Chapter 3. The Discrete Fourier Transform The Discrete Fourier Series Definition: Periodic sequence N: the fundamental period of the sequences From FT analysis we know that the periodic functions can be synthesized as a bination plex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency (2pi/N). From the frequency-domain periodicity of the DTFT, we conclude that there are a finite number of harmonics; the frequencies are {2pi/N*k,k=0,1,…,N-1}. Engineering college, Linyi Normal University Fourier series of periodic continuous signals Ω0—period of x(t) in radian; Let T---sampling period; ω0----smpling period in radian Engineering college, Linyi Normal University So X(k) is also a periodic function with N Engineering college, Linyi Normal University DFS pair Engineering college, Linyi Normal University Properties of DFS Suppose the following 3 sequences’s period is N Linearity Engineering college, Linyi Normal University Shifting Symmetry Engineering college, Linyi Normal University Periodic convolution Distinction with convolution sum Engineering college, Linyi Normal University Engineering college, Linyi Normal University The Discrete Fourier Transform Suppose: x(n)------finite-length sequence, N-----length; . , x(n)=0 when n<0 or n>N-1 Let x(n) be a period sequence of a periodic sequence Then we have Engineering college, Linyi Normal University
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