1. Digitize the following systems to find $H_d(z)$ by using the bilinear transformation where $H_d(z) = H\left(\frac{2}{T}\left(\frac{z-1}{z+1}\right)\right)$. Assume that $T = 0.2$ seconds. The answers should have all terms in their simplest integer form over a common denominator. a. $H(s) = \frac{2}{s+2}$ b. $H(s) = \frac{s}{s+4}$ c. $H(s) = \frac{s}{(s^2+4s+4)}$
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$H(s) = \frac{2}{s+2}$ $s = \frac{2}{T}\frac{z-1}{z+1} = \frac{2}{0.2}\frac{z-1}{z+1} = 10\frac{z-1}{z+1}$ $H_d(z) = \frac{2}{10\frac{z-1}{z+1}+2} = \frac{2}{\frac{10z-10+2z+2}{z+1}} = \frac{2(z+1)}{12z-8} = \frac{z+1}{6z-4} = \frac{z+1}{2(3z-2)} = Show more…
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a) In general, which designed method (impulse invariance or bilinear transformation) will guarantee H(e^jw)|w=0 = Hc(jO)|O=0? b) A continuous-time filter with impulse response hc(t) and frequency-response magnitude |Hc(jO)| = { |O|, |O| < 10pi; 0, |O| > 10pi } is to be used as the prototype for the design of a discrete-time filter. A discrete-time system with impulse response h1[n] and system function H1(z) is obtained from the prototype continuous-time system by impulse invariance with Td = 0.01, i.e., h1[n] = 0.01hc(0.01n). Plot the magnitude of the frequency-response of the discrete-time system, i.e., H1(e^jw). c) Alternatively, suppose that a discrete-time system with impulse response h2[n] and system function H2(z) is obtained from the same prototype continuous-time system by the bilinear transformation with Td = 2, i.e., H2(z) = Hc(s)|s=(1-z^-1)/(1+z^-1). Plot the magnitude of the frequency-response of the discrete-time system, i.e., H2(e^jw).
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