Transform-Domain Representation of Discrete-Time Signals Three useful representations of discrete-time sequences in the transform domain v Discrete-time Fourier Transform DTFT Discrete Fourier Transform DFT z-ransform 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(eJo)of a sequence xn is given by (e1)=∑xnle1on In general, X(eJo) is a complex function of the real variable o and can be written as X(e10)=Xe(e/0)+jm(e0) 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(e /o)are, respectively, the real and imaginary parts of X(e/o), and are real functions of o X(eJo)can alternately be expressed as X(ejo)=r(ejo )eje(o) where 6(0)=arg{X(e/0)} 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(eJo )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 x[n1￥x(e0) and X.((e0) are even functions of o, whereas 0(o) anu im(ejo) are odd functions of o Note X￥(e)=XY(e10)e (+2k) X(e)|e(0o) for any integer k The phase function A(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 =
Discrete-Time Fourier Transform We will assume that the phase function A(o) is restricted to the following range of values π≤(0)<兀 called the principal value Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 6 Discrete-Time Fourier Transform • We will assume that the phase function (w) is restricted to the following range of values: called the principal value − (w)
Discrete-Time Fourier Transform The dtFts of some sequences exhibit discontinuities of 2T in their phase responses An alternate type o of phase function that is a continuous function of o is often used It is derived from the original phase function by removing the discontinuities of 2兀 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 7 Discrete-Time Fourier 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 w is often used • It is derived from the original phase function by removing the discontinuities of 2
Discrete-Time Fourier Transform The process of removing the discontinuities 2 is calle ed unwrapping The continuous phase function generated by unwrapping is denoted as 0c(o) ° In some cases, discontinuities ofπ may be present alter unwrapping Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 8 Discrete-Time Fourier 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 (w) c
Discrete-Time Fourier Transform Example- The dtft of the unit sample sequence 8[n is given by △(e0)=∑8 n]e on=80]=1 n=-0 Example - Consider the causal sequence xm=∞m],a<1 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 9 Discrete-Time Fourier Transform • Example - The DTFT of the unit sample sequence d[n] is given by • Example - Consider the causal sequence ( ) = d[ ] = d[0] =1 − w =− w j n n j e n e x[n] = [n], 1 n
Discrete-Time Fourier Transform Its dtFT is given by XY(e10)=∑oune-~0n=∑ae~on n=- n=0 =∑(e n=0 1-ae J as ae j0=0<1 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 10 Discrete-Time Fourier Transform • Its DTFT is given by as = = = − w =− w − w 0 ( ) [ ] n n j n n j n j n X e n e e − w − = − w = = j e n j n e 1 1 0 ( ) = 1 − jw e
