Digital Filters - Signal Processing of Random Physiological Signals - page 114

A Ideal low-pass filter

A 0

Practical low-pass filter

Ripple width

w

w c

FIGURE 11.2 : Chebyshev low-pass characteristic transfer function response curves. The ideal
low-pass filter is shown as the dash-line trace, and the normal practical response trace is shown
as the dotted-line trace. Note the ripple in the band-pass region

or the input is large. Hence, the term that the “filter rings,” meaning it oscillates during
it damping phase. The Chebyshev filter response best approaches the “Ideal filter” and
is most accurate near the filter cutoff frequency, but at the expense of having ripples in
the band-pass region. It should be noted that the filter is monotonic in the stop-band as
is the Butterworth low-pass filter. Figure 11.2 shows the ideal response of a Chebyshev
low-pass filter with its cutoff frequency at

ω c and the practical response for the same

cutoff frequency.
The “Transfer Function” or describing equation for the Butterworth low-pass filter
is given by (11.4) in terms of frequency.

H ( j

) =

A lp

ω 1 (11.4)

2 C n ω c

+ ε

where ε is a constant, and Cn is the Chebyshev polynomial, Cn ( x ) = cos[ n cos − 1( x )]

The ripple factor is calculated from (11.5).

1

RW = 1 − √ 1 (11.5)

+ ε

2

Figure 11.3 shows the response of a third-order Chebyshev Voltage Controlled
Voltage Source (VCVS) low-pass filter.