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420 Chapter 8 IIR FILTER DESIGN
|H a (j Ω)| 2 N Odd
1
1
N Even
1
A 2
0
Ω c
Ω
|H a (j Ω)| 2 Ω c
1
1 +
1
1 +
1
A 2
0
Ω
8.3.11 COMPUTATION OF FILTER ORDER N
Even though the analysis of (8.61) is difficult, the order calculation formula
is very compact and is available in many textbooks [18, 23, 24]. It
is given by
N =
(√ )
K(k)K 1 − k
2
1
K (k 1 ) K (√ (8.62)
1 − k2) where
and
k = Ω p
ɛ
, k 1 = √
Ω s A2 − 1
K(x) =
∫ π/2
0
dθ
√
1 − x2 sin 2 θ
is the complete elliptic integral of the first kind. MATLAB provides the
function ellipke to numerically compute the above integral, which we
will use to compute N and to design elliptic filters.
8.3.12 MATLAB IMPLEMENTATION
MATLAB provides a function called [z,p,k]=ellipap(N,Rp,As) to design
a normalized elliptic analog prototype filter of order N, passband
ripple Rp, and stopband attenuation As, and that returns zeros in z array,
poles in p array, and the gain value k. Weneed an unnormalized elliptic
filter with arbitrary Ω c . This is achieved by scaling the arrays p and z of
the normalized filter by Ω c and the gain k by the ratio of the unnormalized
to the normalized rational functions evaluated at s =0.Inthe following
function, called U elipap(N,Rp,As,Omegac), wedesign an unnormalized
elliptic analog prototype filter that returns H a (s) inthe direct form.
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