Transform-Domain Representation of Discrete-Time Signals Three useful representations of discrete-time sequences in the transform domain Y Discrete-time Fourier Transform (DTFT v Discrete Fourier Transform() √z- Transform Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 Transform-Domain Representation of Discrete-Time Signals • Three useful representations of discrete-time sequences in the transform domain: ✓Discrete-time Fourier Transform (DTFT) ✓Discrete Fourier Transform (DFT) ✓z-Transform
Discrete-Time Fourier Transform Definition- The discrete-time fourier transform dtFt)X(e/o)of a sequence is given X(e10)=∑ xInle joi In general. x(o jo) is a complex function of the real variable o and can be written as X(e/0)=X2(e0)+jXm(e) ime Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 2 Discrete-Time Fourier 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 w and can be written as ( ) jw X e ( ) jw X e =− − = n j j n X e x n e w w ( ) [ ] ( ) ( ) ( ) w w w = + j im j re j X e X e j X e
Discrete-Time Fourier Transform(DTFT) Xre(eJo) and Xim(ejo) are, respectively, the real and imaginary parts of X(eJo), and are real functions of o X(e/o)can alternately be expressed as X(eo)=X(e jo )e jo( o) where 6(0)=ag{X(e/) Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 3 Discrete-Time Fourier Transform (DTFT) • and are, respectively, the real and imaginary parts of , and are real functions of w • can alternately be expressed as where ( ) jw X e ( ) jw re X e ( ) jw Xim e ( ) jw X e ( ) ( ) ( ) w w w = j j j X e X e e ( ) arg{ ( )} w w = j X e
Discrete-Time Fourier Transform X(eo )is called the magnitude function e(o)is called the phase function Both quantities are again real functions of o In many applications, the dtfT is called the Fourier spectrum Likewise, X(eJo ) and 0(o)are called the magnitude and phase spectra Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Discrete-Time Fourier Transform • is called the magnitude function • is called the phase function • Both quantities are again real functions of w • In many applications, the DTFT is called the Fourier spectrum • Likewise, and are called the magnitude and phase spectra ( ) jw X e (w) ( ) jw X e (w)
Discrete-Time Fourier Transform For a real sequence xn] X(e/o)land Xre(e Jo are even functions of @, whereas 0(o) and Xm(ejo )are odd functions of o Note: X(ejo)=X(eo)lee(o+27k =X(e yo 6() for any integer k The phase function 0(o) cannot be uniquely specified for any DTFT Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 5 Discrete-Time Fourier Transform • For a real sequence x[n], and are even functions of w, whereas and are odd functions of w • Note: for any integer k • The phase function (w) cannot be uniquely specified for any DTFT | ( ) | j X e w (w) ( ) jw re X e ( ) jw Xim e ( 2 ) ( ) | ( ) | j j j k X e X e e w w w + = ( ) | ( ) | j j X e e w w =