Unformatted text preview: EL6113 Assignment 2 1. Determine if the following systems are: a. Memoryless b. Causal c. Invertible d. Stable e. Linear f. Time invariant i. ! ! = cos ! ! − 1 ii. ! ! = ! [!] ! ! ! (! − ! ) !!
iii. ! ! = ! !!! ! [! ] !!
iv. ! ! = !
!
!! 5! !" v. ! ! = ! ! ! [!] 2. You are given an LTI system where the input !! ! = ! ! − ! ! − 1 produces the output !! ! .What is the response, !! ! to the given !! ! in terms of !! (! ). x2(t) 3 2 1 0
2
1 0 1 2 3. The system L is known to be time invariant. Below, when x1(t), x2(t) and x3(t) are put through the system L, they produce y1(t), y2(t) and y3(t) respectively. a. Could this system be linear? b. What is the impulse response, h(n), of this system? x1[n] y1[n] 3 4 2 2 1 0
2
1 0 0 1 2
2
1 0 1 2 x2[n] y2[n] 3 6 2 4 1 2 0
2
1 0 0 1 2
2
1 0 x3[n] 1 2 4 y3[n] 3 4 2 2 1 0
2
1 0 3 0 1 2 3 4 5
3
2
1 0 1 4. By direct evaluation of the convolution sum, determine the step response of an LTI system whose impulse response is ℎ ! = ! !! ! −! 0 < ! < 1 5. Consider the system below and answer the questions below: u[n] x[n] ! ℎ ! = ( )! ! [! + 10] ! a. Is this overall system LTI? b. Is the overall system causal? c. Is this overall system BIBO stable? x y[n] 6. The impulse response of a LTI system is show below. Sketch the response of the system to an input ! ! = ! ! − 4 . h[n] 2 1 0
1
1 0 1 2 3 4 5 6
2
3 7. The system L is known to be linear. Below, when x1(t), x2(t) and x3(t) are put through the system L, they produce y1(t), y2(t) and y3(t) respectively. a. Could this system be TI? b. What is the system response if the input is an impulse? x1[n] y1[n] 2 0
2
1 0 1 2
2
2
4 4 3 2 1 0
1
1 0
2 x2[n]
2
4 3 4 2 0
1 2 y2[n] 2
2 1 0 1 2
2 0
1 0
2
4 1 2 3 4 x3[n] y3[n] 2 1
3 0
2
1 0 1 2 3 4 5
2 3 2 1 0
1
1 0
2
3
4 1 2 3 ...
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 Spring '12
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