电子科技大学电子工程学院：《信号与系统》课程教学资源（PPT课件讲稿，英文版）Chapter 5 THE DISCRETE-TIME FOURIER TRANSFORM

Chapter 5 The discrete-Time Fourier transform CHAPTER 5 THE DISCRETE-TIME FOURIER TRANSFORM

1 CHAPTER 5 THE DISCRETE-TIME FOURIER TRANSFORM Chapter 5 The Discrete-Time Fourier Transform

Chapter 5 The discrete-Time Fourier transform Consider a discrete-time lti system: Eigenfunction 特征函数 H H(x)=∑kn— Eigenvalue(特征值) xn]=xn+n@%=2r/N x∑a”=∑% k= 2兀 j y=∑ S keko).iko,n ∑ ahle k

2 Chapter 5 The Discrete-Time Fourier Transform hn yn n z n z ( ) n H z z Eigenfunction 特征函数 ( )   ——Eigenvalue (特征值） n n H z h n z − + =− =  Consider a discrete-time LTI system: xn= xn+ N 0 = 2 / N   n N jk k k N jk n k k N x n a e a e   2  0  =  =  = =   ( ) N jk N jk k k N jk jk n k k N y n a H e e a H e e     2 2 0 0         =  =  =  = 

Chapter 5 The discrete-Time Fourier transform N N2 N 0 Discrete-time Fourier transform pair x{] tejo lejon do Synthesis equation 2丌J2丌 eo)=∑xlp Xle Analysis equation n=-

3 Chapter 5 The Discrete-Time Fourier Transform Discrete-time Fourier transform pair   ( )      x n X e e d j j n  = 2 2 1 ( )   j n n j X e x n e  −  + =− =  Synthesis equation Analysis equation

Chapter 5 The discrete-Time Fourier transform onvergence ssues Associated with the Discrete-Time Fourier Transform 1. xn is absolutely summable, ∑x[|<∞ 2. xin] has finite energy. <OO n=-0 Discrete-time r(eio)=x(eio H(eio) Fourier Analysis F-Y 4

4 Chapter 5 The Discrete-Time Fourier Transform Convergence Issues Associated with the Discrete-Time Fourier Transform   n x n + =−    1. is absolutely summable, xn 2. has finite energy, xn      + =− 2 x n n ( ) ( ) ( ) j j j Y e = X e H e    ( )  1 F - j y n Y e  = Discrete-time Fourier Analysis

Chapter 10 The z-transform CHAPTER 10 THE Z-TRANSFORM

5 Chapter 10 The Z-Transform CHAPTER 10 THE Z-TRANSFORM

Chapter 10 The z-transform §10.1TheZ- Transform X(z)=∑xk n=-00 x小]X(z) X(z=Fkn/- unit circle Xle z-e ROC=z=1 Z-plane 6

6 ( )   n n X z x n z − + =−  =  §10.1 The Z-Transform xn⎯→ X(z) Z ( )     n X z x n r − =F Chapter 10 The Z-Transform ROC  z =1 ( ) ( )   j z e j X e X z = = 0 1 Z-plane z =1 unit circle

Chapter 10 The z-transform Example 10.1 x/n]=a"un na z Example 10.2 xn]=-a"ul-n-1 n"l-n-<2)1-0nrk c< a 7

7 Chapter 10 The Z-Transform Example 10.1 xn a un n =   1 1 1 − − ⎯→ az a u n n Z z  a 0 a Example 10.2 xn= −a u− n −1 n   1 1 1 1 − − − − − ⎯→ az a u n n Z z  a 0 a z  a

Chapter 10 The z-transform x[n]2 ,X×

8 Chapter 10 The Z-Transform Example 10.3 xn ( ) un ( ) un n n = 7 1/3 −6 1/ 2 ( ) ( ) ( 1/ 3)( 1/ 2) 3/ 2 − − − = z z z z X z 2 1 z  0 2 1 3 1 2 3   ( ) Z x n X z ROC ⎯→ ;

Chapter 10 The z-transform 510.2 The Region of Convergence for the Z-Transform Property 1: The roc of X(z consists of a ring in the z-plane centered about the origin x(a)=FkInI Property 2: The roc does not contain any poles

9 Chapter 10 The Z-Transform §10.2 The Region of Convergence for the Z-Transform Property 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin. ( )     n X z x n r − =F   n n x n r + − =−    Property 2: The ROC does not contain any poles

Chapter 10 The z-transform Property 3: If xIn is of finite duration, then the Roc is the entire z-plane, except possibly F0 and/or zoo x{四]=0,nN2 xnr“0z=0女ROC X(z)=∑ -n nz+ ∑|[lz n=M n=0 positive powers ofz negative powers of乙

10 Chapter 10 The Z-Transform Property 3: If is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and/or z=∞. xn   1 2 x n = 0 , n  N ;n  N   2 1 N n n N x n r − =    ① N1  0 z =   ROC ② N2  0 z = 0  ROC ( )     n N n n n N X z x n z x n z − = − − = =  + 2 1 0 1 positive powers of z negative powers of z