(.)are the Chebyshev polynomials of the first kind encountered before in the analog design. The ripple

parameter remains the same as in the analog design (since it does not depend on frequency):

ε=( 10 0.1αmax− 1 )^1 /^2 (11.35)

while using the warping relation between the continuous and discrete frequencies gives that the minimal

order of the filter is

N≥

cosh−^1 [( 10 0.1αmin− 1 )/( 10 0.1αmax− 1 )]^1 /^2

cosh−^1 [tan(0.5ωst)/tan(0.5ωp)]

(11.36)

and that the half-power frequency can be found to be

ωhp=2 tan−^1

[

tan(0.5ωp)cosh

(

1

N

cosh−^1

(

1

ε

))]

(11.37)

After calculating these parameters, the transfer function of the Chebyshev discrete filter is found by

transforming the Chebyshev analog filter of orderNinto a discrete filter using the bilinear transformation

HN(z)=HN(s)|s=Kc( 1 −z− (^1) )/( 1 +z− (^1) ) (11.38)
Remarks
n Just as with the Butterworth filter, the equations for the filter parameters(N,ωhp)can be obtained from
the analog formulas by substituting
st
p
=Kctan(0.5ωst)=
tan(0.5ωst)
tan(0.5ωp)
n The filter parameters(N,ωhp,ε)can also be found from the loss function, obtained from the discrete
Chebyshev squared magnitude,
α(ejω)=10 log 10

[

1 +ε^2 C^2 N

(

tan(0.5ω)

tan(0.5ωp)

)]

(11.39)

This is done by following a similar approach to the one in the analog case.