113_1_113-final2010ppt

113_1_113-final2010ppt - Discrete-Time Transforms (contd)...

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1 Discrete-Time Transforms (cont’d) 2) DTFT = = ω π 2 1 ) ( ) ( ) ( ) ( d e e X n x e n x e X n j j n j n j 2 DTFT is in general a complex function and is conjugate symmetric for real functions, continuous and periodic. •Recall: When adding , real and imaginary parts add: ( a +j b ) + ( c +j d ) = ( a + c ) + j( b + d ) •When multiplying , magnitudes multiply and phases add. •Phases modulo 2 π Discrete-Time Transforms (cont’d) 3) DFT: The N-point DFT is defined as: 1 2/ N jk n N 0 () , 01 2 ( ) where Spectral resolution is 2 /N n j Xk xne kN k Xe N = = ≤≤ == in Hz: Fs/N
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2 Discrete-Time Transforms (cont’d) X(z) Z-plane 1 -1 X( w ) X(k) Angular frequencies close to ± pi are high frequencies Angular frequencies close to 0 are low frequencies The Inverse DFT = = where N n N kn j e N k k X N n p x 1 0 , / 2 1 0 ) ( 1 ) ( π −∞ = + = l lN n x n p x ) ( ) (
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3 The DFT • One has to careful about the size of the DFT (N) in relation to the length of the segment of the signal being analyzed (L). To avoid time aliasing: L N The DFT can be computed efficiently using the Fast Fourier Transform (FFT). Radix-2 FFT is quite popular where N is a power of 2 where N is a power of 2.
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113_1_113-final2010ppt - Discrete-Time Transforms (contd)...

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