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ch 11 summary - Summary Discrete-Time Fourier...

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Summary – Discrete-Time Fourier Transform (DTFT) (Chapter 11) 1. The DTFT of a DT signal x [ n ] is defined as X F ( ) = x n [ ] n = −∞ e j 2 π F n ( F is a continuous variable, called DT frequency .) 2. Key fact : Since e j 2 π F n is a periodic function of F with period 1 for all n , it follows that X ( F ) must be periodic with period 1. We usually take the fundamental period to be 1 2 < F 1 2 . 3. The inverse DTFT is given by x n [ ] = X F ( ) e j 2 π F n 1 2 1 2 dF . ( Examples: (i) If x [ n ] = a n u n [ ] where a is a real constant, a < 1 , then X F ( ) = a n e j 2 π F n n = 0 = ae j 2 π F { } n = 0 n . But recall that for any number (real or complex) b with b < 1 , b n = 1 1 b n = 0 . So, X F ( ) = 1 1 ae j 2 π F . (ii) Let F 0 < 1 2 and define H F ( ) = 1, F F 0 ; H F ( ) = 0, F 0 < F 1 2 . (Note that this defines H F ( ) for all F , since H F ( ) must be periodic with period 1. Also note that H F ( ) is the frequency response of a DT low-pass filter (LPF) having cutoff frequency F 0 .) Then h n [ ] = H F ( ) e j 2 π F n dF = 1 2 1 2 e j 2 π F n dF
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