Which of the following is NOT actually true of ReLU (and the…
Questions
Which оf the fоllоwing is NOT аctuаlly true of ReLU (аnd therefore not a real reason it's preferred over sigmoid/tanh for hidden layers)?
Prоblem 1 - True/Fаlse (10 pоints) Mаrk eаch statement as either True оr False. Partial credit will be awarded for the last part (only) of this problem. Please show your work clearly to receive partial credit. (a) The Continuous-Time Fourier Transform (CTFT) does not exhibit dualitybetween its analysis and synthesis equations. (b) The Continuous-Time Fourier Series (CTFS) exhibits duality between itsanalysis and synthesis equations. (c) The Discrete-Time Fourier Transform (DTFT) exhibits duality between itsanalysis and synthesis equations. (d) The Discrete-Time Fourier Series (DTFS) exhibits duality between itsanalysis and synthesis equations. (e) If (X(e^{jomega}) = X(e^{j(omega-1)})), then (x[n]=0) for (|n|>0). Problem 2. Multiple Choice (10 points) Select the correct answer. Partial credit will be awarded for this problem. Please show your work clearly to receive partial credit. Consider the causal LTI system described by the difference equation [y[n]-frac{3}{4}y[n-1]-frac{1}{8}y[n-2]=2x[n].] The impulse response of the system is: (A) [ h[n] = 4left(frac{1}{2}right)^n u[n] - 2left(frac{1}{4}right)^n u[n] ] (B) [ h[n] = 2left(frac{1}{2}right)^n u[n] - 4left(frac{1}{4}right)^n u[n] ] (C) [ h[n] = 4left(frac{1}{2}right)^n u[n] + 2left(frac{1}{4}right)^n u[n] ] (D) [ h[n] = 2left[ left(frac{1}{2}right)^n - left(frac{1}{4}right)^n right]u[n] ] (E) None of the above. Problem 3. (25 points) (a) [5 points] Consider a discrete-time LTI system with unit-sample response [h[n]=left(frac{1}{2}right)^n u[n]+frac{1}{2}left(frac{1}{4}right)^n u[n].] Determine a linear constant-coefficient difference equation relating the input (x[n]) and the output (y[n]) of the system. (b) The following figure depicts a block-diagram implementation of a causal LTI system. Fig. 1: Block-diagram implementation of the causal LTI system. For the system shown in Fig. 1: (i) (10 points) Find a difference equation relating (x[n]) and (y[n]). (ii) (5 points) Determine the frequency response (H(e^{jomega})) of the system. (iii) (5 points) Determine the impulse response (h[n]) of the system. Problem 4. (15 points) (a) A particular discrete-time system has input (x[n]) and output (y[n]). The Fourier transforms of these signals are related by [Y(e^{jomega})=2X(e^{jomega})+e^{-jomega}X(e^{jomega})-frac{dX(e^{jomega})}{domega}.] Answer the following questions. (i) (5 points) Is the system linear? Clearly justify your answer. (ii) (5 points) Is the system time invariant? Clearly justify your answer. (b) (5 points) Consider a discrete-time system for which the transform (Y(e^{jomega})) of the output is related to the transform of the input by [Y(e^{jomega})=int_{omega-pi/4}^{omega+pi/4}X(e^{jomega}),domega .] Find an expression for (y[n]) in terms of (x[n]). Problem 5. (25 points) The system shown in Fig. 5 consists of a continuous-time LTI system followed by a sampler, conversion to a sequence, and a discrete-time LTI system. The continuous-time LTI system is causal and satisfies the linear, constant-coefficient differential equation [frac{d y_c(t)}{dt}+y_c(t)=x_c(t).] The input to the continuous-time system is a unit impulse: [x_c(t)=delta(t).] Fig. 2: Continuous-time LTI system followed by sampling and a discrete-time LTI system. (a) [10 points] Determine (y_c(t)). (b) [15 points] Determine the frequency response (H(e^{jomega})) and the impulse response (h[n]) of the discrete-time LTI system such that [ w[n]=delta[n]. ] Problem 6. (15 points) Determine the Nyquist rate corresponding to each of the following signals. (a) [5 points] [x(t)=1+cos(2000pi t)+sin(4000pi t)] (b) [5 points] [x(t)=frac{sin(4000pi t)}{pi t}] (c) [5 points] [x(t)=left(frac{sin(4000pi t)}{pi t}right)^2] Congratulations, you are almost done with this exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Exam 3 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.