电子科技大学：《数字信号处理》课程教学资源（PPT课件讲稿，英文版）Chapter 3 TransformDomain-Representation of Discrete-Time Signals

Chapter 3 Transform-Domain Representation of Discrete-Time Signals

Chapter 3 Transform-Domain Representation of Discrete-Time Signals

83.1 Discrete-Time Fourier Transform Definition The discrete-time fourier transform TFT)X(eJo ) of a sequence xn is given by X(e O )=∑xnle on 1=-00 n general, X(ejo )is a complex function of the real variable o and can be written as X(ejo)=xre(eJo)+j Xim(ejo)

§3.1 Discrete-Time Fourier Transform • Definition - The discrete-time Fourier transform (DTFT) X(ej) of a sequence x[n] is given by   =− − = n j j n X e x n e   ( ) [ ] X(ej) = Xre(ej) + j Xim(ej) In general, X(ej) is a complex function of the real variable  and can be written as

83.1 Discrete-Time Fourier Transform Xr(eo)and xim(eo)are, respectively, the real and imaginary parts of x(eJo), and are real functions of o 。x(e°) can alternately be expressed as X(ejo)= X(ejo) eje(@) where 6(0)=arg{X(e)}

§3.1 Discrete-Time Fourier Transform • Xre(ej) and Xim(ej) are, respectively, the real and imaginary parts of X(ej) , and are real functions of  • X(ej) can alternately be expressed as X(ej) = | X(ej) |ej() where () = arg{X(ej) }

83.1 Discrete-Time Fourier Transform X(eo) is called the magnitude function e(o)is called the phase function Both quantities are again real functions In many applications, the dtft is called the fourier spectrum Likewise, X(ejo)l and e(@) are called the magnitude and phase spectra

§3.1 Discrete-Time Fourier Transform • | X(ej) | 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, | X(ej) | and () are called the magnitude and phase spectra

83.1 Discrete-Time Fourier Transform For a real sequence xn, X(ejo) and Xre(ejo) are even functions of o, whereas, H(o)and Xim(ejo)are odd functions of a Note: X(ejo)= X(ejo) ejb(o+Tk) I X(ejo)jeje(o) for any integer k uniquely specified for any DTA o The phase function A(@)cannot be

§3.1 Discrete-Time Fourier Transform • For a real sequence x[n], | X(ej) | and Xre(ej) are even functions of , whereas, () and Xim(ej) are odd functions of  • Note: X(ej) = | X(ej) |ej(+2k) = | X(ej) |ej() for any integer k • The phase function () cannot be uniquely specified for any DTFT

83.1 Discrete-Time Fourier Transform Unless otherwise stated, we shall assume that the phase function ((o) is restricted to the following range of values π≤θ(0)≤ called the principal value

§3.1 Discrete-Time Fourier Transform • Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: - ()   called the principal value

83.1 Discrete-Time Fourier Transform The DTFTs 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 of2π

§3.1 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  is often used • It is derived from the original phase function by removing the discontinuities of 2

83.1 Discrete-Time Fourier Transform Example-The dtFT of the unit sample sequence 8n is given by △(e)=∑8[neon=8O]=1 Example- Consider the causal sequence x[n]=a"u[n]a<1

§3.1 Discrete-Time Fourier Transform • Example - The DTFT of the unit sample sequence d[n] is given by ( ) = d[ ] = d[0] =1 −   =−  j n n j e n e x[n] =  [n],  1 n • Example - Consider the causal sequence

83.1 Discrete-Time Fourier Transform · Its DTFT is given by 1= n=o e on X(e/0)=∑ un]e j=∑a ∑(oe-)=.1 0 1-ae o ae Jo=a <1

§3.1 Discrete-Time Fourier Transform • Its DTFT is given by =   =   = −   =−  −  0 ( ) [ ] n n j n n j n j n X e n e e −  −  = −  =   = j e n j n e 1 1 0 ( )  =  1 − j as e

83.1 Discrete-Time Fourier Transform The magnitude and phase of the dtft ⅹ(e°) 1/(1-0.5e-Jo)are shown below 04 0.4

§3.1 Discrete-Time Fourier Transform • The magnitude and phase of the DTFT X(ej) = 1/(1 – 0.5e-j) are shown below