1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation, or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,565 ocw.mit.edu. 8 00:00:20,565 --> 00:00:56,890 [MUSIC PLAYING] 9 00:00:56,890 --> 00:00:59,510 PROFESSOR: In the last several lectures, we've talked about a 10 00:00:59,510 --> 00:01:02,930 generalization of the continuous-time Fourier 11 00:01:02,930 --> 00:01:07,490 transform and a very similar strategy also applies to 12 00:01:07,490 --> 00:01:10,610 discrete-time, and that's what we want to begin to deal with 13 00:01:10,610 --> 00:01:12,750 in today's lecture. 14 00:01:12,750 --> 00:01:17,400 So what we want to talk about is generalizing the Fourier 15 00:01:17,400 --> 00:01:21,650 transform, and what this will lead to in discrete-time is a 16 00:01:21,650 --> 00:01:24,380 notion referred to as the z-transform. 17 00:01:24,380 --> 00:01:27,810 Now, just as in continuous-time, in 18 00:01:27,810 --> 00:01:32,670 discrete-time the Fourier transform corresponded to a 19 00:01:32,670 --> 00:01:38,830 representation of a sequence as a linear combination of 20 00:01:38,830 --> 00:01:41,750 complex exponentials. 21 00:01:41,750 --> 00:01:45,430 So this was the synthesis equation. 22 00:01:45,430 --> 00:01:46,810 And, of course, there is the 23 00:01:46,810 --> 00:01:50,100 corresponding analysis equation. 24 00:01:50,100 --> 00:01:55,310 And as you recall, and as is the same for continuous-time, 25 00:01:55,310 --> 00:01:59,300 the reason that we picked complex exponentials was 26 00:01:59,300 --> 00:02:02,900 because of the fact that they are eigenfunctions of linear 27 00:02:02,900 --> 00:02:04,730 time-invariant systems. 28 00:02:04,730 --> 00:02:09,229 In other words, if you have a complex exponential into a 29 00:02:09,229 --> 00:02:12,560 linear time-invariant system, the output is a complex 30 00:02:12,560 --> 00:02:13,820 exponential. 31 00:02:13,820 --> 00:02:19,620 And the change in complex amplitude, which corresponds 32 00:02:19,620 --> 00:02:23,150 to the frequency response, in fact is what led to the 33 00:02:23,150 --> 00:02:25,640 definition of the Fourier transform. 34 00:02:25,640 --> 00:02:29,070 In particular, it is the Fourier transform of the 35 00:02:29,070 --> 00:02:31,610 impulse response. 36 00:02:31,610 --> 00:02:36,450 Well, that set of notions is, more or less, identical to the 37 00:02:36,450 --> 00:02:40,410 way we motivated the Laplace transform in the 38 00:02:40,410 --> 00:02:43,030 continuous-time case, in the Fourier transform in the 39 00:02:43,030 --> 00:02:44,680 continuous-time case. 40 00:02:44,680 --> 00:02:49,710 And just as in continuous-time, there are a 41 00:02:49,710 --> 00:02:55,050 set of signals more general than the complex exponentials, 42 00:02:55,050 --> 00:02:57,760 which are also eigenfunctions of linear 43 00:02:57,760 --> 00:02:59,420 time-invariant systems. 44 00:02:59,420 --> 00:03:03,810 In particular, in discrete-time, if we had 45 00:03:03,810 --> 00:03:08,790 instead of an exponential e to the j omega n, we had a more 46 00:03:08,790 --> 00:03:16,220 general complex number z, that the signal z to the n is also 47 00:03:16,220 --> 00:03:21,090 an eigenfunction of a linear time-invariant system for any 48 00:03:21,090 --> 00:03:22,630 particular z. 49 00:03:22,630 --> 00:03:26,770 So we can see that by substituting that into the 50 00:03:26,770 --> 00:03:30,580 convolution sum and recognizing, again, very 51 00:03:30,580 --> 00:03:34,820 strongly paralleling the continuous-time argument, that 52 00:03:34,820 --> 00:03:42,670 we can rewrite this factor as z to the n z to the minus k. 53 00:03:42,670 --> 00:03:48,960 And because of the fact that it's a sum on k and this term 54 00:03:48,960 --> 00:03:52,460 doesn't depend on k, we can take that term out. 55 00:03:52,460 --> 00:03:57,010 And the conclusion is that if we have z to the n as an 56 00:03:57,010 --> 00:04:04,640 input, that the output is of the same form times a factor 57 00:04:04,640 --> 00:04:06,980 which depends on z. 58 00:04:06,980 --> 00:04:10,630 But of course, doesn't depend on k because that is summed 59 00:04:10,630 --> 00:04:13,260 out as we form the summation. 60 00:04:13,260 --> 00:04:18,149 So, in fact, this summation corresponds to a complex 61 00:04:18,149 --> 00:04:26,490 number, which we'll denote as H of z, where z will represent 62 00:04:26,490 --> 00:04:28,750 a more general complex number. 63 00:04:28,750 --> 00:04:34,870 Namely, z is r e to the j omega, r being the magnitude 64 00:04:34,870 --> 00:04:38,020 of this complex number and omega, of 65 00:04:38,020 --> 00:04:40,530 course, being the angle. 66 00:04:40,530 --> 00:04:46,950 So for a linear time-invariant system, a more general complex 67 00:04:46,950 --> 00:04:51,900 exponential sequence of this form generates as an output a 68 00:04:51,900 --> 00:04:55,990 complex exponential sequence of the same form with a change 69 00:04:55,990 --> 00:05:00,480 in amplitude which we're representing as H of z, 70 00:05:00,480 --> 00:05:04,010 recognizing the fact that it's going to be a function of what 71 00:05:04,010 --> 00:05:06,080 that complex number is. 72 00:05:06,080 --> 00:05:11,150 And this amplitude factor is given by this summation. 73 00:05:11,150 --> 00:05:18,130 And it is this summation which is defined as the z-transform 74 00:05:18,130 --> 00:05:22,100 of the sequence H of n. 75 00:05:22,100 --> 00:05:24,240 Now, let me stress that-- 76 00:05:24,240 --> 00:05:27,650 and I'll continue to stress this as the lecture goes on. 77 00:05:27,650 --> 00:05:33,260 That much of what we've said is directly parallel to what 78 00:05:33,260 --> 00:05:35,790 we said in the continuous-time case. 79 00:05:35,790 --> 00:05:41,310 And what we've simply done is to expand from complex 80 00:05:41,310 --> 00:05:46,060 exponentials with a purely imaginary exponent, complex 81 00:05:46,060 --> 00:05:50,950 exponential time functions or sequences of that form, to 82 00:05:50,950 --> 00:05:56,990 ones that have more general complex exponential factors. 83 00:05:56,990 --> 00:06:03,830 Now, we have a mapping here from the impulse response to 84 00:06:03,830 --> 00:06:08,660 the amplitude or the eigenvalue associated with 85 00:06:08,660 --> 00:06:14,320 that input, and this mapping is what is referred to as the 86 00:06:14,320 --> 00:06:16,130 z-transform. 87 00:06:16,130 --> 00:06:21,120 So H of z is, in fact, the z-transform 88 00:06:21,120 --> 00:06:23,060 of the impulse response. 89 00:06:23,060 --> 00:06:27,010 And if we consider applying this mapping as we did in 90 00:06:27,010 --> 00:06:30,420 continuous-time in a similar argument, applying this 91 00:06:30,420 --> 00:06:35,490 mapping to a sequence whether or not it corresponds to the 92 00:06:35,490 --> 00:06:39,180 impulse response or a linear time-invariant system, that 93 00:06:39,180 --> 00:06:46,270 leads then to the z-transform of a general sequence x of n. 94 00:06:46,270 --> 00:06:51,410 The z-transform being defined by this relationship. 95 00:06:51,410 --> 00:06:56,460 And again, notationally, we'll often represent a time 96 00:06:56,460 --> 00:07:00,740 function and a z-transform through a shorthand notation, 97 00:07:00,740 --> 00:07:06,690 just indicating that x of z is the z-transform of x of n. 98 00:07:09,260 --> 00:07:12,750 So we've kind of motivated the development in a manner 99 00:07:12,750 --> 00:07:14,760 exactly identical to what we had done 100 00:07:14,760 --> 00:07:16,570 with the Laplace transform. 101 00:07:16,570 --> 00:07:19,010 Kind of the idea that if you look at the eigenvalue 102 00:07:19,010 --> 00:07:23,620 associated with a linear time-invariant system, that 103 00:07:23,620 --> 00:07:29,220 essentially generates a mapping between the sequence-- 104 00:07:29,220 --> 00:07:33,750 system impulse response and a function of z. 105 00:07:33,750 --> 00:07:36,690 And that corresponds to the z-transform here, it 106 00:07:36,690 --> 00:07:39,020 corresponded to the Laplace transform in the 107 00:07:39,020 --> 00:07:41,760 continuous-time case. 108 00:07:41,760 --> 00:07:46,850 That same argument is also the kind of argument that we use 109 00:07:46,850 --> 00:07:51,200 to lead us into the Fourier transform originally. 110 00:07:51,200 --> 00:07:57,400 And once again, what you would expect is that the z-transform 111 00:07:57,400 --> 00:08:00,170 has a very close and important relationship 112 00:08:00,170 --> 00:08:01,810 to the Fourier transform. 113 00:08:01,810 --> 00:08:06,730 And indeed, that relationship turns out to be, more or less, 114 00:08:06,730 --> 00:08:11,630 identical to the relationship between the Laplace transform 115 00:08:11,630 --> 00:08:15,400 and the Fourier transform in continuous-time. 116 00:08:15,400 --> 00:08:18,650 Well, let's look at the relationship. 117 00:08:18,650 --> 00:08:23,240 First of all, what we recognize is that if we 118 00:08:23,240 --> 00:08:30,960 compare the Fourier transform expression for a sequence and 119 00:08:30,960 --> 00:08:36,299 the z-transform expression for the same sequence that they 120 00:08:36,299 --> 00:08:39,750 involve, essentially, the same operations. 121 00:08:39,750 --> 00:08:47,230 And in fact, since z is of the form r e to the j omega, if we 122 00:08:47,230 --> 00:08:52,300 want this sum to look like this sum, then that would mean 123 00:08:52,300 --> 00:08:56,900 that we would choose z equal to e to the j omega. 124 00:08:56,900 --> 00:09:01,740 Said another way, the z-transform, when z is e to 125 00:09:01,740 --> 00:09:04,990 the j omega, is going to reduce 126 00:09:04,990 --> 00:09:07,900 to the Fourier transform. 127 00:09:07,900 --> 00:09:12,290 So we have a relationship like the one, again, that we had 128 00:09:12,290 --> 00:09:15,140 between the Laplace transform and the Fourier transform in 129 00:09:15,140 --> 00:09:16,710 continuous-time. 130 00:09:16,710 --> 00:09:22,020 Namely that for a certain set of values of the complex 131 00:09:22,020 --> 00:09:26,220 variable, the transform, the z-transform, reduces to the 132 00:09:26,220 --> 00:09:28,470 Fourier transform. 133 00:09:28,470 --> 00:09:32,970 So if we have x of z, the z-transform, and we look at 134 00:09:32,970 --> 00:09:40,100 that for z equal to e to the j omega, and z equal to e to the 135 00:09:40,100 --> 00:09:43,850 j omega is similar to saying that we're looking at that for 136 00:09:43,850 --> 00:09:46,840 the magnitude of z equal to 1. 137 00:09:46,840 --> 00:09:49,740 We're specifically choosing r equal to 1, which is the 138 00:09:49,740 --> 00:09:52,050 magnitude of z. 139 00:09:52,050 --> 00:09:54,550 Then this is equal to the Fourier 140 00:09:54,550 --> 00:09:57,290 transform of the sequence. 141 00:09:57,290 --> 00:10:02,580 So the z-transform for z equal to e to the j omega is the 142 00:10:02,580 --> 00:10:07,020 Fourier transform, and so this then 143 00:10:07,020 --> 00:10:11,290 corresponds to x of omega. 144 00:10:11,290 --> 00:10:12,540 Namely, the Fourier transform. 145 00:10:15,000 --> 00:10:19,510 Well, we now have ourselves in a similar situation, again, to 146 00:10:19,510 --> 00:10:22,840 what we had when we talked about the Laplace transform. 147 00:10:22,840 --> 00:10:28,050 Namely, a notational awkwardness, or inconvenience, 148 00:10:28,050 --> 00:10:30,510 which we can resolve by simply 149 00:10:30,510 --> 00:10:33,840 redefining some of our notation. 150 00:10:33,840 --> 00:10:36,960 In particular, the awkwardness relates to the fact that 151 00:10:36,960 --> 00:10:40,990 whereas we've been writing our Fourier transforms this way, 152 00:10:40,990 --> 00:10:46,620 as x of omega, if we were to express x of z and look at it, 153 00:10:46,620 --> 00:10:48,510 it's equal to e to the j omega. 154 00:10:48,510 --> 00:10:52,320 We end up with the independent variable being e to the j 155 00:10:52,320 --> 00:10:56,240 omega rather than omega. 156 00:10:56,240 --> 00:10:58,920 Well, in fact, the Fourier transform is 157 00:10:58,920 --> 00:11:00,020 a function of omega. 158 00:11:00,020 --> 00:11:02,620 It's also a function of e to the j omega. 159 00:11:02,620 --> 00:11:06,620 And now what we can see is that given the fact that we 160 00:11:06,620 --> 00:11:12,310 want to generalize the Fourier transform to the z-transform, 161 00:11:12,310 --> 00:11:17,870 it's convenient now to use as notation for the Fourier 162 00:11:17,870 --> 00:11:22,120 transform x of z with z equal to e to the j omega. 163 00:11:22,120 --> 00:11:25,030 Namely, our Fourier transforms will now be written as I've 164 00:11:25,030 --> 00:11:26,640 indicated here. 165 00:11:26,640 --> 00:11:31,970 So just summarizing that, our new notation is that the 166 00:11:31,970 --> 00:11:37,980 independent variable on the Fourier transform is now going 167 00:11:37,980 --> 00:11:42,480 to be expressed as e to the j omega rather than as omega. 168 00:11:42,480 --> 00:11:47,320 It's a minor notational change, but I recognize the 169 00:11:47,320 --> 00:11:51,630 fact that it's somewhat confusing initially, and takes 170 00:11:51,630 --> 00:11:55,200 a few minutes to sit down and just get it straightened out. 171 00:11:55,200 --> 00:11:56,580 It's very similar to what we did 172 00:11:56,580 --> 00:11:58,440 with the Laplace transform. 173 00:11:58,440 --> 00:12:02,560 But let me draw your attention to the fact that in the 174 00:12:02,560 --> 00:12:06,090 Laplace transform, the independent variable that we 175 00:12:06,090 --> 00:12:11,520 ended up with in talking about the Fourier transform is 176 00:12:11,520 --> 00:12:14,640 different than what we're ending up with here. 177 00:12:14,640 --> 00:12:18,160 In particular, before we had j omega, now we 178 00:12:18,160 --> 00:12:21,430 have e to the j omega. 179 00:12:21,430 --> 00:12:27,350 And the reason for that is simply that whereas in 180 00:12:27,350 --> 00:12:29,530 continuous time we were talking about functions of the 181 00:12:29,530 --> 00:12:34,530 form e to the st, now we're talking about sequences of the 182 00:12:34,530 --> 00:12:38,100 form z to the n. 183 00:12:38,100 --> 00:12:41,940 So we have one relationship between the Fourier transform 184 00:12:41,940 --> 00:12:43,060 and the z-transform. 185 00:12:43,060 --> 00:12:47,330 Namely, the fact that for the magnitude of z equal to 1, the 186 00:12:47,330 --> 00:12:53,390 z-transform reduces to the Fourier transform. 187 00:12:53,390 --> 00:12:56,690 Now, in the Laplace transform, we also had another important 188 00:12:56,690 --> 00:13:03,250 relationship and observation, which was the fact that the 189 00:13:03,250 --> 00:13:07,100 Laplace transform was the Fourier 190 00:13:07,100 --> 00:13:11,870 transform of x of t modified. 191 00:13:11,870 --> 00:13:12,970 And how was it modified? 192 00:13:12,970 --> 00:13:18,270 It was modified by multiplying by a decaying or growing 193 00:13:18,270 --> 00:13:21,210 exponential, depending on what the real part of s is. 194 00:13:21,210 --> 00:13:23,820 Well, we have a very similar situation with the 195 00:13:23,820 --> 00:13:25,430 z-transform. 196 00:13:25,430 --> 00:13:31,560 In particular, in addition to the fact that the z-transform 197 00:13:31,560 --> 00:13:34,910 for z equal to e to the j omega reduces to the Fourier 198 00:13:34,910 --> 00:13:39,670 transform, we'll see that the z-transform for other values 199 00:13:39,670 --> 00:13:45,330 of z is the Fourier transform of the sequence with an 200 00:13:45,330 --> 00:13:46,800 exponential weighting. 201 00:13:46,800 --> 00:13:50,180 And let's see where that comes from. 202 00:13:50,180 --> 00:13:53,620 Here we have the general expression for the 203 00:13:53,620 --> 00:13:55,230 z-transform. 204 00:13:55,230 --> 00:13:59,680 And recognizing that z is a complex number which we're 205 00:13:59,680 --> 00:14:05,440 expressing in polar form as r e to the j omega, substituting 206 00:14:05,440 --> 00:14:10,400 that in, this summation now becomes x of n, r e to the j 207 00:14:10,400 --> 00:14:12,240 omega to the minus n. 208 00:14:12,240 --> 00:14:17,355 We can factor out these two terms, r to the minus n and e 209 00:14:17,355 --> 00:14:19,670 to the minus j omega n. 210 00:14:19,670 --> 00:14:26,060 And combining the r to the minus n with x of n and the e 211 00:14:26,060 --> 00:14:30,130 to the minus j omega n being treated separately, what we 212 00:14:30,130 --> 00:14:35,530 end up with is the summation that I have here. 213 00:14:35,530 --> 00:14:39,840 Well, what this says is that the z-transform, which is 214 00:14:39,840 --> 00:14:46,430 this, at z equal to r e to the j omega, is in fact the 215 00:14:46,430 --> 00:14:53,200 Fourier transform of what? 216 00:14:53,200 --> 00:14:58,670 It's the Fourier transform of x of n multiplied by r 217 00:14:58,670 --> 00:15:00,610 to the minus n. 218 00:15:00,610 --> 00:15:06,200 So that is the expression that we have here. 219 00:15:06,200 --> 00:15:10,150 And in continuous-time, we had the Laplace transform as the 220 00:15:10,150 --> 00:15:15,710 Fourier transform of x of t e to the minus sigma t. 221 00:15:15,710 --> 00:15:18,820 Here we have the Fourier transform of x of n r 222 00:15:18,820 --> 00:15:21,790 to the minus n. 223 00:15:21,790 --> 00:15:25,095 Now, something to just reflect on for a minute is-- 224 00:15:27,730 --> 00:15:31,980 because it tends to cause a little bit of problem with the 225 00:15:31,980 --> 00:15:35,640 algebra later on if you're attention isn't drawn to it, 226 00:15:35,640 --> 00:15:39,670 is that we're talking about multiplying x of n times r to 227 00:15:39,670 --> 00:15:41,410 the minus n. 228 00:15:41,410 --> 00:15:50,120 The question is, for r greater than 1, does r to the minus n 229 00:15:50,120 --> 00:15:55,220 increase exponentially as n increases or does it decrease? 230 00:15:55,220 --> 00:15:57,670 We're talking about r to the minus n. 231 00:15:57,670 --> 00:16:00,860 If r is greater than 1, if the magnitude of r is 232 00:16:00,860 --> 00:16:01,370 greater than 1. 233 00:16:01,370 --> 00:16:04,760 For example, if it's equal to 2, r to the minus n 234 00:16:04,760 --> 00:16:06,800 is 1/2 to the n. 235 00:16:06,800 --> 00:16:10,440 And so, in fact, that decreases exponentially. 236 00:16:10,440 --> 00:16:14,900 Or more generally, the larger r is, the faster r to the 237 00:16:14,900 --> 00:16:17,930 minus n decays with increasing n. 238 00:16:22,060 --> 00:16:28,490 Well, let's just look at some examples of the z-transform. 239 00:16:28,490 --> 00:16:33,930 And examples that I've picked, again, are examples directly 240 00:16:33,930 --> 00:16:34,820 out of the text. 241 00:16:34,820 --> 00:16:40,900 And so the details of the algebra you can look at more 242 00:16:40,900 --> 00:16:44,280 leisurely as you sit with the textbook. 243 00:16:44,280 --> 00:16:48,200 Let's consider, first of all, an exponential sequence x of n 244 00:16:48,200 --> 00:16:50,960 equals a to the n times the unit step. 245 00:16:50,960 --> 00:16:56,460 So 0 for negative time and an exponential for positive time. 246 00:16:56,460 --> 00:17:00,060 And the Fourier transform, as we've seen in earlier 247 00:17:00,060 --> 00:17:05,210 lectures, is 1 over 1 minus a e to the minus j omega. 248 00:17:05,210 --> 00:17:08,540 But this doesn't always converge. 249 00:17:08,540 --> 00:17:12,210 In particular, for convergence of the Fourier transform, we 250 00:17:12,210 --> 00:17:14,810 would require absolute summability of 251 00:17:14,810 --> 00:17:16,770 the original sequence. 252 00:17:16,770 --> 00:17:20,319 And that, in turn, requires that the magnitude of a be 253 00:17:20,319 --> 00:17:22,250 less than 1. 254 00:17:22,250 --> 00:17:25,730 So the Fourier transform is this, provided that the 255 00:17:25,730 --> 00:17:28,140 magnitude of a is less than 1. 256 00:17:28,140 --> 00:17:31,530 And what is the Fourier transform if the magnitude of 257 00:17:31,530 --> 00:17:34,330 a is not less than 1? 258 00:17:34,330 --> 00:17:36,380 Well, the answer is that, in that 259 00:17:36,380 --> 00:17:37,630 case, it doesn't converge. 260 00:17:40,310 --> 00:17:42,740 Now, let's look at the z-transform. 261 00:17:42,740 --> 00:17:47,770 The z-transform is the sum from minus infinity to plus 262 00:17:47,770 --> 00:17:54,420 infinity of a to the n z to the minus n 263 00:17:54,420 --> 00:17:56,480 times the unit step. 264 00:17:56,480 --> 00:17:59,130 The unit step will change the lower limit to 0. 265 00:17:59,130 --> 00:18:01,460 So it's the sum from 0 to infinity. 266 00:18:01,460 --> 00:18:08,720 And this is of the form a times z to the 267 00:18:08,720 --> 00:18:11,490 minus 1 to the n. 268 00:18:11,490 --> 00:18:14,740 So we're summing from 0 to infinity a times z to the 269 00:18:14,740 --> 00:18:16,360 minus 1 to the n. 270 00:18:16,360 --> 00:18:22,320 That sum is 1 over 1 minus a z to the minus 1. 271 00:18:22,320 --> 00:18:29,000 But in order for that sum to converge, we require that the 272 00:18:29,000 --> 00:18:35,710 magnitude of a times z to the minus 1 be less than 1. 273 00:18:35,710 --> 00:18:43,320 Now, the z-transform is the Fourier transform of the 274 00:18:43,320 --> 00:18:48,090 sequence a to the n times r to the minus n. 275 00:18:48,090 --> 00:18:55,530 And this statement about the z-transform converging is 276 00:18:55,530 --> 00:18:58,710 exactly identical to the statement that what we're 277 00:18:58,710 --> 00:19:04,280 requiring is that the magnitude of a times r to the 278 00:19:04,280 --> 00:19:11,350 minus 1 be less than 1, where this represents the 279 00:19:11,350 --> 00:19:17,440 exponential factor that we have that in effect is applied 280 00:19:17,440 --> 00:19:21,800 to the sequence, so that the Fourier transform becomes the 281 00:19:21,800 --> 00:19:23,430 z-transform. 282 00:19:23,430 --> 00:19:27,560 And so, if we put this condition, we can interpret 283 00:19:27,560 --> 00:19:32,100 this condition in exactly the same way that we interpret the 284 00:19:32,100 --> 00:19:36,920 condition on convergence of the Fourier transform. 285 00:19:36,920 --> 00:19:44,230 So from what we've worked out here then, what we have is the 286 00:19:44,230 --> 00:19:50,160 z-transform of a to the n times u of n is 1 over 1 minus 287 00:19:50,160 --> 00:19:53,600 a z to the minus 1. 288 00:19:53,600 --> 00:19:58,940 That works for any value of a provided that we pick the 289 00:19:58,940 --> 00:20:01,550 value of z correctly. 290 00:20:01,550 --> 00:20:05,120 In particular, we have to pick the set of values 291 00:20:05,120 --> 00:20:07,900 of z, so that what? 292 00:20:07,900 --> 00:20:11,870 So that the magnitude of a times z to the minus 1 293 00:20:11,870 --> 00:20:13,590 is less than 1. 294 00:20:13,590 --> 00:20:17,640 Or equivalently, so that the magnitude of z is greater than 295 00:20:17,640 --> 00:20:20,040 the magnitude of a. 296 00:20:20,040 --> 00:20:25,260 So associated with the z-transform of this sequence 297 00:20:25,260 --> 00:20:29,890 is this algebraic expression, and this set of values on z 298 00:20:29,890 --> 00:20:32,820 for which that algebraic expression is valid. 299 00:20:32,820 --> 00:20:38,600 And just as with the Laplace transform, this range of 300 00:20:38,600 --> 00:20:44,430 values is referred to as the region of convergence of the 301 00:20:44,430 --> 00:20:45,680 z-transform. 302 00:20:47,530 --> 00:20:52,950 Now, again, as we saw with the Laplace transform, it's 303 00:20:52,950 --> 00:20:58,590 important to recognize that in specifying or having worked 304 00:20:58,590 --> 00:21:02,870 out the z-transform of a sequence, it's not just the 305 00:21:02,870 --> 00:21:07,160 algebraic expression, but also the region of convergence 306 00:21:07,160 --> 00:21:11,700 that's required to uniquely specify it. 307 00:21:11,700 --> 00:21:15,420 To emphasize that further, here is Example 308 00:21:15,420 --> 00:21:17,500 10.2 from the text. 309 00:21:17,500 --> 00:21:20,910 And if you work that one through, what you find is 310 00:21:20,910 --> 00:21:26,190 that, algebraically, the z-transform of this sequence 311 00:21:26,190 --> 00:21:30,770 is 1 over 1 minus a z to the minus 1. 312 00:21:30,770 --> 00:21:34,240 Identical algebraically to what we had up here. 313 00:21:34,240 --> 00:21:42,360 But now with a region of convergence, which is the 314 00:21:42,360 --> 00:21:45,640 magnitude of z less than the magnitude of a. 315 00:21:45,640 --> 00:21:49,470 In contrast to this example, where the region of 316 00:21:49,470 --> 00:21:53,040 convergence was the magnitude of z greater than the 317 00:21:53,040 --> 00:21:56,160 magnitude of a. 318 00:21:56,160 --> 00:22:01,170 So again, it requires not just the algebraic expression, but 319 00:22:01,170 --> 00:22:06,230 also requires a specification of the region of convergence. 320 00:22:06,230 --> 00:22:12,380 And also, as with the Laplace transform, it's convenient in 321 00:22:12,380 --> 00:22:19,870 looking at the z-transform to represent it 322 00:22:19,870 --> 00:22:22,330 in the complex plane. 323 00:22:22,330 --> 00:22:25,710 In this case, the complex plane referred to as the 324 00:22:25,710 --> 00:22:28,790 z-plane, whereas in continuous-time when we talked 325 00:22:28,790 --> 00:22:31,400 about the Laplace transform, it was the s-plane. 326 00:22:31,400 --> 00:22:34,660 z, of course, because z is the complex variable in terms of 327 00:22:34,660 --> 00:22:37,910 which we're representing the z-transform. 328 00:22:37,910 --> 00:22:44,090 So we will be representing the z-transform in terms of 329 00:22:44,090 --> 00:22:48,400 representations in the complex plane, real part 330 00:22:48,400 --> 00:22:50,530 and imaginary part. 331 00:22:50,530 --> 00:22:54,630 But I've also identified a circle here. 332 00:22:54,630 --> 00:22:56,830 And you could wonder, well, what's the 333 00:22:56,830 --> 00:22:59,750 significance of the circle? 334 00:22:59,750 --> 00:23:05,710 Recall that in the discussion that we just came from, when 335 00:23:05,710 --> 00:23:09,380 we talked about the relationship between the 336 00:23:09,380 --> 00:23:13,460 z-transform and the Fourier transform, the z-transform 337 00:23:13,460 --> 00:23:16,290 reduces to the Fourier transform when the magnitude 338 00:23:16,290 --> 00:23:19,620 of z is equal to 1. 339 00:23:19,620 --> 00:23:21,420 The magnitude of z equal to 1 in the 340 00:23:21,420 --> 00:23:23,610 complex plane is a circle. 341 00:23:23,610 --> 00:23:28,880 And that circle, in fact, is a circle of radius 1. 342 00:23:28,880 --> 00:23:33,370 And so it's on this contour in the z-plane that the 343 00:23:33,370 --> 00:23:38,820 z-transform reduces to the Fourier transform. 344 00:23:38,820 --> 00:23:40,760 And we'll see some additional significance of 345 00:23:40,760 --> 00:23:43,550 that as we go along. 346 00:23:43,550 --> 00:23:47,010 Just again to emphasize the relationships and differences 347 00:23:47,010 --> 00:23:51,150 with continuous-time, with the Laplace transform it's the 348 00:23:51,150 --> 00:23:56,430 behavior in the s-plane on the j omega axis that corresponds 349 00:23:56,430 --> 00:23:58,440 to the Fourier transform. 350 00:23:58,440 --> 00:24:02,440 Here it's the behavior on the unit circle where the 351 00:24:02,440 --> 00:24:07,570 z-transform corresponds to the Fourier transform. 352 00:24:07,570 --> 00:24:08,930 Now, we'll be talking-- 353 00:24:08,930 --> 00:24:12,460 as we did with the Laplace transform, we'll be talking 354 00:24:12,460 --> 00:24:18,140 very often about transforms which are rational, and 355 00:24:18,140 --> 00:24:23,820 rational transforms as we'll see, represent systems which 356 00:24:23,820 --> 00:24:25,960 are characterized by linear constant coefficient 357 00:24:25,960 --> 00:24:27,700 difference equations. 358 00:24:27,700 --> 00:24:35,510 And so for the rational z-transforms, we'll again find 359 00:24:35,510 --> 00:24:38,530 it convenient to use a representation in terms of 360 00:24:38,530 --> 00:24:40,870 poles and zeroes in the z-plane. 361 00:24:40,870 --> 00:24:45,420 So let's look at our example as we've worked it out 362 00:24:45,420 --> 00:24:47,920 previously, Example 10.1. 363 00:24:47,920 --> 00:24:54,170 And with this sequence, the z-transform is 1 divided by 1 364 00:24:54,170 --> 00:24:56,360 minus a z to the minus 1. 365 00:24:56,360 --> 00:24:59,350 And we happen to have written it as a function of z 366 00:24:59,350 --> 00:25:00,950 to the minus 1. 367 00:25:00,950 --> 00:25:06,200 Clearly, we can rewrite this by multiplying numerator and 368 00:25:06,200 --> 00:25:09,900 denominator by z, and this would equivalently then be z 369 00:25:09,900 --> 00:25:13,470 divided by z minus a. 370 00:25:13,470 --> 00:25:19,220 And so if we were to represent this through a pole-zero plot, 371 00:25:19,220 --> 00:25:22,370 we would have a 0 at the origin corresponding to this 372 00:25:22,370 --> 00:25:27,340 factor and a pole at z equals a corresponding to the 373 00:25:27,340 --> 00:25:29,270 denominator factor. 374 00:25:29,270 --> 00:25:34,730 And so the pole-zero pattern for this is then a pole at z 375 00:25:34,730 --> 00:25:40,190 equal to a and a 0 at the origin. 376 00:25:40,190 --> 00:25:44,415 Now, let me just comment quickly about the fact that we 377 00:25:44,415 --> 00:25:48,440 had written this as 1 over 1 minus a z to the minus 1, and 378 00:25:48,440 --> 00:25:51,730 that seems kind of strange because perhaps we should have 379 00:25:51,730 --> 00:25:54,160 multiplied through by z. 380 00:25:54,160 --> 00:25:58,350 Let me just indicate that as you'll see as you work 381 00:25:58,350 --> 00:26:03,680 examples, it's very typical for the z-transform to come 382 00:26:03,680 --> 00:26:06,910 out as a function of z to the minus 1. 383 00:26:06,910 --> 00:26:11,820 And so very typically, you'll get to recognize that things 384 00:26:11,820 --> 00:26:15,980 will be expressed in terms of factors involving terms like 1 385 00:26:15,980 --> 00:26:20,380 minus a z to the minus 1, rather than factors of the 386 00:26:20,380 --> 00:26:23,320 form z minus a. 387 00:26:23,320 --> 00:26:28,740 Well, here is the one example that we had referred to. 388 00:26:28,740 --> 00:26:33,910 And if we consider another example, the other example, 389 00:26:33,910 --> 00:26:40,780 which was example 10.2 consists of an algebraic 390 00:26:40,780 --> 00:26:43,640 expression as I indicate here. 391 00:26:43,640 --> 00:26:47,660 But its region of validity is the magnitude of z less than 392 00:26:47,660 --> 00:26:49,240 the magnitude of a. 393 00:26:49,240 --> 00:26:53,850 And that corresponds to the same pole-zero plot, but a 394 00:26:53,850 --> 00:26:59,230 region of convergence which is inside this circle. 395 00:26:59,230 --> 00:27:05,400 Whereas, in the previous case, with the pole-zero plot, we 396 00:27:05,400 --> 00:27:09,010 had a region of convergence which was for the magnitude of 397 00:27:09,010 --> 00:27:11,740 z greater than the magnitude of a. 398 00:27:11,740 --> 00:27:15,610 So these two examples, this one and the other one, have 399 00:27:15,610 --> 00:27:18,350 exactly the same pole-zero pattern and they're 400 00:27:18,350 --> 00:27:22,530 distinguished by their region of convergence. 401 00:27:22,530 --> 00:27:29,210 Now, notice incidentally that in this particular case, the 402 00:27:29,210 --> 00:27:32,810 region of convergence includes the unit circle provided that 403 00:27:32,810 --> 00:27:35,240 the magnitude of a is less than 1. 404 00:27:35,240 --> 00:27:41,470 And so, in fact, that would say that the sequence has a 405 00:27:41,470 --> 00:27:43,460 Fourier transform that converges. 406 00:27:43,460 --> 00:27:47,410 Namely, with the magnitude of z equal to 1. 407 00:27:47,410 --> 00:27:52,020 Whereas, in this example, the region of convergence does not 408 00:27:52,020 --> 00:27:54,140 include the unit circle. 409 00:27:54,140 --> 00:27:57,540 And so, in fact, we cannot look at x of z for the 410 00:27:57,540 --> 00:27:59,570 magnitude of z equal to 1. 411 00:27:59,570 --> 00:28:03,850 And so this example, with the magnitude of a less than 1, 412 00:28:03,850 --> 00:28:07,950 does not have a Fourier transform that converges. 413 00:28:07,950 --> 00:28:12,160 Well, assuming that the magnitude of a is less than 1 414 00:28:12,160 --> 00:28:16,010 and the Fourier transform converges, we can, in fact, 415 00:28:16,010 --> 00:28:22,080 look at the Fourier transform by observing what happens as 416 00:28:22,080 --> 00:28:24,460 we go around the unit circle. 417 00:28:24,460 --> 00:28:26,910 We had seen this with the Laplace transform in terms of 418 00:28:26,910 --> 00:28:31,010 observing what happened as we move along the j omega axis. 419 00:28:31,010 --> 00:28:39,620 And here again, we can use the vectors as we trace out the 420 00:28:39,620 --> 00:28:41,060 unit circle. 421 00:28:41,060 --> 00:28:43,790 And in particular, what we would be looking at in this 422 00:28:43,790 --> 00:28:51,510 case is the ratio of the zero vector to the pole vector. 423 00:28:51,510 --> 00:28:53,960 For example, if we were looking at the magnitude of 424 00:28:53,960 --> 00:28:58,120 the z-transform, the magnitude of the z-transform would be 425 00:28:58,120 --> 00:29:02,520 the ratio of the length of this vector to the length of 426 00:29:02,520 --> 00:29:04,200 this vector. 427 00:29:04,200 --> 00:29:08,130 And to observe the Fourier transform, we would observe 428 00:29:08,130 --> 00:29:13,090 how those vectors change in length as we move around the 429 00:29:13,090 --> 00:29:15,030 unit circle. 430 00:29:15,030 --> 00:29:18,030 And as we move around the unit circle, what we would trace 431 00:29:18,030 --> 00:29:21,730 out in terms of the ratio of the lengths of those vectors 432 00:29:21,730 --> 00:29:23,835 is the Fourier transform. 433 00:29:26,910 --> 00:29:36,690 Well, let's focus on that also in the context of a slightly 434 00:29:36,690 --> 00:29:37,855 different z-transform. 435 00:29:37,855 --> 00:29:43,260 In the z-transform here as we'll see in a later lecture, 436 00:29:43,260 --> 00:29:45,800 is the z-transform associated with a second 437 00:29:45,800 --> 00:29:48,070 order difference equation. 438 00:29:48,070 --> 00:29:52,460 It has a denominator factor which has two poles 439 00:29:52,460 --> 00:29:54,810 associated with it. 440 00:29:54,810 --> 00:30:00,730 And so here, if we assumed that the Fourier transform of 441 00:30:00,730 --> 00:30:05,360 the associated sequence converged, then again we would 442 00:30:05,360 --> 00:30:08,080 look at the behavior of this as we moved 443 00:30:08,080 --> 00:30:10,040 around the unit circle. 444 00:30:10,040 --> 00:30:14,730 And the ratio of the lengths of the appropriate vectors 445 00:30:14,730 --> 00:30:19,060 would describe for us the frequency response. 446 00:30:19,060 --> 00:30:21,220 I'm sorry, the Fourier transform. 447 00:30:21,220 --> 00:30:27,800 So the Fourier transform magnitude would consist of the 448 00:30:27,800 --> 00:30:32,830 ratio of the lengths of the zero vectors divided by the 449 00:30:32,830 --> 00:30:34,930 lengths of the pole vectors. 450 00:30:34,930 --> 00:30:39,350 And one thing that we observe is that as we move in 451 00:30:39,350 --> 00:30:44,290 frequency omega in the vicinity of this pole, this 452 00:30:44,290 --> 00:30:49,260 pole vector, in fact, reaches a minimum length. 453 00:30:49,260 --> 00:30:54,170 That would mean that it's reciprocal would be maximum. 454 00:30:54,170 --> 00:30:57,290 And then, as we sweep past, the lengths of these two 455 00:30:57,290 --> 00:30:58,380 vectors would increase. 456 00:30:58,380 --> 00:31:02,040 The zero vectors, of course, would retain the same length 457 00:31:02,040 --> 00:31:05,940 no matter where we were on the unit circle. 458 00:31:05,940 --> 00:31:08,500 So, in fact, if we looked at the Fourier transform 459 00:31:08,500 --> 00:31:12,320 associated with this pole-zero pattern, if this was, for 460 00:31:12,320 --> 00:31:20,760 example, represented the z-transform of the impulse 461 00:31:20,760 --> 00:31:24,220 response or a linear time-invariant system, the 462 00:31:24,220 --> 00:31:27,040 corresponding frequency response would be what I 463 00:31:27,040 --> 00:31:28,970 plotted out below. 464 00:31:28,970 --> 00:31:31,810 And so it would peak. 465 00:31:31,810 --> 00:31:35,560 And in fact, where it would peak is in the vicinity of the 466 00:31:35,560 --> 00:31:42,220 frequency location of the pole as I indicate up here. 467 00:31:45,950 --> 00:31:51,530 So as we sweep past this pole then, in fact, this Fourier 468 00:31:51,530 --> 00:31:52,560 transform . 469 00:31:52,560 --> 00:31:54,550 Peaks. 470 00:31:54,550 --> 00:31:59,060 Well, this notion of looking at the frequency response as 471 00:31:59,060 --> 00:32:04,210 we move around the unit circle is a very important notion. 472 00:32:04,210 --> 00:32:06,610 And it's important to recognize it's the unit circle 473 00:32:06,610 --> 00:32:10,060 we're talking about here, whereas before we were talking 474 00:32:10,060 --> 00:32:12,740 about the j omega axis. 475 00:32:12,740 --> 00:32:18,470 And to emphasize this further, let me just show this example. 476 00:32:18,470 --> 00:32:22,900 And in fact, the previous example with the computer 477 00:32:22,900 --> 00:32:26,730 displays, so that we can see the frequency response as it 478 00:32:26,730 --> 00:32:31,100 sweeps out as we go around the unit circle. 479 00:32:31,100 --> 00:32:36,350 So here we have the pole-zero pattern for the 480 00:32:36,350 --> 00:32:38,640 second order example. 481 00:32:38,640 --> 00:32:42,780 And to generate the Fourier transform, we want to look at 482 00:32:42,780 --> 00:32:46,360 the behavior of the pole and zero vectors as we move around 483 00:32:46,360 --> 00:32:47,970 the unit circle. 484 00:32:47,970 --> 00:32:51,200 So first, let's display the vectors. 485 00:32:51,200 --> 00:32:55,030 And here we have them displayed to the point 486 00:32:55,030 --> 00:32:58,250 corresponding to zero frequency. 487 00:32:58,250 --> 00:33:03,750 And the magnitude of the Fourier transform will be, as 488 00:33:03,750 --> 00:33:06,650 we discussed, the magnitude of the length of the zero vector 489 00:33:06,650 --> 00:33:08,380 is divided by the magnitude of the 490 00:33:08,380 --> 00:33:10,580 length of the pole vectors. 491 00:33:10,580 --> 00:33:14,760 Shown below will be the Fourier transform. 492 00:33:14,760 --> 00:33:18,440 And we have the Fourier transform displayed here from 493 00:33:18,440 --> 00:33:23,300 0 to 2 pi, rather than from minus pi to pi as it was 494 00:33:23,300 --> 00:33:25,640 displayed in the transparency. 495 00:33:25,640 --> 00:33:29,130 Because of the periodicity of the Fourier transform, both of 496 00:33:29,130 --> 00:33:32,440 those are equivalent. 497 00:33:32,440 --> 00:33:38,400 Now we're sweeping away from omega equals 0 and the lengths 498 00:33:38,400 --> 00:33:40,590 of the pole vectors have changed. 499 00:33:40,590 --> 00:33:43,100 And that, of course, generates a change 500 00:33:43,100 --> 00:33:46,390 in the Fourier transform. 501 00:33:46,390 --> 00:33:51,530 And as we continued the process further, if we 502 00:33:51,530 --> 00:33:57,440 increase frequency, as we sweep closer to the location 503 00:33:57,440 --> 00:34:02,060 of the pole, the pole vector decreases in length 504 00:34:02,060 --> 00:34:03,370 dramatically. 505 00:34:03,370 --> 00:34:07,400 And that generates a residence in the Fourier transform, very 506 00:34:07,400 --> 00:34:11,040 similar to what we saw in continuous-time. 507 00:34:11,040 --> 00:34:16,875 Now as we continue to sweep further, what will happen is 508 00:34:16,875 --> 00:34:19,949 that that pole vector will begin to 509 00:34:19,949 --> 00:34:22,050 increase in length again. 510 00:34:22,050 --> 00:34:25,300 And so, in fact, the magnitude of the Fourier 511 00:34:25,300 --> 00:34:28,730 transform will decrease. 512 00:34:28,730 --> 00:34:32,889 And we see that here as we sweep toward 513 00:34:32,889 --> 00:34:35,250 omega equal to pi. 514 00:34:35,250 --> 00:34:39,830 Now, notice in this process that the length of the zero 515 00:34:39,830 --> 00:34:44,929 vectors has stayed the same because of the fact that the 516 00:34:44,929 --> 00:34:49,429 zeroes are at the origin, and no matter where we are in the 517 00:34:49,429 --> 00:34:52,420 unit circle, the length of those vectors is unity. 518 00:34:52,420 --> 00:34:55,650 So they don't influence in this example the magnitude, 519 00:34:55,650 --> 00:34:59,420 but they would, of course, influence the phase. 520 00:34:59,420 --> 00:35:02,600 Now we want to continue sweeping from omega equal to 521 00:35:02,600 --> 00:35:04,960 pi around to 2 pi. 522 00:35:04,960 --> 00:35:08,280 And because of the symmetry in the Fourier transform, what we 523 00:35:08,280 --> 00:35:12,310 will see in the magnitude is identical to what we would see 524 00:35:12,310 --> 00:35:16,880 if we swept from omega equal to pi back clockwise to 525 00:35:16,880 --> 00:35:19,630 omega equals 0. 526 00:35:19,630 --> 00:35:23,670 In particular now, as we're increasing frequency, notice 527 00:35:23,670 --> 00:35:27,550 that the length of the pole vector associated with the 528 00:35:27,550 --> 00:35:30,740 lower half plane pole is decreasing. 529 00:35:30,740 --> 00:35:35,370 And so, in fact, that corresponds to generating a 530 00:35:35,370 --> 00:35:39,540 resonance as we sweep past that pole 531 00:35:39,540 --> 00:35:42,630 location as we are here. 532 00:35:42,630 --> 00:35:46,380 And then finally, that pole vector increases in length as 533 00:35:46,380 --> 00:35:49,760 we begin to approach omega equal to 2 pi. 534 00:35:49,760 --> 00:35:53,700 Or equivalently, as we approach omega equal to 0. 535 00:35:59,400 --> 00:36:05,610 Now finally, let's also look at the Fourier transform 536 00:36:05,610 --> 00:36:08,670 associated with the first order example that we 537 00:36:08,670 --> 00:36:11,370 discussed earlier in the lecture. 538 00:36:11,370 --> 00:36:16,140 And so what we'll want to look at is the Fourier transform as 539 00:36:16,140 --> 00:36:18,990 the pole and zero vectors change. 540 00:36:18,990 --> 00:36:22,150 Once again, the Fourier transform will be displayed on 541 00:36:22,150 --> 00:36:25,950 a scale from 0 to 2 pi, a frequency scale from 0 to 2 542 00:36:25,950 --> 00:36:29,060 pi, rather than minus pi to pi. 543 00:36:29,060 --> 00:36:32,600 And we want to observe the pole and zero vectors as we 544 00:36:32,600 --> 00:36:34,010 sweep around the unit circle. 545 00:36:38,670 --> 00:36:42,420 We display first the pole and zero vectors at 546 00:36:42,420 --> 00:36:44,960 omega equal to 0. 547 00:36:44,960 --> 00:36:52,140 And as the frequency increases, the pole vector 548 00:36:52,140 --> 00:36:53,910 increases in length. 549 00:36:53,910 --> 00:36:57,640 The zero vector, since the zero is at the origin, will 550 00:36:57,640 --> 00:37:00,190 have constant length no matter where we 551 00:37:00,190 --> 00:37:01,430 are on the unit circle. 552 00:37:01,430 --> 00:37:04,470 Although it would affect the phase, which we are not 553 00:37:04,470 --> 00:37:06,000 displaying here. 554 00:37:06,000 --> 00:37:09,520 And so the principle effect, the only effect really on the 555 00:37:09,520 --> 00:37:12,540 magnitude, is due to the pole vector. 556 00:37:12,540 --> 00:37:17,470 As the frequency continues to increase, the pole vector 557 00:37:17,470 --> 00:37:21,890 increases in length, monotonically in fact. 558 00:37:21,890 --> 00:37:25,190 And so that means that the magnitude of the Fourier 559 00:37:25,190 --> 00:37:30,700 transform will decrease monotonically until we get 560 00:37:30,700 --> 00:37:33,030 past omega equal to pi. 561 00:37:44,820 --> 00:37:48,910 Here we are now at omega equal to pi. 562 00:37:48,910 --> 00:37:54,360 And when we continue sweeping past this frequency around to 563 00:37:54,360 --> 00:37:59,860 2 pi, then we will see basically the same curve swept 564 00:37:59,860 --> 00:38:01,320 out in reverse. 565 00:38:01,320 --> 00:38:04,210 Since because of the symmetry, again, of the Fourier 566 00:38:04,210 --> 00:38:10,190 transform magnitude, sweeping from pi to 2 pi is going to be 567 00:38:10,190 --> 00:38:13,740 equivalent with regard to the magnitude to sweeping 568 00:38:13,740 --> 00:38:17,400 from pi back to 0. 569 00:38:17,400 --> 00:38:21,730 And so now the pole vector begins to decrease in length 570 00:38:21,730 --> 00:38:25,140 and correspondingly, the magnitude of the Fourier 571 00:38:25,140 --> 00:38:29,070 transform will increase. 572 00:38:29,070 --> 00:38:33,130 And that will continue until we get around to omega equal 573 00:38:33,130 --> 00:38:36,265 to 2 pi, which is equivalent, of course, to 574 00:38:36,265 --> 00:38:38,020 omega equal to 0. 575 00:38:38,020 --> 00:38:41,430 And obviously, if we continue to sweep around again, we 576 00:38:41,430 --> 00:38:46,500 would simply trace out other periods associated with the 577 00:38:46,500 --> 00:38:47,750 Fourier transform. 578 00:38:55,550 --> 00:39:00,270 Well, that hopefully gives you kind of some feel for the 579 00:39:00,270 --> 00:39:02,760 notion of sweeping around the unit circle. 580 00:39:02,760 --> 00:39:06,430 And of course, you can see that because the circle is 581 00:39:06,430 --> 00:39:09,520 periodic as we go around and around, of course, what we'll 582 00:39:09,520 --> 00:39:13,130 get is a periodic Fourier transform, which is the way 583 00:39:13,130 --> 00:39:16,990 Fourier transforms are supposed to be. 584 00:39:16,990 --> 00:39:26,830 Now, just as with the Laplace transform, the region of 585 00:39:26,830 --> 00:39:30,780 convergence of the z-transform, as we've seen in 586 00:39:30,780 --> 00:39:36,240 this example, is a very important part of the 587 00:39:36,240 --> 00:39:39,430 specification of the z-transform. 588 00:39:39,430 --> 00:39:42,690 And we can, in talking about sequences and their 589 00:39:42,690 --> 00:39:46,100 transforms, either specify the region of convergence 590 00:39:46,100 --> 00:39:49,730 implicitly, or we can specify it explicitly. 591 00:39:49,730 --> 00:39:52,870 We can, for example, say what it is, as let's say the 592 00:39:52,870 --> 00:39:56,020 magnitude of z being greater than the magnitude of a. 593 00:39:56,020 --> 00:40:00,830 Or we can recognize that the region of convergence has 594 00:40:00,830 --> 00:40:06,020 certain constraints associated with certain properties of the 595 00:40:06,020 --> 00:40:07,480 time function. 596 00:40:07,480 --> 00:40:14,370 And in particular, there are some important properties of 597 00:40:14,370 --> 00:40:20,060 the region of convergence which allow us, given that we 598 00:40:20,060 --> 00:40:23,540 know certain characteristics of the time function, to then 599 00:40:23,540 --> 00:40:26,440 identify the region of convergence by looking at the 600 00:40:26,440 --> 00:40:27,940 pole-zero pattern. 601 00:40:27,940 --> 00:40:32,420 For example, we recognize that the region of convergence does 602 00:40:32,420 --> 00:40:37,800 not contain any poles because of the fact that at poles, the 603 00:40:37,800 --> 00:40:42,010 z-transform, in fact, blows up and, of course, can't converge 604 00:40:42,010 --> 00:40:44,060 at that point. 605 00:40:44,060 --> 00:40:48,480 Furthermore, the region of convergence consists of a ring 606 00:40:48,480 --> 00:40:52,420 in the z-plane centered about the origin. 607 00:40:52,420 --> 00:40:55,750 Recall that with the Laplace transform, the region of 608 00:40:55,750 --> 00:40:59,220 convergence consisted of strips in the s-plane. 609 00:41:01,720 --> 00:41:04,380 With the z-transform, the region of convergence consists 610 00:41:04,380 --> 00:41:07,880 of a ring, basically because of the fact that the region of 611 00:41:07,880 --> 00:41:12,490 convergence is dependent on the magnitude of z. 612 00:41:12,490 --> 00:41:14,990 Whereas, with the Laplace transform, the region of 613 00:41:14,990 --> 00:41:18,310 convergence was dependent on the real part of s. 614 00:41:18,310 --> 00:41:21,510 The fact that it's the magnitude of z says, in 615 00:41:21,510 --> 00:41:25,580 effect, that all values of z that have the same magnitude 616 00:41:25,580 --> 00:41:27,350 lie on a circle. 617 00:41:27,350 --> 00:41:29,900 And so the region of convergence you would expect 618 00:41:29,900 --> 00:41:36,420 to be a concentric ring in the z-plane. 619 00:41:36,420 --> 00:41:40,000 Furthermore, as we've already talked about and exploited 620 00:41:40,000 --> 00:41:44,130 actually, convergence of the Fourier transform is 621 00:41:44,130 --> 00:41:47,720 equivalent to the statement that the region of convergence 622 00:41:47,720 --> 00:41:49,530 includes the unit circle in the z-plane. 623 00:41:52,130 --> 00:41:57,420 Now, we can also associate the region of convergence with 624 00:41:57,420 --> 00:42:01,700 issues about whether the sequence is of finite duration 625 00:42:01,700 --> 00:42:04,610 or right-sided or left-sided. 626 00:42:04,610 --> 00:42:09,920 And let me sort of quickly indicate again what the style 627 00:42:09,920 --> 00:42:12,550 of the argument is. 628 00:42:12,550 --> 00:42:17,840 If we have a finite duration sequence, so that the sequence 629 00:42:17,840 --> 00:42:20,580 is absolutely summable, and therefore has a Fourier 630 00:42:20,580 --> 00:42:25,590 transform that converges, then because of the fact that it's 631 00:42:25,590 --> 00:42:30,210 0 outside some interval, I can multiply it by an 632 00:42:30,210 --> 00:42:35,070 exponentially decaying sequence or by an 633 00:42:35,070 --> 00:42:37,340 exponentially growing sequence. 634 00:42:37,340 --> 00:42:41,170 And since I'm only doing this over a finite interval, no 635 00:42:41,170 --> 00:42:45,030 matter how I choose that exponential, we'll end up with 636 00:42:45,030 --> 00:42:47,130 an absolutely summable product. 637 00:42:47,130 --> 00:42:51,610 So if x of n is a finite duration, then in fact the 638 00:42:51,610 --> 00:42:56,590 region of convergence is the entire z-plane, possibly with 639 00:42:56,590 --> 00:42:58,935 the exception of the origin or infinity. 640 00:43:01,820 --> 00:43:06,490 On the other hand, if the sequence is a right-sided 641 00:43:06,490 --> 00:43:12,390 sequence, then we have to be careful that we don't multiply 642 00:43:12,390 --> 00:43:17,290 by an exponential that grows too fast for positive time. 643 00:43:17,290 --> 00:43:22,300 Or equivalently, we might have to choose the exponential so 644 00:43:22,300 --> 00:43:26,720 that it decays sufficiently fast for positive time. 645 00:43:26,720 --> 00:43:31,240 As a consequence of that, for a right-sided sequence, if we 646 00:43:31,240 --> 00:43:36,730 have a value which is in the region of convergence, as long 647 00:43:36,730 --> 00:43:40,950 as I multiply by exponentials that decay faster than that 648 00:43:40,950 --> 00:43:47,020 for positive time, then I'll also have convergence. 649 00:43:47,020 --> 00:43:50,630 In other words, all finite values of z for which the 650 00:43:50,630 --> 00:43:53,710 magnitude of z is greater than this, so that the exponentials 651 00:43:53,710 --> 00:43:56,760 die off even faster will also be in the region of 652 00:43:56,760 --> 00:43:59,300 convergence. 653 00:43:59,300 --> 00:44:01,810 If we combine that statement with the fact that there are 654 00:44:01,810 --> 00:44:05,320 no poles in the region of convergence, then we end up 655 00:44:05,320 --> 00:44:07,150 with a statement similar to what we had 656 00:44:07,150 --> 00:44:09,000 with the Laplace transform. 657 00:44:09,000 --> 00:44:14,430 Here, the statement is that if the sequence is right-sided, 658 00:44:14,430 --> 00:44:19,160 then the region of convergence has to be outside the 659 00:44:19,160 --> 00:44:20,560 outermost pole. 660 00:44:20,560 --> 00:44:23,610 Essentially, because it has to be outside someplace and can't 661 00:44:23,610 --> 00:44:24,860 include any poles. 662 00:44:27,220 --> 00:44:36,110 Finally, if we have a left-sided sequence, then if 663 00:44:36,110 --> 00:44:39,230 we have a value which is in the region of convergence, all 664 00:44:39,230 --> 00:44:42,390 values for which the magnitude of z is less than that will 665 00:44:42,390 --> 00:44:44,630 also be in the region of convergence. 666 00:44:44,630 --> 00:44:48,680 Or if x of z is rational, then the region of convergence must 667 00:44:48,680 --> 00:44:51,130 be inside the innermost pole. 668 00:44:51,130 --> 00:44:56,610 And finally, if we have a two-sided sequence, then 669 00:44:56,610 --> 00:45:00,130 there's kind of a balance between the exponential factor 670 00:45:00,130 --> 00:45:01,590 that we use. 671 00:45:01,590 --> 00:45:05,980 And so in that case, then the region of convergence will be 672 00:45:05,980 --> 00:45:09,880 a ring in the z-plane, and essentially will extend 673 00:45:09,880 --> 00:45:12,880 outward to a pole and inward to a pole. 674 00:45:12,880 --> 00:45:19,700 So if we had an algebraic expression, let's say as we 675 00:45:19,700 --> 00:45:28,150 have here, then we could associate with that a region 676 00:45:28,150 --> 00:45:30,980 of convergence outside this pole. 677 00:45:30,980 --> 00:45:34,770 And that would correspond to a right-sided sequence. 678 00:45:34,770 --> 00:45:40,010 Or we can associate with it a region of convergence, which 679 00:45:40,010 --> 00:45:45,120 is inside the innermost pole. 680 00:45:45,120 --> 00:45:48,440 And that would correspond to a left-sided sequence. 681 00:45:48,440 --> 00:45:53,890 And the third and only other possibility is a region of 682 00:45:53,890 --> 00:45:58,330 convergence which lies between these two poles. 683 00:45:58,330 --> 00:46:02,980 And that would then correspond to a two-sided sequence. 684 00:46:02,980 --> 00:46:05,070 And notice incidentally because of where I've placed 685 00:46:05,070 --> 00:46:08,700 these poles, that this is the only one for which the region 686 00:46:08,700 --> 00:46:11,130 of convergence includes the unit circle. 687 00:46:11,130 --> 00:46:14,670 In other words, it's the only one for which the Fourier 688 00:46:14,670 --> 00:46:17,220 transform converges. 689 00:46:17,220 --> 00:46:19,830 OK, now we've moved through that fairly quickly. 690 00:46:19,830 --> 00:46:24,560 And I've emphasized the fact that it parallels very closely 691 00:46:24,560 --> 00:46:27,570 what we did with the Laplace transform. 692 00:46:27,570 --> 00:46:32,140 What I'd like to do is just conclude with a discussion of 693 00:46:32,140 --> 00:46:37,600 how we get the time function back again when we have the 694 00:46:37,600 --> 00:46:42,150 z-transform including its region of convergence. 695 00:46:42,150 --> 00:46:45,780 Well, we can, first of all, develop a more or less formal 696 00:46:45,780 --> 00:46:46,580 expression. 697 00:46:46,580 --> 00:46:50,910 And the algebra for this is gone through in the text, and 698 00:46:50,910 --> 00:46:53,300 you went through something similar to this with the 699 00:46:53,300 --> 00:46:56,510 Laplace transform in the video course manual. 700 00:46:56,510 --> 00:46:59,350 So I won't carry through the details. 701 00:46:59,350 --> 00:47:04,130 But basically, what we can use to develop a formal expression 702 00:47:04,130 --> 00:47:08,320 is the fact that the z-transform is the Fourier 703 00:47:08,320 --> 00:47:11,550 transform of the sequence exponentially weighted. 704 00:47:11,550 --> 00:47:16,550 So we can apply the inverse transform to that, and that 705 00:47:16,550 --> 00:47:19,720 gives us not x of n, but x of n exponentially weighted. 706 00:47:19,720 --> 00:47:24,540 And if we track that through, then what we'll end up with is 707 00:47:24,540 --> 00:47:26,020 an expression. 708 00:47:26,020 --> 00:47:28,950 After we've taken care of a few of the epsilons and 709 00:47:28,950 --> 00:47:33,410 deltas, we'll end up with an expression that expresses 710 00:47:33,410 --> 00:47:41,000 formally the sequence x of n in terms of the z-transform, 711 00:47:41,000 --> 00:47:43,830 where this, in fact, is a contour integral in the 712 00:47:43,830 --> 00:47:45,390 complex plane. 713 00:47:45,390 --> 00:47:49,300 And so there's a formal expression, just as there's a 714 00:47:49,300 --> 00:47:52,280 formal expression for the Laplace transform. 715 00:47:52,280 --> 00:47:58,900 But in fact, the more typical procedure is to use 716 00:47:58,900 --> 00:48:03,630 essentially transformed pairs that we know together with the 717 00:48:03,630 --> 00:48:07,350 idea of using a partial fraction expansion. 718 00:48:07,350 --> 00:48:15,460 So if we had a z-transform as I indicate here, and if we 719 00:48:15,460 --> 00:48:21,120 expand it out in a partial fraction expansion, then we 720 00:48:21,120 --> 00:48:26,580 can recognize, as we did in a similar style with the Laplace 721 00:48:26,580 --> 00:48:27,860 transform-- 722 00:48:27,860 --> 00:48:33,090 we can recognize that each term, together with the 723 00:48:33,090 --> 00:48:38,450 identified region of convergence corresponds to an 724 00:48:38,450 --> 00:48:40,240 exponential factor. 725 00:48:40,240 --> 00:48:44,420 And so this term, together with the fact we know that the 726 00:48:44,420 --> 00:48:49,960 magnitude of z must be greater than 2, allows us to recognize 727 00:48:49,960 --> 00:48:54,800 this as similar to the Example 10.1. 728 00:48:54,800 --> 00:48:58,800 And in particular then, the sequence associated with that 729 00:48:58,800 --> 00:49:00,970 is what I indicate here. 730 00:49:00,970 --> 00:49:04,230 And for the second term, the sequence is 731 00:49:04,230 --> 00:49:06,070 what I indicate here. 732 00:49:06,070 --> 00:49:11,260 So what we're simply doing is using the fact that we've 733 00:49:11,260 --> 00:49:15,850 worked out the example going one direction before, and now 734 00:49:15,850 --> 00:49:19,350 we use that together with the partial fraction expansion to 735 00:49:19,350 --> 00:49:23,240 get the individual sequences back again, and 736 00:49:23,240 --> 00:49:24,720 then add them together. 737 00:49:24,720 --> 00:49:27,790 There's one other method which I'll just point to, which is 738 00:49:27,790 --> 00:49:31,370 also elaborated on a little more in the text. 739 00:49:31,370 --> 00:49:35,860 But it's kind of the idea of developing the inverse 740 00:49:35,860 --> 00:49:42,570 z-transform by recognizing that this z-transform formula, 741 00:49:42,570 --> 00:49:46,900 in fact, is a power series. 742 00:49:46,900 --> 00:49:54,130 So if we take x of z and expand it in a power series, 743 00:49:54,130 --> 00:49:59,640 then we can pick off the values of x of n by 744 00:49:59,640 --> 00:50:01,350 identifying the individual 745 00:50:01,350 --> 00:50:02,945 coefficients in this expansion. 746 00:50:06,070 --> 00:50:11,780 And so by simply doing long division, for example, we can 747 00:50:11,780 --> 00:50:14,750 also get the inverse transform. 748 00:50:14,750 --> 00:50:17,650 And that, by the way, is very useful. 749 00:50:17,650 --> 00:50:23,330 Particularly if we want to get the inverse z-transform for a 750 00:50:23,330 --> 00:50:27,700 z-transform expression, which is not rational. 751 00:50:27,700 --> 00:50:30,330 Now we've moved through this fairly quickly. 752 00:50:30,330 --> 00:50:33,080 On the other hand, I've stressed that it's very 753 00:50:33,080 --> 00:50:36,220 similar to what we went through for the Laplace 754 00:50:36,220 --> 00:50:41,320 transform, except for a very important difference. 755 00:50:41,320 --> 00:50:44,390 The principal difference really being that with the 756 00:50:44,390 --> 00:50:48,580 Laplace transform, it was the j omega axis in the s-plane 757 00:50:48,580 --> 00:50:51,300 that we focused attention on when we were thinking about 758 00:50:51,300 --> 00:50:52,970 the Fourier transform. 759 00:50:52,970 --> 00:50:56,230 Here, the unit circle in the z-plane plays 760 00:50:56,230 --> 00:50:58,580 an important role. 761 00:50:58,580 --> 00:51:01,900 What we'll see when we continue this in the next 762 00:51:01,900 --> 00:51:07,250 lecture is that there are properties of the z-transform, 763 00:51:07,250 --> 00:51:10,630 just as there were properties of the Laplace transform. 764 00:51:10,630 --> 00:51:16,910 And those properties allow us to develop and exploit the 765 00:51:16,910 --> 00:51:21,860 z-transform in the context of systems describable by linear 766 00:51:21,860 --> 00:51:24,250 constant coefficient difference equations. 767 00:51:24,250 --> 00:51:28,420 So in the next lecture, we'll focus on some properties of 768 00:51:28,420 --> 00:51:31,970 the z-transform, and then we'll see how to use those 769 00:51:31,970 --> 00:51:37,030 properties to help us in getting further insight and 770 00:51:37,030 --> 00:51:38,870 working with systems describable 771 00:51:38,870 --> 00:51:40,580 by difference equations. 772 00:51:40,580 --> 00:51:41,830 Thank you.