2. System function of a discrete-time and causal LTI system has a zero at z = 0 and two poles at pi = 0.5e /4 and p2= 2e i4. Write the system function H(z). a) Show zero, poles and ROC of this system on Z-plane. Is this system stable? Explain. b) Find the impulse response of the system. c) Give the input / output difference equation of the system. d) If the system is stable solve (a), (b), (c) again.

The pathway for a binary electrical signal between gates in an integrated circuit can be modeled as an RC circuit, as shown in the figure to the right; the voltage source models the transmitting gate, and the capacitor models the receiving gate. Suppose the resistance is 500 ohms and the capacitance is 10^-12 F (1 picofarad, pF. If the capacitor is initially uncharged and the transmitting gate changes instantaneously from 0 to 6 V, how long will it take for the voltage at the receiving gate to reach 5 V?

BC:3.4 Consider the LTI (linear time invariant) dis- crete time systems with the following impulse re- sponses, h[n]. For each system, determine whether or not the system is causal or noncausal and whether or not the system is FIR (finite impulse response) or IIR (infinite impulse response). Give a brief justifi- cation. a.) b.) h[n] = (1.25e-j0.35)2—nu[n – 10] h[n] = (1.25e-j0.35π)2—nu[n + 14] c.) h[n] = (1.25e-j0.35π)4-n (u[n + 4] - u[n – 4]) d.) h[n] = (1.25e¯j0.35″)4—nu[n-4]-(1.25e-j0.35″)4—nu[n—6] e.) h[n] = (1.25e¯j0.35″)4—nu[n-4]-(1.25e¯j0.35π)6—nu[n—6]