ELEC212 Chapter 8 Discrete Fourier Transform

ELEC212 Chapter 8 Discrete Fourier Transform - Discrete...

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1 The Hong Kong University of Science and Technology Department of Electronic and Computer Engineering ELEC 212 Chapter 8 Discrete Fourier Transform 2 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) Discrete Fourier Transform (DFT) ± Recall what we have learnt in Chapter 02, 3 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) Discrete Fourier Transform (DFT) ± In discrete-time signal processing, the DTFT is not directly applicable to the digital analysis because: – We cannot use a computer / digital processing unit / DSP chip to evaluate the DTFT, since X ( e j ω ) is a continuous function of ω ± For practical applications, we use a closely related transform known as Discrete Fourier transform (DFT). –U s e d t o evaluate the frequency response of a finite sequence – Can be used to evaluate the samples of the continuous spectrum (ie. we can regard the DFT as a discrete version of the DTFT) – Can be evaluated using efficient algorithms (next lecture) ± DFT is defined based on the Discrete Fourier Series (DFS) 4 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) ± DFS gives a discrete frequency domain representation for periodic signals ± For periodic signals, the DTFT and z -transform do not uniformly converge – For a periodic sequence , the sequence is not absolutely summable for any ± DFS is based on fact that any periodic sequence with period N can be represented as a finite weighted sum of N harmonically-related complex exponentials of the form ± Definition: ] [ ~ n x r n r n x ] [ ~ ] [ ~ n x ] [ ~ of (IDFS) DFS Inverse ] [ ~ 1 ] [ ~ ] [ ~ of DFS ] [ ~ ] [ ~ 1 0 / 2 1 0 / 2 k X e k X N n x n x e n x k X N k N kn j N n N kn j = = = = π integers are and , ] [ ] [ ) / 2 ( l k n e e n e lN k kn N j k + = = Discrete Fourier Series (DFS)
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5 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) ] [ ~ of (IDFS) DFS Inverse ] [ ~ 1 ] [ ~ ] [ ~ of DFS ] [ ~ ] [ ~ 1 0 / 2 1 0 / 2 k X e k X N n x n x e n x k X N k N kn j N n N kn j = = = = π ± Point to note: The periodic sequence is represented by only N coefficients , can be defined arbitrarily for k outside this range – Convention is to define to be periodic with period N, since it leads to simple duality relations between time and frequency domains for DFS ] [ ~ n x ] [ ~ k X 1 0 ] [ ~ N k k X for ] [ ~ k X Discrete Fourier Series (DFS) 6 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) −∞ = = = = r rN n rN n n x otherwise , 0 , 1 ] [ ] [ ~ δ Discrete Fourier Series (DFS) ± Example 8.1: DFS of a periodic impulse train – Consider the periodic impulse train – Since for , the DFS coefficients are found as – In this case, is the same for all k – Substituting , into the IDFS formula leads to the equivalent time-domain representation 1 ] [ ] [ ~ 1 0 / 2 = = = N n N kn j e n k X ] [ ] [ ~ n n x = 1 0 N n ] [ ~ k X 1 0 N k 1 ] [ ~ = k X = = 1 0 / 2 1 ] [ ~ N n N kn j e N n x 7 09-10 Fall ELEC 212 Discrete Fourier Transform (DFT) Properties of the DFS ± Linearity ± Shift of a Sequence (Time Domain) – Any shift greater than the period (ie. m N ) cannot be distinguished in the time-domain from a shorter shift m 1 such that m = m 1 + rN , where m 1 and r are
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ELEC212 Chapter 8 Discrete Fourier Transform - Discrete...

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