Fall 2017- Problem Set 2 ECE 301: Signals and Systems Prof.Aly El Gamal Due: 29th September, 2017 Instructions . Please write clearly and legibly Your solutions must include detailed steps and/or explanations. Do not simply state the answer Write your full name(first,last), PUID on your homework submission All problems carry almost equal weight Problem 1. For each of the below systems with input x and output y, determine if they are Linear or Non Linear Causal or Non causal Time Variant or Time Invariant Stable or Unstable a. y[n] = x[n-1] - 4x[n-8] b. y[n] = (n 1)2 x[n-2] R 4t c. y(t) = x( )d d. y(t) = x(t-4) + x(4-t) e. y(t) = sin(6t)x(t) Problem 2. a. If x(t) ? h(t) = y(t) where ? indicates convolution, then show that x(t t1 ) ? h(t t2 ) = y(t (t1 + t2 )) 1 F17-ECE 301-HW 2 Prof. Aly El Gamal b. Find Z Due: September 29, 2017 x(t)(t t0 )dt c. Find Z x( )(t t0 )d d. Find Z x( )(t )d Problem 3. Continuous Time Convolution: Find x(t) ? y(t) a. x(t) = u(t) u(t T1 ) y(t) = u(t) u(t T2 ) with T1 < T2 Note: u(t) is unit-step b. ( e(1+j)t , if t 1 x(t) = 0, if t < 1 ( 0, if t 3 y(t) = cos(t 2 ) jsin(t 2 ), if t < 3 c. For x(t) and y(t) in the figure 1, (Note: x(t) is periodic) Figure 1: x(t) and y(t) Problem 4. Discrete Time Convolution: a. x[n] = [n + 1] + [n] + [n 1] + [n 2] y[n] = u[n] u[n 2] Find x[n] ? y[n]. Note: u[n] is unit-step 2 F17-ECE 301-HW 2 b. Prof. Aly El Gamal ( 1, x[n] = 0, ( 1, y[n] = 0, Due: September 29, 2017 if 0 n A otherwise if 0 n B otherwise Find the number of non - zero terms in x[n] ? y[n] c. Calculate x1 [n] ? x2 [n] ? x3 [n]. Note: u[n] is unit-step 1 x1 [n] = ( )n u[n] 2 x2 [n] = u[n] x3 [n] = [n 2] Problem 5. The output at time 'n' is obtained by accumulating the values of the input from 'n+3' to ''. 1. What is the impulse response? 2. What is y[n] when x[n] = u[n] - u[n-3] where u[n] indicates unit step 3. Is this system Causal? Memoryless? Stable? Invertible? If so, what is the impulse response of the inverse system 3