WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high-quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation, or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.565 ocw.mit.edu. 00:00:20.565 --> 00:00:56.890 [MUSIC PLAYING] 00:00:56.890 --> 00:00:59.510 PROFESSOR: In the last several lectures, we've talked about a 00:00:59.510 --> 00:01:02.930 generalization of the continuous-time Fourier 00:01:02.930 --> 00:01:07.490 transform and a very similar strategy also applies to 00:01:07.490 --> 00:01:10.610 discrete-time, and that's what we want to begin to deal with 00:01:10.610 --> 00:01:12.750 in today's lecture. 00:01:12.750 --> 00:01:17.400 So what we want to talk about is generalizing the Fourier 00:01:17.400 --> 00:01:21.650 transform, and what this will lead to in discrete-time is a 00:01:21.650 --> 00:01:24.380 notion referred to as the z-transform. 00:01:24.380 --> 00:01:27.810 Now, just as in continuous-time, in 00:01:27.810 --> 00:01:32.670 discrete-time the Fourier transform corresponded to a 00:01:32.670 --> 00:01:38.830 representation of a sequence as a linear combination of 00:01:38.830 --> 00:01:41.750 complex exponentials. 00:01:41.750 --> 00:01:45.430 So this was the synthesis equation. 00:01:45.430 --> 00:01:46.810 And, of course, there is the 00:01:46.810 --> 00:01:50.100 corresponding analysis equation. 00:01:50.100 --> 00:01:55.310 And as you recall, and as is the same for continuous-time, 00:01:55.310 --> 00:01:59.300 the reason that we picked complex exponentials was 00:01:59.300 --> 00:02:02.900 because of the fact that they are eigenfunctions of linear 00:02:02.900 --> 00:02:04.730 time-invariant systems. 00:02:04.730 --> 00:02:09.229 In other words, if you have a complex exponential into a 00:02:09.229 --> 00:02:12.560 linear time-invariant system, the output is a complex 00:02:12.560 --> 00:02:13.820 exponential. 00:02:13.820 --> 00:02:19.620 And the change in complex amplitude, which corresponds 00:02:19.620 --> 00:02:23.150 to the frequency response, in fact is what led to the 00:02:23.150 --> 00:02:25.640 definition of the Fourier transform. 00:02:25.640 --> 00:02:29.070 In particular, it is the Fourier transform of the 00:02:29.070 --> 00:02:31.610 impulse response. 00:02:31.610 --> 00:02:36.450 Well, that set of notions is, more or less, identical to the 00:02:36.450 --> 00:02:40.410 way we motivated the Laplace transform in the 00:02:40.410 --> 00:02:43.030 continuous-time case, in the Fourier transform in the 00:02:43.030 --> 00:02:44.680 continuous-time case. 00:02:44.680 --> 00:02:49.710 And just as in continuous-time, there are a 00:02:49.710 --> 00:02:55.050 set of signals more general than the complex exponentials, 00:02:55.050 --> 00:02:57.760 which are also eigenfunctions of linear 00:02:57.760 --> 00:02:59.420 time-invariant systems. 00:02:59.420 --> 00:03:03.810 In particular, in discrete-time, if we had 00:03:03.810 --> 00:03:08.790 instead of an exponential e to the j omega n, we had a more 00:03:08.790 --> 00:03:16.220 general complex number z, that the signal z to the n is also 00:03:16.220 --> 00:03:21.090 an eigenfunction of a linear time-invariant system for any 00:03:21.090 --> 00:03:22.630 particular z. 00:03:22.630 --> 00:03:26.770 So we can see that by substituting that into the 00:03:26.770 --> 00:03:30.580 convolution sum and recognizing, again, very 00:03:30.580 --> 00:03:34.820 strongly paralleling the continuous-time argument, that 00:03:34.820 --> 00:03:42.670 we can rewrite this factor as z to the n z to the minus k. 00:03:42.670 --> 00:03:48.960 And because of the fact that it's a sum on k and this term 00:03:48.960 --> 00:03:52.460 doesn't depend on k, we can take that term out. 00:03:52.460 --> 00:03:57.010 And the conclusion is that if we have z to the n as an 00:03:57.010 --> 00:04:04.640 input, that the output is of the same form times a factor 00:04:04.640 --> 00:04:06.980 which depends on z. 00:04:06.980 --> 00:04:10.630 But of course, doesn't depend on k because that is summed 00:04:10.630 --> 00:04:13.260 out as we form the summation. 00:04:13.260 --> 00:04:18.149 So, in fact, this summation corresponds to a complex 00:04:18.149 --> 00:04:26.490 number, which we'll denote as H of z, where z will represent 00:04:26.490 --> 00:04:28.750 a more general complex number. 00:04:28.750 --> 00:04:34.870 Namely, z is r e to the j omega, r being the magnitude 00:04:34.870 --> 00:04:38.020 of this complex number and omega, of 00:04:38.020 --> 00:04:40.530 course, being the angle. 00:04:40.530 --> 00:04:46.950 So for a linear time-invariant system, a more general complex 00:04:46.950 --> 00:04:51.900 exponential sequence of this form generates as an output a 00:04:51.900 --> 00:04:55.990 complex exponential sequence of the same form with a change 00:04:55.990 --> 00:05:00.480 in amplitude which we're representing as H of z, 00:05:00.480 --> 00:05:04.010 recognizing the fact that it's going to be a function of what 00:05:04.010 --> 00:05:06.080 that complex number is. 00:05:06.080 --> 00:05:11.150 And this amplitude factor is given by this summation. 00:05:11.150 --> 00:05:18.130 And it is this summation which is defined as the z-transform 00:05:18.130 --> 00:05:22.100 of the sequence H of n. 00:05:22.100 --> 00:05:24.240 Now, let me stress that-- 00:05:24.240 --> 00:05:27.650 and I'll continue to stress this as the lecture goes on. 00:05:27.650 --> 00:05:33.260 That much of what we've said is directly parallel to what 00:05:33.260 --> 00:05:35.790 we said in the continuous-time case. 00:05:35.790 --> 00:05:41.310 And what we've simply done is to expand from complex 00:05:41.310 --> 00:05:46.060 exponentials with a purely imaginary exponent, complex 00:05:46.060 --> 00:05:50.950 exponential time functions or sequences of that form, to 00:05:50.950 --> 00:05:56.990 ones that have more general complex exponential factors. 00:05:56.990 --> 00:06:03.830 Now, we have a mapping here from the impulse response to 00:06:03.830 --> 00:06:08.660 the amplitude or the eigenvalue associated with 00:06:08.660 --> 00:06:14.320 that input, and this mapping is what is referred to as the 00:06:14.320 --> 00:06:16.130 z-transform. 00:06:16.130 --> 00:06:21.120 So H of z is, in fact, the z-transform 00:06:21.120 --> 00:06:23.060 of the impulse response. 00:06:23.060 --> 00:06:27.010 And if we consider applying this mapping as we did in 00:06:27.010 --> 00:06:30.420 continuous-time in a similar argument, applying this 00:06:30.420 --> 00:06:35.490 mapping to a sequence whether or not it corresponds to the 00:06:35.490 --> 00:06:39.180 impulse response or a linear time-invariant system, that 00:06:39.180 --> 00:06:46.270 leads then to the z-transform of a general sequence x of n. 00:06:46.270 --> 00:06:51.410 The z-transform being defined by this relationship. 00:06:51.410 --> 00:06:56.460 And again, notationally, we'll often represent a time 00:06:56.460 --> 00:07:00.740 function and a z-transform through a shorthand notation, 00:07:00.740 --> 00:07:06.690 just indicating that x of z is the z-transform of x of n. 00:07:09.260 --> 00:07:12.750 So we've kind of motivated the development in a manner 00:07:12.750 --> 00:07:14.760 exactly identical to what we had done 00:07:14.760 --> 00:07:16.570 with the Laplace transform. 00:07:16.570 --> 00:07:19.010 Kind of the idea that if you look at the eigenvalue 00:07:19.010 --> 00:07:23.620 associated with a linear time-invariant system, that 00:07:23.620 --> 00:07:29.220 essentially generates a mapping between the sequence-- 00:07:29.220 --> 00:07:33.750 system impulse response and a function of z. 00:07:33.750 --> 00:07:36.690 And that corresponds to the z-transform here, it 00:07:36.690 --> 00:07:39.020 corresponded to the Laplace transform in the 00:07:39.020 --> 00:07:41.760 continuous-time case. 00:07:41.760 --> 00:07:46.850 That same argument is also the kind of argument that we use 00:07:46.850 --> 00:07:51.200 to lead us into the Fourier transform originally. 00:07:51.200 --> 00:07:57.400 And once again, what you would expect is that the z-transform 00:07:57.400 --> 00:08:00.170 has a very close and important relationship 00:08:00.170 --> 00:08:01.810 to the Fourier transform. 00:08:01.810 --> 00:08:06.730 And indeed, that relationship turns out to be, more or less, 00:08:06.730 --> 00:08:11.630 identical to the relationship between the Laplace transform 00:08:11.630 --> 00:08:15.400 and the Fourier transform in continuous-time. 00:08:15.400 --> 00:08:18.650 Well, let's look at the relationship. 00:08:18.650 --> 00:08:23.240 First of all, what we recognize is that if we 00:08:23.240 --> 00:08:30.960 compare the Fourier transform expression for a sequence and 00:08:30.960 --> 00:08:36.299 the z-transform expression for the same sequence that they 00:08:36.299 --> 00:08:39.750 involve, essentially, the same operations. 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 00:08:47.230 --> 00:08:52.300 want this sum to look like this sum, then that would mean 00:08:52.300 --> 00:08:56.900 that we would choose z equal to e to the j omega. 00:08:56.900 --> 00:09:01.740 Said another way, the z-transform, when z is e to 00:09:01.740 --> 00:09:04.990 the j omega, is going to reduce 00:09:04.990 --> 00:09:07.900 to the Fourier transform. 00:09:07.900 --> 00:09:12.290 So we have a relationship like the one, again, that we had 00:09:12.290 --> 00:09:15.140 between the Laplace transform and the Fourier transform in 00:09:15.140 --> 00:09:16.710 continuous-time. 00:09:16.710 --> 00:09:22.020 Namely that for a certain set of values of the complex 00:09:22.020 --> 00:09:26.220 variable, the transform, the z-transform, reduces to the 00:09:26.220 --> 00:09:28.470 Fourier transform. 00:09:28.470 --> 00:09:32.970 So if we have x of z, the z-transform, and we look at 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 00:09:40.100 --> 00:09:43.850 j omega is similar to saying that we're looking at that for 00:09:43.850 --> 00:09:46.840 the magnitude of z equal to 1. 00:09:46.840 --> 00:09:49.740 We're specifically choosing r equal to 1, which is the 00:09:49.740 --> 00:09:52.050 magnitude of z. 00:09:52.050 --> 00:09:54.550 Then this is equal to the Fourier 00:09:54.550 --> 00:09:57.290 transform of the sequence. 00:09:57.290 --> 00:10:02.580 So the z-transform for z equal to e to the j omega is the 00:10:02.580 --> 00:10:07.020 Fourier transform, and so this then 00:10:07.020 --> 00:10:11.290 corresponds to x of omega. 00:10:11.290 --> 00:10:12.540 Namely, the Fourier transform. 00:10:15.000 --> 00:10:19.510 Well, we now have ourselves in a similar situation, again, to 00:10:19.510 --> 00:10:22.840 what we had when we talked about the Laplace transform. 00:10:22.840 --> 00:10:28.050 Namely, a notational awkwardness, or inconvenience, 00:10:28.050 --> 00:10:30.510 which we can resolve by simply 00:10:30.510 --> 00:10:33.840 redefining some of our notation. 00:10:33.840 --> 00:10:36.960 In particular, the awkwardness relates to the fact that 00:10:36.960 --> 00:10:40.990 whereas we've been writing our Fourier transforms this way, 00:10:40.990 --> 00:10:46.620 as x of omega, if we were to express x of z and look at it, 00:10:46.620 --> 00:10:48.510 it's equal to e to the j omega. 00:10:48.510 --> 00:10:52.320 We end up with the independent variable being e to the j 00:10:52.320 --> 00:10:56.240 omega rather than omega. 00:10:56.240 --> 00:10:58.920 Well, in fact, the Fourier transform is 00:10:58.920 --> 00:11:00.020 a function of omega. 00:11:00.020 --> 00:11:02.620 It's also a function of e to the j omega. 00:11:02.620 --> 00:11:06.620 And now what we can see is that given the fact that we 00:11:06.620 --> 00:11:12.310 want to generalize the Fourier transform to the z-transform, 00:11:12.310 --> 00:11:17.870 it's convenient now to use as notation for the Fourier 00:11:17.870 --> 00:11:22.120 transform x of z with z equal to e to the j omega. 00:11:22.120 --> 00:11:25.030 Namely, our Fourier transforms will now be written as I've 00:11:25.030 --> 00:11:26.640 indicated here. 00:11:26.640 --> 00:11:31.970 So just summarizing that, our new notation is that the 00:11:31.970 --> 00:11:37.980 independent variable on the Fourier transform is now going 00:11:37.980 --> 00:11:42.480 to be expressed as e to the j omega rather than as omega. 00:11:42.480 --> 00:11:47.320 It's a minor notational change, but I recognize the 00:11:47.320 --> 00:11:51.630 fact that it's somewhat confusing initially, and takes 00:11:51.630 --> 00:11:55.200 a few minutes to sit down and just get it straightened out. 00:11:55.200 --> 00:11:56.580 It's very similar to what we did 00:11:56.580 --> 00:11:58.440 with the Laplace transform. 00:11:58.440 --> 00:12:02.560 But let me draw your attention to the fact that in the 00:12:02.560 --> 00:12:06.090 Laplace transform, the independent variable that we 00:12:06.090 --> 00:12:11.520 ended up with in talking about the Fourier transform is 00:12:11.520 --> 00:12:14.640 different than what we're ending up with here. 00:12:14.640 --> 00:12:18.160 In particular, before we had j omega, now we 00:12:18.160 --> 00:12:21.430 have e to the j omega. 00:12:21.430 --> 00:12:27.350 And the reason for that is simply that whereas in 00:12:27.350 --> 00:12:29.530 continuous time we were talking about functions of the 00:12:29.530 --> 00:12:34.530 form e to the st, now we're talking about sequences of the 00:12:34.530 --> 00:12:38.100 form z to the n. 00:12:38.100 --> 00:12:41.940 So we have one relationship between the Fourier transform 00:12:41.940 --> 00:12:43.060 and the z-transform. 00:12:43.060 --> 00:12:47.330 Namely, the fact that for the magnitude of z equal to 1, the 00:12:47.330 --> 00:12:53.390 z-transform reduces to the Fourier transform. 00:12:53.390 --> 00:12:56.690 Now, in the Laplace transform, we also had another important 00:12:56.690 --> 00:13:03.250 relationship and observation, which was the fact that the 00:13:03.250 --> 00:13:07.100 Laplace transform was the Fourier 00:13:07.100 --> 00:13:11.870 transform of x of t modified. 00:13:11.870 --> 00:13:12.970 And how was it modified? 00:13:12.970 --> 00:13:18.270 It was modified by multiplying by a decaying or growing 00:13:18.270 --> 00:13:21.210 exponential, depending on what the real part of s is. 00:13:21.210 --> 00:13:23.820 Well, we have a very similar situation with the 00:13:23.820 --> 00:13:25.430 z-transform. 00:13:25.430 --> 00:13:31.560 In particular, in addition to the fact that the z-transform 00:13:31.560 --> 00:13:34.910 for z equal to e to the j omega reduces to the Fourier 00:13:34.910 --> 00:13:39.670 transform, we'll see that the z-transform for other values 00:13:39.670 --> 00:13:45.330 of z is the Fourier transform of the sequence with an 00:13:45.330 --> 00:13:46.800 exponential weighting. 00:13:46.800 --> 00:13:50.180 And let's see where that comes from. 00:13:50.180 --> 00:13:53.620 Here we have the general expression for the 00:13:53.620 --> 00:13:55.230 z-transform. 00:13:55.230 --> 00:13:59.680 And recognizing that z is a complex number which we're 00:13:59.680 --> 00:14:05.440 expressing in polar form as r e to the j omega, substituting 00:14:05.440 --> 00:14:10.400 that in, this summation now becomes x of n, r e to the j 00:14:10.400 --> 00:14:12.240 omega to the minus n. 00:14:12.240 --> 00:14:17.355 We can factor out these two terms, r to the minus n and e 00:14:17.355 --> 00:14:19.670 to the minus j omega n. 00:14:19.670 --> 00:14:26.060 And combining the r to the minus n with x of n and the e 00:14:26.060 --> 00:14:30.130 to the minus j omega n being treated separately, what we 00:14:30.130 --> 00:14:35.530 end up with is the summation that I have here. 00:14:35.530 --> 00:14:39.840 Well, what this says is that the z-transform, which is 00:14:39.840 --> 00:14:46.430 this, at z equal to r e to the j omega, is in fact the 00:14:46.430 --> 00:14:53.200 Fourier transform of what? 00:14:53.200 --> 00:14:58.670 It's the Fourier transform of x of n multiplied by r 00:14:58.670 --> 00:15:00.610 to the minus n. 00:15:00.610 --> 00:15:06.200 So that is the expression that we have here. 00:15:06.200 --> 00:15:10.150 And in continuous-time, we had the Laplace transform as the 00:15:10.150 --> 00:15:15.710 Fourier transform of x of t e to the minus sigma t. 00:15:15.710 --> 00:15:18.820 Here we have the Fourier transform of x of n r 00:15:18.820 --> 00:15:21.790 to the minus n. 00:15:21.790 --> 00:15:25.095 Now, something to just reflect on for a minute is-- 00:15:27.730 --> 00:15:31.980 because it tends to cause a little bit of problem with the 00:15:31.980 --> 00:15:35.640 algebra later on if you're attention isn't drawn to it, 00:15:35.640 --> 00:15:39.670 is that we're talking about multiplying x of n times r to 00:15:39.670 --> 00:15:41.410 the minus n. 00:15:41.410 --> 00:15:50.120 The question is, for r greater than 1, does r to the minus n 00:15:50.120 --> 00:15:55.220 increase exponentially as n increases or does it decrease? 00:15:55.220 --> 00:15:57.670 We're talking about r to the minus n. 00:15:57.670 --> 00:16:00.860 If r is greater than 1, if the magnitude of r is 00:16:00.860 --> 00:16:01.370 greater than 1. 00:16:01.370 --> 00:16:04.760 For example, if it's equal to 2, r to the minus n 00:16:04.760 --> 00:16:06.800 is 1/2 to the n. 00:16:06.800 --> 00:16:10.440 And so, in fact, that decreases exponentially. 00:16:10.440 --> 00:16:14.900 Or more generally, the larger r is, the faster r to the 00:16:14.900 --> 00:16:17.930 minus n decays with increasing n. 00:16:22.060 --> 00:16:28.490 Well, let's just look at some examples of the z-transform. 00:16:28.490 --> 00:16:33.930 And examples that I've picked, again, are examples directly 00:16:33.930 --> 00:16:34.820 out of the text. 00:16:34.820 --> 00:16:40.900 And so the details of the algebra you can look at more 00:16:40.900 --> 00:16:44.280 leisurely as you sit with the textbook. 00:16:44.280 --> 00:16:48.200 Let's consider, first of all, an exponential sequence x of n 00:16:48.200 --> 00:16:50.960 equals a to the n times the unit step. 00:16:50.960 --> 00:16:56.460 So 0 for negative time and an exponential for positive time. 00:16:56.460 --> 00:17:00.060 And the Fourier transform, as we've seen in earlier 00:17:00.060 --> 00:17:05.210 lectures, is 1 over 1 minus a e to the minus j omega. 00:17:05.210 --> 00:17:08.540 But this doesn't always converge. 00:17:08.540 --> 00:17:12.210 In particular, for convergence of the Fourier transform, we 00:17:12.210 --> 00:17:14.810 would require absolute summability of 00:17:14.810 --> 00:17:16.770 the original sequence. 00:17:16.770 --> 00:17:20.319 And that, in turn, requires that the magnitude of a be 00:17:20.319 --> 00:17:22.250 less than 1. 00:17:22.250 --> 00:17:25.730 So the Fourier transform is this, provided that the 00:17:25.730 --> 00:17:28.140 magnitude of a is less than 1. 00:17:28.140 --> 00:17:31.530 And what is the Fourier transform if the magnitude of 00:17:31.530 --> 00:17:34.330 a is not less than 1? 00:17:34.330 --> 00:17:36.380 Well, the answer is that, in that 00:17:36.380 --> 00:17:37.630 case, it doesn't converge. 00:17:40.310 --> 00:17:42.740 Now, let's look at the z-transform. 00:17:42.740 --> 00:17:47.770 The z-transform is the sum from minus infinity to plus 00:17:47.770 --> 00:17:54.420 infinity of a to the n z to the minus n 00:17:54.420 --> 00:17:56.480 times the unit step. 00:17:56.480 --> 00:17:59.130 The unit step will change the lower limit to 0. 00:17:59.130 --> 00:18:01.460 So it's the sum from 0 to infinity. 00:18:01.460 --> 00:18:08.720 And this is of the form a times z to the 00:18:08.720 --> 00:18:11.490 minus 1 to the n. 00:18:11.490 --> 00:18:14.740 So we're summing from 0 to infinity a times z to the 00:18:14.740 --> 00:18:16.360 minus 1 to the n. 00:18:16.360 --> 00:18:22.320 That sum is 1 over 1 minus a z to the minus 1. 00:18:22.320 --> 00:18:29.000 But in order for that sum to converge, we require that the 00:18:29.000 --> 00:18:35.710 magnitude of a times z to the minus 1 be less than 1. 00:18:35.710 --> 00:18:43.320 Now, the z-transform is the Fourier transform of the 00:18:43.320 --> 00:18:48.090 sequence a to the n times r to the minus n. 00:18:48.090 --> 00:18:55.530 And this statement about the z-transform converging is 00:18:55.530 --> 00:18:58.710 exactly identical to the statement that what we're 00:18:58.710 --> 00:19:04.280 requiring is that the magnitude of a times r to the 00:19:04.280 --> 00:19:11.350 minus 1 be less than 1, where this represents the 00:19:11.350 --> 00:19:17.440 exponential factor that we have that in effect is applied 00:19:17.440 --> 00:19:21.800 to the sequence, so that the Fourier transform becomes the 00:19:21.800 --> 00:19:23.430 z-transform. 00:19:23.430 --> 00:19:27.560 And so, if we put this condition, we can interpret 00:19:27.560 --> 00:19:32.100 this condition in exactly the same way that we interpret the 00:19:32.100 --> 00:19:36.920 condition on convergence of the Fourier transform. 00:19:36.920 --> 00:19:44.230 So from what we've worked out here then, what we have is the 00:19:44.230 --> 00:19:50.160 z-transform of a to the n times u of n is 1 over 1 minus 00:19:50.160 --> 00:19:53.600 a z to the minus 1. 00:19:53.600 --> 00:19:58.940 That works for any value of a provided that we pick the 00:19:58.940 --> 00:20:01.550 value of z correctly. 00:20:01.550 --> 00:20:05.120 In particular, we have to pick the set of values 00:20:05.120 --> 00:20:07.900 of z, so that what? 00:20:07.900 --> 00:20:11.870 So that the magnitude of a times z to the minus 1 00:20:11.870 --> 00:20:13.590 is less than 1. 00:20:13.590 --> 00:20:17.640 Or equivalently, so that the magnitude of z is greater than 00:20:17.640 --> 00:20:20.040 the magnitude of a. 00:20:20.040 --> 00:20:25.260 So associated with the z-transform of this sequence 00:20:25.260 --> 00:20:29.890 is this algebraic expression, and this set of values on z 00:20:29.890 --> 00:20:32.820 for which that algebraic expression is valid. 00:20:32.820 --> 00:20:38.600 And just as with the Laplace transform, this range of 00:20:38.600 --> 00:20:44.430 values is referred to as the region of convergence of the 00:20:44.430 --> 00:20:45.680 z-transform. 00:20:47.530 --> 00:20:52.950 Now, again, as we saw with the Laplace transform, it's 00:20:52.950 --> 00:20:58.590 important to recognize that in specifying or having worked 00:20:58.590 --> 00:21:02.870 out the z-transform of a sequence, it's not just the 00:21:02.870 --> 00:21:07.160 algebraic expression, but also the region of convergence 00:21:07.160 --> 00:21:11.700 that's required to uniquely specify it. 00:21:11.700 --> 00:21:15.420 To emphasize that further, here is Example 00:21:15.420 --> 00:21:17.500 10.2 from the text. 00:21:17.500 --> 00:21:20.910 And if you work that one through, what you find is 00:21:20.910 --> 00:21:26.190 that, algebraically, the z-transform of this sequence 00:21:26.190 --> 00:21:30.770 is 1 over 1 minus a z to the minus 1. 00:21:30.770 --> 00:21:34.240 Identical algebraically to what we had up here. 00:21:34.240 --> 00:21:42.360 But now with a region of convergence, which is the 00:21:42.360 --> 00:21:45.640 magnitude of z less than the magnitude of a. 00:21:45.640 --> 00:21:49.470 In contrast to this example, where the region of 00:21:49.470 --> 00:21:53.040 convergence was the magnitude of z greater than the 00:21:53.040 --> 00:21:56.160 magnitude of a. 00:21:56.160 --> 00:22:01.170 So again, it requires not just the algebraic expression, but 00:22:01.170 --> 00:22:06.230 also requires a specification of the region of convergence. 00:22:06.230 --> 00:22:12.380 And also, as with the Laplace transform, it's convenient in 00:22:12.380 --> 00:22:19.870 looking at the z-transform to represent it 00:22:19.870 --> 00:22:22.330 in the complex plane. 00:22:22.330 --> 00:22:25.710 In this case, the complex plane referred to as the 00:22:25.710 --> 00:22:28.790 z-plane, whereas in continuous-time when we talked 00:22:28.790 --> 00:22:31.400 about the Laplace transform, it was the s-plane. 00:22:31.400 --> 00:22:34.660 z, of course, because z is the complex variable in terms of 00:22:34.660 --> 00:22:37.910 which we're representing the z-transform. 00:22:37.910 --> 00:22:44.090 So we will be representing the z-transform in terms of 00:22:44.090 --> 00:22:48.400 representations in the complex plane, real part 00:22:48.400 --> 00:22:50.530 and imaginary part. 00:22:50.530 --> 00:22:54.630 But I've also identified a circle here. 00:22:54.630 --> 00:22:56.830 And you could wonder, well, what's the 00:22:56.830 --> 00:22:59.750 significance of the circle? 00:22:59.750 --> 00:23:05.710 Recall that in the discussion that we just came from, when 00:23:05.710 --> 00:23:09.380 we talked about the relationship between the 00:23:09.380 --> 00:23:13.460 z-transform and the Fourier transform, the z-transform 00:23:13.460 --> 00:23:16.290 reduces to the Fourier transform when the magnitude 00:23:16.290 --> 00:23:19.620 of z is equal to 1. 00:23:19.620 --> 00:23:21.420 The magnitude of z equal to 1 in the 00:23:21.420 --> 00:23:23.610 complex plane is a circle. 00:23:23.610 --> 00:23:28.880 And that circle, in fact, is a circle of radius 1. 00:23:28.880 --> 00:23:33.370 And so it's on this contour in the z-plane that the 00:23:33.370 --> 00:23:38.820 z-transform reduces to the Fourier transform. 00:23:38.820 --> 00:23:40.760 And we'll see some additional significance of 00:23:40.760 --> 00:23:43.550 that as we go along. 00:23:43.550 --> 00:23:47.010 Just again to emphasize the relationships and differences 00:23:47.010 --> 00:23:51.150 with continuous-time, with the Laplace transform it's the 00:23:51.150 --> 00:23:56.430 behavior in the s-plane on the j omega axis that corresponds 00:23:56.430 --> 00:23:58.440 to the Fourier transform. 00:23:58.440 --> 00:24:02.440 Here it's the behavior on the unit circle where the 00:24:02.440 --> 00:24:07.570 z-transform corresponds to the Fourier transform. 00:24:07.570 --> 00:24:08.930 Now, we'll be talking-- 00:24:08.930 --> 00:24:12.460 as we did with the Laplace transform, we'll be talking 00:24:12.460 --> 00:24:18.140 very often about transforms which are rational, and 00:24:18.140 --> 00:24:23.820 rational transforms as we'll see, represent systems which 00:24:23.820 --> 00:24:25.960 are characterized by linear constant coefficient 00:24:25.960 --> 00:24:27.700 difference equations. 00:24:27.700 --> 00:24:35.510 And so for the rational z-transforms, we'll again find 00:24:35.510 --> 00:24:38.530 it convenient to use a representation in terms of 00:24:38.530 --> 00:24:40.870 poles and zeroes in the z-plane. 00:24:40.870 --> 00:24:45.420 So let's look at our example as we've worked it out 00:24:45.420 --> 00:24:47.920 previously, Example 10.1. 00:24:47.920 --> 00:24:54.170 And with this sequence, the z-transform is 1 divided by 1 00:24:54.170 --> 00:24:56.360 minus a z to the minus 1. 00:24:56.360 --> 00:24:59.350 And we happen to have written it as a function of z 00:24:59.350 --> 00:25:00.950 to the minus 1. 00:25:00.950 --> 00:25:06.200 Clearly, we can rewrite this by multiplying numerator and 00:25:06.200 --> 00:25:09.900 denominator by z, and this would equivalently then be z 00:25:09.900 --> 00:25:13.470 divided by z minus a. 00:25:13.470 --> 00:25:19.220 And so if we were to represent this through a pole-zero plot, 00:25:19.220 --> 00:25:22.370 we would have a 0 at the origin corresponding to this 00:25:22.370 --> 00:25:27.340 factor and a pole at z equals a corresponding to the 00:25:27.340 --> 00:25:29.270 denominator factor. 00:25:29.270 --> 00:25:34.730 And so the pole-zero pattern for this is then a pole at z 00:25:34.730 --> 00:25:40.190 equal to a and a 0 at the origin. 00:25:40.190 --> 00:25:44.415 Now, let me just comment quickly about the fact that we 00:25:44.415 --> 00:25:48.440 had written this as 1 over 1 minus a z to the minus 1, and 00:25:48.440 --> 00:25:51.730 that seems kind of strange because perhaps we should have 00:25:51.730 --> 00:25:54.160 multiplied through by z. 00:25:54.160 --> 00:25:58.350 Let me just indicate that as you'll see as you work 00:25:58.350 --> 00:26:03.680 examples, it's very typical for the z-transform to come 00:26:03.680 --> 00:26:06.910 out as a function of z to the minus 1. 00:26:06.910 --> 00:26:11.820 And so very typically, you'll get to recognize that things 00:26:11.820 --> 00:26:15.980 will be expressed in terms of factors involving terms like 1 00:26:15.980 --> 00:26:20.380 minus a z to the minus 1, rather than factors of the 00:26:20.380 --> 00:26:23.320 form z minus a. 00:26:23.320 --> 00:26:28.740 Well, here is the one example that we had referred to. 00:26:28.740 --> 00:26:33.910 And if we consider another example, the other example, 00:26:33.910 --> 00:26:40.780 which was example 10.2 consists of an algebraic 00:26:40.780 --> 00:26:43.640 expression as I indicate here. 00:26:43.640 --> 00:26:47.660 But its region of validity is the magnitude of z less than 00:26:47.660 --> 00:26:49.240 the magnitude of a. 00:26:49.240 --> 00:26:53.850 And that corresponds to the same pole-zero plot, but a 00:26:53.850 --> 00:26:59.230 region of convergence which is inside this circle. 00:26:59.230 --> 00:27:05.400 Whereas, in the previous case, with the pole-zero plot, we 00:27:05.400 --> 00:27:09.010 had a region of convergence which was for the magnitude of 00:27:09.010 --> 00:27:11.740 z greater than the magnitude of a. 00:27:11.740 --> 00:27:15.610 So these two examples, this one and the other one, have 00:27:15.610 --> 00:27:18.350 exactly the same pole-zero pattern and they're 00:27:18.350 --> 00:27:22.530 distinguished by their region of convergence. 00:27:22.530 --> 00:27:29.210 Now, notice incidentally that in this particular case, the 00:27:29.210 --> 00:27:32.810 region of convergence includes the unit circle provided that 00:27:32.810 --> 00:27:35.240 the magnitude of a is less than 1. 00:27:35.240 --> 00:27:41.470 And so, in fact, that would say that the sequence has a 00:27:41.470 --> 00:27:43.460 Fourier transform that converges. 00:27:43.460 --> 00:27:47.410 Namely, with the magnitude of z equal to 1. 00:27:47.410 --> 00:27:52.020 Whereas, in this example, the region of convergence does not 00:27:52.020 --> 00:27:54.140 include the unit circle. 00:27:54.140 --> 00:27:57.540 And so, in fact, we cannot look at x of z for the 00:27:57.540 --> 00:27:59.570 magnitude of z equal to 1. 00:27:59.570 --> 00:28:03.850 And so this example, with the magnitude of a less than 1, 00:28:03.850 --> 00:28:07.950 does not have a Fourier transform that converges. 00:28:07.950 --> 00:28:12.160 Well, assuming that the magnitude of a is less than 1 00:28:12.160 --> 00:28:16.010 and the Fourier transform converges, we can, in fact, 00:28:16.010 --> 00:28:22.080 look at the Fourier transform by observing what happens as 00:28:22.080 --> 00:28:24.460 we go around the unit circle. 00:28:24.460 --> 00:28:26.910 We had seen this with the Laplace transform in terms of 00:28:26.910 --> 00:28:31.010 observing what happened as we move along the j omega axis. 00:28:31.010 --> 00:28:39.620 And here again, we can use the vectors as we trace out the 00:28:39.620 --> 00:28:41.060 unit circle. 00:28:41.060 --> 00:28:43.790 And in particular, what we would be looking at in this 00:28:43.790 --> 00:28:51.510 case is the ratio of the zero vector to the pole vector. 00:28:51.510 --> 00:28:53.960 For example, if we were looking at the magnitude of 00:28:53.960 --> 00:28:58.120 the z-transform, the magnitude of the z-transform would be 00:28:58.120 --> 00:29:02.520 the ratio of the length of this vector to the length of 00:29:02.520 --> 00:29:04.200 this vector. 00:29:04.200 --> 00:29:08.130 And to observe the Fourier transform, we would observe 00:29:08.130 --> 00:29:13.090 how those vectors change in length as we move around the 00:29:13.090 --> 00:29:15.030 unit circle. 00:29:15.030 --> 00:29:18.030 And as we move around the unit circle, what we would trace 00:29:18.030 --> 00:29:21.730 out in terms of the ratio of the lengths of those vectors 00:29:21.730 --> 00:29:23.835 is the Fourier transform. 00:29:26.910 --> 00:29:36.690 Well, let's focus on that also in the context of a slightly 00:29:36.690 --> 00:29:37.855 different z-transform. 00:29:37.855 --> 00:29:43.260 In the z-transform here as we'll see in a later lecture, 00:29:43.260 --> 00:29:45.800 is the z-transform associated with a second 00:29:45.800 --> 00:29:48.070 order difference equation. 00:29:48.070 --> 00:29:52.460 It has a denominator factor which has two poles 00:29:52.460 --> 00:29:54.810 associated with it. 00:29:54.810 --> 00:30:00.730 And so here, if we assumed that the Fourier transform of 00:30:00.730 --> 00:30:05.360 the associated sequence converged, then again we would 00:30:05.360 --> 00:30:08.080 look at the behavior of this as we moved 00:30:08.080 --> 00:30:10.040 around the unit circle. 00:30:10.040 --> 00:30:14.730 And the ratio of the lengths of the appropriate vectors 00:30:14.730 --> 00:30:19.060 would describe for us the frequency response. 00:30:19.060 --> 00:30:21.220 I'm sorry, the Fourier transform. 00:30:21.220 --> 00:30:27.800 So the Fourier transform magnitude would consist of the 00:30:27.800 --> 00:30:32.830 ratio of the lengths of the zero vectors divided by the 00:30:32.830 --> 00:30:34.930 lengths of the pole vectors. 00:30:34.930 --> 00:30:39.350 And one thing that we observe is that as we move in 00:30:39.350 --> 00:30:44.290 frequency omega in the vicinity of this pole, this 00:30:44.290 --> 00:30:49.260 pole vector, in fact, reaches a minimum length. 00:30:49.260 --> 00:30:54.170 That would mean that it's reciprocal would be maximum. 00:30:54.170 --> 00:30:57.290 And then, as we sweep past, the lengths of these two 00:30:57.290 --> 00:30:58.380 vectors would increase. 00:30:58.380 --> 00:31:02.040 The zero vectors, of course, would retain the same length 00:31:02.040 --> 00:31:05.940 no matter where we were on the unit circle. 00:31:05.940 --> 00:31:08.500 So, in fact, if we looked at the Fourier transform 00:31:08.500 --> 00:31:12.320 associated with this pole-zero pattern, if this was, for 00:31:12.320 --> 00:31:20.760 example, represented the z-transform of the impulse 00:31:20.760 --> 00:31:24.220 response or a linear time-invariant system, the 00:31:24.220 --> 00:31:27.040 corresponding frequency response would be what I 00:31:27.040 --> 00:31:28.970 plotted out below. 00:31:28.970 --> 00:31:31.810 And so it would peak. 00:31:31.810 --> 00:31:35.560 And in fact, where it would peak is in the vicinity of the 00:31:35.560 --> 00:31:42.220 frequency location of the pole as I indicate up here. 00:31:45.950 --> 00:31:51.530 So as we sweep past this pole then, in fact, this Fourier 00:31:51.530 --> 00:31:52.560 transform . 00:31:52.560 --> 00:31:54.550 Peaks. 00:31:54.550 --> 00:31:59.060 Well, this notion of looking at the frequency response as 00:31:59.060 --> 00:32:04.210 we move around the unit circle is a very important notion. 00:32:04.210 --> 00:32:06.610 And it's important to recognize it's the unit circle 00:32:06.610 --> 00:32:10.060 we're talking about here, whereas before we were talking 00:32:10.060 --> 00:32:12.740 about the j omega axis. 00:32:12.740 --> 00:32:18.470 And to emphasize this further, let me just show this example. 00:32:18.470 --> 00:32:22.900 And in fact, the previous example with the computer 00:32:22.900 --> 00:32:26.730 displays, so that we can see the frequency response as it 00:32:26.730 --> 00:32:31.100 sweeps out as we go around the unit circle. 00:32:31.100 --> 00:32:36.350 So here we have the pole-zero pattern for the 00:32:36.350 --> 00:32:38.640 second order example. 00:32:38.640 --> 00:32:42.780 And to generate the Fourier transform, we want to look at 00:32:42.780 --> 00:32:46.360 the behavior of the pole and zero vectors as we move around 00:32:46.360 --> 00:32:47.970 the unit circle. 00:32:47.970 --> 00:32:51.200 So first, let's display the vectors. 00:32:51.200 --> 00:32:55.030 And here we have them displayed to the point 00:32:55.030 --> 00:32:58.250 corresponding to zero frequency. 00:32:58.250 --> 00:33:03.750 And the magnitude of the Fourier transform will be, as 00:33:03.750 --> 00:33:06.650 we discussed, the magnitude of the length of the zero vector 00:33:06.650 --> 00:33:08.380 is divided by the magnitude of the 00:33:08.380 --> 00:33:10.580 length of the pole vectors. 00:33:10.580 --> 00:33:14.760 Shown below will be the Fourier transform. 00:33:14.760 --> 00:33:18.440 And we have the Fourier transform displayed here from 00:33:18.440 --> 00:33:23.300 0 to 2 pi, rather than from minus pi to pi as it was 00:33:23.300 --> 00:33:25.640 displayed in the transparency. 00:33:25.640 --> 00:33:29.130 Because of the periodicity of the Fourier transform, both of 00:33:29.130 --> 00:33:32.440 those are equivalent. 00:33:32.440 --> 00:33:38.400 Now we're sweeping away from omega equals 0 and the lengths 00:33:38.400 --> 00:33:40.590 of the pole vectors have changed. 00:33:40.590 --> 00:33:43.100 And that, of course, generates a change 00:33:43.100 --> 00:33:46.390 in the Fourier transform. 00:33:46.390 --> 00:33:51.530 And as we continued the process further, if we 00:33:51.530 --> 00:33:57.440 increase frequency, as we sweep closer to the location 00:33:57.440 --> 00:34:02.060 of the pole, the pole vector decreases in length 00:34:02.060 --> 00:34:03.370 dramatically. 00:34:03.370 --> 00:34:07.400 And that generates a residence in the Fourier transform, very 00:34:07.400 --> 00:34:11.040 similar to what we saw in continuous-time. 00:34:11.040 --> 00:34:16.875 Now as we continue to sweep further, what will happen is 00:34:16.875 --> 00:34:19.949 that that pole vector will begin to 00:34:19.949 --> 00:34:22.050 increase in length again. 00:34:22.050 --> 00:34:25.300 And so, in fact, the magnitude of the Fourier 00:34:25.300 --> 00:34:28.730 transform will decrease. 00:34:28.730 --> 00:34:32.889 And we see that here as we sweep toward 00:34:32.889 --> 00:34:35.250 omega equal to pi. 00:34:35.250 --> 00:34:39.830 Now, notice in this process that the length of the zero 00:34:39.830 --> 00:34:44.929 vectors has stayed the same because of the fact that the 00:34:44.929 --> 00:34:49.429 zeroes are at the origin, and no matter where we are in the 00:34:49.429 --> 00:34:52.420 unit circle, the length of those vectors is unity. 00:34:52.420 --> 00:34:55.650 So they don't influence in this example the magnitude, 00:34:55.650 --> 00:34:59.420 but they would, of course, influence the phase. 00:34:59.420 --> 00:35:02.600 Now we want to continue sweeping from omega equal to 00:35:02.600 --> 00:35:04.960 pi around to 2 pi. 00:35:04.960 --> 00:35:08.280 And because of the symmetry in the Fourier transform, what we 00:35:08.280 --> 00:35:12.310 will see in the magnitude is identical to what we would see 00:35:12.310 --> 00:35:16.880 if we swept from omega equal to pi back clockwise to 00:35:16.880 --> 00:35:19.630 omega equals 0. 00:35:19.630 --> 00:35:23.670 In particular now, as we're increasing frequency, notice 00:35:23.670 --> 00:35:27.550 that the length of the pole vector associated with the 00:35:27.550 --> 00:35:30.740 lower half plane pole is decreasing. 00:35:30.740 --> 00:35:35.370 And so, in fact, that corresponds to generating a 00:35:35.370 --> 00:35:39.540 resonance as we sweep past that pole 00:35:39.540 --> 00:35:42.630 location as we are here. 00:35:42.630 --> 00:35:46.380 And then finally, that pole vector increases in length as 00:35:46.380 --> 00:35:49.760 we begin to approach omega equal to 2 pi. 00:35:49.760 --> 00:35:53.700 Or equivalently, as we approach omega equal to 0. 00:35:59.400 --> 00:36:05.610 Now finally, let's also look at the Fourier transform 00:36:05.610 --> 00:36:08.670 associated with the first order example that we 00:36:08.670 --> 00:36:11.370 discussed earlier in the lecture. 00:36:11.370 --> 00:36:16.140 And so what we'll want to look at is the Fourier transform as 00:36:16.140 --> 00:36:18.990 the pole and zero vectors change. 00:36:18.990 --> 00:36:22.150 Once again, the Fourier transform will be displayed on 00:36:22.150 --> 00:36:25.950 a scale from 0 to 2 pi, a frequency scale from 0 to 2 00:36:25.950 --> 00:36:29.060 pi, rather than minus pi to pi. 00:36:29.060 --> 00:36:32.600 And we want to observe the pole and zero vectors as we 00:36:32.600 --> 00:36:34.010 sweep around the unit circle. 00:36:38.670 --> 00:36:42.420 We display first the pole and zero vectors at 00:36:42.420 --> 00:36:44.960 omega equal to 0. 00:36:44.960 --> 00:36:52.140 And as the frequency increases, the pole vector 00:36:52.140 --> 00:36:53.910 increases in length. 00:36:53.910 --> 00:36:57.640 The zero vector, since the zero is at the origin, will 00:36:57.640 --> 00:37:00.190 have constant length no matter where we 00:37:00.190 --> 00:37:01.430 are on the unit circle. 00:37:01.430 --> 00:37:04.470 Although it would affect the phase, which we are not 00:37:04.470 --> 00:37:06.000 displaying here. 00:37:06.000 --> 00:37:09.520 And so the principle effect, the only effect really on the 00:37:09.520 --> 00:37:12.540 magnitude, is due to the pole vector. 00:37:12.540 --> 00:37:17.470 As the frequency continues to increase, the pole vector 00:37:17.470 --> 00:37:21.890 increases in length, monotonically in fact. 00:37:21.890 --> 00:37:25.190 And so that means that the magnitude of the Fourier 00:37:25.190 --> 00:37:30.700 transform will decrease monotonically until we get 00:37:30.700 --> 00:37:33.030 past omega equal to pi. 00:37:44.820 --> 00:37:48.910 Here we are now at omega equal to pi. 00:37:48.910 --> 00:37:54.360 And when we continue sweeping past this frequency around to 00:37:54.360 --> 00:37:59.860 2 pi, then we will see basically the same curve swept 00:37:59.860 --> 00:38:01.320 out in reverse. 00:38:01.320 --> 00:38:04.210 Since because of the symmetry, again, of the Fourier 00:38:04.210 --> 00:38:10.190 transform magnitude, sweeping from pi to 2 pi is going to be 00:38:10.190 --> 00:38:13.740 equivalent with regard to the magnitude to sweeping 00:38:13.740 --> 00:38:17.400 from pi back to 0. 00:38:17.400 --> 00:38:21.730 And so now the pole vector begins to decrease in length 00:38:21.730 --> 00:38:25.140 and correspondingly, the magnitude of the Fourier 00:38:25.140 --> 00:38:29.070 transform will increase. 00:38:29.070 --> 00:38:33.130 And that will continue until we get around to omega equal 00:38:33.130 --> 00:38:36.265 to 2 pi, which is equivalent, of course, to 00:38:36.265 --> 00:38:38.020 omega equal to 0. 00:38:38.020 --> 00:38:41.430 And obviously, if we continue to sweep around again, we 00:38:41.430 --> 00:38:46.500 would simply trace out other periods associated with the 00:38:46.500 --> 00:38:47.750 Fourier transform. 00:38:55.550 --> 00:39:00.270 Well, that hopefully gives you kind of some feel for the 00:39:00.270 --> 00:39:02.760 notion of sweeping around the unit circle. 00:39:02.760 --> 00:39:06.430 And of course, you can see that because the circle is 00:39:06.430 --> 00:39:09.520 periodic as we go around and around, of course, what we'll 00:39:09.520 --> 00:39:13.130 get is a periodic Fourier transform, which is the way 00:39:13.130 --> 00:39:16.990 Fourier transforms are supposed to be. 00:39:16.990 --> 00:39:26.830 Now, just as with the Laplace transform, the region of 00:39:26.830 --> 00:39:30.780 convergence of the z-transform, as we've seen in 00:39:30.780 --> 00:39:36.240 this example, is a very important part of the 00:39:36.240 --> 00:39:39.430 specification of the z-transform. 00:39:39.430 --> 00:39:42.690 And we can, in talking about sequences and their 00:39:42.690 --> 00:39:46.100 transforms, either specify the region of convergence 00:39:46.100 --> 00:39:49.730 implicitly, or we can specify it explicitly. 00:39:49.730 --> 00:39:52.870 We can, for example, say what it is, as let's say the 00:39:52.870 --> 00:39:56.020 magnitude of z being greater than the magnitude of a. 00:39:56.020 --> 00:40:00.830 Or we can recognize that the region of convergence has 00:40:00.830 --> 00:40:06.020 certain constraints associated with certain properties of the 00:40:06.020 --> 00:40:07.480 time function. 00:40:07.480 --> 00:40:14.370 And in particular, there are some important properties of 00:40:14.370 --> 00:40:20.060 the region of convergence which allow us, given that we 00:40:20.060 --> 00:40:23.540 know certain characteristics of the time function, to then 00:40:23.540 --> 00:40:26.440 identify the region of convergence by looking at the 00:40:26.440 --> 00:40:27.940 pole-zero pattern. 00:40:27.940 --> 00:40:32.420 For example, we recognize that the region of convergence does 00:40:32.420 --> 00:40:37.800 not contain any poles because of the fact that at poles, the 00:40:37.800 --> 00:40:42.010 z-transform, in fact, blows up and, of course, can't converge 00:40:42.010 --> 00:40:44.060 at that point. 00:40:44.060 --> 00:40:48.480 Furthermore, the region of convergence consists of a ring 00:40:48.480 --> 00:40:52.420 in the z-plane centered about the origin. 00:40:52.420 --> 00:40:55.750 Recall that with the Laplace transform, the region of 00:40:55.750 --> 00:40:59.220 convergence consisted of strips in the s-plane. 00:41:01.720 --> 00:41:04.380 With the z-transform, the region of convergence consists 00:41:04.380 --> 00:41:07.880 of a ring, basically because of the fact that the region of 00:41:07.880 --> 00:41:12.490 convergence is dependent on the magnitude of z. 00:41:12.490 --> 00:41:14.990 Whereas, with the Laplace transform, the region of 00:41:14.990 --> 00:41:18.310 convergence was dependent on the real part of s. 00:41:18.310 --> 00:41:21.510 The fact that it's the magnitude of z says, in 00:41:21.510 --> 00:41:25.580 effect, that all values of z that have the same magnitude 00:41:25.580 --> 00:41:27.350 lie on a circle. 00:41:27.350 --> 00:41:29.900 And so the region of convergence you would expect 00:41:29.900 --> 00:41:36.420 to be a concentric ring in the z-plane. 00:41:36.420 --> 00:41:40.000 Furthermore, as we've already talked about and exploited 00:41:40.000 --> 00:41:44.130 actually, convergence of the Fourier transform is 00:41:44.130 --> 00:41:47.720 equivalent to the statement that the region of convergence 00:41:47.720 --> 00:41:49.530 includes the unit circle in the z-plane. 00:41:52.130 --> 00:41:57.420 Now, we can also associate the region of convergence with 00:41:57.420 --> 00:42:01.700 issues about whether the sequence is of finite duration 00:42:01.700 --> 00:42:04.610 or right-sided or left-sided. 00:42:04.610 --> 00:42:09.920 And let me sort of quickly indicate again what the style 00:42:09.920 --> 00:42:12.550 of the argument is. 00:42:12.550 --> 00:42:17.840 If we have a finite duration sequence, so that the sequence 00:42:17.840 --> 00:42:20.580 is absolutely summable, and therefore has a Fourier 00:42:20.580 --> 00:42:25.590 transform that converges, then because of the fact that it's 00:42:25.590 --> 00:42:30.210 0 outside some interval, I can multiply it by an 00:42:30.210 --> 00:42:35.070 exponentially decaying sequence or by an 00:42:35.070 --> 00:42:37.340 exponentially growing sequence. 00:42:37.340 --> 00:42:41.170 And since I'm only doing this over a finite interval, no 00:42:41.170 --> 00:42:45.030 matter how I choose that exponential, we'll end up with 00:42:45.030 --> 00:42:47.130 an absolutely summable product. 00:42:47.130 --> 00:42:51.610 So if x of n is a finite duration, then in fact the 00:42:51.610 --> 00:42:56.590 region of convergence is the entire z-plane, possibly with 00:42:56.590 --> 00:42:58.935 the exception of the origin or infinity. 00:43:01.820 --> 00:43:06.490 On the other hand, if the sequence is a right-sided 00:43:06.490 --> 00:43:12.390 sequence, then we have to be careful that we don't multiply 00:43:12.390 --> 00:43:17.290 by an exponential that grows too fast for positive time. 00:43:17.290 --> 00:43:22.300 Or equivalently, we might have to choose the exponential so 00:43:22.300 --> 00:43:26.720 that it decays sufficiently fast for positive time. 00:43:26.720 --> 00:43:31.240 As a consequence of that, for a right-sided sequence, if we 00:43:31.240 --> 00:43:36.730 have a value which is in the region of convergence, as long 00:43:36.730 --> 00:43:40.950 as I multiply by exponentials that decay faster than that 00:43:40.950 --> 00:43:47.020 for positive time, then I'll also have convergence. 00:43:47.020 --> 00:43:50.630 In other words, all finite values of z for which the 00:43:50.630 --> 00:43:53.710 magnitude of z is greater than this, so that the exponentials 00:43:53.710 --> 00:43:56.760 die off even faster will also be in the region of 00:43:56.760 --> 00:43:59.300 convergence. 00:43:59.300 --> 00:44:01.810 If we combine that statement with the fact that there are 00:44:01.810 --> 00:44:05.320 no poles in the region of convergence, then we end up 00:44:05.320 --> 00:44:07.150 with a statement similar to what we had 00:44:07.150 --> 00:44:09.000 with the Laplace transform. 00:44:09.000 --> 00:44:14.430 Here, the statement is that if the sequence is right-sided, 00:44:14.430 --> 00:44:19.160 then the region of convergence has to be outside the 00:44:19.160 --> 00:44:20.560 outermost pole. 00:44:20.560 --> 00:44:23.610 Essentially, because it has to be outside someplace and can't 00:44:23.610 --> 00:44:24.860 include any poles. 00:44:27.220 --> 00:44:36.110 Finally, if we have a left-sided sequence, then if 00:44:36.110 --> 00:44:39.230 we have a value which is in the region of convergence, all 00:44:39.230 --> 00:44:42.390 values for which the magnitude of z is less than that will 00:44:42.390 --> 00:44:44.630 also be in the region of convergence. 00:44:44.630 --> 00:44:48.680 Or if x of z is rational, then the region of convergence must 00:44:48.680 --> 00:44:51.130 be inside the innermost pole. 00:44:51.130 --> 00:44:56.610 And finally, if we have a two-sided sequence, then 00:44:56.610 --> 00:45:00.130 there's kind of a balance between the exponential factor 00:45:00.130 --> 00:45:01.590 that we use. 00:45:01.590 --> 00:45:05.980 And so in that case, then the region of convergence will be 00:45:05.980 --> 00:45:09.880 a ring in the z-plane, and essentially will extend 00:45:09.880 --> 00:45:12.880 outward to a pole and inward to a pole. 00:45:12.880 --> 00:45:19.700 So if we had an algebraic expression, let's say as we 00:45:19.700 --> 00:45:28.150 have here, then we could associate with that a region 00:45:28.150 --> 00:45:30.980 of convergence outside this pole. 00:45:30.980 --> 00:45:34.770 And that would correspond to a right-sided sequence. 00:45:34.770 --> 00:45:40.010 Or we can associate with it a region of convergence, which 00:45:40.010 --> 00:45:45.120 is inside the innermost pole. 00:45:45.120 --> 00:45:48.440 And that would correspond to a left-sided sequence. 00:45:48.440 --> 00:45:53.890 And the third and only other possibility is a region of 00:45:53.890 --> 00:45:58.330 convergence which lies between these two poles. 00:45:58.330 --> 00:46:02.980 And that would then correspond to a two-sided sequence. 00:46:02.980 --> 00:46:05.070 And notice incidentally because of where I've placed 00:46:05.070 --> 00:46:08.700 these poles, that this is the only one for which the region 00:46:08.700 --> 00:46:11.130 of convergence includes the unit circle. 00:46:11.130 --> 00:46:14.670 In other words, it's the only one for which the Fourier 00:46:14.670 --> 00:46:17.220 transform converges. 00:46:17.220 --> 00:46:19.830 OK, now we've moved through that fairly quickly. 00:46:19.830 --> 00:46:24.560 And I've emphasized the fact that it parallels very closely 00:46:24.560 --> 00:46:27.570 what we did with the Laplace transform. 00:46:27.570 --> 00:46:32.140 What I'd like to do is just conclude with a discussion of 00:46:32.140 --> 00:46:37.600 how we get the time function back again when we have the 00:46:37.600 --> 00:46:42.150 z-transform including its region of convergence. 00:46:42.150 --> 00:46:45.780 Well, we can, first of all, develop a more or less formal 00:46:45.780 --> 00:46:46.580 expression. 00:46:46.580 --> 00:46:50.910 And the algebra for this is gone through in the text, and 00:46:50.910 --> 00:46:53.300 you went through something similar to this with the 00:46:53.300 --> 00:46:56.510 Laplace transform in the video course manual. 00:46:56.510 --> 00:46:59.350 So I won't carry through the details. 00:46:59.350 --> 00:47:04.130 But basically, what we can use to develop a formal expression 00:47:04.130 --> 00:47:08.320 is the fact that the z-transform is the Fourier 00:47:08.320 --> 00:47:11.550 transform of the sequence exponentially weighted. 00:47:11.550 --> 00:47:16.550 So we can apply the inverse transform to that, and that 00:47:16.550 --> 00:47:19.720 gives us not x of n, but x of n exponentially weighted. 00:47:19.720 --> 00:47:24.540 And if we track that through, then what we'll end up with is 00:47:24.540 --> 00:47:26.020 an expression. 00:47:26.020 --> 00:47:28.950 After we've taken care of a few of the epsilons and 00:47:28.950 --> 00:47:33.410 deltas, we'll end up with an expression that expresses 00:47:33.410 --> 00:47:41.000 formally the sequence x of n in terms of the z-transform, 00:47:41.000 --> 00:47:43.830 where this, in fact, is a contour integral in the 00:47:43.830 --> 00:47:45.390 complex plane. 00:47:45.390 --> 00:47:49.300 And so there's a formal expression, just as there's a 00:47:49.300 --> 00:47:52.280 formal expression for the Laplace transform. 00:47:52.280 --> 00:47:58.900 But in fact, the more typical procedure is to use 00:47:58.900 --> 00:48:03.630 essentially transformed pairs that we know together with the 00:48:03.630 --> 00:48:07.350 idea of using a partial fraction expansion. 00:48:07.350 --> 00:48:15.460 So if we had a z-transform as I indicate here, and if we 00:48:15.460 --> 00:48:21.120 expand it out in a partial fraction expansion, then we 00:48:21.120 --> 00:48:26.580 can recognize, as we did in a similar style with the Laplace 00:48:26.580 --> 00:48:27.860 transform-- 00:48:27.860 --> 00:48:33.090 we can recognize that each term, together with the 00:48:33.090 --> 00:48:38.450 identified region of convergence corresponds to an 00:48:38.450 --> 00:48:40.240 exponential factor. 00:48:40.240 --> 00:48:44.420 And so this term, together with the fact we know that the 00:48:44.420 --> 00:48:49.960 magnitude of z must be greater than 2, allows us to recognize 00:48:49.960 --> 00:48:54.800 this as similar to the Example 10.1. 00:48:54.800 --> 00:48:58.800 And in particular then, the sequence associated with that 00:48:58.800 --> 00:49:00.970 is what I indicate here. 00:49:00.970 --> 00:49:04.230 And for the second term, the sequence is 00:49:04.230 --> 00:49:06.070 what I indicate here. 00:49:06.070 --> 00:49:11.260 So what we're simply doing is using the fact that we've 00:49:11.260 --> 00:49:15.850 worked out the example going one direction before, and now 00:49:15.850 --> 00:49:19.350 we use that together with the partial fraction expansion to 00:49:19.350 --> 00:49:23.240 get the individual sequences back again, and 00:49:23.240 --> 00:49:24.720 then add them together. 00:49:24.720 --> 00:49:27.790 There's one other method which I'll just point to, which is 00:49:27.790 --> 00:49:31.370 also elaborated on a little more in the text. 00:49:31.370 --> 00:49:35.860 But it's kind of the idea of developing the inverse 00:49:35.860 --> 00:49:42.570 z-transform by recognizing that this z-transform formula, 00:49:42.570 --> 00:49:46.900 in fact, is a power series. 00:49:46.900 --> 00:49:54.130 So if we take x of z and expand it in a power series, 00:49:54.130 --> 00:49:59.640 then we can pick off the values of x of n by 00:49:59.640 --> 00:50:01.350 identifying the individual 00:50:01.350 --> 00:50:02.945 coefficients in this expansion. 00:50:06.070 --> 00:50:11.780 And so by simply doing long division, for example, we can 00:50:11.780 --> 00:50:14.750 also get the inverse transform. 00:50:14.750 --> 00:50:17.650 And that, by the way, is very useful. 00:50:17.650 --> 00:50:23.330 Particularly if we want to get the inverse z-transform for a 00:50:23.330 --> 00:50:27.700 z-transform expression, which is not rational. 00:50:27.700 --> 00:50:30.330 Now we've moved through this fairly quickly. 00:50:30.330 --> 00:50:33.080 On the other hand, I've stressed that it's very 00:50:33.080 --> 00:50:36.220 similar to what we went through for the Laplace 00:50:36.220 --> 00:50:41.320 transform, except for a very important difference. 00:50:41.320 --> 00:50:44.390 The principal difference really being that with the 00:50:44.390 --> 00:50:48.580 Laplace transform, it was the j omega axis in the s-plane 00:50:48.580 --> 00:50:51.300 that we focused attention on when we were thinking about 00:50:51.300 --> 00:50:52.970 the Fourier transform. 00:50:52.970 --> 00:50:56.230 Here, the unit circle in the z-plane plays 00:50:56.230 --> 00:50:58.580 an important role. 00:50:58.580 --> 00:51:01.900 What we'll see when we continue this in the next 00:51:01.900 --> 00:51:07.250 lecture is that there are properties of the z-transform, 00:51:07.250 --> 00:51:10.630 just as there were properties of the Laplace transform. 00:51:10.630 --> 00:51:16.910 And those properties allow us to develop and exploit the 00:51:16.910 --> 00:51:21.860 z-transform in the context of systems describable by linear 00:51:21.860 --> 00:51:24.250 constant coefficient difference equations. 00:51:24.250 --> 00:51:28.420 So in the next lecture, we'll focus on some properties of 00:51:28.420 --> 00:51:31.970 the z-transform, and then we'll see how to use those 00:51:31.970 --> 00:51:37.030 properties to help us in getting further insight and 00:51:37.030 --> 00:51:38.870 working with systems describable 00:51:38.870 --> 00:51:40.580 by difference equations. 00:51:40.580 --> 00:51:41.830 Thank you.