George Mason University Signals and Systems I Spring 2016

Transcription

1 George Mason University Signals and Systems I Spring 206 Problem Set #6 Assigned: March, 206 Due Date: March 5, 206 Reading: This problem set is on Fourier series representations of periodic signals. The relevant sections in the text Linear Systems and Signals by Lathi are Sections Your lecture notes will also be helpful. Assignment: Given below are two sets of problems. The first, Practice Problems, are optional and for those of you who would like extra practice in solving problems using some of the concepts developed in class. s are provided. The second set, Regular Problems, which are starred, are to be turned in for grading. Detailed solutions will be given for the starred problems after the due date. Practice Problems Problem 6. If x(t) is periodic with a period T and fundamental frequency f 0, and y(t) = x(at), (a) What is the period and fundamental frequency of y(t)? (b) If the Fourier series coefficients of x(t) are x n, what are the Fourier series coefficients of y(t)? (a) Period is T/ a and the fundamental frequency is f 0/ a. (b) They are the same, y n = x n. Problem 6.2 For each of the following signals, determine whether or not they are periodic and, if they are periodic, find the fundamental period. (a) x (t) = cos(t/3) + cos(t/4) (b) x 2 (t) = cos(2πt) sin(6πt) (c) x 3 (t) = cos(0t) + cos(0 + π)t (a) Periodic with period T = 24π. (b) Periodic with period T =. (c) Not periodic. Problem 6.3 If x(t) is periodic with period T = and has (complex) Fourier series coefficients x n, what are the Fourier series coefficients of y(t) = ax(t 0.5) + b Express your answer in terms of x n. y 0 = ax 0 + b and y n = a( ) n x n for n 0.

2 Problem 6.4 Find the complex Fourier series coefficients of the signal x(t) that is defined as follows, x(t) = t 2 ; π < t < π x(t) = x(t + 2π) for all t x 0 = 3 π2 and x n = 4 πn 3 j. Problem 6.5 A continuous-time signal x(t) is real-valued and has a fundamental period T = 8. The non-zero Fourier series coefficients for x(t) are Express x(t) in the form x(t) = 4 cos(πt/4) + 8 cos(3πt/4 + π/2) x = x = 2 ; x 3 = x 3 = 4j x(t) = A k cos(ω k t + φ k ) k=0 Problem 6.6 Suppose that you are given the following information about a signal x(t). x(t) is real and odd, 2. x(t) is periodic with period T = 2 and has Fourier series coefficients x n, 3. x n = 0 for n >, x(t) 2 dt =. Find two different signals that satisfy these conditions. One of the signals is x(t) = 2 sin(πt). Problem 6.7 Let x(t) be a signal that is periodic with a fundamental frequency ω 0 and Fourier series coefficients x k. Given that y(t) = x(t ) + x( t) how is the fundamental frequency of y(t) related to ω 0? Also, find a relationship between the Fourier series coefficients y k of y(t) and the coefficients x k. Same frequency and y k = e jkω 0 [x k + x k ].

3 Problem 6.8 Consider a causal LTI system implemented as the RL circuit shown in the figure below, x(t) L R y(t) A current source produces an input current x(t), and the system output is considered to be the current y(t) flowing through the inductor. The differential equation relating x(t) to y(t) is given by L dy(t) + y(t) = x(t) R dt (a) Determine the frequency response of this system by considering the output of the system to inputs of the form x(t) = e jωt. (b) Determine the output y(t) if x(t) = cos(t) if R = Ω and L = H. The frequency response is H(jω) = (L/R)jω + and y(t) = 2 cos(t π 4 ).

4 Regular Problems Problem 6. Find the complex Fourier series coefficients of the signal [ ( x(t) = ( + cos(2πt)) sin 0πt + π )] 6 Problem 6.2 Consider the following function that is periodic with a period of 2π, (a) Find the complex Fourier series x n of x(t). x(t) = e t + e t ; π < t < π (b) Express the Fourier series expansion of x(t) in terms of sines and cosines. (c) Use the derivative property to find the Fourier series coefficients of y(t) = e t e t ; π < t < π Problem 6.3 Find the Fourier series coefficients for the following signal that is periodic with a period T = 6 (only one period of x(t) is shown. x(t) t Problem 6.4 A periodic signal has a Fourier series expansion given by x(t) = n= x n e jnω0t Find the Fourier series coefficients for the following signals, (a) x (t) = x(t ) (b) x 2 (t) = d dt x(t) (c) x 3 (t) = x(t)e j2πt/t (d) x 4 (t) = x(t) cos(2πt/t )

5 George Mason University Signals and Systems Spring 206 Problem Set #6 Homework Cover Sheet Name: Lecture Section: (, Hayes) or (2, Griffiths) Names of other students I discussed this problem set with: Estimated percentage of how much of each problem has been completed: ) % 2) % 3) % 4) % Total amount of time spent on this problem set: (Hours)