DTFT - Discrete-time Fourier Analysis (DTFT) Discrete-time...

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Unformatted text preview: Discrete-time Fourier Analysis (DTFT) Discrete-time Fourier analysis (transform) is a powerful tool to analyze the frequency characteristics of discrete-time signals and systems. If the discrete-time signal is periodic, discrete-time Fourier series will be used instead. Continuous-time Fourier Transform If a continuous-time signal x ( t ) is non-periodic, 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 ) ( ) ( Discrete-time Fourier Transform The frequency analysis of a discrete-time 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 discrete-time 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 discrete-time 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. Discrete-time 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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DTFT - Discrete-time Fourier Analysis (DTFT) Discrete-time...

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