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406 Chapter 8 IIR FILTER DESIGN
□ EXAMPLE 8.2 Design a 3rd-order Butterworth analog prototype filter with Ω c =0.5 given in
Example 8.1.
Solution
MATLAB script:
>> N = 3; OmegaC = 0.5; [b,a] = u_buttap(N,OmegaC);
>> [C,B,A] = sdir2cas(b,a)
C = 0.1250
B = 0 0 1
A = 1.0000 0.5000 0.2500
0 1.0000 0.5000
The cascade form coefficients agree with those in Example 8.1.
□
8.3.3 DESIGN EQUATIONS
The analog lowpass filter is specified by the parameters Ω p , R p ,Ω s , and
A s . Therefore the essence of the design in the case of Butterworth filter
is to obtain the order N and the cutoff frequency Ω c , given these specifications.
We want
• at Ω=Ω p , −10 log 10 |H a (jΩ)| 2 = R p or
⎛
⎞
1
−10 log 10 ⎜ ( )
⎝
2N
⎟
Ωp ⎠ = R p
1+
Ω c
and
• at Ω=Ω s , −10 log 10 |H a (jΩ)| 2 = A s or
⎛
⎞
1
−10 log 10 ⎜ ( )
⎝
2N
⎟
Ωs ⎠ = A s
1+
Ω c
Solving these two equations for N and Ω c ,wehave
N = log [(
10 10
R p/10 − 1 ) / ( 10 As/10 − 1 )]
2 log 10 (Ω p /Ω s )
In general, N will not be an integer. Since we want N to be an integer,
we must choose
⌈ [(
log10 10
R p/10 − 1 ) / ( 10 As/10 − 1 )] ⌉
N =
2 log 10 (Ω p /Ω s )
(8.49)
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