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 00:00:56,260 --> 00:00:58,910 PROFESSOR: Last time, we began to address the issue of 9 00:00:58,910 --> 00:01:02,610 building continuous time signals out of a linear 10 00:01:02,610 --> 00:01:05,560 combination of complex exponentials. 11 00:01:05,560 --> 00:01:10,110 And for the class of periodic signals specifically, what 12 00:01:10,110 --> 00:01:13,350 this led to was the Fourier series representation for 13 00:01:13,350 --> 00:01:15,020 periodic signals. 14 00:01:15,020 --> 00:01:17,270 Let me just summarize the results that we 15 00:01:17,270 --> 00:01:19,110 developed last time. 16 00:01:19,110 --> 00:01:23,880 For periodic signals, we had the continuous-time Fourier 17 00:01:23,880 --> 00:01:29,220 series, where we built the periodic signal out of a 18 00:01:29,220 --> 00:01:33,250 linear combination of harmonically related complex 19 00:01:33,250 --> 00:01:34,750 exponentials. 20 00:01:34,750 --> 00:01:39,030 And what that led to was what we referred to as the 21 00:01:39,030 --> 00:01:41,050 synthesis equation. 22 00:01:41,050 --> 00:01:45,900 And we briefly addressed the issue of when this in fact 23 00:01:45,900 --> 00:01:50,360 builds, when this in fact is a complete representation of the 24 00:01:50,360 --> 00:01:55,970 periodic signal, and in essence, what we presented was 25 00:01:55,970 --> 00:02:00,540 conditions either for x of t being square integrable or x 26 00:02:00,540 --> 00:02:03,090 of t being absolutely integrable. 27 00:02:03,090 --> 00:02:06,880 Then, the other side of the Fourier series is what I 28 00:02:06,880 --> 00:02:10,380 referred to as the analysis equation. 29 00:02:10,380 --> 00:02:14,120 And the analysis equation was the equation that told us how 30 00:02:14,120 --> 00:02:19,110 we get the Fourier series coefficients from x of t. 31 00:02:19,110 --> 00:02:23,430 And so this equation together with the synthesis equation 32 00:02:23,430 --> 00:02:26,600 represent the Fourier series description 33 00:02:26,600 --> 00:02:29,470 for periodic signals. 34 00:02:29,470 --> 00:02:34,580 Now what we'd like to do is extend this idea to provide a 35 00:02:34,580 --> 00:02:39,460 mechanism for building non-periodic signals also out 36 00:02:39,460 --> 00:02:42,500 of a linear combination of complex exponentials. 37 00:02:42,500 --> 00:02:47,310 And the basic idea behind doing this is very simple and 38 00:02:47,310 --> 00:02:50,680 also very clever as I indicated last time. 39 00:02:50,680 --> 00:02:54,850 Essentially, the thought is the following, if we have a 40 00:02:54,850 --> 00:02:59,220 non-periodic signal or aperiodic signal, we can think 41 00:02:59,220 --> 00:03:04,720 of constructing a periodic signal by simply periodically 42 00:03:04,720 --> 00:03:08,230 replicating that aperiodic signal. 43 00:03:08,230 --> 00:03:12,520 So for example, if I have an aperiodic signal as I've 44 00:03:12,520 --> 00:03:18,870 indicated here, I can consider building a periodic signal, 45 00:03:18,870 --> 00:03:23,700 where I simply take this original signal and repeat it 46 00:03:23,700 --> 00:03:27,910 at multiples of some period t 0. 47 00:03:27,910 --> 00:03:30,380 Now, two things to recognize about this. 48 00:03:30,380 --> 00:03:36,155 One is that the periodic signal is equal to the 49 00:03:36,155 --> 00:03:39,150 aperiodic signal over one period. 50 00:03:39,150 --> 00:03:45,020 And the second is that as the period goes to infinity then, 51 00:03:45,020 --> 00:03:52,490 in fact, the periodic signal goes to the aperiodic signal. 52 00:03:52,490 --> 00:03:59,360 So the basic idea then is to use the Fourier series to 53 00:03:59,360 --> 00:04:03,920 represent the periodic signal, and then examine the Fourier 54 00:04:03,920 --> 00:04:08,880 series expression as we let the period go to infinity. 55 00:04:08,880 --> 00:04:14,150 Well, let's quickly see how this develops in terms of the 56 00:04:14,150 --> 00:04:16,130 associated equations. 57 00:04:16,130 --> 00:04:22,079 Here, again, we have the periodic signal. 58 00:04:22,079 --> 00:04:26,920 And what we want to inquire into is what happens to the 59 00:04:26,920 --> 00:04:30,690 Fourier series expression for this as we let the period go 60 00:04:30,690 --> 00:04:31,750 to infinity. 61 00:04:31,750 --> 00:04:35,140 As that happens, whatever Fourier series representation 62 00:04:35,140 --> 00:04:40,260 we end up with will correspond also to a representation for 63 00:04:40,260 --> 00:04:42,850 this aperiodic signal. 64 00:04:42,850 --> 00:04:44,220 Well, let's see. 65 00:04:44,220 --> 00:04:48,040 The Fourier series synthesis expression for the periodic 66 00:04:48,040 --> 00:04:53,650 signal expresses x tilde of t, the periodic signal as a 67 00:04:53,650 --> 00:04:57,230 linear combination of harmonically related complex 68 00:04:57,230 --> 00:05:01,510 exponentials with the fundamental frequency omega 0 69 00:05:01,510 --> 00:05:04,370 equaled to 2 pi divided by the period. 70 00:05:04,370 --> 00:05:11,530 And the analysis equation tells us what the relationship 71 00:05:11,530 --> 00:05:16,180 is for the coefficients in terms of the periodic signal. 72 00:05:16,180 --> 00:05:20,215 Now, I indicated that the periodic signal and the 73 00:05:20,215 --> 00:05:23,370 aperiodic signal are equal over one period. 74 00:05:23,370 --> 00:05:27,640 We recognize that this integration, in fact, only 75 00:05:27,640 --> 00:05:29,460 occurs over one period. 76 00:05:29,460 --> 00:05:33,840 And so we can re-express this in terms of our original 77 00:05:33,840 --> 00:05:35,510 aperiodic signal. 78 00:05:35,510 --> 00:05:38,730 So this tells us the Fourier series coefficients in 79 00:05:38,730 --> 00:05:40,012 terms of x of t. 80 00:05:43,320 --> 00:05:49,360 Now, if we look at this expression, which is the 81 00:05:49,360 --> 00:05:53,020 expression for the Fourier coefficients of the aperiodic 82 00:05:53,020 --> 00:05:58,690 signal, one of the things to recognize is that in effect 83 00:05:58,690 --> 00:06:06,420 what this represents are samples of an integral, where 84 00:06:06,420 --> 00:06:12,350 we can think of the variable omega taking on values that 85 00:06:12,350 --> 00:06:14,830 are integer multiples of omega 0. 86 00:06:14,830 --> 00:06:18,600 Said another way, let's define a function, as I've indicated 87 00:06:18,600 --> 00:06:24,540 here, which is this integral, where we may think of omega as 88 00:06:24,540 --> 00:06:27,930 being a continuous variable and then the Fourier series 89 00:06:27,930 --> 00:06:35,890 coefficients correspond to substituting for 90 00:06:35,890 --> 00:06:38,660 omega k omega 0. 91 00:06:38,660 --> 00:06:42,710 Now, one reason for doing that, as we'll see, is that in 92 00:06:42,710 --> 00:06:47,480 fact, this will turn out to provide us with a mechanism 93 00:06:47,480 --> 00:06:51,250 for a Fourier representation of x of t. 94 00:06:51,250 --> 00:06:56,360 And this, in fact, then, is an envelope of the Fourier series 95 00:06:56,360 --> 00:06:57,340 coefficients. 96 00:06:57,340 --> 00:07:03,530 In other words, t 0 the period times the coefficients is 97 00:07:03,530 --> 00:07:07,850 equal to this integral add integer multiples of omega 0. 98 00:07:10,540 --> 00:07:14,360 So this, in effect, tells us how to get the Fourier series 99 00:07:14,360 --> 00:07:19,490 coefficients of the periodic signal in terms of samples of 100 00:07:19,490 --> 00:07:20,140 an envelope. 101 00:07:20,140 --> 00:07:23,770 And that will become a very important notion shortly. 102 00:07:23,770 --> 00:07:28,370 And that, in effect, will correspond to an analysis 103 00:07:28,370 --> 00:07:32,280 equation to represent the aperiodic signal. 104 00:07:32,280 --> 00:07:35,770 Now, let's look at the synthesis equation. 105 00:07:35,770 --> 00:07:40,200 Recall that in the synthesis our strategy is to build a 106 00:07:40,200 --> 00:07:44,190 periodic signal and let the period go to infinity. 107 00:07:44,190 --> 00:07:50,160 Well, here is the expression for the synthesis of the 108 00:07:50,160 --> 00:07:56,510 periodic signal now expressed in terms of samples of this 109 00:07:56,510 --> 00:08:01,840 envelope function, and where I've simply used the fact or 110 00:08:01,840 --> 00:08:06,870 the substitution that t 0 is 2 pi over omega 0, and so I have 111 00:08:06,870 --> 00:08:10,830 an omega 0 here and a 1 over 2 pi. 112 00:08:10,830 --> 00:08:14,280 And the reason for doing that, as we'll see in a minute, is 113 00:08:14,280 --> 00:08:17,440 that this then turns into an integral. 114 00:08:17,440 --> 00:08:20,880 Specifically, then, the synthesis equation that we 115 00:08:20,880 --> 00:08:26,690 have is what I've indicated here. 116 00:08:26,690 --> 00:08:30,300 We would now want to examine this as the period goes to 117 00:08:30,300 --> 00:08:33,990 infinity, which means that omega 0 becomes 118 00:08:33,990 --> 00:08:36,140 infinitesimally small. 119 00:08:36,140 --> 00:08:39,820 And without dwelling on the details, and with my 120 00:08:39,820 --> 00:08:43,440 suggesting that you give this a fair amount of reflection, 121 00:08:43,440 --> 00:08:47,830 in fact, what happens as the period goes to infinity is 122 00:08:47,830 --> 00:08:53,910 that this summation approaches an integral over omega, where 123 00:08:53,910 --> 00:08:57,650 omega 0 becomes the differential in omega, and the 124 00:08:57,650 --> 00:09:00,020 periodic signal, of course, approach is 125 00:09:00,020 --> 00:09:02,740 the aperiodic signal. 126 00:09:02,740 --> 00:09:08,240 So the resulting equation that we get out of the original 127 00:09:08,240 --> 00:09:14,680 Fourier series synthesis equation is the equation that 128 00:09:14,680 --> 00:09:21,730 I indicate down here, x of t synthesized in terms of this 129 00:09:21,730 --> 00:09:25,720 integral, which is what the Fourier series approaches as 130 00:09:25,720 --> 00:09:28,780 omega 0 goes to 0. 131 00:09:28,780 --> 00:09:34,400 And we had previously that x of omega was in fact an 132 00:09:34,400 --> 00:09:36,900 envelope function. 133 00:09:36,900 --> 00:09:39,660 And we have then the corresponding Fourier 134 00:09:39,660 --> 00:09:45,240 transform analysis equation, which tells us how we arrive 135 00:09:45,240 --> 00:09:49,930 at that envelope in terms of x of t. 136 00:09:49,930 --> 00:09:54,410 So we now have an analysis equation and a synthesis 137 00:09:54,410 --> 00:09:59,150 equation, which in effect expresses for us how to build 138 00:09:59,150 --> 00:10:03,910 x of t in terms of infinitesimally finely spaced 139 00:10:03,910 --> 00:10:06,750 complex exponentials. 140 00:10:06,750 --> 00:10:11,690 The strategy to review it, and which I'd like to illustrate 141 00:10:11,690 --> 00:10:21,090 with a succession of overlays, was to begin with our 142 00:10:21,090 --> 00:10:28,780 aperiodic signal, as I indicate here, and then we 143 00:10:28,780 --> 00:10:34,160 constructed from that a periodic signal. 144 00:10:34,160 --> 00:10:39,880 And this periodic signal has a Fourier series, and we express 145 00:10:39,880 --> 00:10:43,550 the Fourier series coefficients of this as 146 00:10:43,550 --> 00:10:46,630 samples of an envelope function. 147 00:10:46,630 --> 00:10:51,520 The envelope function is what I indicate on the curve below. 148 00:10:51,520 --> 00:10:55,390 So this is the envelope of the Fourier series coefficients. 149 00:10:55,390 --> 00:11:02,640 For example, if the period t 0 was four times t1, then the 150 00:11:02,640 --> 00:11:06,270 Fourier series coefficients that we would end up with is 151 00:11:06,270 --> 00:11:10,480 this set of samples of the envelope. 152 00:11:10,480 --> 00:11:15,500 If instead we doubled that period, then the Fourier 153 00:11:15,500 --> 00:11:18,470 series coefficients that we end up with 154 00:11:18,470 --> 00:11:22,180 are more finely spaced. 155 00:11:22,180 --> 00:11:28,470 And as t 0 continues to increase, we get more and more 156 00:11:28,470 --> 00:11:33,290 finely spaced samples of this envelope function, and as t 0 157 00:11:33,290 --> 00:11:37,030 goes to infinity in fact, what we get is every single point 158 00:11:37,030 --> 00:11:41,080 on the envelope, and that provides us with the 159 00:11:41,080 --> 00:11:44,780 representation for the aperiodic signal. 160 00:11:44,780 --> 00:11:50,530 Let me, just to really emphasize the point, show this 161 00:11:50,530 --> 00:11:52,520 example once again. 162 00:11:52,520 --> 00:11:55,800 But now, let's look at it dynamically on 163 00:11:55,800 --> 00:11:58,440 the computer display. 164 00:11:58,440 --> 00:12:01,220 So here we have the square wave, and below it, the 165 00:12:01,220 --> 00:12:03,010 Fourier series coefficients. 166 00:12:03,010 --> 00:12:06,490 And we now want to look at the Fourier series coefficients as 167 00:12:06,490 --> 00:12:09,550 the period of the square wave starts to increase. 168 00:12:17,760 --> 00:12:21,260 And what we see is that these look like 169 00:12:21,260 --> 00:12:24,100 samples of an envelope. 170 00:12:24,100 --> 00:12:26,970 And in fact, the envelope of the Fourier series 171 00:12:26,970 --> 00:12:30,090 coefficients is shown in the bottom [? trace, ?] 172 00:12:30,090 --> 00:12:33,300 and to emphasize in fact that it is the envelope let's 173 00:12:33,300 --> 00:12:37,400 superimpose it on top of the Fourier series coefficients 174 00:12:37,400 --> 00:12:38,840 that we've generated so far. 175 00:12:48,340 --> 00:12:48,710 OK. 176 00:12:48,710 --> 00:12:54,020 Now, let's increase the period even further, and we'll see 177 00:12:54,020 --> 00:12:57,910 the Fourier series coefficients fill in under 178 00:12:57,910 --> 00:13:02,780 that envelope function even more. 179 00:13:02,780 --> 00:13:07,980 And in fact, as the period gets large enough, what we 180 00:13:07,980 --> 00:13:13,570 begin to get a sense of is that we're sampling more and 181 00:13:13,570 --> 00:13:15,940 more finely this envelope. 182 00:13:15,940 --> 00:13:19,060 And in fact, in the limit, as the period goes off to 183 00:13:19,060 --> 00:13:23,160 infinity, the samples basically will represent every 184 00:13:23,160 --> 00:13:25,050 single point on the envelope. 185 00:13:25,050 --> 00:13:28,030 Well, this is about as far as we want to go. 186 00:13:28,030 --> 00:13:33,130 Let's once again, plot the envelope function, and again, 187 00:13:33,130 --> 00:13:36,160 to emphasize that we've generated samples of that, 188 00:13:36,160 --> 00:13:39,175 let's superimpose that on the Fourier series coefficients. 189 00:13:46,300 --> 00:13:52,000 So what we have then is now our Fourier transform 190 00:13:52,000 --> 00:13:57,530 representation, the continuous time Fourier transform with 191 00:13:57,530 --> 00:14:01,960 the synthesis equation expressed as an integral, as 192 00:14:01,960 --> 00:14:06,870 I've indicated here, and this integral is what the Fourier 193 00:14:06,870 --> 00:14:12,820 series sum went to as we let the period go to infinity or 194 00:14:12,820 --> 00:14:14,720 the frequency go to zero. 195 00:14:14,720 --> 00:14:19,080 The corresponding analysis equation, which we have here, 196 00:14:19,080 --> 00:14:23,670 the analysis equation being the expression for the 197 00:14:23,670 --> 00:14:26,970 envelope of the Fourier series coefficients for the 198 00:14:26,970 --> 00:14:30,620 periodically replicated signal. 199 00:14:30,620 --> 00:14:33,690 And in shorthand notation, we would think 200 00:14:33,690 --> 00:14:35,370 of x of t and [? its ?] 201 00:14:35,370 --> 00:14:40,380 Fourier transform as a pair, as I've indicated here. 202 00:14:40,380 --> 00:14:45,830 And the Fourier transform, as we'll emphasize in several 203 00:14:45,830 --> 00:14:49,800 examples, and certainly as is consistent with the Fourier 204 00:14:49,800 --> 00:14:54,590 Series, is a complex valued function even 205 00:14:54,590 --> 00:14:55,730 when x of t is real. 206 00:14:55,730 --> 00:14:59,690 So with x of t real, we end up with a Fourier transform, 207 00:14:59,690 --> 00:15:01,770 which is a complex function. 208 00:15:01,770 --> 00:15:06,060 Just as the Fourier series coefficients were complex for 209 00:15:06,060 --> 00:15:08,250 a real value time function. 210 00:15:08,250 --> 00:15:12,420 So we could alternatively, as with the Fourier series, 211 00:15:12,420 --> 00:15:18,150 express the Fourier transform in terms of it's real part and 212 00:15:18,150 --> 00:15:23,530 imaginary part, or alternatively, in terms of its 213 00:15:23,530 --> 00:15:25,520 magnitude and its angle. 214 00:15:28,270 --> 00:15:33,560 All right, now let's look at an example of a time function 215 00:15:33,560 --> 00:15:35,750 in its Fourier transform. 216 00:15:35,750 --> 00:15:39,680 And so let's consider an example, which in fact is an 217 00:15:39,680 --> 00:15:41,350 example worked out in the text. 218 00:15:41,350 --> 00:15:44,500 It's example 4.7 in the text. 219 00:15:44,500 --> 00:15:49,520 And this is our old familiar friend the exponential. 220 00:15:49,520 --> 00:15:53,230 It's Fourier transform is the integral from minus infinity 221 00:15:53,230 --> 00:15:57,950 to plus infinity, x of t, e to the minus j omega t dt. 222 00:15:57,950 --> 00:16:03,960 And so, if we substitute in x of t and combine these two 223 00:16:03,960 --> 00:16:09,200 exponentials together, these two exponentials combined are 224 00:16:09,200 --> 00:16:15,130 e to the minus t times a plus j omega. 225 00:16:15,130 --> 00:16:21,070 And if we carry out the integration of this, we end up 226 00:16:21,070 --> 00:16:25,790 with the expression indicated here and provided now, and 227 00:16:25,790 --> 00:16:32,830 this is important, provided that a is greater than 0, then 228 00:16:32,830 --> 00:16:36,310 at the upper limit, this exponential becomes 0. 229 00:16:36,310 --> 00:16:38,870 At the lower limit, of course, it's one. 230 00:16:38,870 --> 00:16:44,460 And so what we have finally is for the Fourier transform 231 00:16:44,460 --> 00:16:48,780 expression 1 over a plus j omega. 232 00:16:52,700 --> 00:16:56,970 Now, this Fourier transform as I indicated is a complex 233 00:16:56,970 --> 00:16:58,380 valued function. 234 00:16:58,380 --> 00:17:04,210 Let's just take a look at what it looks like graphically. 235 00:17:04,210 --> 00:17:10,640 We have the expression for the Fourier transform pair, e to 236 00:17:10,640 --> 00:17:11,910 the minus a t times [? a ?] 237 00:17:11,910 --> 00:17:12,839 [? step ?]. 238 00:17:12,839 --> 00:17:16,980 And its Fourier transform is 1 over a plus j omega. 239 00:17:16,980 --> 00:17:22,609 And I indicated that that's true for a greater than 0. 240 00:17:22,609 --> 00:17:26,710 Now, in the expression that we just worked out, if a is less 241 00:17:26,710 --> 00:17:33,360 than 0, in fact, the expression doesn't converge e 242 00:17:33,360 --> 00:17:38,800 to the minus a t for a negative as t goes to 243 00:17:38,800 --> 00:17:44,170 infinity, blows up, and so in fact the Fourier transform 244 00:17:44,170 --> 00:17:48,990 doesn't converge except for the case where a is greater 245 00:17:48,990 --> 00:17:52,930 than 0 And in fact, there is a more detailed discussion of 246 00:17:52,930 --> 00:17:54,990 convergence issues in the text. 247 00:17:54,990 --> 00:17:58,000 The convergence issues are very much the same for the 248 00:17:58,000 --> 00:18:00,860 Fourier transform as they are for the Fourier series. 249 00:18:00,860 --> 00:18:03,510 And in fact, that's not surprising, because we 250 00:18:03,510 --> 00:18:05,360 developed the Fourier transform out of a 251 00:18:05,360 --> 00:18:07,410 consideration of the Fourier series. 252 00:18:07,410 --> 00:18:10,970 So the convergence conditions as you'll see as you refer in 253 00:18:10,970 --> 00:18:15,310 detail to the text relate to whether the time function is 254 00:18:15,310 --> 00:18:18,400 absolutely integrable under one set of conditions and 255 00:18:18,400 --> 00:18:22,290 square integrable under another set of conditions. 256 00:18:22,290 --> 00:18:28,110 OK, now, if we plot the Fourier transform, let's first 257 00:18:28,110 --> 00:18:30,870 consider the shape of the time function. 258 00:18:30,870 --> 00:18:34,580 And as I indicated, we're restricting the time function 259 00:18:34,580 --> 00:18:37,630 so that the exponential factor a is positive. 260 00:18:37,630 --> 00:18:40,730 In other words, e to the minus a t decays 261 00:18:40,730 --> 00:18:42,920 as t goes to infinity. 262 00:18:42,920 --> 00:18:48,830 The magnitude of the Fourier transform is as I indicate 263 00:18:48,830 --> 00:18:53,970 here and the phase below it. 264 00:18:53,970 --> 00:18:59,350 And there are a number of things we can see about the 265 00:18:59,350 --> 00:19:02,630 magnitude and phase of the Fourier transform for this 266 00:19:02,630 --> 00:19:07,990 example, which in fact we'll see in the next lecture are 267 00:19:07,990 --> 00:19:10,210 properties that apply more generally. 268 00:19:10,210 --> 00:19:13,810 For example, the fact that the Fourier transform magnitude is 269 00:19:13,810 --> 00:19:17,440 an even function of frequency, and the phase is an odd 270 00:19:17,440 --> 00:19:19,870 function of frequency. 271 00:19:19,870 --> 00:19:26,810 Now, let me also draw your attention to the fact that on 272 00:19:26,810 --> 00:19:31,220 this curve we have both positive frequencies and 273 00:19:31,220 --> 00:19:32,830 negative frequencies. 274 00:19:32,830 --> 00:19:35,950 In other words, in our expression for the Fourier 275 00:19:35,950 --> 00:19:40,220 transform, it requires both omega 276 00:19:40,220 --> 00:19:43,920 positive and omega negative. 277 00:19:43,920 --> 00:19:46,700 This, of course, was exactly the same in the case of the 278 00:19:46,700 --> 00:19:48,160 Fourier series. 279 00:19:48,160 --> 00:19:53,530 And the reason you should recall and keep in mind is 280 00:19:53,530 --> 00:19:56,390 related to the fact that we're building our signals out of 281 00:19:56,390 --> 00:20:01,010 complex exponentials, which require both positive values 282 00:20:01,010 --> 00:20:03,900 of omega and negative values of omega. 283 00:20:03,900 --> 00:20:07,000 Alternatively, if we chosen other representation, which 284 00:20:07,000 --> 00:20:10,060 turns out notationally to be much more difficult, namely 285 00:20:10,060 --> 00:20:13,300 sines and cosines, then we would in fact only consider 286 00:20:13,300 --> 00:20:14,700 positive frequencies. 287 00:20:14,700 --> 00:20:19,100 So it's important to keep in mind that, in our case, both 288 00:20:19,100 --> 00:20:23,050 with the Fourier series and the Fourier transform, we deal 289 00:20:23,050 --> 00:20:26,400 and require both positive and negative frequencies in order 290 00:20:26,400 --> 00:20:27,650 to build our signals. 291 00:20:29,910 --> 00:20:34,220 Now, in the graphical representation that I've shown 292 00:20:34,220 --> 00:20:39,210 here, I've chosen a linear amplitude scale and a linear 293 00:20:39,210 --> 00:20:40,360 frequency scale. 294 00:20:40,360 --> 00:20:44,070 And that's one graphical representation for the Fourier 295 00:20:44,070 --> 00:20:47,550 transform that we'll typically use. 296 00:20:47,550 --> 00:20:54,110 There's another one that very commonly arises, which I'll 297 00:20:54,110 --> 00:20:57,450 just briefly indicate for this example. 298 00:20:57,450 --> 00:21:04,240 And that is what's referred to as a bode plot in which the 299 00:21:04,240 --> 00:21:07,640 magnitude is displayed on a log amplitude and log 300 00:21:07,640 --> 00:21:09,160 frequency scale. 301 00:21:09,160 --> 00:21:11,940 And the phase is displayed on a log frequency scale. 302 00:21:11,940 --> 00:21:14,140 Let me show you what I mean. 303 00:21:14,140 --> 00:21:17,560 Here is the general expression for the bode plot. 304 00:21:17,560 --> 00:21:23,080 The bode plot expresses for us the amplitude in terms of the 305 00:21:23,080 --> 00:21:26,430 logarithm to the base 10 of the magnitude. 306 00:21:26,430 --> 00:21:33,020 And it also expresses the angle in both cases expressed 307 00:21:33,020 --> 00:21:37,810 as a function of a logarithmic frequency axis. 308 00:21:37,810 --> 00:21:45,240 So here is the amplitude as I've displayed it. 309 00:21:45,240 --> 00:21:50,110 And this is a log magnitude scale, a logarithmic frequency 310 00:21:50,110 --> 00:21:54,280 scale as indicated by the fact that as we move in equal 311 00:21:54,280 --> 00:21:58,190 increments along this axis, we change frequency 312 00:21:58,190 --> 00:22:00,050 by a factor of 10. 313 00:22:00,050 --> 00:22:07,440 And similarly, what we have is a display for the phase again 314 00:22:07,440 --> 00:22:10,190 on a log frequency scale. 315 00:22:10,190 --> 00:22:15,000 And I indicated that there is a symmetry to the Fourier 316 00:22:15,000 --> 00:22:19,570 transform, and so in fact, we can infer from this particular 317 00:22:19,570 --> 00:22:23,360 picture what it looks like for the negative frequencies as 318 00:22:23,360 --> 00:22:24,695 well as for the positive frequencies. 319 00:22:29,560 --> 00:22:39,450 Now, what we've done so far is to develop the Fourier 320 00:22:39,450 --> 00:22:44,285 transform on the basis, the Fourier transform of an 321 00:22:44,285 --> 00:22:48,170 aperiodic signal on the basis of periodically repeating it 322 00:22:48,170 --> 00:22:52,420 and recognizing that the Fourier series coefficients 323 00:22:52,420 --> 00:22:57,200 are samples of an envelope and that these become more finely 324 00:22:57,200 --> 00:22:59,990 spaced as frequency increases. 325 00:22:59,990 --> 00:23:08,080 And in fact, we can go back to our original equation in which 326 00:23:08,080 --> 00:23:13,810 we developed an envelope function, and what we had 327 00:23:13,810 --> 00:23:19,200 indicated is that the Fourier series coefficients were 328 00:23:19,200 --> 00:23:22,920 samples of this envelope. 329 00:23:22,920 --> 00:23:31,290 We then defined this envelope as the Fourier transform of 330 00:23:31,290 --> 00:23:33,760 this aperiodic signal. 331 00:23:33,760 --> 00:23:38,670 So that provided us with a way-- and it was a mechanism-- 332 00:23:38,670 --> 00:23:44,810 for getting a representation for an aperiodic signal. 333 00:23:44,810 --> 00:23:50,820 Now, suppose that we have instead a periodic signal, are 334 00:23:50,820 --> 00:23:54,660 there, in fact, some statements that we can make 335 00:23:54,660 --> 00:23:58,050 about how the Fourier series coefficients of that are 336 00:23:58,050 --> 00:24:02,240 related to the Fourier transform of something. 337 00:24:02,240 --> 00:24:05,780 Well, in fact, this statement tells us 338 00:24:05,780 --> 00:24:07,390 exactly how to do that. 339 00:24:07,390 --> 00:24:11,000 What this statement says is that, in fact, the Fourier 340 00:24:11,000 --> 00:24:16,140 series coefficients are samples of the Fourier 341 00:24:16,140 --> 00:24:19,180 transform of one period? 342 00:24:19,180 --> 00:24:28,450 So if we now consider a periodic signal, we can in 343 00:24:28,450 --> 00:24:33,080 fact get the Fourier series coefficients of that periodic 344 00:24:33,080 --> 00:24:35,980 signal by considering the Fourier 345 00:24:35,980 --> 00:24:38,630 transform of one period. 346 00:24:38,630 --> 00:24:42,400 Said another way, the Fourier series coefficients are 347 00:24:42,400 --> 00:24:45,510 proportional to samples of the Fourier 348 00:24:45,510 --> 00:24:47,670 transform of one period. 349 00:24:47,670 --> 00:24:53,030 So if we consider this a periodic signal, computed as 350 00:24:53,030 --> 00:25:00,380 Fourier transform, and selected these samples that I 351 00:25:00,380 --> 00:25:06,780 indicate here, namely samples equally spaced in omega by 352 00:25:06,780 --> 00:25:12,260 integer multiples of omega 0, then in fact, those would be 353 00:25:12,260 --> 00:25:14,680 the Fourier series coefficients. 354 00:25:14,680 --> 00:25:22,890 So we can go back to our example previously that 355 00:25:22,890 --> 00:25:27,190 involved the square wave. 356 00:25:27,190 --> 00:25:31,910 And now, in this case, we could argue that if in fact it 357 00:25:31,910 --> 00:25:36,490 was the periodic signal that we started with, we could get 358 00:25:36,490 --> 00:25:40,610 the Fourier series coefficients of that by 359 00:25:40,610 --> 00:25:46,950 thinking about the Fourier transform of one period, which 360 00:25:46,950 --> 00:25:49,590 I indicate here. 361 00:25:49,590 --> 00:25:53,920 And then the Fourier series coefficients of the periodic 362 00:25:53,920 --> 00:26:00,070 signal, in fact, are the appropriate set of samples of 363 00:26:00,070 --> 00:26:01,320 this envelope. 364 00:26:04,710 --> 00:26:11,400 All right, now, we have a way of getting the Fourier series 365 00:26:11,400 --> 00:26:14,930 coefficients from the Fourier transform of one period. 366 00:26:14,930 --> 00:26:18,100 We originally derived the Fourier transform of one 367 00:26:18,100 --> 00:26:21,420 period from the Fourier series. 368 00:26:21,420 --> 00:26:25,820 What would, in fact, be nice is if we could incorporate the 369 00:26:25,820 --> 00:26:28,790 Fourier series and the Fourier transform 370 00:26:28,790 --> 00:26:30,530 within a common framework. 371 00:26:30,530 --> 00:26:34,390 And in fact, it turns out that there is a very convenient way 372 00:26:34,390 --> 00:26:38,040 of doing that almost by definition. 373 00:26:38,040 --> 00:26:44,380 Essentially, if we consider what the equation for the 374 00:26:44,380 --> 00:26:51,640 synthesis looks like in both cases, we can in effect define 375 00:26:51,640 --> 00:26:56,820 a Fourier transform for the periodic signal, which we know 376 00:26:56,820 --> 00:26:59,360 is represented by its Fourier series coefficients. 377 00:26:59,360 --> 00:27:04,150 We can define a Fourier transform, and the definition 378 00:27:04,150 --> 00:27:09,960 of the Fourier transform is as an impulse train, where the 379 00:27:09,960 --> 00:27:14,220 coefficients in the impulse train are proportional, with a 380 00:27:14,220 --> 00:27:17,200 proportionality factor of 2 pi for a more or less a 381 00:27:17,200 --> 00:27:20,140 bookkeeping reason, proportional to the Fourier 382 00:27:20,140 --> 00:27:22,220 series coefficients. 383 00:27:22,220 --> 00:27:28,430 And the validity of this is, more or less, can be seen 384 00:27:28,430 --> 00:27:30,860 essentially by substitution. 385 00:27:30,860 --> 00:27:37,210 Specifically, here is then the synthesis equation for the 386 00:27:37,210 --> 00:27:42,030 Fourier transform if we substitute this definition for 387 00:27:42,030 --> 00:27:45,290 the Fourier transform of the periodic signal into this 388 00:27:45,290 --> 00:27:51,310 expression then when we do the appropriate bookkeeping and 389 00:27:51,310 --> 00:27:54,850 interchange the order of summation and integration the 390 00:27:54,850 --> 00:27:58,690 impulse integrates out to the exponential 391 00:27:58,690 --> 00:28:02,530 factor that we want. 392 00:28:02,530 --> 00:28:05,820 So we have the exponential factor. 393 00:28:05,820 --> 00:28:07,990 We have the Fourier series coefficients. 394 00:28:07,990 --> 00:28:11,450 The 2 pis take care of each other, and what we're left 395 00:28:11,450 --> 00:28:16,620 with is the synthesis equation for aperiodic signal in terms 396 00:28:16,620 --> 00:28:21,160 of the Fourier transform, or in terms of its Fourier series 397 00:28:21,160 --> 00:28:23,600 coefficients. 398 00:28:23,600 --> 00:28:30,230 Now, we can just see this in terms of a simple example. 399 00:28:30,230 --> 00:28:36,570 If we consider the example of a symmetric square wave, then 400 00:28:36,570 --> 00:28:40,320 in effect what we're saying is that for this symmetric square 401 00:28:40,320 --> 00:28:45,240 wave, this has a set of Fourier series coefficients, 402 00:28:45,240 --> 00:28:50,020 which we worked out previously and which I indicate on this 403 00:28:50,020 --> 00:28:52,440 figure with a bar graph. 404 00:28:52,440 --> 00:28:55,930 And really all that we're saying is that, whereas these 405 00:28:55,930 --> 00:28:59,840 Fourier series coefficients are indexed on an integer 406 00:28:59,840 --> 00:29:04,790 variable k, and [? they're ?] bars not impulses. 407 00:29:04,790 --> 00:29:10,020 If we simply redefine or define the Fourier transform 408 00:29:10,020 --> 00:29:15,360 of the periodic signal as an impulse train, where the 409 00:29:15,360 --> 00:29:20,840 weights of the impulses are 2 pi times the corresponding 410 00:29:20,840 --> 00:29:25,750 Fourier series coefficients, then this, in fact, is what we 411 00:29:25,750 --> 00:29:29,690 would use as the Fourier transform of 412 00:29:29,690 --> 00:29:30,940 the periodic signal. 413 00:29:34,010 --> 00:29:39,080 Now, we've kind of gone back and forth, and maybe even it 414 00:29:39,080 --> 00:29:42,180 might seem like we've gone around in circles. 415 00:29:42,180 --> 00:29:45,880 So let me just try to summarize the various 416 00:29:45,880 --> 00:29:49,490 relationships and steps that we've gone through, keeping in 417 00:29:49,490 --> 00:29:52,670 mind that one of our objectives was first to 418 00:29:52,670 --> 00:29:56,690 develop a representation for aperiodic signals and then 419 00:29:56,690 --> 00:30:01,910 attempt to incorporate within one framework both periodic 420 00:30:01,910 --> 00:30:05,530 and aperiodic signals. 421 00:30:05,530 --> 00:30:11,250 We began with an aperiodic signal. 422 00:30:11,250 --> 00:30:15,290 And the strategy was to develop a Fourier 423 00:30:15,290 --> 00:30:21,150 representation by constructing a periodic signal for which 424 00:30:21,150 --> 00:30:22,860 that was one period. 425 00:30:22,860 --> 00:30:25,770 And then we let the period go to infinity, 426 00:30:25,770 --> 00:30:27,580 as I indicate here. 427 00:30:27,580 --> 00:30:32,770 So we have an aperiodic signal. 428 00:30:32,770 --> 00:30:37,760 We construct a periodic signal, x tilde of t for which 429 00:30:37,760 --> 00:30:41,500 one period is the aperiodic signal. 430 00:30:41,500 --> 00:30:45,550 X tilde of t, the periodic signal, has a Fourier series, 431 00:30:45,550 --> 00:30:50,700 and as its period increases that approaches the aperiodic 432 00:30:50,700 --> 00:30:57,050 signal, and the Fourier series of that approaches the Fourier 433 00:30:57,050 --> 00:31:00,720 transform of the original aperiodic signal. 434 00:31:00,720 --> 00:31:05,700 So that was the first step we took. 435 00:31:05,700 --> 00:31:10,940 Now, the second thing that we recognize is that once we have 436 00:31:10,940 --> 00:31:15,280 the concept of the Fourier transform, we can, in fact, 437 00:31:15,280 --> 00:31:19,570 relate the Fourier series coefficients to the Fourier 438 00:31:19,570 --> 00:31:22,210 transform of one period. 439 00:31:22,210 --> 00:31:27,330 So the second statement that we made was that if in fact 440 00:31:27,330 --> 00:31:33,070 we're trying to represent a periodic signal, we can get 441 00:31:33,070 --> 00:31:35,900 the Fourier series coefficients of that by 442 00:31:35,900 --> 00:31:42,340 computing the Fourier transform of one period and 443 00:31:42,340 --> 00:31:47,380 then samples of that Fourier transform are, in fact, the 444 00:31:47,380 --> 00:31:50,110 Fourier series coefficients for the periodic signal. 445 00:31:53,210 --> 00:31:58,320 Then, the third step that we took was to inquire as to 446 00:31:58,320 --> 00:32:02,340 whether there is a Fourier transform that can 447 00:32:02,340 --> 00:32:06,420 appropriately be defined for the periodic signal, and the 448 00:32:06,420 --> 00:32:11,260 mechanism for doing that was to recognize that if we simply 449 00:32:11,260 --> 00:32:14,840 defined the Fourier transform of the periodic signal as an 450 00:32:14,840 --> 00:32:19,420 impulse train, where the impulse heights or areas were 451 00:32:19,420 --> 00:32:22,580 proportional to the Fourier series coefficients, then, in 452 00:32:22,580 --> 00:32:29,630 fact, the Fourier transform synthesis equation reduced to 453 00:32:29,630 --> 00:32:32,140 the Fourier series synthesis equation. 454 00:32:32,140 --> 00:32:37,960 So the third step, then, was with a periodic signal. 455 00:32:37,960 --> 00:32:42,280 The Fourier transform of that periodic signal, defined as an 456 00:32:42,280 --> 00:32:44,610 impulse train, where the heights or areas of the 457 00:32:44,610 --> 00:32:47,810 impulses are proportional to the Fourier series 458 00:32:47,810 --> 00:32:53,040 coefficients, provides us with a mechanism for combining it 459 00:32:53,040 --> 00:32:57,825 together the concepts or notation of the Fourier series 460 00:32:57,825 --> 00:33:00,900 and Fourier transform. 461 00:33:00,900 --> 00:33:07,340 So if we just took a very simple example, here is an 462 00:33:07,340 --> 00:33:12,950 example in which we have an aperiodic signal, which is 463 00:33:12,950 --> 00:33:16,620 just an impulse, and its Fourier 464 00:33:16,620 --> 00:33:20,670 transform is just a constant. 465 00:33:20,670 --> 00:33:24,730 We can think of a periodic signal associated with this, 466 00:33:24,730 --> 00:33:29,820 which is this signal periodically replicated with a 467 00:33:29,820 --> 00:33:32,300 spacing t 0. 468 00:33:32,300 --> 00:33:35,720 The Fourier transform of this is a constant. 469 00:33:35,720 --> 00:33:38,970 And this, of course, has a Fourier series representation. 470 00:33:38,970 --> 00:33:42,800 So the Fourier transform of the original 471 00:33:42,800 --> 00:33:46,170 impulse is just a constant. 472 00:33:46,170 --> 00:33:51,830 The Fourier transform of the periodic signal is an impulse 473 00:33:51,830 --> 00:33:56,170 train, where the heights of the impulses are proportional 474 00:33:56,170 --> 00:33:58,830 to the Fourier series coefficients. 475 00:33:58,830 --> 00:34:02,700 And, of course, we could previously have computed the 476 00:34:02,700 --> 00:34:06,680 Fourier series coefficients for that impulse train, and 477 00:34:06,680 --> 00:34:09,040 those Fourier series coefficients are 478 00:34:09,040 --> 00:34:10,370 as I've shown here. 479 00:34:10,370 --> 00:34:14,290 So in both of these cases, these in effect represent just 480 00:34:14,290 --> 00:34:16,909 a change in notation, where here we have a bar graph, and 481 00:34:16,909 --> 00:34:19,199 here we have an impulse train. 482 00:34:19,199 --> 00:34:23,980 And both of these simply represent samples of what we 483 00:34:23,980 --> 00:34:30,690 have above, which is the Fourier transform of the 484 00:34:30,690 --> 00:34:31,940 original aperiodic signal. 485 00:34:35,179 --> 00:34:39,239 Once again, I suspect that kind of moving back and forth 486 00:34:39,239 --> 00:34:42,239 and trying to straighten out when we're talking about 487 00:34:42,239 --> 00:34:46,989 periodic and aperiodic signals may require a little mental 488 00:34:46,989 --> 00:34:48,810 gymnastics initially. 489 00:34:48,810 --> 00:34:53,310 Basically, what we've tried to do is incorporate within one 490 00:34:53,310 --> 00:34:58,200 framework a representation for both aperiodic and periodic 491 00:34:58,200 --> 00:35:01,450 signals, and the Fourier transform provides us with a 492 00:35:01,450 --> 00:35:04,410 mechanism to do that. 493 00:35:04,410 --> 00:35:07,640 In the next lecture, I'll continue with the discussion 494 00:35:07,640 --> 00:35:11,310 of the continuous-time Fourier transform in particular 495 00:35:11,310 --> 00:35:15,260 focusing on a number of its properties, some of which 496 00:35:15,260 --> 00:35:18,000 we've already seen, namely the symmetry properties. 497 00:35:18,000 --> 00:35:21,360 We'll see lots of other properties that relate, of 498 00:35:21,360 --> 00:35:23,780 course, both to the Fourier transform and 499 00:35:23,780 --> 00:35:24,970 to the Fourier series. 500 00:35:24,970 --> 00:35:26,220 Thank you.