华南理工大学：《数字信号处理》（双语版） Transform-Domain Representation of Discrete-Time Signals

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 =

Discrete-Time Fourier Transform We will assume that the phase function 0(o) is restricted to the following range of values π≤6(0)<T 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 tFTs of some sequences exhibit discontinuities of 2T in their phase responses An alternate type 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丌 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 is called unwrapping The continuous phase function generated by unwrapping is denoted as ec(o) In some cases, discontinuities of Tt may be present after unwrapping 8 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 △(e)=∑8[ n]e=80]=1 1=-00 Example- Consider the causal sequence x[n]=a"u[n] 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 X(e0)=∑a' u[n]e j=∑a'e-~0n n=-0 n=0 =∑ejoy=e/o n=0 as ae Jo=a<1 10 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