Discrete_Fourier_Transform_w2010

Discrete_Fourier_Transform_w2010 - S Y S C 4 4 0 5 : D i g...

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SYSC4405: Digital Signal Processing Discrete_Fourier_Transform_w2010.fm MOHAMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 1 / 13 Discrete-Time Fourier Series Discrete Time Fourier Series (DTFS) provides a means to express a periodic sequence x(n), with period N, in terms of a finite set of complex exponentials n is a time index, and k is a frequency index. The set of exponentials { φ k (n)} are periodic both in k and in n. They are also orthogonal in the sense that: The DTFS is of the form In the expression above n is a time index and k is the frequency index. {C kd } are the DTFS coefficients and are given by: φ k n () j 2 π kn N ------------ ⎝⎠ ⎛⎞ exp = φ m n φ k n n 0 = N 1 Nm k 0 mk = xn C kd φ k n k C j 2 π N exp k == n ≤≤
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SYSC4405: Digital Signal Processing Discrete_Fourier_Transform_w2010.fm MOHAMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 2 / 13 where Based on the relationship above, it can be shown that {C kd } are periodic in k with period N. Also note that C kd is associated with the complex exponential The exponential can be rewritten as: where T is the sampling period. In other words, may be viewed as a discrete-time version of a continuous-time exponential obtained by taking uniformly spaced samples every T sec interval, where the frequency f is given by: . The frequency index k is best chosen in the range Time shifting property of the DTFS Assume x(n) is periodic with period N: C kd 1 N --- xn () j 2 π kn N ------------ ⎝⎠ ⎛⎞ exp n 0 = N 1 = k 0 kk 0 N 1 + ≤≤ j 2 π N exp φ k n j 2 π N exp = φ k n j 2 π N exp j 2 π N f s T exp j 2 π kf s N ------ nT exp == = φ k n j 2 π ft exp fk f s N = N 1 2 m N 1 2
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SYSC4405: Digital Signal Processing Discrete_Fourier_Transform_w2010.fm MOHAMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 3 / 13 Let y(n)=x(n-n 0 ); i.e. y(n) is a time shifted version of x(n). Y(n) is still periodic with period N and can be written as a DTFS in the form: It can be shown that the DTFS coefficients for x(n) and y(n) are related by: In other words, time shifting does not change the magnitude of the DTFS coefficents. The phase changes resulting from time shifting are dependent on the frequency index m. Example The continuos time signal is sampled at a rate to produce a sequence . Determine the DFS coefficients of the sequence .
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Discrete_Fourier_Transform_w2010 - S Y S C 4 4 0 5 : D i g...

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