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 hn yn 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: xn= xn+ 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, xn 2. has finite energy, xn + =− 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 xn⎯→ 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 xn a un n = 1 1 1 − − ⎯→ az a u n n Z z a 0 a Example 10.2 xn= −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 xn ( ) un ( ) un 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=∞. xn 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