Page 412 - Analog and Digital Filter Design

APPENDIX

DESIGN EQUATIONS

Bessel Transfer Function

The Bessel response is produced by a transfer function that is derived from
Bessel polynomials, and using the Bessel transfer function produced the graph
in Figure 2.6.

As previously stated, the Bessel response is produced from a time-delay func-
tion. The time-delay for all filter orders is normalized to one second, which
results in a frequency response that is dependent on the order, n. The transfer
function for a pure delay is given by:

H(s) = e-.'', and where normalization gives T = 1, and H(sj = e-.'
1
H(s) = e-' =
sinh(s) + coslz(s)

Hyperbolic sine and cosine functions can be expressed as a series, with the sine
functions having even powers of s and the cosine function having odd powers
of s. The transfer function, H(s) then becomes a simple polynomial.

n
B,,(s) is the Bessel polynomial, B,, (s) = a,s'
1 4
This looks complex but consider values of B,,(s) for orders up to three:

Bo(s) = 1
B,(s) = s+l
B2(S) = s1+ 3s + 3
B3(s)=s3+6s'+15~+15