582 Practical MATLAB® Applications for Engineers
The variables used in the preceding function fi les are defi ned as follows:
z = Zeros of the transfer function
p = Poles of the transfer function
k = Gain of the transfer function
num = Numerator coeffi cients of the transfer function given as vectors in descend-
ing powers of s
den = Denominator coeffi cients of the transfer function given as vectors in descend-
ing powers of s
n = Order of fi lter
Rs = Minimum stop-band ripple attenuation (dB)
wn = Pass-band edge (rad/s)
Rp = Maximum pass-band ripple attenuation (dB)
wp = Pass-band edge (rad/s)
‘fi ltertype’ = High or stop (for either HP or BP)
The type-2 Chebyshev fi lter functions are self-explanatory since they are similar in
syntax and content to the function defi ned in R.6.71 for the Chebyshev type-1 fi lter.R.6.73 The following function fi les can be used in the analysis and synthesis of the analog
elliptic fi lters:
a. [z, p, k] = elliap (n, Rs)
b. [num, den] = ellip (n, Rs, wn, ‘s’)
c. [num, den] = ellip (n, Rs, ‘fi ltertype’, ‘s’)
d. [n, wn] = ellipord (wp, ws, Rp, Rs, ‘s’)
Note that the elliptic fi lter functions follow the syntax and content similar to
the function fi les defi ned for the Butterworth and Chebyshev fi lters, and no
further function description is given. The workout examples provide insight of the
specifi c function and applications.
R.6.74 The function fi les used in the analysis and synthesis of the Bessel fi lters are defi ned
as follows:
a. [z, p, k] = besselap (n)
b. [num, den] = besself (n, wn)
c. [num, den] = besself (n, wn, ‘fi ltertype’)
The functions and variables used in the Bessel functions are self-explanatory and in
case of doubt refer to the Chebyshev and elliptic fi lter function descriptions (R.6.71).
R.6.75 In most fi lter problems, the MATLAB functions that return the zeros, poles, and gain
forms provide a more accurate response than the ones using the transfer function.
For fi lters of order 15 or less, the function fi les that return the num (numerator)
and den (denominator) of the transfer function can safely be used, but for fi lters of
orders greater than 15, the zeros, poles, and gain returns a more reliable version.
R.6.76 Some important observations of the different fi lter’s phase prototypes are summa-
rized as follows:
a. Butterworth and Chebyshev fi lters present a linear-phase response over a good
portion of the pass-band region.
b. Bessel fi lters are generally used when the desired phase response is linear over
a larger portion of the pass band.