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Unformatted text preview: ECE 3250 CONTINUOUS-TIME FOURIER TRANSFORMS Fall 2008 1. Motivation, definition, and “derivation” The notion of frequency content makes perfect sense for periodic signals. If x is a peri- odic signal that has T o as a period, it might have frequency content at any number of frequencies. All those frequencies are of the form k Ω o , where Ω o = 2 π/T o and k ∈ Z . The coefficients in a Fourier series for x show how the frequency content in x is distributed over these frequencies. Why might one expect to be able to make sense of the notion of frequency content for non-periodic signals? An example I enjoy pondering is a finite-duration A-440. Recall that a true A-440 is a periodic signal with fundamental period T o = 1 / 440 and fundamental frequency Ω o = 880 π . If I play an A-440 on an instrument and you listen to it, what you hear is not a true A-440 but a finite-duration signal that “sounds like an A-440” while I’m playing it. That signal, one would like to think, has significant “frequency content” around frequency 880 π . A reasonable mathematical model for a particular finite-duration A-440 is the signal x with specification y ( t ) = p a ( t ) cos(880 πt ) = cos(880 πt )- a/ 2 ≤ t < a/ 2 otherwise . A more general model would replace the cosine with an arbitrary periodic signal x with fundamental frequency 880 π . The Fourier transform enables us to make mathematical sense of frequency content for non-periodic signals like the finite-duration A-440. I’ll try first to motivate the definition with a sketchy sort of pseudo-derivation. Suppose first that x is a finite-duration signal and T o > 0 is such that x ( t ) = 0 when | t | ≥ T o / 2. x is about as non-periodic as you can imagine, but we can extend x to form a periodic signal x r that has T as a period by adding infinitely many shifted replicas of x . Technically, x r ( t ) = ∞ X n =-∞ x ( t- nT o ) for all t ∈ R . For any specific t , at most one term in the infinite series is nonzero due to the finite- duration condition on x , so the sum converges trivially. Note that x r ( t ) = x ( t ) for- T o / 2 ≤ t ≤ T o / 2 . Now let Ω o = 2 π/T and expand x r in a Fourier series x r ( t ) = ∞ X k =-∞ c k e jk Ω o t for all t ∈ R . For every k ∈ Z , c k = 1 T o Z T o / 2- T o / 2 x r ( t ) e- jk Ω o t dt = 1 T o Z T o / 2- T o / 2 x ( t ) e- jk Ω o t dt , where the last equality holds because x r = x on the interval [- T o / 2 ,T o / 2]. Accordingly, x r ( t ) = ∞ X k =-∞ 1 T o Z T o / 2- T o / 2 x ( t ) e- jk Ω o t dt ! e jk Ω o t for all t ∈ R and therefore x ( t ) = ∞ X k =-∞ 1 T o Z T o / 2- T o / 2 x ( t ) e- jk Ω o t dt ! e jk Ω o t for- T o / 2 ≤ t ≤ T o / 2 ....
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell.