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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 28 Tue 04/28 Lecture Topics: further analogies between Fourier series and the DFT Key Points: Circular operations (reversal and shifts) on a finite vector are equivalent to non-circular operations on the infinite periodic extension of that vector. Time Domain Signal Fourier Series Coefficients x ( t ) X k x * ( t ) X *- k x (- t ) X- k x ( t- T ) e- jk T X k ( K Z ) x ( t ) e jK t X k- K ( > 0) x ( t ) X k Theory and Examples: 1 . The Fourier series and the discrete Fourier transform are parallel concepts. They both provide a decomposition of a periodic signal into mutually orthogonal, harmonically related sinusoids. The coefficients of these sinusoids collectively form the spectrum of the periodic signal. We have already seen many similarities between the two concepts, notably in the way the spectrum is obtained from the time-domain signal (i.e., by means of an inner product). The two concepts also exhibit similar properties in regard to signal transformations and time- frequency duality. 2 . In this lecture we will explore some of these similarities, but also point out some important differences, between the Fourier series and the DFT. Most notably: In the case of the DFT, the two domainstime and frequencyhave a common struc- ture, i.e., they are index sets of the same finite size. Periodicity is an optional feature: thus the DFT applies to both a finite-dimensional time-domain vector and its infinite periodic extension; while the set of Fourier frequencies has a counterpart in every interval of the form [2 r, 2( r + 1) ), where r Z . In the case of the Fourier series, the two domains are very different. The time domain is continuous and inherently periodic. (Like the DFT, the Fourier series describes both a signal over a finite interval of time and its infinite periodic extension outside that interval.) On the other hand, the frequency domain is discrete, infinite (indexed by Z ) and distinctly non-periodic: we recall that, unlike discrete-time sinusoids, continuous- time sinusoids can have arbitrarily high frequencies. As a result of these differences, duality between time and frequency does not manifest itself as plainly as in the case of the DFT....
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- Spring '08