1、1CHAPTER 5THE DISCRETE-TIMEFOURIER TRANSFORM Chapter 5 The Discrete-Time Fourier Transform2 Chapter 5 The Discrete-Time Fourier Transform nh nynznz nzzH Eigenfunction特征函数特征函数Eigenvalue (特征值）特征值） nnznhzHConsider a discrete-time LTI system: NnxnxN/20 nNjkkNknjkkNkeaeanx 20 NjkNjkkNknjkjkkNkeeHaeeHany
2、22003 Chapter 5 The Discrete-Time Fourier TransformDiscrete-time Fourier transform pair deeXnxnjj221 n jnjenxeX Synthesis equationAnalysis equation4 Chapter 5 The Discrete-Time Fourier Transformxnak5 Chapter 5 The Discrete-Time Fourier Transform6 Chapter 5 The Discrete-Time Fourier TransformConverge
3、nce Issues Associated with the Discrete-Time Fourier Transform nxn1. is absolutely summable, nx2. has finite energy, nx 2 nxn jjjeHeXeY j-eYny1F FDiscrete-time Fourier Analysis7 Chapter 10 The Z-TransformCHAPTER 10THE Z-TRANSFORM8 nnznxzX10.1 The Z-Transform zXnxZ Z jrez njnrenxzX njnnernx nrnxzXF F
4、 Chapter 10 The Z-Transform n jnjenxeX 9 Chapter 10 The Z-Transform zROC1 jezjzXeX01Z-plane1zunit circleExample 10.1 nuanxn nnnznuazXnnaz10 111azzX11az 111aznuanZ Zaz 0aaz 10 Chapter 10 The Z-TransformExample 10.2 1nuanxn nnnznuazX1nnaz11mmmnza11 1111za 111azzX11za1111aznuanZ Zaz 0aaz 11 Chapter 10
5、The Z-TransformExample 10.3 nununxnn2/163/17 131173/17znunZ Z 121162/16znunZ Z31z21z 1121311617zzzX 2/13/12/3zzzzzX21z021312321z12 Chapter 10 The Z-TransformExample 10.4 nunnxn4/sin3/1 nuejejnxnjnnjn4/4/3/1213/121 nuejnj4/3121 14/3112/1zejj Z nuejnj4/3121 14/3112/1zejj Z31314/ jez31314/ jez 4/4/3131
6、23/ jjezezzzX31z13 Chapter 10 The Z-Transform10.2 The Region of Convergence for the Z-TransformProperty 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin. nrnxzXF F nnrnxProperty 2: The ROC does not contain any poles.14 Chapter 10 The Z-TransformProperty 3: If is of fini
7、te duration, then the ROC is the entire z-plane, except possibly z=0 and/or z=. nx 21;,0NnNn nx nNNnrnx21ROC zN01ROC 002zN nNnnNnznxznxzX2101positive powers of znegative powers of z15 Chapter 10 The Z-TransformExample 10.5 1nnznn Z0znnznn11 Z1 z0zzznnnn11 Zzz and 016 Chapter 10 The Z-TransformExampl
8、e 10.6 Nnunuanxn nnNnzazX10 1111azazzXN azzazzXNNN1 kjNNjeare2Nkjark/2 Zeros:(N-1)st order polea0z17Property 4: If is right sided, 1,0Nnnx nx Chapter 10 The Z-TransformROC 0rROC0rzmaxrz If is right sided, nx01NFurthermore, ifROCz01NIf is not causal, nx nnnNnznxznxzX011positive powers of z18 Chapter
9、10 The Z-TransformExample Nnuanxn nnNnzazXNmmNmnaz10 111azazzXNazN 0zazN and 019Property 5: If is left sided, 1,0Nnnx nx Chapter 10 The Z-TransformROC 0rROC0rzminrz If is left sided, nx01NFurthermore, ifROCz 001NIf is not anticausal, nx nNnnnznxznxzX110negative powers of z20 Chapter 10 The Z-Transfo
10、rmExample Nnuanxn nnNnzazXmNmmnza1 zazazXN111 00NNNkkNkmza10az 00NNaz 0 21 Chapter 10 The Z-TransformProperty 6: If is two sided, nxROC 0r021rrzrThe ROC of X(z) isExample 10.7 0bbnxn , 1nubnubnxnn1111zbbz1111bzbz 10 b 1111111zbbzzX1b X(z) does not exist.bzb122 Chapter 10 The Z-Transform 11211131zzzX
11、Example 23/12zzz2z is right sided nx023102310231231 z is two sided nxis left sided nx31z 23 Chapter 10 The Z-TransformBasic Z-Transform pairs: 01zn Z 111aznuanZ Zaz 1111aznuanZ Zaz 111znuZ1z1111znuZ1z24 Chapter 10 The Z-Transform10.3 The Inverse Z-Transform deeXnxnjj221 nrnxzXF F dezXrnxnjn221 derzX
12、nxnjn221 jrez jzddjredzj dzzzXjnxn 121 nz25 Chapter 10 The Z-Transform1. Partial-Fraction ExpansionExample 10.9 1113/114/116/53zzzzXDetermine for all possible ROC. nx 113/114/11zzzXSolution 141z31z12Solution 2 3/14/16/532zzzzzX 3/14/16/53zzzzzX3/124/11zz26 Chapter 10 The Z-Transform 113/1124/111zzzX
13、31z 3141 z 41z nunxn4/1 nun3/12 nunxn4/113/12nun 14/1nunxn13/12nun27 Chapter 10 The Z-Transform2. Power-Series Expansion (幂级数展开法）幂级数展开法） nnznxzX 1021012zxzxzxzxzXPower-Series of z-1 12324zzzXExample 10.12 z0 otherwise , 0 1n , 3 0n , 2 -2n , 4 nx 13224nnnnx 28 Chapter 10 The Z-Transform 111azzXaz Ex
14、ample 10.13 Consider 11az111az11az1az221zaaz22za22za3322zaza zX33 za nnanxn000 nuanxn29 Chapter 10 The Z-Transform 111azzXaz Example 10.14 Consider 11az1za11za1221zaza22za3322zaza zX33za nnanxn001 - 1nuanxnza122za33za30 Chapter 10 The Z-Transform azazzX ,1ln1Example 10.14 Consider the z-transform11a
15、zaz nxxxxxnn 1321321ln11x if nzazXnnnn111az otherwise 0111 nnanxnn 1nunanxn31 Chapter 10 The Z-Transform10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot zROC1 jezjzHeH iniimizzMzH11 ijniijmijeeMeH 11iiije Pole vector:ijiijeAe Zero vector:ijiijeBe iAije iBii1z je32 Chapter 1
16、0 The Z-Transform11AeHj Example Consider a first-order system 12/111zzH21z nmjAAABBBMeH2121 nmjH 212111 jH2/101A11B1z33 Chapter 10 The Z-Transform10.5 Properties of the z-Transform10.5.1 Linearity 222111R;zXnxR;zXnx ZZ zbXzaXnbxnax2121Z21RRROC 1111aznuanxnZaz 12nuanxnnnnza1Z1111azaz nxnx211Z0z34 Cha
17、pter 10 The Z-Transform10.5.2 Time Shifting RROCzXnx Z RROCzXznnxn Z00Consider z=0 or z= 111aznuanxnZaz 111nuanxn11azzZz and az 1111aznuanxnZaz 122nuanxn121azzZaz 035 Chapter 10 The Z-Transform mNnnxm 0Example 0zzmNnmN Z NmNmzzzX110 1NNzzzXZeros: z=01102,N-, kezNkjk Poles:1z1Nst order zero 36 Chapte
18、r 10 The Z-Transform10.5.3 Scaling in the z-Domain RROCzXnx Z RzROCzzXnxzn000/ ZPoles of : zXkjkkerz Poles of :0/ zzX000 kjkkerrzzSpecially ,00 jez nxenj0 RROCzeXj Z0 37 Chapter 10 The Z-Transform 111znuZ1z nuenj0 Z e-zj-0111 1z nuenj0 Z e-zj-0111 1z nun0cos Z e-ze-zj-j-001112/112/1 zz-z-20101cos21c
19、os1 1z nunan0cos Z zaaz-az-220101cos21cos1 az 38 Chapter 10 The Z-Transform10.5.4 Time Reversal RROCzXnx ZRROCzXnx1/1 ZPoles of : zXkjkkerz Poles of :zX/1kjkkerz 1139 Chapter 10 The Z-Transform 111znuZ1zznu11Zzznu11Z1z111z1zTime ReversalTime ShiftingScaling in the z-Domain1nuan111azZaz 40 Chapter 10
20、 The Z-Transform10.5.5 Time Expansion otherwise , 0 of multiple a is if knknxnxk/ RROCzXnx Z kkkRROCzXnx/1 Z Znxk nknznx kmmzmx mkmzmx kzXkmn kkRz/1/10Rz 041 Chapter 10 The Z-Transform10.5.6 Conjugation RROCzXnx Z RROCzXnx Z nnznxzX nnznxzXIf is real , nx nxnx zXzX实序列实序列 的复极点共轭成对出现。的复极点共轭成对出现。 nx42
21、Chapter 10 The Z-Transform10.5.7 The Convolution Property 222111R;zXnxR;zXnx ZZ zXzXnxnx2121Z21RRROC nxnx21Z Z nnnnznxznx21Example 10.15 1nnnh First Differencing一阶差分一阶差分 011z zzH R OC ZRzXz11 1nxnxExceptz=0 ，z=43 Chapter 10 The Z-TransformExample 10.16 nunhConsider a summation kxnunxnk 1111z zzH nun
22、x Z111zzX1zRROC44 Chapter 10 The Z-Transform 11anuanxn nunx2Example Determine nxnx21 1121111zazzXzX 1,max az 1111zaz1aaa11 nxnx21 nuanuaan1111 nuaan111 nxnx2145 Chapter 10 The Z-Transform10.5.8 Differentiation in the z-Domain RROCzXnx Z RROCdzzdXznnx Z azazzX ,1ln1Example 10.17 dzzdXznnx Z111azaz 11
23、nuaannxn 1nunanxn46 Chapter 10 The Z-Transform 111aznuanZ Zaz nunan2111azazZaz azaznuann Z21111More generally , azaznuammnnnmn Z1111!2147 Chapter 10 The Z-TransformExample nunnnx21 nunn2213111zZ1z121nunn3111zzZ nunn213111zzZ1z48 Chapter 10 The Z-Transform10.5.9 The Initial-Value Theorem , 0,0nnxIf t
24、hen zXxz lim0 , 0,0nnnxIf then zXznxnz0lim0Example 0/1zezXz Z nx CausalDetermine the initial-value 1,0 xx 1zzex/1lim0 1lim1/1zzezx149 Chapter 10 The Z-Transform1a 11111limlim1azznxzn1a 1a 终值不存在。终值不存在。 azazzX 111Example 0 11111limlim1zznxzn11a 终值不存在。终值不存在。10.5.10 The Final-Value Theorem zXzxz111lim如果
25、如果 的极点均在单位圆内（允许在的极点均在单位圆内（允许在z=1有一个一阶极点）有一个一阶极点） zX zXnxZ因果序列因果序列50 Chapter 10 The Z-Transform10.7 Analysis and Characterization of LTI Systems using z-Transforms nh ny nx zX zH zY nhnxny zHzXzY10.7.1 CausalityA discrete-time system is causalincluding infinity. max:rzzHROC of If is rational function
26、, zH系统因果系统因果分子阶数不大于分母阶数分子阶数不大于分母阶数 max:rzzHROC of 51 Chapter 10 The Z-TransformExample 10.20 81412232zzzzzzHThis system is not causal.Example 10.21 2211111121zzzzH , It is a causal system. nunhnn22152 Chapter 10 The Z-Transform10.7.2 Stability nhnA stable systemA discrete-time system is stable 1 zzH
27、ROC of Example 10.21 112111121zzzH2z 221 z 21z The system is causal but not stable.The system is not causal but stable.The system is anticausal and not stable.53 Chapter 10 The Z-Transform如果如果 为有理函数为有理函数, zH系统因果、稳定系统因果、稳定maxrz zH的极点均在单位圆内的极点均在单位圆内 H e-rze-rzzj-j-0011111 H zrrz-z-2201cos211 Example 1
28、0.24stable.not is system The 1rstable. is system The 1r54 Chapter 10 The Z-Transform10.7.3 Linear Constant-Coefficient Difference EquationsknxbknyakMkkNk00 zXzbzYzakkMkkkNk00 kkNkkkMkzazbzXzYzH00ROC55 Chapter 10 The Z-TransformExample Consider an causal system for which the input and output satisfy
29、the linear constant-coefficient equation ny nx(a) Determine the unit impulse response .(b) Determine the unit step response .(c) Determine the unit impulse response of another system whichsatisfy the following linear constant -coefficient equation. nh ns 1213261165nxnxnxnynyny nxnynyny26116556 Chapt
30、er 10 The Z-Transform例例 已知一因果已知一因果LTI系统的单位阶跃响应系统的单位阶跃响应 ，当输入，当输入为为 时，其零状态响应时，其零状态响应 ，求输入，求输入 ns nx isnyni0 nx nunnx1 1nyny nsisisnini100 zYz11 zHz111 nhnu zHzXzY 2111zzX1z57 Chapter 10 The Z-TransformExample 10.26 Suppose that we are given the following information about an LTI system:2. If ，then th
31、e output isDetermine the system function for this system, and deduce the causality and stability of this system. zH nunxn61. 11 nuanynn3110211 nnx12 nny14/72Write the difference equation characterizes the system.58 Chapter 10 The Z-Transform9a 212161651316131zzzzzH 2311613261165nxnxnxnynyny59 Chapte
32、r 10 The Z-TransformExample 10.27 一具有有理系统函数一具有有理系统函数 的因果、稳定系统，的因果、稳定系统，在在 有一极点，在单位圆上某处有一零点，有一极点，在单位圆上某处有一零点，其余零极点未知，试判断下列说法是否正确。其余零极点未知，试判断下列说法是否正确。 zH2/1z1. 收敛。收敛。 nnhF2/12. 对某一对某一值有值有0 jeH nnnh2 2zzHROC of 60 Chapter 10 The Z-Transform3. 为有限长序列为有限长序列 nh单位脉冲响应单位脉冲响应5. 是一因果、稳定系统的是一因果、稳定系统的 nhnhnng4.
33、 为实信号。为实信号。 nh无法判断。无法判断。61Three basic operations1. Addition nx1 nx2 nxnx212. Multiplication by a coefficient nx naxa3. Delay nx1nxZ-1 Chapter 10 The Z-Transform10.8 System Function Algebra and Block Diagram Representations11 nxnx21 nx2 nx1a nax nx1z nx1nx62 Chapter 10 The Z-TransformBlock Diagram Re
34、presentationsExample 10.29 Consider the causal LTI system 1141121zzzH 12141nxnxnyny 14112nynxnxny nwS1S2 nw1z-2 nx1/4 ny1z63交换交换S1和和S2的连接顺序的连接顺序S1S2输入相同输入相同输出相同输出相同系统等价为：系统等价为： Chapter 10 The Z-Transform1/4-2 ny nx1z1z1/4-21z ny nx64 Chapter 10 The Z-TransformExample 10.30 Consider the causal LTI sy
35、stem 218/14/111zzzH nxnynyny281141 281141nynynxny-1/41z ny nx1z1/865 Chapter 10 The Z-TransformExample 10.31 Consider the causal LTI system 21218/14/112/14/71zzzzzH 212121471814111zzzzzH/ 1z1z-1/4 nx1/8 nwS2-7/4-1/2 ny1nw2nw66 Chapter 10 The Z-Transform信号流图模拟信号流图模拟两个基本约定：两个基本约定：1. 假定所有的环路均相互接触；假定所有的
36、环路均相互接触；iiL12. 假定每一前向通路与所有的环路相互接触；假定每一前向通路与所有的环路相互接触；1kiikkkkkLggHov1167 Chapter 10 The Z-TransformExample 10.31 Consider the causal LTI system 21218/14/112/14/71zzzzzH zY-1/2Z-1Z-1 zX1公共点公共点-1/41/8-7/411. 直接模拟直接模拟68 Chapter 10 The Z-Transform2. 级联模拟级联模拟 111141121211411zzzzzH zX1Z-11-1/21/41/4-2 zY1
37、Z-11169 Chapter 10 The Z-Transform2. 并联模拟并联模拟 114113/142115/34zzzH13/5 zX zY4-14/311121z141z170 Chapter 10 The Z-Transform10.9 The Unilateral z-Transform nnIznxzX0 Z ZnunxzXImaxrz 0211ndzzzXjnxnI zzXIXIf is causal, nx71 Chapter 10 The Z-TransformExamples nuanxn 1. 111azzXzXIaz Z1nx11azzzaz and IZ1nx
38、nnnznua11011azaaz nanxn , 2.It does not exist bilateral z-Transform. 111zzXI1z72 Chapter 10 The Z-Transform10.9.2 Properties of the Unilateral z-Transform1. Time-Shifting RROCzXnxI ZI0m If ZI kmkImzkxzXzmnx10 ZI kmkImzkxzXzmnx173 Chapter 10 The Z-Transform10.9.3 Solving Difference Equations Using th
39、e Unilateral z-Transform时域解时域解经典解法经典解法零输入、零状态解法零输入、零状态解法频域解频域解 jjjeHeXeY j-feYny1F nynynypc复频域解复频域解双边双边Z变换变换初始状态为零初始状态为零 zHzXzY zYny-f1Z Z单边单边Z变换变换初始状态不为零初始状态不为零差分方程的求解差分方程的求解 nynynyfx nynynyfx ,74 Chapter 10 The Z-TransformExample 10.36Consider a causal LTI system nxnyny13If the input ,nunx Determi
40、ne the output . ny1. If the system is initial rest, 0 0nnx, 0 0nny,Using bilateral z-Transform 11131zzzHzXzY 3z nunyn 4343 75 Chapter 10 The Z-Transform nxnyny13 ynunx1 2,. Using unilateral z-Transform 11113zzyzYzzYII 1113113113zzzyzYI nyxIZ Z nyfIZ Z3z 0 0n n 434333 nnny nyx nyf76 Chapter 10 The Z-
41、Transform10.10 Z变换与拉氏变换的关系变换与拉氏变换的关系一一 Z平面与平面与S平面的关系平面的关系 nTxnx txnTx采样采样 nTtnTxnTttxtxnns snTnsenTxsX nnznTxzXnxZ sTezssXzX77 Chapter 10 The Z-TransformTjTjsteereez TerT 10 . 1rjs 10 . 2r 10 . 3r 00 . 4 2 . 5sconstantconstant . 6 sT 278二二 连续时间系统和离散时间系统的关系连续时间系统和离散时间系统的关系 Chapter 10 The Z-Transform
42、nThnh thnTh采样采样冲激响应不变法冲激响应不变法 nTtnThnTtththnns snTnsenThsH nnznThzHnhZ stezssHzH79 Chapter 10 The Z-Transform kkNkssAsH1因果系统因果系统 tueAthtskNkk1 nueAnueAnThnhnTskNknTskNkkk11 111zeAzHTskNkk因果稳定因果稳定因果稳定因果稳定 th nh80Chap.10: 10.2 10.6 10.7 10.9 10.10 10.16 10.18 10.23 10.24 10.27 Chapter 10 The Z-Transform
