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. 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 System Properties 30 2 Linear and Time-Invariant System (Convolution) 30 3 Continuous-Time Fourier Transform 40
Prob. 1. [30 pts] We covered a number of general properties of systems in the lecture. In particular, a system may or may not be (1) Memoryless (2) Time invariant (3) Linear (4) Causal (5) Stable. Determine which of these properties hold and which do not hold for each of the following continuoustime systems. Justify you answers. In each example, denotes the system output and is the system input. (5pts each) (a) (b) (c) (e) (d) (f) ANSWER You may either prove each properties or justify answers in words. (a) (1) To be memoryless, a system should have an impulse response satisfying The impulse response of the system is The system is not memoryless since depends on the past. does not satisfy the memoryless condition and the output (2) This system is not time invariant, since the output to a time shifted input is while the time shifted output is (3) The system is linear since it satisfies where the system has input/output pairs, (4) Not causal as (5) The system is stable since (b)
(1) The system is memoryless since (2) It is not time invariant since (3) Linear as it satisfies (4) Causal since it's memoryless. (5) Stable since (c) (1) Not memoryless since it depends on the past. (2) Not time invariant as (3) Linear as it satisfies (4) Not causal. For example, depends on the time 2 which is future input. (5) Not stable since the integral takes values from. (d) (1) Not memoryless since, for example, y which means that the system memorize past inputs. (2) Not time invariant as (3) Linear as it satisfies
(4) Not causal since, for example, y depends on the value of which indexes a point in time -1 greater than -3. (5) It is easy to see that the system is stable because input is mapped to output one-to-one. OR you can say it's stable since (e) We see that the system is not memoryless because output depends on the past, linear as it satisfies, causal since output only depends on the present and the past, and stable since. This system is not time invariant as the output of is different from the time shifted output. Input is mapped to the function while the time shifted output (f) We see that the system is not memoryless because output depends on the past input, causal since output only depends on the present and the past, and stable since. To test its linearity, see that and. The input, however, maps to zero function, showing that the system is not linear. This system is time invariant since input is mapped to the function which is equal to the time shifted output.
Prob. 2. [30 pts] (a) Determine as the convolution of the two signals below. You can either do a plot or write down as an equation for your answer. Show all your work and clearly indicate your answer. (10pts) See lecture notes on 6/23 and 6/24 for more detail.
(b) Consider the causal discrete-time LTI system below. Determine and write a closed-form expression for the output for the input. (10pts) ANSWER The best way to find the output is to first find the impulse response of the system and convolve it with the input x[n]. You may want to find the pattern of the output manually but it would be hard to express in closed-form equation. You get full credit only when you have the correct closed-form expression of. since the system is causal. We learned the convolution result of the two geometric sequences as below. (See lecture notes on 7/1 for details) Thus we have
(c) Consider the same system as part (b), but now determine a closed-form expressions for the output y[n] for the input. (10 pts) ANSWER Impulse response remain the same as part (b), but input has been amplitude-scaled by and time-shifted by 2 from the input in part (b), Thus,
Prob. 3. [40 pts] Consider the input signal defined as below, (a) Plot the magnitude of, the Fourier Transform of this signal, as a function of frequency. Show all work. (HINT: You may use multiplication property of Fourier Transform.) (10 pts) For EACH of the remaining parts from (b) to (f), the signal is input to an LTI system whose impulse response is given below as. For EACH part, (i) plot the magnitude of the Fourier Transform of and (ii) plot the magnitude of the Fourier Transform of. (6 pts each) (b) (c) (d) (e) (f) ANSWER Basic Fourier Transform pairs to be used are: (a)
From part (b) to (f) use the Fourier Transform property, so take FT on and then simply do point-wise multiplication.