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Bandpass Signaling Principles and Circuits Chap. 4
TABLE 4–3
Type
Butterworth
Chebyshev
Bessel
SOME FILTER CHARACTERISTICS
Optimization Criterion
Maximally flat: as many
derivatives of |H(f)| as
possible go to zero
as f → 0
For a given peak-to-peak
ripple in the passband of
the |H(f)| characteristic, the
|H(f)| attenuates the fastest
for any filter of nth order
Attempts to maintain
linear phase in the
passband
|H(f)| =
|H(f)| =
Transfer Characteristic for the
Low-Pass Filter a
1
21 + (f/f b ) 2n
1
21 + e 2 C n 2 (f/f b )
e = a design constant; C n (f) is the nth-order
Chebyshev polynomial defined by the recursion
relation C n (x) = 2xC n-1 (x) - C n-2 (x),
where C 0 (x) = 1 and C 1 (x) = x
K n
H(f)
B n (f/f b )
K n is a constant chosen to make H (0) = 1,
and the Bessel recursion relation is
B n (x) = (2n - 1) B n-1 (x) - x 2 B n-2 (x),
where B 0 (x) = 1 and B 1 (x) = 1 + jx
a f b is the cutoff frequency of the filter.
filter characteristics and the optimization criterion that defines each one. The Chebyshev
filter is used when a sharp attenuation characteristic is required for a minimum number of
circuit elements. The Bessel filter is often used in data transmission when the pulse shape is
to be preserved, since it attempts to maintain a linear phase response in the passband.
The Butterworth filter is often used as a compromise between the Chebyshev and Bessel
characteristics.
The topic of filters is immense, and not all aspects of filtering can be covered here. For
example, with the advent of inexpensive microprocessors, digital filtering and digital signal
processing are becoming very important [Oppenheim and Schafer, 1975, 1989].
For additional reading on analog filters with an emphasis on communication system
applications, see Bowron and Stephenson [1979].
Amplifiers
For analysis purposes, electronic circuits and, more specifically, amplifiers can be classified
into two main categories: Nonlinear and linear. Linearity was defined in Sec. 2–6. In practice,
all circuits are nonlinear to some degree, even at low (voltage and current) signal levels, and
become highly nonlinear for high signal levels. Linear circuit analysis is often used for the
low signal levels, since it greatly simplifies the mathematics and gives accurate answers if the
signal level is sufficiently small.