Digital Filters - Signal Processing of Random Physiological Signals - page 116

The “Transfer Function” or describing equation for the first-order Butterworth
high-pass filter is given by (11.6) for the s -domain and rewritten in (11.7) in terms of
frequency. It should be noted that the high-pass filter is not an “All Pole Filter,” since the
equations show a zero or an s in the numerator of the equations. Hence, in the s -plane a
zero exists at the origin, s = 0.

A hp s

Ks

H ( s ) =

+ ω b =

(11.6)

+ ω b s s

A hp s

H ( j ω ) = (11.7)

1
R 1 C

s +

where A hp = high-pass gain.

11.1.4 2nd-Order Butterworth High-Pass Filter
The transfer function for a second-order Butterworth high-pass filter is given by (11.8):

A hp bs 2

H ( s ) = (11.8)

ω

c

a
b

s 2

+ ω c s +

b

11.1.5 Band-Pass Filters
Band-pass filters may be of the Butterworth or of the Chebyshev class. Band-pass filters
are designed to pass a band of frequencies of bandwidth, B , with the center of the
band around a center frequency ( ω 0 rad/sec) or f 0 = ω 0 / 2 π Hz. Band-pass filter must

designate two cutoff frequencies: a lower cutoff, U .

The frequencies that are passed by the filter are those for which the transfer
function, H ( s ), gain is greater than or equal to 0.707 A 0 as shown in Fig. 11.5.

ω L , and an upper cutoff, ω

11.1.5.1 Quality Factor
Quality Factor ( Q ) is a measure of narrowness of the band-pass filter. By definition, the
quality factor, Q , is the ratio of the center frequency, ω 0 , to the bandwidth, B = ω U − ω L

in radians per second; hence, the ratio may be expressed as Q B .If

the quality factor, Q , is greater than 10, the band-pass region is considered to be narrow.
The gain of the band-pass filter is the magnitude at the center frequency. The transfer
function for the band-pass filter is given by (11.9). Band-pass filters must be of second

= ω / B or Q = f 0 / 0