Page 205 - Mechanical Engineers' Handbook (Volume 2)

194 Signal Processing

Figure 4 Frequency response to Butterworth filters of order 2, 5, and 11.

2.2 Tchebyshev
Unlike the Butterworth rational function, the Tchebyshev (Type I) rational function permits
some ripple to occur in the passband (frequencies less than ƒ in the low-pass filter), in
c
exchange for a sharper transition region compared to a Butterworth filter of equal order N,
as seen in Fig. 5:
1
2

H(s)
(10)
22
1 T (s/s )
N
c
Where T is the Tchebyshev polynomial defined as
N
1
T (s) cos[n cos (s)] (11)
N
The pole placements p + j for the Tchebyshev Type I filter are
i
i
i
2 ƒ sinh(v ) cos (N 2i 1) i 0,..., N 1
i
0
c
2 ƒ cosh(v ) sin 2N (N 2i 1) (12)

i c 0 i 0,..., N 1
2N
where
1
1 sinh
v 1 (13)
0
N
1
1 0 r 1 (14)
(1 r) 2
where r is this amplitude of the ripple in proportion to the gain of the passband.