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Unformatted text preview: Discretetime Fourier Analysis (DTFT) Discretetime Fourier analysis (transform) is a powerful tool to analyze the frequency characteristics of discretetime signals and systems. If the discretetime signal is periodic, discretetime Fourier series will be used instead. Continuoustime Fourier Transform If a continuoustime signal x ( t ) is nonperiodic, the frequency characteristics can be expressed in terms of Fourier transform  = dt e t x f X ft j 2 ) ( ) ( The above equation is referred to as analysis equation. The time signal x ( t ) can be obtained (synthesized) by the corresponding inverse transform  = df e f X t x ft j 2 ) ( ) ( Discretetime Fourier Transform The frequency analysis of a discretetime signal x [ n ] is to express the signal sequence in terms of complex exponential sequences { n j e }, where is the frequency variable called as normalized radian frequency. The analysis is called as discretetime Fourier transform (DTFT) and defined by  = = n n j j e n x e X ] [ ) ( , ) , (  The DTFT is a continuous periodic function of with period of 2 . Please note that { n j e } are periodic functions with period of 2 Example: DTFT of an exponential sequence Consider the causal sequence . 1   ], [ ] [ < = n u n x n The DTFT of x [ n ] is given by =  =  = = = = ] [ ] [ ) ( n n j n n n j n n n j j e e n u e n x e X 1 1 ) ( j n n j e e = = = Note: The DTFT of most practical discretetime sequences is expressed in using geometric series. The time signal x [ n ] can be computed from the following inverse integral  = ) ( 2 1 ] [ d e e X n x n j j Proof: Substituting the DTFT definition into the inverse transform gives   =  = = 2 1 ] [ ] [ 2 1 ) ( d e m x d e e m x m n j m n j m m j We have n m n m d e m n j = =  1 ) ( 2 1 Using the above result into the inverse transform, we obtain   = = ] [ ] [ 2 1 n x d e e m x n j m m j The inverse transform is called as synthesis equation because the time signal can be synthesized by using the Fourier coefficients and complex exponential functions. Discretetime Fourier transform pair:  = = n n j j e n x e X ] [ ) (  = ) ( 2 1 ] [ d e e X n x n j j Note: If the DTFT is expressed in terms of a rational function of n j e , to find the inverse transform, we can express the rational function in terms of partial fraction expansion and use binomial expansion to find the inverse. For example: 1 1 ) 1 )( 1 ( 1 ) ( j j j j j e e e e e X = = = = = n n j n n n j n e e...
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Frequency, Signal Processing

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