Question

please provide a complete solution with the correct answer.

In the following questions, a discrete-time filter is to be designed using the impulse invariance method. The sample rate of the digital system is 2 samples/second. The discrete-time filter is to replace the causal continuous-time filter below, with H(s) = Vo(s)/Vi(s) . L-2H Vo(t) R=1 ohm 31. The z-transform of the discrete-time filter is H(z)= 1 -1 )-- 0. 25 , , -0.25 a) c) e) none above 0.2 -0.2 0.5 0.5 1-e-0.5--

Answer 1

At convert the circuit into Laplace domain (R rightarrow R L rightarrow SL) V_x(s)/V_i(s) = H(s) = 1/2s + 1 (Voltage division rule) H(s) = 0.5/s + 1/2 Apply inverse Laplace transform h(t) = 0.5 e^-1/2t U(t) substitute t = nT_s h(nT_s) = 0.5 e^-1/2nT_s U(nT_s) h(n) = 0.5 e^-1/2(2n) U(n) T_s = 2s/sec h(n) = 0.5 e^-n U(n) Apply z-Transform on the both sides H(z) = 0.5 z/z.e^-1Vertical-bar z Vertical-bar > e^-1 = 0.5/1-z^-1 e^-1 Vertical-bar z Vertical-bar > e^-1 z^n U(n) rightarrow z/z-a Vertical-bar z Vertical-bar > a