The following statements are true except one. Which one is…
Questions
The fоllоwing stаtements аre true except оne. Which one is the exception?
Prоblem 1. (9 pts) A system hаs аn input-оutput relаtiоnship $$y(t) = int_{-infty}^{infty} x(tau - 3) cos^2(t - tau) dtau$$ (a) Is the system linear? Justify your answer. (b) Is the system time-invariant? Justify your answer. Problem 2. (6 pts) A discrete-time LTI system has an impulse response described by $$h[5 - n] = u[n] - u[n - 5]$$ Is the system causal? Justify your answer. Problem 3. (9 pts) A discrete-time LTI system has impulse response (h[n] = u[n]). Determine the output (y[n]) for an input$$x[n] = begin{cases} 0, & n < -1 \ 2n, & -1 leq n le 3 \ 0, & n > 3end{cases}$$ Solve this problem in the time domain and show all steps for the computation. Problem 4. (9 pts) A periodic signal with fundamental frequency (omega_0 = pi) is known to have the following Fourier series coefficients (magnitude (|a_k|) and phase (theta_{a_k})): The coefficients are (0) outside of the given range. (a) Is this a continuous-time or discrete-time signal? Explain in 1-2 sentences. (b) Is this signal purely real, purely imaginary, or complex? Explain in 1-2 sentences. (c) Express the signal as a Fourier series summation. For full credit, combine the result where possible to have less than 4 terms. Problem 5. (10 pts) The system described in the block diagram below: is a special case of a negative feedback system where the output (y(t)) is subtracted from the input (x(t)) and fed into the forward system (S). In this problem, assume (S) is an LTI system with impulse response (h_S(t) = e^{2t} u(-t)). (a) Find the frequency response (H(jomega)) of the overall system (i.e., the system (x(t) rightarrow h(t) rightarrow y(t))). (b) Find the output (y(t)) for an input (x(t) = e^{2t} u(-t - 1)). Problem 6. (10 pts) A continuous-time stable LTI system is described by the following differential equation:$$2 frac{d y^2(t)}{dt^2} + 6 frac{d y(t)}{dt} + 4 y(t) = 2frac{dx(t)}{dt} + 6 x(t)$$ (a) Find the frequency response (H(jomega)) of the system. (b) Find the impulse response (h(t)) of the system. Problem 7. (9 pts) Consider the four signals given below: Determine which, if any, among these signals have Fourier transforms (DTFT) that satisfy each of these conditions (individually): (a) (mathcal{I}m{X(e^{jw})} = 0) (b) (int_{-pi}^{pi} X(e^{jomega}) domega = 0) (c) (X(e^{jomega})) periodic (d) (X(e^{j0}) =0 ) Problem 8. (10 pts) The following amplitude modulation system is used to transmit a sinusoidal message (x(t) = A_m cos(w_mt)) over a communication channel. Let (x_{TX}(t)) and (x_{RX}(t)) be the modulated signal transmitted over the channel and the demodulated signal at the receiver, respectively. (y(t)) is the final output signal after the lowpass filter (H(jomega)). Assume that the communication channel is an ideal, zero-noise channel (i.e., (x_{TX}(t)) is the input to the demodulator). (a) Sketch (X_{TX}(jw)), the spectrum of (x_{TX}(t)). (b) Assuming the carrier generated at receiver is fully phase-synchronized with the transmitter (i.e., (phi=0)), sketch (X_{RX}(jw)). (c) Design the amplitude (A_{cf}) and cut-off frequency (w_{cf}) of the low-pass filter (H(jw)) so that the output of the system matches the input (i.e., to have (Y(jw) = X(jw))). Comment in 1-2 sentences on a practical range of choice for (w_{cf}) to deal with the fact that no lowpass filter is ideal. Problem 9. (9 pts) In the system below, a signal (x(t) = sin(40pi t)) is sampled with a pulse-train sampler (p(t) = sum_{n = -infty}^{infty}delta(t - nT_s)). Later on, a lowpass filter (H(jomega)) with cutoff frequency (omega_f) slightly larger than (40pi) and amplitude (T_s) is applied to reconstruct the signal. In this problem, we investigate the impact of the sampling frequency (omega_s = 2pi / T). (a) Sketch the spectrum (X(jomega)) of the input (x(t)). (b) Sketch the spectrum (X_p(jomega)) of (x_p(t)) if (T_s = 1 / 50) (in seconds). Determine the recovered signal (x_r(t)) at the output of the lowpass filter, assuming it is ideal. Problem 10. (9 pts) Recall (e^{-at}u(t)) from the laplace transform tables. (a) Find the Laplace Transform of (x(t) = 3e^{-2t} u(t) - e^tu(t)). Sketch the pole-zero plot (with the ROC region). (b) Verify Initial and final value theorem for $$x(t) = e^{-at}u(t), quad a>0$$ Problem 11. (10 pts) The Input and output of an LTI system is given by,$$ x[n] = left(frac{1}{6}right)^nu[n], quad y[n] = left[left(frac{1}{2}right)^n+10left(frac{1}{3}right)^nright]u[n]$$ (a) Find the difference equation governing the system. (b) Find the impulse response of the system if it is causal. Discuss stability of the system in this case. (c) Find the impulse response of the system if it is non-causal. Discuss stability in this case too. Congratulations, you are almost done with the Final Exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. 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Select аll thаt wоuld оccur with pаrasympathetic activatiоn: