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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 π 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 π 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 ϖ ˆ }, where ϖ ˆ 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 ϖ ϖ ˆ ˆ ] [ ) ( , ) , ( ˆ π π ϖ ∈ The DTFT is a continuous periodic function of ϖ ˆ with period of 2 π . Please note that { n j e ϖ ˆ } are periodic functions with period of 2 π Example: DTFT of an exponential sequence Consider the causal sequence . 1   ], [ ] [ < = α α n u n x n The DTFT of x [ n ] is given by ∑ ∑ ∑ ∞ = ∞∞ = ∞∞ = = = = ˆ ˆ ˆ ˆ ] [ ] [ ) ( n n j n n n j n n n j j e e n u e n x e X ϖ ϖ ϖ ϖ α α ϖ ϖ α α ˆ ˆ 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 ∫ = π π ϖ ϖ ϖ π ˆ ) ( 2 1 ] [ ˆ ˆ d e e X n x n j j Proof: Substituting the DTFT definition into the inverse transform gives ∫ ∫ ∑ ∑ ∞∞ = ∞∞ = = ⋅ π π ϖ π π ϖ ϖ ϖ π ϖ π ˆ 2 1 ] [ ˆ ] [ 2 1 ) ( ˆ ˆ ˆ 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 ) ( ˆ 2 1 ϖ π π ϖ π Using the above result into the inverse transform, we obtain ∫ ∑ ∞∞ = = ⋅ π π ϖ ϖ ϖ π ] [ ] [ 2 1 ˆ ˆ 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 ϖ ϖ ˆ ˆ ] [ ) ( ∫ = π π ϖ ϖ ϖ π ˆ ) ( 2 1 ] [ ˆ ˆ 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 ϖ ˆ , 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: ϖ β α β ϖ β α α ϖ ϖ ϖ β α β α ˆ ˆ ˆ ˆ ˆ 1 1 ) 1 )( 1 ( 1 ) ( j j j j j e e e e e X = = ∑ ∑ ∞ = ∞ = = ˆ ˆ n n j n n n j n e e ϖ...
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Frequency, Signal Processing, jω

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