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Unformatted text preview: ECE 313 Fall 2010 Homework 8 – Due Thursday, November 4 (All book problems are from Chapter 8
copies of the problems from the book are attached. In every problem that refers to Fourier Series coefficients you should use the complex exponential version, not the trigonometric (sine and cosine) version.) 1.
Suppose we have signals defined over the interval −1 ≤ t ≤ 1. For each of the cases below: plot the signal x ( t ) over the interval −1 ≤ t ≤ 1; find a general expression for the kth Fourier Series coefficient X [ k ] (with T = 2 sec); and graph €
{ X [k ], − 8 ≤ k ≤ 8} . €
ȹ€ȹ
t
(a)
x ( t ) = rectȹ ȹ €
€
ȹ 2 Ⱥ
(b)
(c) €
x ( t ) = rect ( t ) x ( t ) = rect (2 t ) €
P€
roblem 22, p. 286. (Note: “complex CTFS description” means Fourier Series coefficients.) € 2. 3. Problem 26, parts b, c, p. 287. (Note: “CTFS harmonic function” also means Fourier Series coefficients, and “representation time” means the time T taken as one period in calculating the Fourier Series.) 4. Problem 31, p. 288. 5. Problem 32, p. 288. 6. Consider the periodic rectangular pulse train (with period T0 ) shown below. € € (a) Find an expression for the average power Px in the signal as a function w
of the pulse width fraction . (Hint: Do this in the time domain.) T0
€ € 2 (b) Find an expression for the power Px [ k ] = X ( k ) in the kth harmonic w
component of this signal as a function of the pulse width fraction . T0
(c) Find an expression for the fraction of the total average signal power that €
is contained in the harmonic components −2 ≤ k ≤ 2 as a function of the pulse 2
w
2
width fraction . (That is, find an expression for ∑ X€ ]
Px as a [k
T0
k = −2
€
w
function of .) T0
w
€
€
(d) Evaluate your result from part (c) for pulse width fractions = 0.1 and T0
w
€= 0.5 . T0 €
Consider a periodic signal x p ( t ) that has a fundamental frequency of 10 Hz and Fourier Series coefficients given by: X [ k ] = 1, k = 0 7.
€ k €X [ k ] = 0.5( −1) , k = ±1, ±2, ±3, ±4 X [ k ] = 0, otherwise (a) Use MATLAB to graph x p ( t ) for −0.3 ≤ t ≤ 0.3 . (b) Suppose that x p ( t ) is input to a LTI system that has transfer function € 0.01s2
. Graph the spectrum (that is, the frequency domain €
404 π 2€ s2
+
representation) of the output signal y p ( t ) . €
(c) Again using MATLAB, graph y p ( t ) for −0.3 ≤ t ≤ 0.3 . € H ( s) = € € € Problems from Textbook ...
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 Fall '08
 MUSCHINSKI

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