1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:25,516 --> 00:00:28,492 [MUSIC PLAYING] 9 00:00:55,772 --> 00:00:57,770 ALAN OPPENHEIM: Hi. 10 00:00:57,770 --> 00:01:00,680 In this lecture, there are two sets 11 00:01:00,680 --> 00:01:04,790 of ideas that I'd like to discuss, both of which 12 00:01:04,790 --> 00:01:09,200 are related to our topic of the last several lectures, namely, 13 00:01:09,200 --> 00:01:11,540 the Z-transform. 14 00:01:11,540 --> 00:01:15,290 The first topic that we will focus on 15 00:01:15,290 --> 00:01:20,840 is what I'll referred to as the geometric interpretation 16 00:01:20,840 --> 00:01:23,330 of the frequency response. 17 00:01:23,330 --> 00:01:28,320 Now, you recall that in lecture 5 when I introduced 18 00:01:28,320 --> 00:01:33,350 the Z-transform, we observed the relationship 19 00:01:33,350 --> 00:01:37,760 between the Z-transform and the Fourier transform. 20 00:01:37,760 --> 00:01:41,330 The notion behind the geometric interpretation of the frequency 21 00:01:41,330 --> 00:01:47,870 response is to relate, in terms of a geometric picture, 22 00:01:47,870 --> 00:01:53,300 the rough characteristics of the frequency response of a system 23 00:01:53,300 --> 00:01:56,720 and the pole-zero pattern for the system function 24 00:01:56,720 --> 00:01:58,650 for the system. 25 00:01:58,650 --> 00:02:02,870 The second topic that we'll focus on and eventually tie 26 00:02:02,870 --> 00:02:05,870 back into this geometric interpretation 27 00:02:05,870 --> 00:02:12,410 is the general issue of some of the properties 28 00:02:12,410 --> 00:02:14,420 of the Z-transform. 29 00:02:14,420 --> 00:02:16,850 And we'll illustrate some of these properties 30 00:02:16,850 --> 00:02:19,010 with some examples. 31 00:02:19,010 --> 00:02:22,160 But first of all, let's turn our attention 32 00:02:22,160 --> 00:02:26,990 to the subject of the geometric interpretation of the frequency 33 00:02:26,990 --> 00:02:28,880 response. 34 00:02:28,880 --> 00:02:33,680 And let me begin by reminding you 35 00:02:33,680 --> 00:02:40,280 of one of the important properties of the Z-transform 36 00:02:40,280 --> 00:02:44,870 for a linear shift invariant system, namely the fact 37 00:02:44,870 --> 00:02:50,240 that if we have a linear shift invariant system with a unit 38 00:02:50,240 --> 00:02:57,020 sample response, H of n, input x of n, and output y of n, 39 00:02:57,020 --> 00:03:00,560 then of course, y of n is the convolution of H of n 40 00:03:00,560 --> 00:03:02,120 with x of n. 41 00:03:02,120 --> 00:03:05,420 And the Z-transform of the output 42 00:03:05,420 --> 00:03:08,720 is the product of the Z-transform of the input 43 00:03:08,720 --> 00:03:13,730 and H of Z, the Z-transform of the unit sample response, which 44 00:03:13,730 --> 00:03:18,090 is what we've been referring to as the system function. 45 00:03:18,090 --> 00:03:23,070 And in first developing the Z-transform, 46 00:03:23,070 --> 00:03:27,420 we tie together the notion of the Z-transform and the Fourier 47 00:03:27,420 --> 00:03:31,920 transform, in particular, the important interpretation 48 00:03:31,920 --> 00:03:37,200 that the Fourier transform is the Z-transform evaluated 49 00:03:37,200 --> 00:03:38,740 on the unit circle. 50 00:03:38,740 --> 00:03:44,070 So in terms of the system function H of z, 51 00:03:44,070 --> 00:03:48,160 if we evaluate that on the unit circle 52 00:03:48,160 --> 00:03:53,800 for z equal to e to the j omega, the result 53 00:03:53,800 --> 00:03:59,530 is the frequency response of the system, H of e to the j omega. 54 00:03:59,530 --> 00:04:01,720 Simply a statement that the Z-transform 55 00:04:01,720 --> 00:04:05,050 evaluated on the unit circle is the frequency response 56 00:04:05,050 --> 00:04:07,130 of the system. 57 00:04:07,130 --> 00:04:11,090 Well, in terms of a simple example, 58 00:04:11,090 --> 00:04:15,270 we drag out our usual simple example. 59 00:04:15,270 --> 00:04:21,620 The system function is 1 over 1 minus az to the minus 1 or z 60 00:04:21,620 --> 00:04:24,880 divided by z minus a. 61 00:04:24,880 --> 00:04:30,110 And in terms of the z plane representation, 62 00:04:30,110 --> 00:04:33,800 or pole-zero representation in the z-plane, 63 00:04:33,800 --> 00:04:39,380 we then have a representation of this rational function in terms 64 00:04:39,380 --> 00:04:40,820 of one pole-- 65 00:04:40,820 --> 00:04:42,485 the pole at z equals a-- 66 00:04:42,485 --> 00:04:47,910 and one zero, that is the zero at z equals 0. 67 00:04:47,910 --> 00:04:51,300 Well, to get the frequency response associated 68 00:04:51,300 --> 00:04:53,700 with that system function, we want 69 00:04:53,700 --> 00:04:59,780 to look at the Z-transform evaluated on the unit circle. 70 00:04:59,780 --> 00:05:04,700 We can do that geometrically by interpreting 71 00:05:04,700 --> 00:05:09,140 this complex number as a vector in the z plane 72 00:05:09,140 --> 00:05:14,090 and this complex number as a vector in the z plane, in which 73 00:05:14,090 --> 00:05:18,020 case the magnitude of H of z will 74 00:05:18,020 --> 00:05:22,280 be the ratio of the magnitudes of those vectors. 75 00:05:22,280 --> 00:05:27,620 And the angle of H of z will be the angle of this vector 76 00:05:27,620 --> 00:05:30,240 minus the angle of this vector. 77 00:05:30,240 --> 00:05:34,270 In other words, suppose that we want to look at H of z, 78 00:05:34,270 --> 00:05:35,780 at some value of z. 79 00:05:35,780 --> 00:05:40,130 And let's say, for the moment, not on the unit circle, 80 00:05:40,130 --> 00:05:42,110 or it could be on the unit circle. 81 00:05:42,110 --> 00:05:47,630 Let's just pick a general value z equals z1. 82 00:05:47,630 --> 00:05:52,730 We have a vector corresponding to this complex number, which 83 00:05:52,730 --> 00:05:57,170 is the vector, or complex number, z. 84 00:05:57,170 --> 00:06:03,170 And that's a vector going from the origin out to this point. 85 00:06:03,170 --> 00:06:10,750 We have one vector, which is the vector z or z1 86 00:06:10,750 --> 00:06:14,320 because we're evaluating this at z equals z1. 87 00:06:14,320 --> 00:06:17,830 So this is a vector z1. 88 00:06:17,830 --> 00:06:24,440 And the vector z corresponding to the complex number z minus a 89 00:06:24,440 --> 00:06:30,840 is the vector z or z1 minus the vector a. 90 00:06:30,840 --> 00:06:34,900 The vector z1 is this vector. 91 00:06:34,900 --> 00:06:39,390 The vector minus a is this vector. 92 00:06:39,390 --> 00:06:41,900 And the sum of those two, the vector 93 00:06:41,900 --> 00:06:45,800 that corresponds to the complex number z minus a, 94 00:06:45,800 --> 00:06:49,460 is the vector whose tail is at this pole 95 00:06:49,460 --> 00:06:52,370 and whose head is at the value of z, at which we 96 00:06:52,370 --> 00:06:55,280 want to evaluate H of z. 97 00:06:55,280 --> 00:07:02,610 And so we end up then with an interpretation 98 00:07:02,610 --> 00:07:07,390 of the value of H of z in polar form 99 00:07:07,390 --> 00:07:10,990 with its magnitude equal to the magnitude 100 00:07:10,990 --> 00:07:16,630 of this vector divided by the magnitude or length of the pole 101 00:07:16,630 --> 00:07:18,070 vector. 102 00:07:18,070 --> 00:07:22,232 And the angle of this complex number, H of z at z 103 00:07:22,232 --> 00:07:26,410 equals z1 is the angle of the 0 vector 104 00:07:26,410 --> 00:07:30,010 minus the angle of the pole vector. 105 00:07:30,010 --> 00:07:32,430 That's a general statement. 106 00:07:32,430 --> 00:07:36,500 And z1 can be any place in the z plane. 107 00:07:36,500 --> 00:07:42,310 So we have then the statement as I've just made it, 108 00:07:42,310 --> 00:07:49,030 which we would now like to apply to the interpretation 109 00:07:49,030 --> 00:07:56,060 or the generation of the frequency response of a system. 110 00:07:56,060 --> 00:08:01,880 So we have, then, our example that is one pole at equals a 111 00:08:01,880 --> 00:08:06,920 and 0 at z equals 0. 112 00:08:06,920 --> 00:08:11,930 We have the 0 vector and the pole vector. 113 00:08:11,930 --> 00:08:16,460 And as we generate the frequency response for z1 114 00:08:16,460 --> 00:08:21,860 equal to e to the j omega, we have the point 115 00:08:21,860 --> 00:08:26,000 on the unit circle at which we're evaluating 116 00:08:26,000 --> 00:08:29,240 the Z-transform changing. 117 00:08:29,240 --> 00:08:33,350 If we think of angular distance around the unit circle 118 00:08:33,350 --> 00:08:37,940 as omega, then as omega varies and we 119 00:08:37,940 --> 00:08:44,570 consider the relative behavior of these two vectors, what 120 00:08:44,570 --> 00:08:47,090 we essentially trace out is the frequency 121 00:08:47,090 --> 00:08:50,430 response of the system. 122 00:08:50,430 --> 00:08:57,510 So let's look at this and for this example see if we can get, 123 00:08:57,510 --> 00:09:01,850 for this example, a rough idea of what the frequency 124 00:09:01,850 --> 00:09:04,540 response should look like. 125 00:09:04,540 --> 00:09:07,000 Well, first of all we observe. 126 00:09:07,000 --> 00:09:10,140 And this is an important observation, since it's 127 00:09:10,140 --> 00:09:13,230 a point that comes up frequently, 128 00:09:13,230 --> 00:09:17,160 is that as we travel around the unit 129 00:09:17,160 --> 00:09:24,190 circle, the vector from the 0 to the unit circle 130 00:09:24,190 --> 00:09:27,670 changes in angle, of course. 131 00:09:27,670 --> 00:09:30,250 But it doesn't change in length. 132 00:09:30,250 --> 00:09:33,040 The 0 vector for this example, or in fact 133 00:09:33,040 --> 00:09:36,220 any vector from the origin out to the unit circle, 134 00:09:36,220 --> 00:09:41,210 as we vary omega and, therefore, trace around the unit circle, 135 00:09:41,210 --> 00:09:43,520 this vector doesn't change in length. 136 00:09:43,520 --> 00:09:47,470 So if we're interested in, say, the magnitude of the frequency 137 00:09:47,470 --> 00:09:51,880 response, any poles or zeros at the origin, of course, 138 00:09:51,880 --> 00:09:54,430 have no effect on the magnitude. 139 00:09:54,430 --> 00:09:56,590 The only effect that they have is 140 00:09:56,590 --> 00:09:59,800 on the phase of the frequency response 141 00:09:59,800 --> 00:10:02,950 that is on the angle of H of z. 142 00:10:02,950 --> 00:10:06,640 And in fact, as we sweep around the unit circle, 143 00:10:06,640 --> 00:10:08,470 what happens to this angle? 144 00:10:08,470 --> 00:10:12,760 Well, that angle is just simply equal to the value omega 145 00:10:12,760 --> 00:10:16,660 that we're looking at that is the frequency value omega. 146 00:10:16,660 --> 00:10:20,270 So then, in fact, poles or zeros at the origin as, of course, 147 00:10:20,270 --> 00:10:24,610 we could see from a strictly algebraic argument. 148 00:10:24,610 --> 00:10:26,240 But from a geometric argument, it 149 00:10:26,240 --> 00:10:30,630 should be clear that poles or zeros at the origin introduce, 150 00:10:30,630 --> 00:10:35,020 in the frequency response, a linear phase term. 151 00:10:35,020 --> 00:10:37,510 And they have no effect on the magnitude 152 00:10:37,510 --> 00:10:40,540 since the length of a vector from the origin to the unit 153 00:10:40,540 --> 00:10:43,690 circle is obviously 1. 154 00:10:43,690 --> 00:10:45,530 As a consequence of that, by the way, 155 00:10:45,530 --> 00:10:49,390 often when talking about the frequency response 156 00:10:49,390 --> 00:10:55,050 or equivalently when looking at pole-zero patterns, 157 00:10:55,050 --> 00:10:59,050 it's common to minimize or ignore 158 00:10:59,050 --> 00:11:01,970 the presence of zeros or poles at the origin 159 00:11:01,970 --> 00:11:06,460 since they just correspond to a linear phase term. 160 00:11:06,460 --> 00:11:10,540 Now, let's look at what the behavior of the pole vector is. 161 00:11:10,540 --> 00:11:16,750 The pole vector as we start, say, at omega equals 0, 162 00:11:16,750 --> 00:11:19,720 we have a vector going from the pole 163 00:11:19,720 --> 00:11:24,620 to this point on the unit circle. 164 00:11:24,620 --> 00:11:29,050 And as we sweep around in frequency 165 00:11:29,050 --> 00:11:33,970 until we get to omega equals pi halfway around the unit circle, 166 00:11:33,970 --> 00:11:38,830 the length of this vector is monotonically increasing. 167 00:11:38,830 --> 00:11:42,880 Well, what does that mean about the magnitude of the frequency 168 00:11:42,880 --> 00:11:44,050 response? 169 00:11:44,050 --> 00:11:47,710 It means that the magnitude of the frequency response, 170 00:11:47,710 --> 00:11:50,890 which should be the magnitude of that vector-- 171 00:11:50,890 --> 00:11:53,350 the magnitude of the frequency response 172 00:11:53,350 --> 00:11:57,460 is monotonically decreasing as we sweep from omega 173 00:11:57,460 --> 00:12:02,810 equals zero around to omega equals pi. 174 00:12:02,810 --> 00:12:08,430 At omega equal to pi, we have exactly the reverse situation. 175 00:12:08,430 --> 00:12:10,325 The length of the vector from the pole-- 176 00:12:10,325 --> 00:12:12,600 and let me draw that vector-- 177 00:12:12,600 --> 00:12:15,990 from the pole to omega equals pi as we 178 00:12:15,990 --> 00:12:17,880 sweep around back to omega equals 179 00:12:17,880 --> 00:12:20,970 0, the length of that vector decreases. 180 00:12:20,970 --> 00:12:25,390 So the magnitude of the frequency response increases. 181 00:12:25,390 --> 00:12:29,220 So for this particular example, by just observing 182 00:12:29,220 --> 00:12:32,340 what happens to the length of this pole vector 183 00:12:32,340 --> 00:12:34,920 as we trace around the unit circle, 184 00:12:34,920 --> 00:12:39,240 we can see that the frequency response starts at some value. 185 00:12:39,240 --> 00:12:41,150 That's obvious. 186 00:12:41,150 --> 00:12:49,470 And as it sweeps around to pi, it monotonically decreases. 187 00:12:49,470 --> 00:12:54,110 And when we come back from pi around to 0, 188 00:12:54,110 --> 00:12:59,160 the frequency response increases again. 189 00:12:59,160 --> 00:13:03,870 And I don't know if I've drawn it so that it looks clearly 190 00:13:03,870 --> 00:13:04,910 this way. 191 00:13:04,910 --> 00:13:11,990 But in fact, this piece is just this piece reflected over. 192 00:13:11,990 --> 00:13:16,160 There's another important point that we 193 00:13:16,160 --> 00:13:17,870 can observe geometrically. 194 00:13:17,870 --> 00:13:21,290 It's a point about the frequency response that, of course, 195 00:13:21,290 --> 00:13:24,860 we've emphasized several times in several ways. 196 00:13:24,860 --> 00:13:27,950 This just offers an opportunity to re-emphasise it 197 00:13:27,950 --> 00:13:30,380 in yet another way. 198 00:13:30,380 --> 00:13:38,270 As the frequency variable goes from 0 around to 2pi, 199 00:13:38,270 --> 00:13:41,000 we trace out a magnitude and then, 200 00:13:41,000 --> 00:13:44,420 of course, also a phase for the frequency response. 201 00:13:44,420 --> 00:13:51,710 When we get back to 0 or 2pi, if we go from 2pi around to 4pi, 202 00:13:51,710 --> 00:13:54,200 what should we see for the frequency response? 203 00:13:54,200 --> 00:13:56,570 Well, we should see exactly the same thing 204 00:13:56,570 --> 00:14:00,140 that we saw before because what we're doing literally, 205 00:14:00,140 --> 00:14:03,020 actually, is going around in circles. 206 00:14:03,020 --> 00:14:05,960 We started here, went around there to get 207 00:14:05,960 --> 00:14:08,600 the 2pi, omega equals 2pi. 208 00:14:08,600 --> 00:14:11,860 For omega from 2pi to 4pi, we go around again. 209 00:14:11,860 --> 00:14:14,780 For 4pi to 6pi, we go around again, et cetera. 210 00:14:14,780 --> 00:14:18,500 So just geometrically by looking at 211 00:14:18,500 --> 00:14:22,460 this geometric interpretation of the frequency response, 212 00:14:22,460 --> 00:14:24,440 it gives us an opportunity to emphasize 213 00:14:24,440 --> 00:14:28,040 once again that the frequency response is a periodic function 214 00:14:28,040 --> 00:14:30,690 of frequency. 215 00:14:30,690 --> 00:14:35,200 Now, this is for one specific example. 216 00:14:35,200 --> 00:14:37,750 The more general statement, then, 217 00:14:37,750 --> 00:14:45,310 is that the magnitude of the frequency response 218 00:14:45,310 --> 00:14:53,180 is equal to the product of the length of the 0 vectors-- 219 00:14:53,180 --> 00:14:57,890 the vectors from the zeros to the unit circle-- 220 00:14:57,890 --> 00:15:02,330 divided by the product of the length of the pole vectors-- 221 00:15:02,330 --> 00:15:06,380 the vectors from the poles to the unit circle. 222 00:15:06,380 --> 00:15:09,740 And the angle of the frequency response 223 00:15:09,740 --> 00:15:14,150 is the sum of the angles of the 0 vectors 224 00:15:14,150 --> 00:15:18,050 minus the sum of the angles of the pole vectors. 225 00:15:18,050 --> 00:15:21,230 Incidentally, it usually is the case-- or it is for me, 226 00:15:21,230 --> 00:15:22,280 anyway-- 227 00:15:22,280 --> 00:15:29,270 that often, it's possible to get a rough picture of what 228 00:15:29,270 --> 00:15:31,730 the magnitude of the frequency response 229 00:15:31,730 --> 00:15:34,790 is like by looking at this geometric picture. 230 00:15:34,790 --> 00:15:38,570 It's usually somewhat more difficult except, perhaps, 231 00:15:38,570 --> 00:15:40,820 for linear phase terms, et cetera, 232 00:15:40,820 --> 00:15:47,450 to get a very clear picture of what the phase looks like. 233 00:15:47,450 --> 00:15:51,770 Well let's just look at one more example, which 234 00:15:51,770 --> 00:15:56,990 is a kind of example that we haven't discussed explicitly 235 00:15:56,990 --> 00:15:57,980 up to this point. 236 00:15:57,980 --> 00:16:00,950 And this also provides us with an opportunity 237 00:16:00,950 --> 00:16:04,580 to introduce this idea. 238 00:16:04,580 --> 00:16:09,050 Let's consider the case of a Z-transform 239 00:16:09,050 --> 00:16:13,220 which has a pole-zero pattern consisting 240 00:16:13,220 --> 00:16:17,240 of a pair of complex conjugate poles in the z-plane. 241 00:16:17,240 --> 00:16:20,090 These are complex conjugate poles 242 00:16:20,090 --> 00:16:22,940 with a radius equal to some value, 243 00:16:22,940 --> 00:16:28,440 say r, and an angular spacing equal to omega 0. 244 00:16:28,440 --> 00:16:31,520 So this pole is at omega 0. 245 00:16:31,520 --> 00:16:36,410 This pole is at an angle of minus omega 0. 246 00:16:36,410 --> 00:16:39,830 And using this notion of interpreting the frequency 247 00:16:39,830 --> 00:16:44,780 response geometrically, let's just sketch out 248 00:16:44,780 --> 00:16:47,630 roughly what we would expect the frequency 249 00:16:47,630 --> 00:16:51,370 response to look like. 250 00:16:51,370 --> 00:16:52,980 Well, let's see. 251 00:16:52,980 --> 00:16:57,540 If we start at omega equals zero, 252 00:16:57,540 --> 00:17:02,550 then we have a vector from this pole 253 00:17:02,550 --> 00:17:06,119 and a vector from this pole. 254 00:17:06,119 --> 00:17:08,670 The resulting value of the frequency response 255 00:17:08,670 --> 00:17:12,090 is 1 over the product of those two vectors. 256 00:17:14,660 --> 00:17:22,010 As we move around the unit circle from omega equals 0, 257 00:17:22,010 --> 00:17:28,580 let's say, to some point that's closer to omega 0, 258 00:17:28,580 --> 00:17:33,670 then this vector gets changed to that one. 259 00:17:33,670 --> 00:17:38,590 And this vector gets changed to this one. 260 00:17:38,590 --> 00:17:41,980 Well, we can see what's happening to the vector 261 00:17:41,980 --> 00:17:43,210 lengths. 262 00:17:43,210 --> 00:17:48,490 This vector is getting shorter as we approach 263 00:17:48,490 --> 00:17:51,160 omega equal to omega 0. 264 00:17:51,160 --> 00:17:53,680 This vector is getting longer. 265 00:17:53,680 --> 00:17:56,320 But you can imagine-- and I think it should be relatively 266 00:17:56,320 --> 00:17:57,880 clear from the picture-- 267 00:17:57,880 --> 00:18:00,880 that this vector is getting shorter faster than this one 268 00:18:00,880 --> 00:18:04,010 is getting longer. 269 00:18:04,010 --> 00:18:08,570 It seems relatively clear from a geometric picture 270 00:18:08,570 --> 00:18:12,440 that the product of the lengths of these two vectors 271 00:18:12,440 --> 00:18:18,170 is smaller when omega is closer to omega 0 than, let's say, 272 00:18:18,170 --> 00:18:21,710 when omega is equal to 0. 273 00:18:21,710 --> 00:18:26,240 So consequently, we would expect the magnitude of the frequency 274 00:18:26,240 --> 00:18:30,200 response to start at some value. 275 00:18:30,200 --> 00:18:34,160 As we get in the vicinity of omega equal to omega 0, 276 00:18:34,160 --> 00:18:37,640 we would expect that frequency response to peak. 277 00:18:37,640 --> 00:18:41,090 As we pass that pole, the frequency response 278 00:18:41,090 --> 00:18:43,640 will now begin to decrease. 279 00:18:43,640 --> 00:18:53,160 And clearly, when we are at omega equal to pi, 280 00:18:53,160 --> 00:18:57,030 the product of the lengths of these two vectors 281 00:18:57,030 --> 00:18:59,610 is considerably longer than the products 282 00:18:59,610 --> 00:19:02,970 of the lengths of the vectors at omega equals 0. 283 00:19:02,970 --> 00:19:05,460 So on the basis of that argument, 284 00:19:05,460 --> 00:19:09,240 we can see that roughly what we expect the frequency 285 00:19:09,240 --> 00:19:13,950 response to do is begin at some value, 286 00:19:13,950 --> 00:19:19,170 increase as we approach the pole at omega equals omega 0, 287 00:19:19,170 --> 00:19:25,740 decrease as we pass the pole heading toward pi. 288 00:19:25,740 --> 00:19:30,270 And then from pi around to pi, or equivalently from 0 289 00:19:30,270 --> 00:19:34,530 around to minus pi, we would expect the reciprocal behavior. 290 00:19:34,530 --> 00:19:38,130 That is, the frequency response would increase, 291 00:19:38,130 --> 00:19:41,850 peak in the vicinity of this pole in the lower 292 00:19:41,850 --> 00:19:46,260 half of the z-plane, and then return to the same value 293 00:19:46,260 --> 00:19:47,610 that it started from. 294 00:19:47,610 --> 00:19:50,280 And of course, it'll be periodic as we 295 00:19:50,280 --> 00:19:53,070 run around the unit circle. 296 00:19:53,070 --> 00:19:57,180 Well, this is a resonant characteristic, 297 00:19:57,180 --> 00:20:01,410 reminiscent in the analog case of what we would expect 298 00:20:01,410 --> 00:20:05,700 from a complex conjugate pole pair close to the j 299 00:20:05,700 --> 00:20:08,760 omega axis that's in the s-plane. 300 00:20:08,760 --> 00:20:11,130 And in fact, of course, in the s-plane, 301 00:20:11,130 --> 00:20:14,490 we have exactly the same kinds of geometrical arguments 302 00:20:14,490 --> 00:20:18,570 to allow us to roughly sketch out the frequency response. 303 00:20:18,570 --> 00:20:20,880 The only real important difference 304 00:20:20,880 --> 00:20:23,220 is that in the discrete time case, 305 00:20:23,220 --> 00:20:26,760 it's the unit circle, which is the locus in the z-plane 306 00:20:26,760 --> 00:20:28,020 that we're looking at. 307 00:20:28,020 --> 00:20:30,090 In the continuous time case, it's 308 00:20:30,090 --> 00:20:33,120 the vertical, or j omega axis in the s-plane 309 00:20:33,120 --> 00:20:35,230 that we're looking at. 310 00:20:35,230 --> 00:20:38,400 Well, this shouldn't be particularly evident from what 311 00:20:38,400 --> 00:20:39,270 we've done here. 312 00:20:39,270 --> 00:20:45,120 But by inference or by carrying your intuition 313 00:20:45,120 --> 00:20:48,670 from the continuous time to the discrete time, 314 00:20:48,670 --> 00:20:53,770 can you guess at what you would expect the unit sample 315 00:20:53,770 --> 00:21:00,460 response of a system with this pole-zero pattern to look like? 316 00:21:00,460 --> 00:21:03,510 Well, it would look like the discrete time 317 00:21:03,510 --> 00:21:07,110 counterpart of what happens in the continuous time case. 318 00:21:07,110 --> 00:21:12,810 That is, it would look like a damped sinusoidal sequence 319 00:21:12,810 --> 00:21:17,640 with the damping influenced by the distance of these poles 320 00:21:17,640 --> 00:21:19,230 from the unit circle. 321 00:21:19,230 --> 00:21:21,990 The closer the pole is to the unit circle, 322 00:21:21,990 --> 00:21:23,700 the sharper this resonant peak will 323 00:21:23,700 --> 00:21:26,760 be and the less damping on the sinusoid. 324 00:21:29,710 --> 00:21:35,200 One additional point to remind you of-- 325 00:21:35,200 --> 00:21:38,350 I made reference to this in an earlier lecture. 326 00:21:38,350 --> 00:21:43,500 But notice that in talking about the Z-transform poles 327 00:21:43,500 --> 00:21:47,520 and zeros, et cetera here, I didn't say what 328 00:21:47,520 --> 00:21:49,560 the region of convergence was. 329 00:21:49,560 --> 00:21:51,060 Or at least, I didn't say explicitly 330 00:21:51,060 --> 00:21:53,220 what the region of convergence was. 331 00:21:53,220 --> 00:21:57,060 But did I say implicitly what it is? 332 00:21:57,060 --> 00:22:01,800 Well, sure I did because what I said or what I've been assuming 333 00:22:01,800 --> 00:22:05,100 is that the systems that we've been talking about 334 00:22:05,100 --> 00:22:08,640 can be described in terms of a frequency response. 335 00:22:08,640 --> 00:22:10,740 In other words, the unit sample response 336 00:22:10,740 --> 00:22:13,190 has a Fourier transform. 337 00:22:13,190 --> 00:22:16,200 Well for that to be the case, the region of convergence 338 00:22:16,200 --> 00:22:18,850 has to include the unit circle. 339 00:22:18,850 --> 00:22:20,700 And then, we can use all those other rules 340 00:22:20,700 --> 00:22:25,170 of regions of convergences to allow us to figure out 341 00:22:25,170 --> 00:22:30,144 from there how far on both sides of the unit circle 342 00:22:30,144 --> 00:22:31,560 the region of convergence extends. 343 00:22:34,390 --> 00:22:39,370 All right, well this is, then, a geometrical interpretation 344 00:22:39,370 --> 00:22:41,320 of the frequency response. 345 00:22:41,320 --> 00:22:43,930 It's often useful as a rough guide 346 00:22:43,930 --> 00:22:47,590 in getting a general picture of what the frequency 347 00:22:47,590 --> 00:22:49,240 response might look like. 348 00:22:49,240 --> 00:22:52,720 Although, for complicated cases, it's often 349 00:22:52,720 --> 00:22:55,930 difficult to get precise details about the frequency 350 00:22:55,930 --> 00:23:01,210 response, which of course, we could also get algebraically. 351 00:23:01,210 --> 00:23:06,300 Well now, I'd like to, at least temporarily, 352 00:23:06,300 --> 00:23:15,390 change gears or topics and talk about the issue of properties 353 00:23:15,390 --> 00:23:19,920 of the Z-transform in talking about the properties 354 00:23:19,920 --> 00:23:22,950 and working some examples to illustrate 355 00:23:22,950 --> 00:23:25,620 the use of the properties of the Z-transform. 356 00:23:25,620 --> 00:23:29,400 In fact, I'll have occasion to make reference back 357 00:23:29,400 --> 00:23:33,600 to this geometric interpretation of the frequency response 358 00:23:33,600 --> 00:23:37,180 to help with at least one of the examples. 359 00:23:37,180 --> 00:23:41,490 So now I'd like to turn our attention, then, 360 00:23:41,490 --> 00:23:46,969 to the question or the topic of the properties 361 00:23:46,969 --> 00:23:47,760 of the Z-transform. 362 00:23:47,760 --> 00:23:52,070 Well, first of all, why do we want 363 00:23:52,070 --> 00:23:53,360 properties of the Z-transform? 364 00:23:53,360 --> 00:23:55,680 The Z-transform has properties. 365 00:23:55,680 --> 00:23:57,780 Why do we want these properties? 366 00:23:57,780 --> 00:23:58,830 One of the reasons-- 367 00:23:58,830 --> 00:24:01,100 it's a very practical reason-- 368 00:24:01,100 --> 00:24:04,760 is that the properties of the Z-transform 369 00:24:04,760 --> 00:24:09,290 help us in calculating Z-transforms and inverse 370 00:24:09,290 --> 00:24:11,210 Z-transforms. 371 00:24:11,210 --> 00:24:15,590 And they also, obviously, provide a certain amount 372 00:24:15,590 --> 00:24:18,260 of intuition and insight with regard 373 00:24:18,260 --> 00:24:21,500 to generally dealing with Z-transforms 374 00:24:21,500 --> 00:24:24,500 and their inverses. 375 00:24:24,500 --> 00:24:26,670 Well, there are a lot of properties. 376 00:24:26,670 --> 00:24:28,820 In fact, there are trivial properties. 377 00:24:28,820 --> 00:24:31,370 There are very complicated properties. 378 00:24:31,370 --> 00:24:34,850 There are properties that we more commonly 379 00:24:34,850 --> 00:24:36,650 tend to carry around. 380 00:24:36,650 --> 00:24:41,900 And I've listed a few that shouldn't be considered 381 00:24:41,900 --> 00:24:44,930 to be exhaustive but generally tend 382 00:24:44,930 --> 00:24:49,280 to be the properties that turn out to be the handiest. 383 00:24:49,280 --> 00:24:51,170 That is, these are the properties, 384 00:24:51,170 --> 00:24:53,780 at a minimum, that you should carry around 385 00:24:53,780 --> 00:24:56,820 in your back pocket. 386 00:24:56,820 --> 00:24:58,380 Well, let's see. 387 00:24:58,380 --> 00:25:02,700 We're talking about a sequence, x of n with a Z-transform, 388 00:25:02,700 --> 00:25:04,580 x of z. 389 00:25:04,580 --> 00:25:09,150 One of the properties that we've taken advantage of already 390 00:25:09,150 --> 00:25:12,120 throughout the entire discussion of the Z-transform 391 00:25:12,120 --> 00:25:15,060 is the fact that the Z-transform maps 392 00:25:15,060 --> 00:25:19,310 convolution to multiplication. 393 00:25:19,310 --> 00:25:23,390 If I have the convolution of two sequences, 394 00:25:23,390 --> 00:25:28,610 then the resulting Z-transform is the product 395 00:25:28,610 --> 00:25:31,650 of the Z-transforms. 396 00:25:31,650 --> 00:25:35,640 The second property, which is often very useful, 397 00:25:35,640 --> 00:25:40,500 is referred to as the shifting property, which says that if I 398 00:25:40,500 --> 00:25:44,130 shift x of n by an amount n0-- 399 00:25:46,800 --> 00:25:49,980 and you should think, by the way, of if n0 is positive, 400 00:25:49,980 --> 00:25:53,500 does that mean shifting to the right or shifting to the left? 401 00:25:53,500 --> 00:25:55,080 Well, I'll let you think about that. 402 00:25:55,080 --> 00:25:57,990 It's something that you should nail down. 403 00:25:57,990 --> 00:26:02,400 If I shift the sequence x of n by replacing the argument by n 404 00:26:02,400 --> 00:26:06,480 plus n0, then the resulting Z-transform 405 00:26:06,480 --> 00:26:09,750 is z to the n0 times x of z. 406 00:26:09,750 --> 00:26:14,040 That is, shifting corresponds to multiplying by z to the n0. 407 00:26:16,690 --> 00:26:21,700 Another useful property has to do with taking a sequence 408 00:26:21,700 --> 00:26:24,100 and turning it around in n. 409 00:26:24,100 --> 00:26:28,060 That is, replacing n by minus n. 410 00:26:28,060 --> 00:26:30,640 The resulting effect on the Z-transform 411 00:26:30,640 --> 00:26:36,140 is to replace z by 1 over z. 412 00:26:36,140 --> 00:26:39,500 Another useful property is the result 413 00:26:39,500 --> 00:26:44,570 of multiplying a sequence x of n by an exponential, a to the n. 414 00:26:44,570 --> 00:26:45,740 a might be complex. 415 00:26:45,740 --> 00:26:47,360 Or it might be real. 416 00:26:47,360 --> 00:26:52,850 And the result there is that the z transform is x of 8 417 00:26:52,850 --> 00:26:55,610 to the minus 1 times z. 418 00:26:55,610 --> 00:26:59,180 Another one which is useful is multiplication 419 00:26:59,180 --> 00:27:04,040 of a sequence by n which results in a z transform, which 420 00:27:04,040 --> 00:27:08,690 is minus z times the derivative of x of z. 421 00:27:08,690 --> 00:27:10,940 And then I've indicated-- 422 00:27:10,940 --> 00:27:12,890 tried to be somewhat explicit-- 423 00:27:12,890 --> 00:27:15,170 that that's not the end of the list. 424 00:27:15,170 --> 00:27:17,360 There are lots of other properties-- 425 00:27:17,360 --> 00:27:20,490 a number of others that are presented in the text, 426 00:27:20,490 --> 00:27:23,150 others besides that that aren't presented in the text, ones 427 00:27:23,150 --> 00:27:25,640 that you can dream up yourself, ones that your friends 428 00:27:25,640 --> 00:27:29,040 know that you don't, et cetera. 429 00:27:29,040 --> 00:27:33,930 Now, we could, of course, go through the proof of all these. 430 00:27:33,930 --> 00:27:39,300 The proofs of properties tend to all 431 00:27:39,300 --> 00:27:43,200 be in somewhat of a similar vein, as a matter of fact. 432 00:27:43,200 --> 00:27:47,790 And the style, once you see what the trick is or roughly 433 00:27:47,790 --> 00:27:49,374 how you go about proving properties, 434 00:27:49,374 --> 00:27:50,790 then you can just prove properties 435 00:27:50,790 --> 00:27:52,740 and prove properties. 436 00:27:52,740 --> 00:27:56,070 And we won't do that in this lecture, 437 00:27:56,070 --> 00:27:59,430 with the exception of illustrating 438 00:27:59,430 --> 00:28:04,480 the style of proving properties with a couple of examples. 439 00:28:04,480 --> 00:28:08,340 And the two examples that I've picked, somewhat arbitrarily 440 00:28:08,340 --> 00:28:11,940 as a matter of fact, is the Shifting Property, 441 00:28:11,940 --> 00:28:16,260 that is Property 2, and the result of multiplication 442 00:28:16,260 --> 00:28:21,420 by an exponential, which is Property 4. 443 00:28:21,420 --> 00:28:25,230 But this is only to illustrate the style 444 00:28:25,230 --> 00:28:27,090 of proving properties. 445 00:28:27,090 --> 00:28:30,060 And you can guess where, actually, you 446 00:28:30,060 --> 00:28:34,030 get a chance to see the proof of some of the others. 447 00:28:34,030 --> 00:28:36,820 All right, let's take a look at Property 2, 448 00:28:36,820 --> 00:28:39,960 that is the Shifting Property. 449 00:28:39,960 --> 00:28:46,110 Well, to prove that replacing n by n plus n0 450 00:28:46,110 --> 00:28:49,830 results in z to the n0 times x of z, 451 00:28:49,830 --> 00:28:57,150 we want to consider a sequence, x1 of n equal to x of n plus n0 452 00:28:57,150 --> 00:29:00,500 so that its Z-transform, x1 of z, 453 00:29:00,500 --> 00:29:05,990 is the sum of x of n plus n0 times z to the minus n. 454 00:29:05,990 --> 00:29:11,030 Well, a simple idea here is a substitution of variables. 455 00:29:11,030 --> 00:29:18,560 Let's replace n plus n0 by a new variable, m, or n 456 00:29:18,560 --> 00:29:22,890 is equal to m minus n0, in which case, 457 00:29:22,890 --> 00:29:28,430 we can rewrite this expression as x1 of z 458 00:29:28,430 --> 00:29:35,300 is the sum on m of x of m, because this is now m. 459 00:29:35,300 --> 00:29:38,570 n is replaced by m minus n0. 460 00:29:38,570 --> 00:29:45,590 So we have z to the n0 times z to the minus m. 461 00:29:45,590 --> 00:29:54,750 Well, the z to the n0 can come outside the sum. 462 00:29:54,750 --> 00:29:56,540 The limits on the sum, incidentally, 463 00:29:56,540 --> 00:29:59,370 are m equals minus infinity to plus infinity 464 00:29:59,370 --> 00:30:04,910 because if n0 is finite, as n runs from minus infinity 465 00:30:04,910 --> 00:30:08,510 to plus infinity, so does m. 466 00:30:08,510 --> 00:30:11,150 That comes outside the sum. 467 00:30:11,150 --> 00:30:16,040 And what's left, then, is the sum of x of m z to the minus m, 468 00:30:16,040 --> 00:30:17,930 which is just x of z. 469 00:30:17,930 --> 00:30:22,910 So consequently, then, what we end up with is that x of z 470 00:30:22,910 --> 00:30:30,030 is z to the n0 times x of z. 471 00:30:30,030 --> 00:30:33,630 x1 of z is z to the n0 times x of z, 472 00:30:33,630 --> 00:30:37,650 which is, of course, the way I advertised. 473 00:30:37,650 --> 00:30:41,010 Well, there are lots of times when, in fact, the Shifting 474 00:30:41,010 --> 00:30:44,770 Property comes into play. 475 00:30:44,770 --> 00:30:47,430 One thing that we can use it for immediately 476 00:30:47,430 --> 00:30:51,390 is to tie together a couple of things 477 00:30:51,390 --> 00:30:54,330 that have been floating, more or less, in the background. 478 00:30:57,090 --> 00:31:01,850 I have alluded several times to the fact 479 00:31:01,850 --> 00:31:09,330 that systems whose system function is rational 480 00:31:09,330 --> 00:31:13,230 correspond to systems that are characterized 481 00:31:13,230 --> 00:31:16,140 by linear constant coefficient difference equations. 482 00:31:16,140 --> 00:31:18,600 That is, linear constant coefficient difference 483 00:31:18,600 --> 00:31:21,690 equations, those are the systems that 484 00:31:21,690 --> 00:31:25,770 end up with system functions that are rational functions. 485 00:31:25,770 --> 00:31:30,360 And in fact, we can see that in a straightforward way 486 00:31:30,360 --> 00:31:34,950 by just simply applying the Shifting Property. 487 00:31:34,950 --> 00:31:42,320 Well, let's consider a linear constant coefficient difference 488 00:31:42,320 --> 00:31:48,190 equation of the general form is the sum from k equals 0 to n. 489 00:31:48,190 --> 00:31:54,860 a sub k y of n minus k is equal to a sum of b sub k times x 490 00:31:54,860 --> 00:31:55,850 of n minus k. 491 00:31:59,250 --> 00:32:05,640 If we take the Z-transform of both sides of this equation 492 00:32:05,640 --> 00:32:08,430 and use the fact-- incidentally, this is another property 493 00:32:08,430 --> 00:32:12,000 that I've more or less been using, 494 00:32:12,000 --> 00:32:13,710 although I've never stated explicitly-- 495 00:32:13,710 --> 00:32:17,430 that the Z-transform of a sum is the sum of the Z-transforms. 496 00:32:17,430 --> 00:32:21,480 Taking, then, the Z-transform of this equation, 497 00:32:21,480 --> 00:32:26,590 y of n minus k using the Shifting Property 498 00:32:26,590 --> 00:32:34,170 will give us z to the minus k times y of z. 499 00:32:34,170 --> 00:32:44,770 And x of n minus k will give us z to the minus k times x of z. 500 00:32:44,770 --> 00:32:50,320 Consequently, the sum of a sub k z to the minus k y of z 501 00:32:50,320 --> 00:32:55,960 is equal to the sum of b sub k z to the minus k x of z. 502 00:32:55,960 --> 00:32:59,170 If we solve that equation for y of z 503 00:32:59,170 --> 00:33:03,730 over x of z, which is the system function, 504 00:33:03,730 --> 00:33:07,570 then we end up with the sum of b sub k z 505 00:33:07,570 --> 00:33:13,090 to the minus k divided by the sum of a sub k times z 506 00:33:13,090 --> 00:33:14,770 to the minus k. 507 00:33:14,770 --> 00:33:17,210 And there are a couple of important observations. 508 00:33:17,210 --> 00:33:21,130 One is we ended up with a rational function, which 509 00:33:21,130 --> 00:33:26,090 is what we expected, or what I said we were going to get. 510 00:33:26,090 --> 00:33:32,680 And a second is that the coefficients in the numerator 511 00:33:32,680 --> 00:33:37,540 polynomial are exactly the same as the coefficients 512 00:33:37,540 --> 00:33:41,230 on the right hand side of the difference equation. 513 00:33:41,230 --> 00:33:44,530 And the coefficients in the denominator polynomial 514 00:33:44,530 --> 00:33:48,580 are exactly the same as the coefficients on the left hand 515 00:33:48,580 --> 00:33:51,070 side of the difference equation. 516 00:33:51,070 --> 00:33:54,730 This by the way, is exactly consistent, or analogous, 517 00:33:54,730 --> 00:33:59,800 with what happens when we apply the Laplace transform 518 00:33:59,800 --> 00:34:02,620 to linear constant coefficient differential equation. 519 00:34:02,620 --> 00:34:05,530 Exactly the same thing happens. 520 00:34:05,530 --> 00:34:07,950 It should be clear then, incidentally, 521 00:34:07,950 --> 00:34:10,420 that if I give you a system function that's 522 00:34:10,420 --> 00:34:14,260 a rational function of z, that you could construct, 523 00:34:14,260 --> 00:34:17,139 in a straightforward way, the difference equation that 524 00:34:17,139 --> 00:34:19,389 characterizes that system because you 525 00:34:19,389 --> 00:34:22,420 can pick the coefficients off from the numerator. 526 00:34:22,420 --> 00:34:24,500 Those are on the right hand side. 527 00:34:24,500 --> 00:34:26,380 And you can pick the coefficients off 528 00:34:26,380 --> 00:34:27,730 from the denominator. 529 00:34:27,730 --> 00:34:32,980 Those are on the left hand side of the difference equation. 530 00:34:32,980 --> 00:34:37,750 The Shifting Property is a property that arises very often 531 00:34:37,750 --> 00:34:39,989 and, in fact, is a very, very useful property. 532 00:34:42,690 --> 00:34:50,360 Now, the second property that I want to outline the proof for 533 00:34:50,360 --> 00:34:58,490 is the property related to multiplication of the sequence 534 00:34:58,490 --> 00:35:03,890 by an exponential a to the n. 535 00:35:03,890 --> 00:35:09,230 I am forming a new sequence x1 of n, which is a to the n times 536 00:35:09,230 --> 00:35:11,530 x of n. 537 00:35:11,530 --> 00:35:17,440 And to derive the relationship between x1 of z and x of n, 538 00:35:17,440 --> 00:35:22,750 again, we can look at the Z-transform, x1 of z, 539 00:35:22,750 --> 00:35:28,330 which is the sum of a to the n, x of n, times z to the minus n. 540 00:35:28,330 --> 00:35:30,640 Well, it's straightforward to rewrite 541 00:35:30,640 --> 00:35:41,590 what's inside the sum as x of n times a to the minus 1 times 542 00:35:41,590 --> 00:35:44,440 z to the minus n. 543 00:35:47,340 --> 00:35:53,070 x1 of z is the sum of x of n times a to the minus 1z-- 544 00:35:53,070 --> 00:35:56,180 all that raised to the minus n. 545 00:35:56,180 --> 00:35:59,390 Well, that looks just like the Z-transform of x of n 546 00:35:59,390 --> 00:36:03,200 but with z replaced by a to the minus 1 times z. 547 00:36:03,200 --> 00:36:11,270 So this says, consequently, that the Z-transform of x1 of n 548 00:36:11,270 --> 00:36:16,700 is equal to the Z-transform of x of n 549 00:36:16,700 --> 00:36:21,800 but with z replaced by a to the minus 1 times z. 550 00:36:21,800 --> 00:36:25,850 So we stick in here a to the minus 1 times z. 551 00:36:25,850 --> 00:36:28,250 And that then relates the Z-transform 552 00:36:28,250 --> 00:36:30,650 of the original sequence and the Z-transform 553 00:36:30,650 --> 00:36:35,740 of the sequence multiplied by a decaying exponential. 554 00:36:35,740 --> 00:36:43,590 Well, it's interesting to look at what this property implies 555 00:36:43,590 --> 00:36:49,290 in terms of the movement of the poles and zeros in the z-plane. 556 00:36:49,290 --> 00:36:54,120 That is, a useful notion or a useful fact 557 00:36:54,120 --> 00:36:57,900 to have, again, stored away in your hip pocket 558 00:36:57,900 --> 00:37:03,240 is the effect on the poles and zeros of a system function 559 00:37:03,240 --> 00:37:08,580 or a Z-transform of multiplying the sequence 560 00:37:08,580 --> 00:37:11,670 by an exponential-- maybe a complex exponential, 561 00:37:11,670 --> 00:37:14,530 maybe a real exponential. 562 00:37:14,530 --> 00:37:19,520 Well, let's focus on a pole or a 0. 563 00:37:19,520 --> 00:37:22,670 And the result that we get by considering 564 00:37:22,670 --> 00:37:25,790 just a simple pole or 0 will, of course, 565 00:37:25,790 --> 00:37:28,580 generalize to all the poles and zeros. 566 00:37:28,580 --> 00:37:32,390 So let's consider x of z to have a factor 567 00:37:32,390 --> 00:37:35,390 either in the numerator and denominator of the form 568 00:37:35,390 --> 00:37:38,500 z minus z0. 569 00:37:38,500 --> 00:37:42,880 Well, x1 of z will then have a factor 570 00:37:42,880 --> 00:37:47,140 derived from that one, but for which z is replaced 571 00:37:47,140 --> 00:37:51,220 by 8 to the minus 1 times z. 572 00:37:51,220 --> 00:37:53,630 And then we have the minus z0. 573 00:37:53,630 --> 00:37:57,440 Or if we pull the a to the minus outside, 574 00:37:57,440 --> 00:38:04,950 we have a to the minus 1 times z minus a times z0. 575 00:38:04,950 --> 00:38:06,780 So there are two effects here. 576 00:38:06,780 --> 00:38:11,040 One effect-- if we think of x of z as a product of zeros divided 577 00:38:11,040 --> 00:38:12,930 by a product of poles-- 578 00:38:12,930 --> 00:38:16,200 one effect is that there is a constant that, 579 00:38:16,200 --> 00:38:19,380 perhaps, collects out in front. 580 00:38:19,380 --> 00:38:23,730 But we never see that constant anyway in the pole-0 pattern. 581 00:38:23,730 --> 00:38:27,840 The more important point is that whereas here we 582 00:38:27,840 --> 00:38:36,210 had, say, 0 at z equals z0, that 0 is now shifted to a times z0. 583 00:38:36,210 --> 00:38:41,910 So the 0 at z0 gets replaced by a 0 at eight times z0. 584 00:38:41,910 --> 00:38:46,670 A pole at z0 gets replaced by a pole at a times z0. 585 00:38:52,230 --> 00:38:57,330 Well, more specifically, then, here's 586 00:38:57,330 --> 00:39:01,920 our polar 0 which gets converted to that polar 0 multiplied 587 00:39:01,920 --> 00:39:04,915 by a. 588 00:39:04,915 --> 00:39:10,860 And if we write this or think of it 589 00:39:10,860 --> 00:39:15,930 in polar form, as r0 e to the j theta 0, 590 00:39:15,930 --> 00:39:21,930 then the result is a times r0 times e to the j theta 0. 591 00:39:21,930 --> 00:39:28,090 Well obviously then, if this number a is a real number, 592 00:39:28,090 --> 00:39:33,750 then the only effect on the location of the pole 593 00:39:33,750 --> 00:39:38,990 is to change its radial value and not change its angle. 594 00:39:38,990 --> 00:39:44,540 More generally, if a is complex, then to write the resulting 595 00:39:44,540 --> 00:39:48,980 polar zero in polar form, we would 596 00:39:48,980 --> 00:39:54,560 replace this by its magnitude and add to the phase-- 597 00:39:54,560 --> 00:39:57,750 the phase that corresponds to that number a. 598 00:40:00,270 --> 00:40:04,880 Well consequently, first of all, if a is real and positive, 599 00:40:04,880 --> 00:40:08,450 actually, if a is real and positive, then 600 00:40:08,450 --> 00:40:12,380 if we had a pole of x of n say here, 601 00:40:12,380 --> 00:40:17,460 here being any place, then if a is real, 602 00:40:17,460 --> 00:40:25,080 what happens to that pole is that it moves either in or out. 603 00:40:25,080 --> 00:40:30,180 But it moves radially as we vary the value of a. 604 00:40:30,180 --> 00:40:33,615 So that's for a real. 605 00:40:36,900 --> 00:40:41,890 What's the movement of the pole if a is pure imaginary? 606 00:40:41,890 --> 00:40:48,840 Well, if a is pure imaginary, then the magnitude of a is-- 607 00:40:48,840 --> 00:40:50,490 I'm sorry. 608 00:40:50,490 --> 00:40:54,120 Not if a is pure imaginary, but if the magnitude of a 609 00:40:54,120 --> 00:40:58,920 is equal to 1 and it has only a phase component, in that case 610 00:40:58,920 --> 00:41:01,290 there's no effect on the radial value. 611 00:41:01,290 --> 00:41:03,660 There's only an effect on the angle. 612 00:41:03,660 --> 00:41:06,990 And in that case, the movement of the pole 613 00:41:06,990 --> 00:41:12,660 is such that the radial value stays the same. 614 00:41:12,660 --> 00:41:16,420 But the angle of the pole changes. 615 00:41:16,420 --> 00:41:19,456 So in general, of course, if a is complex, 616 00:41:19,456 --> 00:41:21,330 the poles can have a little movement that way 617 00:41:21,330 --> 00:41:24,150 and also a little movement that way. 618 00:41:24,150 --> 00:41:30,090 In some cases, if a is pure real, 619 00:41:30,090 --> 00:41:32,470 the movement will just be this way. 620 00:41:32,470 --> 00:41:35,580 And if the magnitude of a is equal to 1, 621 00:41:35,580 --> 00:41:39,700 the movement of the pole will just be that way. 622 00:41:39,700 --> 00:41:43,420 All right, so we have first of all, a list of properties. 623 00:41:43,420 --> 00:41:45,940 But in particular, there are two that we've 624 00:41:45,940 --> 00:41:49,080 spent a little time on. 625 00:41:49,080 --> 00:41:52,690 Finally, let's look at, actually, 626 00:41:52,690 --> 00:41:56,880 one example or one and a half examples 627 00:41:56,880 --> 00:41:59,460 to see how some of the properties 628 00:41:59,460 --> 00:42:05,900 might be useful in obtaining the frequency response, 629 00:42:05,900 --> 00:42:09,050 or the Z-transform of a system. 630 00:42:09,050 --> 00:42:12,740 And the sequence that I want to focus on 631 00:42:12,740 --> 00:42:17,810 is a sequence that will play an important role 632 00:42:17,810 --> 00:42:19,940 throughout digital signal processing 633 00:42:19,940 --> 00:42:22,440 and in particular, in some lectures coming up, 634 00:42:22,440 --> 00:42:26,920 which is a sequence that I'll refer to as a boxcar sequence. 635 00:42:26,920 --> 00:42:32,000 It's the sequence which is equal to unity for n between 0 636 00:42:32,000 --> 00:42:34,160 and capital N minus 1. 637 00:42:34,160 --> 00:42:37,250 And it's equal to 0 otherwise. 638 00:42:37,250 --> 00:42:41,300 It's basically a rectangular sequence. 639 00:42:41,300 --> 00:42:44,790 0 for n negative, 0 for n greater than 640 00:42:44,790 --> 00:42:47,820 or equal to capital N and unity otherwise. 641 00:42:47,820 --> 00:42:53,980 It's the counterpart of the rectangular time function. 642 00:42:53,980 --> 00:42:57,660 Well, there are a couple of ways of getting at Z-transform. 643 00:42:57,660 --> 00:43:01,560 Since we had some properties, let's use one of them, 644 00:43:01,560 --> 00:43:04,320 in particular, the Shifting Property. 645 00:43:04,320 --> 00:43:08,000 We can think of a boxcar sequence 646 00:43:08,000 --> 00:43:10,310 as the sum of two sequences. 647 00:43:10,310 --> 00:43:13,580 One is a unit step. 648 00:43:13,580 --> 00:43:17,830 And the second is the negative of a unit step starting 649 00:43:17,830 --> 00:43:20,540 at n equals capital N to subtract off 650 00:43:20,540 --> 00:43:23,420 these other values. 651 00:43:23,420 --> 00:43:28,010 We can express x of n, the boxcar sequence, 652 00:43:28,010 --> 00:43:34,460 as a unit step minus a unit step delayed by capital N. 653 00:43:34,460 --> 00:43:38,750 Well, if you don't see this graphical picture exactly, 654 00:43:38,750 --> 00:43:41,120 you can just see quickly that this is true 655 00:43:41,120 --> 00:43:45,740 since for n greater than or equal to capital N, 656 00:43:45,740 --> 00:43:48,530 both of these arguments are non-negative. 657 00:43:48,530 --> 00:43:51,380 So the value of both of these units steps is unity. 658 00:43:51,380 --> 00:43:52,780 And they subtract off to zero. 659 00:43:55,320 --> 00:43:58,140 All right, then to get the Z-transform, 660 00:43:58,140 --> 00:44:02,390 we can add the Z-transform or this piece and this piece. 661 00:44:02,390 --> 00:44:04,790 The Z-transform of a unit step, well that's 662 00:44:04,790 --> 00:44:07,760 our old friend a to the n times a unit step, 663 00:44:07,760 --> 00:44:10,340 except in this case, a equals 1. 664 00:44:10,340 --> 00:44:14,180 So the Z-transform of this piece is 1 over 1 minus z 665 00:44:14,180 --> 00:44:17,550 to the minus 1. 666 00:44:17,550 --> 00:44:20,610 This one is this one shifted. 667 00:44:20,610 --> 00:44:25,110 So we can apply our Shifting Property to multiply this by z 668 00:44:25,110 --> 00:44:29,960 to the minus capital N, since that's the amount of our shift, 669 00:44:29,960 --> 00:44:33,890 so that this piece, then, has a Z-transform z to the minus 670 00:44:33,890 --> 00:44:37,810 n over 1 minus z to the minus 1. 671 00:44:37,810 --> 00:44:45,860 Or if we add these two together, we have 1 minus z 672 00:44:45,860 --> 00:44:53,110 to the minus capital N divided by 1 minus z to the minus 1. 673 00:44:53,110 --> 00:44:56,500 Or we can rewrite that, just to focus on something 674 00:44:56,500 --> 00:45:02,750 a little more clearly, multiplying top and bottom by z 675 00:45:02,750 --> 00:45:04,620 to the n minus 1. 676 00:45:04,620 --> 00:45:07,010 We can rewrite that in this form so 677 00:45:07,010 --> 00:45:09,260 that we have in the numerator z to the capital 678 00:45:09,260 --> 00:45:13,180 N minus 1, in the denominator, this factor times 679 00:45:13,180 --> 00:45:14,240 z to the minus 1. 680 00:45:17,030 --> 00:45:19,390 Well, let's look at the pole-zero pattern. 681 00:45:22,860 --> 00:45:33,180 First of all, at z equals 1, we have a pole. 682 00:45:33,180 --> 00:45:35,190 So there's that pole. 683 00:45:35,190 --> 00:45:39,990 Second of all, at z equals 0, we have n minus1 poles. 684 00:45:39,990 --> 00:45:43,710 Well, let's stick in the n minus 1 poles. 685 00:45:43,710 --> 00:45:47,430 And let me just draw that with an asterisk and an indication 686 00:45:47,430 --> 00:45:51,330 that that's n minus 1 poles. 687 00:45:51,330 --> 00:45:53,820 That's from that term. 688 00:45:53,820 --> 00:45:56,510 And the zeros, well, where are the zeros? 689 00:45:56,510 --> 00:45:58,320 They're the roots of the numerator, which 690 00:45:58,320 --> 00:46:03,680 are at the N roots of unity. 691 00:46:03,680 --> 00:46:04,750 Where are they? 692 00:46:04,750 --> 00:46:05,630 They're distributed. 693 00:46:05,630 --> 00:46:07,790 The N roots of unity are distributed 694 00:46:07,790 --> 00:46:12,800 around the unit circle equally spaced in angle 695 00:46:12,800 --> 00:46:16,070 starting at z equals 1. 696 00:46:16,070 --> 00:46:18,515 So there's a zero at z equals 1. 697 00:46:18,515 --> 00:46:21,170 But there's also a pole it equals 1. 698 00:46:21,170 --> 00:46:24,500 So in fact at z equals 1, there's neither 0 nor a pole 699 00:46:24,500 --> 00:46:27,570 because the two cancel out. 700 00:46:27,570 --> 00:46:30,330 If we look at the other zeros, then 701 00:46:30,330 --> 00:46:33,900 let's take a specific case, that is n equals 8. 702 00:46:33,900 --> 00:46:37,980 We'd expect to see eight zeros, except for the 1 at z 703 00:46:37,980 --> 00:46:40,080 equals 1 that got canceled out. 704 00:46:40,080 --> 00:46:42,820 So there are seven left-- 705 00:46:42,820 --> 00:46:52,160 one there, there, there, there, here, here, and here. 706 00:46:52,160 --> 00:46:54,740 Then, there was the one at the origin. 707 00:46:54,740 --> 00:47:00,240 Let me just indicate that and the fact 708 00:47:00,240 --> 00:47:02,940 that it got canceled out by a pole. 709 00:47:02,940 --> 00:47:06,330 So in fact, that one isn't there. 710 00:47:06,330 --> 00:47:09,480 So the pole-zero pattern for the boxcar sequence, 711 00:47:09,480 --> 00:47:13,830 then, is n minus 1 poles at the origin 712 00:47:13,830 --> 00:47:18,570 plus zeros equally spaced in angle, but with the 1 713 00:47:18,570 --> 00:47:21,650 at z equals 1 missing. 714 00:47:21,650 --> 00:47:25,130 Now, what does this mean in terms of the frequency 715 00:47:25,130 --> 00:47:26,630 response? 716 00:47:26,630 --> 00:47:30,620 Well, we can very quickly generate the frequency 717 00:47:30,620 --> 00:47:34,760 response, or a rough idea of the frequency response, 718 00:47:34,760 --> 00:47:39,290 geometrically by referring back to the set of ideas 719 00:47:39,290 --> 00:47:42,740 that we introduced at the beginning of the lecture 720 00:47:42,740 --> 00:47:48,110 and ask what the behavior of the pole-zero vectors 721 00:47:48,110 --> 00:47:51,860 are as we go around the unit circle. 722 00:47:51,860 --> 00:47:55,760 The pole vectors, first of all, introduce only a linear phase 723 00:47:55,760 --> 00:47:58,160 term and have no effect on the magnitude. 724 00:47:58,160 --> 00:48:01,520 We had agreed on that before. 725 00:48:01,520 --> 00:48:08,260 And so for the magnitude, we only need focus on the zeros. 726 00:48:08,260 --> 00:48:11,110 Well, one thing is obvious and that 727 00:48:11,110 --> 00:48:18,810 is that as we go around the unit circle, 728 00:48:18,810 --> 00:48:20,820 the frequency response is obviously 729 00:48:20,820 --> 00:48:25,110 0 when we hit each one of these zeros. 730 00:48:25,110 --> 00:48:31,200 That corresponds to pi over 4, pi over 2, pi over 2 731 00:48:31,200 --> 00:48:33,480 plus pi over 4-- whatever that is, 732 00:48:33,480 --> 00:48:38,640 pi, and the next increment of pi over 4, et cetera. 733 00:48:38,640 --> 00:48:41,970 Furthermore, you can see that at least, it's not implausible. 734 00:48:41,970 --> 00:48:46,200 Actually, we really could argue this somewhat precisely, 735 00:48:46,200 --> 00:48:52,080 that at omega equals 0, that's the place where 736 00:48:52,080 --> 00:48:56,280 we're the farthest away from all the zeros. 737 00:48:56,280 --> 00:48:59,040 As we start moving around the unit circle, 738 00:48:59,040 --> 00:49:01,265 if we're in-between two of these zeros, 739 00:49:01,265 --> 00:49:03,390 maybe we're a little farther away from one of them, 740 00:49:03,390 --> 00:49:05,530 but we're closer to another one. 741 00:49:05,530 --> 00:49:08,850 And in terms of the product of the length of the zero vectors, 742 00:49:08,850 --> 00:49:11,790 that will tend to stay smaller than the product 743 00:49:11,790 --> 00:49:16,020 of the lengths of the zero vectors at z equals 1. 744 00:49:16,020 --> 00:49:21,030 Consequently, the frequency response starts at some value 745 00:49:21,030 --> 00:49:26,160 and of course, goes down to 0 at pi over 4. 746 00:49:26,160 --> 00:49:29,870 That's because of this 0. 747 00:49:29,870 --> 00:49:34,610 Then, it comes back up again, but not quite as far, 748 00:49:34,610 --> 00:49:38,780 and then goes back down to 0 at pi over 2. 749 00:49:38,780 --> 00:49:42,610 Then it goes up again and comes down again. 750 00:49:42,610 --> 00:49:44,450 It goes up not quite as far. 751 00:49:44,450 --> 00:49:47,720 And that's not particularly obvious geometrically. 752 00:49:47,720 --> 00:49:52,850 And then, the same thing again going to 0 at pi. 753 00:49:52,850 --> 00:49:55,370 And then of course as we come back around the unit circle, 754 00:49:55,370 --> 00:49:58,450 we see the same thing, the same type of behavior, 755 00:49:58,450 --> 00:49:59,784 repeated again. 756 00:50:04,220 --> 00:50:08,740 So roughly, we can get a geometric picture 757 00:50:08,740 --> 00:50:13,090 of the frequency response for a boxcar sequence 758 00:50:13,090 --> 00:50:17,410 by looking at the location of the poles 759 00:50:17,410 --> 00:50:21,640 and zeros in the z-plane and the behavior of the vectors 760 00:50:21,640 --> 00:50:25,230 as we travel around the unit circle. 761 00:50:25,230 --> 00:50:31,500 Well finally, let's just look at the Fourier transform 762 00:50:31,500 --> 00:50:35,640 of the boxcar somewhat more formally because in fact, it's 763 00:50:35,640 --> 00:50:37,200 an important sequence. 764 00:50:37,200 --> 00:50:42,720 And it's important to have a fairly complete 765 00:50:42,720 --> 00:50:45,510 precise statement of the Fourier transform 766 00:50:45,510 --> 00:50:49,360 and a complete picture of what it looks like. 767 00:50:49,360 --> 00:50:51,060 Well, let's see. 768 00:50:51,060 --> 00:50:55,120 We had the Z-transform as 1 minus z 769 00:50:55,120 --> 00:50:59,192 to the minus n over 1 minus z to the minus 1. 770 00:50:59,192 --> 00:51:01,890 Substituting in z equals z to the j omega, 771 00:51:01,890 --> 00:51:07,600 then we have x of e to the j omega as this. 772 00:51:07,600 --> 00:51:11,560 We can factor out a factor e to the minus j omega capital 773 00:51:11,560 --> 00:51:17,220 N over 2, leaving e to the j omega capital N over 2 minus 774 00:51:17,220 --> 00:51:22,010 e to the minus j omega capital N over 2, and a denominator 775 00:51:22,010 --> 00:51:26,840 factor e to the minus j omega over 2 times e to the j omega-- 776 00:51:26,840 --> 00:51:31,490 over 2 minus e to the minus j omega over 2. 777 00:51:31,490 --> 00:51:40,890 This piece we recognize as 2j sine omega capital N over to 2. 778 00:51:40,890 --> 00:51:50,430 And this piece we recognize as 2-j times sine omega over 2. 779 00:51:50,430 --> 00:51:53,910 And consequently, putting these two terms together 780 00:51:53,910 --> 00:51:58,440 and inserting this substitution, the Fourier transform 781 00:51:58,440 --> 00:52:03,210 is e to the minus j omega capital N minus 1 over 2-- 782 00:52:03,210 --> 00:52:06,780 that's a linear phase term, by the way-- 783 00:52:06,780 --> 00:52:13,980 times sine omega capital N over 2 divided by sine omega over 2. 784 00:52:13,980 --> 00:52:18,510 And this function, sine omega capital N 785 00:52:18,510 --> 00:52:22,290 over 2 divided by sine omega over 2, 786 00:52:22,290 --> 00:52:27,830 is the discrete time counterpart of what we usually 787 00:52:27,830 --> 00:52:33,320 find in the continuous time case as sine x over x. 788 00:52:33,320 --> 00:52:37,910 That is, this is a sine nx over sine x kind of function. 789 00:52:37,910 --> 00:52:41,690 And it plays exactly the same role in the discrete time case 790 00:52:41,690 --> 00:52:45,800 that the sine x over x function plays in the continuous time 791 00:52:45,800 --> 00:52:47,610 case. 792 00:52:47,610 --> 00:52:49,880 And that's not unreasonable, actually, 793 00:52:49,880 --> 00:52:54,810 because this arose by looking at the Fourier 794 00:52:54,810 --> 00:52:58,530 transform of a rectangular sequence, 795 00:52:58,530 --> 00:53:01,830 whereas the sine x over x function 796 00:53:01,830 --> 00:53:05,970 arises by looking at the Fourier transform of a continuous time 797 00:53:05,970 --> 00:53:08,020 rectangle. 798 00:53:08,020 --> 00:53:12,550 So the sine nx over sine x function, 799 00:53:12,550 --> 00:53:17,280 which is what we have here, is an extremely important 800 00:53:17,280 --> 00:53:21,540 function-- as important as sine x over x in the continuous time 801 00:53:21,540 --> 00:53:22,780 case. 802 00:53:22,780 --> 00:53:25,590 And consequently, let me just show you 803 00:53:25,590 --> 00:53:29,160 this function plotted out a little more precisely 804 00:53:29,160 --> 00:53:31,170 than I would be able to do at the board. 805 00:53:31,170 --> 00:53:40,220 In particular, let me show you a viewgraph which illustrates 806 00:53:40,220 --> 00:53:44,030 the function sine omega capital N over 2 807 00:53:44,030 --> 00:53:47,150 divided by sine omega over 2. 808 00:53:47,150 --> 00:53:52,160 And I've sketched this now over more than just a 0 to 2 pi 809 00:53:52,160 --> 00:53:58,040 interval to, again, stress the fact that this is periodic-- 810 00:53:58,040 --> 00:54:00,320 as all Fourier transforms are-- 811 00:54:00,320 --> 00:54:04,460 periodic with a period of 2 pi. 812 00:54:04,460 --> 00:54:06,770 The important characteristics of it-- 813 00:54:06,770 --> 00:54:08,144 some important characteristics. 814 00:54:08,144 --> 00:54:08,810 There are a lot. 815 00:54:08,810 --> 00:54:11,600 But some important characteristics of it 816 00:54:11,600 --> 00:54:15,270 are that it has an envelope. 817 00:54:15,270 --> 00:54:18,620 It's basically a sinusoidal function 818 00:54:18,620 --> 00:54:24,920 with an envelope that is the reciprocal of a sinusoid. 819 00:54:24,920 --> 00:54:32,330 The period of this sinusoid is from 0 to 2pi, 820 00:54:32,330 --> 00:54:34,790 whereas this one wiggles faster, depending 821 00:54:34,790 --> 00:54:38,180 on the value of capital N. I've sketched it here 822 00:54:38,180 --> 00:54:40,370 for n equals 15. 823 00:54:40,370 --> 00:54:44,120 But in fact, it has a lot of the character 824 00:54:44,120 --> 00:54:48,950 of a sine x over x function. 825 00:54:48,950 --> 00:54:52,880 That is, it has a big central lobe, 826 00:54:52,880 --> 00:54:56,330 decays down and wiggles and gets smaller as it's decaying. 827 00:54:56,330 --> 00:54:59,180 But then, of course, the fact that it has to be periodic 828 00:54:59,180 --> 00:55:01,970 is what distinguishes it in the discrete time 829 00:55:01,970 --> 00:55:06,090 case from the sine x over x function in the continuous time 830 00:55:06,090 --> 00:55:06,590 case. 831 00:55:13,380 --> 00:55:13,880 OK. 832 00:55:13,880 --> 00:55:20,450 Well, this concludes our discussion of the Z-transform. 833 00:55:20,450 --> 00:55:23,094 We've now talked about two transforms. 834 00:55:23,094 --> 00:55:24,760 We've talked about the Fourier transform 835 00:55:24,760 --> 00:55:30,710 and the Z-transform spread out over about five lectures. 836 00:55:30,710 --> 00:55:34,490 And in the next lecture, the next set of two or three 837 00:55:34,490 --> 00:55:39,740 lectures, we'll be talking about yet another transform, which 838 00:55:39,740 --> 00:55:42,980 is a transform that's really somewhat special 839 00:55:42,980 --> 00:55:47,570 and linked very closely to the notion of discrete time 840 00:55:47,570 --> 00:55:50,420 signals and discrete time signal processing. 841 00:55:50,420 --> 00:55:54,260 That transform is the Discrete Fourier transform. 842 00:55:54,260 --> 00:55:58,520 And besides being a mathematical tool, 843 00:55:58,520 --> 00:56:02,230 as the Fourier transform and the Z-transform have been, 844 00:56:02,230 --> 00:56:07,970 Discrete Fourier transform has some important computational 845 00:56:07,970 --> 00:56:11,930 realizations and computational implications 846 00:56:11,930 --> 00:56:14,990 that will be one of the important things that 847 00:56:14,990 --> 00:56:18,110 will want to capitalize on in applying digital signal 848 00:56:18,110 --> 00:56:20,650 processing to real problems. 849 00:56:20,650 --> 00:56:22,540 Thank you. 850 00:56:22,540 --> 00:56:25,590 [MUSIC PLAYING]