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Unformatted text preview: ECE 325 THE DTFT AND SAMPLING Fall 2007 1. The discrete-time Fourier transform The whole frequency-domain concept makes ample sense in continuous time. It starts with periodic signals such as t 7→ cos Ω o t , t 7→ sin Ω o t , and t 7→ e j Ω o t , where Ω o is a given real number. Each of these signals is a pure sinusoid with frequency Ω o ; in other words, each of these signals has “all of its frequency content” concentrated at the frequency Ω o . Fourier series enable us to express any periodic signal as a superposition of such pure sinusoids. The continuous-time Fourier transform generalizes the Fourier-series idea to continuous- time signals that aren’t necessarily periodic. A non-periodic signal can’t be expanded in a Fourier series, i.e., as a discrete superposition of pure sinusoids. Still, we can often expand it as a “continuous superposition” of pure sinusoids using the Fourier transform, notably equation F- 1 . Like Fourier coefficients for periodic signals, the Fourier transform b X of a continuous-time signal x indicates how the frequency content, also known as spectral content, of x is distributed over “Ω-space,” which we call the frequency domain. It’s convenient to regard pure sinusoids and more general periodic signals as having Fourier transforms. To make that extension, we allowed for impulses in the frequency domain, which led to the formulas t 7→ e j Ω o t F ←→ Ω 7→ 2 πδ (Ω- Ω o ) and by extension t 7→ ∞ X k =-∞ c k e jk Ω o t F ←→ Ω 7→ 2 π ∞ X k =-∞ c k δ (Ω- k Ω o ) . How do we make sense of the frequency-domain concept for discrete-time signals? Let’s see what happens if we attempt to define the discrete-time Fourier by imitating the continuous-time theory. Let x ∈ F Z be a real- or complex-valued discrete-time signal. The equation b X ( ω ) = ∞ X n =-∞ x ( n ) e- jnω for all ω ∈ R (DTFT) is the analogue of equation F in continuous time. The sum in (DTFT) need not converge — it depends on properties of the signal x . Note that if it does converge for all ω ∈ R , the function ω 7→ b X ( ω ) is periodic in ω , and has 2 π as a period. Equation (DTFT) is tantamount to a Fourier-series expansion of the function b X . If you compare (DTFT) with the Fourier series b X ( ω ) = ∞ X k =-∞ c k e jkω for all ω ∈ R , for b X , you see that x ( k ) = c- k for all k ∈ Z . Thus x ( n ) = 1 2 π Z π- π b X ( ω ) e- j (- n ) ω dω for all n ∈ Z , which is the same as x ( n ) = 1 2 π Z π- π b X ( ω ) e jnω dω for all n ∈ Z . (DTFT- 1 ) Here is the formal definition of the discrete-time Fourier transform. Definition 1: Let x ∈ F Z be a discrete-time signal and let ω 7→ b X ( ω ) be a complex- valued function of the real variable ω that has 2 π as a period. We say that x and b X are a 1 2 discrete-time Fourier transform pair if and only if one or both of the equations (DTFT) or (DTFT- 1 ) holds. In this case, we also say that) holds....
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell University (Engineering School).