ch3_1.pdf - Transform-Domain Representation of Discrete-Time Signals Discrete-Time Fourier Transform • Definition The discrete-time Fourier

ch3_1.pdf - Transform-Domain Representation of...

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1 Copyright © 2001, S. K. Mitra Transform Transform- Domain Domain Representation Representation of Discrete of Discrete- Time Signals Time Signals Three useful representations of discrete- time sequences in the transform domain: - Discrete-time Fourier Transform - Discrete Fourier Transform - z Transform 2 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform • Definition - The discrete-time Fourier transform ( DTFT ) of a sequence x [ n ] is given by In general, is a complex function of the real variable and can be written as ) ( j e X ) ( j e X n n j j e n x e X ) ( ) ( ) ( j im j re j e X j e X e X 3 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform and are, respectively, the real and imaginary parts of , and are real functions of can alternately be expressed as where ) ( j e X ) ( j re e X ) ( j im e X ) ( j e X ) ( ) ( ) ( j j j e e X e X )} ( arg{ ) ( j e X 4 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform is called the magnitude function is called the phase function Both quantities are again real functions of In many applications, the DTFT is called the Fourier spectrum • Likewise, and are called the magnitude and phase spectra ) ( j e X ) ( ) ( j e X ) (
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5 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform For a real sequence x [ n ] , and are even functions of , whereas, and are odd functions of • Note : for any integer k The phase function ( ) cannot be uniquely specified for any DTFT ) ( j e X ) ( ( j re e X ) ( j im e X ) 2 ( ) ( ) ( k j j j e e X e X ) ( ) ( j j e e X ) 6 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform Unless otherwise stated, we shall assume that the phase function ( ) is restricted to the following range of values: called the principal value ) ( 7 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform The DTFTs of some sequences exhibit discontinuities of 2 in their phase responses An alternate type of phase function that is a continuous function of is often used It is derived from the original phase function by removing the discontinuities of 2 8 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform The process of removing the discontinuities is called “ unwrapping The continuous phase function generated by unwrapping is denoted as In some cases, discontinuities of may be present after unwrapping ) ( c
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9 Copyright © 2001, S. K. Mitra Discrete Discrete- Time Fourier Time Fourier Transform Transform • Example - The DTFT of the unit sample sequence [ n ] is given by
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