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# 291 Class Note for ECE 43800 at Purdue

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Date Created: 02/06/15
Sec 1 6 Transform 105 I 16 Z Transform The z tmnsform is an important tool for lter design and for analyzing the stability of systems It is de ned as Xe 2 my 144 n7oo where z is a complex variable ie z lzlew Figure 1 70 Ztransform DTFT can be interpreted as the z transform evaluated on the unit circle Xej Xzl 145 zej I 161 Rational Z Transform We will mostly be interested in z transforms that are rational functions of 2 ie ratios of two polynomials 146 where and are polynomials Rational z transforms are transfer functions of LTl systems described by linear constant coe icient difference equations such as N71 M yn Emma izakmik 147 i0 k1 We now introduce some terminology o If there is no second term in the right hand side of Eq 147 then the system is called non recursive or nite duration impulse response FIR system 106 CHAPTER 1 ANALYSIS OF DISCRETErTIME LINEAR TIMErlNVARIANT SYSTEMS o Otherwise7 it is recursive because the current output sample is expressed in terms of the past output samples If it cannot be written as non recursive7 then it is an in nite duration impulse response system ZT mm 71 2 iXz 148 N71 I M Yz Z biz lX z 7 Z akz kY z i0 M N71 I k 1 Yz 1 Zakzikgt Xz Z bizquotl k1 i0 Therefore7 the transfer function can be written as 1112 k1 2 EH 3 l 3 assum bweo boniNJrlil 149 Zips w H H where the last equality comes from the fact that polynomials of degrees N 7 1 and M have N71 and M roots7 respectively The values ofz for which 0 ie7 the roots of the numerator 21 22 7ZNA are called the zeros of whereas the values for which is in nite ie7 zeros of the denominator 191192 7pM are called the poles H ifMgt N71thenthereareM7 N71 zerosatz0 Of In addltlon if M lt EN 7 1 then there are N 71 7 NI poles at z O Poles and zeros may also occur at z oo zero at 0077 means lim O7 M700 poles at 0077 means lim oo M700 Sec 1 6 Transform 107 Example 130 Let us consider the z transform of a un ZTanun Zanz n n0 co 2 0271 n0 1 1 7 azfl if 12quot1 lt17 7 z 7 z 7a The region of convergence R00 is the set of all values of z for which the z transform converges In this example it is gt a 0 Zero gtlt Pole El ROG A Tmz Rez Figure 171 The region of convergence of the series in Ebrample 130 Let us now consider the lter Whose transfer function is the z transform of Exam ple 130 Example 131 Let 1112 2 2 7 a a Difference Equation To see how the z transform is related to the time domain representation we expand it 39 Yz 1 Xz 1 7 az l Yz 7 1271142 7 Yz 1271142 Xz Inuerting the z transfor39m we have 108 CHAPTER 1 ANALYSIS OF DISCRETErTIME LINEAR TIMErlNVARIANT SYSTEMS b System Diagram The system diagram is depicted in Fig 172 9671 1101 Figure 172 The system diagram of Ebrample 131 z 1 delays a signal by one time unit C Frequency Response The frequency response of the system can be obtained by replacing z with 57quot in the transfer function 1 We lHlejwll liae dxliaew i 1 7 172acoswa2 1 17 a2 2a17 cosw Hej O NOOAU39ICDNOJCOO Figure 173 Magnitude response of the lter in Example 131 for several Values of the parameter a Fig 173 shows the magnitude response of the lter for di erent ualues of a In Sec 16 Transform particular note that the height of the peak is determined only by a since the term with the cosine is removed when w 0 The closer a is to the unit circle the sharper the peak and the thinner the passband As we have seen from Example 131 in general a pole near the unit circle will cause the frequency response to increase in the neighborhood of that pole a zero will cause the frequency response to decrease in the neighborhood of that zero Fig 174 11112 Eje Rez a Zeros on the unit circle 11112 c Poles near the unit circle lHe l 771 719 19 71 b A bandstop lter corresponding to a lHe l 771 719 19 71 d A bandpass lter corresponding to Figure 174 The e ects of zeros and poles near the unit Circle I 162 Region of Convergence ROC of the Z Transform Example 132 Let Then Xe 2qu i When can we sum this seriesf2 In other words when does this series convergef2 Consider these two cases 1 If g 2 then Z 1 This means that every term in the series has an absolute value greater than or equal to 1 If it is greater than 1 then every successive terms grows larger If it is equal to 1 then we are just adding 1 an in nite number of n0 times In both cases the series diverges 110 CHAPTER 1 ANALYSIS OF DISCRETErTIME LINEAR TIMErlNVARIANT SYSTEMS 2 On the other hand if gt 2 then the geometric series conuerges because lt 1 and we have 1 Xa Z 150 1 717 gt 2 Xe unde ned g 2 Roc m2 Figure 175 The ROC of Example 132 Usually gt 2 is called the region of conuergencequot R00 of the z transform because when 2 lies in this region the series actually conuerges to the function 150 A slightly more accurate term would be the region of de nitionquot since the z transfor39m is unde ned outside of this region Example 133 Let 72 u7n 71 Then 700 Yz 2727327 n71 0 7 Z 27milzm1 where m in 71 m0 o m 1 If 2 2 the series diuerges Sec 1 6 Transform 1 1 1 2 if lt 2 the series conuerges and Yz 7 Putting it all together unde ned Z 2 YZ 1 lt 2 1 We get the same expression for the z transform as in Example 132 but the R00 is di erent and in fact does not intersect the ROC from Example 132 Example 134 Let wn 2 for 7 00 lt n lt 00 Then 2nun 2nu7n 71 7 yn7 where and are from Examples 132 and 133 respectively Hence Xz7Yz 1 1 1727172 7 O What is wrong with this deriuation As we saw in the two preuious examples Xz and Yz have no common ROC Xz is unde ned for 2 g 2 Yz is unde ned for 2 2 2 This means that is not de ned for any 2 De nition 112 The region of conuergence of Z xnz n n7oo is the set of all z for which this series is absolutely conuergent ie 00 Z lt oo n7oo 112 CHAPTER 1 ANALYSIS OF DISCRETErTIME LINEAR TIMErlNVARIANT SYSTEMS We use absolute convergence to avoid certain pathological series which converge7 but do not converge absolutely An example of this is 00 Z 71 un27 at z 1 I 163 Properties of ROC Poles and Zeros The following is a list of several important properties of the z transform 1 The R00 is a ring or a disc centered at the origin 71 lt z lt r2 Note that 71 or r2 could be 0 or 00 El ROG 1m 1m 1m Rez Rea Rea a leftesided b Generic c rightisided Figure 176 The geometries of the ROC 2 The ROC cannot contain any poles 3 If is a niteduration sequence ie7 0 except for N1 n N27 then the R00 is the whole z plane except possibly 2 O 4 If is a right sided sequence 0 for n lt N17 then the R00 is gt 1pmam17 Where pm is the outermost nite pole of 5 If is a left sided sequence 0 for n 2 N27 then the R00 is 121 lt 1pmin17 where pmm is the innermost non zero pole of 6 Generalizing Example 132 and Example 1337 we have 1 where the R00 is gt la 151 1702 anun lt gt 17 7a u7n 71 lt gt 1 71 where the R00 is lt la 152 704239 Sec 1 6 Transform 1 1 3 7 An LTI system is BlBO stable if and only if the ROC of its transfer function includes 2 1 ie includes the unit circle lt 00 21 11 Zlhnl lt oo cgt BIBO stability n Note that this stability criterion is applicable to LTl systems only Example 135 Find all sequences whose z transform is 17 4271 X Z 17 3271 2272 Solution We decompose Xz into partial fractions 7 71 1 42 A1 A2 153 Xlzl 17271172271 17271 17227139 Then we solve for A1 and A2 Method 1 Rearranging the equation we have X i A1lt172271A217271 i A1A272A1A2271 Z 172 117 2271 172 117 2271 Comparing terms we solve for A1 and A2 39 A1A2 1gt A1 3 2A1A2 4 7 Method 2 Using 153 X21 2 1lz1 11 A1 sigggwk 3 A1 lX2122 1lz2 A1114LZ 7 2 114 CHAPTER 1 ANALYSIS OF DISCRETErTIME LINEAR TIMErlNVARIANT SYSTEMS Thus we have 3 2 m 7 W 39 We now consider the three possible ROC s that this z tmnsform can have Case 1 ROC gt 2 Using 151 3 1 un 7 2 2nun 3 7 2n1un Xz Case 2 R00 1 lt lt 2 Using 151 and 152 3 1 un 2 2nu7n 71 Case 3 ROClt1 Using 152 73 2n1u7n 71 Imz Case 1 Rez Case 2 Figure 177 ROC for the cases in Ekample 135 10 5 1 5 3 3 10 A A 1 A 1 530 5 5 gtlt gtlt gtlt 5o 3 7o 5 3 5 3 1 13 5 5 3 1 13 5 5 3 1 13 5 n n n a1z1gt 2 b 1 lt 121 lt 2 C1Z1lt 1 Figure 178 The inverse Ztransforms for the 3 possible ROC S of Example 135 Sec 1 6 Transform 1 1 5 I 164 Discrete Time Exponentials 23 are Eigenfunctions of Discrete Time LTI systems Suppose that a discrete time complex exponential 23 is the input signal to an LTl system with impulse response Then the output is 00 W Z hltkgtzltnekgt k7eo 23 Z hkzak k7eo If 20 is in the ROC of H27 we can write 2471 23 39 ZT 71W lzzo 2311190 as shown in Fig 179 101 H20237 zn 23 LTI system if 20 6 ROC of Hz gt with impulse gt response hn Figure 179 An LTI system with ZS as the input The transfer function is the z transform of the impulse response It is also the scaling factor of 2 when 2 goes through the system Recall that we have already considered the case of 20 57m The frequency response H 57 is o the DTFT of the impulse response Mn 0 the scaling factor when 57 is the input7 as shown in Fig 180 giwon LTl system Hgiwo5jwon gt with impulse gt response hn Figure 180 An LTI system with 67mm as the input 116 CHAPTER 1 ANALYSIS OF DISCRETETIME LINEAR TIMEINVARIANT SYSTEMS X67 f9 I I Q 39 27r w a Input signal spectrum before interpolation Xuejw f9 I I I I I I I I I I Q 39 2 2 27r w b The spectrum of the upsampled signal HLPF5W L I I 727 27r LA 6 Frequency response of the lowipass lter th6 st I I I I I I 727 27r LA d The spectrum of the interpolated signal Figure 165 The e ect of interpolation on the spectrum of the input signal x

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