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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 11 37 Copyright © Orchard Publications Low-Pass Analog Filter Prototypes Figure 11.26. Bode plots for the filter of Example 11.9 On the Bode plots shown in Figure 11.26, the ripple is not so obvious. The reason is that this is a Bode plot with straight line approximations. To see the ripple, we use the MATLAB script below. w=0:0.01:10; [z,p,k]=cheb1ap(2,3); [b,a]=zp2tf(z,p,k); Gs=freqs(b,a,w);. .. xlabel('Frequency in rad/s'), ylabel('Magnitude of G(s) (absolute values)');. .. semilogx(w,abs(Gs)); title('Type 1 Chebyshev Low Pass Filter'); grid The generated plot is shown in Figure 11.27. Figure 11.27. Magnitude characteristics for the Chebyshev Type I filter of Example 11.9 -80 -60 -40 -20 0 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -180 -135 -90 -45 0 Phase (deg) Bode Plot for Type 1 Chebyshev Low-Pass Filter Frequency (rad/sec) 10 -2 10 -1 10 0 10 1 0 0.2 0.4 0.6 0.8 1 Type 1 Chebyshev Low-Pass Filter

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Chapter 11 Analog and Digital Filters 11 38 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 11.3.3 Chebyshev Type II Analog Low Pass Filter Design The Chebyshev Type II , also known as Inverted Chebyshev filters, are characterized by the following magnitude square approximation. (11.68) and has the ripple in the stop band as opposed to Type I which has the ripple in the pass band as shown in Figure 11.28. Figure 11.28. Chebyshev Type II low pass filter In relation (11.68), the frequency defines the beginning of the stop band. We can design Chebyshev Type II low pass filters with the MATLAB cheb2ap function. Thus, the statement [z,p,k] = cheb2ap(N,Rs) where N denotes the order of the filter, returns the zeros, poles, and gain of an order normalized prototype Chebyshev Type II analog low pass filter with ripple Rs decibels in the stop band. Example 11.10 Using the MATLAB cheb2ap function, design a third order Chebyshev Type II analog filter with ripple in the stop band. Solution: We begin with the MATLAB script below. w=0:0.01:1000; [z,p,k]=cheb2ap(3,3); [b,a]=zp2tf(z,p,k); Gs=freqs(b,a,w);. .. semilogx(w,abs(Gs)); xlabel('Frequency in rad/sec log scale');. .. ylabel('Magnitude of G(s) (absolute values)');. .. title('Type 2 Chebyshev Low Pass Filter, k=3, 3 dB ripple in stop band'); grid The plot for this filter is shown in Figure 11.29. A 2 ω () ε 2 C k 2 ω C ω 1 ε 2 C k 2 ω C ω + ------------------------------------------ = k = 4 1 dB ripple in stop band A ω ω ω C Nth 3 dB
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 11 39 Copyright © Orchard Publications Low-Pass Analog Filter Prototypes Figure 11.29. Plot for the Chebyshev Type II filter of Example 11.10 11.3.4 Elliptic Analog Low Pass Filter Design The elliptic, also known as Cauer

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