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10Fall_hw06soln - ECE210 Fall 2010 Homework 06 Solutions...

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Unformatted text preview: ECE210 - Fall 2010 - Homework 06 Solutions Textbook problems 4.1,4.2,4.3,4.9,4.10,4.11 1. Consider the nth-order homogeneous ODE in p. 114 of the textbook and its general solution in terms of the characteristic polynomial. y y (a) Solve the ODE by the phasor method (i.e. solve d n + a1 d n−1 + · · · + an y (t) = f (t) and nd the phasor dt dt output Y as a response to a phasor input F , where y (t) = Re{Y ejωt } and f (t) = Re{F ejωt }). n n−1 Solution: dn−1 y dn y + a1 n−1 + · · · + an y (t) = f (t) dtn dt n n−1 (jω ) Y + a1 (jω ) Y + · · · + an Y = F (jω ) + a1 (jω ) n n−1 + · · · + an Y = F Y= F (jω ) + a1 (jω ) n n−1 + · · · + an (b) Where does the characteristic polynomial appear in the output expression? Solution: The characteristic polynomial, P (jω ) appears in the denominator of the output. Y= F P (jω ) (c) Does the frequency of the output dier from the frequency of the input? Solution: No. For a linear, time-invariant system, the frequency of the signal will remain the same in the output as in the input. (d) Why doesn't the term (ωt) appear in the phasor solution? Solution: For an LTI system, only the phase and magnitude of the input signal may be modied. The frequency of the signal cannot be modied, so the (ωt) term will remain the same at the output as the input. 2. Textbook problem 4.1 1 3. Textbook problem 4.2 2 4. Textbook problem 4.3 5. Textbook problem 4.9 3 6. Textbook problem 4.10 4 7. Textbook problem 4.11 5 6 ...
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