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Lecture Digital signal processing - Lecture 4 introduce the discrete fourier transform. This lesson presents the following content: The discrete Fourier series, the Fourier transform of periodic signals, sampling the Fourier transform, the discrete Fourier transform, properties of the DFT, linear convolution using the DFT. | Digital Signal Processing Fall 2006 Lecture 9 The Discrete Fourier Transform Zheng-Hua Tan Department of Electronic Systems Aalborg University Denmark zt@kom.aau.dk 1 Digital Signal Processing IX Zheng-Hua Tan 2006 Course at a glance MM3 MM9 MM10 2 Digital Signal Processing IX Zheng-Hua Tan 2006 1 The discrete-time Fourier transform DTFT The DTFT is useful for the theoretical analysis of signals and systems. But according to its definition x X eja xn n - computation of DTFT by computer has several problems The summation over n is infinite The independent variable w is continuous 3 Digital Signal Processing IX Zheng-Hua Tan 2006 AALBORG UNIVERSITY The discrete Fourier transform DFT In many cases only finite duration is of concern The signal itself is finite duration Only a segment is of interest at a time Signal is periodic and thus only finite unique values For finite duration sequences an alternative Fourier representation is DFT The summation over n is finite DFT itself is a sequence rather than a function of a continuous variable Therefore DFT is computable and important for the implementation of DSP systems DFT corresponds to samples of the Fourier transform 4 Digital Signal Processing IX Zheng-Hua Tan 2006 AALBORG UNIVERSITY 2 Part I The discrete Fourier series The discrete Fourier series The Fourier transform of periodic signals Sampling the Fourier transform The discrete Fourier transform Properties of the DFT Linear convolution using the DFT 5 Digital Signal Processing IX Zheng-Hua Tan 2006 AALBORG UNIVERSITY The discrete Fourier series A periodic sequence with period N x n n rN Periodic sequence can be represented by a Fourier series i.e. a sum of complex exponential sequences with frequencies being integer multiples of the fundamental frequency 2 n N associated with the n 1 j 2n N kn The frequency of the periodic sequence. x n N yX k e Only N unique harmonically related complex exponentials since gj 2n N k mN n _gj 7.n N kngj2nmn _gj 2n N kn a so n -1 y1
