1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation, or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,540 ocw.mit.edu. 8 00:00:55,600 --> 00:00:58,360 PROFESSOR: In the last two lectures, we saw how periodic 9 00:00:58,360 --> 00:01:01,300 and non periodic signals could be represented as linear 10 00:01:01,300 --> 00:01:04,440 combinations of complex exponentials. 11 00:01:04,440 --> 00:01:08,570 And this led to the Fourier series representation, in the 12 00:01:08,570 --> 00:01:12,280 periodic case, and it led to the Fourier transform 13 00:01:12,280 --> 00:01:14,950 representation in the aperiodic case. 14 00:01:14,950 --> 00:01:19,340 And then, in fact, what we did was to incorporate the Fourier 15 00:01:19,340 --> 00:01:25,480 series within the framework of the Fourier transform. 16 00:01:25,480 --> 00:01:29,350 What I'd like to do in today's lecture is look at the Fourier 17 00:01:29,350 --> 00:01:33,620 transform more closely, in particular with regard to some 18 00:01:33,620 --> 00:01:35,050 of its properties. 19 00:01:35,050 --> 00:01:39,460 So let me begin by reminding you of the analysis and 20 00:01:39,460 --> 00:01:43,020 synthesis equations for the Fourier transform, as I've 21 00:01:43,020 --> 00:01:45,350 summarized them here. 22 00:01:45,350 --> 00:01:49,550 The synthesis equation being an equation the tells us how 23 00:01:49,550 --> 00:01:53,970 to build the time function out of, in essence, a linear 24 00:01:53,970 --> 00:01:57,060 combination of complex exponentials. 25 00:01:57,060 --> 00:02:01,390 And the analysis equation telling us how to get the 26 00:02:01,390 --> 00:02:05,820 amplitudes of those complex exponentials from the 27 00:02:05,820 --> 00:02:07,970 associated time function. 28 00:02:07,970 --> 00:02:12,360 So essentially, in the decomposition of x of t as a 29 00:02:12,360 --> 00:02:16,270 linear combination of complex exponentials, the complex 30 00:02:16,270 --> 00:02:20,560 amplitudes of those are, in effect, the Fourier transform 31 00:02:20,560 --> 00:02:26,670 scaled by the differential and scaled by 1 over 2 pi. 32 00:02:26,670 --> 00:02:33,730 As I indicated last time, the Fourier transform is a complex 33 00:02:33,730 --> 00:02:35,810 function of frequency. 34 00:02:35,810 --> 00:02:40,490 And in particular, the complex function of frequency has an 35 00:02:40,490 --> 00:02:44,770 important and very useful symmetry property. 36 00:02:44,770 --> 00:02:49,030 The symmetry of the Fourier transform, when x of t is 37 00:02:49,030 --> 00:02:53,450 real, is what is referred to as conjugate symmetric. 38 00:02:53,450 --> 00:02:57,940 In other words, if we take the complex function, f of omega, 39 00:02:57,940 --> 00:03:01,970 and its complex conjugate, that's equivalent to replacing 40 00:03:01,970 --> 00:03:04,930 omega by minus omega. 41 00:03:04,930 --> 00:03:10,300 And a consequence of that, if we think in terms of the real 42 00:03:10,300 --> 00:03:14,890 part of the Fourier transform, the real part is an even 43 00:03:14,890 --> 00:03:19,730 function of frequency, and the magnitude is an even function 44 00:03:19,730 --> 00:03:21,370 of frequency. 45 00:03:21,370 --> 00:03:26,550 Whereas the imaginary part is an odd function of frequency, 46 00:03:26,550 --> 00:03:31,170 and the phase angle is an odd function of frequency. 47 00:03:31,170 --> 00:03:35,450 So we have this symmetry relationship that, for x of t 48 00:03:35,450 --> 00:03:39,390 real, if we think of either the real part or the magnitude 49 00:03:39,390 --> 00:03:42,880 of the Fourier transform, it's even symmetric. 50 00:03:42,880 --> 00:03:48,100 And the imaginary part, or the phase angle, either one, is 51 00:03:48,100 --> 00:03:49,310 odd symmetric. 52 00:03:49,310 --> 00:03:53,740 In other words, if we flip it, we multiply by a minus sign. 53 00:03:53,740 --> 00:03:58,260 Let's look at an example, in the context of an example that 54 00:03:58,260 --> 00:04:02,870 we worked last time for the Fourier transform. 55 00:04:02,870 --> 00:04:08,320 We took the case of a real exponential of the form e to 56 00:04:08,320 --> 00:04:11,550 the minus at times the step. 57 00:04:11,550 --> 00:04:17,120 And the Fourier transform, as we found, was of the algebraic 58 00:04:17,120 --> 00:04:20,089 form 1 over a plus j omega. 59 00:04:20,089 --> 00:04:23,580 And, incidentally, the Fourier transform integral only 60 00:04:23,580 --> 00:04:26,790 converged for a greater than 0. 61 00:04:26,790 --> 00:04:31,250 In other words, for this exponential decaying. 62 00:04:31,250 --> 00:04:34,810 And I illustrated the magnitude and angle, and what 63 00:04:34,810 --> 00:04:40,460 we see is that e to the minus at is a real time function, 64 00:04:40,460 --> 00:04:44,110 therefore its magnitude should be an even function of 65 00:04:44,110 --> 00:04:46,630 frequency, and indeed it is. 66 00:04:46,630 --> 00:04:50,800 And its phase angle, shown below, is an 67 00:04:50,800 --> 00:04:54,050 odd function of frequency. 68 00:04:54,050 --> 00:05:02,170 So in fact, although I stressed last time that the 69 00:05:02,170 --> 00:05:04,800 complex exponentials is required to build a time 70 00:05:04,800 --> 00:05:09,000 function require exponentials of both positive and negative 71 00:05:09,000 --> 00:05:14,160 frequencies, for x of t real what we see is that, because 72 00:05:14,160 --> 00:05:16,270 of these symmetry properties, either for the real and 73 00:05:16,270 --> 00:05:20,960 imaginary or magnitude and angle, we can specify the 74 00:05:20,960 --> 00:05:24,630 Fourier transform for, let's say only positive frequencies, 75 00:05:24,630 --> 00:05:28,470 and the symmetry, then, implies, or tells us, what the 76 00:05:28,470 --> 00:05:30,850 Fourier transform, then, would be for the negative 77 00:05:30,850 --> 00:05:32,100 frequencies. 78 00:05:34,560 --> 00:05:39,480 This same example, the decaying exponential, 79 00:05:39,480 --> 00:05:44,320 demonstrates another important and often useful property of 80 00:05:44,320 --> 00:05:47,370 the Fourier transform. 81 00:05:47,370 --> 00:05:52,410 Specifically, let's rewrite, algebraically, this example as 82 00:05:52,410 --> 00:05:54,350 I indicate here. 83 00:05:54,350 --> 00:05:57,720 So we have, again, the exponential, whose Fourier 84 00:05:57,720 --> 00:06:01,800 transform is 1 over a plus j omega. 85 00:06:01,800 --> 00:06:06,020 And if I just simply divide numerator and denominator by 86 00:06:06,020 --> 00:06:10,370 a, I can rewrite it in the form shown here. 87 00:06:10,370 --> 00:06:15,800 And what we notice is that in the time function we have a 88 00:06:15,800 --> 00:06:20,020 term of the form a times t, and in the frequency function 89 00:06:20,020 --> 00:06:24,630 we have a term of the form omega divided by a. 90 00:06:24,630 --> 00:06:30,390 Or equivalently, we could think of the time function, 91 00:06:30,390 --> 00:06:37,280 which I show here, and as the parameter a gets smaller, the 92 00:06:37,280 --> 00:06:41,300 exponential gets spread out in time. 93 00:06:41,300 --> 00:06:45,880 Whereas its Fourier transform, or the magnitude of its 94 00:06:45,880 --> 00:06:50,080 Fourier transform, has the inverse property that as a 95 00:06:50,080 --> 00:06:56,840 gets smaller, in fact, this scales down in frequency. 96 00:06:56,840 --> 00:07:00,430 Well, this is a general property of the Fourier 97 00:07:00,430 --> 00:07:06,350 transform, namely the fact that a linear scaling in time 98 00:07:06,350 --> 00:07:10,590 generates the inverse linear scaling in frequency. 99 00:07:10,590 --> 00:07:14,190 And the general statement of this time frequency scaling is 100 00:07:14,190 --> 00:07:18,100 what I show at the top of the transparency, namely the 101 00:07:18,100 --> 00:07:23,570 equation that if we scale the time function in time, then we 102 00:07:23,570 --> 00:07:26,140 apply an inverse scaling in frequency. 103 00:07:28,720 --> 00:07:33,430 This, in fact, is probably a result that you're already 104 00:07:33,430 --> 00:07:35,580 possibly familiar with, in somewhat 105 00:07:35,580 --> 00:07:37,190 of a different context. 106 00:07:37,190 --> 00:07:42,250 Essentially, you could think of this as an example, or a 107 00:07:42,250 --> 00:07:47,700 generalization, rather, of the notion that if, let's say I 108 00:07:47,700 --> 00:07:53,610 had a signal that was recorded on a tape player, and if I 109 00:07:53,610 --> 00:07:57,140 play the tape back at, let's say, twice the speed, which 110 00:07:57,140 --> 00:08:00,380 means that I'm compressing the time axis linearly 111 00:08:00,380 --> 00:08:02,240 by a factor of 2. 112 00:08:02,240 --> 00:08:05,280 Then, in fact, what happens is that the frequencies that we 113 00:08:05,280 --> 00:08:08,930 observe get pushed up by a factor of 2. 114 00:08:08,930 --> 00:08:12,450 And, in fact, let me illustrate that. 115 00:08:12,450 --> 00:08:18,670 I have here a glockenspiel, and anyone who loves a parade 116 00:08:18,670 --> 00:08:22,610 certainly knows what a glockenspiel is. 117 00:08:22,610 --> 00:08:28,700 And this particular glockenspiel has three a's, 118 00:08:28,700 --> 00:08:30,360 separated each by an octave. 119 00:08:30,360 --> 00:08:35,500 There's a middle a, a high a, and a low a. 120 00:08:38,960 --> 00:08:46,240 And what I've done is to record the middle a on a tape 121 00:08:46,240 --> 00:08:48,950 at 7 and 1/2 inches per second. 122 00:08:48,950 --> 00:08:54,680 And what I'd like to demonstrate is that as we play 123 00:08:54,680 --> 00:08:59,790 that back, either at twice or half speed, the effective note 124 00:08:59,790 --> 00:09:03,650 gets moved down or up by an octave. 125 00:09:03,650 --> 00:09:07,340 So let me first play the note at the speed at 126 00:09:07,340 --> 00:09:08,760 which it was recorded. 127 00:09:08,760 --> 00:09:12,250 And so what we'll hear is the middle a as I've recorded it. 128 00:09:12,250 --> 00:09:15,200 And let me just start the tape player. 129 00:09:26,190 --> 00:09:30,190 Let me stop it, and hopefully what you heard is the same 130 00:09:30,190 --> 00:09:33,930 note from the tape recorder as the note that I played on the 131 00:09:33,930 --> 00:09:35,980 glockenspiel. 132 00:09:35,980 --> 00:09:40,740 Now I'll rewind the tape, and we'll go back to the beginning 133 00:09:40,740 --> 00:09:43,700 of that portion. 134 00:09:43,700 --> 00:09:49,930 And now, if I change the tape speed to half the speed, so 135 00:09:49,930 --> 00:09:53,050 from 7 and 1/2 inches per second, I'll change the tape 136 00:09:53,050 --> 00:09:57,980 speed to 3 and 3/4 inches per second. 137 00:09:57,980 --> 00:10:04,820 And now when I play the tape back, because of the inverse 138 00:10:04,820 --> 00:10:08,060 relationship between time and frequency scaling, we're now 139 00:10:08,060 --> 00:10:10,750 scaling in time by stretching out, we would expect the 140 00:10:10,750 --> 00:10:13,770 frequencies to be lowered by a factor of 2. 141 00:10:13,770 --> 00:10:19,170 We should now expect the taped note to be an octave lower, 142 00:10:19,170 --> 00:10:20,390 matching this lower a. 143 00:10:20,390 --> 00:10:22,010 So let's just play that. 144 00:10:39,300 --> 00:10:43,795 Let me stop it, and, again, we'll rewind the tape. 145 00:10:46,410 --> 00:10:50,020 Go back to the beginning, and now we'll play this 146 00:10:50,020 --> 00:10:51,770 at twice the speed. 147 00:10:51,770 --> 00:10:57,310 So I'll change from 3 and 3/4 to 15 inches per second. 148 00:10:57,310 --> 00:11:00,680 And now when I play it, we would expect that to match the 149 00:11:00,680 --> 00:11:01,180 upper note. 150 00:11:01,180 --> 00:11:02,430 And let's just do that. 151 00:11:15,100 --> 00:11:18,980 Although that's a result that, intuitively, probably makes 152 00:11:18,980 --> 00:11:22,720 considerable sense, in fact, what that is is an 153 00:11:22,720 --> 00:11:27,130 illustration of the inverse relationship between time 154 00:11:27,130 --> 00:11:29,250 scaling and frequency scaling. 155 00:11:29,250 --> 00:11:31,970 And also, by the way, it was finally my opportunity to play 156 00:11:31,970 --> 00:11:33,220 the glockenspiel on television. 157 00:11:36,560 --> 00:11:42,300 In addition, there is another very important relationship 158 00:11:42,300 --> 00:11:45,810 between the time and frequency domains, namely what is 159 00:11:45,810 --> 00:11:50,150 referred to as a duality relationship. 160 00:11:50,150 --> 00:11:56,660 And the duality relationship between time and frequency 161 00:11:56,660 --> 00:12:01,220 falls out, more or less directly, from the equations, 162 00:12:01,220 --> 00:12:03,790 the analysis and synthesis equations. 163 00:12:03,790 --> 00:12:08,470 In particular, if we look at the synthesis equation, which 164 00:12:08,470 --> 00:12:12,760 I repeat here, and the analysis equation, which I 165 00:12:12,760 --> 00:12:17,510 repeat below it, what we observe is that, in fact, 166 00:12:17,510 --> 00:12:23,420 these equations are basically identical, except for the fact 167 00:12:23,420 --> 00:12:26,870 that in the top integral we have things as a function of 168 00:12:26,870 --> 00:12:30,320 omega, in the bottom integral as a function of t, and 169 00:12:30,320 --> 00:12:32,960 there's a factor of 1 over 2 pi, and, by the 170 00:12:32,960 --> 00:12:34,590 way, a minus sign. 171 00:12:34,590 --> 00:12:39,500 You can look at the algebra more carefully at your 172 00:12:39,500 --> 00:12:43,920 leisure, but essentially what this says is that if x of 173 00:12:43,920 --> 00:12:51,420 omega is the Fourier transform of a time function x of t, 174 00:12:51,420 --> 00:12:58,120 then, in fact, x of t is very much like the Fourier 175 00:12:58,120 --> 00:13:00,180 transform of x of omega. 176 00:13:00,180 --> 00:13:04,060 In fact, it's the Fourier transform of x of minus omega 177 00:13:04,060 --> 00:13:06,060 to account for this minus sign. 178 00:13:06,060 --> 00:13:07,800 And, by the way, there's just an additional 179 00:13:07,800 --> 00:13:10,550 factor of 1 over 2 pi. 180 00:13:10,550 --> 00:13:14,070 So the duality relationship which follows from these two 181 00:13:14,070 --> 00:13:21,450 equations, in fact, says that if x of t and x of omega are a 182 00:13:21,450 --> 00:13:26,360 Fourier transform pair, if x and X are a Fourier transform 183 00:13:26,360 --> 00:13:32,150 pair, then X, in fact, has a Fourier transform which is 184 00:13:32,150 --> 00:13:35,910 proportional to x turned around. 185 00:13:35,910 --> 00:13:40,060 This duality in the continuous time Fourier 186 00:13:40,060 --> 00:13:42,560 transform is very important. 187 00:13:42,560 --> 00:13:44,110 It's very useful. 188 00:13:44,110 --> 00:13:48,590 It, by the way, is not a duality that surfaced in the 189 00:13:48,590 --> 00:13:52,620 Fourier series, because, as you recall, the Fourier series 190 00:13:52,620 --> 00:13:58,250 begins with a continuous time function and in the frequency 191 00:13:58,250 --> 00:14:02,470 domain generates a sequence, which would just naturally 192 00:14:02,470 --> 00:14:06,630 have problems associated with it if we attempted to 193 00:14:06,630 --> 00:14:08,580 interpret a duality. 194 00:14:08,580 --> 00:14:12,130 And we'll see, also, that in the discrete time case, one of 195 00:14:12,130 --> 00:14:14,560 the important differences between continuous time and 196 00:14:14,560 --> 00:14:18,250 discrete time Fourier transforms is the fact that in 197 00:14:18,250 --> 00:14:23,950 continuous time we have duality, in the discrete time 198 00:14:23,950 --> 00:14:27,330 Fourier transform we don't. 199 00:14:27,330 --> 00:14:32,400 Let's illustrate this with an example. 200 00:14:32,400 --> 00:14:37,000 Here are, in fact, two examples of Fourier transform 201 00:14:37,000 --> 00:14:40,460 pairs taken from examples in the text. 202 00:14:40,460 --> 00:14:46,690 The top one being example 4.11 from the text, and it's a time 203 00:14:46,690 --> 00:14:52,310 function which is a sine x over x type of function. 204 00:14:52,310 --> 00:14:59,710 And its Fourier transform corresponds to a rectangular 205 00:14:59,710 --> 00:15:03,830 shape in the frequency domain. 206 00:15:03,830 --> 00:15:07,320 There's also another example in the text, the example that 207 00:15:07,320 --> 00:15:11,590 precedes this one, which is example 4.10. 208 00:15:11,590 --> 00:15:17,460 And in example 4.10, we begin with a rectangle, and its 209 00:15:17,460 --> 00:15:21,340 Fourier transform is of the form of a 210 00:15:21,340 --> 00:15:24,140 sine x over x function. 211 00:15:24,140 --> 00:15:28,980 So in fact, if we look at these two examples together, 212 00:15:28,980 --> 00:15:33,260 what we see is the duality very evident. 213 00:15:33,260 --> 00:15:38,390 In other words, if we take this time function and instead 214 00:15:38,390 --> 00:15:42,260 think of a frequency function that has the same form, then 215 00:15:42,260 --> 00:15:45,360 we simply interchange the roles of time and frequency in 216 00:15:45,360 --> 00:15:46,630 the other domains. 217 00:15:46,630 --> 00:15:50,810 So the fact that these two correspond means that these 218 00:15:50,810 --> 00:15:52,330 two correspond. 219 00:15:52,330 --> 00:15:57,910 Of course in this particular example, because of the fact 220 00:15:57,910 --> 00:16:01,290 that we picked a symmetric function, an even function, in 221 00:16:01,290 --> 00:16:05,530 fact, the additional twist of the time axis being reversed 222 00:16:05,530 --> 00:16:10,830 didn't show up in duality with this example. 223 00:16:10,830 --> 00:16:13,930 One thing this says, of course, is that essentially 224 00:16:13,930 --> 00:16:19,390 any time you've calculated the Fourier transform of one time 225 00:16:19,390 --> 00:16:23,290 function, then you've actually calculated the Fourier 226 00:16:23,290 --> 00:16:25,990 transform of two time functions. 227 00:16:25,990 --> 00:16:28,970 Another one being the dual example to the one that you 228 00:16:28,970 --> 00:16:30,220 just calculated. 229 00:16:32,370 --> 00:16:37,805 Also somewhat related to duality is what is referred to 230 00:16:37,805 --> 00:16:41,650 as Parseval's relation for the continuous 231 00:16:41,650 --> 00:16:43,620 time Fourier transform. 232 00:16:43,620 --> 00:16:47,880 And essentially, what Parseval's relationship says, 233 00:16:47,880 --> 00:16:55,800 as a summary of it, says that the energy in a time function 234 00:16:55,800 --> 00:17:00,260 and the energy in its Fourier transform are proportional, 235 00:17:00,260 --> 00:17:04,430 the proportionality factor being a factor of 2 pi. 236 00:17:04,430 --> 00:17:06,770 That's summarized here. 237 00:17:06,770 --> 00:17:10,190 What's meant by the energy is, of course, the integral of the 238 00:17:10,190 --> 00:17:13,010 magnitude squared of x of t. 239 00:17:13,010 --> 00:17:16,819 And the statement of Parseval's relation is that 240 00:17:16,819 --> 00:17:21,670 that integral, the energy in x of t, is proportional to this 241 00:17:21,670 --> 00:17:25,770 integral, which is the energy in x of omega. 242 00:17:25,770 --> 00:17:31,660 Although we've incorporated the Fourier series within a 243 00:17:31,660 --> 00:17:36,460 framework of the Fourier transform, Parseval's relation 244 00:17:36,460 --> 00:17:40,830 needs to be modified slightly for Fourier series, because of 245 00:17:40,830 --> 00:17:43,840 the fact that a periodic signal has an infinite amount 246 00:17:43,840 --> 00:17:46,930 of energy in it, and, essentially, that form of 247 00:17:46,930 --> 00:17:50,380 Parseval's relationship for the periodic case would say 248 00:17:50,380 --> 00:17:53,560 infinity equals infinity, which isn't too useful. 249 00:17:53,560 --> 00:17:57,320 However, it can be modified so that Parseval's relationship 250 00:17:57,320 --> 00:18:02,760 to the periodic case says, essentially, that the energy 251 00:18:02,760 --> 00:18:09,740 in one period of the periodic time function is proportional 252 00:18:09,740 --> 00:18:12,160 with, this is the proportionality factor, 253 00:18:12,160 --> 00:18:15,870 proportional to the sum of the magnitude squared of the 254 00:18:15,870 --> 00:18:17,020 coefficients. 255 00:18:17,020 --> 00:18:21,300 In other words, the energy in one period is proportional to 256 00:18:21,300 --> 00:18:25,580 the energy in the sequence that represents the Fourier 257 00:18:25,580 --> 00:18:26,830 series coefficients. 258 00:18:29,660 --> 00:18:34,070 There are lots of other properties, and they're 259 00:18:34,070 --> 00:18:37,180 developed in the text and in the study guide. 260 00:18:37,180 --> 00:18:42,170 A number of properties that we want to make particular use of 261 00:18:42,170 --> 00:18:46,100 during this lecture, and in later lectures, are ones that 262 00:18:46,100 --> 00:18:48,080 I summarize here. 263 00:18:48,080 --> 00:18:54,200 And I won't demonstrate the proofs, but principally focus 264 00:18:54,200 --> 00:18:58,630 on some of the interpretation as the lecture goes on. 265 00:18:58,630 --> 00:19:01,760 The first property that I have listed here is what's referred 266 00:19:01,760 --> 00:19:04,600 to as the time shifting property. 267 00:19:04,600 --> 00:19:08,470 And the time shifting property says, if I have a time 268 00:19:08,470 --> 00:19:13,610 function with a Fourier transform x of omega, if I 269 00:19:13,610 --> 00:19:19,950 shift that time function in time, then that corresponds to 270 00:19:19,950 --> 00:19:24,440 multiplying the Fourier transform by this factor. 271 00:19:24,440 --> 00:19:29,310 As you examine this factor, what you can see is that this 272 00:19:29,310 --> 00:19:34,300 factor has magnitude unity and it has a phase, which is 273 00:19:34,300 --> 00:19:39,740 linear with frequency, and a slope of minus t0. 274 00:19:39,740 --> 00:19:43,730 So a statement to remember, that will come up many times 275 00:19:43,730 --> 00:19:49,010 throughout the course, is that a time shift, or a 276 00:19:49,010 --> 00:19:58,190 displacement in time, corresponds to a linear change 277 00:19:58,190 --> 00:20:01,110 in phase and frequency. 278 00:20:01,110 --> 00:20:06,020 Another property and, in fact, a pair of properties that 279 00:20:06,020 --> 00:20:09,840 we'll make reference to as we turn our attention toward the 280 00:20:09,840 --> 00:20:14,660 end of this lecture to solving differential equations using 281 00:20:14,660 --> 00:20:18,920 the Fourier transform, is what's referred to as the 282 00:20:18,920 --> 00:20:23,790 differentiation property and its companion, which is the 283 00:20:23,790 --> 00:20:26,600 integration property. 284 00:20:26,600 --> 00:20:30,700 The differentiation property says, again, if we have a time 285 00:20:30,700 --> 00:20:34,950 function with Fourier transform x of omega, the 286 00:20:34,950 --> 00:20:41,470 Fourier transform of the time derivative of that corresponds 287 00:20:41,470 --> 00:20:45,950 to multiplying the Fourier transform by a linear function 288 00:20:45,950 --> 00:20:46,940 of frequency. 289 00:20:46,940 --> 00:20:51,460 So here it's a linear amplitude change that 290 00:20:51,460 --> 00:20:55,470 corresponds to differentiation. 291 00:20:55,470 --> 00:20:59,500 At first glance, what you might think is that the 292 00:20:59,500 --> 00:21:02,230 integration property is just the reverse of that. 293 00:21:02,230 --> 00:21:05,030 If for the differentiation property you multiply by j 294 00:21:05,030 --> 00:21:09,150 omega, then for integration you must divide by j omega. 295 00:21:09,150 --> 00:21:12,460 And that's almost correct, except not quite. 296 00:21:12,460 --> 00:21:16,930 And the reason for the not quite is that recall that if 297 00:21:16,930 --> 00:21:20,400 you differentiate, what happens, of course, is that 298 00:21:20,400 --> 00:21:21,970 you lose a constant. 299 00:21:21,970 --> 00:21:25,450 And if we have a time function that's some finite energy 300 00:21:25,450 --> 00:21:29,520 signal plus a constant, differentiating will destroy 301 00:21:29,520 --> 00:21:32,250 the constant. 302 00:21:32,250 --> 00:21:34,000 The integration property, in essence, tries 303 00:21:34,000 --> 00:21:35,300 to bring that back. 304 00:21:35,300 --> 00:21:39,730 So the integration property, which is the inverse of the 305 00:21:39,730 --> 00:21:44,450 differentiation property, says that we divide the transform 306 00:21:44,450 --> 00:21:50,610 by j omega, and then if, in fact, there was a constant 307 00:21:50,610 --> 00:21:54,600 added to x of t, we have to account for that by inserting 308 00:21:54,600 --> 00:21:58,220 an impulse into the Fourier transform. 309 00:21:58,220 --> 00:22:03,370 And the final property that I want to draw your attention to 310 00:22:03,370 --> 00:22:08,090 on this view graph is the linearity property, which is 311 00:22:08,090 --> 00:22:11,320 very straightforward to demonstrate from the analysis 312 00:22:11,320 --> 00:22:17,010 and synthesis equations, which simply says if x1 of omega is 313 00:22:17,010 --> 00:22:21,120 the Fourier transform x1 of t, and x2 of omega is the Fourier 314 00:22:21,120 --> 00:22:25,080 transform of x2 of t, then the Fourier transform of a linear 315 00:22:25,080 --> 00:22:29,370 combination is a linear combination of the Fourier 316 00:22:29,370 --> 00:22:30,620 transforms. 317 00:22:32,700 --> 00:22:36,370 Let me emphasize, also, that these properties, for the most 318 00:22:36,370 --> 00:22:40,310 part, apply both to Fourier series and Fourier transforms 319 00:22:40,310 --> 00:22:44,530 because, in fact, what we've done is to incorporate the 320 00:22:44,530 --> 00:22:48,260 Fourier series within the framework 321 00:22:48,260 --> 00:22:49,510 of the Fourier transform. 322 00:22:51,900 --> 00:22:54,970 We'll be using a number of these properties shortly, when 323 00:22:54,970 --> 00:22:57,740 we turn our attention to linear constant coefficient 324 00:22:57,740 --> 00:22:59,350 differential equations. 325 00:22:59,350 --> 00:23:03,670 However, before we do that I'd like to focus on two 326 00:23:03,670 --> 00:23:10,010 additional major properties, and these are what I refer to 327 00:23:10,010 --> 00:23:13,780 as the convolution property and the modulation property. 328 00:23:13,780 --> 00:23:17,610 And in fact, the convolution property, as I'm about to 329 00:23:17,610 --> 00:23:22,090 introduce it, forms the mathematical and conceptual 330 00:23:22,090 --> 00:23:27,250 basis for the whole notion of filtering, which, in fact, 331 00:23:27,250 --> 00:23:32,810 will be a topic by itself in a set of lectures, and, in fact, 332 00:23:32,810 --> 00:23:36,700 is a chapter by itself in the textbook. 333 00:23:36,700 --> 00:23:40,220 Similarly, what I'll refer to as the modulation property, 334 00:23:40,220 --> 00:23:45,130 again, will occupy its own set of lectures as we go through 335 00:23:45,130 --> 00:23:49,060 the course, and, in fact, has its own 336 00:23:49,060 --> 00:23:52,160 chapter in the textbook. 337 00:23:52,160 --> 00:23:58,450 Let me just indicate what the convolution property is. 338 00:23:58,450 --> 00:24:08,870 And what the convolution property tells us is that the 339 00:24:08,870 --> 00:24:13,050 Fourier transform of the convolution of two time 340 00:24:13,050 --> 00:24:19,620 functions is the product of their Fourier transforms. 341 00:24:19,620 --> 00:24:25,340 So it says, for example, that if I have a linear time 342 00:24:25,340 --> 00:24:30,500 invariant system, and I have an input x of t, an impulse 343 00:24:30,500 --> 00:24:33,550 response h of t, and the output, of course, being the 344 00:24:33,550 --> 00:24:38,280 convolution, then, in fact, if I look at this in the 345 00:24:38,280 --> 00:24:45,200 frequency domain, the Fourier transform of the output is the 346 00:24:45,200 --> 00:24:49,920 Fourier transform of the input times the Fourier transform of 347 00:24:49,920 --> 00:24:53,190 the impulse response. 348 00:24:53,190 --> 00:24:58,340 You can demonstrate this property algebraically by 349 00:24:58,340 --> 00:25:00,990 essentially taking the convolution integral and 350 00:25:00,990 --> 00:25:04,270 applying the Fourier transform and doing the appropriate 351 00:25:04,270 --> 00:25:08,510 interchanging of the order of integration, et cetera. 352 00:25:08,510 --> 00:25:13,000 But what I'd like to draw your attention to is a somewhat 353 00:25:13,000 --> 00:25:16,560 more intuitive interpretation of the property. 354 00:25:16,560 --> 00:25:20,070 And the intuitive interpretation stems from the 355 00:25:20,070 --> 00:25:23,720 relationship between the Fourier transform of the 356 00:25:23,720 --> 00:25:26,890 impulse response and what we've referred to as the 357 00:25:26,890 --> 00:25:28,840 frequency response. 358 00:25:28,840 --> 00:25:34,610 Recall that one of the things that led us to use complex 359 00:25:34,610 --> 00:25:37,910 exponentials as building blocks was the fact that 360 00:25:37,910 --> 00:25:39,610 they're eigenfunctions of linear 361 00:25:39,610 --> 00:25:41,000 time and variant systems. 362 00:25:41,000 --> 00:25:45,640 In other words, if we have a linear time invariant system, 363 00:25:45,640 --> 00:25:48,990 and I have an input which is a complex exponential, the 364 00:25:48,990 --> 00:25:51,770 output is a complex exponential of the same 365 00:25:51,770 --> 00:25:55,790 frequency multiplied by what we call 366 00:25:55,790 --> 00:25:57,510 the frequency response. 367 00:25:57,510 --> 00:26:02,410 And, in fact, the expression for the frequency response is 368 00:26:02,410 --> 00:26:05,970 identical to the expression for the Fourier transform of 369 00:26:05,970 --> 00:26:07,710 the impulse response. 370 00:26:07,710 --> 00:26:11,670 In other words, the frequency response is the Fourier 371 00:26:11,670 --> 00:26:15,750 transform of the impulse response. 372 00:26:15,750 --> 00:26:20,350 Now in that context, how can we interpret 373 00:26:20,350 --> 00:26:22,260 the convolution property? 374 00:26:22,260 --> 00:26:26,630 Well, remember what I said at the beginning of the lecture, 375 00:26:26,630 --> 00:26:30,850 when I pointed to the synthesis equation and I said, 376 00:26:30,850 --> 00:26:35,010 in essence, the synthesis equation tells us how to 377 00:26:35,010 --> 00:26:38,490 decompose x of t as a linear combination of complex 378 00:26:38,490 --> 00:26:40,740 exponentials. 379 00:26:40,740 --> 00:26:43,460 What are the complex amplitudes of those complex 380 00:26:43,460 --> 00:26:45,020 exponentials? 381 00:26:45,020 --> 00:26:50,250 In terms of our notation here, the complex amplitude of those 382 00:26:50,250 --> 00:26:55,590 complex exponentials is x of omega, or proportional to x of 383 00:26:55,590 --> 00:26:58,650 omega, in particular it's x of omega, d omega, and then a 384 00:26:58,650 --> 00:27:01,160 factor of 2 pi. 385 00:27:01,160 --> 00:27:06,480 As this signal goes through this linear time invariant 386 00:27:06,480 --> 00:27:12,100 system, what happens to each of those exponential is each 387 00:27:12,100 --> 00:27:15,610 one gets multiplied by the frequency response at the 388 00:27:15,610 --> 00:27:18,090 associated frequency. 389 00:27:18,090 --> 00:27:22,770 What comes out is the amplitude of the complex 390 00:27:22,770 --> 00:27:28,610 exponentials that are used to build the output. 391 00:27:28,610 --> 00:27:35,990 So in fact, the convolution property simply is telling us 392 00:27:35,990 --> 00:27:39,730 that, in terms of the decomposition of the signal, 393 00:27:39,730 --> 00:27:42,940 in terms of complex exponentials, as we push that 394 00:27:42,940 --> 00:27:46,230 signal through a linear time invariant system, we're 395 00:27:46,230 --> 00:27:52,780 separately multiplying by the frequency response, the 396 00:27:52,780 --> 00:27:56,590 amplitudes of the exponential components used 397 00:27:56,590 --> 00:27:58,100 to build the input. 398 00:27:58,100 --> 00:28:03,860 And that sum, in turn, is the decomposition of the output in 399 00:28:03,860 --> 00:28:05,270 terms of complex exponentials. 400 00:28:08,630 --> 00:28:12,310 I understand that going through that involves a little 401 00:28:12,310 --> 00:28:16,640 bit of sorting out, and I strongly encourage you to try 402 00:28:16,640 --> 00:28:20,940 to understand and interpret the convolution property in 403 00:28:20,940 --> 00:28:25,400 those conceptual terms, rather than simply by applying the 404 00:28:25,400 --> 00:28:29,050 mathematics to the convolution integral and seeing the terms 405 00:28:29,050 --> 00:28:30,300 match up on both sides. 406 00:28:33,930 --> 00:28:38,190 As I indicated, the convolution property forms the 407 00:28:38,190 --> 00:28:41,580 basis for what's referred to as filtering. 408 00:28:41,580 --> 00:28:46,700 And this is a topic that we'll be treating in a considerable 409 00:28:46,700 --> 00:28:51,330 amount of detail after we've also gone through a discussion 410 00:28:51,330 --> 00:28:54,400 of the discrete time Fourier transform in the 411 00:28:54,400 --> 00:28:56,050 next several lectures. 412 00:28:56,050 --> 00:28:59,600 However, what I'd like to do is just indicate, now, a 413 00:28:59,600 --> 00:29:04,560 little bit of the conceptual ideas involved. 414 00:29:04,560 --> 00:29:08,990 Essentially, conceptually, what filtering, as it's 415 00:29:08,990 --> 00:29:15,250 typically referred to, corresponds to is modifying 416 00:29:15,250 --> 00:29:18,220 separately the individual frequency 417 00:29:18,220 --> 00:29:20,030 components in a signal. 418 00:29:20,030 --> 00:29:25,360 The convolution property told us that if we look at the 419 00:29:25,360 --> 00:29:28,330 individual frequency components, they get 420 00:29:28,330 --> 00:29:30,970 multiplied by the frequency response, and so what that 421 00:29:30,970 --> 00:29:34,710 says is that we can amplify or attenuate any of those 422 00:29:34,710 --> 00:29:39,570 components separately using a linear time invariant system. 423 00:29:39,570 --> 00:29:43,050 For example, what I've illustrated here is the 424 00:29:43,050 --> 00:29:47,320 frequency response of what is commonly referred to as an 425 00:29:47,320 --> 00:29:50,220 ideal low pass filter. 426 00:29:50,220 --> 00:29:55,990 What an ideal low pass filter does is to pass exactly 427 00:29:55,990 --> 00:30:01,200 frequencies in one frequency range and eliminate totally 428 00:30:01,200 --> 00:30:04,350 frequencies outside that range. 429 00:30:04,350 --> 00:30:08,890 Another filter which is not so ideal might, for example, 430 00:30:08,890 --> 00:30:15,980 attenuate components in this band but not 431 00:30:15,980 --> 00:30:17,230 totally eliminate them. 432 00:30:19,650 --> 00:30:22,190 In terms of filtering, we can think back to the 433 00:30:22,190 --> 00:30:27,390 differentiation property and, in fact, interpret 434 00:30:27,390 --> 00:30:29,680 differentiator as a filter. 435 00:30:29,680 --> 00:30:32,710 Recall that the differentiation property said 436 00:30:32,710 --> 00:30:39,330 that the Fourier transform of the differentiated signal is 437 00:30:39,330 --> 00:30:41,180 the Fourier transform of the original signal 438 00:30:41,180 --> 00:30:43,600 multiplied by j omega. 439 00:30:43,600 --> 00:30:47,060 So what that says, then, is that if we have a 440 00:30:47,060 --> 00:30:52,490 differentiator, the frequency response of that is j omega. 441 00:30:52,490 --> 00:30:55,640 In other words, the Fourier transform of the output is j 442 00:30:55,640 --> 00:31:00,120 omega times the Fourier transform of the input. 443 00:31:00,120 --> 00:31:04,640 And so the frequency response of the differentiator looks 444 00:31:04,640 --> 00:31:07,140 like this, in terms of its magnitude. 445 00:31:07,140 --> 00:31:10,700 And what it does, of course, is it amplifies high 446 00:31:10,700 --> 00:31:14,950 frequencies and attenuates low frequencies. 447 00:31:17,670 --> 00:31:23,520 Let me just, to cement some of these ideas, illustrate them 448 00:31:23,520 --> 00:31:29,140 in the context of one kind of signal, namely a signal which, 449 00:31:29,140 --> 00:31:34,170 in fact, is a spatial signal rather than a time signal. 450 00:31:34,170 --> 00:31:39,270 And this also gives me an opportunity to introduce you 451 00:31:39,270 --> 00:31:43,080 to our colleague, J. B. J. Fourier. 452 00:31:43,080 --> 00:31:47,790 So if we could look at our colleague, Mr. Fourier, who, 453 00:31:47,790 --> 00:31:53,500 by the way, is not only a person who had tremendously 454 00:31:53,500 --> 00:31:59,140 brilliant insights, and his insights, in fact, have led to 455 00:31:59,140 --> 00:32:02,670 forming the foundation of the developments that is the basis 456 00:32:02,670 --> 00:32:04,040 for this course. 457 00:32:04,040 --> 00:32:05,730 He was also a very 458 00:32:05,730 --> 00:32:07,560 interesting, fascinating person. 459 00:32:07,560 --> 00:32:11,830 And there's a certain amount of historical discussion about 460 00:32:11,830 --> 00:32:15,300 Fourier and his background, which you might enjoy reading 461 00:32:15,300 --> 00:32:17,120 in the text. 462 00:32:17,120 --> 00:32:18,740 In any case, what you're looking at, of 463 00:32:18,740 --> 00:32:20,500 course, is a signal. 464 00:32:20,500 --> 00:32:25,160 And the signal is a spatial signal, and it has high 465 00:32:25,160 --> 00:32:26,880 frequencies and low frequencies. 466 00:32:26,880 --> 00:32:31,030 High frequencies corresponding to things that are varying 467 00:32:31,030 --> 00:32:34,310 rapidly spatially, and low frequencies 468 00:32:34,310 --> 00:32:37,810 varying slowly spatially. 469 00:32:37,810 --> 00:32:43,250 And so, for example, we could low pass filter this picture 470 00:32:43,250 --> 00:32:48,090 simply by asking the video crew if they could be slightly 471 00:32:48,090 --> 00:32:50,270 defocus it. 472 00:32:50,270 --> 00:32:53,540 And what you see as the picture is defocused, if you 473 00:32:53,540 --> 00:32:58,560 could hold it there, is that we've lost edges, which is the 474 00:32:58,560 --> 00:33:00,230 rapid variation. 475 00:33:00,230 --> 00:33:05,020 And what we've retained is the broader, slow variation. 476 00:33:05,020 --> 00:33:09,170 And now let's take out the defocusing low pass filter and 477 00:33:09,170 --> 00:33:11,550 go back to a focused image. 478 00:33:19,480 --> 00:33:23,260 What we can also consider is what would happen if we looked 479 00:33:23,260 --> 00:33:25,090 at the differentiated image. 480 00:33:25,090 --> 00:33:28,210 And there are several ways we can think about this. 481 00:33:28,210 --> 00:33:30,580 One is that a differentiator-- 482 00:33:30,580 --> 00:33:35,400 Of course the output of a differentiator is larger where 483 00:33:35,400 --> 00:33:40,460 the discontinuity, or where the variation, is faster. 484 00:33:40,460 --> 00:33:44,890 And so we would expect the edges to be enhanced if, in 485 00:33:44,890 --> 00:33:47,490 fact, we differentiated the image. 486 00:33:47,490 --> 00:33:51,200 Or if we interpret differentiation in the context 487 00:33:51,200 --> 00:33:59,190 of our filter, then what we're saying is that, in effect, 488 00:33:59,190 --> 00:34:04,730 what's happening is that the differentiator is accentuating 489 00:34:04,730 --> 00:34:08,130 the high frequencies because of the frequency shape of the 490 00:34:08,130 --> 00:34:09,790 differentiator. 491 00:34:09,790 --> 00:34:13,940 Recall that this all fits together as a nice package. 492 00:34:13,940 --> 00:34:16,120 We expect intuitively that differentiation 493 00:34:16,120 --> 00:34:18,020 will enhance edges. 494 00:34:18,020 --> 00:34:21,250 When we talked about square waves and we saw how the 495 00:34:21,250 --> 00:34:25,380 Fourier series built up a square wave, we saw that it 496 00:34:25,380 --> 00:34:29,420 was the high frequencies that were required in order to 497 00:34:29,420 --> 00:34:32,610 build up the sharp edges. 498 00:34:32,610 --> 00:34:36,250 And so either viewed as a filter, or viewed intuitively, 499 00:34:36,250 --> 00:34:40,070 we would expect that the differentiated image would, in 500 00:34:40,070 --> 00:34:46,560 fact, attenuate this slowly varying background and amplify 501 00:34:46,560 --> 00:34:48,320 the rapidly varying edges. 502 00:34:48,320 --> 00:34:52,170 So let's look again at our original image, just to remind 503 00:34:52,170 --> 00:34:55,829 you of the fact that there are edges, of course, and there is 504 00:34:55,829 --> 00:34:59,480 a more slowly varying background. 505 00:34:59,480 --> 00:35:05,860 And now let's look at the result of passing that through 506 00:35:05,860 --> 00:35:07,730 a differentiator. 507 00:35:07,730 --> 00:35:11,790 And, as I think is very evident in the resulting 508 00:35:11,790 --> 00:35:17,610 image, clearly it's the edges that are retained and the 509 00:35:17,610 --> 00:35:21,590 slower background variations are destroyed, which is 510 00:35:21,590 --> 00:35:23,220 consistent with everything that we've said. 511 00:35:27,270 --> 00:35:31,790 I've emphasized that we'll be returning to a much broader 512 00:35:31,790 --> 00:35:35,540 discussion of filtering at a later point in the course. 513 00:35:35,540 --> 00:35:40,690 I'd now like to comment on another property, which is 514 00:35:40,690 --> 00:35:44,110 also, as I indicated, a topic in its own right, and which 515 00:35:44,110 --> 00:35:48,740 really is the dual property to the convolution property, and, 516 00:35:48,740 --> 00:35:53,360 in fact, could be argued directly from duality. 517 00:35:53,360 --> 00:35:59,230 And that is what's referred to as the modulation property. 518 00:35:59,230 --> 00:36:04,210 The convolution property told us that if we convolve in the 519 00:36:04,210 --> 00:36:08,940 time domain, we multiply in the frequency domain. 520 00:36:08,940 --> 00:36:10,920 And we know that time and frequency domains are 521 00:36:10,920 --> 00:36:13,920 interchangeable because of duality, so what that would 522 00:36:13,920 --> 00:36:19,860 suggest is that if we multiply in the time domain, that would 523 00:36:19,860 --> 00:36:23,140 correspond to convolution in the frequency domain. 524 00:36:23,140 --> 00:36:25,430 And, in fact, that is exactly what the 525 00:36:25,430 --> 00:36:27,610 modulation property is. 526 00:36:27,610 --> 00:36:31,940 I have it summarized here that if we multiply a time function 527 00:36:31,940 --> 00:36:34,340 by another time function, then in the 528 00:36:34,340 --> 00:36:37,880 frequency domain we convolve. 529 00:36:37,880 --> 00:36:40,850 Whereas the convolution property is just the dual of 530 00:36:40,850 --> 00:36:45,870 that, namely convolving in the time domain corresponds to 531 00:36:45,870 --> 00:36:49,040 multiplication in the frequency domain. 532 00:36:49,040 --> 00:36:52,170 The convolution property is the basis, as I 533 00:36:52,170 --> 00:36:54,390 indicated, for filtering. 534 00:36:54,390 --> 00:36:58,620 The modulation property, as I've summarized it here, in 535 00:36:58,620 --> 00:37:06,000 fact, is the entire basis for amplitude modulation systems 536 00:37:06,000 --> 00:37:10,080 as used almost universally in communications. 537 00:37:10,080 --> 00:37:13,110 And what the modulation property, as we'll see when we 538 00:37:13,110 --> 00:37:18,300 explore it in more detail, tells us is that if we have a 539 00:37:18,300 --> 00:37:22,560 signal with a certain spectrum, and we multiply by a 540 00:37:22,560 --> 00:37:27,020 sinusoidal signal whose Fourier transform is a set of 541 00:37:27,020 --> 00:37:31,500 impulses, then in a frequency domain we convolve. 542 00:37:31,500 --> 00:37:35,570 And that corresponds to taking the original spectrum and 543 00:37:35,570 --> 00:37:41,160 translating it, shifting it in frequency up to the frequency 544 00:37:41,160 --> 00:37:45,060 of the carrier, namely the sinusoidal signal. 545 00:37:45,060 --> 00:37:49,330 And as I said, we'll come to that in much more detail in a 546 00:37:49,330 --> 00:37:50,580 number of lectures. 547 00:37:52,960 --> 00:37:56,780 We've seen a number of properties, and I indicated 548 00:37:56,780 --> 00:38:00,100 sometime earlier when we talked about differential 549 00:38:00,100 --> 00:38:03,350 equations, that, in fact, it's the properties of the Fourier 550 00:38:03,350 --> 00:38:08,730 transform that provide us with a very useful and important 551 00:38:08,730 --> 00:38:13,420 mechanism for solving linear constant coefficient 552 00:38:13,420 --> 00:38:15,350 differential equations. 553 00:38:15,350 --> 00:38:21,340 And what I'd like to do now is illustrate the procedure, the 554 00:38:21,340 --> 00:38:24,780 basis for that, and I think what we'll do is illustrate it 555 00:38:24,780 --> 00:38:26,985 simply in the context of several examples. 556 00:38:31,120 --> 00:38:39,380 What I've indicated is a system with an impulse 557 00:38:39,380 --> 00:38:43,550 response h of t, or frequency response h of omega. 558 00:38:43,550 --> 00:38:46,630 And, of course, we know that in the time domain it's 559 00:38:46,630 --> 00:38:50,880 described through convolution, in the frequency domain it's 560 00:38:50,880 --> 00:38:54,020 described through multiplication. 561 00:38:54,020 --> 00:38:59,040 And I, in essence, am assuming that we're talking about a 562 00:38:59,040 --> 00:39:01,400 linear time invariant system. 563 00:39:01,400 --> 00:39:05,560 And we're also going to assume then it's characterized by a 564 00:39:05,560 --> 00:39:09,540 linear constant coefficient differential equation, where 565 00:39:09,540 --> 00:39:13,730 we're going to impose the condition that it's causal 566 00:39:13,730 --> 00:39:17,760 linear and time invariant, or equivalently that the initial 567 00:39:17,760 --> 00:39:22,320 conditions are consistent with the initial rest. 568 00:39:22,320 --> 00:39:26,160 And it's because of the fact that we're assuming that it's 569 00:39:26,160 --> 00:39:30,380 a linear time invariant system that we can describe it in the 570 00:39:30,380 --> 00:39:34,560 frequency domain through the convolution property, and we 571 00:39:34,560 --> 00:39:38,940 can use the properties of the Fourier transform. 572 00:39:38,940 --> 00:39:43,190 So let's take, as our example, a first order differential 573 00:39:43,190 --> 00:39:46,670 equation as I indicate here. 574 00:39:46,670 --> 00:39:49,560 So the derivative of the output plus a times the output 575 00:39:49,560 --> 00:39:52,330 is equal to the input. 576 00:39:52,330 --> 00:39:57,080 And now we can use the differentiation property. 577 00:39:57,080 --> 00:40:01,250 If we Fourier transform this entire expression, the 578 00:40:01,250 --> 00:40:06,690 differentiation property tells us that the Fourier transform 579 00:40:06,690 --> 00:40:10,290 of the derivative of the output is the Fourier 580 00:40:10,290 --> 00:40:14,560 transform of the output multiplied by j omega. 581 00:40:14,560 --> 00:40:20,330 And linearity will let us write the Fourier transform of 582 00:40:20,330 --> 00:40:22,810 this as a times y of omega. 583 00:40:22,810 --> 00:40:25,690 And since these are added together, and since we have 584 00:40:25,690 --> 00:40:29,330 the linearity property, these are added together. 585 00:40:29,330 --> 00:40:34,600 And x of omega is the Fourier transform of x of t. 586 00:40:34,600 --> 00:40:40,630 So what we've used is the differentiation property, and 587 00:40:40,630 --> 00:40:43,185 we've used the linearity property. 588 00:40:48,290 --> 00:40:52,980 We can solve this equation for y of omega. 589 00:40:52,980 --> 00:40:57,420 The Fourier transform of the output in terms of x of omega, 590 00:40:57,420 --> 00:41:01,970 the Fourier transform of the input, and a simple algebraic 591 00:41:01,970 --> 00:41:05,430 step gets us to this expression. 592 00:41:05,430 --> 00:41:08,550 So the Fourier transform of the output is 1 over j omega 593 00:41:08,550 --> 00:41:12,110 plus a times the Fourier transform of the input. 594 00:41:12,110 --> 00:41:16,220 And I've just simply repeated that equation up here. 595 00:41:19,980 --> 00:41:25,470 So far this is algebra, and the question is, now, how do 596 00:41:25,470 --> 00:41:27,540 we interpret this? 597 00:41:27,540 --> 00:41:32,880 Well, we know that the Fourier transform of the output is the 598 00:41:32,880 --> 00:41:38,060 Fourier transform of the input times the Fourier transform of 599 00:41:38,060 --> 00:41:40,120 the impulse response of the system, namely 600 00:41:40,120 --> 00:41:41,830 the frequency response. 601 00:41:41,830 --> 00:41:47,170 So, in fact, if we think of h of t and h of omega as a 602 00:41:47,170 --> 00:41:51,980 Fourier transform pair, it's the convolution property that 603 00:41:51,980 --> 00:41:58,295 lets us equate this term with h of omega. 604 00:41:58,295 --> 00:42:02,230 So here we're using the convolution property. 605 00:42:06,170 --> 00:42:10,750 So we know what the Fourier transform of the impulse 606 00:42:10,750 --> 00:42:14,030 response is, namely 1 over j omega plus a. 607 00:42:17,270 --> 00:42:20,870 We may have, for example, wanted in our problem, instead 608 00:42:20,870 --> 00:42:23,420 of getting the frequency response, to 609 00:42:23,420 --> 00:42:25,790 get the impulse response. 610 00:42:25,790 --> 00:42:28,690 And there are a variety of ways that we can do this. 611 00:42:28,690 --> 00:42:32,400 We can attempt to go through the inverse Fourier transform 612 00:42:32,400 --> 00:42:33,570 expression. 613 00:42:33,570 --> 00:42:38,780 But in fact, one of the most useful ways is formally called 614 00:42:38,780 --> 00:42:40,490 the inspection method. 615 00:42:40,490 --> 00:42:46,190 Informally it's called, if you worked it out going one way, 616 00:42:46,190 --> 00:42:48,630 then you ought to remember the answer so that you know how to 617 00:42:48,630 --> 00:42:51,000 get back method. 618 00:42:51,000 --> 00:42:56,140 So what that says is, remember that we worked an example, and 619 00:42:56,140 --> 00:42:58,520 in fact I showed you the example earlier in the 620 00:42:58,520 --> 00:43:03,540 lecture, that the Fourier transform of e to the minus at 621 00:43:03,540 --> 00:43:07,460 times the step is 1 over j omega plus a? 622 00:43:07,460 --> 00:43:11,550 So what is the inverse Fourier transfer of 1 over 623 00:43:11,550 --> 00:43:13,130 j omega plus a? 624 00:43:13,130 --> 00:43:19,890 Well, it's e to the minus at times a unit step. 625 00:43:19,890 --> 00:43:22,840 And that's just simply remembering, in essence, this 626 00:43:22,840 --> 00:43:24,090 particular transform pair. 627 00:43:26,940 --> 00:43:32,070 I've drawn, graphically, the magnitude of the Fourier 628 00:43:32,070 --> 00:43:35,890 transform here. 629 00:43:35,890 --> 00:43:41,100 And below it we have the impulse response. 630 00:43:41,100 --> 00:43:45,610 The impulse response is, as I just indicated, e to the minus 631 00:43:45,610 --> 00:43:48,450 at times u of t. 632 00:43:48,450 --> 00:43:53,400 Now let's go back up and look at the magnitude of the 633 00:43:53,400 --> 00:43:56,240 frequency response. 634 00:43:56,240 --> 00:44:00,030 And given just the little bit of discussion that we had 635 00:44:00,030 --> 00:44:05,680 previously about filtering, you should be able to infer 636 00:44:05,680 --> 00:44:09,940 something about the filtering characteristics of this 637 00:44:09,940 --> 00:44:13,680 simple, first order differential equation. 638 00:44:13,680 --> 00:44:17,740 In particular, if you look at that frequency response, the 639 00:44:17,740 --> 00:44:21,550 frequency response falls off with frequency and so what it 640 00:44:21,550 --> 00:44:27,080 tends to do is attenuate high frequencies and retain low 641 00:44:27,080 --> 00:44:28,160 frequencies. 642 00:44:28,160 --> 00:44:32,950 So in fact, you could think of the defocusing that we did on 643 00:44:32,950 --> 00:44:35,640 the image of Fourier, you could think of that, 644 00:44:35,640 --> 00:44:39,390 approximately, as similar to the kind of filtering action 645 00:44:39,390 --> 00:44:43,630 that you would get by passing a signal through a first order 646 00:44:43,630 --> 00:44:44,880 differential equation. 647 00:44:47,620 --> 00:44:52,640 Just to illustrate one additional step in both 648 00:44:52,640 --> 00:44:56,090 evaluating inverse transforms and using Fourier transform 649 00:44:56,090 --> 00:44:59,690 properties to solve linear constant coefficient 650 00:44:59,690 --> 00:45:04,250 differential equations, let's take the same example and, 651 00:45:04,250 --> 00:45:07,830 rather than finding the impulse response, let's find 652 00:45:07,830 --> 00:45:11,910 the response to another exponential input. 653 00:45:11,910 --> 00:45:13,830 We could, of course, do that using 654 00:45:13,830 --> 00:45:14,950 the convolution integral. 655 00:45:14,950 --> 00:45:17,090 We've just gotten the impulse response, and we could put 656 00:45:17,090 --> 00:45:20,710 that through the convolution integral to get the response 657 00:45:20,710 --> 00:45:21,930 to this input. 658 00:45:21,930 --> 00:45:25,160 But let's do it, instead, by going back to the 659 00:45:25,160 --> 00:45:27,890 differential equation. 660 00:45:27,890 --> 00:45:32,330 And so here I'm taking a differential equation. 661 00:45:32,330 --> 00:45:35,920 I'll choose, just to have some numbers to work 662 00:45:35,920 --> 00:45:38,400 with, a equal to 2. 663 00:45:38,400 --> 00:45:41,690 And now I'll choose an exponential on the right hand 664 00:45:41,690 --> 00:45:46,030 side, e to the minus t times u of t. 665 00:45:46,030 --> 00:45:51,700 And again, we Fourier transform the equation. 666 00:45:51,700 --> 00:45:55,510 And we can remember this particular 667 00:45:55,510 --> 00:45:57,490 Fourier transform pair. 668 00:45:57,490 --> 00:46:00,690 It's just the one we worked out previously, now with a 669 00:46:00,690 --> 00:46:01,650 equal to 1. 670 00:46:01,650 --> 00:46:05,430 Clearly we're getting a lot of mileage out of that example. 671 00:46:05,430 --> 00:46:09,660 And now, if we want to determine what the output y of 672 00:46:09,660 --> 00:46:14,920 t is, we can do that by solving for y of omega and 673 00:46:14,920 --> 00:46:18,520 then generating the inverse Fourier transform. 674 00:46:18,520 --> 00:46:22,850 Let's solve this algebraically for y of omega, and that gets 675 00:46:22,850 --> 00:46:25,460 us to this expression. 676 00:46:25,460 --> 00:46:29,420 And this is not a Fourier transform that we've worked 677 00:46:29,420 --> 00:46:30,830 out before. 678 00:46:30,830 --> 00:46:36,190 And this is the second part to the inspection procedure. 679 00:46:36,190 --> 00:46:39,430 What we have is a Fourier transform which is a product 680 00:46:39,430 --> 00:46:42,940 of two terms, each of which we can recognize. 681 00:46:42,940 --> 00:46:48,660 And what we can consider doing is expanding that out in a 682 00:46:48,660 --> 00:46:52,160 partial fraction expansion, namely as a sum of terms. 683 00:46:52,160 --> 00:46:54,650 Because of the linearity property associated with the 684 00:46:54,650 --> 00:46:59,040 Fourier transform, the inverse transform is then the sum of 685 00:46:59,040 --> 00:47:01,030 the inverse transform of each of those terms. 686 00:47:03,750 --> 00:47:08,020 So if we expand this out in a partial fraction expansion, 687 00:47:08,020 --> 00:47:12,990 and you can just verify that if you add these two together 688 00:47:12,990 --> 00:47:15,440 you'll get back to where we started. 689 00:47:15,440 --> 00:47:22,670 We now have the sum of two terms, and if we now 690 00:47:22,670 --> 00:47:26,740 recognize, by inspection, the inverse Fourier transform of 691 00:47:26,740 --> 00:47:32,520 this, we see that it's simply minus e to the minus 2t times 692 00:47:32,520 --> 00:47:33,720 the unit step. 693 00:47:33,720 --> 00:47:38,320 This one is plus e to the minus t times the unit step. 694 00:47:38,320 --> 00:47:43,280 And so, in fact, it's the sum of these two terms which are 695 00:47:43,280 --> 00:47:47,970 the inverse transforms of the individual terms in the 696 00:47:47,970 --> 00:48:02,820 partial fraction expansion that then give us the output. 697 00:48:02,820 --> 00:48:08,170 So this is y of t, which is the sum of these two 698 00:48:08,170 --> 00:48:14,350 exponentials, and this is the inverse Fourier transform of y 699 00:48:14,350 --> 00:48:16,690 of omega as we calculated it previously. 700 00:48:21,110 --> 00:48:24,940 Hopefully you're beginning to get some sense, now, of how 701 00:48:24,940 --> 00:48:29,190 powerful and also beautiful the Fourier transform is. 702 00:48:29,190 --> 00:48:33,980 We've seen already a glimpse of how it plays a role in 703 00:48:33,980 --> 00:48:38,050 filtering, modulation, how its properties help us with linear 704 00:48:38,050 --> 00:48:39,510 constant coefficient differential 705 00:48:39,510 --> 00:48:42,456 equations, et cetera. 706 00:48:42,456 --> 00:48:48,520 What we will do, beginning with the next lecture, is 707 00:48:48,520 --> 00:48:52,310 develop a similar set of tools for the discrete time case. 708 00:48:52,310 --> 00:48:56,650 And there are some very strong similarities to what we've 709 00:48:56,650 --> 00:48:58,850 done in continuous time, also some very important 710 00:48:58,850 --> 00:49:00,190 differences. 711 00:49:00,190 --> 00:49:04,150 And then, after we have the continuous time and discrete 712 00:49:04,150 --> 00:49:10,270 time Fourier transforms, we'll then see how the concepts 713 00:49:10,270 --> 00:49:14,030 involved and the properties involved lead to very 714 00:49:14,030 --> 00:49:18,690 important and powerful notions of filtering, modulation, 715 00:49:18,690 --> 00:49:22,250 sampling, and other signal processing ideas. 716 00:49:22,250 --> 00:49:23,500 Thank you.