midterm1.solutions - ECE 2704: MIDTERM EXAM 1 Name:

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Unformatted text preview: ECE 2704: MIDTERM EXAM 1 Name: _______________________________SOLUTION_____________________ Signature:__________________________________________________________ NOTES: 1. One hand‐written 8.5x11 sheet of paper is permitted. It must be your own handwriting. Electronic printing and photocopying is strictly prohibited. 2. Scores are based on correctness and clarity of presentation. 3. No calculators are allowed. 4. The exam will end promptly at 2:15 pm. 5. Put a box around your answers. 1. For and , compute (a) the product in Cartesian coordinates Solution and , thus (b) the sum Solution and in polar coordinates , thus 2. Express the function of shown in Figure 1 compactly using unit step functions 4 3 2 1 2 Figure 1 4 6 Solution The diagonal line begins at function in Figure 1 is expressed and ends at and is described by the line . Thus the 3. The step‐response of a system is the output of the system when the input is a unit step . Suppose the step response of a linear time‐invariant system is the signal shown in Figure 2. Carefully sketch the output of the system to the pulse shown in Figure 3. 1 Figure 2: Unite step response of an LTI system 1 2 3 2 5 Solution The signal in Figure 3 can be expressed . Let the step response in Figure 2 be the signal input is is , which is shown in Figure 3a. 7 Figure 3: Input pulse . Then the output of the system when the 2 2 4 Figure 3a 6 4. Let the signal shown in Figure 4 be , and define 1 ‐1 (a) Compute 0 Figure 4. 1 in it. as a function of . That is, your answer should have Solution The problem is most easily addressed by determining how a time‐shifted integral. That is, Thus, (b) Clearly sketch Solution . behaves under an 2 ‐2 0 2 5. (a) Compute the average power of the periodic unit‐amplitude signal shown in Figure 5. 1 ‐2 2 0 ‐1 Figure 5. Solution Since for all , the average power is 1. (b) Compute the energy in the signal . See discussion in class. (c) What is the average power? (hint: since the signal has finite duration, you should be able to immediately write down the average power) See discussion in class. ...
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