DTFT - Discrete-time Fourier Analysis(DTFT Discrete-time Fourier analysis(transform is a powerful tool to analyze the frequency characteristics of

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

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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 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 ) ( ) ( 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 ϖ ˆ }, where ϖ ˆ 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 ϖ ϖ ˆ ˆ ] [ ) ( , ) , ( ˆ π π ϖ -
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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 = - -∞ = - -∞ = - = = = 0 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 0 ˆ 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 - = 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
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n m n m d e m n j = = - - 0 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. Discrete-time 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 - - - - - - - - - = - - = = - - = - - - = 0 ˆ 0 ˆ n n j n n n j n e e ϖ β α β ϖ β α α β α Comparing the expansion with the DTFT, -∞ = - = n n j j e n x e X ϖ ϖ ˆ ˆ ] [ ) ( gives F
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] [ ] [ ] [ n u n x n n β α β α β β α α - - - = Magnitude Spectrum and Phase Spectrum The magnitude spectrum and phase spectrum of a discrete-time signal x [ n
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