HW4.2008.SOLUTIONS - ECE 2704 Homework Number 4 Due at the...

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Unformatted text preview: ECE 2704 Homework Number 4 Due at the start of class on Wednesday, 13 February 2008 1. Exercise 1.77 (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 1.79 parts (a), (b), (c), and (d) Solutions (a) Linear. Let and scalar constants. Then (b) Memoryless. The output depends on only at time . , and let . Since be a , the and then draw a sketch. (c) Causal. For the same reason that it is memoryless. (d) Timevarying (not timeinvariant). Let delay of . Then and , a time in the past. , a time in the future. . . for , then , then , and let and be complexvalued system is not timeinvariant. 3. Exercise 1.713 (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 . (b) Since the system is linear timeinvariant, the corresponding output is . A sketch of the output is shown in Figure 1. 6 4 2 2 4 6 Figure 1: Sketch of 4. 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 timeinvariant , 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 timeinvariant. Timeshifting the output yields While timeshifting 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. ) 5. Exercise 2.45 (do the first two only, omit last expression Solutions (i) Note that when . Thus Similarly, when , or equivalently when . Thus (ii) Be careful here. When we put the two conditions together and express the later condition differently, when when We see that whenever , then only when . But when . Thus when and Using the same argument as in Part (i), As in Part (i), when , thus 6. Exercise 2.47 (a) (b) Solutions (a) The output of the system is . We compute (b) ...
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