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10Fall_hw06soln

# 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 dier 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 modied. The frequency of the signal cannot be modied, 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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