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Unformatted text preview: ECE 301: Homework 3
Landis Huﬀman Purdue University, Summer 2009 Due July 9, 2009 in class
1. Chapter 3 Book problems 3.3, 3.8, 3.22(b), 3.30, 3.37(a) 2. Chapter 4 Book problems 4.4(a), 4.6(a) 3. Rhea Contribution Add a page to the Rhea which shows the derivation of any one of the Fourier properties outlined in Tables 3.1 (p.206), 3.2 (p. 221), or 4.1 (p. 328). In other words, the property you show may be for either Continous-Time Fourier Series, Discrete-Time Fourier Series, or Continuous-Time Fourier Transform. The text shows these derivations in Chapters 3 and 4. 4. Fourier Approximation Fourier showed that any periodic signal can be expressed as an (inﬁnite) series of complex exponentials. What happens, however, if we only use a ﬁnite number of complex exponentials when inﬁnite are necessary? This problem investigates this question. Let x(t) be periodic with period T = 4. For t ∈ [−2, 2), let x(t) = 1, −2 ≤ t < 0 0, 0 ≤ t < 2 (a) Find the Fourier series coeﬃcients ak of x(t). (b) Using the Fourier series coeﬃcients computed in (a), write a Matlab program which estimates x(t) using a ﬁnite number (2K + 1) of terms in the Fourier series; i.e. compute
K xK (t) = ˆ
k =−K ak ejk T t , 2π 1 where K is an input to the Matlab function you write. Let your program compute and plot 3 periods of this signal, i.e. compute xK (t) for t ∈ [−4, 4] using ﬁne sampling in this range. ˆ (c) Run your program to compute and plot xK (t) for K = 0, 1, 5, 10, 50. ˆ As K increases, xK (t) should more and more closely resemble the ˆ true signal x(t). Submit printouts of your plots with your written homework. You are not required to submit your code. 2 ...
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- Summer '09