Signals and Systems - Electrical Engineering

6.5 Analog Filtering 397

definition is used. Likewise, whenever|′|≤1, the definition based in the hyperbolic cosine would
not be possible since the inverse of this function only exists for values of′bigger or equal to 1 and
so the cos(.)definition is used. From the definition it is not clear thatCN(′)is anNth-order poly-
nomial in′. However, if we letθ=cos−^1 (′)or′=cos(θ)when|′|≤1, we have thatCN(′)=
cos(Nθ)and

CN+ 1 (′)=cos((N+ 1 )θ)=cos(Nθ)cos(θ)−sin(Nθ)sin(θ)

CN− 1 (′)=cos((N− 1 )θ)=cos(Nθ)cos(θ)+sin(Nθ)sin(θ)

so that adding them we get

CN+ 1 (′)+CN− 1 (′)=2 cos(θ)cos(Nθ)= 2 ′CN(′)

This gives a three-term expression for computingCN(′), or a difference equation

CN+ 1 (′)+CN− 1 (′)= 2 ′CN(′) N≥ 0 (6.48)

with initial conditions

C 0 (′)=cos( 0 )= 1

C 1 (′)=cos(cos−^1 (′))=′

We can then see that

C 0 (′)= 1

C 1 (′)=′

C 2 (′)=− 1 + 2 ′^2

C 3 (′)=− 3 ′+ 4 ′^3

..

.

which are polynomials in′of orderN=0, 1, 2, 3,.... In Chapter 0 we gave a script to compute
and plot these polynomials using symbolic MATLAB.

Remarks

n Two fundamental characteristics of the CN(′)polynomials are: (1) they vary between 0 and 1 in the
range′∈[−1, 1], and (2) they grow outside this range (according to their definition, the Chebyshev
polynomials outside this range becomecosh(.)functions, which are functions always bigger than 1). The
first characteristic generates ripples in the passband, while the second makes these filters have a magnitude
response that goes to zero faster than Butterworth’s.
n There are other characteristics of interest for the Chebyshev polynomials. The Chebyshev polynomials are
unity at′= 1 (i.e., CN( 1 )= 1 for all N). In fact, C 0 ( 1 )=1,C 1 ( 1 )= 1 , and if we assume that
CN− 1 ( 1 )=CN( 1 )= 1 , we then have that CN+ 1 ( 1 )= 1 according to the three-term recursion. This
indicates that the magnitude-square function is|HN(j 1 )|^2 = 1 /( 1 +ε^2 )for any N.