Signals and Systems - Electrical Engineering

398 C H A P T E R 6: Application to Control and Communications

n Different from the Butterworth filter that has a unit dc gain, the dc gain of the Chebyshev filter depends

on the order of the filter. This is due to the property of the Chebyshev polynomial of being|CN( 0 )|= 0 if

N is odd and 1 if N is even. Thus, the dc gain is 1 when N is odd, but 1 /

√

1 +ε^2 when N is even. This

is due to the fact that the Chebyshev polynomials of odd order do not have a constant term, and those of

even order have 1 or− 1 as the constant term.

n Finally, the polynomials CN(′)have N real roots between− 1 and 1. Thus, the Chebyshev filter displays

N/ 2 ripples between 1 and

√

1 +ε^2 for normalized frequencies between 0 and 1.

Design

The loss function for the Chebyshev filter is

α(′)=10 log 10

[

1 +ε^2 C^2 N(′)

]

′=



p

(6.49)

The design equations for the Chebyshev filter are obtained as follows:

n Ripple factorεand ripple width (RW): FromCN( 1 )=1, and letting the loss equalαmaxat that

normalized frequency, we have that

ε=

√

10 0.1αmax− 1

RW= 1 −

1

√

1 +ε^2

(6.50)

n Minimum order: The loss function at

′

sis bigger or equal toαmin, so that solving for the Chebyshev

polynomial we get after replacingε,

CN(

′

s)=cosh(Ncosh

− (^1) (′
s))

≥

(

10 .1αmin− 1

10 .1αmax− 1

)0.5

where we used the cosh(.)definition of the Chebyshev polynomials since

′

s>1. Solving forN

we get

N≥

cosh−^1

([

10 0.1αmin− 1

10 0.1αmax− 1

]0.5)

cosh−^1

(

s

p

) (6.51)

n Half-power frequency: Letting the loss at the half-power frequency equal 3 dB and using that 100.3≈

2, we obtain from Equation 6.49 the Chebyshev polynomial at that normalized frequency to be

CN(

′

hp)=

1

ε

=cosh

(

Ncosh−^1 (

′

hp)

)