Problem 2. Suppose the impulse response function h[n] of a DT system is: h[n] = (-−0.5)^u[n]. Determine the transfer function / eigenvalue H(z) of this DT system through its definition: assuming the input signal is the eigenfunction x[n] = z^, and calculate the absolute value and the phase angle of the eigenvalue H(z) at: 1 (a) z = √(1 + j) (Q5, Q6); (b) z = −0.5 + j; (Q7, Q8)

y(n) is output. x(n) is input

(a) The analog signal x(t) = sin(60rt) is sampled at 20 Hz. Derive and write down an expression for the sampled signal r[n]. What is the Nyquist frequency for this r(t)?

We have seen that transfer functions can be frequency scaled by replacing s by s/kf each time that it occurs. Alternately, circuits can also be frequency scaled by dividing each capacitance and each inductance by the frequency scaling factor kf. Either way, the effect is the same. The frequency response is shifted to the right by kf. In particular, all cutoff, corner, and resonant frequencies are multiplied by kf. Suppose that we want to change the cutoff frequency of a filter circuit from wold to wnew. We set the frequency scale factor to Wnew ke = Wold and then divide each capacitance and each inductance by kf. Use frequency scaling to change the cutoff frequency of the circuit in Figure to 150 rad/s. Determine the scaling coefficient. 25 2 10mH vi(t) 3 µF volt) the tolerance is +/-2%

4+ Page 4. For two systems with impulse responses hi[n] = (0.3)" u[n] h2[n] = (0.7)" u[n] pud (a) Write down their transfer functions, H1(z) and H2(z). Include the ROCS. (b)Determine the transfer function for the parallel interconnection of these two systems. Indicate the ROC and all the poles and zeros. Plot the pole-zero diagram as well. (c)Using the result of part (b) above: Determine all possible inverse systems, H1(z). (i) Indicate which inverse is causal, and whether this inverse is stable and/or FIR? Explain. O words

Q1: For the pole-zero diagram shown in Figure below, the lengths of poles and zeros from the origin are 0.5 and 0.8 respectively: a. Find the transfer function H(z), impulse response h(n) and the difference equation. b. Calculate and plot the magnitude and phase frequency responses. Im(z) Re(z)

Given the difference equation: y(n) = 0.2y(n-1) + 0.2y(n-2) + 0.1u(n-1) 1-Find the discrete transfer function H(z) = 2-Draw a block diagram of this discrete system using delays and gains. Y(z) (2) U(z)

The question is in the photo

3. Consider the closed-loop system in the following figure where 14 G(s) = K and H(s) = s + 10 s² + 5s + 6 T«(s) + R(s) G(s) Y(s) H(s) Fig. Feedback system.

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