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. 00:00:56.260 --> 00:00:58.910 PROFESSOR: Last time, we began to address the issue of 00:00:58.910 --> 00:01:02.610 building continuous time signals out of a linear 00:01:02.610 --> 00:01:05.560 combination of complex exponentials. 00:01:05.560 --> 00:01:10.110 And for the class of periodic signals specifically, what 00:01:10.110 --> 00:01:13.350 this led to was the Fourier series representation for 00:01:13.350 --> 00:01:15.020 periodic signals. 00:01:15.020 --> 00:01:17.270 Let me just summarize the results that we 00:01:17.270 --> 00:01:19.110 developed last time. 00:01:19.110 --> 00:01:23.880 For periodic signals, we had the continuous-time Fourier 00:01:23.880 --> 00:01:29.220 series, where we built the periodic signal out of a 00:01:29.220 --> 00:01:33.250 linear combination of harmonically related complex 00:01:33.250 --> 00:01:34.750 exponentials. 00:01:34.750 --> 00:01:39.030 And what that led to was what we referred to as the 00:01:39.030 --> 00:01:41.050 synthesis equation. 00:01:41.050 --> 00:01:45.900 And we briefly addressed the issue of when this in fact 00:01:45.900 --> 00:01:50.360 builds, when this in fact is a complete representation of the 00:01:50.360 --> 00:01:55.970 periodic signal, and in essence, what we presented was 00:01:55.970 --> 00:02:00.540 conditions either for x of t being square integrable or x 00:02:00.540 --> 00:02:03.090 of t being absolutely integrable. 00:02:03.090 --> 00:02:06.880 Then, the other side of the Fourier series is what I 00:02:06.880 --> 00:02:10.380 referred to as the analysis equation. 00:02:10.380 --> 00:02:14.120 And the analysis equation was the equation that told us how 00:02:14.120 --> 00:02:19.110 we get the Fourier series coefficients from x of t. 00:02:19.110 --> 00:02:23.430 And so this equation together with the synthesis equation 00:02:23.430 --> 00:02:26.600 represent the Fourier series description 00:02:26.600 --> 00:02:29.470 for periodic signals. 00:02:29.470 --> 00:02:34.580 Now what we'd like to do is extend this idea to provide a 00:02:34.580 --> 00:02:39.460 mechanism for building non-periodic signals also out 00:02:39.460 --> 00:02:42.500 of a linear combination of complex exponentials. 00:02:42.500 --> 00:02:47.310 And the basic idea behind doing this is very simple and 00:02:47.310 --> 00:02:50.680 also very clever as I indicated last time. 00:02:50.680 --> 00:02:54.850 Essentially, the thought is the following, if we have a 00:02:54.850 --> 00:02:59.220 non-periodic signal or aperiodic signal, we can think 00:02:59.220 --> 00:03:04.720 of constructing a periodic signal by simply periodically 00:03:04.720 --> 00:03:08.230 replicating that aperiodic signal. 00:03:08.230 --> 00:03:12.520 So for example, if I have an aperiodic signal as I've 00:03:12.520 --> 00:03:18.870 indicated here, I can consider building a periodic signal, 00:03:18.870 --> 00:03:23.700 where I simply take this original signal and repeat it 00:03:23.700 --> 00:03:27.910 at multiples of some period t 0. 00:03:27.910 --> 00:03:30.380 Now, two things to recognize about this. 00:03:30.380 --> 00:03:36.155 One is that the periodic signal is equal to the 00:03:36.155 --> 00:03:39.150 aperiodic signal over one period. 00:03:39.150 --> 00:03:45.020 And the second is that as the period goes to infinity then, 00:03:45.020 --> 00:03:52.490 in fact, the periodic signal goes to the aperiodic signal. 00:03:52.490 --> 00:03:59.360 So the basic idea then is to use the Fourier series to 00:03:59.360 --> 00:04:03.920 represent the periodic signal, and then examine the Fourier 00:04:03.920 --> 00:04:08.880 series expression as we let the period go to infinity. 00:04:08.880 --> 00:04:14.150 Well, let's quickly see how this develops in terms of the 00:04:14.150 --> 00:04:16.130 associated equations. 00:04:16.130 --> 00:04:22.079 Here, again, we have the periodic signal. 00:04:22.079 --> 00:04:26.920 And what we want to inquire into is what happens to the 00:04:26.920 --> 00:04:30.690 Fourier series expression for this as we let the period go 00:04:30.690 --> 00:04:31.750 to infinity. 00:04:31.750 --> 00:04:35.140 As that happens, whatever Fourier series representation 00:04:35.140 --> 00:04:40.260 we end up with will correspond also to a representation for 00:04:40.260 --> 00:04:42.850 this aperiodic signal. 00:04:42.850 --> 00:04:44.220 Well, let's see. 00:04:44.220 --> 00:04:48.040 The Fourier series synthesis expression for the periodic 00:04:48.040 --> 00:04:53.650 signal expresses x tilde of t, the periodic signal as a 00:04:53.650 --> 00:04:57.230 linear combination of harmonically related complex 00:04:57.230 --> 00:05:01.510 exponentials with the fundamental frequency omega 0 00:05:01.510 --> 00:05:04.370 equaled to 2 pi divided by the period. 00:05:04.370 --> 00:05:11.530 And the analysis equation tells us what the relationship 00:05:11.530 --> 00:05:16.180 is for the coefficients in terms of the periodic signal. 00:05:16.180 --> 00:05:20.215 Now, I indicated that the periodic signal and the 00:05:20.215 --> 00:05:23.370 aperiodic signal are equal over one period. 00:05:23.370 --> 00:05:27.640 We recognize that this integration, in fact, only 00:05:27.640 --> 00:05:29.460 occurs over one period. 00:05:29.460 --> 00:05:33.840 And so we can re-express this in terms of our original 00:05:33.840 --> 00:05:35.510 aperiodic signal. 00:05:35.510 --> 00:05:38.730 So this tells us the Fourier series coefficients in 00:05:38.730 --> 00:05:40.012 terms of x of t. 00:05:43.320 --> 00:05:49.360 Now, if we look at this expression, which is the 00:05:49.360 --> 00:05:53.020 expression for the Fourier coefficients of the aperiodic 00:05:53.020 --> 00:05:58.690 signal, one of the things to recognize is that in effect 00:05:58.690 --> 00:06:06.420 what this represents are samples of an integral, where 00:06:06.420 --> 00:06:12.350 we can think of the variable omega taking on values that 00:06:12.350 --> 00:06:14.830 are integer multiples of omega 0. 00:06:14.830 --> 00:06:18.600 Said another way, let's define a function, as I've indicated 00:06:18.600 --> 00:06:24.540 here, which is this integral, where we may think of omega as 00:06:24.540 --> 00:06:27.930 being a continuous variable and then the Fourier series 00:06:27.930 --> 00:06:35.890 coefficients correspond to substituting for 00:06:35.890 --> 00:06:38.660 omega k omega 0. 00:06:38.660 --> 00:06:42.710 Now, one reason for doing that, as we'll see, is that in 00:06:42.710 --> 00:06:47.480 fact, this will turn out to provide us with a mechanism 00:06:47.480 --> 00:06:51.250 for a Fourier representation of x of t. 00:06:51.250 --> 00:06:56.360 And this, in fact, then, is an envelope of the Fourier series 00:06:56.360 --> 00:06:57.340 coefficients. 00:06:57.340 --> 00:07:03.530 In other words, t 0 the period times the coefficients is 00:07:03.530 --> 00:07:07.850 equal to this integral add integer multiples of omega 0. 00:07:10.540 --> 00:07:14.360 So this, in effect, tells us how to get the Fourier series 00:07:14.360 --> 00:07:19.490 coefficients of the periodic signal in terms of samples of 00:07:19.490 --> 00:07:20.140 an envelope. 00:07:20.140 --> 00:07:23.770 And that will become a very important notion shortly. 00:07:23.770 --> 00:07:28.370 And that, in effect, will correspond to an analysis 00:07:28.370 --> 00:07:32.280 equation to represent the aperiodic signal. 00:07:32.280 --> 00:07:35.770 Now, let's look at the synthesis equation. 00:07:35.770 --> 00:07:40.200 Recall that in the synthesis our strategy is to build a 00:07:40.200 --> 00:07:44.190 periodic signal and let the period go to infinity. 00:07:44.190 --> 00:07:50.160 Well, here is the expression for the synthesis of the 00:07:50.160 --> 00:07:56.510 periodic signal now expressed in terms of samples of this 00:07:56.510 --> 00:08:01.840 envelope function, and where I've simply used the fact or 00:08:01.840 --> 00:08:06.870 the substitution that t 0 is 2 pi over omega 0, and so I have 00:08:06.870 --> 00:08:10.830 an omega 0 here and a 1 over 2 pi. 00:08:10.830 --> 00:08:14.280 And the reason for doing that, as we'll see in a minute, is 00:08:14.280 --> 00:08:17.440 that this then turns into an integral. 00:08:17.440 --> 00:08:20.880 Specifically, then, the synthesis equation that we 00:08:20.880 --> 00:08:26.690 have is what I've indicated here. 00:08:26.690 --> 00:08:30.300 We would now want to examine this as the period goes to 00:08:30.300 --> 00:08:33.990 infinity, which means that omega 0 becomes 00:08:33.990 --> 00:08:36.140 infinitesimally small. 00:08:36.140 --> 00:08:39.820 And without dwelling on the details, and with my 00:08:39.820 --> 00:08:43.440 suggesting that you give this a fair amount of reflection, 00:08:43.440 --> 00:08:47.830 in fact, what happens as the period goes to infinity is 00:08:47.830 --> 00:08:53.910 that this summation approaches an integral over omega, where 00:08:53.910 --> 00:08:57.650 omega 0 becomes the differential in omega, and the 00:08:57.650 --> 00:09:00.020 periodic signal, of course, approach is 00:09:00.020 --> 00:09:02.740 the aperiodic signal. 00:09:02.740 --> 00:09:08.240 So the resulting equation that we get out of the original 00:09:08.240 --> 00:09:14.680 Fourier series synthesis equation is the equation that 00:09:14.680 --> 00:09:21.730 I indicate down here, x of t synthesized in terms of this 00:09:21.730 --> 00:09:25.720 integral, which is what the Fourier series approaches as 00:09:25.720 --> 00:09:28.780 omega 0 goes to 0. 00:09:28.780 --> 00:09:34.400 And we had previously that x of omega was in fact an 00:09:34.400 --> 00:09:36.900 envelope function. 00:09:36.900 --> 00:09:39.660 And we have then the corresponding Fourier 00:09:39.660 --> 00:09:45.240 transform analysis equation, which tells us how we arrive 00:09:45.240 --> 00:09:49.930 at that envelope in terms of x of t. 00:09:49.930 --> 00:09:54.410 So we now have an analysis equation and a synthesis 00:09:54.410 --> 00:09:59.150 equation, which in effect expresses for us how to build 00:09:59.150 --> 00:10:03.910 x of t in terms of infinitesimally finely spaced 00:10:03.910 --> 00:10:06.750 complex exponentials. 00:10:06.750 --> 00:10:11.690 The strategy to review it, and which I'd like to illustrate 00:10:11.690 --> 00:10:21.090 with a succession of overlays, was to begin with our 00:10:21.090 --> 00:10:28.780 aperiodic signal, as I indicate here, and then we 00:10:28.780 --> 00:10:34.160 constructed from that a periodic signal. 00:10:34.160 --> 00:10:39.880 And this periodic signal has a Fourier series, and we express 00:10:39.880 --> 00:10:43.550 the Fourier series coefficients of this as 00:10:43.550 --> 00:10:46.630 samples of an envelope function. 00:10:46.630 --> 00:10:51.520 The envelope function is what I indicate on the curve below. 00:10:51.520 --> 00:10:55.390 So this is the envelope of the Fourier series coefficients. 00:10:55.390 --> 00:11:02.640 For example, if the period t 0 was four times t1, then the 00:11:02.640 --> 00:11:06.270 Fourier series coefficients that we would end up with is 00:11:06.270 --> 00:11:10.480 this set of samples of the envelope. 00:11:10.480 --> 00:11:15.500 If instead we doubled that period, then the Fourier 00:11:15.500 --> 00:11:18.470 series coefficients that we end up with 00:11:18.470 --> 00:11:22.180 are more finely spaced. 00:11:22.180 --> 00:11:28.470 And as t 0 continues to increase, we get more and more 00:11:28.470 --> 00:11:33.290 finely spaced samples of this envelope function, and as t 0 00:11:33.290 --> 00:11:37.030 goes to infinity in fact, what we get is every single point 00:11:37.030 --> 00:11:41.080 on the envelope, and that provides us with the 00:11:41.080 --> 00:11:44.780 representation for the aperiodic signal. 00:11:44.780 --> 00:11:50.530 Let me, just to really emphasize the point, show this 00:11:50.530 --> 00:11:52.520 example once again. 00:11:52.520 --> 00:11:55.800 But now, let's look at it dynamically on 00:11:55.800 --> 00:11:58.440 the computer display. 00:11:58.440 --> 00:12:01.220 So here we have the square wave, and below it, the 00:12:01.220 --> 00:12:03.010 Fourier series coefficients. 00:12:03.010 --> 00:12:06.490 And we now want to look at the Fourier series coefficients as 00:12:06.490 --> 00:12:09.550 the period of the square wave starts to increase. 00:12:17.760 --> 00:12:21.260 And what we see is that these look like 00:12:21.260 --> 00:12:24.100 samples of an envelope. 00:12:24.100 --> 00:12:26.970 And in fact, the envelope of the Fourier series 00:12:26.970 --> 00:12:30.090 coefficients is shown in the bottom [? trace, ?] 00:12:30.090 --> 00:12:33.300 and to emphasize in fact that it is the envelope let's 00:12:33.300 --> 00:12:37.400 superimpose it on top of the Fourier series coefficients 00:12:37.400 --> 00:12:38.840 that we've generated so far. 00:12:48.340 --> 00:12:48.710 OK. 00:12:48.710 --> 00:12:54.020 Now, let's increase the period even further, and we'll see 00:12:54.020 --> 00:12:57.910 the Fourier series coefficients fill in under 00:12:57.910 --> 00:13:02.780 that envelope function even more. 00:13:02.780 --> 00:13:07.980 And in fact, as the period gets large enough, what we 00:13:07.980 --> 00:13:13.570 begin to get a sense of is that we're sampling more and 00:13:13.570 --> 00:13:15.940 more finely this envelope. 00:13:15.940 --> 00:13:19.060 And in fact, in the limit, as the period goes off to 00:13:19.060 --> 00:13:23.160 infinity, the samples basically will represent every 00:13:23.160 --> 00:13:25.050 single point on the envelope. 00:13:25.050 --> 00:13:28.030 Well, this is about as far as we want to go. 00:13:28.030 --> 00:13:33.130 Let's once again, plot the envelope function, and again, 00:13:33.130 --> 00:13:36.160 to emphasize that we've generated samples of that, 00:13:36.160 --> 00:13:39.175 let's superimpose that on the Fourier series coefficients. 00:13:46.300 --> 00:13:52.000 So what we have then is now our Fourier transform 00:13:52.000 --> 00:13:57.530 representation, the continuous time Fourier transform with 00:13:57.530 --> 00:14:01.960 the synthesis equation expressed as an integral, as 00:14:01.960 --> 00:14:06.870 I've indicated here, and this integral is what the Fourier 00:14:06.870 --> 00:14:12.820 series sum went to as we let the period go to infinity or 00:14:12.820 --> 00:14:14.720 the frequency go to zero. 00:14:14.720 --> 00:14:19.080 The corresponding analysis equation, which we have here, 00:14:19.080 --> 00:14:23.670 the analysis equation being the expression for the 00:14:23.670 --> 00:14:26.970 envelope of the Fourier series coefficients for the 00:14:26.970 --> 00:14:30.620 periodically replicated signal. 00:14:30.620 --> 00:14:33.690 And in shorthand notation, we would think 00:14:33.690 --> 00:14:35.370 of x of t and [? its ?] 00:14:35.370 --> 00:14:40.380 Fourier transform as a pair, as I've indicated here. 00:14:40.380 --> 00:14:45.830 And the Fourier transform, as we'll emphasize in several 00:14:45.830 --> 00:14:49.800 examples, and certainly as is consistent with the Fourier 00:14:49.800 --> 00:14:54.590 Series, is a complex valued function even 00:14:54.590 --> 00:14:55.730 when x of t is real. 00:14:55.730 --> 00:14:59.690 So with x of t real, we end up with a Fourier transform, 00:14:59.690 --> 00:15:01.770 which is a complex function. 00:15:01.770 --> 00:15:06.060 Just as the Fourier series coefficients were complex for 00:15:06.060 --> 00:15:08.250 a real value time function. 00:15:08.250 --> 00:15:12.420 So we could alternatively, as with the Fourier series, 00:15:12.420 --> 00:15:18.150 express the Fourier transform in terms of it's real part and 00:15:18.150 --> 00:15:23.530 imaginary part, or alternatively, in terms of its 00:15:23.530 --> 00:15:25.520 magnitude and its angle. 00:15:28.270 --> 00:15:33.560 All right, now let's look at an example of a time function 00:15:33.560 --> 00:15:35.750 in its Fourier transform. 00:15:35.750 --> 00:15:39.680 And so let's consider an example, which in fact is an 00:15:39.680 --> 00:15:41.350 example worked out in the text. 00:15:41.350 --> 00:15:44.500 It's example 4.7 in the text. 00:15:44.500 --> 00:15:49.520 And this is our old familiar friend the exponential. 00:15:49.520 --> 00:15:53.230 It's Fourier transform is the integral from minus infinity 00:15:53.230 --> 00:15:57.950 to plus infinity, x of t, e to the minus j omega t dt. 00:15:57.950 --> 00:16:03.960 And so, if we substitute in x of t and combine these two 00:16:03.960 --> 00:16:09.200 exponentials together, these two exponentials combined are 00:16:09.200 --> 00:16:15.130 e to the minus t times a plus j omega. 00:16:15.130 --> 00:16:21.070 And if we carry out the integration of this, we end up 00:16:21.070 --> 00:16:25.790 with the expression indicated here and provided now, and 00:16:25.790 --> 00:16:32.830 this is important, provided that a is greater than 0, then 00:16:32.830 --> 00:16:36.310 at the upper limit, this exponential becomes 0. 00:16:36.310 --> 00:16:38.870 At the lower limit, of course, it's one. 00:16:38.870 --> 00:16:44.460 And so what we have finally is for the Fourier transform 00:16:44.460 --> 00:16:48.780 expression 1 over a plus j omega. 00:16:52.700 --> 00:16:56.970 Now, this Fourier transform as I indicated is a complex 00:16:56.970 --> 00:16:58.380 valued function. 00:16:58.380 --> 00:17:04.210 Let's just take a look at what it looks like graphically. 00:17:04.210 --> 00:17:10.640 We have the expression for the Fourier transform pair, e to 00:17:10.640 --> 00:17:11.910 the minus a t times [? a ?] 00:17:11.910 --> 00:17:12.839 [? step ?]. 00:17:12.839 --> 00:17:16.980 And its Fourier transform is 1 over a plus j omega. 00:17:16.980 --> 00:17:22.609 And I indicated that that's true for a greater than 0. 00:17:22.609 --> 00:17:26.710 Now, in the expression that we just worked out, if a is less 00:17:26.710 --> 00:17:33.360 than 0, in fact, the expression doesn't converge e 00:17:33.360 --> 00:17:38.800 to the minus a t for a negative as t goes to 00:17:38.800 --> 00:17:44.170 infinity, blows up, and so in fact the Fourier transform 00:17:44.170 --> 00:17:48.990 doesn't converge except for the case where a is greater 00:17:48.990 --> 00:17:52.930 than 0 And in fact, there is a more detailed discussion of 00:17:52.930 --> 00:17:54.990 convergence issues in the text. 00:17:54.990 --> 00:17:58.000 The convergence issues are very much the same for the 00:17:58.000 --> 00:18:00.860 Fourier transform as they are for the Fourier series. 00:18:00.860 --> 00:18:03.510 And in fact, that's not surprising, because we 00:18:03.510 --> 00:18:05.360 developed the Fourier transform out of a 00:18:05.360 --> 00:18:07.410 consideration of the Fourier series. 00:18:07.410 --> 00:18:10.970 So the convergence conditions as you'll see as you refer in 00:18:10.970 --> 00:18:15.310 detail to the text relate to whether the time function is 00:18:15.310 --> 00:18:18.400 absolutely integrable under one set of conditions and 00:18:18.400 --> 00:18:22.290 square integrable under another set of conditions. 00:18:22.290 --> 00:18:28.110 OK, now, if we plot the Fourier transform, let's first 00:18:28.110 --> 00:18:30.870 consider the shape of the time function. 00:18:30.870 --> 00:18:34.580 And as I indicated, we're restricting the time function 00:18:34.580 --> 00:18:37.630 so that the exponential factor a is positive. 00:18:37.630 --> 00:18:40.730 In other words, e to the minus a t decays 00:18:40.730 --> 00:18:42.920 as t goes to infinity. 00:18:42.920 --> 00:18:48.830 The magnitude of the Fourier transform is as I indicate 00:18:48.830 --> 00:18:53.970 here and the phase below it. 00:18:53.970 --> 00:18:59.350 And there are a number of things we can see about the 00:18:59.350 --> 00:19:02.630 magnitude and phase of the Fourier transform for this 00:19:02.630 --> 00:19:07.990 example, which in fact we'll see in the next lecture are 00:19:07.990 --> 00:19:10.210 properties that apply more generally. 00:19:10.210 --> 00:19:13.810 For example, the fact that the Fourier transform magnitude is 00:19:13.810 --> 00:19:17.440 an even function of frequency, and the phase is an odd 00:19:17.440 --> 00:19:19.870 function of frequency. 00:19:19.870 --> 00:19:26.810 Now, let me also draw your attention to the fact that on 00:19:26.810 --> 00:19:31.220 this curve we have both positive frequencies and 00:19:31.220 --> 00:19:32.830 negative frequencies. 00:19:32.830 --> 00:19:35.950 In other words, in our expression for the Fourier 00:19:35.950 --> 00:19:40.220 transform, it requires both omega 00:19:40.220 --> 00:19:43.920 positive and omega negative. 00:19:43.920 --> 00:19:46.700 This, of course, was exactly the same in the case of the 00:19:46.700 --> 00:19:48.160 Fourier series. 00:19:48.160 --> 00:19:53.530 And the reason you should recall and keep in mind is 00:19:53.530 --> 00:19:56.390 related to the fact that we're building our signals out of 00:19:56.390 --> 00:20:01.010 complex exponentials, which require both positive values 00:20:01.010 --> 00:20:03.900 of omega and negative values of omega. 00:20:03.900 --> 00:20:07.000 Alternatively, if we chosen other representation, which 00:20:07.000 --> 00:20:10.060 turns out notationally to be much more difficult, namely 00:20:10.060 --> 00:20:13.300 sines and cosines, then we would in fact only consider 00:20:13.300 --> 00:20:14.700 positive frequencies. 00:20:14.700 --> 00:20:19.100 So it's important to keep in mind that, in our case, both 00:20:19.100 --> 00:20:23.050 with the Fourier series and the Fourier transform, we deal 00:20:23.050 --> 00:20:26.400 and require both positive and negative frequencies in order 00:20:26.400 --> 00:20:27.650 to build our signals. 00:20:29.910 --> 00:20:34.220 Now, in the graphical representation that I've shown 00:20:34.220 --> 00:20:39.210 here, I've chosen a linear amplitude scale and a linear 00:20:39.210 --> 00:20:40.360 frequency scale. 00:20:40.360 --> 00:20:44.070 And that's one graphical representation for the Fourier 00:20:44.070 --> 00:20:47.550 transform that we'll typically use. 00:20:47.550 --> 00:20:54.110 There's another one that very commonly arises, which I'll 00:20:54.110 --> 00:20:57.450 just briefly indicate for this example. 00:20:57.450 --> 00:21:04.240 And that is what's referred to as a bode plot in which the 00:21:04.240 --> 00:21:07.640 magnitude is displayed on a log amplitude and log 00:21:07.640 --> 00:21:09.160 frequency scale. 00:21:09.160 --> 00:21:11.940 And the phase is displayed on a log frequency scale. 00:21:11.940 --> 00:21:14.140 Let me show you what I mean. 00:21:14.140 --> 00:21:17.560 Here is the general expression for the bode plot. 00:21:17.560 --> 00:21:23.080 The bode plot expresses for us the amplitude in terms of the 00:21:23.080 --> 00:21:26.430 logarithm to the base 10 of the magnitude. 00:21:26.430 --> 00:21:33.020 And it also expresses the angle in both cases expressed 00:21:33.020 --> 00:21:37.810 as a function of a logarithmic frequency axis. 00:21:37.810 --> 00:21:45.240 So here is the amplitude as I've displayed it. 00:21:45.240 --> 00:21:50.110 And this is a log magnitude scale, a logarithmic frequency 00:21:50.110 --> 00:21:54.280 scale as indicated by the fact that as we move in equal 00:21:54.280 --> 00:21:58.190 increments along this axis, we change frequency 00:21:58.190 --> 00:22:00.050 by a factor of 10. 00:22:00.050 --> 00:22:07.440 And similarly, what we have is a display for the phase again 00:22:07.440 --> 00:22:10.190 on a log frequency scale. 00:22:10.190 --> 00:22:15.000 And I indicated that there is a symmetry to the Fourier 00:22:15.000 --> 00:22:19.570 transform, and so in fact, we can infer from this particular 00:22:19.570 --> 00:22:23.360 picture what it looks like for the negative frequencies as 00:22:23.360 --> 00:22:24.695 well as for the positive frequencies. 00:22:29.560 --> 00:22:39.450 Now, what we've done so far is to develop the Fourier 00:22:39.450 --> 00:22:44.285 transform on the basis, the Fourier transform of an 00:22:44.285 --> 00:22:48.170 aperiodic signal on the basis of periodically repeating it 00:22:48.170 --> 00:22:52.420 and recognizing that the Fourier series coefficients 00:22:52.420 --> 00:22:57.200 are samples of an envelope and that these become more finely 00:22:57.200 --> 00:22:59.990 spaced as frequency increases. 00:22:59.990 --> 00:23:08.080 And in fact, we can go back to our original equation in which 00:23:08.080 --> 00:23:13.810 we developed an envelope function, and what we had 00:23:13.810 --> 00:23:19.200 indicated is that the Fourier series coefficients were 00:23:19.200 --> 00:23:22.920 samples of this envelope. 00:23:22.920 --> 00:23:31.290 We then defined this envelope as the Fourier transform of 00:23:31.290 --> 00:23:33.760 this aperiodic signal. 00:23:33.760 --> 00:23:38.670 So that provided us with a way-- and it was a mechanism-- 00:23:38.670 --> 00:23:44.810 for getting a representation for an aperiodic signal. 00:23:44.810 --> 00:23:50.820 Now, suppose that we have instead a periodic signal, are 00:23:50.820 --> 00:23:54.660 there, in fact, some statements that we can make 00:23:54.660 --> 00:23:58.050 about how the Fourier series coefficients of that are 00:23:58.050 --> 00:24:02.240 related to the Fourier transform of something. 00:24:02.240 --> 00:24:05.780 Well, in fact, this statement tells us 00:24:05.780 --> 00:24:07.390 exactly how to do that. 00:24:07.390 --> 00:24:11.000 What this statement says is that, in fact, the Fourier 00:24:11.000 --> 00:24:16.140 series coefficients are samples of the Fourier 00:24:16.140 --> 00:24:19.180 transform of one period? 00:24:19.180 --> 00:24:28.450 So if we now consider a periodic signal, we can in 00:24:28.450 --> 00:24:33.080 fact get the Fourier series coefficients of that periodic 00:24:33.080 --> 00:24:35.980 signal by considering the Fourier 00:24:35.980 --> 00:24:38.630 transform of one period. 00:24:38.630 --> 00:24:42.400 Said another way, the Fourier series coefficients are 00:24:42.400 --> 00:24:45.510 proportional to samples of the Fourier 00:24:45.510 --> 00:24:47.670 transform of one period. 00:24:47.670 --> 00:24:53.030 So if we consider this a periodic signal, computed as 00:24:53.030 --> 00:25:00.380 Fourier transform, and selected these samples that I 00:25:00.380 --> 00:25:06.780 indicate here, namely samples equally spaced in omega by 00:25:06.780 --> 00:25:12.260 integer multiples of omega 0, then in fact, those would be 00:25:12.260 --> 00:25:14.680 the Fourier series coefficients. 00:25:14.680 --> 00:25:22.890 So we can go back to our example previously that 00:25:22.890 --> 00:25:27.190 involved the square wave. 00:25:27.190 --> 00:25:31.910 And now, in this case, we could argue that if in fact it 00:25:31.910 --> 00:25:36.490 was the periodic signal that we started with, we could get 00:25:36.490 --> 00:25:40.610 the Fourier series coefficients of that by 00:25:40.610 --> 00:25:46.950 thinking about the Fourier transform of one period, which 00:25:46.950 --> 00:25:49.590 I indicate here. 00:25:49.590 --> 00:25:53.920 And then the Fourier series coefficients of the periodic 00:25:53.920 --> 00:26:00.070 signal, in fact, are the appropriate set of samples of 00:26:00.070 --> 00:26:01.320 this envelope. 00:26:04.710 --> 00:26:11.400 All right, now, we have a way of getting the Fourier series 00:26:11.400 --> 00:26:14.930 coefficients from the Fourier transform of one period. 00:26:14.930 --> 00:26:18.100 We originally derived the Fourier transform of one 00:26:18.100 --> 00:26:21.420 period from the Fourier series. 00:26:21.420 --> 00:26:25.820 What would, in fact, be nice is if we could incorporate the 00:26:25.820 --> 00:26:28.790 Fourier series and the Fourier transform 00:26:28.790 --> 00:26:30.530 within a common framework. 00:26:30.530 --> 00:26:34.390 And in fact, it turns out that there is a very convenient way 00:26:34.390 --> 00:26:38.040 of doing that almost by definition. 00:26:38.040 --> 00:26:44.380 Essentially, if we consider what the equation for the 00:26:44.380 --> 00:26:51.640 synthesis looks like in both cases, we can in effect define 00:26:51.640 --> 00:26:56.820 a Fourier transform for the periodic signal, which we know 00:26:56.820 --> 00:26:59.360 is represented by its Fourier series coefficients. 00:26:59.360 --> 00:27:04.150 We can define a Fourier transform, and the definition 00:27:04.150 --> 00:27:09.960 of the Fourier transform is as an impulse train, where the 00:27:09.960 --> 00:27:14.220 coefficients in the impulse train are proportional, with a 00:27:14.220 --> 00:27:17.200 proportionality factor of 2 pi for a more or less a 00:27:17.200 --> 00:27:20.140 bookkeeping reason, proportional to the Fourier 00:27:20.140 --> 00:27:22.220 series coefficients. 00:27:22.220 --> 00:27:28.430 And the validity of this is, more or less, can be seen 00:27:28.430 --> 00:27:30.860 essentially by substitution. 00:27:30.860 --> 00:27:37.210 Specifically, here is then the synthesis equation for the 00:27:37.210 --> 00:27:42.030 Fourier transform if we substitute this definition for 00:27:42.030 --> 00:27:45.290 the Fourier transform of the periodic signal into this 00:27:45.290 --> 00:27:51.310 expression then when we do the appropriate bookkeeping and 00:27:51.310 --> 00:27:54.850 interchange the order of summation and integration the 00:27:54.850 --> 00:27:58.690 impulse integrates out to the exponential 00:27:58.690 --> 00:28:02.530 factor that we want. 00:28:02.530 --> 00:28:05.820 So we have the exponential factor. 00:28:05.820 --> 00:28:07.990 We have the Fourier series coefficients. 00:28:07.990 --> 00:28:11.450 The 2 pis take care of each other, and what we're left 00:28:11.450 --> 00:28:16.620 with is the synthesis equation for aperiodic signal in terms 00:28:16.620 --> 00:28:21.160 of the Fourier transform, or in terms of its Fourier series 00:28:21.160 --> 00:28:23.600 coefficients. 00:28:23.600 --> 00:28:30.230 Now, we can just see this in terms of a simple example. 00:28:30.230 --> 00:28:36.570 If we consider the example of a symmetric square wave, then 00:28:36.570 --> 00:28:40.320 in effect what we're saying is that for this symmetric square 00:28:40.320 --> 00:28:45.240 wave, this has a set of Fourier series coefficients, 00:28:45.240 --> 00:28:50.020 which we worked out previously and which I indicate on this 00:28:50.020 --> 00:28:52.440 figure with a bar graph. 00:28:52.440 --> 00:28:55.930 And really all that we're saying is that, whereas these 00:28:55.930 --> 00:28:59.840 Fourier series coefficients are indexed on an integer 00:28:59.840 --> 00:29:04.790 variable k, and [? they're ?] bars not impulses. 00:29:04.790 --> 00:29:10.020 If we simply redefine or define the Fourier transform 00:29:10.020 --> 00:29:15.360 of the periodic signal as an impulse train, where the 00:29:15.360 --> 00:29:20.840 weights of the impulses are 2 pi times the corresponding 00:29:20.840 --> 00:29:25.750 Fourier series coefficients, then this, in fact, is what we 00:29:25.750 --> 00:29:29.690 would use as the Fourier transform of 00:29:29.690 --> 00:29:30.940 the periodic signal. 00:29:34.010 --> 00:29:39.080 Now, we've kind of gone back and forth, and maybe even it 00:29:39.080 --> 00:29:42.180 might seem like we've gone around in circles. 00:29:42.180 --> 00:29:45.880 So let me just try to summarize the various 00:29:45.880 --> 00:29:49.490 relationships and steps that we've gone through, keeping in 00:29:49.490 --> 00:29:52.670 mind that one of our objectives was first to 00:29:52.670 --> 00:29:56.690 develop a representation for aperiodic signals and then 00:29:56.690 --> 00:30:01.910 attempt to incorporate within one framework both periodic 00:30:01.910 --> 00:30:05.530 and aperiodic signals. 00:30:05.530 --> 00:30:11.250 We began with an aperiodic signal. 00:30:11.250 --> 00:30:15.290 And the strategy was to develop a Fourier 00:30:15.290 --> 00:30:21.150 representation by constructing a periodic signal for which 00:30:21.150 --> 00:30:22.860 that was one period. 00:30:22.860 --> 00:30:25.770 And then we let the period go to infinity, 00:30:25.770 --> 00:30:27.580 as I indicate here. 00:30:27.580 --> 00:30:32.770 So we have an aperiodic signal. 00:30:32.770 --> 00:30:37.760 We construct a periodic signal, x tilde of t for which 00:30:37.760 --> 00:30:41.500 one period is the aperiodic signal. 00:30:41.500 --> 00:30:45.550 X tilde of t, the periodic signal, has a Fourier series, 00:30:45.550 --> 00:30:50.700 and as its period increases that approaches the aperiodic 00:30:50.700 --> 00:30:57.050 signal, and the Fourier series of that approaches the Fourier 00:30:57.050 --> 00:31:00.720 transform of the original aperiodic signal. 00:31:00.720 --> 00:31:05.700 So that was the first step we took. 00:31:05.700 --> 00:31:10.940 Now, the second thing that we recognize is that once we have 00:31:10.940 --> 00:31:15.280 the concept of the Fourier transform, we can, in fact, 00:31:15.280 --> 00:31:19.570 relate the Fourier series coefficients to the Fourier 00:31:19.570 --> 00:31:22.210 transform of one period. 00:31:22.210 --> 00:31:27.330 So the second statement that we made was that if in fact 00:31:27.330 --> 00:31:33.070 we're trying to represent a periodic signal, we can get 00:31:33.070 --> 00:31:35.900 the Fourier series coefficients of that by 00:31:35.900 --> 00:31:42.340 computing the Fourier transform of one period and 00:31:42.340 --> 00:31:47.380 then samples of that Fourier transform are, in fact, the 00:31:47.380 --> 00:31:50.110 Fourier series coefficients for the periodic signal. 00:31:53.210 --> 00:31:58.320 Then, the third step that we took was to inquire as to 00:31:58.320 --> 00:32:02.340 whether there is a Fourier transform that can 00:32:02.340 --> 00:32:06.420 appropriately be defined for the periodic signal, and the 00:32:06.420 --> 00:32:11.260 mechanism for doing that was to recognize that if we simply 00:32:11.260 --> 00:32:14.840 defined the Fourier transform of the periodic signal as an 00:32:14.840 --> 00:32:19.420 impulse train, where the impulse heights or areas were 00:32:19.420 --> 00:32:22.580 proportional to the Fourier series coefficients, then, in 00:32:22.580 --> 00:32:29.630 fact, the Fourier transform synthesis equation reduced to 00:32:29.630 --> 00:32:32.140 the Fourier series synthesis equation. 00:32:32.140 --> 00:32:37.960 So the third step, then, was with a periodic signal. 00:32:37.960 --> 00:32:42.280 The Fourier transform of that periodic signal, defined as an 00:32:42.280 --> 00:32:44.610 impulse train, where the heights or areas of the 00:32:44.610 --> 00:32:47.810 impulses are proportional to the Fourier series 00:32:47.810 --> 00:32:53.040 coefficients, provides us with a mechanism for combining it 00:32:53.040 --> 00:32:57.825 together the concepts or notation of the Fourier series 00:32:57.825 --> 00:33:00.900 and Fourier transform. 00:33:00.900 --> 00:33:07.340 So if we just took a very simple example, here is an 00:33:07.340 --> 00:33:12.950 example in which we have an aperiodic signal, which is 00:33:12.950 --> 00:33:16.620 just an impulse, and its Fourier 00:33:16.620 --> 00:33:20.670 transform is just a constant. 00:33:20.670 --> 00:33:24.730 We can think of a periodic signal associated with this, 00:33:24.730 --> 00:33:29.820 which is this signal periodically replicated with a 00:33:29.820 --> 00:33:32.300 spacing t 0. 00:33:32.300 --> 00:33:35.720 The Fourier transform of this is a constant. 00:33:35.720 --> 00:33:38.970 And this, of course, has a Fourier series representation. 00:33:38.970 --> 00:33:42.800 So the Fourier transform of the original 00:33:42.800 --> 00:33:46.170 impulse is just a constant. 00:33:46.170 --> 00:33:51.830 The Fourier transform of the periodic signal is an impulse 00:33:51.830 --> 00:33:56.170 train, where the heights of the impulses are proportional 00:33:56.170 --> 00:33:58.830 to the Fourier series coefficients. 00:33:58.830 --> 00:34:02.700 And, of course, we could previously have computed the 00:34:02.700 --> 00:34:06.680 Fourier series coefficients for that impulse train, and 00:34:06.680 --> 00:34:09.040 those Fourier series coefficients are 00:34:09.040 --> 00:34:10.370 as I've shown here. 00:34:10.370 --> 00:34:14.290 So in both of these cases, these in effect represent just 00:34:14.290 --> 00:34:16.909 a change in notation, where here we have a bar graph, and 00:34:16.909 --> 00:34:19.199 here we have an impulse train. 00:34:19.199 --> 00:34:23.980 And both of these simply represent samples of what we 00:34:23.980 --> 00:34:30.690 have above, which is the Fourier transform of the 00:34:30.690 --> 00:34:31.940 original aperiodic signal. 00:34:35.179 --> 00:34:39.239 Once again, I suspect that kind of moving back and forth 00:34:39.239 --> 00:34:42.239 and trying to straighten out when we're talking about 00:34:42.239 --> 00:34:46.989 periodic and aperiodic signals may require a little mental 00:34:46.989 --> 00:34:48.810 gymnastics initially. 00:34:48.810 --> 00:34:53.310 Basically, what we've tried to do is incorporate within one 00:34:53.310 --> 00:34:58.200 framework a representation for both aperiodic and periodic 00:34:58.200 --> 00:35:01.450 signals, and the Fourier transform provides us with a 00:35:01.450 --> 00:35:04.410 mechanism to do that. 00:35:04.410 --> 00:35:07.640 In the next lecture, I'll continue with the discussion 00:35:07.640 --> 00:35:11.310 of the continuous-time Fourier transform in particular 00:35:11.310 --> 00:35:15.260 focusing on a number of its properties, some of which 00:35:15.260 --> 00:35:18.000 we've already seen, namely the symmetry properties. 00:35:18.000 --> 00:35:21.360 We'll see lots of other properties that relate, of 00:35:21.360 --> 00:35:23.780 course, both to the Fourier transform and 00:35:23.780 --> 00:35:24.970 to the Fourier series. 00:35:24.970 --> 00:35:26.220 Thank you.