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 00:00:22,940 --> 00:00:26,440 [MUSIC PLAYING] 9 00:00:50,180 --> 00:00:51,300 ALAN OPPENHEIM: Hi. 10 00:00:51,300 --> 00:00:54,510 As you recall, last time we developed 11 00:00:54,510 --> 00:00:57,810 the discrete-time Fourier transform. 12 00:00:57,810 --> 00:01:01,830 And one of the issues that I alluded to, 13 00:01:01,830 --> 00:01:04,379 although I didn't say it explicitly, 14 00:01:04,379 --> 00:01:07,680 is that the Fourier transform doesn't 15 00:01:07,680 --> 00:01:09,720 exist for all sequences. 16 00:01:09,720 --> 00:01:13,710 In fact, as we'll see today, one of the difficulties 17 00:01:13,710 --> 00:01:15,930 is the issue of convergence. 18 00:01:15,930 --> 00:01:18,300 That is, the Fourier transform doesn't 19 00:01:18,300 --> 00:01:20,910 converge for all sequences. 20 00:01:20,910 --> 00:01:24,360 And to get around this, we'll generalize 21 00:01:24,360 --> 00:01:27,390 the notion of the Fourier transform as we introduced it 22 00:01:27,390 --> 00:01:33,750 last time to what's called the Z-transform, the Z-transform 23 00:01:33,750 --> 00:01:37,170 being the discrete time counterpart of the Laplace 24 00:01:37,170 --> 00:01:41,050 transform for continuous time systems. 25 00:01:41,050 --> 00:01:47,160 Well, let me begin by examining just a little this issue 26 00:01:47,160 --> 00:01:50,550 of convergence. 27 00:01:50,550 --> 00:01:54,240 We have here the Fourier transform relationship 28 00:01:54,240 --> 00:01:59,430 as we developed it last time, x of e to the j omega given 29 00:01:59,430 --> 00:02:04,890 by the sum of x of 10 times e to the minus j omega n. 30 00:02:04,890 --> 00:02:08,190 And as we discussed the Fourier transform last time, 31 00:02:08,190 --> 00:02:13,800 we assumed that the sum was convergent, 32 00:02:13,800 --> 00:02:16,530 and-- or at least we didn't particularly raise the issue 33 00:02:16,530 --> 00:02:18,840 as to whether it is or isn't. 34 00:02:18,840 --> 00:02:25,800 Well, we can look at the magnitude of x, or equivalently 35 00:02:25,800 --> 00:02:29,310 the magnitude of this sum, and ask 36 00:02:29,310 --> 00:02:33,840 under what conditions this sum will converge. 37 00:02:33,840 --> 00:02:39,240 That is, under what conditions will this sum be finite. 38 00:02:39,240 --> 00:02:44,220 Well, here we have the magnitude of a sum of things 39 00:02:44,220 --> 00:02:47,580 from what's called the triangle inequality, 40 00:02:47,580 --> 00:02:50,430 or you can just kind of imagine that this 41 00:02:50,430 --> 00:02:55,740 is true from common sense, is that the magnitude of a sum 42 00:02:55,740 --> 00:03:00,600 has to be less than or equal to the sum of the magnitudes. 43 00:03:00,600 --> 00:03:04,920 So the magnitude of this sum has to be less than 44 00:03:04,920 --> 00:03:10,320 or equal to the sum of the magnitude of x of n e 45 00:03:10,320 --> 00:03:12,720 to the minus j omega n. 46 00:03:12,720 --> 00:03:16,110 The magnitude of a product is the product of the magnitudes, 47 00:03:16,110 --> 00:03:17,190 of course. 48 00:03:17,190 --> 00:03:22,650 So this says that the magnitude of the Fourier transform 49 00:03:22,650 --> 00:03:27,060 has to be less than or equal to the sum of the magnitude of x 50 00:03:27,060 --> 00:03:31,670 of n times the magnitude of the complex exponential, e 51 00:03:31,670 --> 00:03:34,450 to the minus j omega n. 52 00:03:34,450 --> 00:03:38,530 Well, this of course we can't say a lot about. 53 00:03:38,530 --> 00:03:44,170 This we can recognize as unity. 54 00:03:44,170 --> 00:03:45,670 It's a complex exponential. 55 00:03:45,670 --> 00:03:49,600 The magnitude of a complex exponential is one. 56 00:03:49,600 --> 00:03:54,550 So consequently, the magnitude of the Fourier transform 57 00:03:54,550 --> 00:03:59,950 is less than or equal to the sum of the magnitudes of x 58 00:03:59,950 --> 00:04:03,280 of n summed over minus infinity-- n equals 59 00:04:03,280 --> 00:04:07,060 minus infinity to plus infinity. 60 00:04:07,060 --> 00:04:11,290 Well, that tells us then that the Fourier transform 61 00:04:11,290 --> 00:04:16,360 will converge, converge meaning that the sum ends up finite, 62 00:04:16,360 --> 00:04:18,670 if this converges. 63 00:04:18,670 --> 00:04:21,200 That is, certainly if this sum is finite, 64 00:04:21,200 --> 00:04:25,310 then the magnitude of x of e to the j omega is finite. 65 00:04:25,310 --> 00:04:31,430 So we can say that x of e to the j omega, the Fourier transform, 66 00:04:31,430 --> 00:04:37,210 converges if the sum of the magnitude of x of n 67 00:04:37,210 --> 00:04:40,040 is less than infinity. 68 00:04:40,040 --> 00:04:44,690 Well, this is a condition that's commonly referred 69 00:04:44,690 --> 00:04:47,390 to as absolutely summable. 70 00:04:47,390 --> 00:04:49,700 The condition is that the sequence x of n 71 00:04:49,700 --> 00:04:52,070 be absolutely summable. 72 00:04:52,070 --> 00:04:55,070 The way we've gone through this, the statement 73 00:04:55,070 --> 00:04:59,190 is that, if x of n is absolutely summable, 74 00:04:59,190 --> 00:05:02,900 then the Fourier transform converges 75 00:05:02,900 --> 00:05:05,540 under some sorts of conditions. 76 00:05:05,540 --> 00:05:08,390 And depending on how you interpret convergence, 77 00:05:08,390 --> 00:05:10,520 this can be modified somewhat. 78 00:05:10,520 --> 00:05:12,770 But basically-- and that, in fact, 79 00:05:12,770 --> 00:05:16,980 is discussed in some detail in the text. 80 00:05:16,980 --> 00:05:22,020 But basically, the condition for convergence of the Fourier 81 00:05:22,020 --> 00:05:27,260 transform is that the sequence whose Fourier transform we're 82 00:05:27,260 --> 00:05:31,980 talking about be absolutely summable. 83 00:05:31,980 --> 00:05:35,840 Now, we've seen a sum like this someplace else. 84 00:05:35,840 --> 00:05:38,470 In fact, a couple lectures ago, we 85 00:05:38,470 --> 00:05:42,490 talked about stability of discrete time systems. 86 00:05:42,490 --> 00:05:46,300 And the statement about stability 87 00:05:46,300 --> 00:05:49,210 was that the unit sample response of the system 88 00:05:49,210 --> 00:05:51,040 be absolutely summable. 89 00:05:51,040 --> 00:05:55,000 That is, or was, that, if the unit sample response 90 00:05:55,000 --> 00:05:58,660 is absolutely summable, then the system is stable. 91 00:05:58,660 --> 00:06:03,580 So one conclusion, which you should tuck away for now 92 00:06:03,580 --> 00:06:07,720 and we'll be bringing into play later on in this lecture, 93 00:06:07,720 --> 00:06:16,310 is that a stable system implies that the system-- 94 00:06:16,310 --> 00:06:20,820 that the frequency response, or the Fourier transform 95 00:06:20,820 --> 00:06:24,330 of the unit sample response, converges. 96 00:06:24,330 --> 00:06:28,070 So this is convergence, then. 97 00:06:28,070 --> 00:06:32,120 And we can look at a couple of examples just 98 00:06:32,120 --> 00:06:37,910 to see how this looks in some specific cases. 99 00:06:37,910 --> 00:06:42,860 Let's first of all take the example of the sequence x of n 100 00:06:42,860 --> 00:06:49,010 is 1/2 to the n u of n as an exponential sequence which 101 00:06:49,010 --> 00:06:52,400 is zero for n less than zero because of the unit step. 102 00:06:52,400 --> 00:06:59,420 It's exponentially decaying for n greater than zero. 103 00:06:59,420 --> 00:07:02,960 If we sum the magnitude of x of n 104 00:07:02,960 --> 00:07:05,575 from minus infinity to plus infinity, 105 00:07:05,575 --> 00:07:11,510 we get that the magnitude, the sum of the magnitude of x of n 106 00:07:11,510 --> 00:07:13,290 is equal to two. 107 00:07:13,290 --> 00:07:16,790 Well, this is something, a type of sum 108 00:07:16,790 --> 00:07:20,210 that hopefully you're familiar with carrying out. 109 00:07:20,210 --> 00:07:25,090 In any case, what we're summing here is the series 110 00:07:25,090 --> 00:07:28,220 1, 1/2, 1/4, 1/8, et cetera. 111 00:07:28,220 --> 00:07:30,470 That sums to two. 112 00:07:30,470 --> 00:07:34,640 So we can say that, for this sequence, certainly, 113 00:07:34,640 --> 00:07:37,025 the Fourier transform converges. 114 00:07:37,025 --> 00:07:39,800 It's a perfectly well-behaved sequence. 115 00:07:39,800 --> 00:07:41,510 A Fourier transform converges, we 116 00:07:41,510 --> 00:07:45,850 can compute the Fourier transform, et cetera. 117 00:07:45,850 --> 00:07:50,560 Well, another sequence we can try is the sequence x of n 118 00:07:50,560 --> 00:07:56,160 equals 2 to the n times a unit step. 119 00:07:56,160 --> 00:07:58,710 Well, if we just think about the values 120 00:07:58,710 --> 00:08:02,440 that we have, at n equals zero we have 1, at n equals 1, 121 00:08:02,440 --> 00:08:05,320 we have 2, at n equals 2, we have 4, at n equals 3, 122 00:08:05,320 --> 00:08:07,600 we have 8. 123 00:08:07,600 --> 00:08:11,740 And obviously, then, the sum of the absolute values 124 00:08:11,740 --> 00:08:18,470 of x of n for that particular sequence diverges. 125 00:08:18,470 --> 00:08:20,900 That is, the sum is infinite. 126 00:08:20,900 --> 00:08:26,240 So here's an example of a sequence for which the Fourier 127 00:08:26,240 --> 00:08:28,070 transform doesn't converge. 128 00:08:28,070 --> 00:08:30,830 In other words, the Fourier transform for this sequence 129 00:08:30,830 --> 00:08:32,090 doesn't exist. 130 00:08:32,090 --> 00:08:34,750 And you can imagine the generalization here 131 00:08:34,750 --> 00:08:38,450 that, if we talk about a sequence which 132 00:08:38,450 --> 00:08:40,490 decays exponentially-- by the way, 133 00:08:40,490 --> 00:08:43,419 in either direction, negative n or positive n-- then 134 00:08:43,419 --> 00:08:46,160 the Fourier transform will exist, 135 00:08:46,160 --> 00:08:49,760 whereas if it's an exponential sequence that 136 00:08:49,760 --> 00:08:54,110 grows exponentially, then the Fourier transform is not 137 00:08:54,110 --> 00:08:58,040 going to exist, or is not going to converge. 138 00:08:58,040 --> 00:09:01,200 Well, what do we do about that? 139 00:09:01,200 --> 00:09:07,410 There is a way around it by essentially using 140 00:09:07,410 --> 00:09:13,830 the artifice of multiplying a sequence that 141 00:09:13,830 --> 00:09:17,880 doesn't converge, or say that grows exponentially 142 00:09:17,880 --> 00:09:21,510 by a decaying exponential, and making 143 00:09:21,510 --> 00:09:25,230 the decay on the exponential fast enough so that the product 144 00:09:25,230 --> 00:09:27,660 has a Fourier transform. 145 00:09:27,660 --> 00:09:31,380 That is, we can talk about the Fourier transform 146 00:09:31,380 --> 00:09:36,390 in general, not of x of n, but of x 147 00:09:36,390 --> 00:09:46,620 of n multiplied by a sequence r to the minus n, for which we 148 00:09:46,620 --> 00:09:50,730 pick r so that the product of these two 149 00:09:50,730 --> 00:09:52,950 is absolutely summable. 150 00:09:52,950 --> 00:09:57,240 And since we've introduced another parameter r, then 151 00:09:57,240 --> 00:10:01,830 the Fourier transform of x of n times r to the minus n, 152 00:10:01,830 --> 00:10:04,880 we don't want to call x of e to the j omega. 153 00:10:04,880 --> 00:10:11,070 Let's for now think of it as x of r times e to the j omega, 154 00:10:11,070 --> 00:10:14,880 the r denoting the complex exponential 155 00:10:14,880 --> 00:10:17,790 that we're multiplying by. 156 00:10:17,790 --> 00:10:20,940 And then, of course, for some values of r, 157 00:10:20,940 --> 00:10:23,640 this sum is going to converge. 158 00:10:23,640 --> 00:10:28,110 For other values of r, this sum is not going to converge. 159 00:10:28,110 --> 00:10:30,540 For example, if we look at our example 160 00:10:30,540 --> 00:10:37,800 two here, if I multiply this sequence by a decaying 161 00:10:37,800 --> 00:10:42,270 exponential that decays fast enough so 162 00:10:42,270 --> 00:10:45,120 that the product decays-- 163 00:10:45,120 --> 00:10:46,990 and what do I have to multiply it by? 164 00:10:46,990 --> 00:10:50,520 I have to multiply it by something that decays faster 165 00:10:50,520 --> 00:10:53,710 than 1/2 to the n. 166 00:10:53,710 --> 00:11:01,560 So if I stick in here an r that is greater than 2-- 167 00:11:01,560 --> 00:11:05,520 2 because there's a minus sign here, r to the minus n-- 168 00:11:05,520 --> 00:11:10,360 then I'll end up with, for this sequence, 169 00:11:10,360 --> 00:11:13,190 with a sum here that converges. 170 00:11:13,190 --> 00:11:15,450 So for this example, if r-- 171 00:11:15,450 --> 00:11:19,120 any choice of r greater than two, this sum 172 00:11:19,120 --> 00:11:20,170 is going to converge. 173 00:11:20,170 --> 00:11:23,300 And I can talk about x of r of e to the j omega. 174 00:11:23,300 --> 00:11:24,250 So that's the idea. 175 00:11:24,250 --> 00:11:28,150 The idea is to multiply by an exponential that 176 00:11:28,150 --> 00:11:30,220 decays fast enough or doesn't grow 177 00:11:30,220 --> 00:11:35,140 so fast that the product x of n times r to the minus n 178 00:11:35,140 --> 00:11:37,150 converges. 179 00:11:37,150 --> 00:11:40,030 Well, we can rewrite this a little differently. 180 00:11:40,030 --> 00:11:44,200 We can combine the r to the minus n and the e to the minus 181 00:11:44,200 --> 00:11:46,240 j omega n. 182 00:11:46,240 --> 00:11:49,420 And what results is the sum, minus infinity 183 00:11:49,420 --> 00:11:53,590 to plus infinity, of x of n times 184 00:11:53,590 --> 00:11:56,500 re to the j omega to the minus n, 185 00:11:56,500 --> 00:12:00,490 just simply combining these two terms together. 186 00:12:00,490 --> 00:12:07,210 And it's convenient to think of this as a new complex variable, 187 00:12:07,210 --> 00:12:12,490 magnitude r and angle omega, which we'll denote 188 00:12:12,490 --> 00:12:18,050 by z, z as in the Z-transform. 189 00:12:18,050 --> 00:12:20,140 The Z-transform. 190 00:12:20,140 --> 00:12:26,560 So, this essentially leads us to the notion of the Z-transform, 191 00:12:26,560 --> 00:12:31,870 where the Z-transform of a sequence x of n 192 00:12:31,870 --> 00:12:38,950 is defined as x of z is equal to the sum of x of n times z 193 00:12:38,950 --> 00:12:46,120 to the minus n, where z is equal to r times e to the j omega. 194 00:12:46,120 --> 00:12:51,610 In other words, x of z is just simply the x sub r of e 195 00:12:51,610 --> 00:12:55,110 to the j omega that we finished talking about. 196 00:12:55,110 --> 00:12:59,400 Well, how is the Z-transform related to the Fourier 197 00:12:59,400 --> 00:13:03,540 transform if the Fourier transform exists? 198 00:13:03,540 --> 00:13:07,420 Obviously, from what we just finished saying, 199 00:13:07,420 --> 00:13:11,320 the Z-transform is the Fourier transform for this weighting 200 00:13:11,320 --> 00:13:13,660 factor r equal to unity. 201 00:13:13,660 --> 00:13:27,600 That is, x of e to the j omega is equal to x of z where? 202 00:13:27,600 --> 00:13:34,410 Well, for z equal to e to the j omega-- 203 00:13:34,410 --> 00:13:39,150 or another way of saying that is for the magnitude of z 204 00:13:39,150 --> 00:13:40,470 equal to unity. 205 00:13:40,470 --> 00:13:43,530 With z equal to e to the j omega, the magnitude of z 206 00:13:43,530 --> 00:13:46,170 obviously is equal to unity. 207 00:13:46,170 --> 00:13:50,310 So we have now the Z-transform. 208 00:13:50,310 --> 00:13:53,190 It's a function of a complex variable 209 00:13:53,190 --> 00:14:00,240 z, which has a magnitude r and an angle omega. 210 00:14:00,240 --> 00:14:03,120 And it's just simply a generalization 211 00:14:03,120 --> 00:14:04,860 of the Fourier transform. 212 00:14:04,860 --> 00:14:06,870 It, in fact, is equal to the Fourier 213 00:14:06,870 --> 00:14:10,690 transform for the magnitude of z equal to 1. 214 00:14:10,690 --> 00:14:15,450 And it, in effect, corresponds to the Fourier transform 215 00:14:15,450 --> 00:14:21,090 of the sequence multiplied by an exponential sequence r 216 00:14:21,090 --> 00:14:24,360 to the minus n. 217 00:14:24,360 --> 00:14:27,450 When does the Z-transform converge? 218 00:14:27,450 --> 00:14:32,820 Well, the Z-transform converges when this x sub r of e 219 00:14:32,820 --> 00:14:36,660 to the j omega that we were talking about converges. 220 00:14:36,660 --> 00:14:40,320 And that converges when the sequence x of n r to the minus 221 00:14:40,320 --> 00:14:42,870 n is absolutely summable. 222 00:14:42,870 --> 00:14:47,610 So the Z-transform converges if the sum 223 00:14:47,610 --> 00:14:52,920 of x of n times r to the minus n is finite. 224 00:14:52,920 --> 00:14:56,490 And for some values of r, given a certain sequence, 225 00:14:56,490 --> 00:15:01,930 for some values of r the Z-transform will converge, 226 00:15:01,930 --> 00:15:06,720 for some values of r the Z-transform won't converge. 227 00:15:06,720 --> 00:15:09,390 Well let's look at an example. 228 00:15:09,390 --> 00:15:13,740 And it's an example, I guess, that you're 229 00:15:13,740 --> 00:15:15,270 seeing fairly frequently. 230 00:15:15,270 --> 00:15:19,880 In fact, it'll keep cropping up in a variety of ways. 231 00:15:19,880 --> 00:15:22,710 It's kind of a simple one, and nice to use. 232 00:15:22,710 --> 00:15:25,470 That is the sequence-- our old friend again-- 233 00:15:25,470 --> 00:15:31,530 x of n is 1/2 to the n times u of n. 234 00:15:31,530 --> 00:15:40,110 Well, the Z-transform of x of n, x of z is the sum of x of n 235 00:15:40,110 --> 00:15:42,360 z to the minus n. 236 00:15:42,360 --> 00:15:45,060 Since this is multiplied by a unit step, 237 00:15:45,060 --> 00:15:48,640 the limits on the sum are from zero to infinity. 238 00:15:48,640 --> 00:15:53,580 So this is the sum from zero to infinity of 1/2 to the n z 239 00:15:53,580 --> 00:15:55,940 to the minus n. 240 00:15:55,940 --> 00:16:00,230 And that's equal to the sum from n equals 0 to infinity of 1/2 241 00:16:00,230 --> 00:16:04,550 times z to the minus 1-- just combining these together-- 242 00:16:04,550 --> 00:16:06,130 to the n. 243 00:16:06,130 --> 00:16:08,040 Now, let me point out, incidentally-- 244 00:16:08,040 --> 00:16:12,350 and I think that by now you've seen this-- 245 00:16:12,350 --> 00:16:15,650 this is a kind of sum, this geometric series 246 00:16:15,650 --> 00:16:19,860 is a kind of sum that comes up over and over and over again 247 00:16:19,860 --> 00:16:21,920 in dealing with sequences. 248 00:16:21,920 --> 00:16:25,970 And it's the kind of summation that you should carry around 249 00:16:25,970 --> 00:16:27,290 in your hip pocket. 250 00:16:27,290 --> 00:16:30,140 That is, you should keep in mind, because it keeps 251 00:16:30,140 --> 00:16:32,050 coming up like standard integrals do 252 00:16:32,050 --> 00:16:35,150 in the continuous time case, that the sum from 0 253 00:16:35,150 --> 00:16:37,410 to infinity of a to the n-- 254 00:16:37,410 --> 00:16:41,510 a has to be less than 1 for convergence-- of a to the n 255 00:16:41,510 --> 00:16:44,060 is equal to 1 over 1 minus a. 256 00:16:44,060 --> 00:16:46,330 Just store that away. 257 00:16:46,330 --> 00:16:52,490 Anyway, summing this, then, we end up with 1 over 1 258 00:16:52,490 --> 00:16:54,960 minus this thing. 259 00:16:54,960 --> 00:16:59,390 So the Z-transform is 1 over 1 minus 1/2 times z 260 00:16:59,390 --> 00:17:03,020 to the minus 1, which we can leave in this form. 261 00:17:03,020 --> 00:17:04,640 And sometimes, we will. 262 00:17:04,640 --> 00:17:08,270 Or, we can multiply numerator and denominator 263 00:17:08,270 --> 00:17:13,695 by z, in which case we have z divided by z minus 1/2. 264 00:17:17,290 --> 00:17:19,690 Well, when does this converge? 265 00:17:19,690 --> 00:17:26,020 It converges when the magnitude of 1/2 times z to the minus 1 266 00:17:26,020 --> 00:17:29,780 raised to the n is finite. 267 00:17:29,780 --> 00:17:31,850 And that requires that this thing 268 00:17:31,850 --> 00:17:36,710 that we're taking the absolute magnitude of be-- 269 00:17:36,710 --> 00:17:38,030 raised to the n-- 270 00:17:38,030 --> 00:17:42,460 be less than unity, or that implies 271 00:17:42,460 --> 00:17:48,010 that z, the magnitude of z, must be greater than 1/2. 272 00:17:48,010 --> 00:17:51,850 The magnitude of z must be greater than 1/2, 273 00:17:51,850 --> 00:17:55,000 or the magnitude of z minus z to the minus 1 274 00:17:55,000 --> 00:17:56,590 must be less than two. 275 00:17:56,590 --> 00:17:58,840 Magnitude z to the minus 1 less than two, 276 00:17:58,840 --> 00:18:02,530 or the magnitude of z greater than 1/2. 277 00:18:02,530 --> 00:18:08,020 And under those conditions, the Z-transform will converge. 278 00:18:08,020 --> 00:18:13,270 Now, for this example, does the Fourier transform exist? 279 00:18:13,270 --> 00:18:15,700 Does the Fourier transform converge? 280 00:18:15,700 --> 00:18:16,940 How do we answer that? 281 00:18:16,940 --> 00:18:19,990 Well, the Fourier transform converges 282 00:18:19,990 --> 00:18:23,680 if the convergence of the Z-transform 283 00:18:23,680 --> 00:18:27,820 includes the magnitude of z equal to 1. 284 00:18:27,820 --> 00:18:28,870 Does it? 285 00:18:28,870 --> 00:18:29,980 Sure it does. 286 00:18:29,980 --> 00:18:32,620 The magnitude of z must be greater than 1/2 287 00:18:32,620 --> 00:18:34,830 for convergence of the Z-transform. 288 00:18:34,830 --> 00:18:37,940 That includes the magnitude of z equal to 1. 289 00:18:37,940 --> 00:18:41,170 Therefore, for this sequence, the Fourier transform 290 00:18:41,170 --> 00:18:42,430 converges. 291 00:18:42,430 --> 00:18:44,470 We had already decided that before, of course. 292 00:18:44,470 --> 00:18:47,800 That is obviously a better turnout that way 293 00:18:47,800 --> 00:18:49,480 this time also. 294 00:18:49,480 --> 00:18:53,320 But an important notion, then, is interpreting 295 00:18:53,320 --> 00:18:55,480 the existence of the Fourier transform 296 00:18:55,480 --> 00:19:01,900 in terms of the region of convergence of the Z-transform. 297 00:19:01,900 --> 00:19:04,630 This then defines the region of convergence, 298 00:19:04,630 --> 00:19:11,330 the set of values of z for which the Z-transform converges. 299 00:19:11,330 --> 00:19:15,690 Another interesting observation about this example 300 00:19:15,690 --> 00:19:20,490 is that we ended up with a Z-transform that 301 00:19:20,490 --> 00:19:23,545 is a ratio of polynomials. 302 00:19:23,545 --> 00:19:27,110 In this case, z to the minus 1 or, If we rewrite it, 303 00:19:27,110 --> 00:19:29,970 a ratio of polynomials in z. 304 00:19:29,970 --> 00:19:32,580 And that's an important observation. 305 00:19:32,580 --> 00:19:36,510 In fact, it is a consequence of the fact 306 00:19:36,510 --> 00:19:38,610 that we're talking about the Z-transform 307 00:19:38,610 --> 00:19:41,330 of an exponential sequence. 308 00:19:41,330 --> 00:19:44,730 Any exponential sequence will have a Z-transform that's 309 00:19:44,730 --> 00:19:47,550 a ratio of polynomials in z. 310 00:19:47,550 --> 00:19:52,260 Furthermore, any sequence that's a sum of exponentials 311 00:19:52,260 --> 00:19:56,250 will have a Z-transform that's a ratio of polynomials in z, 312 00:19:56,250 --> 00:19:59,340 since the Z-transform of a sum is the sum of the Z-transforms, 313 00:19:59,340 --> 00:20:01,290 and the sum of a ratio of polynomials 314 00:20:01,290 --> 00:20:03,670 is a ratio of polynomials and all that. 315 00:20:03,670 --> 00:20:06,180 But basically, the point to remember 316 00:20:06,180 --> 00:20:10,410 is that sums of exponentials as sequences 317 00:20:10,410 --> 00:20:15,780 result in Z-transforms that are ratios of polynomials in z, 318 00:20:15,780 --> 00:20:19,200 or in z to the minus 1. 319 00:20:19,200 --> 00:20:24,300 Well, for those class of sequences, which, in fact, are 320 00:20:24,300 --> 00:20:25,740 most of the sequences we're going 321 00:20:25,740 --> 00:20:28,470 to be talking about in these lectures, 322 00:20:28,470 --> 00:20:36,380 it's convenient to look at the Z-transform graphically, 323 00:20:36,380 --> 00:20:41,750 in terms of a plot in the z-plane. 324 00:20:41,750 --> 00:20:44,390 So I've indicated here the z-plane. 325 00:20:47,480 --> 00:20:51,370 This is the real part of z, this axis. 326 00:20:51,370 --> 00:20:54,580 This axis is the imaginary part of z. 327 00:20:54,580 --> 00:20:57,250 And I've drawn a circle, which I've 328 00:20:57,250 --> 00:21:02,680 called the unit circle, which is convenient to focus on. 329 00:21:02,680 --> 00:21:06,310 It's the circle in the z-plane for which the magnitude of z 330 00:21:06,310 --> 00:21:08,200 is equal to 1. 331 00:21:08,200 --> 00:21:11,110 What's important about the magnitude of z equal to 1? 332 00:21:11,110 --> 00:21:13,600 Well, it's on that circle, when we 333 00:21:13,600 --> 00:21:15,670 look at the Z-transform on that circle, 334 00:21:15,670 --> 00:21:17,770 that we see the Fourier transform. 335 00:21:17,770 --> 00:21:21,790 So it's often convenient to make reference 336 00:21:21,790 --> 00:21:24,970 to the unit circle in the z-plane corresponding 337 00:21:24,970 --> 00:21:28,960 to the place in the z-plane where the Z-transform is 338 00:21:28,960 --> 00:21:32,580 equal to the Fourier transform. 339 00:21:32,580 --> 00:21:39,750 When we have a Z-transform that is a ratio of polynomials in z, 340 00:21:39,750 --> 00:21:44,070 it's convenient to represent it in terms 341 00:21:44,070 --> 00:21:48,080 of the roots of the numerator polynomial 342 00:21:48,080 --> 00:21:51,320 and the roots of the denominator polynomial. 343 00:21:51,320 --> 00:21:53,930 The roots of the numerator polynomial 344 00:21:53,930 --> 00:21:58,910 we'll refer to as the zeros of the Z-transform. 345 00:21:58,910 --> 00:22:02,900 They're those values of z for which the Z-transform 346 00:22:02,900 --> 00:22:05,570 is going to be zero. 347 00:22:05,570 --> 00:22:08,100 And the roots of the denominator polynomial 348 00:22:08,100 --> 00:22:11,320 we'll refer to as the poles of the Z-transform. 349 00:22:11,320 --> 00:22:13,480 Those are the values of z at which 350 00:22:13,480 --> 00:22:17,870 the Z-transform blows up, just like poles and zeros 351 00:22:17,870 --> 00:22:22,180 of the Laplace transform in the continuous time case. 352 00:22:22,180 --> 00:22:30,710 We'll denote in the z-plane roots, the zeros, by a circle, 353 00:22:30,710 --> 00:22:36,680 and we'll denote the poles by a cross. 354 00:22:36,680 --> 00:22:41,360 So we have, for that particular example, a zero at the origin, 355 00:22:41,360 --> 00:22:44,740 since the numerator polynomial was simply z, 356 00:22:44,740 --> 00:22:48,080 and we have a pole at z equal to 1/2. 357 00:22:50,610 --> 00:22:53,610 Well, what else do we know about this Z-transform? 358 00:22:53,610 --> 00:22:56,700 We know that the Z-transform doesn't 359 00:22:56,700 --> 00:22:59,580 exist for all values of z. 360 00:22:59,580 --> 00:23:02,190 In particular, for convergence we 361 00:23:02,190 --> 00:23:06,480 require that the magnitude of z is greater than 1/2. 362 00:23:06,480 --> 00:23:11,760 So I've indicated that region in the z-plane, 363 00:23:11,760 --> 00:23:16,570 with these slanted lines, the region of convergence 364 00:23:16,570 --> 00:23:20,920 for this particular example are those values of z that 365 00:23:20,920 --> 00:23:24,940 lie outside the circle that's bounded by the pole at z 366 00:23:24,940 --> 00:23:26,020 equal to 1/2. 367 00:23:26,020 --> 00:23:28,810 And we'll see that that is, in fact, the way regions 368 00:23:28,810 --> 00:23:32,320 of convergences tend to go. 369 00:23:32,320 --> 00:23:35,620 They'll tend to be outside and inside poles. 370 00:23:35,620 --> 00:23:38,860 So the important aspects of this picture, 371 00:23:38,860 --> 00:23:42,280 there are the zeroes and the poles, 372 00:23:42,280 --> 00:23:45,430 there's the unit circle where the Z-transform is 373 00:23:45,430 --> 00:23:48,190 equal to the Fourier transform, and there's 374 00:23:48,190 --> 00:23:52,450 the region of convergence, which tells us where in the z-plane 375 00:23:52,450 --> 00:23:54,260 the Z-transform makes sense. 376 00:23:56,850 --> 00:23:57,350 All right. 377 00:23:57,350 --> 00:24:00,850 Let's try another example. 378 00:24:00,850 --> 00:24:06,490 Let's take the example of x of n equal to minus 1/2 379 00:24:06,490 --> 00:24:11,000 to the n u of minus n minus 1. 380 00:24:11,000 --> 00:24:15,470 Now, you saw that sequence a couple of lectures ago. 381 00:24:15,470 --> 00:24:20,120 It's a sequence that is like 1/2 to the n, 382 00:24:20,120 --> 00:24:23,180 but not for positive values of n, 383 00:24:23,180 --> 00:24:24,650 but for negative values of n. 384 00:24:24,650 --> 00:24:27,770 In other words, it's zero for positive values of n, 385 00:24:27,770 --> 00:24:29,630 and it does something exponentially 386 00:24:29,630 --> 00:24:30,840 for negative values of n. 387 00:24:33,940 --> 00:24:38,230 If you evaluate the Z-transform sum-- 388 00:24:38,230 --> 00:24:40,910 and I'll leave it to you to do this-- 389 00:24:40,910 --> 00:24:44,500 then we end up with x of z, the Z-transform, 390 00:24:44,500 --> 00:24:48,820 is 1 over 1 minus 1/2 z to the minus 1, 391 00:24:48,820 --> 00:24:50,150 just like we saw before. 392 00:24:50,150 --> 00:24:53,290 In fact, that's exactly the same expression 393 00:24:53,290 --> 00:24:57,960 as a ratio of polynomials that we saw before, 394 00:24:57,960 --> 00:25:00,450 which, of course, we can rewrite, 395 00:25:00,450 --> 00:25:05,430 as we did before, as z divided by z minus 1/2. 396 00:25:07,821 --> 00:25:08,320 All right. 397 00:25:08,320 --> 00:25:10,760 Do you think that the Fourier transform of this sequence 398 00:25:10,760 --> 00:25:11,260 exists? 399 00:25:13,780 --> 00:25:19,640 Well, it's 1/2 to the n, which dies out 400 00:25:19,640 --> 00:25:21,890 for negative n-- for positive n. 401 00:25:21,890 --> 00:25:23,600 But for negative n, that's a sequence 402 00:25:23,600 --> 00:25:25,530 that grows exponentially. 403 00:25:25,530 --> 00:25:28,840 So in fact, if you examine convergence of this, 404 00:25:28,840 --> 00:25:31,910 this sequence is not absolutely summable. 405 00:25:31,910 --> 00:25:33,755 The Fourier transform does not exist. 406 00:25:36,370 --> 00:25:40,300 What's the region of convergence of the Z-transform? 407 00:25:40,300 --> 00:25:47,160 Well, to decide that, we need to look at the sum of 1/2 times e 408 00:25:47,160 --> 00:25:54,190 to the minus 1 to the n, and ask for what values of z this sum 409 00:25:54,190 --> 00:25:55,490 is less than infinity. 410 00:25:55,490 --> 00:25:58,960 In other words, for what values of z that sum converges. 411 00:25:58,960 --> 00:26:04,210 And to do that, if this was a sum over positive values of n, 412 00:26:04,210 --> 00:26:09,440 then we would want the magnitude of this to be less than 1. 413 00:26:09,440 --> 00:26:12,010 Since we're looking at this for negative values of n, 414 00:26:12,010 --> 00:26:16,400 we want the magnitude of this to be greater than 1. 415 00:26:16,400 --> 00:26:18,860 And the conclusion is-- 416 00:26:18,860 --> 00:26:20,590 and if you don't see this right away, 417 00:26:20,590 --> 00:26:23,650 you can just fiddle with it on the back of an envelope-- 418 00:26:23,650 --> 00:26:30,040 that this converges provided that the magnitude of z 419 00:26:30,040 --> 00:26:35,062 is less than 1/2. 420 00:26:37,930 --> 00:26:41,530 So we can look at this example in the z-plane, 421 00:26:41,530 --> 00:26:43,990 just as we looked at the previous example 422 00:26:43,990 --> 00:26:45,800 in the z-plane. 423 00:26:45,800 --> 00:26:48,970 We first of all have poles and zeroes. 424 00:26:48,970 --> 00:26:51,580 We have-- well, where do we have a zero? 425 00:26:51,580 --> 00:26:55,490 We have a zero at z equals 0. 426 00:26:55,490 --> 00:26:58,120 So that's here. 427 00:26:58,120 --> 00:27:01,619 We have a pole at z equal to 1/2. 428 00:27:01,619 --> 00:27:02,285 And that's here. 429 00:27:05,040 --> 00:27:10,020 Just like in the previous example, the zero and the pole 430 00:27:10,020 --> 00:27:13,030 fall in exactly the same place. 431 00:27:13,030 --> 00:27:14,990 What's the region of convergence? 432 00:27:14,990 --> 00:27:17,650 Well, the region of convergence is for the magnitude 433 00:27:17,650 --> 00:27:20,440 of z less than 1/2. 434 00:27:20,440 --> 00:27:26,110 And now, in this case, that's inside this circle, 435 00:27:26,110 --> 00:27:28,000 rather than in the previous case, 436 00:27:28,000 --> 00:27:29,980 where it was outside that circle. 437 00:27:29,980 --> 00:27:35,320 So the region of convergence now is the region 438 00:27:35,320 --> 00:27:40,280 that is inside the circle, which is bounded by that pole. 439 00:27:44,130 --> 00:27:44,630 All right. 440 00:27:44,630 --> 00:27:48,170 This raises an important point. 441 00:27:48,170 --> 00:27:51,590 Here we had an example where we got a Z-transform ratio 442 00:27:51,590 --> 00:27:53,870 of polynomials. 443 00:27:53,870 --> 00:27:58,820 The previous example, we got exactly the same ratio 444 00:27:58,820 --> 00:28:00,980 of polynomials. 445 00:28:00,980 --> 00:28:02,750 How did these two examples differ 446 00:28:02,750 --> 00:28:04,670 in terms of the Z-transform? 447 00:28:04,670 --> 00:28:08,270 They differed in terms of the region of convergence 448 00:28:08,270 --> 00:28:10,760 that we associate with the Z-transform. 449 00:28:10,760 --> 00:28:12,890 The other case, the region of convergence 450 00:28:12,890 --> 00:28:14,480 was outside this circle. 451 00:28:14,480 --> 00:28:16,250 In this case, the region of convergence 452 00:28:16,250 --> 00:28:18,260 is inside this circle. 453 00:28:18,260 --> 00:28:20,900 The ratio of polynomials for the Z-transforms 454 00:28:20,900 --> 00:28:22,860 is the same in both cases. 455 00:28:22,860 --> 00:28:25,430 It's the region of convergence that's different. 456 00:28:25,430 --> 00:28:28,910 So an important point to keep in mind 457 00:28:28,910 --> 00:28:32,690 is that the Z-transform is not specified just 458 00:28:32,690 --> 00:28:35,975 by the function of z, it's also specified-- 459 00:28:35,975 --> 00:28:40,100 it has to have attached to it a little tag that tells you 460 00:28:40,100 --> 00:28:42,830 for what values of z it's legitimate to look 461 00:28:42,830 --> 00:28:45,050 at that expression. 462 00:28:45,050 --> 00:28:48,260 So it's specified by the ratio of polynomials, 463 00:28:48,260 --> 00:28:53,480 and also by the region of convergence of the Z-transform. 464 00:28:53,480 --> 00:28:56,810 So here, for example, that's the ratio of polynomials, 465 00:28:56,810 --> 00:28:59,060 and that's the Z-transform. 466 00:28:59,060 --> 00:29:00,860 That's the ratio of polynomials, and that's 467 00:29:00,860 --> 00:29:02,180 the region of convergence. 468 00:29:02,180 --> 00:29:05,870 And the two together are the Z-transform. 469 00:29:05,870 --> 00:29:10,850 Now, it often happens that the region of convergence 470 00:29:10,850 --> 00:29:16,460 of the Z-transform is specified in a somewhat indirect way, 471 00:29:16,460 --> 00:29:21,050 by saying something about the sequence that basically implies 472 00:29:21,050 --> 00:29:23,720 what the region of convergence has to be. 473 00:29:23,720 --> 00:29:29,180 And we can lead into that by stating 474 00:29:29,180 --> 00:29:32,390 what some of the properties are that the region of convergence 475 00:29:32,390 --> 00:29:33,500 has. 476 00:29:33,500 --> 00:29:36,800 And I'll always be talking now about sequences 477 00:29:36,800 --> 00:29:39,530 that are sums of exponentials, or Z-transforms 478 00:29:39,530 --> 00:29:42,560 that are ratios of polynomials, and consequently that 479 00:29:42,560 --> 00:29:46,730 can be described by poles and zeroes in the z-plane. 480 00:29:46,730 --> 00:29:50,840 OK, well, I have here the first thing you can say. 481 00:29:50,840 --> 00:29:52,550 Actually, there's a 0 thing you can 482 00:29:52,550 --> 00:29:58,890 say that follows fairly straightforwardly, and that is, 483 00:29:58,890 --> 00:30:02,080 obviously, there can't be any poles 484 00:30:02,080 --> 00:30:05,110 in the region of convergence because, at a pole 485 00:30:05,110 --> 00:30:09,310 of the Z-transform, that's where the Z-transform blows up. 486 00:30:09,310 --> 00:30:11,140 And we know that the region of convergence 487 00:30:11,140 --> 00:30:13,990 is where the Z-transform converges. 488 00:30:13,990 --> 00:30:17,810 So obviously, there are no poles in the region of convergence. 489 00:30:17,810 --> 00:30:19,610 However, we can make another statement, 490 00:30:19,610 --> 00:30:26,920 which is justified in some detail in the text, 491 00:30:26,920 --> 00:30:29,530 that the region of convergence will always 492 00:30:29,530 --> 00:30:36,740 be bounded by either poles or zero or infinity. 493 00:30:36,740 --> 00:30:40,300 For example, in this example that we worked out here, 494 00:30:40,300 --> 00:30:43,330 the region of convergence goes to zero-- 495 00:30:43,330 --> 00:30:45,160 I mean, it includes zero-- 496 00:30:45,160 --> 00:30:48,610 and it's bounded at the outside, in this particular case, 497 00:30:48,610 --> 00:30:51,150 by a pole. 498 00:30:51,150 --> 00:30:53,880 In the example that we had previously, 499 00:30:53,880 --> 00:30:58,180 if we look at this example, the region of convergence 500 00:30:58,180 --> 00:31:02,110 is bounded on the inside by a pole, 501 00:31:02,110 --> 00:31:05,170 and it's bounded on the outside by infinity. 502 00:31:05,170 --> 00:31:07,900 And that's a statement that we can generally make. 503 00:31:07,900 --> 00:31:10,390 I won't try to justify that right now, 504 00:31:10,390 --> 00:31:14,410 but the region of convergence is bounded by poles, 505 00:31:14,410 --> 00:31:17,200 and in fact is bounded by a circle that 506 00:31:17,200 --> 00:31:19,960 goes through a pole. 507 00:31:19,960 --> 00:31:20,460 All right. 508 00:31:20,460 --> 00:31:22,590 That's one statement we can make. 509 00:31:22,590 --> 00:31:25,230 Now some other statements that we can make, 510 00:31:25,230 --> 00:31:29,130 which are all developed in the text. 511 00:31:29,130 --> 00:31:31,380 That is, the proof of them is worked out. 512 00:31:31,380 --> 00:31:35,790 First of all, let's consider finite length sequences. 513 00:31:35,790 --> 00:31:39,870 A finite length sequence, sequenced at 0, 514 00:31:39,870 --> 00:31:44,910 except for a finite number of values of n, 515 00:31:44,910 --> 00:31:51,240 has a region of convergence that is the entire z-plane, perhaps 516 00:31:51,240 --> 00:31:53,452 with the exception of the point z 517 00:31:53,452 --> 00:31:56,730 equals 0, or perhaps with the exception of the point z 518 00:31:56,730 --> 00:31:58,260 equals infinity. 519 00:31:58,260 --> 00:32:00,730 And that follows in a pretty straightforward way, 520 00:32:00,730 --> 00:32:05,580 basically because the sum, in examining absolute convergence, 521 00:32:05,580 --> 00:32:07,980 the sum just includes finite limits. 522 00:32:07,980 --> 00:32:10,890 So as long as each of the sequence values is finite, 523 00:32:10,890 --> 00:32:14,450 the sum has to be finite. 524 00:32:14,450 --> 00:32:18,980 A little more involved, we have right-sided sequences. 525 00:32:18,980 --> 00:32:20,990 What I mean by a right-sided sequence 526 00:32:20,990 --> 00:32:26,900 is a sequence which is 0 for n less than some value of n. 527 00:32:26,900 --> 00:32:32,820 And then goes on up to n equals plus infinity. 528 00:32:32,820 --> 00:32:36,200 And what you can say about right-sided sequences, 529 00:32:36,200 --> 00:32:40,550 as it turns out, is that the region of convergence 530 00:32:40,550 --> 00:32:46,910 is outside some value, which I've denoted by r sub x minus, 531 00:32:46,910 --> 00:32:48,480 and goes to infinity. 532 00:32:48,480 --> 00:32:51,590 It may include infinity or it may not include infinity, 533 00:32:51,590 --> 00:32:55,350 but it goes at least up to infinity. 534 00:32:55,350 --> 00:32:57,530 So it begins at some finite value 535 00:32:57,530 --> 00:33:00,890 in the z-plane, bounded by some circle with radius 536 00:33:00,890 --> 00:33:05,170 r sub x minus, and goes up to infinity. 537 00:33:05,170 --> 00:33:08,090 Well, look, if we had a right-sided sequence, 538 00:33:08,090 --> 00:33:12,080 how are you going to find what r sub x minus is? 539 00:33:12,080 --> 00:33:14,540 Just by looking at the pole-zero pattern, 540 00:33:14,540 --> 00:33:18,710 because we know that the region of convergence 541 00:33:18,710 --> 00:33:20,870 is bounded by poles. 542 00:33:20,870 --> 00:33:23,870 We know also that the region of convergence 543 00:33:23,870 --> 00:33:27,680 can't have any poles, so r sub x minus 544 00:33:27,680 --> 00:33:32,570 has to be the circle that goes through the outermost pole 545 00:33:32,570 --> 00:33:34,140 in the z-plane. 546 00:33:34,140 --> 00:33:37,430 So if we have a right-sided sequence, that 547 00:33:37,430 --> 00:33:39,440 implies a region of convergence, it 548 00:33:39,440 --> 00:33:42,050 implies that the region of convergence 549 00:33:42,050 --> 00:33:47,220 is outside the outermost pole in the z-plane. 550 00:33:47,220 --> 00:33:51,540 We can talk also about left-sided sequences. 551 00:33:51,540 --> 00:33:54,180 Left-sided sequences are ones for which 552 00:33:54,180 --> 00:33:57,630 the sequence is zero for n greater than some value n1. 553 00:33:57,630 --> 00:34:05,610 One And here, the region of convergence begins at zero-- 554 00:34:05,610 --> 00:34:10,500 may or may not include zero, depending on some issues about 555 00:34:10,500 --> 00:34:15,690 the left-sided sequence, but at least goes up to zero-- 556 00:34:15,690 --> 00:34:20,600 and then goes out to some value r sub x plus. 557 00:34:20,600 --> 00:34:22,920 How do we find r sub x plus? 558 00:34:22,920 --> 00:34:24,540 We do it the same way. 559 00:34:24,540 --> 00:34:27,090 That is, we know that the region of convergence 560 00:34:27,090 --> 00:34:30,659 is bounded by poles and doesn't contain any poles, 561 00:34:30,659 --> 00:34:35,940 so this must be a circle whose radius is equal-- 562 00:34:35,940 --> 00:34:39,810 who's a circle that goes through the innermost pole, 563 00:34:39,810 --> 00:34:43,090 other than the one at z equals 0 in the z plane. 564 00:34:43,090 --> 00:34:45,139 So if I have a left-sided sequence, 565 00:34:45,139 --> 00:34:46,739 if I know it's a left-sided sequence, 566 00:34:46,739 --> 00:34:49,500 then by looking at the pole-zero pattern 567 00:34:49,500 --> 00:34:53,699 I can immediately infer what the region of convergence is. 568 00:34:53,699 --> 00:34:57,480 Finally, we have two-sided sequences, sequences 569 00:34:57,480 --> 00:35:02,490 that go on to minus infinity and they go on the plus infinity. 570 00:35:02,490 --> 00:35:05,280 In that case, the region of convergence 571 00:35:05,280 --> 00:35:12,090 lies between two circles, which go through two poles. 572 00:35:12,090 --> 00:35:15,540 And obviously, again, they have to be two poles that 573 00:35:15,540 --> 00:35:18,100 are adjacent in the z-plane. 574 00:35:18,100 --> 00:35:21,870 In other words, again, to avoid having poles that 575 00:35:21,870 --> 00:35:26,220 are inside the region of convergence. 576 00:35:26,220 --> 00:35:30,330 And out of all this, incidentally, 577 00:35:30,330 --> 00:35:34,290 if I tell you that I have a sequence whose 578 00:35:34,290 --> 00:35:37,020 Fourier transform exists, and I don't tell you 579 00:35:37,020 --> 00:35:39,670 anything else about the sequence, 580 00:35:39,670 --> 00:35:43,210 can you infer what the region of convergence is? 581 00:35:43,210 --> 00:35:45,190 Well, sure you can, because you know 582 00:35:45,190 --> 00:35:48,380 that the region of convergence includes the unit circle. 583 00:35:48,380 --> 00:35:53,020 And then where it's got to go from there is to the first pole 584 00:35:53,020 --> 00:35:56,060 that you run into heading toward the origin, 585 00:35:56,060 --> 00:35:58,960 and to the first pole that you run into heading out 586 00:35:58,960 --> 00:36:00,230 toward infinity. 587 00:36:00,230 --> 00:36:03,910 So in fact, if I tell you that the sequence is absolutely 588 00:36:03,910 --> 00:36:08,230 summable, or that the Fourier transform exists, that also, 589 00:36:08,230 --> 00:36:10,870 in effect, lets you construct the region 590 00:36:10,870 --> 00:36:13,600 of convergence in the z-plane. 591 00:36:13,600 --> 00:36:18,780 Well, let's just cement this with an example. 592 00:36:18,780 --> 00:36:23,720 Let's say that I have a Z-transform which 593 00:36:23,720 --> 00:36:34,000 has a pole-zero pattern that has a pole at z equal a and a pole 594 00:36:34,000 --> 00:36:36,000 at z equal b. 595 00:36:36,000 --> 00:36:41,760 So this is a, and this is b. 596 00:36:41,760 --> 00:36:45,720 And let's consider what turn out to be the only three 597 00:36:45,720 --> 00:36:49,680 possible choices for the region of convergence, given 598 00:36:49,680 --> 00:36:51,380 this pole-zero pattern. 599 00:36:51,380 --> 00:36:54,420 No zeroes, to make the example easy. 600 00:36:54,420 --> 00:36:58,230 Well first of all, let's suppose that we 601 00:36:58,230 --> 00:37:01,740 picked the region of convergence to be less than a. 602 00:37:01,740 --> 00:37:05,460 So the region of convergence for this first case 603 00:37:05,460 --> 00:37:10,784 is inside a circle bounded by that pole. 604 00:37:10,784 --> 00:37:11,450 Well, let's see. 605 00:37:11,450 --> 00:37:15,190 Let's answer the easy question first. 606 00:37:15,190 --> 00:37:18,030 Does the Fourier transform exist? 607 00:37:18,030 --> 00:37:20,740 To answer that, we ask, is the unit circle 608 00:37:20,740 --> 00:37:22,480 inside the region of convergence? 609 00:37:22,480 --> 00:37:24,340 Well, it's not if the region of convergence 610 00:37:24,340 --> 00:37:26,050 is inside that pole. 611 00:37:26,050 --> 00:37:32,410 So for this case, the Fourier transform doesn't exist. 612 00:37:32,410 --> 00:37:35,010 That is, it doesn't converge. 613 00:37:35,010 --> 00:37:36,330 How about which sided? 614 00:37:36,330 --> 00:37:39,310 Left-sided, right-sided, two-sided? 615 00:37:39,310 --> 00:37:45,160 Well, we can answer that, again, because we 616 00:37:45,160 --> 00:37:47,860 know, from what we said previously, 617 00:37:47,860 --> 00:37:55,460 that a left-sided sequence has a region of convergence that 618 00:37:55,460 --> 00:37:59,960 starts at zero and goes out to some value r sub x plus. 619 00:37:59,960 --> 00:38:01,640 Well, that's the region of convergence 620 00:38:01,640 --> 00:38:03,630 that we're talking about here. 621 00:38:03,630 --> 00:38:07,130 So this particular example is a sequence 622 00:38:07,130 --> 00:38:11,020 for a sequence that's left-sided. 623 00:38:11,020 --> 00:38:15,360 How about the region of convergence between a and b? 624 00:38:15,360 --> 00:38:19,080 Think of two circles with radii a and b. 625 00:38:19,080 --> 00:38:21,810 And now the region of convergence is between them. 626 00:38:21,810 --> 00:38:23,630 Does that include the unit circle? 627 00:38:23,630 --> 00:38:25,720 Yeah, it includes the unit circle. 628 00:38:25,720 --> 00:38:31,500 So for that case, the Fourier transform exists. 629 00:38:31,500 --> 00:38:33,780 Is it left-sided, right-sided, or two-sided? 630 00:38:33,780 --> 00:38:36,120 Well, it's two-sided, because the region of convergence 631 00:38:36,120 --> 00:38:39,790 doesn't make it to zero, and it doesn't make it to infinity. 632 00:38:39,790 --> 00:38:44,500 So this is a two-sided sequence. 633 00:38:44,500 --> 00:38:46,660 The final example, or the final choice 634 00:38:46,660 --> 00:38:48,740 for the region of convergence, the magnitude of z 635 00:38:48,740 --> 00:38:51,150 greater than b. 636 00:38:51,150 --> 00:38:54,540 Does the Fourier transform exist? 637 00:38:54,540 --> 00:38:57,420 No, because the region of convergence 638 00:38:57,420 --> 00:38:59,430 is outside this pole. 639 00:38:59,430 --> 00:39:01,660 It doesn't include the unit circle, 640 00:39:01,660 --> 00:39:06,130 so the Fourier transform doesn't exist. 641 00:39:06,130 --> 00:39:10,630 Well, we can figure out which sided 642 00:39:10,630 --> 00:39:15,240 it is, either by using what we know, or using quizmanship. 643 00:39:15,240 --> 00:39:17,070 Here, we had left, here we had two-sided, 644 00:39:17,070 --> 00:39:18,900 so obviously it's right-sided. 645 00:39:18,900 --> 00:39:21,120 In fact, it is right-sided, because the region 646 00:39:21,120 --> 00:39:25,860 of convergence starts at some value r sub x minus, 647 00:39:25,860 --> 00:39:27,520 and goes out to infinity. 648 00:39:27,520 --> 00:39:33,390 So this case, then, is a right-sided sequence. 649 00:39:33,390 --> 00:39:36,540 This stresses, again, the point that, 650 00:39:36,540 --> 00:39:40,710 if I just specify the pole-zero pattern, that 651 00:39:40,710 --> 00:39:44,832 doesn't specify the Z-transform or the sequence uniquely. 652 00:39:47,730 --> 00:39:51,990 Now, one of the properties of the Z-transform, 653 00:39:51,990 --> 00:39:55,020 which follows essentially directly from the fact that 654 00:39:55,020 --> 00:39:58,710 the Z-transform is interpretable in terms of the Fourier 655 00:39:58,710 --> 00:40:02,700 transform-- or this could be developed more formally, 656 00:40:02,700 --> 00:40:05,040 and we won't do that here-- 657 00:40:05,040 --> 00:40:06,900 is the convolution property. 658 00:40:06,900 --> 00:40:09,600 Namely, if y of n is the convolution 659 00:40:09,600 --> 00:40:16,700 of x of n with h of n, then the Z-transform of y of n 660 00:40:16,700 --> 00:40:21,770 is equal to the product of the Z-transform of x of n 661 00:40:21,770 --> 00:40:24,650 and the Z-transform of h of n. 662 00:40:24,650 --> 00:40:29,750 That is, z transformation, or the Z-transform, maps 663 00:40:29,750 --> 00:40:34,660 convolution into multiplication, just like the Fourier transform 664 00:40:34,660 --> 00:40:35,160 did. 665 00:40:35,160 --> 00:40:37,930 And that's not surprising. 666 00:40:37,930 --> 00:40:44,500 We typically refer to h of z, the Z-transform 667 00:40:44,500 --> 00:40:47,320 of the unit sample response of the system, 668 00:40:47,320 --> 00:40:51,260 as the system function. 669 00:40:51,260 --> 00:40:53,380 Well, on the basis of the discussion that we just 670 00:40:53,380 --> 00:40:55,870 finished, there are some things that we 671 00:40:55,870 --> 00:40:59,350 can say about the system function and the unit sample 672 00:40:59,350 --> 00:41:03,830 response and regions of convergence, et cetera. 673 00:41:03,830 --> 00:41:10,380 First of all, suppose that we're talking about a stable system. 674 00:41:10,380 --> 00:41:13,710 Stable system, the unit sample response 675 00:41:13,710 --> 00:41:15,660 is absolutely summable. 676 00:41:15,660 --> 00:41:18,450 What does that imply about the Z-transform? 677 00:41:18,450 --> 00:41:24,410 Well, it implies that the Z-transform, 678 00:41:24,410 --> 00:41:26,690 the region of convergence of the Z-transform 679 00:41:26,690 --> 00:41:28,490 includes the unit circle. 680 00:41:28,490 --> 00:41:30,650 That is, it implies the Fourier transform exists, 681 00:41:30,650 --> 00:41:33,980 or, equivalently, that the Z-transform region 682 00:41:33,980 --> 00:41:36,660 of convergence includes the unit circle. 683 00:41:36,660 --> 00:41:40,610 So this is a statement that we can make. 684 00:41:40,610 --> 00:41:44,360 Suppose that we were talking about a stable system. 685 00:41:44,360 --> 00:41:47,510 Sorry, suppose that we were talking about a causal system. 686 00:41:47,510 --> 00:41:51,640 If the system is causal, well, what does causal mean? 687 00:41:51,640 --> 00:41:55,470 It means that the unit sample response 688 00:41:55,470 --> 00:41:57,720 is zero for n less than zero. 689 00:41:57,720 --> 00:42:02,400 In particular, the unit sample response is right-sided. 690 00:42:02,400 --> 00:42:06,480 If it's right-sided, then the region of convergence 691 00:42:06,480 --> 00:42:09,990 must be outside the outermost pole, 692 00:42:09,990 --> 00:42:13,200 because that's what we said about what happens 693 00:42:13,200 --> 00:42:14,490 with a right-sided sequence. 694 00:42:14,490 --> 00:42:19,080 The region of convergence has to be outside the outermost pole. 695 00:42:19,080 --> 00:42:22,590 So you can get the notion, then, that when 696 00:42:22,590 --> 00:42:25,960 we talk about the Z-transform of a system, in essence 697 00:42:25,960 --> 00:42:28,920 we can imply the region of convergence 698 00:42:28,920 --> 00:42:33,450 by making some statements about stability or causality. 699 00:42:33,450 --> 00:42:33,950 All right. 700 00:42:33,950 --> 00:42:37,380 Let's try this out on an example. 701 00:42:37,380 --> 00:42:40,230 Let's take an example which, in fact, 702 00:42:40,230 --> 00:42:42,960 we've talked about in some lectures-- in a lecture 703 00:42:42,960 --> 00:42:45,040 previously. 704 00:42:45,040 --> 00:42:48,150 Let's take the example of a system that 705 00:42:48,150 --> 00:42:53,130 is represented by a linear constant coefficient difference 706 00:42:53,130 --> 00:42:59,600 equation, y of n minus 1/2, y of n minus 1 equals x of n. 707 00:42:59,600 --> 00:43:05,156 To get the system function for this, we'll take a shortcut. 708 00:43:05,156 --> 00:43:08,370 And in particular, we'll use a property 709 00:43:08,370 --> 00:43:13,350 that we haven't derived, at least not yet, 710 00:43:13,350 --> 00:43:17,500 but in fact you'll be deriving in the study guide. 711 00:43:17,500 --> 00:43:20,616 Notice how I put everything off to the study guide. 712 00:43:20,616 --> 00:43:21,990 When we get the study guide, I'll 713 00:43:21,990 --> 00:43:24,840 tell you we did it in lecture. 714 00:43:24,840 --> 00:43:30,240 Well, the property is that if the Z-transform of y of n 715 00:43:30,240 --> 00:43:35,850 is y of z, then the Z-transform of y of n plus n0 716 00:43:35,850 --> 00:43:40,090 is z to the n0 times y and z. 717 00:43:40,090 --> 00:43:44,470 So if we take the Z-transform of this difference equation, we 718 00:43:44,470 --> 00:43:50,470 have, then, y of z, the Z-transform of that minus 1/2 719 00:43:50,470 --> 00:43:53,820 z to the minus 1, since we have y of n minus 1, z 720 00:43:53,820 --> 00:43:58,210 to the minus 1 y of z is equal to the Z-transform 721 00:43:58,210 --> 00:44:02,230 of the right side of the equation, which is x of z. 722 00:44:02,230 --> 00:44:06,970 Well, the system function is equal to y of z 723 00:44:06,970 --> 00:44:09,670 divided by x of z. 724 00:44:09,670 --> 00:44:14,370 And so we can simply solve this equation for y of z 725 00:44:14,370 --> 00:44:15,900 over x of z. 726 00:44:15,900 --> 00:44:20,160 And what we get is 1 over 1 minus 1/2 times z 727 00:44:20,160 --> 00:44:22,510 to the minus 1. 728 00:44:22,510 --> 00:44:25,668 So that's the system function for that system. 729 00:44:28,950 --> 00:44:33,050 Well, what's the region of convergence? 730 00:44:33,050 --> 00:44:36,010 We don't know what the region of convergence is. 731 00:44:36,010 --> 00:44:37,930 In fact, we're going to have to-- 732 00:44:37,930 --> 00:44:40,030 in fact, there might be several choices 733 00:44:40,030 --> 00:44:42,400 for the region of convergence. 734 00:44:42,400 --> 00:44:45,160 But that's not too surprising, because remember 735 00:44:45,160 --> 00:44:46,780 when we talked about this equation 736 00:44:46,780 --> 00:44:52,090 a couple of lectures ago and we showed that, in fact, there 737 00:44:52,090 --> 00:44:54,370 is more than one choice for the unit sample 738 00:44:54,370 --> 00:44:57,280 response of this system. 739 00:44:57,280 --> 00:45:03,090 And that is related to the fact that all we got out of here 740 00:45:03,090 --> 00:45:08,550 so far was the ratio of polynomials that h of z is, 741 00:45:08,550 --> 00:45:12,135 and not what the region of convergence is. 742 00:45:12,135 --> 00:45:14,690 All right, so this ratio of polynomials 743 00:45:14,690 --> 00:45:17,960 is 1 over 1 minus 1/2 times z to the minus 1. 744 00:45:17,960 --> 00:45:20,390 We've been staring at that ratio of polynomials 745 00:45:20,390 --> 00:45:23,030 through a couple of previous examples. 746 00:45:23,030 --> 00:45:28,640 It has a zero at z equals 0, and a pole at z equals 1/2. 747 00:45:28,640 --> 00:45:32,240 So let's take a look at that in the z-plane. 748 00:45:37,650 --> 00:45:44,730 We have then the z-plane here, the unit circle. 749 00:45:44,730 --> 00:45:50,460 We have, for the example we're talking about, a zero there, 750 00:45:50,460 --> 00:45:53,640 and a pole there. 751 00:45:53,640 --> 00:45:55,930 And what's the region of convergence? 752 00:45:55,930 --> 00:45:58,980 Well, it seems like we can take the region of convergence 753 00:45:58,980 --> 00:46:02,670 to be whatever we want to consistent with what regions 754 00:46:02,670 --> 00:46:05,460 of convergences tend to do. 755 00:46:05,460 --> 00:46:09,660 So let's say, for example, that the region of convergence 756 00:46:09,660 --> 00:46:11,610 was outside that pole. 757 00:46:11,610 --> 00:46:17,020 So let's take a region of convergence first that 758 00:46:17,020 --> 00:46:20,530 looks like that. 759 00:46:20,530 --> 00:46:24,520 Well, for that case, if that's the region of convergence, 760 00:46:24,520 --> 00:46:28,250 there are some things that we could say about the system. 761 00:46:28,250 --> 00:46:31,120 For example, is the system stable? 762 00:46:31,120 --> 00:46:35,350 Is the system stable if the unit sample response 763 00:46:35,350 --> 00:46:36,790 has a Fourier transform? 764 00:46:36,790 --> 00:46:40,660 And it has that if the region of convergence 765 00:46:40,660 --> 00:46:44,280 includes the unit circle, which it does here. 766 00:46:44,280 --> 00:46:50,300 So this particular system is stable. 767 00:46:50,300 --> 00:46:53,100 And is it causal? 768 00:46:53,100 --> 00:46:56,580 Well, what we want to examine for that region of convergence 769 00:46:56,580 --> 00:47:00,760 is does it imply a right-sided sequence. 770 00:47:00,760 --> 00:47:02,760 Well, it does, because the region of convergence 771 00:47:02,760 --> 00:47:04,530 is outside this pole. 772 00:47:04,530 --> 00:47:11,820 So in fact, for this example, the system is also causal. 773 00:47:11,820 --> 00:47:14,490 And in fact, look, we know what the unit sample response 774 00:47:14,490 --> 00:47:18,060 is because we've been working with that example previously. 775 00:47:18,060 --> 00:47:22,700 The unit sample response here, in fact, is h of n 776 00:47:22,700 --> 00:47:26,970 is 1/2 to the n times a unit step. 777 00:47:29,890 --> 00:47:33,560 But that's not the only choice for the region of convergence. 778 00:47:33,560 --> 00:47:37,330 We could, alternatively, pick a region of convergence 779 00:47:37,330 --> 00:47:39,070 which is inside that pole. 780 00:47:39,070 --> 00:47:42,690 That is-- let's look at this again. 781 00:47:42,690 --> 00:47:47,830 Zero at the origin, a pole at equals z equals 1/2. 782 00:47:47,830 --> 00:47:50,980 But now let's take the region of convergence 783 00:47:50,980 --> 00:47:54,740 to be inside that pole. 784 00:47:54,740 --> 00:47:58,340 If it's inside that pole, then, first of all, 785 00:47:58,340 --> 00:47:59,840 is the system stable? 786 00:47:59,840 --> 00:48:02,420 No, it's not stable, because the region of convergence 787 00:48:02,420 --> 00:48:04,440 didn't include the unit circle. 788 00:48:04,440 --> 00:48:05,540 So it's not stable. 789 00:48:08,550 --> 00:48:10,439 And furthermore, it's not causal. 790 00:48:10,439 --> 00:48:11,730 In fact, it's just the reverse. 791 00:48:11,730 --> 00:48:13,950 It's anti-causal. 792 00:48:13,950 --> 00:48:17,470 It's not causal. 793 00:48:17,470 --> 00:48:20,700 And in fact, we know, again, what the unit sample 794 00:48:20,700 --> 00:48:23,400 response is for that case because we worked it out 795 00:48:23,400 --> 00:48:31,800 previously, namely h of n is equal to minus 1/2 to the n, 796 00:48:31,800 --> 00:48:34,710 but for n negative, not for n positive. 797 00:48:34,710 --> 00:48:41,820 That is, multiplied by u of minus n minus 1. 798 00:48:41,820 --> 00:48:43,950 And this, by the way, if you refer back 799 00:48:43,950 --> 00:48:46,830 several lectures ago, is consistent with the two 800 00:48:46,830 --> 00:48:50,310 solutions that we got for a linear constant coefficient 801 00:48:50,310 --> 00:48:52,140 difference equation. 802 00:48:52,140 --> 00:48:55,000 And now you can see that, for example, 803 00:48:55,000 --> 00:48:58,860 if I specified the difference equation, 804 00:48:58,860 --> 00:49:01,710 and I said that, in addition, what we're talking about 805 00:49:01,710 --> 00:49:04,500 is, say, a stable system, then you 806 00:49:04,500 --> 00:49:07,260 could construct the region of convergence by saying, 807 00:49:07,260 --> 00:49:08,760 all right, the unit circle has to be 808 00:49:08,760 --> 00:49:10,980 in the region of convergence, and now I'll 809 00:49:10,980 --> 00:49:14,700 take the region of convergence to go in toward a pole 810 00:49:14,700 --> 00:49:16,170 and out toward a pole. 811 00:49:16,170 --> 00:49:18,390 Or if I said, the system was causal, 812 00:49:18,390 --> 00:49:20,830 then you could construct the region of convergence. 813 00:49:20,830 --> 00:49:24,270 So sometimes, in stating a linear constant coefficient 814 00:49:24,270 --> 00:49:27,390 difference equation, we'll basically [AUDIO OUT] 815 00:49:27,390 --> 00:49:31,620 convergence by making an additional statement 816 00:49:31,620 --> 00:49:32,860 about the system. 817 00:49:32,860 --> 00:49:34,710 For [AUDIO OUT] system is causal, 818 00:49:34,710 --> 00:49:39,330 or the system is stable, or both if it can be both. 819 00:49:39,330 --> 00:49:44,130 [AUDIO OUT] So this is the Z-transform. 820 00:49:44,130 --> 00:49:47,910 Just to summarize the key points first, 821 00:49:47,910 --> 00:49:51,540 the Z-transform was introduced to get around 822 00:49:51,540 --> 00:49:55,360 the convergence problem with the Fourier transform. 823 00:49:55,360 --> 00:49:59,760 The Z-transform of sequences that are exponentials 824 00:49:59,760 --> 00:50:01,530 are ratios of polynomials. 825 00:50:01,530 --> 00:50:05,460 That is, they can be described in terms of poles and zeros. 826 00:50:05,460 --> 00:50:09,150 But the Z-transform is not specified just 827 00:50:09,150 --> 00:50:11,220 by the ratio of polynomials. 828 00:50:11,220 --> 00:50:14,000 You also need to know what the region of convergence 829 00:50:14,000 --> 00:50:18,630 is of the Z-transform so that, in fact, 830 00:50:18,630 --> 00:50:25,370 for several different sequences, you 831 00:50:25,370 --> 00:50:27,701 can end up with the same ratio of polynomials, 832 00:50:27,701 --> 00:50:29,450 but of course you'll get different regions 833 00:50:29,450 --> 00:50:30,650 of convergence. 834 00:50:30,650 --> 00:50:32,390 Finally, the region of convergence 835 00:50:32,390 --> 00:50:33,950 includes the unit circle. 836 00:50:33,950 --> 00:50:38,240 That tells us that the Fourier transform converges. 837 00:50:38,240 --> 00:50:40,820 And also, we're able to make statements 838 00:50:40,820 --> 00:50:43,340 about a region of convergence as it 839 00:50:43,340 --> 00:50:45,830 relates to right-sided, left-sided, and two-sided 840 00:50:45,830 --> 00:50:47,540 sequences. 841 00:50:47,540 --> 00:50:51,950 Next time, we'll continue the discussion of the Z-transform, 842 00:50:51,950 --> 00:50:55,750 and in particular develop the inverse Z-transform. 843 00:50:55,750 --> 00:50:56,980 Thank you. 844 00:50:56,980 --> 00:50:59,982 [MUSIC PLAYING]