Solution 7 August 2015 ECE301 Signals and Systems: Final Exam. Cover Sheet

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1 Solution 7 August 2015 ECE301 Signals and Systems: Final Exam Cover Sheet Test Duration: 120 minutes Coverage: Chap. 1, 2, 3, 4, 5, 7 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes are not allowed. Fourier Transform Tables are attached at the end. All work should be done on the provided sheets. You must show work or explain answer for each problem to receive full credit. Problem No. Topics Points 1 Linear and Time-Invariant system 20 2 Fourier Series Representation 25 3 Discrete-Time Fourier Transform 30 4 Sampling Theory 25

2 Prob. 1. [20 pts] Consider the cascade interconnection of three causal LTI systems as below: The impulse response is and the impulse response of the overall system is shown below: (a) Find the impulse resposne of. (Answer) (The values are 0 when and ) where Now since we have the impulse response of the overall system, Pluggin in the values of h[n] on the given plot,

3 Since each system is causal, the overall system is also causal meaning that we have Solving for the seven equations,

4 (b) Find the response of the overall system to the input (Answer) When the input is the output is Since we have where indicates that it is the value when n=0. Your answer can be either in a sequence form, a plot, or an equation.

5 Prob. 2. [25 pts] Let be a periodic signal with fundamental period and Fourier coefficients. (a) Determine the value of. (Answer) (b) Determine the Fourier series representation of. (Answer) Let, then with period T=2. Then, let the coefficients of. where are defined above. (c) Use the result of part (b) and the differentiation property of the continuous-time Fourier series to determine the Fourier series (FS) coefficients of. (Hint: If has FS coefficients, then has FS coefficients.) (Answer) Since has FS coefficients,

6 Solving the equation,

Prob. 3. [30 pts] The signal below is defined for all parts of this problem. For each sub-part, answer the three questions (i),(ii), and (iii). (a) For, (i) Compute the energy. Show work below.

7 Prob. 3. [30 pts] The signal below is defined for all parts of this problem. For each sub-part, answer the three questions (i),(ii), and (iii). (a) For, (i) Compute the energy. Show work below. CTFT of x(t),, is a rectangle. From Parseval's Theorem, (ii) What is the Nyquist sampling rate (in angular frequency, w) for x? (iii) Define. Plot the magnitude of the DTFT of, where. Plot over and properly label the values on the plot. is mapped to in DT frequency.

8 (b) For (i) Compute the energy. Show work below. So the energy is same as (ii) What is the Nyquist sampling rate (in angular frequency, w) for y? (iii) Define. Plot the magnitude of the DTFT of, where. Plot over and properly label the values on the plot. Similarly as in (a), is mapped to in DT frequency.

9 (c) For (i) Compute the energy. Show work below. Using the differentiation property of CTFT, (ii) What is the Nyquist sampling rate (in angular frequency, w) for y? (iii) Define. Plot the magnitude of the DTFT of, where. Plot over and properly label the values on the plot. is mapped to in DT frequency.

10 (d) For (i) Compute the energy. Show work below. From Parseval's Theorem, (ii) What is the Nyquist sampling rate (in angular frequency, w) for y? (iii) Define. Plot the magnitude of the DTFT of, where. Plot over and properly label the values on the plot. is mapped to in DT frequency.

11 (e) For (i) Compute the energy. Show work below. From Parseval's Theorem, (ii) What is the Nyquist sampling rate (in angular frequency, w) for y? (iii) Define. Plot the magnitude of the DTFT of, where. Plot over and properly label the values on the plot. Since, aliasing exists. is mapped to and is mapped to.

12 Prob. 4. [25 pts] The signal is sampled every to form. Then the discrete-time (DT) signal is input to the DT system which has impulse response as below: Note that is a constant real number. (Answer) (a) Check if there is an aliasing when sampling the signal. Plot to find the maximum frequency Since, there is an aliasing. (b) Plot where is the DTFT of.

13 There are overlaps between triangles so can be plotted as below: Using the relationship where, (For example, is mapped to ) (c) Plot where Y is the DTFT of and y[n] is the output of x[n] through the DT system. Since

14 Then since. (d) Determine Using Parseval's Theorem,

Solution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet

Solution 10 July 2015 ECE301 Signals and Systems: Midterm Cover Sheet Test Duration: 60 minutes Coverage: Chap. 1,2,3,4 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes are not allowed.