Ch3(1) Transform-Domain Representation of Discrete-Time Signals.pdf

Transform-Domain Discrete-Time Fourier
Representation of Transform
Discrete-Time Signals • Definition - The discrete-time Fourier
jw
• Three useful representations of discrete- transform (DTFT) X ( e ) of a sequence
x[n] is given by
time sequences in the transform domain: ¥
- Discrete-time Fourier Transform X(ejω) = å x[n]e­ jωn
- Discrete Fourier Transform n=­¥
jw
- z Transform • In general, X ( e ) is plex function
of the real variable w and can be written as
jw jw jw
X(e ) = Xre(e ) + j Xim(e )
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Copyright © 2001, S. K. Mitra Copyright © 2001, S. K. Mitra
Discrete-Time Fourier Discrete-Time Fourier
Transform Transform
jw jw jw
• X re ( e ) and X im ( e ) are, respectively, • X ( e ) is called the magnitude function
jw
the real and imaginary parts of X ( e ) , and • q ( w ) is called the phase function
are real functions of w • Both quantities are again real functions of w
jw
• X ( e ) can alternately be expressed as • In many applications, the DTFT is called
jw jw jq(w)
X(e ) = X (e ) e the Fourier spectrum
where • Likewise, X ( e j w ) and q ( w ) are called the
q(w) = arg{X (e jw)} magnitude and phase spectra
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Copyright © 2001, S. K. Mitra Copyright © 2001, S. K. Mitra
Discrete-Time Fourier
Discrete-Time Fourier
Transform
jw jw Transform
• For a real sequence x[n], X ( e ) and Xre(e )
are even funct