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 --> 00:00:20.540 ocw.mit.edu. 00:00:54.685 --> 00:00:57.730 PROFESSOR: Over the last several lectures, we've dealt 00:00:57.730 --> 00:01:01.370 with the representation of linear time-invariant systems 00:01:01.370 --> 00:01:03.080 through convolution. 00:01:03.080 --> 00:01:07.850 And just to remind you of our basic strategy, essentially, 00:01:07.850 --> 00:01:13.230 the idea was to exploit the notion of linearity by 00:01:13.230 --> 00:01:17.250 decomposing the input into a sum of basic inputs and then 00:01:17.250 --> 00:01:20.580 using linearity to tell us that the output can be 00:01:20.580 --> 00:01:23.200 represented as the corresponding linear 00:01:23.200 --> 00:01:25.880 combination of the associated outputs. 00:01:25.880 --> 00:01:30.420 So, if we have a linear system, either continuous-time 00:01:30.420 --> 00:01:34.570 or discrete-time, for example, with continuous time, if the 00:01:34.570 --> 00:01:38.690 input is decomposed as a linear combination of basic 00:01:38.690 --> 00:01:42.250 inputs, with each of these basic inputs generating an 00:01:42.250 --> 00:01:46.900 associated output, and if the system is linear, then the 00:01:46.900 --> 00:01:51.850 output of the system is the same linear combination of the 00:01:51.850 --> 00:01:53.530 associated outputs. 00:01:53.530 --> 00:01:57.450 And the same statement is identical both for continuous 00:01:57.450 --> 00:02:01.170 time and discrete time. 00:02:01.170 --> 00:02:06.600 So the strategy is to decompose the input into these 00:02:06.600 --> 00:02:08.440 basic inputs. 00:02:08.440 --> 00:02:14.120 And the inputs were chosen also with some particular 00:02:14.120 --> 00:02:15.960 strategy in mind. 00:02:15.960 --> 00:02:19.390 In particular, for both continuous time or discrete 00:02:19.390 --> 00:02:25.120 time, in this representation, the basic inputs used in the 00:02:25.120 --> 00:02:30.890 decomposition are chosen, first of all, so that a broad 00:02:30.890 --> 00:02:34.440 class of signals could be represented in terms of these 00:02:34.440 --> 00:02:39.840 basic inputs, and second of all, so that the response to 00:02:39.840 --> 00:02:45.660 these basic inputs is, in some sense, easy to compute. 00:02:45.660 --> 00:02:52.420 Now, in the representation which led us to convolution, 00:02:52.420 --> 00:02:56.930 the particular choice that we made in the discrete-time case 00:02:56.930 --> 00:03:02.630 for our basic inputs was a decomposition of the input in 00:03:02.630 --> 00:03:06.960 terms of delayed impulses. 00:03:06.960 --> 00:03:11.780 And the associated outputs that that generated were 00:03:11.780 --> 00:03:15.580 delayed versions of the impulse response. 00:03:15.580 --> 00:03:19.100 Decomposing the input into a linear combination of these, 00:03:19.100 --> 00:03:21.510 the output into the corresponding linear 00:03:21.510 --> 00:03:26.190 combination of these, then led to the convolution sum in the 00:03:26.190 --> 00:03:28.430 discrete time case. 00:03:28.430 --> 00:03:33.080 And in the continuous-time case, a similar kind of 00:03:33.080 --> 00:03:36.330 decomposition, in terms of impulses, and associated 00:03:36.330 --> 00:03:39.740 representation of the output, in terms of the impulse 00:03:39.740 --> 00:03:44.220 response, led to the convolution integral. 00:03:44.220 --> 00:03:48.380 Now, in this lecture, and for a number of the succeeding 00:03:48.380 --> 00:03:52.450 lectures, we'll want to turn our attention to a very 00:03:52.450 --> 00:03:56.120 different set of basic building blocks. 00:03:56.120 --> 00:04:01.150 And in particular, the signals that we'll be using as the 00:04:01.150 --> 00:04:04.740 building blocks for our more general signals, rather than 00:04:04.740 --> 00:04:09.870 impulses, as we've dealt with before, will be, in general, 00:04:09.870 --> 00:04:12.230 complex exponentials. 00:04:12.230 --> 00:04:17.810 So, in a general sense, in the continuous-time case, we'll be 00:04:17.810 --> 00:04:21.339 thinking in terms of a decomposition of our signals 00:04:21.339 --> 00:04:26.160 as a linear combination of complex exponentials, 00:04:26.160 --> 00:04:31.740 continuous-time, or, in the discrete-time case, complex 00:04:31.740 --> 00:04:37.020 exponentials, where z_k is complex here in discrete time 00:04:37.020 --> 00:04:42.500 and s sub k is complex here in continuous time. 00:04:42.500 --> 00:04:47.860 Now, the basic strategy, of course, requires that we 00:04:47.860 --> 00:04:52.220 choose a set of inputs, basic building blocks, which have 00:04:52.220 --> 00:04:53.580 two properties. 00:04:53.580 --> 00:04:57.320 One is that the system response be straightforward to 00:04:57.320 --> 00:05:00.150 compute, or in some sense, easy to compute. 00:05:00.150 --> 00:05:03.280 And second is that it be a fairly general set of building 00:05:03.280 --> 00:05:09.290 blocks so that we can build lots of signals out of them. 00:05:09.290 --> 00:05:12.690 What we'll find with complex exponentials, either 00:05:12.690 --> 00:05:14.740 continuous-time or discrete-time, is that they 00:05:14.740 --> 00:05:17.870 very nicely have those two properties. 00:05:17.870 --> 00:05:23.280 In particular, the notion that the output of a linear 00:05:23.280 --> 00:05:27.320 time-invariant system is easy to compute is tied to what's 00:05:27.320 --> 00:05:29.250 referred to as the Eigenfunction function 00:05:29.250 --> 00:05:32.280 property of complex exponentials, which we'll 00:05:32.280 --> 00:05:36.780 focus on shortly in a little more detail. 00:05:36.780 --> 00:05:41.600 And second of all, the fact that we can, in fact, 00:05:41.600 --> 00:05:44.880 represent very broad classes of signals as linear 00:05:44.880 --> 00:05:50.010 combinations of these will be a topic and an issue that 00:05:50.010 --> 00:05:53.780 we'll develop in detail over, in fact, the next set of 00:05:53.780 --> 00:05:57.860 lectures, this lecture, and the next set of lectures. 00:05:57.860 --> 00:06:03.960 Now, in doing this, although we could, in fact, begin with 00:06:03.960 --> 00:06:10.040 our attention focused on, in general, complex exponentials, 00:06:10.040 --> 00:06:14.330 what we'll choose to do is first focus on the case in 00:06:14.330 --> 00:06:19.150 which the exponent in the continuous-time case is purely 00:06:19.150 --> 00:06:25.420 imaginary, as I indicate here, and in the discrete-time case, 00:06:25.420 --> 00:06:27.990 where the magnitude of the complex number 00:06:27.990 --> 00:06:30.590 z_k is equal to 1. 00:06:30.590 --> 00:06:35.320 So what that corresponds to in the continuous-time case is a 00:06:35.320 --> 00:06:41.280 set of building blocks of the form e^(j omega_k t), and in 00:06:41.280 --> 00:06:44.800 the discrete-time case, a set of building blocks of the form 00:06:44.800 --> 00:06:46.050 e^(j Omega_k n). 00:06:47.930 --> 00:06:52.730 What we'll see is a representation in these terms 00:06:52.730 --> 00:06:57.570 leads to what's referred to as Fourier analysis. 00:06:57.570 --> 00:07:00.150 And that's what will be dealing with over the next set 00:07:00.150 --> 00:07:02.870 of lectures. 00:07:02.870 --> 00:07:06.290 We'll then be exploiting this representation actually 00:07:06.290 --> 00:07:07.710 through most of the course. 00:07:07.710 --> 00:07:11.700 And then toward the end of the course, we'll return to 00:07:11.700 --> 00:07:16.110 generalizing the Fourier representation to a discussion 00:07:16.110 --> 00:07:19.210 Laplace transforms and Z-transforms. 00:07:19.210 --> 00:07:22.900 So for now, we want to restrict ourselves to complex 00:07:22.900 --> 00:07:27.100 exponentials of a particular form, and in fact, also 00:07:27.100 --> 00:07:30.980 initially to continuous-time signals and systems. 00:07:30.980 --> 00:07:34.410 So let's begin with the continuous-time case and the 00:07:34.410 --> 00:07:39.020 complex exponentials that we want to deal with and focus, 00:07:39.020 --> 00:07:43.050 first of all, on what I refer to as the Eigenfunction 00:07:43.050 --> 00:07:48.000 property of this particular set of building blocks. 00:07:48.000 --> 00:07:51.313 We're talking about basic signals of the form e^(j 00:07:51.313 --> 00:07:52.563 omega_k t). 00:07:54.020 --> 00:07:58.370 And the statement is that for a linear time-invariant 00:07:58.370 --> 00:08:03.770 system, the response to one of these is of exactly the same 00:08:03.770 --> 00:08:09.340 form, just simply multiplied by a complex factor, that 00:08:09.340 --> 00:08:12.680 complex factor depending on what the 00:08:12.680 --> 00:08:16.020 frequency, omega_k, is. 00:08:16.020 --> 00:08:18.820 Now more or less, the justification for this, or the 00:08:18.820 --> 00:08:24.170 proof, follows by simply looking at the response to a 00:08:24.170 --> 00:08:28.140 complex exponential, using the convolution integral. 00:08:28.140 --> 00:08:33.750 So if we put a complex exponentials into a linear 00:08:33.750 --> 00:08:39.409 time-invariant system with impulse response h(t), then we 00:08:39.409 --> 00:08:42.780 can express the response as I've indicated here. 00:08:42.780 --> 00:08:46.320 We can then recognize that this complex exponentials can 00:08:46.320 --> 00:08:48.680 be factored into two terms. 00:08:48.680 --> 00:08:53.180 And so we can rewrite this complex 00:08:53.180 --> 00:08:56.080 exponential as this product. 00:08:56.080 --> 00:09:02.280 Second, recognize that this term can be taken outside the 00:09:02.280 --> 00:09:07.120 integral, over here, because of the fact that it depends 00:09:07.120 --> 00:09:10.360 only on t and not on Tau. 00:09:10.360 --> 00:09:14.640 And so what we're left with, when we track this through, is 00:09:14.640 --> 00:09:19.070 that, with a complex exponential input, we get an 00:09:19.070 --> 00:09:23.120 output which is the same complex exponential, namely 00:09:23.120 --> 00:09:26.950 this factor, times this integral. 00:09:26.950 --> 00:09:33.620 And this integral is what I refer to above as H(omega_k). 00:09:37.040 --> 00:09:42.200 And so, in fact, we put in a complex exponential, we get 00:09:42.200 --> 00:09:46.680 out a complex exponentials of the same frequency, multiplied 00:09:46.680 --> 00:09:48.650 by a complex constant. 00:09:48.650 --> 00:09:54.030 And that is what's referred to as the Eigenfunction property, 00:09:54.030 --> 00:09:57.840 Eigenfunction meaning that an Eigenfunction of a system, or 00:09:57.840 --> 00:10:01.370 mathematical expression, is a function which, when you put 00:10:01.370 --> 00:10:04.980 it through the system, comes out looking exactly the same 00:10:04.980 --> 00:10:07.950 except for a change in amplitude, the change in 00:10:07.950 --> 00:10:11.040 amplitude being the Eigenvalue. 00:10:11.040 --> 00:10:15.440 So in fact, this function is the Eigenfunction. 00:10:17.980 --> 00:10:23.800 And this value is the Eigenvalue. 00:10:27.540 --> 00:10:31.920 OK, now it's because of the Eigenfunction property that 00:10:31.920 --> 00:10:34.800 complex exponentials are particularly convenient as 00:10:34.800 --> 00:10:35.950 building blocks. 00:10:35.950 --> 00:10:38.900 Namely you put it through the system, they come out with the 00:10:38.900 --> 00:10:42.980 same form and simply scale. 00:10:42.980 --> 00:10:45.780 The other part to the question, related to the 00:10:45.780 --> 00:10:51.480 strategy that we've been pursuing, is to hope that 00:10:51.480 --> 00:10:57.370 these signals can be used as building blocks to represent a 00:10:57.370 --> 00:11:01.030 very broad class of signals through a linear combination. 00:11:01.030 --> 00:11:04.230 And in fact, that turns out to be the case with complex 00:11:04.230 --> 00:11:06.820 exponentials. 00:11:06.820 --> 00:11:10.420 As we work our way through that, we'll first consider the 00:11:10.420 --> 00:11:13.950 case of periodic signals. 00:11:13.950 --> 00:11:17.710 And what that leads to is a representation of periodic 00:11:17.710 --> 00:11:22.360 signals through what's called the Fourier series. 00:11:22.360 --> 00:11:26.690 Following that, we'll turn our attention to non-periodic, or 00:11:26.690 --> 00:11:29.770 as I refer to it, aperiodic signals. 00:11:29.770 --> 00:11:34.830 And the representation that's developed in terms of linear 00:11:34.830 --> 00:11:38.270 combinations of complex exponentials is what's 00:11:38.270 --> 00:11:41.240 referred to as the Fourier transform. 00:11:41.240 --> 00:11:45.550 So the first thing we want to deal with are periodic signals 00:11:45.550 --> 00:11:47.090 and the Fourier series. 00:11:49.690 --> 00:11:54.190 So what we're talking about then is the continuous-time 00:11:54.190 --> 00:11:57.110 Fourier series. 00:11:57.110 --> 00:12:02.180 And the Fourier series is a representation for periodic 00:12:02.180 --> 00:12:05.090 continuous-time signals. 00:12:05.090 --> 00:12:08.730 We have a signal, then, which is periodic. 00:12:08.730 --> 00:12:13.310 And we're choosing T_0 to denote the period. 00:12:13.310 --> 00:12:19.380 So it's T_0 that corresponds to the period of 00:12:19.380 --> 00:12:21.660 our periodic signal. 00:12:21.660 --> 00:12:29.550 omega_0 is 2 pi / T_0, as you recall from our discussion pf 00:12:29.550 --> 00:12:32.410 periodic signals and sinusoids before. 00:12:32.410 --> 00:12:34.500 And that's 2 pi f_0. 00:12:34.500 --> 00:12:36.930 So this is the fundamental frequency. 00:12:39.640 --> 00:12:45.240 Now let's examine, first of all, complex exponentials, and 00:12:45.240 --> 00:12:49.860 recognize, first of all, that there is a complex exponential 00:12:49.860 --> 00:12:54.060 that has exactly the same period and fundamental 00:12:54.060 --> 00:12:58.680 frequency as our more general periodic signal, namely the 00:12:58.680 --> 00:13:05.550 complex exponential e^(j omega_0 t), where omega_0 is 2 00:13:05.550 --> 00:13:12.640 pi / T_0, or equivalently, T_0 is 2 pi / omega_0. 00:13:12.640 --> 00:13:17.820 Now that's the complex exponential which has T_0 as 00:13:17.820 --> 00:13:19.760 the fundamental period. 00:13:19.760 --> 00:13:24.950 But there are harmonically related complex exponentials 00:13:24.950 --> 00:13:30.130 that also have T_0 as a period, although in fact, 00:13:30.130 --> 00:13:32.520 their fundamental period is shorter. 00:13:32.520 --> 00:13:37.017 So we can also look at complex exponentials of the form e^(j 00:13:37.017 --> 00:13:38.267 k omega_0 t). 00:13:39.710 --> 00:13:46.590 These likewise are periodic with a period of T_0. 00:13:46.590 --> 00:13:52.060 Although, in fact, their fundamental period is T_0 / k, 00:13:52.060 --> 00:13:55.480 or equivalently, 2 pi divided by their fundamental 00:13:55.480 --> 00:13:58.520 frequency, k omega_0. 00:13:58.520 --> 00:14:04.070 So as k, an integer, varies, these correspond to 00:14:04.070 --> 00:14:09.140 harmonically related complex exponentials. 00:14:09.140 --> 00:14:14.060 Now what the Fourier series says, and we'll justify this 00:14:14.060 --> 00:14:18.170 bit by bit as the discussion goes on, what the Fourier 00:14:18.170 --> 00:14:23.030 series says, and in fact, what Fourier said, which was 00:14:23.030 --> 00:14:27.150 essentially his brilliant insight, is that, if I have a 00:14:27.150 --> 00:14:31.670 very general periodic signal, I can represent it as a linear 00:14:31.670 --> 00:14:34.940 combination of these harmonically-related complex 00:14:34.940 --> 00:14:36.900 exponentials. 00:14:36.900 --> 00:14:41.950 So that representation is what I've indicated here. 00:14:41.950 --> 00:14:49.580 And this summation is what will be referred to as the 00:14:49.580 --> 00:14:52.890 Fourier series. 00:14:55.990 --> 00:14:59.320 And as we proceed with the discussion, there are two 00:14:59.320 --> 00:15:02.010 issues that will develop. 00:15:02.010 --> 00:15:06.670 One is, assuming that our periodic signal can be 00:15:06.670 --> 00:15:11.040 represented this way, how do we determine the Fourier 00:15:11.040 --> 00:15:14.820 series coefficients, as they're referred to, a_k. 00:15:14.820 --> 00:15:16.720 That's one question. 00:15:16.720 --> 00:15:20.580 And the second question will be how broad a class of 00:15:20.580 --> 00:15:24.070 signals, in fact, can be represented this way. 00:15:24.070 --> 00:15:27.340 And that's another question that we'll deal with 00:15:27.340 --> 00:15:28.590 separately. 00:15:30.120 --> 00:15:35.430 Now just focusing on this representation for a minute, 00:15:35.430 --> 00:15:43.230 this representation of the Fourier series, which I've 00:15:43.230 --> 00:15:52.000 repeated again here, is what's referred to as the complex 00:15:52.000 --> 00:15:55.910 exponential form of the Fourier series. 00:15:55.910 --> 00:15:59.740 And it's important to note, incidentally, that the 00:15:59.740 --> 00:16:05.740 summation involves frequencies, k omega_0, that 00:16:05.740 --> 00:16:07.800 are both positive and negative. 00:16:07.800 --> 00:16:12.420 In other words, this index k runs over limits that include 00:16:12.420 --> 00:16:16.700 both negative values and positive values. 00:16:16.700 --> 00:16:21.710 Now that complex exponential form is one representation for 00:16:21.710 --> 00:16:23.190 the Fourier series. 00:16:23.190 --> 00:16:27.080 And in fact, it's the one that we will be principally relying 00:16:27.080 --> 00:16:29.300 on in this course. 00:16:29.300 --> 00:16:32.090 There is another representation that perhaps 00:16:32.090 --> 00:16:37.030 you've come across previously and that in a variety of other 00:16:37.030 --> 00:16:40.520 contexts is typically used, which is called the 00:16:40.520 --> 00:16:44.010 trigonometric form for the Fourier series. 00:16:44.010 --> 00:16:47.240 Without really tracking through the algebra, 00:16:47.240 --> 00:16:50.770 essentially we can get to the trigonometric form from the 00:16:50.770 --> 00:16:56.890 complex exponential form by recognizing that if we express 00:16:56.890 --> 00:17:02.290 the complex coefficient in polar form or in rectangular 00:17:02.290 --> 00:17:09.150 form and expand the complex exponential term out in terms 00:17:09.150 --> 00:17:14.099 of cosine plus j sine, using just simply Euler's relation, 00:17:14.099 --> 00:17:21.220 then we will end up with a representation for the 00:17:21.220 --> 00:17:26.230 periodic signal, or a re-expression of the Fourier 00:17:26.230 --> 00:17:31.140 series expression that we had previously, either in the form 00:17:31.140 --> 00:17:35.660 that I indicate here, where now the periodic signal is 00:17:35.660 --> 00:17:41.090 expressed in terms of a summation of cosines with 00:17:41.090 --> 00:17:45.100 appropriate amplitude and phase. 00:17:45.100 --> 00:17:50.020 Or another equivalent trigonometric form involves 00:17:50.020 --> 00:17:54.450 rearranging this in terms of a combination 00:17:54.450 --> 00:17:57.910 of cosines and sines. 00:17:57.910 --> 00:18:02.160 Now in this representation, the frequencies of the 00:18:02.160 --> 00:18:08.790 sinusoids vary only over positive frequencies. 00:18:08.790 --> 00:18:12.860 And typically one thinks of periodic signals as having 00:18:12.860 --> 00:18:16.590 positive frequencies associated with them. 00:18:16.590 --> 00:18:23.910 However, let's look back and the complex exponential form 00:18:23.910 --> 00:18:26.600 for the Fourier series at the top of the board. 00:18:26.600 --> 00:18:30.110 And in that representation, when we use this 00:18:30.110 --> 00:18:35.180 representation, we'll find it convenient to refer to both 00:18:35.180 --> 00:18:38.700 positive frequencies and negative frequencies. 00:18:38.700 --> 00:18:42.800 So the representation that we will most typically be using 00:18:42.800 --> 00:18:45.310 is the complex exponential form. 00:18:45.310 --> 00:18:49.480 And in that form, what we'll find as we think of 00:18:49.480 --> 00:18:55.160 decomposing a periodic signal into its components at 00:18:55.160 --> 00:18:57.880 different frequencies, it will involve both positive 00:18:57.880 --> 00:18:59.610 frequencies and negative frequencies. 00:19:03.710 --> 00:19:09.850 Okay, now we have the Fourier series representation, as I've 00:19:09.850 --> 00:19:11.320 indicated here. 00:19:11.320 --> 00:19:16.050 Again, so far I've sidestepped the issue as to whether this 00:19:16.050 --> 00:19:18.040 in fact represents all the signals that 00:19:18.040 --> 00:19:20.340 we'd like to represent. 00:19:20.340 --> 00:19:24.730 Let's first address the issue of how we determine these 00:19:24.730 --> 00:19:28.350 coefficients a_k, assuming that, in fact, this 00:19:28.350 --> 00:19:30.450 representation is valid. 00:19:30.450 --> 00:19:32.930 And again, I'll kind of move through the 00:19:32.930 --> 00:19:36.490 algebra fairly quickly. 00:19:36.490 --> 00:19:40.460 The algebraic steps are ones that you can pursue more 00:19:40.460 --> 00:19:42.940 leisurely just to kind of verify them and 00:19:42.940 --> 00:19:44.660 step through them. 00:19:44.660 --> 00:19:48.460 But essentially, the algebra develops out of the 00:19:48.460 --> 00:19:53.580 recognition that if we integrate a complex 00:19:53.580 --> 00:19:58.140 exponential over one period, T_0-- 00:19:58.140 --> 00:20:02.610 and I mean by this notation that this is an integral over 00:20:02.610 --> 00:20:05.550 a period, where I don't particularly care where the 00:20:05.550 --> 00:20:09.340 period starts and where the period stops, in other words, 00:20:09.340 --> 00:20:12.040 exactly what period I picked-- 00:20:12.040 --> 00:20:17.140 that this integral is equal to T_0 when m is equal to 0. 00:20:17.140 --> 00:20:21.490 And it's equal to 0 if m is not equal to 0. 00:20:21.490 --> 00:20:26.250 That follows simply from the fact that if we substitute in 00:20:26.250 --> 00:20:29.640 for using or Euler's relation, so that we have the integral 00:20:29.640 --> 00:20:36.540 of a cosine plus j times the sine, if m is not equal to 0, 00:20:36.540 --> 00:20:41.470 then both of these integrals over a period are 0. 00:20:41.470 --> 00:20:45.320 The integral of a of a periodic of a sinusoid, cosine 00:20:45.320 --> 00:20:49.440 or sine, over an integral number of periods is 0. 00:20:49.440 --> 00:20:55.660 Whereas, if m is equal to 0, this integral will be equal to 00:20:55.660 --> 00:20:58.350 T_0, the integral of the cosine. 00:20:58.350 --> 00:21:01.840 And the integral of the sine is equal to 0. 00:21:04.400 --> 00:21:07.480 Okay, well, the next step in developing the expression for 00:21:07.480 --> 00:21:13.180 the coefficient a_k is to refer back to the Fourier 00:21:13.180 --> 00:21:18.170 series expression, which was that x(t) is equal to the sum 00:21:18.170 --> 00:21:19.420 of a_k e^(j k omega_0 t). 00:21:22.310 --> 00:21:28.370 If we multiply both sides of that by e^(-j n omega_0 t), 00:21:28.370 --> 00:21:34.135 and integrate that over a period-- 00:21:36.810 --> 00:21:41.340 both sides of the equation integrated over a period, so 00:21:41.340 --> 00:21:44.160 these two equations are equal-- 00:21:44.160 --> 00:21:47.680 and then in essence, interchange the summation and 00:21:47.680 --> 00:21:52.390 the integration so that this part of the expression comes 00:21:52.390 --> 00:21:57.030 outside the sum, and then we combine these two complex 00:21:57.030 --> 00:22:02.540 exponentials together, where we come out is the expression 00:22:02.540 --> 00:22:04.840 that I've indicated here. 00:22:04.840 --> 00:22:07.930 And then essentially what happens at this point, 00:22:07.930 --> 00:22:12.060 algebraically, is that we use the result that we just 00:22:12.060 --> 00:22:16.140 developed to evaluate this integral. 00:22:16.140 --> 00:22:19.800 So multiplying both sides of the Fourier series and then 00:22:19.800 --> 00:22:23.850 doing the integration leads us, after the appropriate 00:22:23.850 --> 00:22:27.670 manipulation, to the expression 00:22:27.670 --> 00:22:31.140 that I have up here. 00:22:31.140 --> 00:22:42.550 And this integral is equal to T_0 if k is equal to n, 00:22:42.550 --> 00:22:44.920 corresponding to 0 up here. 00:22:44.920 --> 00:22:49.020 And it's 0 otherwise, which is what we had demonstrated or 00:22:49.020 --> 00:22:50.840 argued previously. 00:22:50.840 --> 00:22:53.970 And the upshot of all that, then, is that the right hand 00:22:53.970 --> 00:22:58.620 side of this expression disappears except for the term 00:22:58.620 --> 00:23:01.020 when k is equal to n. 00:23:01.020 --> 00:23:07.400 And so finally, we have what I indicate here, taking T_0 and 00:23:07.400 --> 00:23:12.050 moving it over to the other side of the equation, that 00:23:12.050 --> 00:23:16.580 then tells us how we determine the Fourier series 00:23:16.580 --> 00:23:20.540 coefficients a_n, or a_k. 00:23:20.540 --> 00:23:25.810 So that, in effect, then is what we refer to as the 00:23:25.810 --> 00:23:30.040 analysis equation, the equation that begins with x(t) 00:23:30.040 --> 00:23:34.260 and tells us how to get the Fourier series coefficients. 00:23:34.260 --> 00:23:38.620 What I'll refer to as the Fourier series synthesis 00:23:38.620 --> 00:23:45.330 equation is the equation that tells us how to build x(t) out 00:23:45.330 --> 00:23:48.340 of these complex exponentials. 00:23:48.340 --> 00:23:51.680 So we have the synthesis equation, which is the one we 00:23:51.680 --> 00:23:53.070 started from. 00:23:53.070 --> 00:23:58.250 We have the analysis equation, which is the equation that we 00:23:58.250 --> 00:23:59.500 just developed. 00:24:02.210 --> 00:24:07.950 So we in effect have gone through the issue of, assuming 00:24:07.950 --> 00:24:10.580 that a Fourier series representation is in fact 00:24:10.580 --> 00:24:14.980 valid, how we get the coefficients. 00:24:14.980 --> 00:24:18.440 We'll want to address somewhat the question of how broad a 00:24:18.440 --> 00:24:21.680 class of signals are we talking about. 00:24:21.680 --> 00:24:26.090 And what's in fact amazing, and was Fourier's amazing 00:24:26.090 --> 00:24:29.250 insight, was that it's a very broad class of signals. 00:24:29.250 --> 00:24:34.690 But let's first look at just some examples in which we take 00:24:34.690 --> 00:24:38.170 a signal, assume that it has the Fourier series 00:24:38.170 --> 00:24:41.750 representation, and see what the Fourier series 00:24:41.750 --> 00:24:44.080 coefficients look like. 00:24:44.080 --> 00:24:50.710 So we'll begin with what I refer to as an antisymmetric 00:24:50.710 --> 00:24:53.110 periodic square wave-- 00:24:53.110 --> 00:24:56.010 periodic of course, because we're talking about periodic 00:24:56.010 --> 00:24:59.950 signals; square wave referring to its shape; and 00:24:59.950 --> 00:25:03.660 antisymmetric referring to the fact that it 00:25:03.660 --> 00:25:05.740 is an odd time function. 00:25:05.740 --> 00:25:07.440 In other words, it is 00:25:07.440 --> 00:25:12.030 antisymmetric about the origin. 00:25:12.030 --> 00:25:16.650 Now the expression for the Fourier series coefficients 00:25:16.650 --> 00:25:23.450 tells us that we determine a_k by taking 1 / T0 times the 00:25:23.450 --> 00:25:26.140 integral over a period of x(t), e^(-j k omega_0 t) dt. 00:25:30.490 --> 00:25:35.560 The most convenient thing in this case is to choose a 00:25:35.560 --> 00:25:41.550 period, which let's say goes from -T_0 / 2 to +T_0 / 2. 00:25:41.550 --> 00:25:44.370 So here x(t) is -1. 00:25:44.370 --> 00:25:47.110 Here x(t) is +1. 00:25:47.110 --> 00:25:52.060 And so I've expressed the Fourier series coefficients as 00:25:52.060 --> 00:25:55.950 this integral, that's from -T_0 / 2 to 0. 00:25:55.950 --> 00:26:02.010 And then added to that is the positive part of the cycle. 00:26:02.010 --> 00:26:05.550 And so we have these two integrals. 00:26:05.550 --> 00:26:10.150 Now, I don't want to track through the details of the 00:26:10.150 --> 00:26:11.490 algebra again. 00:26:11.490 --> 00:26:13.980 I guess I've decided that that's much more fun for you 00:26:13.980 --> 00:26:15.620 to do on your own. 00:26:15.620 --> 00:26:19.270 But the way it comes out when you go through it is the 00:26:19.270 --> 00:26:24.100 expression that I finally indicate after suggesting that 00:26:24.100 --> 00:26:26.860 there are few more steps to follow. 00:26:26.860 --> 00:26:31.970 And what develops is that those two integrals together, 00:26:31.970 --> 00:26:38.040 for k not equal to 0, come out to this expression. 00:26:38.040 --> 00:26:42.830 And that expression is not valid for k = 0. 00:26:42.830 --> 00:26:48.160 For k equal to 0, we can go back to the basic expression 00:26:48.160 --> 00:26:53.340 for the Fourier series, which is 1 / T_0, the integral over 00:26:53.340 --> 00:26:58.770 a period, x(t) e^(-j k omega_0 t) dt. 00:26:58.770 --> 00:27:04.170 For k = 0, of course this term just simply becomes 1. 00:27:04.170 --> 00:27:10.520 And so the zeroth coefficient is 1 / T_0 times the integral 00:27:10.520 --> 00:27:13.730 of x(t) over a period. 00:27:13.730 --> 00:27:18.460 Now, going back to the original function that we 00:27:18.460 --> 00:27:23.460 have, what we're saying then is that the zeroth coefficient 00:27:23.460 --> 00:27:29.780 is 1 / T_0 times the integral over one period, which is, in 00:27:29.780 --> 00:27:31.950 effect, the average value. 00:27:31.950 --> 00:27:35.060 And it's straightforward to verify for this case that 00:27:35.060 --> 00:27:37.210 average value is equal to 0. 00:27:40.050 --> 00:27:45.540 Now let's look at these Fourier series coefficients on 00:27:45.540 --> 00:27:47.840 a bar graph. 00:27:47.840 --> 00:27:50.610 And I've indicated that here. 00:27:50.610 --> 00:27:53.090 The expression for the Fourier series 00:27:53.090 --> 00:27:54.980 coefficients we just developed. 00:27:54.980 --> 00:27:58.110 And it involves-- 00:27:58.110 --> 00:28:02.270 it's 0 for k = 0, it's a factor of this 00:28:02.270 --> 00:28:04.990 form for k =/= 0. 00:28:04.990 --> 00:28:10.330 Plotted on a bar graph, then we see values like this, 0 at 00:28:10.330 --> 00:28:13.920 k = 0 and then associated values. 00:28:13.920 --> 00:28:16.820 And there are a number of things to focus on when you 00:28:16.820 --> 00:28:19.890 look at this. 00:28:19.890 --> 00:28:24.210 One is the fact that the Fourier series coefficients 00:28:24.210 --> 00:28:29.870 for this example are purely imaginary. 00:28:29.870 --> 00:28:33.160 A second is that the Fourier series coefficients for this 00:28:33.160 --> 00:28:36.200 example are an odd sequence. 00:28:36.200 --> 00:28:39.160 In other words, if you look at this sequence, what you see 00:28:39.160 --> 00:28:44.460 are these values for -k flipped over. 00:28:44.460 --> 00:28:48.660 So they're imaginary and odd. 00:28:48.660 --> 00:28:57.430 And what that results in, when you look at the trigonometric 00:28:57.430 --> 00:29:02.300 form of the Fourier series, is that in fact, those 00:29:02.300 --> 00:29:06.060 conditions, if you put the terms all together, lead you 00:29:06.060 --> 00:29:10.590 to a trigonometric representation, which involves 00:29:10.590 --> 00:29:13.020 only sine terms-- 00:29:13.020 --> 00:29:15.280 in other words, no cosine terms. 00:29:15.280 --> 00:29:18.470 Let me just draw your attention to the fact that, 00:29:18.470 --> 00:29:23.760 since a_k's are imaginary, this j takes care of that fact 00:29:23.760 --> 00:29:27.530 so that these coefficients are in fact real. 00:29:27.530 --> 00:29:33.350 So what this says is that for the antisymmetric square wave, 00:29:33.350 --> 00:29:36.845 in effect, the Fourier series is a sine series. 00:29:36.845 --> 00:29:40.910 The antisymmetric square wave is an odd function. 00:29:40.910 --> 00:29:43.450 Sinusoids are odd functions. 00:29:43.450 --> 00:29:45.590 And so this is all kind of reasonable, that we're 00:29:45.590 --> 00:29:50.760 building an odd function out of odd functions. 00:29:50.760 --> 00:29:56.320 As an additional aside, which I won't exploit or refer to 00:29:56.320 --> 00:29:59.740 any further here, but just draw your attention to, is 00:29:59.740 --> 00:30:04.530 that another aspect of this periodic square wave, the 00:30:04.530 --> 00:30:08.890 particular one that we chose, is that it is what's referred 00:30:08.890 --> 00:30:11.720 to as an odd harmonic function. 00:30:11.720 --> 00:30:17.480 In other words, for even values of k, the Fourier 00:30:17.480 --> 00:30:19.660 series coefficients are 0. 00:30:19.660 --> 00:30:22.734 They're are only non-zero for odd values of k. 00:30:25.340 --> 00:30:27.970 Now let's look at another example. 00:30:27.970 --> 00:30:31.430 Another example is the symmetric 00:30:31.430 --> 00:30:33.830 periodic square wave. 00:30:33.830 --> 00:30:40.240 And this is in fact example 4.5, worked out in more detail 00:30:40.240 --> 00:30:40.910 in the text. 00:30:40.910 --> 00:30:46.150 Then I won't bother to work this out in detail here, 00:30:46.150 --> 00:30:50.630 except to draw your attention to several points. 00:30:50.630 --> 00:30:54.330 Here is the symmetric periodic square wave. 00:30:54.330 --> 00:30:58.470 And what I mean by symmetric is that it's 00:30:58.470 --> 00:31:00.750 an even time function. 00:31:00.750 --> 00:31:06.430 Now just kind of extrapolating your intuition, what you 00:31:06.430 --> 00:31:10.120 should expect is that if it's only an even time function, it 00:31:10.120 --> 00:31:13.820 should be built up or buildable, if it's buildable 00:31:13.820 --> 00:31:18.110 at all, out of only even sinusoids. 00:31:18.110 --> 00:31:20.630 And in fact, that's the case. 00:31:20.630 --> 00:31:25.720 So if we look at the Fourier series coefficients for this, 00:31:25.720 --> 00:31:29.700 is zeroth coefficient, again, is the average value, which in 00:31:29.700 --> 00:31:31.350 this case, is 1/2. 00:31:31.350 --> 00:31:35.550 Here I've plotted pi times the Fourier series coefficients. 00:31:35.550 --> 00:31:39.590 So the zeroth value is pi / 2. 00:31:39.590 --> 00:31:45.280 The coefficients are now an even sequence, in other words, 00:31:45.280 --> 00:31:48.000 symmetric about k = 0. 00:31:48.000 --> 00:31:51.760 And the consequence of that is that when you take these 00:31:51.760 --> 00:31:57.080 coefficients and put together the equivalent trigonometric 00:31:57.080 --> 00:32:05.290 form, the trigonometric form involves only cosines and no 00:32:05.290 --> 00:32:08.040 sine terms. 00:32:08.040 --> 00:32:12.530 Now you'll see this in other examples, not that we'll do in 00:32:12.530 --> 00:32:15.090 the lecture, but examples in the text and in the video 00:32:15.090 --> 00:32:19.150 manual, if in fact the square wave was neither symmetric or 00:32:19.150 --> 00:32:22.590 antisymmetric, then the trigonometric form would 00:32:22.590 --> 00:32:25.480 involve both sines and cosines. 00:32:25.480 --> 00:32:30.660 And that is, of course, the more general case. 00:32:30.660 --> 00:32:35.250 Furthermore, in the two examples I've shown here, in 00:32:35.250 --> 00:32:40.310 both cases, the signal is odd harmonic. 00:32:40.310 --> 00:32:44.720 In other words, for even values of k, the coefficients 00:32:44.720 --> 00:32:46.710 are equal to 0. 00:32:46.710 --> 00:32:49.260 Although I won't justify that here, that's a consequence of 00:32:49.260 --> 00:32:54.570 the fact that this symmetry is exactly about half a period. 00:32:54.570 --> 00:32:58.570 And if you made the on time of the square wave different in 00:32:58.570 --> 00:33:00.980 relation to the off time, then that 00:33:00.980 --> 00:33:02.575 property would also disappear. 00:33:06.400 --> 00:33:13.240 Now what's kind of amazing, actually, is that if we take a 00:33:13.240 --> 00:33:17.500 square wave, like I have here or as I had in the 00:33:17.500 --> 00:33:21.940 antisymmetric case, the implication is that I can 00:33:21.940 --> 00:33:26.000 build that square wave by adding up 00:33:26.000 --> 00:33:30.110 enough sines or cosines. 00:33:30.110 --> 00:33:33.690 And it really seems kind of amazing because the square 00:33:33.690 --> 00:33:37.700 wave, after all, is a very discontinuous function. 00:33:37.700 --> 00:33:40.080 Sinusoids are very continuous. 00:33:40.080 --> 00:33:43.760 And it seems puzzling that in fact you can do that. 00:33:43.760 --> 00:33:48.690 Well let's look in a little bit of detail how the 00:33:48.690 --> 00:33:55.280 sinusoidal terms add up to build a square wave. 00:33:55.280 --> 00:34:01.750 And to do that, let's first define what I refer to as a 00:34:01.750 --> 00:34:03.480 partial sum. 00:34:03.480 --> 00:34:10.670 So here we have the expression which is the synthesis 00:34:10.670 --> 00:34:15.510 equation, telling us how x(t) could be represented as 00:34:15.510 --> 00:34:18.179 complex exponentials if it can be. 00:34:18.179 --> 00:34:21.280 And let's consider just a finite number of 00:34:21.280 --> 00:34:22.770 terms in this sum. 00:34:22.770 --> 00:34:27.630 And so x_n(t), of course, as n goes to infinity, approaches 00:34:27.630 --> 00:34:30.469 the infinite sum that we're talking about. 00:34:30.469 --> 00:34:32.889 And although we could do this more generally, let's not. 00:34:32.889 --> 00:34:36.760 Let's focus on the symmetric square wave case, where 00:34:36.760 --> 00:34:39.880 because of the symmetry of these coefficients, namely 00:34:39.880 --> 00:34:46.810 that a_k is equal to a_(-k), we can rewrite these terms as 00:34:46.810 --> 00:34:48.389 cosine terms. 00:34:48.389 --> 00:34:52.880 And so this partial sum can be expressed the way I'm 00:34:52.880 --> 00:34:55.900 expressing it here. 00:34:55.900 --> 00:34:58.990 Well let's look at a few of these terms. 00:34:58.990 --> 00:35:05.930 On the graph, I have, first of all, x(t), which is our 00:35:05.930 --> 00:35:09.330 original square wave. 00:35:09.330 --> 00:35:14.400 The term that I indicate here is the factor of 1/2, 00:35:14.400 --> 00:35:19.530 which is this term. 00:35:19.530 --> 00:35:22.080 With n = 1, that would correspond to adding one 00:35:22.080 --> 00:35:23.940 cosine term to that. 00:35:23.940 --> 00:35:28.420 And so the sum of those two would be this, which looks a 00:35:28.420 --> 00:35:32.680 little closer to the square wave, but certainly not very 00:35:32.680 --> 00:35:35.050 close to it at all. 00:35:35.050 --> 00:35:39.570 And in fact, it's somewhat hard to imagine without seeing 00:35:39.570 --> 00:35:43.380 the terms build up how in fact, by adding more and more 00:35:43.380 --> 00:35:47.470 terms, we can generate something that is essentially 00:35:47.470 --> 00:35:50.230 flat, except at the discontinuities. 00:35:50.230 --> 00:35:52.340 So let's look at this example. 00:35:52.340 --> 00:35:57.420 And what I'd like to show is this example, but now as we 00:35:57.420 --> 00:35:59.580 add many more terms to it. 00:35:59.580 --> 00:36:04.640 And let's see in fact how these individual terms add up 00:36:04.640 --> 00:36:07.150 to build up the square wave. 00:36:07.150 --> 00:36:11.190 So this is the square wave that we want to build up 00:36:11.190 --> 00:36:15.280 through the Fourier series as a sum of sinusoids. 00:36:15.280 --> 00:36:19.590 And the term for k = 0 will be a constant which represents 00:36:19.590 --> 00:36:21.550 the DC value of this. 00:36:21.550 --> 00:36:27.380 And so in the partial sum, as we develop it, the first thing 00:36:27.380 --> 00:36:31.450 that we'll show is just the term for k = 0. 00:36:31.450 --> 00:36:38.020 Now for k = 1, we would add to that one sinusoidal term. 00:36:38.020 --> 00:36:43.410 And so the sum of the term for k = 1 and k = 0 00:36:43.410 --> 00:36:44.800 is represented here. 00:36:48.350 --> 00:36:52.250 Now when we go to k = 2, because of the fact that this 00:36:52.250 --> 00:36:58.450 is an odd harmonic function, in fact, the term for k = 2 00:36:58.450 --> 00:37:03.230 will have zero amplitude and so this won't change. 00:37:03.230 --> 00:37:08.480 Here we show the Fourier series with k = 2 00:37:08.480 --> 00:37:10.390 and there's no change. 00:37:10.390 --> 00:37:14.110 And then we will go to k = 3. 00:37:14.110 --> 00:37:16.650 And we will be adding, then, one 00:37:16.650 --> 00:37:18.570 additional sinusoidal term. 00:37:18.570 --> 00:37:20.620 Here is k = 3. 00:37:20.620 --> 00:37:25.120 When we go to k = 4, again, there won't be any change. 00:37:25.120 --> 00:37:31.420 But there will be another term that's added at k = 5 here. 00:37:31.420 --> 00:37:35.560 Then k = 6, again, because it's odd harmonic, no change. 00:37:35.560 --> 00:37:41.350 And finally k = 7 is shown here. 00:37:41.350 --> 00:37:45.040 And we can begin to see that this starts to look somewhat 00:37:45.040 --> 00:37:47.160 like the square wave. 00:37:47.160 --> 00:37:52.500 But now to really emphasize how this builds up, let's more 00:37:52.500 --> 00:37:56.680 rapidly add many more terms, and in fact increase the 00:37:56.680 --> 00:38:00.790 number of terms up to about 100, recognizing that the 00:38:00.790 --> 00:38:05.560 shape will only change on the inclusion of the odd-numbered 00:38:05.560 --> 00:38:08.370 terms, not the even-numbered terms, because it's an odd 00:38:08.370 --> 00:38:10.970 harmonic function. 00:38:10.970 --> 00:38:16.620 So now we're increasing and we're building up toward k = 00:38:16.620 --> 00:38:18.530 100, 100 terms. 00:38:18.530 --> 00:38:24.970 And notice that it is the higher-order terms that tend 00:38:24.970 --> 00:38:30.340 to build up the discontinuity corresponding to the notion 00:38:30.340 --> 00:38:34.940 that the discontinuity, or sharp edges in a signal, in 00:38:34.940 --> 00:38:39.340 fact, are represented through the higher frequencies in the 00:38:39.340 --> 00:38:41.550 Fourier series. 00:38:41.550 --> 00:38:44.900 And here we have a not-too-unreasonable 00:38:44.900 --> 00:38:48.150 approximation to the original square wave. 00:38:48.150 --> 00:38:51.890 There is the artifact of the ripples at the discontinuity. 00:38:51.890 --> 00:38:56.710 And in fact, that rippling behavior at the discontinuity 00:38:56.710 --> 00:39:00.180 is referred to the Gibbs phenomenon. 00:39:00.180 --> 00:39:03.650 And it's an inherent part of the Fourier series 00:39:03.650 --> 00:39:06.380 representation at discontinuities. 00:39:06.380 --> 00:39:10.700 Now to emphasize this, let's decrease the number 00:39:10.700 --> 00:39:13.450 of terms back down. 00:39:13.450 --> 00:39:21.600 And we will carry this down to k = 1, again to emphasize how 00:39:21.600 --> 00:39:26.370 the sinusoids are building up the square wave. 00:39:26.370 --> 00:39:29.100 Here we are back at k = 1. 00:39:29.100 --> 00:39:33.490 And then finally, we will add back in the sinusoids 00:39:33.490 --> 00:39:34.640 that we took out. 00:39:34.640 --> 00:39:39.550 And let's build this back up to 100 terms, showing the 00:39:39.550 --> 00:39:43.380 approximation that we generated with 100 terms to 00:39:43.380 --> 00:39:44.630 the square wave. 00:40:00.900 --> 00:40:05.650 Okay, so what you saw is that, in fact, we got awfully close 00:40:05.650 --> 00:40:06.990 to a square wave. 00:40:06.990 --> 00:40:09.810 And the other thing that was kind of interesting about it 00:40:09.810 --> 00:40:14.610 as it went along was the fact that, with the low 00:40:14.610 --> 00:40:17.650 frequencies, what we were tending to build was the 00:40:17.650 --> 00:40:19.460 general behavior. 00:40:19.460 --> 00:40:26.490 And as the higher frequencies came in, that tended 00:40:26.490 --> 00:40:28.590 contribute to the discontinuity. 00:40:28.590 --> 00:40:34.030 And in fact, something that will stand out more and more 00:40:34.030 --> 00:40:37.150 as we go through our discussion of Fourier series 00:40:37.150 --> 00:40:41.430 and Fourier transforms, is that general statement, that 00:40:41.430 --> 00:40:48.750 it's the low-frequency terms that represent the broad time 00:40:48.750 --> 00:40:52.860 behavior, and it's the high-frequency terms that are 00:40:52.860 --> 00:40:54.940 used to build up the sharp 00:40:54.940 --> 00:40:56.660 transitions in the time domain. 00:41:00.060 --> 00:41:03.670 Now we need to get a little more precise about the 00:41:03.670 --> 00:41:10.210 question of how, in fact, the Fourier series, or when the 00:41:10.210 --> 00:41:13.730 Fourier series represents the functions that we're talking 00:41:13.730 --> 00:41:17.140 about and in what sense they represent them. 00:41:17.140 --> 00:41:26.840 And so if we look again at the synthesis equation, what we 00:41:26.840 --> 00:41:31.750 really want to ask is, if we add up enough of these terms, 00:41:31.750 --> 00:41:37.700 in what sense does this sum represent this time function? 00:41:37.700 --> 00:41:42.660 Well, let's again use the notion of our partial sum. 00:41:42.660 --> 00:41:47.000 So we have the partial sum down here. 00:41:47.000 --> 00:41:51.120 And we can think of the difference between this 00:41:51.120 --> 00:41:59.300 partial sum and the original time function as the error. 00:41:59.300 --> 00:42:02.210 And I've defined the error here. 00:42:02.210 --> 00:42:07.540 And what we would like to know is does this error decrease as 00:42:07.540 --> 00:42:10.150 we add more and more terms? 00:42:10.150 --> 00:42:13.660 And in fact, in what sense, if the error does decrease, in 00:42:13.660 --> 00:42:16.640 what sense does it decrease? 00:42:16.640 --> 00:42:20.420 Now in detail this is a fairly complicated 00:42:20.420 --> 00:42:21.810 and elaborate topic. 00:42:21.810 --> 00:42:23.930 I don't mean to make that sound frightening. 00:42:23.930 --> 00:42:27.130 It's mainly a statement that I don't want to 00:42:27.130 --> 00:42:29.770 explore in a lot of detail. 00:42:29.770 --> 00:42:34.030 But it relates to what it's referred to as the issue of 00:42:34.030 --> 00:42:37.010 convergence of the Fourier series. 00:42:37.010 --> 00:42:41.310 And the convergence of the Fourier series, the bottom 00:42:41.310 --> 00:42:45.220 line on it, the kind of end statement, can be made in 00:42:45.220 --> 00:42:47.710 several ways. 00:42:47.710 --> 00:42:50.090 One statement related to the convergence of the Fourier 00:42:50.090 --> 00:42:52.590 series is the following. 00:42:52.590 --> 00:42:58.590 If I have a time function, which is what is referred to 00:42:58.590 --> 00:43:01.400 as square integrable, namely its integral, over 00:43:01.400 --> 00:43:04.350 a period, is finite. 00:43:04.350 --> 00:43:10.060 Then what you can show, kind of amazingly, is that the 00:43:10.060 --> 00:43:14.440 energy in that error, in other words, the energy and the 00:43:14.440 --> 00:43:17.160 difference between the original function and the 00:43:17.160 --> 00:43:21.370 partial sum, the energy in that, goes to 0 00:43:21.370 --> 00:43:22.770 as n goes to infinity. 00:43:25.430 --> 00:43:30.120 A somewhat tighter condition is a condition referred to as 00:43:30.120 --> 00:43:37.590 it Dirichlet conditions, which says that if the time function 00:43:37.590 --> 00:43:42.170 is absolutely integrable, not square integrable, but 00:43:42.170 --> 00:43:44.170 absolutely integrable-- 00:43:44.170 --> 00:43:47.510 and I've kind of hedged the issue by just simply referring 00:43:47.510 --> 00:43:50.800 to x(t) as being well behaved-- 00:43:50.800 --> 00:43:56.170 then the statement is that the error in fact goes to 0 as n 00:43:56.170 --> 00:44:00.090 increases, except at the discontinuities. 00:44:00.090 --> 00:44:04.690 And what well behaved means in that statement is that, as 00:44:04.690 --> 00:44:07.580 discussed in the book, there are a finite number of maxima 00:44:07.580 --> 00:44:11.310 and minima in any period and a finite number of finite 00:44:11.310 --> 00:44:15.500 discontinuities, which is, essentially, always the case. 00:44:15.500 --> 00:44:19.890 So under square integrability what we have is the statement 00:44:19.890 --> 00:44:23.240 not that the partial sum goes to the right value at every 00:44:23.240 --> 00:44:28.250 point, but that the energy in the error goes to 0. 00:44:28.250 --> 00:44:30.790 Under the Dirichlet conditions, it says that, in 00:44:30.790 --> 00:44:35.800 fact, the signal goes to the right value at every time 00:44:35.800 --> 00:44:40.560 instant except at the discontinuities. 00:44:40.560 --> 00:44:47.770 So going back to the square wave, the square wave 00:44:47.770 --> 00:44:49.880 satisfies either one of those conditions. 00:44:49.880 --> 00:44:55.180 And so what the consequence is is that, with the square wave, 00:44:55.180 --> 00:44:59.680 if we looked at the error, then in fact what we would 00:44:59.680 --> 00:45:04.690 find is that the energy in the error would go to zero as we 00:45:04.690 --> 00:45:07.810 add more and more terms in the partial sum. 00:45:07.810 --> 00:45:11.890 And in fact, since the square wave also satisfies the 00:45:11.890 --> 00:45:16.770 Dirichlet conditions, the actual value of the error, the 00:45:16.770 --> 00:45:22.180 difference between the partial sum and the true value, will 00:45:22.180 --> 00:45:23.610 actually go to 0. 00:45:23.610 --> 00:45:28.330 That difference will go to 0 except at the discontinuities. 00:45:28.330 --> 00:45:34.860 And that, in fact, is kind of evident as we watch the 00:45:34.860 --> 00:45:38.170 function build up by adding up these terms. 00:45:38.170 --> 00:45:44.880 And so in fact, let's go back and see again the development 00:45:44.880 --> 00:45:47.690 of the partial sums in relation to the 00:45:47.690 --> 00:45:49.700 original time function. 00:45:49.700 --> 00:45:53.000 Let's observe, this time again, basically what we saw 00:45:53.000 --> 00:45:56.710 before, which is that it builds up to the right answer. 00:45:56.710 --> 00:46:00.200 And furthermore what we'll plot this time, also as a 00:46:00.200 --> 00:46:03.540 function of time, is the energy in the error. 00:46:03.540 --> 00:46:07.300 And what we'll see is that the energy in the error will be 00:46:07.300 --> 00:46:12.270 tending towards 0 as the number of terms increases. 00:46:12.270 --> 00:46:15.310 So once again, we have the square wave. 00:46:15.310 --> 00:46:21.010 And we want to again show the buildup of the Fourier series, 00:46:21.010 --> 00:46:25.510 this time showing also how the energy in the error decreases 00:46:25.510 --> 00:46:28.120 as we add more and more terms. 00:46:28.120 --> 00:46:32.930 Well, once again, we'll begin with k = 0, corresponding to 00:46:32.930 --> 00:46:34.510 the constant term. 00:46:34.510 --> 00:46:38.060 And what's shown on the bottom trace is the energy in the 00:46:38.060 --> 00:46:40.120 error between those two. 00:46:40.120 --> 00:46:46.870 And we'll then add the term k = 1 to the DC term and we'll 00:46:46.870 --> 00:46:52.150 see that the energy will decrease when we do that. 00:46:52.150 --> 00:46:56.160 Here we have then the sum of k = 0 and k = 1. 00:46:56.160 --> 00:47:01.470 Now with k = 2, the energy won't decrease any further 00:47:01.470 --> 00:47:03.685 because it's an odd harmonic function. 00:47:07.310 --> 00:47:10.210 That's what we've just added in. 00:47:10.210 --> 00:47:13.710 When we add in the term for k = 3, again, we'll see the 00:47:13.710 --> 00:47:16.430 energy in the error decrease as reflected 00:47:16.430 --> 00:47:17.680 in the bottom curve. 00:47:22.440 --> 00:47:25.620 So there we are at k = 3. 00:47:25.620 --> 00:47:30.100 When we go to k = 4, there again is no 00:47:30.100 --> 00:47:32.850 change in the error. 00:47:32.850 --> 00:47:35.203 At k = 5, again the error decreases. 00:47:39.890 --> 00:47:43.940 k = 6, there will be no change again. 00:47:43.940 --> 00:47:47.500 And at k = 7, the energy decreases. 00:47:47.500 --> 00:47:51.020 And now let's show how the error decreases by building up 00:47:51.020 --> 00:47:53.830 the number of terms much more rapidly. 00:47:53.830 --> 00:47:56.610 Already the error has gotten somewhat small on the scale in 00:47:56.610 --> 00:48:00.380 which we're showing it, so let's expand out the error 00:48:00.380 --> 00:48:04.090 scale, the vertical axis displaying the energy in the 00:48:04.090 --> 00:48:09.220 error, so that we could watch how the energy decreases as we 00:48:09.220 --> 00:48:11.090 add more and more terms. 00:48:11.090 --> 00:48:15.070 So here we have the vertical scale expanded. 00:48:15.070 --> 00:48:20.030 And now what we'll do is increase the number of terms 00:48:20.030 --> 00:48:23.490 in the Fourier series and watch the energy in the error 00:48:23.490 --> 00:48:28.750 decreasing, always decreasing, of course, on the inclusion of 00:48:28.750 --> 00:48:33.380 the odd-numbered terms and not on the inclusion of the 00:48:33.380 --> 00:48:36.380 even-numbered terms because of the fact that it's an odd 00:48:36.380 --> 00:48:38.030 harmonic function. 00:48:38.030 --> 00:48:41.230 Now the energy in the error asymptotically will approach 00:48:41.230 --> 00:48:46.580 0, although point by point, the Fourier series will never 00:48:46.580 --> 00:48:48.820 be equal to the square wave. 00:48:48.820 --> 00:48:53.370 It will, at every instant of time, except at the 00:48:53.370 --> 00:48:58.640 discontinuities, where there will always be some ripple 00:48:58.640 --> 00:49:01.370 corresponding to what's referred to as the Gibbs 00:49:01.370 --> 00:49:02.620 phenomenon. 00:49:06.190 --> 00:49:11.870 So what we've seen, then, is a quick look at the Fourier 00:49:11.870 --> 00:49:17.560 series representation of periodic signals. 00:49:17.560 --> 00:49:25.370 We more broadly want to have a more general representation of 00:49:25.370 --> 00:49:28.080 signals in terms of complex exponentials. 00:49:28.080 --> 00:49:32.250 And so our next step will be to move toward a 00:49:32.250 --> 00:49:37.440 representation of nonperiodic or aperiodic signals. 00:49:37.440 --> 00:49:40.950 Now the details of this, I leave for the next lecture. 00:49:40.950 --> 00:49:44.760 The only thought that I want to introduce at this point is 00:49:44.760 --> 00:49:49.420 the basic strategy which is somewhat amazing and kind of 00:49:49.420 --> 00:49:52.810 interesting to reflect on in the interim. 00:49:52.810 --> 00:50:00.060 The basic strategy with an aperiodic signal is to think 00:50:00.060 --> 00:50:05.410 of representing this aperiodic signal as a linear combination 00:50:05.410 --> 00:50:12.590 of complex exponentials by the simple trick of periodically 00:50:12.590 --> 00:50:18.540 replicating this signal, generating a periodic signal 00:50:18.540 --> 00:50:21.590 using a Fourier series representation for that 00:50:21.590 --> 00:50:27.100 periodic signal, and then simply letting the period go 00:50:27.100 --> 00:50:28.650 to infinity. 00:50:28.650 --> 00:50:31.620 As the period goes to infinity, that periodic signal 00:50:31.620 --> 00:50:35.520 becomes the original aperiodic one that we had before. 00:50:35.520 --> 00:50:39.470 And the Fourier series representation then becomes 00:50:39.470 --> 00:50:43.340 what we'll refer to the Fourier transform. 00:50:43.340 --> 00:50:49.060 So that's just a quick look at the basic idea and approach 00:50:49.060 --> 00:50:50.140 that we'll take. 00:50:50.140 --> 00:50:52.620 In the next lecture, we'll develop this a little more 00:50:52.620 --> 00:50:55.500 carefully and more fully, moving from the Fourier 00:50:55.500 --> 00:50:59.670 series, which we've used for periodic signals, to develop 00:50:59.670 --> 00:51:03.740 the Fourier transform, which will then be representation 00:51:03.740 --> 00:51:05.310 for aperiodic signals. 00:51:05.310 --> 00:51:06.560 Thank you.