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 --> 00:00:20.525 ocw.mit.edu. 00:00:20.525 --> 00:00:55.510 [MUSIC PLAYING] 00:00:55.510 --> 00:00:57.900 PROFESSOR: Over the last several lectures, we developed 00:00:57.900 --> 00:01:03.320 the Fourier representation for continuous-time signals. 00:01:03.320 --> 00:01:06.760 What I'd now like to do is develop a similar 00:01:06.760 --> 00:01:09.230 representation for discrete-time. 00:01:09.230 --> 00:01:12.940 And let me begin the discussion by reminding you of 00:01:12.940 --> 00:01:15.280 what our basic motivation was. 00:01:15.280 --> 00:01:19.780 The idea is that what we wanted to do was exploit the 00:01:19.780 --> 00:01:23.640 properties of linearity and time invariance for linear 00:01:23.640 --> 00:01:26.330 time-invariant systems. 00:01:26.330 --> 00:01:31.860 So in the case of linear time-invariant systems, the 00:01:31.860 --> 00:01:38.650 basic idea was to consider decomposing the input as a 00:01:38.650 --> 00:01:42.730 linear combination of basic inputs. 00:01:42.730 --> 00:01:46.620 And then, because of linearity, the output could be 00:01:46.620 --> 00:01:49.960 expressed as a linear combination of corresponding 00:01:49.960 --> 00:01:59.610 outputs where psi sub i is the output due to phi sub i. 00:01:59.610 --> 00:02:03.440 So basically, what we attempted to do was decompose 00:02:03.440 --> 00:02:07.710 the input, and then reconstruct the output through 00:02:07.710 --> 00:02:13.620 a linear combination of the outputs to those basic inputs. 00:02:13.620 --> 00:02:19.020 We then focused on the notion of choosing the basic inputs 00:02:19.020 --> 00:02:20.890 with two criteria in mind. 00:02:20.890 --> 00:02:25.900 One was to choose them so that a broad class of signals could 00:02:25.900 --> 00:02:30.020 be constructed out of those basic inputs. 00:02:30.020 --> 00:02:35.000 And the second was to choose the basic inputs, so that the 00:02:35.000 --> 00:02:38.800 response to those was easy to compute. 00:02:38.800 --> 00:02:41.570 And as you recall, one representation that we ended 00:02:41.570 --> 00:02:47.050 up with, with those basic criteria in mind, was the 00:02:47.050 --> 00:02:49.500 representation through convolution. 00:02:49.500 --> 00:02:52.610 And then in beginning the discussion of the Fourier 00:02:52.610 --> 00:02:55.980 representation of continuous-time signals, we 00:02:55.980 --> 00:03:01.880 chose as another set of basic inputs complex exponentials. 00:03:01.880 --> 00:03:07.860 So for continuous-time, we chose a set of basic inputs 00:03:07.860 --> 00:03:10.380 which were complex exponentials. 00:03:10.380 --> 00:03:15.470 The motivation there was the fact that the complex 00:03:15.470 --> 00:03:18.950 exponentials have what we refer to as the 00:03:18.950 --> 00:03:20.940 eigenfunction property. 00:03:20.940 --> 00:03:24.190 Namely, if we put complex exponentials into our 00:03:24.190 --> 00:03:29.540 continuous-time systems, then the output is a complex 00:03:29.540 --> 00:03:34.830 exponential of the same form with only a 00:03:34.830 --> 00:03:37.980 change in complex amplitude. 00:03:37.980 --> 00:03:42.450 And that change in complex amplitude is what we referred 00:03:42.450 --> 00:03:48.930 to as the frequency response of the system. 00:03:48.930 --> 00:03:52.040 And also, by the way, as it developed later, that 00:03:52.040 --> 00:03:55.470 frequency response, as you should now recognize from this 00:03:55.470 --> 00:03:59.380 expression, is in fact the Fourier transform, the 00:03:59.380 --> 00:04:03.120 continuous-time Fourier transform of the system 00:04:03.120 --> 00:04:06.170 impulse response. 00:04:06.170 --> 00:04:10.790 And the notion of decomposing a signal as a linear 00:04:10.790 --> 00:04:15.960 combination of these complex exponentials is what, first 00:04:15.960 --> 00:04:19.260 the Fourier series representation, and then later 00:04:19.260 --> 00:04:21.130 the Fourier transform representation 00:04:21.130 --> 00:04:22.680 corresponded to. 00:04:22.680 --> 00:04:28.390 And finally, to remind you of one additional point, the fact 00:04:28.390 --> 00:04:30.370 is that because of the eigenfunction 00:04:30.370 --> 00:04:34.640 property, the response-- 00:04:34.640 --> 00:04:37.580 once we have decomposed the input as a linear combination 00:04:37.580 --> 00:04:40.920 of complex exponentials, the response to that linear 00:04:40.920 --> 00:04:44.820 combination is straightforward to compute once we know the 00:04:44.820 --> 00:04:50.110 frequency response because of the eigenfunction property. 00:04:50.110 --> 00:04:54.790 Now, basically the same strategy and many of the same 00:04:54.790 --> 00:05:00.120 ideas work in discrete-time, paralleling almost exactly 00:05:00.120 --> 00:05:02.390 what happened in continuous-time. 00:05:02.390 --> 00:05:05.180 So the similarities between discrete-time and 00:05:05.180 --> 00:05:07.690 continuous-time are very strong. 00:05:07.690 --> 00:05:11.710 Although as we'll see, there are a number of differences. 00:05:11.710 --> 00:05:15.460 And it's important as we go through the discussion to 00:05:15.460 --> 00:05:20.210 illuminate not only the similarities, but obviously 00:05:20.210 --> 00:05:22.160 also the differences. 00:05:22.160 --> 00:05:26.800 Well, let's begin with the eigenfunction property, and 00:05:26.800 --> 00:05:31.900 let me just state that just as in continuous-time, if we 00:05:31.900 --> 00:05:36.580 consider a set of basic signals, which are complex 00:05:36.580 --> 00:05:40.770 exponential rules, then discrete-time linear time-m 00:05:40.770 --> 00:05:45.390 invariant systems have the eigenfunction property. 00:05:45.390 --> 00:05:50.200 Namely, if we put a complex exponential into the system, 00:05:50.200 --> 00:05:53.970 the response is a complex exponential at the same 00:05:53.970 --> 00:06:00.270 complex frequency, and simply multiplied by an appropriate 00:06:00.270 --> 00:06:03.620 complex factor, or constant. 00:06:03.620 --> 00:06:11.510 And just as we did in continuous-time, we will be 00:06:11.510 --> 00:06:15.980 referring to this complex constant, which is a function, 00:06:15.980 --> 00:06:19.910 of course, the frequency of the complex exponential input. 00:06:19.910 --> 00:06:24.400 We'll be referring to this as the frequency response. 00:06:24.400 --> 00:06:27.980 And although it's not particularly evident at this 00:06:27.980 --> 00:06:32.020 point, as the discussion develops through this lecture, 00:06:32.020 --> 00:06:36.620 what in fact will happen is very much paralleling 00:06:36.620 --> 00:06:38.340 continuous-time. 00:06:38.340 --> 00:06:43.990 This particular expression, in fact, will correspond to what 00:06:43.990 --> 00:06:48.550 we'll refer to as the Fourier transform, the discrete-time 00:06:48.550 --> 00:06:52.030 Fourier transform of the system impulse response. 00:06:52.030 --> 00:06:55.190 So there, of course, there's a very strong parallel between 00:06:55.190 --> 00:06:58.330 continuous time and discrete time. 00:06:58.330 --> 00:07:02.750 Now, just as we did in continuous-time, let's begin 00:07:02.750 --> 00:07:07.720 the discussion by first concentrating on periodic-- 00:07:07.720 --> 00:07:11.010 the representation through complex exponentials of 00:07:11.010 --> 00:07:16.030 periodic sequences, and then we'll generalize that 00:07:16.030 --> 00:07:18.270 discussion to the representation 00:07:18.270 --> 00:07:20.570 of aperiodic signals. 00:07:20.570 --> 00:07:26.190 So let's consider first a periodic signal, or in 00:07:26.190 --> 00:07:28.900 general, signals which are periodic. 00:07:28.900 --> 00:07:33.530 Period denoted by capital N. And then, of course, the 00:07:33.530 --> 00:07:39.610 fundamental frequency is 2 pi divided by capital N. 00:07:39.610 --> 00:07:45.200 Now, we can consider exponentials which have this 00:07:45.200 --> 00:07:50.390 as a fundamental frequency, or which are harmonics of that, 00:07:50.390 --> 00:07:53.880 and that would correspond to the class of complex 00:07:53.880 --> 00:07:59.290 exponentials of the form e to the jk omega 0 n. 00:07:59.290 --> 00:08:06.610 So these complex exponentials then, as k varies, are complex 00:08:06.610 --> 00:08:11.930 exponentials that are harmonically related, all of 00:08:11.930 --> 00:08:16.730 which are periodic with the same period capital N. 00:08:16.730 --> 00:08:19.680 Although the fundamental period is 00:08:19.680 --> 00:08:20.960 different for each of these. 00:08:20.960 --> 00:08:26.150 Each of them being related by an integer amount. 00:08:26.150 --> 00:08:29.830 Now, again, just as we did in continuous time, we can 00:08:29.830 --> 00:08:34.390 consider attempting to build our periodic signal out of a 00:08:34.390 --> 00:08:36.780 linear combination of these. 00:08:36.780 --> 00:08:43.280 And so we consider a periodic signal, which is a weighted 00:08:43.280 --> 00:08:48.060 sum of these complex exponentials. 00:08:48.060 --> 00:08:50.980 And, of course, this periodic signal-- 00:08:50.980 --> 00:08:52.420 this is a periodic signal. 00:08:52.420 --> 00:08:56.460 This can be verified, more or less, in a straightforward way 00:08:56.460 --> 00:08:58.150 by substitution. 00:08:58.150 --> 00:09:00.950 And, of course, one of the things that we'll want to 00:09:00.950 --> 00:09:04.930 address shortly is how broad a class of signals, again, can 00:09:04.930 --> 00:09:07.310 be represented by this sum? 00:09:07.310 --> 00:09:10.730 And another question obviously will be, how do we determine 00:09:10.730 --> 00:09:13.860 the coefficients a sub k? 00:09:13.860 --> 00:09:20.010 However, before we do that, let me focus on an important 00:09:20.010 --> 00:09:23.210 distinction between continuous-time and 00:09:23.210 --> 00:09:26.950 discrete-time in the context of these complex exponentials 00:09:26.950 --> 00:09:30.420 and this representation. 00:09:30.420 --> 00:09:34.140 When we talked about complex exponentials and sinusoids 00:09:34.140 --> 00:09:38.100 early in the course, one of the differences that we saw 00:09:38.100 --> 00:09:42.380 between continuous-time and discrete-time is that in 00:09:42.380 --> 00:09:46.990 continuous-time, as we vary the frequency variable, we see 00:09:46.990 --> 00:09:52.100 different complex exponentials as omega varies. 00:09:52.100 --> 00:09:56.580 Whereas, in discrete-time, we saw, in fact, that there was a 00:09:56.580 --> 00:09:57.750 periodicity. 00:09:57.750 --> 00:10:03.670 Or said another way, it's straightforward to verify that 00:10:03.670 --> 00:10:08.880 if we think of this class of complex exponentials. 00:10:08.880 --> 00:10:17.490 That, in fact, if we consider varying k by adding to it 00:10:17.490 --> 00:10:23.670 capital N, where capital N is the period of the fundamental 00:10:23.670 --> 00:10:25.690 complex exponential. 00:10:25.690 --> 00:10:33.700 Then in fact, if we replace k by k plus capital N, we'll see 00:10:33.700 --> 00:10:38.810 exactly the same complex exponentials over again. 00:10:38.810 --> 00:10:39.690 Now, what does that say? 00:10:39.690 --> 00:10:43.430 What it says is that if I consider this class of complex 00:10:43.430 --> 00:10:52.790 exponentials, as k varies from 0 through capital N minus 1, 00:10:52.790 --> 00:10:56.050 we will see all of the ones that there are to see. 00:10:56.050 --> 00:10:58.060 There aren't anymore. 00:10:58.060 --> 00:11:03.160 And so, in fact, if we can build x of n out of this 00:11:03.160 --> 00:11:08.600 linear combination, then we better be able to do it as k 00:11:08.600 --> 00:11:16.230 varies from 0 up to N minus 1. 00:11:16.230 --> 00:11:20.990 Because beyond that, we'll simply see the same complex 00:11:20.990 --> 00:11:22.870 exponentials over again. 00:11:22.870 --> 00:11:27.670 So, for example, if k takes on the value capital N, that will 00:11:27.670 --> 00:11:31.630 be exactly the same complex exponential as if 00:11:31.630 --> 00:11:34.300 k is equal to 0. 00:11:34.300 --> 00:11:41.070 So in fact, this sum ranges only over capital N of the 00:11:41.070 --> 00:11:43.430 distinct complex exponentials. 00:11:43.430 --> 00:11:47.100 Let's say, for example, from 0 to capital N minus 1 . 00:11:47.100 --> 00:11:52.130 Although, in fact, since these complex exponentials repeat in 00:11:52.130 --> 00:11:57.450 k, I could actually consider instead of from 0 to N minus 00:11:57.450 --> 00:12:01.240 1, I could consider from 1 to N, or from 2 to 00:12:01.240 --> 00:12:03.800 N plus 1, or whatever. 00:12:03.800 --> 00:12:09.260 Or said another way, in this representation, I could 00:12:09.260 --> 00:12:16.310 alternatively choose k outside this range, thinking of these 00:12:16.310 --> 00:12:20.430 coefficients simply as periodically repeating in k 00:12:20.430 --> 00:12:24.660 because of the fact that these complex exponentials 00:12:24.660 --> 00:12:27.580 periodically repeat in k. 00:12:27.580 --> 00:12:31.130 So, in fact, in place of this expression, it will be common 00:12:31.130 --> 00:12:35.080 in writing the Fourier series expression to write it as I've 00:12:35.080 --> 00:12:38.910 indicated here, where the implication is that these 00:12:38.910 --> 00:12:44.260 Fourier coefficients periodically repeat as k 00:12:44.260 --> 00:12:47.350 continues to repeat outside the interval from 00:12:47.350 --> 00:12:49.370 0 to N minus 1. 00:12:49.370 --> 00:12:54.520 And so this notation, in fact, says that what we're going to 00:12:54.520 --> 00:13:02.270 use is k ranging over one period of this periodic 00:13:02.270 --> 00:13:04.590 sequence, which is the Fourier series coefficients. 00:13:07.730 --> 00:13:13.330 So the expression that we have then for the Fourier series 00:13:13.330 --> 00:13:15.280 I've repeated here. 00:13:15.280 --> 00:13:18.360 And the implication now is that the 00:13:18.360 --> 00:13:20.650 a sub k's are periodic. 00:13:20.650 --> 00:13:22.820 They periodically repeat because, of course, these 00:13:22.820 --> 00:13:25.150 exponentials periodically repeat. 00:13:25.150 --> 00:13:29.490 This indicates that we only use them over one period. 00:13:29.490 --> 00:13:33.930 And now we can inquire as to how we determine the 00:13:33.930 --> 00:13:36.120 coefficients a sub k. 00:13:36.120 --> 00:13:39.480 Well, we can formally go through this much as we did in 00:13:39.480 --> 00:13:41.550 the continuous-time case. 00:13:41.550 --> 00:13:45.070 And we do, in fact, do that in the text, which involves 00:13:45.070 --> 00:13:48.310 substituting some sums and interchanging the orders of 00:13:48.310 --> 00:13:49.790 summation, et cetera. 00:13:49.790 --> 00:13:54.020 But let me draw your attention to the fact that this, in 00:13:54.020 --> 00:13:59.290 fact, can be thought of as capital N equations and 00:13:59.290 --> 00:14:02.210 capital N unknowns. 00:14:02.210 --> 00:14:08.100 In other words, we know x of n over a period, and so we know 00:14:08.100 --> 00:14:12.280 what the left-hand side of this is for capital N values. 00:14:12.280 --> 00:14:16.070 And we'd like to determine these constants a sub k. 00:14:16.070 --> 00:14:18.860 Well, it turns out that there is a nice convenient 00:14:18.860 --> 00:14:21.050 closed-form expression for that. 00:14:21.050 --> 00:14:26.060 And, in fact, if we evaluate the closed-form expression 00:14:26.060 --> 00:14:29.860 through any of a variety of algebraic manipulations, we 00:14:29.860 --> 00:14:35.290 end up then with the analysis equation. 00:14:35.290 --> 00:14:39.150 And the analysis equation, which tells us how to get the 00:14:39.150 --> 00:14:47.080 coefficients a sub k from x of n is what I've indicated here. 00:14:47.080 --> 00:14:53.020 And so this tells us how from x of n to get the a sub k's. 00:14:53.020 --> 00:14:58.140 And, of course, the first equation tells us how x of n 00:14:58.140 --> 00:15:01.120 is built up out of the a sub k. 00:15:01.120 --> 00:15:05.520 Notice incidentally that there is a strong duality between 00:15:05.520 --> 00:15:07.050 these two equations. 00:15:07.050 --> 00:15:11.590 And that's a duality that we'll return to, actually 00:15:11.590 --> 00:15:13.750 toward the end of the next lecture. 00:15:13.750 --> 00:15:16.340 Now, there is a real difference between the way 00:15:16.340 --> 00:15:19.630 those equations look and the way the continuous-time 00:15:19.630 --> 00:15:21.540 Fourier series looked. 00:15:21.540 --> 00:15:26.650 In the continuous-time case, let me remind you that it 00:15:26.650 --> 00:15:31.630 required an infinite number of coefficients to build up this 00:15:31.630 --> 00:15:33.800 continuous-time function. 00:15:33.800 --> 00:15:37.720 And so this was not simply a matter of identifying how to 00:15:37.720 --> 00:15:41.180 invert capital N or a finite number of equations and a 00:15:41.180 --> 00:15:44.280 finite number of unknowns. 00:15:44.280 --> 00:15:51.210 And the analysis equation was an integration as opposed to 00:15:51.210 --> 00:15:55.510 the synthesis equation, which is a summation. 00:15:55.510 --> 00:15:59.400 So there is a real difference there between the 00:15:59.400 --> 00:16:01.960 continuous-time and discrete-time cases. 00:16:01.960 --> 00:16:06.560 And the difference arises, to a large extent, because of 00:16:06.560 --> 00:16:09.480 this notion that in discrete-time, the complex 00:16:09.480 --> 00:16:16.050 exponentials are periodic in their frequency. 00:16:16.050 --> 00:16:21.470 So we have then to summarize the synthesis equation and the 00:16:21.470 --> 00:16:27.210 analysis equation for the discrete-time Fourier series. 00:16:27.210 --> 00:16:32.250 Again, x of n, our original signal is periodic. 00:16:32.250 --> 00:16:35.400 And, of course, the complex exponentials 00:16:35.400 --> 00:16:38.230 involved are periodic. 00:16:38.230 --> 00:16:40.680 They're periodic obviously in n. 00:16:40.680 --> 00:16:43.530 But in contrast to continuous-time, 00:16:43.530 --> 00:16:46.370 these repeat in k. 00:16:46.370 --> 00:16:51.070 In other words, as k omega 0 goes outside a range that 00:16:51.070 --> 00:16:53.500 covers a 2 pi interval. 00:16:53.500 --> 00:16:57.570 And because of that, we're imposing, in a sense, the 00:16:57.570 --> 00:17:00.450 interpretation that the a sub k's are 00:17:00.450 --> 00:17:02.820 likewise a periodic sequence. 00:17:02.820 --> 00:17:07.890 And in fact, if we look at the analysis equation, as we let k 00:17:07.890 --> 00:17:14.010 vary outside the range from 0 to N minus 1, what you can 00:17:14.010 --> 00:17:18.680 easily verify by substitution in here is that this sequence 00:17:18.680 --> 00:17:22.030 will, in fact, periodically repeat. 00:17:22.030 --> 00:17:26.300 So to underscore the difference between the 00:17:26.300 --> 00:17:30.320 continuous-time and discrete-time cases, we have 00:17:30.320 --> 00:17:33.730 this periodicity in the time domain, and that's a 00:17:33.730 --> 00:17:38.200 periodicity that is, of course, true in discrete-time 00:17:38.200 --> 00:17:43.520 and it's also true in continuous-time if we replace 00:17:43.520 --> 00:17:48.820 the integer variable by the discrete-time time variable. 00:17:48.820 --> 00:17:53.620 And we also, in discrete-time, have this periodicity in k, or 00:17:53.620 --> 00:17:55.430 in k omega 0. 00:17:55.430 --> 00:17:58.700 And correspondingly, a periodicity in the Fourier 00:17:58.700 --> 00:18:00.240 coefficients. 00:18:00.240 --> 00:18:06.980 And that is a set of properties that does not 00:18:06.980 --> 00:18:10.300 happen in continuous-time. 00:18:10.300 --> 00:18:15.520 And it is that that essentially leads to all of 00:18:15.520 --> 00:18:18.650 the important differences between discrete-time Fourier 00:18:18.650 --> 00:18:21.390 representations and continuous-time Fourier 00:18:21.390 --> 00:18:24.250 representations. 00:18:24.250 --> 00:18:29.430 Now, just quickly, let me draw your attention to the issue of 00:18:29.430 --> 00:18:33.400 convergence and when a sequence can and can't be 00:18:33.400 --> 00:18:35.370 represented, et cetera. 00:18:35.370 --> 00:18:39.620 And recall that in the continuous-time case, we 00:18:39.620 --> 00:18:44.530 focused on convergence in the context either of conditions, 00:18:44.530 --> 00:18:48.310 which I referred to as square integrability, or another set 00:18:48.310 --> 00:18:51.510 of conditions, which were the Dirichlet conditions. 00:18:51.510 --> 00:18:55.310 And there was this issue about when the signal does and 00:18:55.310 --> 00:18:58.480 doesn't converge at discontinuities, et cetera. 00:18:58.480 --> 00:19:02.330 Let me just simply draw your attention to the fact that in 00:19:02.330 --> 00:19:07.470 the discrete-time case, what we have is the representation 00:19:07.470 --> 00:19:12.620 of the periodic signal as a sum of a 00:19:12.620 --> 00:19:14.750 finite number of terms. 00:19:14.750 --> 00:19:17.350 This represents capital N equations 00:19:17.350 --> 00:19:19.850 and capital N unknowns. 00:19:19.850 --> 00:19:24.980 If we consider earth the partial sum, namely taking a 00:19:24.980 --> 00:19:31.640 smaller number of terms, then simply what happens is as the 00:19:31.640 --> 00:19:36.480 number of terms increases to the finite number required to 00:19:36.480 --> 00:19:41.620 represent x of n, we simply end up with the partial sum 00:19:41.620 --> 00:19:44.160 representing the finite [? length ?] sequence. 00:19:44.160 --> 00:19:48.600 What all that boils down to is the statement that in 00:19:48.600 --> 00:19:53.090 discrete-time there really are no convergence issues as there 00:19:53.090 --> 00:19:55.670 were in continuous-time. 00:19:55.670 --> 00:20:00.260 OK, well let's look at an example of the Fourier series 00:20:00.260 --> 00:20:03.390 representation for a particular signal. 00:20:03.390 --> 00:20:06.420 And the one that I've picked here is a simple one. 00:20:06.420 --> 00:20:14.430 Namely, a constant, a sine term, and a cosine term. 00:20:14.430 --> 00:20:17.860 Now, for this particular example, we can expand this 00:20:17.860 --> 00:20:22.720 out directly in terms of complex exponentials and 00:20:22.720 --> 00:20:26.710 essentially recognize this as a sum of complex exponentials. 00:20:26.710 --> 00:20:33.370 It's examined in more detail in Example 5.2 in the text. 00:20:33.370 --> 00:20:37.920 And if we look at the Fourier series coefficients, we can 00:20:37.920 --> 00:20:41.430 either look at it in terms of real and imaginary parts or 00:20:41.430 --> 00:20:44.160 magnitude and angle. 00:20:44.160 --> 00:20:48.260 On the left side here, I have the real part of the Fourier 00:20:48.260 --> 00:20:50.150 coefficients. 00:20:50.150 --> 00:20:55.480 And let me draw your attention to the fact that I've drawn 00:20:55.480 --> 00:21:01.330 this to specifically illuminate the periodicity of 00:21:01.330 --> 00:21:04.360 the Fourier series coefficients with a period of 00:21:04.360 --> 00:21:05.570 capital N. 00:21:05.570 --> 00:21:08.740 So here are the Fourier coefficients. 00:21:08.740 --> 00:21:12.920 And, in fact, it's this line that represents the DC, or 00:21:12.920 --> 00:21:17.510 constant term, and these two lines that represent the 00:21:17.510 --> 00:21:19.160 cosine term. 00:21:19.160 --> 00:21:23.540 And of course, these are the three terms that are required. 00:21:23.540 --> 00:21:27.500 Or equivalently, this one, this one, and this one. 00:21:27.500 --> 00:21:29.960 And then because of the periodicity of the Fourier 00:21:29.960 --> 00:21:34.140 series coefficients, this simply periodically repeats. 00:21:34.140 --> 00:21:39.150 So here is the real part and below it I show 00:21:39.150 --> 00:21:41.390 the imaginary part. 00:21:41.390 --> 00:21:45.560 And in the imaginary part, incidentally let me draw your 00:21:45.560 --> 00:21:53.090 attention to the fact that it's this term and this term 00:21:53.090 --> 00:21:56.760 in the imaginary part that represent the sinusoid. 00:21:56.760 --> 00:22:00.280 Whereas it's the symmetric terms in the real part the 00:22:00.280 --> 00:22:03.110 represent the cosine. 00:22:03.110 --> 00:22:07.580 OK, let's look at another example. 00:22:07.580 --> 00:22:12.420 This is another example from the text, and one that we'll 00:22:12.420 --> 00:22:18.000 be making frequent reference to in this particular lecture. 00:22:18.000 --> 00:22:24.090 And what it is, is a square wave. 00:22:24.090 --> 00:22:27.940 And I've expressed the Fourier series coefficients, which are 00:22:27.940 --> 00:22:30.340 algebraically developed in the text. 00:22:30.340 --> 00:22:35.000 I've expressed the Fourier series coefficients as samples 00:22:35.000 --> 00:22:38.616 of an envelope function. 00:22:38.616 --> 00:22:43.120 And so I've expressed it as samples of this particular 00:22:43.120 --> 00:22:46.120 function, which is referred to as a sin 00:22:46.120 --> 00:22:49.020 nx over sin x function. 00:22:49.020 --> 00:22:52.850 And let me just compare it to a continuous-time example, 00:22:52.850 --> 00:22:57.980 which is the continuous-time square wave, where with the 00:22:57.980 --> 00:23:02.840 continuous-times square wave the form of the Fourier series 00:23:02.840 --> 00:23:08.490 coefficients was as samples of what we refer to as a sin x 00:23:08.490 --> 00:23:11.050 over x function. 00:23:11.050 --> 00:23:15.820 Now, the sin nx over sin x function, which is the 00:23:15.820 --> 00:23:19.000 envelope of the Fourier series coefficients for the 00:23:19.000 --> 00:23:24.260 discrete-time periodic square wave plays the role-- 00:23:24.260 --> 00:23:26.490 and we'll see it very often in discrete-time-- 00:23:26.490 --> 00:23:31.080 that sin x over x does in continuous-time. 00:23:31.080 --> 00:23:33.180 And, in fact, we should understand right from the 00:23:33.180 --> 00:23:37.500 beginning that the sin x over x envelope couldn't possibly 00:23:37.500 --> 00:23:40.820 be the envelope of the discrete-time Fourier series 00:23:40.820 --> 00:23:42.220 coefficients. 00:23:42.220 --> 00:23:47.600 And one obvious reason is that it is not periodic. 00:23:47.600 --> 00:23:50.540 What we require, of course, from the discussion that I've 00:23:50.540 --> 00:23:53.530 just gone through is periodicity of the 00:23:53.530 --> 00:23:54.340 coefficients. 00:23:54.340 --> 00:23:58.150 And then consequently, also periodicity of the envelope in 00:23:58.150 --> 00:23:59.710 the discrete-time case. 00:23:59.710 --> 00:24:03.910 So once again, if we look back at the algebraic expression 00:24:03.910 --> 00:24:08.440 that I have, it's samples of the sin nx over sine x 00:24:08.440 --> 00:24:13.130 function that represent the Fourier series coefficients of 00:24:13.130 --> 00:24:16.160 this periodic square wave. 00:24:16.160 --> 00:24:24.650 Now, in the representation in the continuous-time case, we 00:24:24.650 --> 00:24:29.710 essentially had used the concept of an envelope to 00:24:29.710 --> 00:24:33.320 represent the Fourier series coefficients, and the notion 00:24:33.320 --> 00:24:35.990 that the Fourier series coefficients were samples of 00:24:35.990 --> 00:24:36.710 an envelope. 00:24:36.710 --> 00:24:41.500 And that is the same notion that we'll be using in 00:24:41.500 --> 00:24:43.110 discrete-time. 00:24:43.110 --> 00:24:49.560 So again for this square wave example, then what we have is 00:24:49.560 --> 00:24:54.520 an envelope function, the sin nx over sin x envelope 00:24:54.520 --> 00:24:59.720 function for a particular value of the period. 00:24:59.720 --> 00:25:04.280 Here indicated with a period of 10 samples. 00:25:04.280 --> 00:25:08.530 These samples of this envelope function would then represent 00:25:08.530 --> 00:25:12.340 the Fourier series coefficients. 00:25:12.340 --> 00:25:19.280 If we increased the period, then we would simply have a 00:25:19.280 --> 00:25:23.910 finer spacing on the samples of the envelope function to 00:25:23.910 --> 00:25:26.500 get the Fourier series coefficients. 00:25:26.500 --> 00:25:32.380 And likewise, if we increase the period still further, what 00:25:32.380 --> 00:25:37.260 we would have is an even finer spacing. 00:25:37.260 --> 00:25:40.970 So actually, as the period increases, and recall we used 00:25:40.970 --> 00:25:43.810 this in continuous-time also. 00:25:43.810 --> 00:25:47.700 As the period increases, we can view the Fourier series 00:25:47.700 --> 00:25:50.540 coefficients as samples of an envelope. 00:25:50.540 --> 00:25:54.970 And as the period increases, the sample spacing 00:25:54.970 --> 00:25:56.860 gets finer and finer. 00:25:56.860 --> 00:26:00.500 And in fact, as the period goes off essentially to 00:26:00.500 --> 00:26:04.380 infinity, the samples of the envelope, in effect, become 00:26:04.380 --> 00:26:06.140 the envelope. 00:26:06.140 --> 00:26:10.460 And recall also that this was essentially the trick that we 00:26:10.460 --> 00:26:16.850 used in continuous-time to allow us to develop or utilize 00:26:16.850 --> 00:26:21.590 the Fourier series to provide a representation of aperiodic 00:26:21.590 --> 00:26:24.410 signals as a linear combination of complex 00:26:24.410 --> 00:26:26.490 exponentials. 00:26:26.490 --> 00:26:35.220 In particular, what we did in the continuous-time case when 00:26:35.220 --> 00:26:41.280 we had an aperiodic signal was to consider constructing a 00:26:41.280 --> 00:26:44.500 periodic signal for which the aperiodic 00:26:44.500 --> 00:26:47.110 signal was one period. 00:26:47.110 --> 00:26:52.200 And then we developed the notion that since the periodic 00:26:52.200 --> 00:26:58.310 signal has a Fourier series, and since as the period of the 00:26:58.310 --> 00:27:02.760 periodic signal increases and goes to infinity, the periodic 00:27:02.760 --> 00:27:06.520 signal represents the aperiodic signal. 00:27:06.520 --> 00:27:09.930 Then, essentially, the Fourier series provides us with a 00:27:09.930 --> 00:27:11.740 representation. 00:27:11.740 --> 00:27:14.770 Now, we can do exactly the same thing in the 00:27:14.770 --> 00:27:16.420 discrete-time case. 00:27:16.420 --> 00:27:19.780 The statement is exactly the same, except that in the 00:27:19.780 --> 00:27:24.530 discrete-time case, instead of t as the independent variable, 00:27:24.530 --> 00:27:29.450 we simply make exactly the same statement, but with our 00:27:29.450 --> 00:27:31.730 discrete-time variable n. 00:27:31.730 --> 00:27:37.560 So the basic notion then in representing a discrete-time 00:27:37.560 --> 00:27:46.960 aperiodic signal is to first construct a periodic signal. 00:27:46.960 --> 00:27:50.450 Here we have the aperiodic signal. 00:27:50.450 --> 00:27:54.340 We construct a periodic signal by simply periodically 00:27:54.340 --> 00:27:58.840 replicating the aperiodic signal. 00:27:58.840 --> 00:28:04.890 The periodic signal and the aperiodic signal are identical 00:28:04.890 --> 00:28:07.310 for one period. 00:28:07.310 --> 00:28:12.120 And as the period goes off to infinity, it's the Fourier 00:28:12.120 --> 00:28:16.810 series representation of the periodic signal that provides 00:28:16.810 --> 00:28:21.050 a representation of the aperiodic signal. 00:28:21.050 --> 00:28:30.450 Again, to return to the example that we have been kind 00:28:30.450 --> 00:28:31.780 of working through this lecture. 00:28:31.780 --> 00:28:34.730 Namely, the periodic square wave. 00:28:34.730 --> 00:28:38.970 If we have an aperiodic signal, which is a rectangle, 00:28:38.970 --> 00:28:42.330 and we construct a periodic signal. 00:28:42.330 --> 00:28:44.130 And now we consider letting this 00:28:44.130 --> 00:28:46.570 period increase to infinity. 00:28:46.570 --> 00:28:51.320 We would first have this set of samples of the envelope. 00:28:51.320 --> 00:28:55.820 As the period increases, we would decrease the sample 00:28:55.820 --> 00:28:58.660 spacing to this set of samples. 00:28:58.660 --> 00:29:03.140 As the period increases further, it would be this set 00:29:03.140 --> 00:29:04.970 of samples. 00:29:04.970 --> 00:29:08.850 And as the period goes off to infinity, it's every point on 00:29:08.850 --> 00:29:10.050 the envelope. 00:29:10.050 --> 00:29:13.440 In fact, what the representation of the 00:29:13.440 --> 00:29:18.380 aperiodic signal is, is the envelope. 00:29:18.380 --> 00:29:20.670 OK, well, so that's the basic notion. 00:29:20.670 --> 00:29:23.650 It's no different than what we did in the 00:29:23.650 --> 00:29:25.580 continuous-time case. 00:29:25.580 --> 00:29:30.770 And mathematically, it develops in very much the same 00:29:30.770 --> 00:29:35.590 way as in the continuous-time case. 00:29:35.590 --> 00:29:42.130 Specifically, here is our representation through the 00:29:42.130 --> 00:29:47.550 Fourier series of the-- 00:29:47.550 --> 00:29:51.760 here is a representation through the envelope function. 00:29:51.760 --> 00:29:56.500 And this is the Fourier series synthesis equation where the 00:29:56.500 --> 00:30:02.660 equation below tells us how we get these Fourier coefficients 00:30:02.660 --> 00:30:05.970 or the envelope from x of n. 00:30:05.970 --> 00:30:08.860 Now, x tilde of n is the periodic signal. 00:30:08.860 --> 00:30:11.990 And we know that over one period, which is the only 00:30:11.990 --> 00:30:16.080 interval over which we use it, in fact, this is identical to 00:30:16.080 --> 00:30:17.980 the aperiodic signal. 00:30:17.980 --> 00:30:22.610 And so, in fact, we can rewrite this equation simply 00:30:22.610 --> 00:30:27.630 by substituting in instead of x tilde, the original 00:30:27.630 --> 00:30:29.540 aperiodic signal. 00:30:29.540 --> 00:30:33.950 And now we can use infinite limits on this sum. 00:30:33.950 --> 00:30:38.180 And what we would want to examine, mathematically, is 00:30:38.180 --> 00:30:46.950 what happens to the top equation as we let the period 00:30:46.950 --> 00:30:49.250 go off to infinity? 00:30:49.250 --> 00:30:53.090 And what happens is exactly identical, mathematically, to 00:30:53.090 --> 00:30:54.580 continuous-time. 00:30:54.580 --> 00:30:56.970 I won't belabor the details. 00:30:56.970 --> 00:31:01.980 Essentially it's this sum that goes to an integral. 00:31:01.980 --> 00:31:05.590 Omega 0, which is the fundamental frequency, is 00:31:05.590 --> 00:31:06.940 going towards 0. 00:31:06.940 --> 00:31:10.100 In fact, becomes the differential in the integral. 00:31:10.100 --> 00:31:13.200 And in the second equation, of course, this then 00:31:13.200 --> 00:31:15.210 becomes x of omega. 00:31:15.210 --> 00:31:19.980 And as N goes to infinity then, what the Fourier series 00:31:19.980 --> 00:31:25.020 becomes is the Fourier transform as summarized by the 00:31:25.020 --> 00:31:27.610 bottom two equations. 00:31:27.610 --> 00:31:33.490 So although there is a little bit of mathematical trickery. 00:31:33.490 --> 00:31:36.900 Or let's not call it trickery, but subtlety, to be tracked 00:31:36.900 --> 00:31:38.330 through in detail. 00:31:38.330 --> 00:31:43.300 The important conceptual thing to think about is this notion 00:31:43.300 --> 00:31:46.760 that we take the aperiodic signal, form a periodic 00:31:46.760 --> 00:31:49.230 signal, let the period go off to infinity. 00:31:49.230 --> 00:31:52.920 In which case, the Fourier series coefficients become 00:31:52.920 --> 00:31:54.630 these envelopes functions. 00:31:54.630 --> 00:31:56.910 And incidentally, mathematically, one of the 00:31:56.910 --> 00:31:59.890 sums ends up going to an integral. 00:31:59.890 --> 00:32:05.690 So what we have then is the discrete-time Fourier 00:32:05.690 --> 00:32:11.210 transform, which is a representation of 00:32:11.210 --> 00:32:13.080 an aperiodic signal. 00:32:13.080 --> 00:32:17.920 And we have the synthesis equation, which I show as the 00:32:17.920 --> 00:32:21.470 top equation on this transparency. 00:32:21.470 --> 00:32:27.940 And this is the integral that the Fourier series synthesis 00:32:27.940 --> 00:32:32.800 equation went to as the period went off to infinity. 00:32:32.800 --> 00:32:36.730 And we have the corresponding analysis equation, which is 00:32:36.730 --> 00:32:41.180 shown below, where this tells us the Fourier transform. 00:32:41.180 --> 00:32:44.890 In effect, the envelope or the Fourier series coefficients of 00:32:44.890 --> 00:32:47.250 that periodic signal. 00:32:47.250 --> 00:32:53.550 And here represented in terms of the aperiodic signal. 00:32:53.550 --> 00:32:56.215 So we have the analysis equation 00:32:56.215 --> 00:32:58.710 and synthesis equation. 00:32:58.710 --> 00:33:03.160 There are a number of things to focus on 00:33:03.160 --> 00:33:03.980 as you look at this. 00:33:03.980 --> 00:33:06.450 And we'll talk about some of its properties actually in the 00:33:06.450 --> 00:33:07.150 next lecture. 00:33:07.150 --> 00:33:11.120 But some of the points that I'd like you to think about 00:33:11.120 --> 00:33:16.700 and focus on is the fact that now there is somewhat of an 00:33:16.700 --> 00:33:20.420 imbalance or lack of duality between the time domain and 00:33:20.420 --> 00:33:21.890 frequency domain. 00:33:21.890 --> 00:33:24.530 x of n, which is our aperiodic signal, 00:33:24.530 --> 00:33:27.350 is of course, discrete. 00:33:27.350 --> 00:33:32.100 It's Fourier transform, x of omega, is a function of a 00:33:32.100 --> 00:33:33.320 continuous variable. 00:33:33.320 --> 00:33:36.870 Omega is a continuous variable. 00:33:36.870 --> 00:33:41.420 That is essentially what represents the envelope. 00:33:41.420 --> 00:33:46.060 Also, in the time domain x of n is aperiodic. 00:33:46.060 --> 00:33:49.270 It's not a periodic function. 00:33:49.270 --> 00:33:52.440 However, in the frequency domain, remember that the 00:33:52.440 --> 00:33:55.960 Fourier series coefficients were always periodic. 00:33:55.960 --> 00:34:00.090 Well, this envelope function then is also periodic with a 00:34:00.090 --> 00:34:03.820 period in omega of 2 pi. 00:34:03.820 --> 00:34:07.820 Once again, the reason for the periodicity, it all stems back 00:34:07.820 --> 00:34:12.449 to the fact that when we talk about complex exponentials-- 00:34:12.449 --> 00:34:15.199 and recall back to the early lectures. 00:34:15.199 --> 00:34:19.300 In discrete-time, as the frequency variable covers a 00:34:19.300 --> 00:34:25.980 range of 2 pi, when you proceed past that range, you 00:34:25.980 --> 00:34:28.610 simply see the same complex exponentials 00:34:28.610 --> 00:34:30.350 over and over again. 00:34:30.350 --> 00:34:33.300 And so obviously, anything that we do with them would 00:34:33.300 --> 00:34:37.480 have to be periodic in that frequency variable. 00:34:37.480 --> 00:34:42.480 All right, notationally, we'll, again, represent the 00:34:42.480 --> 00:34:44.520 discrete-time Fourier transform pair 00:34:44.520 --> 00:34:46.179 as I indicated here. 00:34:46.179 --> 00:34:49.980 And since it's a complex function of frequency may, on 00:34:49.980 --> 00:34:54.530 occasion, want to either represent it in rectangular 00:34:54.530 --> 00:34:59.180 form as I indicate in this equation, or in polar form as 00:34:59.180 --> 00:35:02.040 I indicate in this equation. 00:35:02.040 --> 00:35:04.930 Let's look at an example. 00:35:04.930 --> 00:35:10.590 And, of course, one example that we can look at is the one 00:35:10.590 --> 00:35:14.480 that has kind of been tracking us through this lecture, which 00:35:14.480 --> 00:35:18.260 is the example of a rectangle. 00:35:18.260 --> 00:35:24.130 Now, the rectangle, if we refer back to our argument of 00:35:24.130 --> 00:35:26.720 how we get a Fourier representation for an 00:35:26.720 --> 00:35:30.550 aperiodic signal, we would form a periodic signal where 00:35:30.550 --> 00:35:31.620 this is repeated. 00:35:31.620 --> 00:35:34.110 And that's our square wave example. 00:35:34.110 --> 00:35:36.740 As the period goes to infinity, the Fourier 00:35:36.740 --> 00:35:40.370 transform of this is represented by the envelope of 00:35:40.370 --> 00:35:44.140 those Fourier series coefficients, and that was our 00:35:44.140 --> 00:35:48.210 sin nx over sin x function, which in this particular case, 00:35:48.210 --> 00:35:54.320 for these particular numbers, is sin 5 omega over 2 divided 00:35:54.320 --> 00:35:57.890 by sin omega over 2. 00:35:57.890 --> 00:36:02.490 And notice, of course, as we would expect-- 00:36:02.490 --> 00:36:07.800 notice that this is a periodic function of the frequency 00:36:07.800 --> 00:36:13.830 variable omega repeating, of course, with a period of 2 pi. 00:36:13.830 --> 00:36:17.680 Whereas, in the time domain, the function was not a 00:36:17.680 --> 00:36:19.020 periodic function, it's aperiodic. 00:36:22.750 --> 00:36:26.170 Now, let's look at another example. 00:36:26.170 --> 00:36:31.880 Let's look at an example which is another signal that has 00:36:31.880 --> 00:36:34.780 kind of popped its head up from time to time as the 00:36:34.780 --> 00:36:36.720 lectures have gone along. 00:36:36.720 --> 00:36:40.110 A signal which is another aperiodic signal, which is a 00:36:40.110 --> 00:36:47.010 decaying exponential of this form with the factor a chosen 00:36:47.010 --> 00:36:49.890 between 0 and 1. 00:36:49.890 --> 00:36:54.670 And you can work out the algebra at your leisure. 00:36:54.670 --> 00:36:58.220 Basically, if we substitute into the Fourier transform 00:36:58.220 --> 00:37:02.830 analysis equation, it's this sum that we evaluate. 00:37:02.830 --> 00:37:07.665 Because we have a unit step here which shuts this off for 00:37:07.665 --> 00:37:11.250 n less than 0, we can change the limits on the sum. 00:37:11.250 --> 00:37:16.790 This then corresponds to the sum over an infinite number of 00:37:16.790 --> 00:37:19.160 terms of a geometric series. 00:37:19.160 --> 00:37:23.690 And that, as we've seen before, is 1 divided by 1 00:37:23.690 --> 00:37:27.340 minus a e to the minus j omega. 00:37:27.340 --> 00:37:32.610 So let's look at what that looks like. 00:37:32.610 --> 00:37:40.100 Here then we have, again, the expression in the time domain 00:37:40.100 --> 00:37:43.330 and the expression in the frequency domain. 00:37:43.330 --> 00:37:47.580 And let's, in particular, focus on what the magnitude of 00:37:47.580 --> 00:37:49.960 the Fourier transform looks like. 00:37:49.960 --> 00:37:54.790 It's as we show here. 00:37:54.790 --> 00:37:59.380 And for the particular values of a that I pick, namely 00:37:59.380 --> 00:38:03.460 between 0 and 1, it's larger at the origin 00:38:03.460 --> 00:38:05.340 than it is at pi. 00:38:05.340 --> 00:38:09.090 And then, of course, it is periodic. 00:38:09.090 --> 00:38:12.080 And the periodicity is inherent in the Fourier 00:38:12.080 --> 00:38:16.150 transform in discrete-time, so we really might only need to 00:38:16.150 --> 00:38:21.160 look at this either from minus pi to pi, or from 0 to 2 pi. 00:38:21.160 --> 00:38:24.710 The periodicity, of course, would imply what the rest of 00:38:24.710 --> 00:38:27.900 this is for other values of omega. 00:38:27.900 --> 00:38:31.290 Let me also draw your attention while we're on it to 00:38:31.290 --> 00:38:33.900 the fact that-- 00:38:33.900 --> 00:38:38.960 observe that if a were, in fact, negative, then this 00:38:38.960 --> 00:38:42.700 value would be less than this value. 00:38:42.700 --> 00:38:47.090 And in fact, for a negative, the magnitude of the frequency 00:38:47.090 --> 00:38:51.630 response would look like this except shifted by an amount in 00:38:51.630 --> 00:38:54.380 omega equal to pi. 00:38:54.380 --> 00:38:59.050 And this example will come up and play an important role in 00:38:59.050 --> 00:39:03.650 our discussion next time, so try to keep it in mind. 00:39:03.650 --> 00:39:06.710 And in fact, work it out more carefully between 00:39:06.710 --> 00:39:08.220 now and next time. 00:39:08.220 --> 00:39:12.800 And also, if you have a chance, focus on this issue of 00:39:12.800 --> 00:39:18.500 how it looks with a positive as compared with a negative. 00:39:18.500 --> 00:39:23.860 Now, we developed the Fourier transform by beginning with 00:39:23.860 --> 00:39:25.370 the Fourier series. 00:39:25.370 --> 00:39:29.350 We did that in continuous-time also. 00:39:29.350 --> 00:39:32.920 What I'd like to do now, just as we did in continuous-time, 00:39:32.920 --> 00:39:38.800 is now absorb the Fourier series within the broader 00:39:38.800 --> 00:39:41.540 framework of the Fourier transform. 00:39:41.540 --> 00:39:44.360 And there are two relationships between the 00:39:44.360 --> 00:39:48.000 Fourier series and the Fourier transform, which are identical 00:39:48.000 --> 00:39:54.250 to relationships that we had in the continuous-time case. 00:39:54.250 --> 00:40:00.980 Let me remind you that in continuous-time we had the 00:40:00.980 --> 00:40:09.290 statement that if we have a periodic signal, that in fact 00:40:09.290 --> 00:40:12.040 the Fourier series coefficients of that periodic 00:40:12.040 --> 00:40:19.340 signal is proportional to samples of the Fourier 00:40:19.340 --> 00:40:23.140 transform of one period. 00:40:23.140 --> 00:40:28.510 Well, in fact, let me remind you flows easily from all the 00:40:28.510 --> 00:40:33.210 things that we built up so far, because of the fact that 00:40:33.210 --> 00:40:36.670 the Fourier transform essentially, by definition, of 00:40:36.670 --> 00:40:44.600 the way we developed it, is what we get as the Fourier 00:40:44.600 --> 00:40:48.780 series coefficients, as we focus on one period, and then 00:40:48.780 --> 00:40:51.050 let the period go off to infinity. 00:40:51.050 --> 00:40:55.510 Well, looking at one period, the Fourier transform of that 00:40:55.510 --> 00:40:59.490 then is the envelope of the Fourier series coefficients. 00:40:59.490 --> 00:41:05.610 And so in continuous-time, we have this relationship. 00:41:05.610 --> 00:41:10.500 And in discrete-time, we have precisely the same 00:41:10.500 --> 00:41:16.240 relationship, except that here we're talking about an integer 00:41:16.240 --> 00:41:20.090 variable as opposed to the continuous variable, and a 00:41:20.090 --> 00:41:28.440 period of capital N as opposed to a period of t0. 00:41:28.440 --> 00:41:38.720 OK, so once again, if we return to our example, or if 00:41:38.720 --> 00:41:40.790 we return to a periodic signal. 00:41:40.790 --> 00:41:46.240 If we have a periodic signal and we consider the Fourier 00:41:46.240 --> 00:41:52.100 transform of one period, the Fourier series coefficients of 00:41:52.100 --> 00:41:56.880 this periodic signal are, in fact, samples-- 00:41:56.880 --> 00:42:01.800 as stated mathematically in the bottom equation, samples 00:42:01.800 --> 00:42:07.340 of the Fourier transform of one period. 00:42:07.340 --> 00:42:11.180 So x of omega is the Fourier transform of one period. 00:42:11.180 --> 00:42:13.760 a sub k's are the Fourier series coefficients of the 00:42:13.760 --> 00:42:15.350 periodic signal. 00:42:15.350 --> 00:42:19.190 And this relationship simply says they're related except 00:42:19.190 --> 00:42:23.030 for scale factor through samples along 00:42:23.030 --> 00:42:25.820 the frequency axis. 00:42:25.820 --> 00:42:33.370 And, of course, we saw this in the context of 00:42:33.370 --> 00:42:36.350 our square wave example. 00:42:36.350 --> 00:42:42.080 In the square wave example, we have a periodic signal, which 00:42:42.080 --> 00:42:45.690 is a periodic square wave. 00:42:45.690 --> 00:42:51.800 And the Fourier transform of one period, in fact, 00:42:51.800 --> 00:42:54.790 represents the envelope. 00:42:54.790 --> 00:42:57.180 And here we have the envelope function. 00:42:57.180 --> 00:43:02.520 Represents the envelope of the Fourier series coefficients. 00:43:02.520 --> 00:43:05.710 And the Fourier series coefficients are samples. 00:43:08.390 --> 00:43:14.460 So what we have then is a relationship back to the 00:43:14.460 --> 00:43:18.120 Fourier series coefficients from the Fourier transform 00:43:18.120 --> 00:43:21.750 that tells us that for a periodic signal now, the 00:43:21.750 --> 00:43:23.420 periodic signal-- 00:43:23.420 --> 00:43:27.170 the Fourier series coefficients are related, are 00:43:27.170 --> 00:43:31.920 samples of the Fourier transform of one period. 00:43:31.920 --> 00:43:36.830 Now, finally, to kind of bring things back in a circle and 00:43:36.830 --> 00:43:40.100 exactly identical to what we did in the continuous-time 00:43:40.100 --> 00:43:46.400 case, we can finally absorb the Fourier series in 00:43:46.400 --> 00:43:47.310 discrete-time. 00:43:47.310 --> 00:43:50.480 We can absorb it into the framework 00:43:50.480 --> 00:43:53.300 of the Fourier transform. 00:43:53.300 --> 00:44:00.800 Now, remember or recall how we did that when we tried to do a 00:44:00.800 --> 00:44:04.970 similar sort of thing in continuous-time. 00:44:04.970 --> 00:44:11.430 In continuous-time, what we essentially did is to develop 00:44:11.430 --> 00:44:15.040 that, more or less, by definition. 00:44:15.040 --> 00:44:18.150 We have a periodic signal. 00:44:18.150 --> 00:44:20.800 The periodic signal is represented through a Fourier 00:44:20.800 --> 00:44:24.360 series and Fourier series coefficients. 00:44:24.360 --> 00:44:27.570 Essentially what I pointed out at that time 00:44:27.570 --> 00:44:30.180 was that if we define-- 00:44:30.180 --> 00:44:33.780 take it as a definition, the Fourier transform of the 00:44:33.780 --> 00:44:39.220 periodic signal as an impulse train where the amplitudes of 00:44:39.220 --> 00:44:42.640 the impulses are proportional to the Fourier series 00:44:42.640 --> 00:44:44.680 coefficients. 00:44:44.680 --> 00:44:50.430 If we take that impulse train representation and simply plug 00:44:50.430 --> 00:44:58.460 it into the Fourier transform synthesis equation, what we 00:44:58.460 --> 00:45:04.860 end up with is the Fourier series synthesis equation. 00:45:04.860 --> 00:45:17.320 So in continuous-time, we had used this definition of the 00:45:17.320 --> 00:45:20.060 continuous-time Fourier transform 00:45:20.060 --> 00:45:22.610 of a periodic signal. 00:45:22.610 --> 00:45:27.670 And again, in discrete-time, it's simply a matter of using 00:45:27.670 --> 00:45:30.110 exactly the same expression. 00:45:30.110 --> 00:45:35.400 And using, instead, the appropriate variables related 00:45:35.400 --> 00:45:38.430 to discrete-time rather than the variables related to 00:45:38.430 --> 00:45:39.810 continuous-time. 00:45:39.810 --> 00:45:43.120 So in discrete-time, if we have a periodic signal, the 00:45:43.120 --> 00:45:47.150 Fourier transform of that periodic signal is defined as 00:45:47.150 --> 00:45:53.290 an impulse train where the amplitudes of the impulses are 00:45:53.290 --> 00:45:57.500 proportional to the Fourier series coefficients. 00:45:57.500 --> 00:46:00.660 If this expression is substituted into the synthesis 00:46:00.660 --> 00:46:05.340 equation for the Fourier transform, that will simply 00:46:05.340 --> 00:46:08.440 then reduce to the synthesis equation 00:46:08.440 --> 00:46:11.240 for the Fourier series. 00:46:11.240 --> 00:46:17.310 So once more returning to our example, which is the square 00:46:17.310 --> 00:46:19.170 wave example that we've carried through these 00:46:19.170 --> 00:46:23.780 lectures, or through this lecture, we can see that 00:46:23.780 --> 00:46:25.870 really what we're talking about really is 00:46:25.870 --> 00:46:27.120 a notational change. 00:46:30.740 --> 00:46:35.290 Here is the periodic signal and below it are the Fourier 00:46:35.290 --> 00:46:39.330 series coefficients, where I've removed the envelope 00:46:39.330 --> 00:46:43.420 function and just indicate the amplitudes of the coefficients 00:46:43.420 --> 00:46:46.610 indexed, of course, on the coefficient number. 00:46:46.610 --> 00:46:49.550 And so this represents a bar graph. 00:46:49.550 --> 00:46:52.760 And if instead of talking about the Fourier series 00:46:52.760 --> 00:46:59.040 coefficients, what I want to talk about is the Fourier 00:46:59.040 --> 00:47:03.300 transform, the Fourier transform, in essence, 00:47:03.300 --> 00:47:08.450 corresponds to simply redrawing that using impulses 00:47:08.450 --> 00:47:14.420 and using an axis that is essentially indexed on the 00:47:14.420 --> 00:47:18.910 fundamental frequency omega 0, rather than on the Fourier 00:47:18.910 --> 00:47:20.433 series coefficient number k. 00:47:23.884 --> 00:47:31.540 OK, so to summarize, what we've done is 00:47:31.540 --> 00:47:34.190 to pretty much parallel-- 00:47:34.190 --> 00:47:38.290 somewhat more quickly, the kind of development that we 00:47:38.290 --> 00:47:41.970 went through for continuous-time representation 00:47:41.970 --> 00:47:45.070 through complex exponentials, paralleled that for the 00:47:45.070 --> 00:47:47.390 discrete-time case. 00:47:47.390 --> 00:47:51.340 And pretty much the conceptual underpinnings of the 00:47:51.340 --> 00:47:55.230 development are identical in discrete-time and in 00:47:55.230 --> 00:47:56.480 continuous-time. 00:47:58.750 --> 00:48:04.510 We saw that there are some major differences, or 00:48:04.510 --> 00:48:06.700 important differences between continuous-time and 00:48:06.700 --> 00:48:08.560 discrete-time. 00:48:08.560 --> 00:48:11.240 And the difference, essentially 00:48:11.240 --> 00:48:14.080 relates to two aspects. 00:48:14.080 --> 00:48:18.790 One aspect is the fact that in discrete-time, we have a 00:48:18.790 --> 00:48:22.690 discrete representation in the time domain, whereas the 00:48:22.690 --> 00:48:25.020 independent variable in the frequency domain is a 00:48:25.020 --> 00:48:26.750 continuous variable. 00:48:26.750 --> 00:48:30.860 Whereas in continuous-time for the Fourier transform, we had 00:48:30.860 --> 00:48:35.120 a duality between the time domain and frequency domain. 00:48:35.120 --> 00:48:38.170 The other very important difference tied back to the 00:48:38.170 --> 00:48:41.310 difference between complex exponentials, continuous-time 00:48:41.310 --> 00:48:43.050 and discrete-time. 00:48:43.050 --> 00:48:45.980 In continuous-time, complex exponentials, as you vary the 00:48:45.980 --> 00:48:52.940 frequency, generate distinct time functions. 00:48:52.940 --> 00:48:56.500 In discrete-time, as you vary the frequency, once you've 00:48:56.500 --> 00:49:01.120 covered a frequency interval of 2 pi, then you've seen all 00:49:01.120 --> 00:49:02.240 the ones there are to see. 00:49:02.240 --> 00:49:03.680 There are no more. 00:49:03.680 --> 00:49:09.610 And this, in effect, imposes a periodicity on the Fourier 00:49:09.610 --> 00:49:13.390 domain representation of discrete-time signals. 00:49:13.390 --> 00:49:15.710 And some of those differences and, of course, lots of the 00:49:15.710 --> 00:49:20.380 similarities will surface, both as we use this 00:49:20.380 --> 00:49:26.120 representation and as we develop further properties. 00:49:26.120 --> 00:49:30.290 In the next lecture, what we'll do is to focus in, 00:49:30.290 --> 00:49:33.540 again, on the Fourier transform, the discrete-time 00:49:33.540 --> 00:49:39.050 Fourier transform, develop or illuminate some of the 00:49:39.050 --> 00:49:42.770 properties of the Fourier transform, and then see how 00:49:42.770 --> 00:49:46.850 these properties can be used for a number of things. 00:49:46.850 --> 00:49:49.360 For example, how the properties as they were in 00:49:49.360 --> 00:49:55.280 continuous-time can be used to efficiently generate the 00:49:55.280 --> 00:49:58.430 solution and analyze linear constant coefficient 00:49:58.430 --> 00:49:59.950 difference equations. 00:49:59.950 --> 00:50:03.680 And then beyond that, the concepts of filtering and 00:50:03.680 --> 00:50:05.090 modulation. 00:50:05.090 --> 00:50:09.220 And both the properties and interpretation, which will 00:50:09.220 --> 00:50:12.750 very strongly parallel the kinds of developments along 00:50:12.750 --> 00:50:14.650 those lines that we did in the last lecture. 00:50:14.650 --> 00:50:15.900 Thank you.