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HW3.solutions.ver2 - ECE 2704 ‐ Homework Number 3 Due at...

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Unformatted text preview: ECE 2704 ‐ Homework Number 3 Due at the start of class on Wednesday, 12 Sept 2007 1. Exercise 1.7‐7 (page 147), parts (a), (b), (c), and (d) Solutions (a) Causal. The signal depends on at time (b) Noncausal. When , depends on and (c) Noncausal. For example, let (d) Noncausal. For example, let 2. Exercise 1.7‐9 parts (a), (b), (c), and (d) Solutions and (a) Linear. Let scalar constants. Then (b) Memoryless. The output depends on only at time . , and let . Since be a , the (c) Causal. For the same reason that it is memoryless. (d) Time‐varying (not time‐invariant). Let delay of . Then system is not time‐invariant. 3. Exercise 1.7‐10 . Ignore the questions and simply write the simplest possible expression for in terms of signals. Solution This is the odd part of the of the signal . and then draw a sketch. 4. Exercise 1.7‐13 (a) and (b). For part (b), write the expression for You may use MATLAB, but it is not required. Solutions (a) Bill is not correct. The correct answer is and then explain what this system does in terms of even and odd parts of and , a time in the past. , a time in the future. . . for , then , then , and let and be complex‐valued . (b) Since the system is linear time‐invariant, the corresponding output is . A sketch of the output is shown in Figure 1. 6 4 2 2 4 6 Figure 1: Sketch of 5. For a given function . , consider a system that generates an output for any input . (a) Show that the system is linear (b) Show that the system in time‐invariant , suggest why (c) By choosing the input response of the system. is sometimes called the impulse Solutions (a) The system is linear since a linear combination of inputs generates the same linear combination of corresponding outputs, (b) The systems is time‐invariant. Time‐shifting the output yields While time‐shifting the input yields To show that (1) and (2) are equivalent, we introduce a change of variables yielding (2) (1) in (2), (c) We compute Thus is the response of the system to an impulse. ...
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