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. 00:00:55.600 --> 00:00:58.360 PROFESSOR: In the last two lectures, we saw how periodic 00:00:58.360 --> 00:01:01.300 and non periodic signals could be represented as linear 00:01:01.300 --> 00:01:04.440 combinations of complex exponentials. 00:01:04.440 --> 00:01:08.570 And this led to the Fourier series representation, in the 00:01:08.570 --> 00:01:12.280 periodic case, and it led to the Fourier transform 00:01:12.280 --> 00:01:14.950 representation in the aperiodic case. 00:01:14.950 --> 00:01:19.340 And then, in fact, what we did was to incorporate the Fourier 00:01:19.340 --> 00:01:25.480 series within the framework of the Fourier transform. 00:01:25.480 --> 00:01:29.350 What I'd like to do in today's lecture is look at the Fourier 00:01:29.350 --> 00:01:33.620 transform more closely, in particular with regard to some 00:01:33.620 --> 00:01:35.050 of its properties. 00:01:35.050 --> 00:01:39.460 So let me begin by reminding you of the analysis and 00:01:39.460 --> 00:01:43.020 synthesis equations for the Fourier transform, as I've 00:01:43.020 --> 00:01:45.350 summarized them here. 00:01:45.350 --> 00:01:49.550 The synthesis equation being an equation the tells us how 00:01:49.550 --> 00:01:53.970 to build the time function out of, in essence, a linear 00:01:53.970 --> 00:01:57.060 combination of complex exponentials. 00:01:57.060 --> 00:02:01.390 And the analysis equation telling us how to get the 00:02:01.390 --> 00:02:05.820 amplitudes of those complex exponentials from the 00:02:05.820 --> 00:02:07.970 associated time function. 00:02:07.970 --> 00:02:12.360 So essentially, in the decomposition of x of t as a 00:02:12.360 --> 00:02:16.270 linear combination of complex exponentials, the complex 00:02:16.270 --> 00:02:20.560 amplitudes of those are, in effect, the Fourier transform 00:02:20.560 --> 00:02:26.670 scaled by the differential and scaled by 1 over 2 pi. 00:02:26.670 --> 00:02:33.730 As I indicated last time, the Fourier transform is a complex 00:02:33.730 --> 00:02:35.810 function of frequency. 00:02:35.810 --> 00:02:40.490 And in particular, the complex function of frequency has an 00:02:40.490 --> 00:02:44.770 important and very useful symmetry property. 00:02:44.770 --> 00:02:49.030 The symmetry of the Fourier transform, when x of t is 00:02:49.030 --> 00:02:53.450 real, is what is referred to as conjugate symmetric. 00:02:53.450 --> 00:02:57.940 In other words, if we take the complex function, f of omega, 00:02:57.940 --> 00:03:01.970 and its complex conjugate, that's equivalent to replacing 00:03:01.970 --> 00:03:04.930 omega by minus omega. 00:03:04.930 --> 00:03:10.300 And a consequence of that, if we think in terms of the real 00:03:10.300 --> 00:03:14.890 part of the Fourier transform, the real part is an even 00:03:14.890 --> 00:03:19.730 function of frequency, and the magnitude is an even function 00:03:19.730 --> 00:03:21.370 of frequency. 00:03:21.370 --> 00:03:26.550 Whereas the imaginary part is an odd function of frequency, 00:03:26.550 --> 00:03:31.170 and the phase angle is an odd function of frequency. 00:03:31.170 --> 00:03:35.450 So we have this symmetry relationship that, for x of t 00:03:35.450 --> 00:03:39.390 real, if we think of either the real part or the magnitude 00:03:39.390 --> 00:03:42.880 of the Fourier transform, it's even symmetric. 00:03:42.880 --> 00:03:48.100 And the imaginary part, or the phase angle, either one, is 00:03:48.100 --> 00:03:49.310 odd symmetric. 00:03:49.310 --> 00:03:53.740 In other words, if we flip it, we multiply by a minus sign. 00:03:53.740 --> 00:03:58.260 Let's look at an example, in the context of an example that 00:03:58.260 --> 00:04:02.870 we worked last time for the Fourier transform. 00:04:02.870 --> 00:04:08.320 We took the case of a real exponential of the form e to 00:04:08.320 --> 00:04:11.550 the minus at times the step. 00:04:11.550 --> 00:04:17.120 And the Fourier transform, as we found, was of the algebraic 00:04:17.120 --> 00:04:20.089 form 1 over a plus j omega. 00:04:20.089 --> 00:04:23.580 And, incidentally, the Fourier transform integral only 00:04:23.580 --> 00:04:26.790 converged for a greater than 0. 00:04:26.790 --> 00:04:31.250 In other words, for this exponential decaying. 00:04:31.250 --> 00:04:34.810 And I illustrated the magnitude and angle, and what 00:04:34.810 --> 00:04:40.460 we see is that e to the minus at is a real time function, 00:04:40.460 --> 00:04:44.110 therefore its magnitude should be an even function of 00:04:44.110 --> 00:04:46.630 frequency, and indeed it is. 00:04:46.630 --> 00:04:50.800 And its phase angle, shown below, is an 00:04:50.800 --> 00:04:54.050 odd function of frequency. 00:04:54.050 --> 00:05:02.170 So in fact, although I stressed last time that the 00:05:02.170 --> 00:05:04.800 complex exponentials is required to build a time 00:05:04.800 --> 00:05:09.000 function require exponentials of both positive and negative 00:05:09.000 --> 00:05:14.160 frequencies, for x of t real what we see is that, because 00:05:14.160 --> 00:05:16.270 of these symmetry properties, either for the real and 00:05:16.270 --> 00:05:20.960 imaginary or magnitude and angle, we can specify the 00:05:20.960 --> 00:05:24.630 Fourier transform for, let's say only positive frequencies, 00:05:24.630 --> 00:05:28.470 and the symmetry, then, implies, or tells us, what the 00:05:28.470 --> 00:05:30.850 Fourier transform, then, would be for the negative 00:05:30.850 --> 00:05:32.100 frequencies. 00:05:34.560 --> 00:05:39.480 This same example, the decaying exponential, 00:05:39.480 --> 00:05:44.320 demonstrates another important and often useful property of 00:05:44.320 --> 00:05:47.370 the Fourier transform. 00:05:47.370 --> 00:05:52.410 Specifically, let's rewrite, algebraically, this example as 00:05:52.410 --> 00:05:54.350 I indicate here. 00:05:54.350 --> 00:05:57.720 So we have, again, the exponential, whose Fourier 00:05:57.720 --> 00:06:01.800 transform is 1 over a plus j omega. 00:06:01.800 --> 00:06:06.020 And if I just simply divide numerator and denominator by 00:06:06.020 --> 00:06:10.370 a, I can rewrite it in the form shown here. 00:06:10.370 --> 00:06:15.800 And what we notice is that in the time function we have a 00:06:15.800 --> 00:06:20.020 term of the form a times t, and in the frequency function 00:06:20.020 --> 00:06:24.630 we have a term of the form omega divided by a. 00:06:24.630 --> 00:06:30.390 Or equivalently, we could think of the time function, 00:06:30.390 --> 00:06:37.280 which I show here, and as the parameter a gets smaller, the 00:06:37.280 --> 00:06:41.300 exponential gets spread out in time. 00:06:41.300 --> 00:06:45.880 Whereas its Fourier transform, or the magnitude of its 00:06:45.880 --> 00:06:50.080 Fourier transform, has the inverse property that as a 00:06:50.080 --> 00:06:56.840 gets smaller, in fact, this scales down in frequency. 00:06:56.840 --> 00:07:00.430 Well, this is a general property of the Fourier 00:07:00.430 --> 00:07:06.350 transform, namely the fact that a linear scaling in time 00:07:06.350 --> 00:07:10.590 generates the inverse linear scaling in frequency. 00:07:10.590 --> 00:07:14.190 And the general statement of this time frequency scaling is 00:07:14.190 --> 00:07:18.100 what I show at the top of the transparency, namely the 00:07:18.100 --> 00:07:23.570 equation that if we scale the time function in time, then we 00:07:23.570 --> 00:07:26.140 apply an inverse scaling in frequency. 00:07:28.720 --> 00:07:33.430 This, in fact, is probably a result that you're already 00:07:33.430 --> 00:07:35.580 possibly familiar with, in somewhat 00:07:35.580 --> 00:07:37.190 of a different context. 00:07:37.190 --> 00:07:42.250 Essentially, you could think of this as an example, or a 00:07:42.250 --> 00:07:47.700 generalization, rather, of the notion that if, let's say I 00:07:47.700 --> 00:07:53.610 had a signal that was recorded on a tape player, and if I 00:07:53.610 --> 00:07:57.140 play the tape back at, let's say, twice the speed, which 00:07:57.140 --> 00:08:00.380 means that I'm compressing the time axis linearly 00:08:00.380 --> 00:08:02.240 by a factor of 2. 00:08:02.240 --> 00:08:05.280 Then, in fact, what happens is that the frequencies that we 00:08:05.280 --> 00:08:08.930 observe get pushed up by a factor of 2. 00:08:08.930 --> 00:08:12.450 And, in fact, let me illustrate that. 00:08:12.450 --> 00:08:18.670 I have here a glockenspiel, and anyone who loves a parade 00:08:18.670 --> 00:08:22.610 certainly knows what a glockenspiel is. 00:08:22.610 --> 00:08:28.700 And this particular glockenspiel has three a's, 00:08:28.700 --> 00:08:30.360 separated each by an octave. 00:08:30.360 --> 00:08:35.500 There's a middle a, a high a, and a low a. 00:08:38.960 --> 00:08:46.240 And what I've done is to record the middle a on a tape 00:08:46.240 --> 00:08:48.950 at 7 and 1/2 inches per second. 00:08:48.950 --> 00:08:54.680 And what I'd like to demonstrate is that as we play 00:08:54.680 --> 00:08:59.790 that back, either at twice or half speed, the effective note 00:08:59.790 --> 00:09:03.650 gets moved down or up by an octave. 00:09:03.650 --> 00:09:07.340 So let me first play the note at the speed at 00:09:07.340 --> 00:09:08.760 which it was recorded. 00:09:08.760 --> 00:09:12.250 And so what we'll hear is the middle a as I've recorded it. 00:09:12.250 --> 00:09:15.200 And let me just start the tape player. 00:09:26.190 --> 00:09:30.190 Let me stop it, and hopefully what you heard is the same 00:09:30.190 --> 00:09:33.930 note from the tape recorder as the note that I played on the 00:09:33.930 --> 00:09:35.980 glockenspiel. 00:09:35.980 --> 00:09:40.740 Now I'll rewind the tape, and we'll go back to the beginning 00:09:40.740 --> 00:09:43.700 of that portion. 00:09:43.700 --> 00:09:49.930 And now, if I change the tape speed to half the speed, so 00:09:49.930 --> 00:09:53.050 from 7 and 1/2 inches per second, I'll change the tape 00:09:53.050 --> 00:09:57.980 speed to 3 and 3/4 inches per second. 00:09:57.980 --> 00:10:04.820 And now when I play the tape back, because of the inverse 00:10:04.820 --> 00:10:08.060 relationship between time and frequency scaling, we're now 00:10:08.060 --> 00:10:10.750 scaling in time by stretching out, we would expect the 00:10:10.750 --> 00:10:13.770 frequencies to be lowered by a factor of 2. 00:10:13.770 --> 00:10:19.170 We should now expect the taped note to be an octave lower, 00:10:19.170 --> 00:10:20.390 matching this lower a. 00:10:20.390 --> 00:10:22.010 So let's just play that. 00:10:39.300 --> 00:10:43.795 Let me stop it, and, again, we'll rewind the tape. 00:10:46.410 --> 00:10:50.020 Go back to the beginning, and now we'll play this 00:10:50.020 --> 00:10:51.770 at twice the speed. 00:10:51.770 --> 00:10:57.310 So I'll change from 3 and 3/4 to 15 inches per second. 00:10:57.310 --> 00:11:00.680 And now when I play it, we would expect that to match the 00:11:00.680 --> 00:11:01.180 upper note. 00:11:01.180 --> 00:11:02.430 And let's just do that. 00:11:15.100 --> 00:11:18.980 Although that's a result that, intuitively, probably makes 00:11:18.980 --> 00:11:22.720 considerable sense, in fact, what that is is an 00:11:22.720 --> 00:11:27.130 illustration of the inverse relationship between time 00:11:27.130 --> 00:11:29.250 scaling and frequency scaling. 00:11:29.250 --> 00:11:31.970 And also, by the way, it was finally my opportunity to play 00:11:31.970 --> 00:11:33.220 the glockenspiel on television. 00:11:36.560 --> 00:11:42.300 In addition, there is another very important relationship 00:11:42.300 --> 00:11:45.810 between the time and frequency domains, namely what is 00:11:45.810 --> 00:11:50.150 referred to as a duality relationship. 00:11:50.150 --> 00:11:56.660 And the duality relationship between time and frequency 00:11:56.660 --> 00:12:01.220 falls out, more or less directly, from the equations, 00:12:01.220 --> 00:12:03.790 the analysis and synthesis equations. 00:12:03.790 --> 00:12:08.470 In particular, if we look at the synthesis equation, which 00:12:08.470 --> 00:12:12.760 I repeat here, and the analysis equation, which I 00:12:12.760 --> 00:12:17.510 repeat below it, what we observe is that, in fact, 00:12:17.510 --> 00:12:23.420 these equations are basically identical, except for the fact 00:12:23.420 --> 00:12:26.870 that in the top integral we have things as a function of 00:12:26.870 --> 00:12:30.320 omega, in the bottom integral as a function of t, and 00:12:30.320 --> 00:12:32.960 there's a factor of 1 over 2 pi, and, by the 00:12:32.960 --> 00:12:34.590 way, a minus sign. 00:12:34.590 --> 00:12:39.500 You can look at the algebra more carefully at your 00:12:39.500 --> 00:12:43.920 leisure, but essentially what this says is that if x of 00:12:43.920 --> 00:12:51.420 omega is the Fourier transform of a time function x of t, 00:12:51.420 --> 00:12:58.120 then, in fact, x of t is very much like the Fourier 00:12:58.120 --> 00:13:00.180 transform of x of omega. 00:13:00.180 --> 00:13:04.060 In fact, it's the Fourier transform of x of minus omega 00:13:04.060 --> 00:13:06.060 to account for this minus sign. 00:13:06.060 --> 00:13:07.800 And, by the way, there's just an additional 00:13:07.800 --> 00:13:10.550 factor of 1 over 2 pi. 00:13:10.550 --> 00:13:14.070 So the duality relationship which follows from these two 00:13:14.070 --> 00:13:21.450 equations, in fact, says that if x of t and x of omega are a 00:13:21.450 --> 00:13:26.360 Fourier transform pair, if x and X are a Fourier transform 00:13:26.360 --> 00:13:32.150 pair, then X, in fact, has a Fourier transform which is 00:13:32.150 --> 00:13:35.910 proportional to x turned around. 00:13:35.910 --> 00:13:40.060 This duality in the continuous time Fourier 00:13:40.060 --> 00:13:42.560 transform is very important. 00:13:42.560 --> 00:13:44.110 It's very useful. 00:13:44.110 --> 00:13:48.590 It, by the way, is not a duality that surfaced in the 00:13:48.590 --> 00:13:52.620 Fourier series, because, as you recall, the Fourier series 00:13:52.620 --> 00:13:58.250 begins with a continuous time function and in the frequency 00:13:58.250 --> 00:14:02.470 domain generates a sequence, which would just naturally 00:14:02.470 --> 00:14:06.630 have problems associated with it if we attempted to 00:14:06.630 --> 00:14:08.580 interpret a duality. 00:14:08.580 --> 00:14:12.130 And we'll see, also, that in the discrete time case, one of 00:14:12.130 --> 00:14:14.560 the important differences between continuous time and 00:14:14.560 --> 00:14:18.250 discrete time Fourier transforms is the fact that in 00:14:18.250 --> 00:14:23.950 continuous time we have duality, in the discrete time 00:14:23.950 --> 00:14:27.330 Fourier transform we don't. 00:14:27.330 --> 00:14:32.400 Let's illustrate this with an example. 00:14:32.400 --> 00:14:37.000 Here are, in fact, two examples of Fourier transform 00:14:37.000 --> 00:14:40.460 pairs taken from examples in the text. 00:14:40.460 --> 00:14:46.690 The top one being example 4.11 from the text, and it's a time 00:14:46.690 --> 00:14:52.310 function which is a sine x over x type of function. 00:14:52.310 --> 00:14:59.710 And its Fourier transform corresponds to a rectangular 00:14:59.710 --> 00:15:03.830 shape in the frequency domain. 00:15:03.830 --> 00:15:07.320 There's also another example in the text, the example that 00:15:07.320 --> 00:15:11.590 precedes this one, which is example 4.10. 00:15:11.590 --> 00:15:17.460 And in example 4.10, we begin with a rectangle, and its 00:15:17.460 --> 00:15:21.340 Fourier transform is of the form of a 00:15:21.340 --> 00:15:24.140 sine x over x function. 00:15:24.140 --> 00:15:28.980 So in fact, if we look at these two examples together, 00:15:28.980 --> 00:15:33.260 what we see is the duality very evident. 00:15:33.260 --> 00:15:38.390 In other words, if we take this time function and instead 00:15:38.390 --> 00:15:42.260 think of a frequency function that has the same form, then 00:15:42.260 --> 00:15:45.360 we simply interchange the roles of time and frequency in 00:15:45.360 --> 00:15:46.630 the other domains. 00:15:46.630 --> 00:15:50.810 So the fact that these two correspond means that these 00:15:50.810 --> 00:15:52.330 two correspond. 00:15:52.330 --> 00:15:57.910 Of course in this particular example, because of the fact 00:15:57.910 --> 00:16:01.290 that we picked a symmetric function, an even function, in 00:16:01.290 --> 00:16:05.530 fact, the additional twist of the time axis being reversed 00:16:05.530 --> 00:16:10.830 didn't show up in duality with this example. 00:16:10.830 --> 00:16:13.930 One thing this says, of course, is that essentially 00:16:13.930 --> 00:16:19.390 any time you've calculated the Fourier transform of one time 00:16:19.390 --> 00:16:23.290 function, then you've actually calculated the Fourier 00:16:23.290 --> 00:16:25.990 transform of two time functions. 00:16:25.990 --> 00:16:28.970 Another one being the dual example to the one that you 00:16:28.970 --> 00:16:30.220 just calculated. 00:16:32.370 --> 00:16:37.805 Also somewhat related to duality is what is referred to 00:16:37.805 --> 00:16:41.650 as Parseval's relation for the continuous 00:16:41.650 --> 00:16:43.620 time Fourier transform. 00:16:43.620 --> 00:16:47.880 And essentially, what Parseval's relationship says, 00:16:47.880 --> 00:16:55.800 as a summary of it, says that the energy in a time function 00:16:55.800 --> 00:17:00.260 and the energy in its Fourier transform are proportional, 00:17:00.260 --> 00:17:04.430 the proportionality factor being a factor of 2 pi. 00:17:04.430 --> 00:17:06.770 That's summarized here. 00:17:06.770 --> 00:17:10.190 What's meant by the energy is, of course, the integral of the 00:17:10.190 --> 00:17:13.010 magnitude squared of x of t. 00:17:13.010 --> 00:17:16.819 And the statement of Parseval's relation is that 00:17:16.819 --> 00:17:21.670 that integral, the energy in x of t, is proportional to this 00:17:21.670 --> 00:17:25.770 integral, which is the energy in x of omega. 00:17:25.770 --> 00:17:31.660 Although we've incorporated the Fourier series within a 00:17:31.660 --> 00:17:36.460 framework of the Fourier transform, Parseval's relation 00:17:36.460 --> 00:17:40.830 needs to be modified slightly for Fourier series, because of 00:17:40.830 --> 00:17:43.840 the fact that a periodic signal has an infinite amount 00:17:43.840 --> 00:17:46.930 of energy in it, and, essentially, that form of 00:17:46.930 --> 00:17:50.380 Parseval's relationship for the periodic case would say 00:17:50.380 --> 00:17:53.560 infinity equals infinity, which isn't too useful. 00:17:53.560 --> 00:17:57.320 However, it can be modified so that Parseval's relationship 00:17:57.320 --> 00:18:02.760 to the periodic case says, essentially, that the energy 00:18:02.760 --> 00:18:09.740 in one period of the periodic time function is proportional 00:18:09.740 --> 00:18:12.160 with, this is the proportionality factor, 00:18:12.160 --> 00:18:15.870 proportional to the sum of the magnitude squared of the 00:18:15.870 --> 00:18:17.020 coefficients. 00:18:17.020 --> 00:18:21.300 In other words, the energy in one period is proportional to 00:18:21.300 --> 00:18:25.580 the energy in the sequence that represents the Fourier 00:18:25.580 --> 00:18:26.830 series coefficients. 00:18:29.660 --> 00:18:34.070 There are lots of other properties, and they're 00:18:34.070 --> 00:18:37.180 developed in the text and in the study guide. 00:18:37.180 --> 00:18:42.170 A number of properties that we want to make particular use of 00:18:42.170 --> 00:18:46.100 during this lecture, and in later lectures, are ones that 00:18:46.100 --> 00:18:48.080 I summarize here. 00:18:48.080 --> 00:18:54.200 And I won't demonstrate the proofs, but principally focus 00:18:54.200 --> 00:18:58.630 on some of the interpretation as the lecture goes on. 00:18:58.630 --> 00:19:01.760 The first property that I have listed here is what's referred 00:19:01.760 --> 00:19:04.600 to as the time shifting property. 00:19:04.600 --> 00:19:08.470 And the time shifting property says, if I have a time 00:19:08.470 --> 00:19:13.610 function with a Fourier transform x of omega, if I 00:19:13.610 --> 00:19:19.950 shift that time function in time, then that corresponds to 00:19:19.950 --> 00:19:24.440 multiplying the Fourier transform by this factor. 00:19:24.440 --> 00:19:29.310 As you examine this factor, what you can see is that this 00:19:29.310 --> 00:19:34.300 factor has magnitude unity and it has a phase, which is 00:19:34.300 --> 00:19:39.740 linear with frequency, and a slope of minus t0. 00:19:39.740 --> 00:19:43.730 So a statement to remember, that will come up many times 00:19:43.730 --> 00:19:49.010 throughout the course, is that a time shift, or a 00:19:49.010 --> 00:19:58.190 displacement in time, corresponds to a linear change 00:19:58.190 --> 00:20:01.110 in phase and frequency. 00:20:01.110 --> 00:20:06.020 Another property and, in fact, a pair of properties that 00:20:06.020 --> 00:20:09.840 we'll make reference to as we turn our attention toward the 00:20:09.840 --> 00:20:14.660 end of this lecture to solving differential equations using 00:20:14.660 --> 00:20:18.920 the Fourier transform, is what's referred to as the 00:20:18.920 --> 00:20:23.790 differentiation property and its companion, which is the 00:20:23.790 --> 00:20:26.600 integration property. 00:20:26.600 --> 00:20:30.700 The differentiation property says, again, if we have a time 00:20:30.700 --> 00:20:34.950 function with Fourier transform x of omega, the 00:20:34.950 --> 00:20:41.470 Fourier transform of the time derivative of that corresponds 00:20:41.470 --> 00:20:45.950 to multiplying the Fourier transform by a linear function 00:20:45.950 --> 00:20:46.940 of frequency. 00:20:46.940 --> 00:20:51.460 So here it's a linear amplitude change that 00:20:51.460 --> 00:20:55.470 corresponds to differentiation. 00:20:55.470 --> 00:20:59.500 At first glance, what you might think is that the 00:20:59.500 --> 00:21:02.230 integration property is just the reverse of that. 00:21:02.230 --> 00:21:05.030 If for the differentiation property you multiply by j 00:21:05.030 --> 00:21:09.150 omega, then for integration you must divide by j omega. 00:21:09.150 --> 00:21:12.460 And that's almost correct, except not quite. 00:21:12.460 --> 00:21:16.930 And the reason for the not quite is that recall that if 00:21:16.930 --> 00:21:20.400 you differentiate, what happens, of course, is that 00:21:20.400 --> 00:21:21.970 you lose a constant. 00:21:21.970 --> 00:21:25.450 And if we have a time function that's some finite energy 00:21:25.450 --> 00:21:29.520 signal plus a constant, differentiating will destroy 00:21:29.520 --> 00:21:32.250 the constant. 00:21:32.250 --> 00:21:34.000 The integration property, in essence, tries 00:21:34.000 --> 00:21:35.300 to bring that back. 00:21:35.300 --> 00:21:39.730 So the integration property, which is the inverse of the 00:21:39.730 --> 00:21:44.450 differentiation property, says that we divide the transform 00:21:44.450 --> 00:21:50.610 by j omega, and then if, in fact, there was a constant 00:21:50.610 --> 00:21:54.600 added to x of t, we have to account for that by inserting 00:21:54.600 --> 00:21:58.220 an impulse into the Fourier transform. 00:21:58.220 --> 00:22:03.370 And the final property that I want to draw your attention to 00:22:03.370 --> 00:22:08.090 on this view graph is the linearity property, which is 00:22:08.090 --> 00:22:11.320 very straightforward to demonstrate from the analysis 00:22:11.320 --> 00:22:17.010 and synthesis equations, which simply says if x1 of omega is 00:22:17.010 --> 00:22:21.120 the Fourier transform x1 of t, and x2 of omega is the Fourier 00:22:21.120 --> 00:22:25.080 transform of x2 of t, then the Fourier transform of a linear 00:22:25.080 --> 00:22:29.370 combination is a linear combination of the Fourier 00:22:29.370 --> 00:22:30.620 transforms. 00:22:32.700 --> 00:22:36.370 Let me emphasize, also, that these properties, for the most 00:22:36.370 --> 00:22:40.310 part, apply both to Fourier series and Fourier transforms 00:22:40.310 --> 00:22:44.530 because, in fact, what we've done is to incorporate the 00:22:44.530 --> 00:22:48.260 Fourier series within the framework 00:22:48.260 --> 00:22:49.510 of the Fourier transform. 00:22:51.900 --> 00:22:54.970 We'll be using a number of these properties shortly, when 00:22:54.970 --> 00:22:57.740 we turn our attention to linear constant coefficient 00:22:57.740 --> 00:22:59.350 differential equations. 00:22:59.350 --> 00:23:03.670 However, before we do that I'd like to focus on two 00:23:03.670 --> 00:23:10.010 additional major properties, and these are what I refer to 00:23:10.010 --> 00:23:13.780 as the convolution property and the modulation property. 00:23:13.780 --> 00:23:17.610 And in fact, the convolution property, as I'm about to 00:23:17.610 --> 00:23:22.090 introduce it, forms the mathematical and conceptual 00:23:22.090 --> 00:23:27.250 basis for the whole notion of filtering, which, in fact, 00:23:27.250 --> 00:23:32.810 will be a topic by itself in a set of lectures, and, in fact, 00:23:32.810 --> 00:23:36.700 is a chapter by itself in the textbook. 00:23:36.700 --> 00:23:40.220 Similarly, what I'll refer to as the modulation property, 00:23:40.220 --> 00:23:45.130 again, will occupy its own set of lectures as we go through 00:23:45.130 --> 00:23:49.060 the course, and, in fact, has its own 00:23:49.060 --> 00:23:52.160 chapter in the textbook. 00:23:52.160 --> 00:23:58.450 Let me just indicate what the convolution property is. 00:23:58.450 --> 00:24:08.870 And what the convolution property tells us is that the 00:24:08.870 --> 00:24:13.050 Fourier transform of the convolution of two time 00:24:13.050 --> 00:24:19.620 functions is the product of their Fourier transforms. 00:24:19.620 --> 00:24:25.340 So it says, for example, that if I have a linear time 00:24:25.340 --> 00:24:30.500 invariant system, and I have an input x of t, an impulse 00:24:30.500 --> 00:24:33.550 response h of t, and the output, of course, being the 00:24:33.550 --> 00:24:38.280 convolution, then, in fact, if I look at this in the 00:24:38.280 --> 00:24:45.200 frequency domain, the Fourier transform of the output is the 00:24:45.200 --> 00:24:49.920 Fourier transform of the input times the Fourier transform of 00:24:49.920 --> 00:24:53.190 the impulse response. 00:24:53.190 --> 00:24:58.340 You can demonstrate this property algebraically by 00:24:58.340 --> 00:25:00.990 essentially taking the convolution integral and 00:25:00.990 --> 00:25:04.270 applying the Fourier transform and doing the appropriate 00:25:04.270 --> 00:25:08.510 interchanging of the order of integration, et cetera. 00:25:08.510 --> 00:25:13.000 But what I'd like to draw your attention to is a somewhat 00:25:13.000 --> 00:25:16.560 more intuitive interpretation of the property. 00:25:16.560 --> 00:25:20.070 And the intuitive interpretation stems from the 00:25:20.070 --> 00:25:23.720 relationship between the Fourier transform of the 00:25:23.720 --> 00:25:26.890 impulse response and what we've referred to as the 00:25:26.890 --> 00:25:28.840 frequency response. 00:25:28.840 --> 00:25:34.610 Recall that one of the things that led us to use complex 00:25:34.610 --> 00:25:37.910 exponentials as building blocks was the fact that 00:25:37.910 --> 00:25:39.610 they're eigenfunctions of linear 00:25:39.610 --> 00:25:41.000 time and variant systems. 00:25:41.000 --> 00:25:45.640 In other words, if we have a linear time invariant system, 00:25:45.640 --> 00:25:48.990 and I have an input which is a complex exponential, the 00:25:48.990 --> 00:25:51.770 output is a complex exponential of the same 00:25:51.770 --> 00:25:55.790 frequency multiplied by what we call 00:25:55.790 --> 00:25:57.510 the frequency response. 00:25:57.510 --> 00:26:02.410 And, in fact, the expression for the frequency response is 00:26:02.410 --> 00:26:05.970 identical to the expression for the Fourier transform of 00:26:05.970 --> 00:26:07.710 the impulse response. 00:26:07.710 --> 00:26:11.670 In other words, the frequency response is the Fourier 00:26:11.670 --> 00:26:15.750 transform of the impulse response. 00:26:15.750 --> 00:26:20.350 Now in that context, how can we interpret 00:26:20.350 --> 00:26:22.260 the convolution property? 00:26:22.260 --> 00:26:26.630 Well, remember what I said at the beginning of the lecture, 00:26:26.630 --> 00:26:30.850 when I pointed to the synthesis equation and I said, 00:26:30.850 --> 00:26:35.010 in essence, the synthesis equation tells us how to 00:26:35.010 --> 00:26:38.490 decompose x of t as a linear combination of complex 00:26:38.490 --> 00:26:40.740 exponentials. 00:26:40.740 --> 00:26:43.460 What are the complex amplitudes of those complex 00:26:43.460 --> 00:26:45.020 exponentials? 00:26:45.020 --> 00:26:50.250 In terms of our notation here, the complex amplitude of those 00:26:50.250 --> 00:26:55.590 complex exponentials is x of omega, or proportional to x of 00:26:55.590 --> 00:26:58.650 omega, in particular it's x of omega, d omega, and then a 00:26:58.650 --> 00:27:01.160 factor of 2 pi. 00:27:01.160 --> 00:27:06.480 As this signal goes through this linear time invariant 00:27:06.480 --> 00:27:12.100 system, what happens to each of those exponential is each 00:27:12.100 --> 00:27:15.610 one gets multiplied by the frequency response at the 00:27:15.610 --> 00:27:18.090 associated frequency. 00:27:18.090 --> 00:27:22.770 What comes out is the amplitude of the complex 00:27:22.770 --> 00:27:28.610 exponentials that are used to build the output. 00:27:28.610 --> 00:27:35.990 So in fact, the convolution property simply is telling us 00:27:35.990 --> 00:27:39.730 that, in terms of the decomposition of the signal, 00:27:39.730 --> 00:27:42.940 in terms of complex exponentials, as we push that 00:27:42.940 --> 00:27:46.230 signal through a linear time invariant system, we're 00:27:46.230 --> 00:27:52.780 separately multiplying by the frequency response, the 00:27:52.780 --> 00:27:56.590 amplitudes of the exponential components used 00:27:56.590 --> 00:27:58.100 to build the input. 00:27:58.100 --> 00:28:03.860 And that sum, in turn, is the decomposition of the output in 00:28:03.860 --> 00:28:05.270 terms of complex exponentials. 00:28:08.630 --> 00:28:12.310 I understand that going through that involves a little 00:28:12.310 --> 00:28:16.640 bit of sorting out, and I strongly encourage you to try 00:28:16.640 --> 00:28:20.940 to understand and interpret the convolution property in 00:28:20.940 --> 00:28:25.400 those conceptual terms, rather than simply by applying the 00:28:25.400 --> 00:28:29.050 mathematics to the convolution integral and seeing the terms 00:28:29.050 --> 00:28:30.300 match up on both sides. 00:28:33.930 --> 00:28:38.190 As I indicated, the convolution property forms the 00:28:38.190 --> 00:28:41.580 basis for what's referred to as filtering. 00:28:41.580 --> 00:28:46.700 And this is a topic that we'll be treating in a considerable 00:28:46.700 --> 00:28:51.330 amount of detail after we've also gone through a discussion 00:28:51.330 --> 00:28:54.400 of the discrete time Fourier transform in the 00:28:54.400 --> 00:28:56.050 next several lectures. 00:28:56.050 --> 00:28:59.600 However, what I'd like to do is just indicate, now, a 00:28:59.600 --> 00:29:04.560 little bit of the conceptual ideas involved. 00:29:04.560 --> 00:29:08.990 Essentially, conceptually, what filtering, as it's 00:29:08.990 --> 00:29:15.250 typically referred to, corresponds to is modifying 00:29:15.250 --> 00:29:18.220 separately the individual frequency 00:29:18.220 --> 00:29:20.030 components in a signal. 00:29:20.030 --> 00:29:25.360 The convolution property told us that if we look at the 00:29:25.360 --> 00:29:28.330 individual frequency components, they get 00:29:28.330 --> 00:29:30.970 multiplied by the frequency response, and so what that 00:29:30.970 --> 00:29:34.710 says is that we can amplify or attenuate any of those 00:29:34.710 --> 00:29:39.570 components separately using a linear time invariant system. 00:29:39.570 --> 00:29:43.050 For example, what I've illustrated here is the 00:29:43.050 --> 00:29:47.320 frequency response of what is commonly referred to as an 00:29:47.320 --> 00:29:50.220 ideal low pass filter. 00:29:50.220 --> 00:29:55.990 What an ideal low pass filter does is to pass exactly 00:29:55.990 --> 00:30:01.200 frequencies in one frequency range and eliminate totally 00:30:01.200 --> 00:30:04.350 frequencies outside that range. 00:30:04.350 --> 00:30:08.890 Another filter which is not so ideal might, for example, 00:30:08.890 --> 00:30:15.980 attenuate components in this band but not 00:30:15.980 --> 00:30:17.230 totally eliminate them. 00:30:19.650 --> 00:30:22.190 In terms of filtering, we can think back to the 00:30:22.190 --> 00:30:27.390 differentiation property and, in fact, interpret 00:30:27.390 --> 00:30:29.680 differentiator as a filter. 00:30:29.680 --> 00:30:32.710 Recall that the differentiation property said 00:30:32.710 --> 00:30:39.330 that the Fourier transform of the differentiated signal is 00:30:39.330 --> 00:30:41.180 the Fourier transform of the original signal 00:30:41.180 --> 00:30:43.600 multiplied by j omega. 00:30:43.600 --> 00:30:47.060 So what that says, then, is that if we have a 00:30:47.060 --> 00:30:52.490 differentiator, the frequency response of that is j omega. 00:30:52.490 --> 00:30:55.640 In other words, the Fourier transform of the output is j 00:30:55.640 --> 00:31:00.120 omega times the Fourier transform of the input. 00:31:00.120 --> 00:31:04.640 And so the frequency response of the differentiator looks 00:31:04.640 --> 00:31:07.140 like this, in terms of its magnitude. 00:31:07.140 --> 00:31:10.700 And what it does, of course, is it amplifies high 00:31:10.700 --> 00:31:14.950 frequencies and attenuates low frequencies. 00:31:17.670 --> 00:31:23.520 Let me just, to cement some of these ideas, illustrate them 00:31:23.520 --> 00:31:29.140 in the context of one kind of signal, namely a signal which, 00:31:29.140 --> 00:31:34.170 in fact, is a spatial signal rather than a time signal. 00:31:34.170 --> 00:31:39.270 And this also gives me an opportunity to introduce you 00:31:39.270 --> 00:31:43.080 to our colleague, J. B. J. Fourier. 00:31:43.080 --> 00:31:47.790 So if we could look at our colleague, Mr. Fourier, who, 00:31:47.790 --> 00:31:53.500 by the way, is not only a person who had tremendously 00:31:53.500 --> 00:31:59.140 brilliant insights, and his insights, in fact, have led to 00:31:59.140 --> 00:32:02.670 forming the foundation of the developments that is the basis 00:32:02.670 --> 00:32:04.040 for this course. 00:32:04.040 --> 00:32:05.730 He was also a very 00:32:05.730 --> 00:32:07.560 interesting, fascinating person. 00:32:07.560 --> 00:32:11.830 And there's a certain amount of historical discussion about 00:32:11.830 --> 00:32:15.300 Fourier and his background, which you might enjoy reading 00:32:15.300 --> 00:32:17.120 in the text. 00:32:17.120 --> 00:32:18.740 In any case, what you're looking at, of 00:32:18.740 --> 00:32:20.500 course, is a signal. 00:32:20.500 --> 00:32:25.160 And the signal is a spatial signal, and it has high 00:32:25.160 --> 00:32:26.880 frequencies and low frequencies. 00:32:26.880 --> 00:32:31.030 High frequencies corresponding to things that are varying 00:32:31.030 --> 00:32:34.310 rapidly spatially, and low frequencies 00:32:34.310 --> 00:32:37.810 varying slowly spatially. 00:32:37.810 --> 00:32:43.250 And so, for example, we could low pass filter this picture 00:32:43.250 --> 00:32:48.090 simply by asking the video crew if they could be slightly 00:32:48.090 --> 00:32:50.270 defocus it. 00:32:50.270 --> 00:32:53.540 And what you see as the picture is defocused, if you 00:32:53.540 --> 00:32:58.560 could hold it there, is that we've lost edges, which is the 00:32:58.560 --> 00:33:00.230 rapid variation. 00:33:00.230 --> 00:33:05.020 And what we've retained is the broader, slow variation. 00:33:05.020 --> 00:33:09.170 And now let's take out the defocusing low pass filter and 00:33:09.170 --> 00:33:11.550 go back to a focused image. 00:33:19.480 --> 00:33:23.260 What we can also consider is what would happen if we looked 00:33:23.260 --> 00:33:25.090 at the differentiated image. 00:33:25.090 --> 00:33:28.210 And there are several ways we can think about this. 00:33:28.210 --> 00:33:30.580 One is that a differentiator-- 00:33:30.580 --> 00:33:35.400 Of course the output of a differentiator is larger where 00:33:35.400 --> 00:33:40.460 the discontinuity, or where the variation, is faster. 00:33:40.460 --> 00:33:44.890 And so we would expect the edges to be enhanced if, in 00:33:44.890 --> 00:33:47.490 fact, we differentiated the image. 00:33:47.490 --> 00:33:51.200 Or if we interpret differentiation in the context 00:33:51.200 --> 00:33:59.190 of our filter, then what we're saying is that, in effect, 00:33:59.190 --> 00:34:04.730 what's happening is that the differentiator is accentuating 00:34:04.730 --> 00:34:08.130 the high frequencies because of the frequency shape of the 00:34:08.130 --> 00:34:09.790 differentiator. 00:34:09.790 --> 00:34:13.940 Recall that this all fits together as a nice package. 00:34:13.940 --> 00:34:16.120 We expect intuitively that differentiation 00:34:16.120 --> 00:34:18.020 will enhance edges. 00:34:18.020 --> 00:34:21.250 When we talked about square waves and we saw how the 00:34:21.250 --> 00:34:25.380 Fourier series built up a square wave, we saw that it 00:34:25.380 --> 00:34:29.420 was the high frequencies that were required in order to 00:34:29.420 --> 00:34:32.610 build up the sharp edges. 00:34:32.610 --> 00:34:36.250 And so either viewed as a filter, or viewed intuitively, 00:34:36.250 --> 00:34:40.070 we would expect that the differentiated image would, in 00:34:40.070 --> 00:34:46.560 fact, attenuate this slowly varying background and amplify 00:34:46.560 --> 00:34:48.320 the rapidly varying edges. 00:34:48.320 --> 00:34:52.170 So let's look again at our original image, just to remind 00:34:52.170 --> 00:34:55.829 you of the fact that there are edges, of course, and there is 00:34:55.829 --> 00:34:59.480 a more slowly varying background. 00:34:59.480 --> 00:35:05.860 And now let's look at the result of passing that through 00:35:05.860 --> 00:35:07.730 a differentiator. 00:35:07.730 --> 00:35:11.790 And, as I think is very evident in the resulting 00:35:11.790 --> 00:35:17.610 image, clearly it's the edges that are retained and the 00:35:17.610 --> 00:35:21.590 slower background variations are destroyed, which is 00:35:21.590 --> 00:35:23.220 consistent with everything that we've said. 00:35:27.270 --> 00:35:31.790 I've emphasized that we'll be returning to a much broader 00:35:31.790 --> 00:35:35.540 discussion of filtering at a later point in the course. 00:35:35.540 --> 00:35:40.690 I'd now like to comment on another property, which is 00:35:40.690 --> 00:35:44.110 also, as I indicated, a topic in its own right, and which 00:35:44.110 --> 00:35:48.740 really is the dual property to the convolution property, and, 00:35:48.740 --> 00:35:53.360 in fact, could be argued directly from duality. 00:35:53.360 --> 00:35:59.230 And that is what's referred to as the modulation property. 00:35:59.230 --> 00:36:04.210 The convolution property told us that if we convolve in the 00:36:04.210 --> 00:36:08.940 time domain, we multiply in the frequency domain. 00:36:08.940 --> 00:36:10.920 And we know that time and frequency domains are 00:36:10.920 --> 00:36:13.920 interchangeable because of duality, so what that would 00:36:13.920 --> 00:36:19.860 suggest is that if we multiply in the time domain, that would 00:36:19.860 --> 00:36:23.140 correspond to convolution in the frequency domain. 00:36:23.140 --> 00:36:25.430 And, in fact, that is exactly what the 00:36:25.430 --> 00:36:27.610 modulation property is. 00:36:27.610 --> 00:36:31.940 I have it summarized here that if we multiply a time function 00:36:31.940 --> 00:36:34.340 by another time function, then in the 00:36:34.340 --> 00:36:37.880 frequency domain we convolve. 00:36:37.880 --> 00:36:40.850 Whereas the convolution property is just the dual of 00:36:40.850 --> 00:36:45.870 that, namely convolving in the time domain corresponds to 00:36:45.870 --> 00:36:49.040 multiplication in the frequency domain. 00:36:49.040 --> 00:36:52.170 The convolution property is the basis, as I 00:36:52.170 --> 00:36:54.390 indicated, for filtering. 00:36:54.390 --> 00:36:58.620 The modulation property, as I've summarized it here, in 00:36:58.620 --> 00:37:06.000 fact, is the entire basis for amplitude modulation systems 00:37:06.000 --> 00:37:10.080 as used almost universally in communications. 00:37:10.080 --> 00:37:13.110 And what the modulation property, as we'll see when we 00:37:13.110 --> 00:37:18.300 explore it in more detail, tells us is that if we have a 00:37:18.300 --> 00:37:22.560 signal with a certain spectrum, and we multiply by a 00:37:22.560 --> 00:37:27.020 sinusoidal signal whose Fourier transform is a set of 00:37:27.020 --> 00:37:31.500 impulses, then in a frequency domain we convolve. 00:37:31.500 --> 00:37:35.570 And that corresponds to taking the original spectrum and 00:37:35.570 --> 00:37:41.160 translating it, shifting it in frequency up to the frequency 00:37:41.160 --> 00:37:45.060 of the carrier, namely the sinusoidal signal. 00:37:45.060 --> 00:37:49.330 And as I said, we'll come to that in much more detail in a 00:37:49.330 --> 00:37:50.580 number of lectures. 00:37:52.960 --> 00:37:56.780 We've seen a number of properties, and I indicated 00:37:56.780 --> 00:38:00.100 sometime earlier when we talked about differential 00:38:00.100 --> 00:38:03.350 equations, that, in fact, it's the properties of the Fourier 00:38:03.350 --> 00:38:08.730 transform that provide us with a very useful and important 00:38:08.730 --> 00:38:13.420 mechanism for solving linear constant coefficient 00:38:13.420 --> 00:38:15.350 differential equations. 00:38:15.350 --> 00:38:21.340 And what I'd like to do now is illustrate the procedure, the 00:38:21.340 --> 00:38:24.780 basis for that, and I think what we'll do is illustrate it 00:38:24.780 --> 00:38:26.985 simply in the context of several examples. 00:38:31.120 --> 00:38:39.380 What I've indicated is a system with an impulse 00:38:39.380 --> 00:38:43.550 response h of t, or frequency response h of omega. 00:38:43.550 --> 00:38:46.630 And, of course, we know that in the time domain it's 00:38:46.630 --> 00:38:50.880 described through convolution, in the frequency domain it's 00:38:50.880 --> 00:38:54.020 described through multiplication. 00:38:54.020 --> 00:38:59.040 And I, in essence, am assuming that we're talking about a 00:38:59.040 --> 00:39:01.400 linear time invariant system. 00:39:01.400 --> 00:39:05.560 And we're also going to assume then it's characterized by a 00:39:05.560 --> 00:39:09.540 linear constant coefficient differential equation, where 00:39:09.540 --> 00:39:13.730 we're going to impose the condition that it's causal 00:39:13.730 --> 00:39:17.760 linear and time invariant, or equivalently that the initial 00:39:17.760 --> 00:39:22.320 conditions are consistent with the initial rest. 00:39:22.320 --> 00:39:26.160 And it's because of the fact that we're assuming that it's 00:39:26.160 --> 00:39:30.380 a linear time invariant system that we can describe it in the 00:39:30.380 --> 00:39:34.560 frequency domain through the convolution property, and we 00:39:34.560 --> 00:39:38.940 can use the properties of the Fourier transform. 00:39:38.940 --> 00:39:43.190 So let's take, as our example, a first order differential 00:39:43.190 --> 00:39:46.670 equation as I indicate here. 00:39:46.670 --> 00:39:49.560 So the derivative of the output plus a times the output 00:39:49.560 --> 00:39:52.330 is equal to the input. 00:39:52.330 --> 00:39:57.080 And now we can use the differentiation property. 00:39:57.080 --> 00:40:01.250 If we Fourier transform this entire expression, the 00:40:01.250 --> 00:40:06.690 differentiation property tells us that the Fourier transform 00:40:06.690 --> 00:40:10.290 of the derivative of the output is the Fourier 00:40:10.290 --> 00:40:14.560 transform of the output multiplied by j omega. 00:40:14.560 --> 00:40:20.330 And linearity will let us write the Fourier transform of 00:40:20.330 --> 00:40:22.810 this as a times y of omega. 00:40:22.810 --> 00:40:25.690 And since these are added together, and since we have 00:40:25.690 --> 00:40:29.330 the linearity property, these are added together. 00:40:29.330 --> 00:40:34.600 And x of omega is the Fourier transform of x of t. 00:40:34.600 --> 00:40:40.630 So what we've used is the differentiation property, and 00:40:40.630 --> 00:40:43.185 we've used the linearity property. 00:40:48.290 --> 00:40:52.980 We can solve this equation for y of omega. 00:40:52.980 --> 00:40:57.420 The Fourier transform of the output in terms of x of omega, 00:40:57.420 --> 00:41:01.970 the Fourier transform of the input, and a simple algebraic 00:41:01.970 --> 00:41:05.430 step gets us to this expression. 00:41:05.430 --> 00:41:08.550 So the Fourier transform of the output is 1 over j omega 00:41:08.550 --> 00:41:12.110 plus a times the Fourier transform of the input. 00:41:12.110 --> 00:41:16.220 And I've just simply repeated that equation up here. 00:41:19.980 --> 00:41:25.470 So far this is algebra, and the question is, now, how do 00:41:25.470 --> 00:41:27.540 we interpret this? 00:41:27.540 --> 00:41:32.880 Well, we know that the Fourier transform of the output is the 00:41:32.880 --> 00:41:38.060 Fourier transform of the input times the Fourier transform of 00:41:38.060 --> 00:41:40.120 the impulse response of the system, namely 00:41:40.120 --> 00:41:41.830 the frequency response. 00:41:41.830 --> 00:41:47.170 So, in fact, if we think of h of t and h of omega as a 00:41:47.170 --> 00:41:51.980 Fourier transform pair, it's the convolution property that 00:41:51.980 --> 00:41:58.295 lets us equate this term with h of omega. 00:41:58.295 --> 00:42:02.230 So here we're using the convolution property. 00:42:06.170 --> 00:42:10.750 So we know what the Fourier transform of the impulse 00:42:10.750 --> 00:42:14.030 response is, namely 1 over j omega plus a. 00:42:17.270 --> 00:42:20.870 We may have, for example, wanted in our problem, instead 00:42:20.870 --> 00:42:23.420 of getting the frequency response, to 00:42:23.420 --> 00:42:25.790 get the impulse response. 00:42:25.790 --> 00:42:28.690 And there are a variety of ways that we can do this. 00:42:28.690 --> 00:42:32.400 We can attempt to go through the inverse Fourier transform 00:42:32.400 --> 00:42:33.570 expression. 00:42:33.570 --> 00:42:38.780 But in fact, one of the most useful ways is formally called 00:42:38.780 --> 00:42:40.490 the inspection method. 00:42:40.490 --> 00:42:46.190 Informally it's called, if you worked it out going one way, 00:42:46.190 --> 00:42:48.630 then you ought to remember the answer so that you know how to 00:42:48.630 --> 00:42:51.000 get back method. 00:42:51.000 --> 00:42:56.140 So what that says is, remember that we worked an example, and 00:42:56.140 --> 00:42:58.520 in fact I showed you the example earlier in the 00:42:58.520 --> 00:43:03.540 lecture, that the Fourier transform of e to the minus at 00:43:03.540 --> 00:43:07.460 times the step is 1 over j omega plus a? 00:43:07.460 --> 00:43:11.550 So what is the inverse Fourier transfer of 1 over 00:43:11.550 --> 00:43:13.130 j omega plus a? 00:43:13.130 --> 00:43:19.890 Well, it's e to the minus at times a unit step. 00:43:19.890 --> 00:43:22.840 And that's just simply remembering, in essence, this 00:43:22.840 --> 00:43:24.090 particular transform pair. 00:43:26.940 --> 00:43:32.070 I've drawn, graphically, the magnitude of the Fourier 00:43:32.070 --> 00:43:35.890 transform here. 00:43:35.890 --> 00:43:41.100 And below it we have the impulse response. 00:43:41.100 --> 00:43:45.610 The impulse response is, as I just indicated, e to the minus 00:43:45.610 --> 00:43:48.450 at times u of t. 00:43:48.450 --> 00:43:53.400 Now let's go back up and look at the magnitude of the 00:43:53.400 --> 00:43:56.240 frequency response. 00:43:56.240 --> 00:44:00.030 And given just the little bit of discussion that we had 00:44:00.030 --> 00:44:05.680 previously about filtering, you should be able to infer 00:44:05.680 --> 00:44:09.940 something about the filtering characteristics of this 00:44:09.940 --> 00:44:13.680 simple, first order differential equation. 00:44:13.680 --> 00:44:17.740 In particular, if you look at that frequency response, the 00:44:17.740 --> 00:44:21.550 frequency response falls off with frequency and so what it 00:44:21.550 --> 00:44:27.080 tends to do is attenuate high frequencies and retain low 00:44:27.080 --> 00:44:28.160 frequencies. 00:44:28.160 --> 00:44:32.950 So in fact, you could think of the defocusing that we did on 00:44:32.950 --> 00:44:35.640 the image of Fourier, you could think of that, 00:44:35.640 --> 00:44:39.390 approximately, as similar to the kind of filtering action 00:44:39.390 --> 00:44:43.630 that you would get by passing a signal through a first order 00:44:43.630 --> 00:44:44.880 differential equation. 00:44:47.620 --> 00:44:52.640 Just to illustrate one additional step in both 00:44:52.640 --> 00:44:56.090 evaluating inverse transforms and using Fourier transform 00:44:56.090 --> 00:44:59.690 properties to solve linear constant coefficient 00:44:59.690 --> 00:45:04.250 differential equations, let's take the same example and, 00:45:04.250 --> 00:45:07.830 rather than finding the impulse response, let's find 00:45:07.830 --> 00:45:11.910 the response to another exponential input. 00:45:11.910 --> 00:45:13.830 We could, of course, do that using 00:45:13.830 --> 00:45:14.950 the convolution integral. 00:45:14.950 --> 00:45:17.090 We've just gotten the impulse response, and we could put 00:45:17.090 --> 00:45:20.710 that through the convolution integral to get the response 00:45:20.710 --> 00:45:21.930 to this input. 00:45:21.930 --> 00:45:25.160 But let's do it, instead, by going back to the 00:45:25.160 --> 00:45:27.890 differential equation. 00:45:27.890 --> 00:45:32.330 And so here I'm taking a differential equation. 00:45:32.330 --> 00:45:35.920 I'll choose, just to have some numbers to work 00:45:35.920 --> 00:45:38.400 with, a equal to 2. 00:45:38.400 --> 00:45:41.690 And now I'll choose an exponential on the right hand 00:45:41.690 --> 00:45:46.030 side, e to the minus t times u of t. 00:45:46.030 --> 00:45:51.700 And again, we Fourier transform the equation. 00:45:51.700 --> 00:45:55.510 And we can remember this particular 00:45:55.510 --> 00:45:57.490 Fourier transform pair. 00:45:57.490 --> 00:46:00.690 It's just the one we worked out previously, now with a 00:46:00.690 --> 00:46:01.650 equal to 1. 00:46:01.650 --> 00:46:05.430 Clearly we're getting a lot of mileage out of that example. 00:46:05.430 --> 00:46:09.660 And now, if we want to determine what the output y of 00:46:09.660 --> 00:46:14.920 t is, we can do that by solving for y of omega and 00:46:14.920 --> 00:46:18.520 then generating the inverse Fourier transform. 00:46:18.520 --> 00:46:22.850 Let's solve this algebraically for y of omega, and that gets 00:46:22.850 --> 00:46:25.460 us to this expression. 00:46:25.460 --> 00:46:29.420 And this is not a Fourier transform that we've worked 00:46:29.420 --> 00:46:30.830 out before. 00:46:30.830 --> 00:46:36.190 And this is the second part to the inspection procedure. 00:46:36.190 --> 00:46:39.430 What we have is a Fourier transform which is a product 00:46:39.430 --> 00:46:42.940 of two terms, each of which we can recognize. 00:46:42.940 --> 00:46:48.660 And what we can consider doing is expanding that out in a 00:46:48.660 --> 00:46:52.160 partial fraction expansion, namely as a sum of terms. 00:46:52.160 --> 00:46:54.650 Because of the linearity property associated with the 00:46:54.650 --> 00:46:59.040 Fourier transform, the inverse transform is then the sum of 00:46:59.040 --> 00:47:01.030 the inverse transform of each of those terms. 00:47:03.750 --> 00:47:08.020 So if we expand this out in a partial fraction expansion, 00:47:08.020 --> 00:47:12.990 and you can just verify that if you add these two together 00:47:12.990 --> 00:47:15.440 you'll get back to where we started. 00:47:15.440 --> 00:47:22.670 We now have the sum of two terms, and if we now 00:47:22.670 --> 00:47:26.740 recognize, by inspection, the inverse Fourier transform of 00:47:26.740 --> 00:47:32.520 this, we see that it's simply minus e to the minus 2t times 00:47:32.520 --> 00:47:33.720 the unit step. 00:47:33.720 --> 00:47:38.320 This one is plus e to the minus t times the unit step. 00:47:38.320 --> 00:47:43.280 And so, in fact, it's the sum of these two terms which are 00:47:43.280 --> 00:47:47.970 the inverse transforms of the individual terms in the 00:47:47.970 --> 00:48:02.820 partial fraction expansion that then give us the output. 00:48:02.820 --> 00:48:08.170 So this is y of t, which is the sum of these two 00:48:08.170 --> 00:48:14.350 exponentials, and this is the inverse Fourier transform of y 00:48:14.350 --> 00:48:16.690 of omega as we calculated it previously. 00:48:21.110 --> 00:48:24.940 Hopefully you're beginning to get some sense, now, of how 00:48:24.940 --> 00:48:29.190 powerful and also beautiful the Fourier transform is. 00:48:29.190 --> 00:48:33.980 We've seen already a glimpse of how it plays a role in 00:48:33.980 --> 00:48:38.050 filtering, modulation, how its properties help us with linear 00:48:38.050 --> 00:48:39.510 constant coefficient differential 00:48:39.510 --> 00:48:42.456 equations, et cetera. 00:48:42.456 --> 00:48:48.520 What we will do, beginning with the next lecture, is 00:48:48.520 --> 00:48:52.310 develop a similar set of tools for the discrete time case. 00:48:52.310 --> 00:48:56.650 And there are some very strong similarities to what we've 00:48:56.650 --> 00:48:58.850 done in continuous time, also some very important 00:48:58.850 --> 00:49:00.190 differences. 00:49:00.190 --> 00:49:04.150 And then, after we have the continuous time and discrete 00:49:04.150 --> 00:49:10.270 time Fourier transforms, we'll then see how the concepts 00:49:10.270 --> 00:49:14.030 involved and the properties involved lead to very 00:49:14.030 --> 00:49:18.690 important and powerful notions of filtering, modulation, 00:49:18.690 --> 00:49:22.250 sampling, and other signal processing ideas. 00:49:22.250 --> 00:49:23.500 Thank you.