1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,400 at ocw.mit.edu. 8 00:00:31,030 --> 00:00:33,400 PROFESSOR 1: OK, let's launch right into it. 9 00:00:33,400 --> 00:00:36,580 Jacob and Uri were inspired by the echo channel 10 00:00:36,580 --> 00:00:39,850 to try out a simulation of what I'd put on the board last time 11 00:00:39,850 --> 00:00:41,480 or on my slides. 12 00:00:41,480 --> 00:00:44,230 So what you're going to hear is Jacob's message 13 00:00:44,230 --> 00:00:45,550 going into the echo channel. 14 00:00:45,550 --> 00:00:48,790 Remember, that was something with a unit sample response 15 00:00:48,790 --> 00:00:52,480 of the type delta n plus. 16 00:00:52,480 --> 00:00:54,050 And actually, I think in their case, 17 00:00:54,050 --> 00:01:01,420 now it's 0.999 delta n minus 1, so it is something like this. 18 00:01:01,420 --> 00:01:02,680 Sorry? 19 00:01:02,680 --> 00:01:07,270 Oh, n minus 4,000, OK. 20 00:01:07,270 --> 00:01:10,000 And so you'll hear the original message, I think. 21 00:01:10,000 --> 00:01:13,330 You'll hear the message going through the echo channel. 22 00:01:13,330 --> 00:01:16,480 And then you'll hear the message cleaned up 23 00:01:16,480 --> 00:01:17,590 with the receiver filter. 24 00:01:17,590 --> 00:01:19,280 That's just the inverse filter. 25 00:01:19,280 --> 00:01:21,670 And then you'll hear what noise does to it and two 26 00:01:21,670 --> 00:01:23,310 flavors of noise, I think. 27 00:01:23,310 --> 00:01:26,920 ECHO VOICE: This is a test of the 602 deconvolution system. 28 00:01:26,920 --> 00:01:28,992 If this is a real deconvolution, you 29 00:01:28,992 --> 00:01:31,180 are instructed to go as quickly as possible 30 00:01:31,180 --> 00:01:33,747 to the frequency domain to make an assessment 31 00:01:33,747 --> 00:01:35,330 of the effectiveness of this approach. 32 00:01:39,400 --> 00:01:42,940 This is a test of the 602 deconvolution system. 33 00:01:42,940 --> 00:01:45,100 If this is a real deconvolution, you 34 00:01:45,100 --> 00:01:47,590 are instructed to go as quickly as possible 35 00:01:47,590 --> 00:01:49,887 to the frequency domain to make an assessment 36 00:01:49,887 --> 00:01:51,470 of the effectiveness of this approach. 37 00:01:55,961 --> 00:01:57,458 [BELL DINGING] 38 00:02:04,497 --> 00:02:06,580 PROFESSOR 2: I'm just going to increase the delay, 39 00:02:06,580 --> 00:02:08,122 so we can hear the echo more clearly. 40 00:02:19,290 --> 00:02:23,010 ECHO VOICE: This is a test of the 602 deconvolution system. 41 00:02:23,010 --> 00:02:24,840 If this is a real deconvolution, you 42 00:02:24,840 --> 00:02:27,400 are instructed to go as quickly as possible 43 00:02:27,400 --> 00:02:29,500 to the frequency domain to make an assessment 44 00:02:29,500 --> 00:02:31,427 of the effectiveness of this approach. 45 00:02:31,427 --> 00:02:33,510 PROFESSOR 2: So you can all hear the echo in that. 46 00:02:38,470 --> 00:02:41,030 And the next one will be it cleaned up-- 47 00:02:41,030 --> 00:02:44,380 ECHO VOICE: This is a test of the 602 deconvolution system. 48 00:02:44,380 --> 00:02:46,465 If this is a real deconvolution, you 49 00:02:46,465 --> 00:02:49,030 are instructed to go as quickly as possible 50 00:02:49,030 --> 00:02:51,400 to the frequency domain to make an assessment 51 00:02:51,400 --> 00:02:53,325 of the effectiveness of this approach. 52 00:02:53,325 --> 00:02:54,700 PROFESSOR 2: So this was just one 53 00:02:54,700 --> 00:02:58,327 in deconvolution, assuming the channel had no noise. 54 00:02:58,327 --> 00:03:00,160 The next one is what happens if there's even 55 00:03:00,160 --> 00:03:03,520 a small amount of noise in the channel-- 56 00:03:03,520 --> 00:03:07,870 so little that you couldn't hear it in the echoed signal. 57 00:03:07,870 --> 00:03:11,340 ECHO VOICE: This is a test of the 602 deconvolution system. 58 00:03:11,340 --> 00:03:13,450 If this is a real deconvolution, you 59 00:03:13,450 --> 00:03:15,960 are instructed to go as quickly as possible 60 00:03:15,960 --> 00:03:18,330 to the frequency domain to make an assessment 61 00:03:18,330 --> 00:03:19,913 of the effectiveness of this approach. 62 00:03:19,913 --> 00:03:22,330 PROFESSOR 2: So you can hear the noise building up because 63 00:03:22,330 --> 00:03:23,130 of the deconvolver. 64 00:03:23,130 --> 00:03:26,338 If you actually had noise at a particular frequency, 65 00:03:26,338 --> 00:03:27,380 you end up with an even-- 66 00:03:27,380 --> 00:03:30,900 ECHO VOICE: This is a test of the 602 deconvolution system. 67 00:03:30,900 --> 00:03:33,030 If this is a real deconvolution, you 68 00:03:33,030 --> 00:03:35,520 are instructed to go as quickly as possible 69 00:03:35,520 --> 00:03:37,617 to the frequency domain to make an assessment 70 00:03:37,617 --> 00:03:39,200 of the effectiveness of this approach. 71 00:03:41,283 --> 00:03:43,700 PROFESSOR 2: So one of the difficulties with deconvolution 72 00:03:43,700 --> 00:03:46,280 is it can be perfect if you have no noise. 73 00:03:46,280 --> 00:03:48,080 But even small amounts of noise can really 74 00:03:48,080 --> 00:03:49,460 mess up the deconvolution. 75 00:03:49,460 --> 00:03:52,162 It magnifies the small noise sources. 76 00:03:52,162 --> 00:03:53,120 PROFESSOR 1: OK, great. 77 00:03:53,120 --> 00:03:53,790 Thank you. 78 00:03:53,790 --> 00:03:54,290 Thank you. 79 00:03:59,380 --> 00:04:01,810 And that was a few lines of MATLAB, right? 80 00:04:01,810 --> 00:04:03,250 Same thing can be done in Python. 81 00:04:06,080 --> 00:04:07,625 All right, we continue. 82 00:04:12,820 --> 00:04:16,149 So we're going to talk today about spectral content 83 00:04:16,149 --> 00:04:19,450 of signals after having spent some time on the frequency 84 00:04:19,450 --> 00:04:21,579 domain-- 85 00:04:21,579 --> 00:04:23,680 the frequency response of LTI systems. 86 00:04:23,680 --> 00:04:24,840 Let's get this up here. 87 00:04:38,560 --> 00:04:40,480 So we've talked about frequency response. 88 00:04:40,480 --> 00:04:42,490 You've seen the definition. 89 00:04:42,490 --> 00:04:46,212 If I give you the unit output response of a system over here, 90 00:04:46,212 --> 00:04:48,170 you know how to compute the frequency response. 91 00:04:48,170 --> 00:04:50,378 And then we went through how you can go the other way 92 00:04:50,378 --> 00:04:53,470 with the DTFT to compute the frequency 93 00:04:53,470 --> 00:04:57,520 response and the inverse DTFT to compute the time domain signal 94 00:04:57,520 --> 00:04:59,230 from the frequency response. 95 00:04:59,230 --> 00:05:02,080 And what we said last time was that 96 00:05:02,080 --> 00:05:04,977 what you're doing with a unit sample response, 97 00:05:04,977 --> 00:05:06,560 you could actually do with any signal. 98 00:05:06,560 --> 00:05:09,610 So you can take any signal x sub n, 99 00:05:09,610 --> 00:05:12,430 compute from it the DTFT, the Discrete Time Fourier 100 00:05:12,430 --> 00:05:15,160 Transform, with the same formula. 101 00:05:15,160 --> 00:05:17,470 What you get is this object that can be used then 102 00:05:17,470 --> 00:05:19,570 to reconstruct the signal-- 103 00:05:19,570 --> 00:05:21,203 again, the same formula, all right? 104 00:05:21,203 --> 00:05:22,870 So there's just a change of perspective. 105 00:05:22,870 --> 00:05:26,320 There's nothing different here. 106 00:05:26,320 --> 00:05:28,390 The key observation now, though, is 107 00:05:28,390 --> 00:05:33,620 that the formula on the left, the inverse DTFT, 108 00:05:33,620 --> 00:05:37,910 is actually allowing us to represent x sub n, the time 109 00:05:37,910 --> 00:05:41,840 domain signal, as a weighted combination of exponentials 110 00:05:41,840 --> 00:05:42,560 of this type. 111 00:05:42,560 --> 00:05:45,157 And the reason that's important is 112 00:05:45,157 --> 00:05:47,240 that we know how to deal with signals of that type 113 00:05:47,240 --> 00:05:49,430 very easily. 114 00:05:49,430 --> 00:05:52,670 We already know that if you have e 115 00:05:52,670 --> 00:05:56,090 to the j omega n going into a system with frequency 116 00:05:56,090 --> 00:05:57,410 response-- 117 00:05:57,410 --> 00:06:02,420 h omega, so an LTI system with that frequency response. 118 00:06:02,420 --> 00:06:05,390 Let me make this a specific frequency omega 0. 119 00:06:05,390 --> 00:06:14,230 What comes out is the frequency response evaluated at omega 0, 120 00:06:14,230 --> 00:06:15,440 multiplying what went in. 121 00:06:15,440 --> 00:06:15,940 All right? 122 00:06:15,940 --> 00:06:17,990 Nothing more complicated than that. 123 00:06:17,990 --> 00:06:21,370 So what now if you had an x sub n 124 00:06:21,370 --> 00:06:24,730 going in that was a weighted combination of terms 125 00:06:24,730 --> 00:06:25,695 of this type? 126 00:06:25,695 --> 00:06:27,820 And I'm going to take a weighted combination that's 127 00:06:27,820 --> 00:06:28,780 actually a continuum. 128 00:06:28,780 --> 00:06:30,820 It's not just a finite number of terms. 129 00:06:30,820 --> 00:06:33,610 I'm going to actually take this particular weighted 130 00:06:33,610 --> 00:06:34,330 combination. 131 00:06:34,330 --> 00:06:37,600 So over some interval of length 2 pi, 132 00:06:37,600 --> 00:06:42,730 I take a weighted combination like this. 133 00:06:45,400 --> 00:06:48,410 So this is going over all frequencies in our interval. 134 00:06:48,410 --> 00:06:49,690 Let's take minus pi to pi. 135 00:06:49,690 --> 00:06:52,480 I might as well write in minus pi to pi-- 136 00:06:52,480 --> 00:06:53,920 keep it explicit. 137 00:06:57,690 --> 00:07:01,020 You can think of this as being approximated 138 00:07:01,020 --> 00:07:03,040 by a sum in the usual fashion. 139 00:07:03,040 --> 00:07:04,780 I'm not going to write this out. 140 00:07:04,780 --> 00:07:09,180 But we know how to approximate integrals by sums. 141 00:07:09,180 --> 00:07:11,950 What the sum will have is, for instance, 142 00:07:11,950 --> 00:07:15,720 a typical term of the type e to the j omega 0n. 143 00:07:15,720 --> 00:07:23,490 And the weight that multiplies it will be x omega 0 d omega, 144 00:07:23,490 --> 00:07:24,510 right? 145 00:07:24,510 --> 00:07:28,040 So we think of x omega 0 d omega as being 146 00:07:28,040 --> 00:07:32,760 the amount of the exponential at frequency omega 0 147 00:07:32,760 --> 00:07:34,810 that's in x sub n. 148 00:07:34,810 --> 00:07:38,400 So here is a representation of how the signal is made up. 149 00:07:38,400 --> 00:07:42,240 So then what would you say is the output of the system? 150 00:07:42,240 --> 00:07:45,930 If that's the input, and this is an LTI system, 151 00:07:45,930 --> 00:07:48,380 what's the output going to be? 152 00:07:48,380 --> 00:07:49,020 Any ideas? 153 00:07:54,450 --> 00:07:56,612 Somebody? 154 00:07:56,612 --> 00:07:57,570 I thought I saw a hand. 155 00:08:01,620 --> 00:08:02,280 No ideas? 156 00:08:07,140 --> 00:08:15,210 What if instead of this, I had a1 e to the j omega 1n plus a2 157 00:08:15,210 --> 00:08:17,670 e to the j omega 2n going in? 158 00:08:20,420 --> 00:08:21,538 Suppose that had gone in? 159 00:08:21,538 --> 00:08:22,580 What would be coming out? 160 00:08:25,973 --> 00:08:27,890 Folks, we're just two weeks from a quiz, here. 161 00:08:27,890 --> 00:08:28,640 Yeah? 162 00:08:28,640 --> 00:08:30,880 STUDENT: [INAUDIBLE] 163 00:08:30,880 --> 00:08:32,630 PROFESSOR 1: So can you tell me explicitly 164 00:08:32,630 --> 00:08:34,679 what I would get in this case? 165 00:08:34,679 --> 00:08:41,083 If this was x sub n, then y of n would be? 166 00:08:41,083 --> 00:08:43,860 STUDENT: [INAUDIBLE] 167 00:08:43,860 --> 00:08:46,010 PROFESSOR 1: a1h. 168 00:08:46,010 --> 00:08:47,513 STUDENT: [INAUDIBLE] 169 00:08:47,513 --> 00:08:48,930 PROFESSOR 1: Is it just omega, or? 170 00:08:48,930 --> 00:08:49,680 STUDENT: It's omega 1. 171 00:08:49,680 --> 00:08:50,847 PROFESSOR 1: Omega 1, right? 172 00:08:50,847 --> 00:08:54,360 It's the frequency response evaluated at the frequency 173 00:08:54,360 --> 00:08:57,030 that you're interested in, and then 174 00:08:57,030 --> 00:09:03,360 e to the j omega 1n, and then the response to the other term. 175 00:09:03,360 --> 00:09:04,920 This is what superposition is about. 176 00:09:12,450 --> 00:09:15,540 So that wasn't so hard. 177 00:09:15,540 --> 00:09:17,910 What if instead, x sub n is given 178 00:09:17,910 --> 00:09:22,140 by a continuum of such exponentials? 179 00:09:22,140 --> 00:09:25,900 Integrals are essentially linear combinations, 180 00:09:25,900 --> 00:09:29,210 but taken to the limit, where it's not just a finite number. 181 00:09:29,210 --> 00:09:31,770 So I'm not asking for a proof of anything. 182 00:09:31,770 --> 00:09:35,778 I'm asking for your conjecture as to what the answer might be. 183 00:09:35,778 --> 00:09:37,320 This is how math is done, by the way. 184 00:09:37,320 --> 00:09:41,010 You conjecture what the result might be based on gut instinct, 185 00:09:41,010 --> 00:09:43,970 based on well-educated intuition. 186 00:09:43,970 --> 00:09:45,720 And then you go back and construct a proof 187 00:09:45,720 --> 00:09:48,090 and hide all your tracks. 188 00:09:48,090 --> 00:09:49,918 But engineers like to work with intuition, 189 00:09:49,918 --> 00:09:51,210 and often will stick with that. 190 00:09:51,210 --> 00:09:51,710 Yeah? 191 00:09:51,710 --> 00:09:54,801 STUDENT: [INAUDIBLE] 192 00:09:57,710 --> 00:09:59,990 PROFESSOR 1: OK, so what would-- 193 00:09:59,990 --> 00:10:04,130 can you give me the explicit expression? 194 00:10:04,130 --> 00:10:06,260 What's your guess? 195 00:10:06,260 --> 00:10:08,150 This is going in. 196 00:10:08,150 --> 00:10:11,600 It's a weighted combination of exponentials with weights that 197 00:10:11,600 --> 00:10:13,370 are given by x omega d omega. 198 00:10:13,370 --> 00:10:14,404 So-- 199 00:10:14,404 --> 00:10:22,308 STUDENT: So you would get x omega d omega [INAUDIBLE].. 200 00:10:26,260 --> 00:10:28,010 PROFESSOR 1: Times what? 201 00:10:28,010 --> 00:10:31,675 I didn't hear the last piece. 202 00:10:31,675 --> 00:10:33,763 STUDENT: Times [INAUDIBLE]. 203 00:10:33,763 --> 00:10:35,430 PROFESSOR 1: Where's the j omega coming? 204 00:10:35,430 --> 00:10:35,995 What's the j? 205 00:10:35,995 --> 00:10:38,480 I'm missing-- oh, e to the j omega? 206 00:10:38,480 --> 00:10:40,400 All right. 207 00:10:40,400 --> 00:10:43,663 So start again. 208 00:10:43,663 --> 00:10:44,580 Oh, you don't want to? 209 00:10:44,580 --> 00:10:45,230 STUDENT: No. 210 00:10:45,230 --> 00:10:46,437 PROFESSOR 1: OK. 211 00:10:46,437 --> 00:10:47,520 You're on the right track. 212 00:10:47,520 --> 00:10:48,312 You got us started. 213 00:10:48,312 --> 00:10:50,722 Anybody else? 214 00:10:50,722 --> 00:10:52,430 I could show it to you on the next slide, 215 00:10:52,430 --> 00:10:54,430 but that will take all the fun out of it, right? 216 00:10:58,870 --> 00:11:01,708 STUDENT: [INAUDIBLE] 217 00:11:02,197 --> 00:11:04,280 PROFESSOR 1: OK, it's probably that I didn't hear. 218 00:11:04,280 --> 00:11:06,434 Can you tell it to me? 219 00:11:06,434 --> 00:11:10,928 STUDENT: x omega [INAUDIBLE]. 220 00:11:10,928 --> 00:11:17,070 PROFESSOR 1: X omega times h omega or e to the j omega n? 221 00:11:17,070 --> 00:11:19,682 What did you say? 222 00:11:19,682 --> 00:11:22,032 STUDENT: [INAUDIBLE] 223 00:11:22,032 --> 00:11:23,240 PROFESSOR 1: I'm willing to-- 224 00:11:23,240 --> 00:11:31,798 STUDENT: [INAUDIBLE] omega n times [INAUDIBLE].. 225 00:11:31,798 --> 00:11:34,090 PROFESSOR 1: That's still the part that went in, right? 226 00:11:34,090 --> 00:11:36,513 Now it's got to get mapped by a frequency response. 227 00:11:36,513 --> 00:11:37,930 STUDENT: Yeah, that's what I said. 228 00:11:37,930 --> 00:11:39,760 PROFESSOR 1: OK, so maybe you'd said that, 229 00:11:39,760 --> 00:11:40,990 and I didn't hear it. 230 00:11:40,990 --> 00:11:42,550 But now we've got to assemble this 231 00:11:42,550 --> 00:11:44,240 over all possible frequencies. 232 00:11:46,750 --> 00:11:48,550 Well, we have the 1 over 2 pi. 233 00:11:55,180 --> 00:11:59,180 Maybe this was said already, and I just couldn't hear it. 234 00:11:59,180 --> 00:12:00,910 All we're doing is we're saying here's 235 00:12:00,910 --> 00:12:03,250 a weighted combination of exponentials going in. 236 00:12:03,250 --> 00:12:05,890 The weights are given by the x omegas or x omega d 237 00:12:05,890 --> 00:12:07,850 omega, if you want to think of it that way. 238 00:12:07,850 --> 00:12:10,240 So what comes out is the combination 239 00:12:10,240 --> 00:12:11,590 of responses to each of those. 240 00:12:11,590 --> 00:12:13,720 These are the things that go in. 241 00:12:13,720 --> 00:12:16,510 For each frequency, you multiply by the corresponding value 242 00:12:16,510 --> 00:12:19,428 of the frequency response and do that for all the frequencies 243 00:12:19,428 --> 00:12:19,970 of the input. 244 00:12:23,280 --> 00:12:28,513 So this is just applying linearity and superposition. 245 00:12:34,200 --> 00:12:49,850 So if I compare that with this expression, which just tells me 246 00:12:49,850 --> 00:12:55,580 how the time domain signal yn relates to its DTFT, right? 247 00:12:55,580 --> 00:12:57,980 This is just writing the same thing for y 248 00:12:57,980 --> 00:13:00,500 that I wrote for x over there. 249 00:13:00,500 --> 00:13:02,270 If you compare the two, what you discover 250 00:13:02,270 --> 00:13:04,940 about the DTFT of the output? 251 00:13:11,170 --> 00:13:12,750 Where's the weighted combination? 252 00:13:12,750 --> 00:13:15,390 It's whatever multiplies e to the j omega 253 00:13:15,390 --> 00:13:18,730 n in this expression, right? 254 00:13:18,730 --> 00:13:21,180 So it's just going to be h omega x omega. 255 00:13:27,730 --> 00:13:29,770 So we've done a complete analysis 256 00:13:29,770 --> 00:13:32,620 of the input-output response of the system 257 00:13:32,620 --> 00:13:35,950 for essentially an arbitrary input with just 258 00:13:35,950 --> 00:13:37,470 a simple multiplication. 259 00:13:37,470 --> 00:13:38,470 So look what we've done. 260 00:13:38,470 --> 00:13:41,710 We've taken the input that we were given, computed 261 00:13:41,710 --> 00:13:45,220 the DTFT of it, which gives us the spectral content, 262 00:13:45,220 --> 00:13:48,130 and I'll spend some time giving you intuition for that. 263 00:13:48,130 --> 00:13:51,460 That's the spectral content of the input signal. 264 00:13:51,460 --> 00:13:54,160 What we've discovered is that the spectral content 265 00:13:54,160 --> 00:13:58,390 of the output is the spectral content of the input scale 266 00:13:58,390 --> 00:14:00,430 by the frequency response. 267 00:14:00,430 --> 00:14:02,800 And once you have the spectral content of the output, 268 00:14:02,800 --> 00:14:06,472 you can reconstruct the time domain signal. 269 00:14:06,472 --> 00:14:08,680 So the big difference here is there's no convolution. 270 00:14:08,680 --> 00:14:10,720 You're just doing a multiplication. 271 00:14:10,720 --> 00:14:22,330 Instead of doing y of n equals h convolved with xn, 272 00:14:22,330 --> 00:14:23,840 we're just doing a multiplication. 273 00:14:23,840 --> 00:14:26,620 So once again, you see the convolution of the time domain 274 00:14:26,620 --> 00:14:29,470 maps to multiplication in the frequency domain. 275 00:14:33,220 --> 00:14:37,330 All right, so we'll build up to the story again. 276 00:14:37,330 --> 00:14:41,830 Let's get some intuition for spectral content. 277 00:14:41,830 --> 00:14:45,550 And let's take a particular example. 278 00:14:45,550 --> 00:14:55,440 So suppose I have an x of n that's a one-sided exponential. 279 00:14:55,440 --> 00:14:58,185 You've probably done things like this in recitation already. 280 00:15:01,580 --> 00:15:05,090 So this is a signal that starts at time 0 281 00:15:05,090 --> 00:15:09,160 and then starts at the value 1 and halves it each time. 282 00:15:09,160 --> 00:15:12,650 So it's a discrete time exponential. 283 00:15:12,650 --> 00:15:18,410 And so what's the DTFT? 284 00:15:18,410 --> 00:15:23,240 Well, you're going to some from m equals 285 00:15:23,240 --> 00:15:24,990 minus infinity to infinity. 286 00:15:24,990 --> 00:15:29,630 But actually, this only exists for positive-- 287 00:15:29,630 --> 00:15:30,730 for non-negative time. 288 00:15:30,730 --> 00:15:34,220 So you're going to get 0.5 to the n. 289 00:15:36,920 --> 00:15:38,217 Or let's keep it at m. 290 00:15:41,750 --> 00:15:43,296 That's just the definition. 291 00:15:47,340 --> 00:15:49,990 Isn't it the definition? 292 00:15:49,990 --> 00:15:51,726 I'll just use the definition. 293 00:15:55,450 --> 00:16:01,337 So you can now sum an infinite series here. 294 00:16:01,337 --> 00:16:02,170 And what do you get? 295 00:16:02,170 --> 00:16:08,530 You get 1 over 1 minus 0.5e to the minus j omega. 296 00:16:08,530 --> 00:16:14,080 This is just summing a geometric series because each term here 297 00:16:14,080 --> 00:16:18,730 follows from the previous one by multiplying by the factor 0.5e 298 00:16:18,730 --> 00:16:22,270 to the minus j omega. 299 00:16:22,270 --> 00:16:26,240 So it's a geometric series with that ratio. 300 00:16:26,240 --> 00:16:28,360 And so this is what the sum works out to be. 301 00:16:31,590 --> 00:16:33,930 So if you wanted to figure out the spectral content, 302 00:16:33,930 --> 00:16:36,090 you first compute the DTFT. 303 00:16:39,193 --> 00:16:41,610 And then the most helpful way to get a feel for the signal 304 00:16:41,610 --> 00:16:43,890 is to look at the magnitude of the DTFT. 305 00:16:43,890 --> 00:16:46,745 And that's what's actually plotted in this case. 306 00:16:46,745 --> 00:16:48,120 This is taken from somewhere that 307 00:16:48,120 --> 00:16:49,530 used slightly different notation, 308 00:16:49,530 --> 00:16:51,900 but we'll talk through it. 309 00:16:51,900 --> 00:16:54,650 What happened to the top of my slide, here? 310 00:16:54,650 --> 00:16:56,400 OK, it doesn't matter. 311 00:16:56,400 --> 00:17:00,300 What I've plotted here is the magnitude of x 312 00:17:00,300 --> 00:17:02,790 and the phase of x. 313 00:17:02,790 --> 00:17:06,060 To get that, you'll actually have to convert this 314 00:17:06,060 --> 00:17:07,589 to magnitude in angle form. 315 00:17:07,589 --> 00:17:08,859 I'm not doing that for you. 316 00:17:08,859 --> 00:17:11,160 I'm assuming you've had practice or will get practice 317 00:17:11,160 --> 00:17:12,839 in recitation. 318 00:17:12,839 --> 00:17:15,450 But the result of that is a magnitude 319 00:17:15,450 --> 00:17:18,300 that looks like this and a phase that looks like this. 320 00:17:18,300 --> 00:17:21,869 The horizontal scale, just to make the point that this 321 00:17:21,869 --> 00:17:24,480 is a periodic object, just like frequency response, 322 00:17:24,480 --> 00:17:28,980 it actually goes from minus 4 pi to plus 4 pi. 323 00:17:28,980 --> 00:17:33,270 But the interval of interest is really just minus pi 324 00:17:33,270 --> 00:17:34,450 to plus pi. 325 00:17:34,450 --> 00:17:34,950 All right? 326 00:17:34,950 --> 00:17:36,325 So it's just that central portion 327 00:17:36,325 --> 00:17:40,410 that's of interest-- similarly here, minus pi to plus pi. 328 00:17:40,410 --> 00:17:42,632 And then it replicates periodically outside of that. 329 00:17:42,632 --> 00:17:45,090 So we didn't really need to show it to you outside of that. 330 00:17:45,090 --> 00:17:46,382 This is just to make the point. 331 00:17:50,200 --> 00:17:52,175 Another thing to observe is that the magnitude 332 00:17:52,175 --> 00:17:54,610 is an even function of frequency, 333 00:17:54,610 --> 00:17:57,170 and the phase is an odd function of frequency. 334 00:17:57,170 --> 00:17:59,440 So these are elementary checks that you should make. 335 00:17:59,440 --> 00:18:01,600 If you get an answer that doesn't 336 00:18:01,600 --> 00:18:04,387 satisfy those properties, you've gone wrong somewhere 337 00:18:04,387 --> 00:18:04,970 along the way. 338 00:18:10,390 --> 00:18:14,230 If you look at the top plot in this set, 339 00:18:14,230 --> 00:18:16,510 the top plot is exactly a signal of that type, 340 00:18:16,510 --> 00:18:18,490 except I've chosen a different number. 341 00:18:18,490 --> 00:18:21,640 Instead of 0.5 to the n times u of n, it's something else. 342 00:18:21,640 --> 00:18:24,220 But here's a geometric-- 343 00:18:24,220 --> 00:18:26,260 sorry, a discrete time exponential or geometric 344 00:18:26,260 --> 00:18:27,590 series. 345 00:18:27,590 --> 00:18:31,210 Now I've just plotted the DTFT from minus pi to pi 346 00:18:31,210 --> 00:18:34,900 to show you what the spectral content of the signal is. 347 00:18:34,900 --> 00:18:36,820 And I'm just plotting the magnitude. 348 00:18:36,820 --> 00:18:38,020 Ignore these labels. 349 00:18:38,020 --> 00:18:39,820 These are the same figures you saw earlier 350 00:18:39,820 --> 00:18:43,660 for little h and big H. I'm using them again, 351 00:18:43,660 --> 00:18:47,950 except now I'm thinking of this as a signal, 352 00:18:47,950 --> 00:18:49,600 and this is its DTFT. 353 00:18:49,600 --> 00:18:54,700 This is a signal, x sub n, and this is its spectral content. 354 00:18:54,700 --> 00:18:58,720 The relationship is exactly what we had with frequency response. 355 00:18:58,720 --> 00:19:02,240 So where is the spectral content concentrated? 356 00:19:02,240 --> 00:19:04,180 Is it at low frequencies or high frequencies 357 00:19:04,180 --> 00:19:08,840 or intermediate frequencies for this first example? 358 00:19:08,840 --> 00:19:10,550 Concentrated around 0, so you'd say 359 00:19:10,550 --> 00:19:12,860 it's concentrated at low frequencies. 360 00:19:12,860 --> 00:19:14,870 There is content on all frequencies, 361 00:19:14,870 --> 00:19:17,480 though, so this doesn't dip down to 0 anywhere else. 362 00:19:17,480 --> 00:19:20,720 You have to assemble a combination of sinusoids 363 00:19:20,720 --> 00:19:24,260 at all frequencies to construct this signal here. 364 00:19:27,900 --> 00:19:29,640 Here is a signal. 365 00:19:29,640 --> 00:19:31,470 I've had to change the horizontal scale 366 00:19:31,470 --> 00:19:34,170 because this is a signal that evolves much more slowly. 367 00:19:34,170 --> 00:19:37,590 Again, I can ask what's the spectral content of it? 368 00:19:37,590 --> 00:19:42,150 I get something which has a peak near 0 frequency. 369 00:19:42,150 --> 00:19:44,100 So it's only got low frequencies in it 370 00:19:44,100 --> 00:19:49,530 and has very little high frequency content. 371 00:19:49,530 --> 00:19:54,090 You can also start to develop ideas for how fast-- 372 00:19:54,090 --> 00:19:56,560 how high a frequency you ought to expect to see here. 373 00:19:56,560 --> 00:19:58,980 So for instance, what's the fastest wiggle 374 00:19:58,980 --> 00:20:02,190 that you see in this signal? 375 00:20:02,190 --> 00:20:04,440 About how long does it take to-- 376 00:20:04,440 --> 00:20:06,840 if you were thinking of underlying sinusoids, 377 00:20:06,840 --> 00:20:10,600 what's the fastest wiggling you're seeing over here? 378 00:20:10,600 --> 00:20:15,180 Well, to my eye, this kind of rises and curves within about 379 00:20:15,180 --> 00:20:17,310 18 or 20 samples, right? 380 00:20:17,310 --> 00:20:20,850 So this might be a half period of an underlying sinusoid. 381 00:20:20,850 --> 00:20:24,983 So if I thought of the period as being-- 382 00:20:24,983 --> 00:20:26,400 these are just rough calculations. 383 00:20:26,400 --> 00:20:32,040 But it helps you understand what we mean by spectral content 384 00:20:32,040 --> 00:20:35,820 and helps you as a way of checking answers. 385 00:20:35,820 --> 00:20:36,450 But let's see. 386 00:20:36,450 --> 00:20:44,490 If I said 18 was approximately a half period of an underlying 387 00:20:44,490 --> 00:20:48,480 sinusoid, and I don't see anything faster than that, 388 00:20:48,480 --> 00:20:51,690 period is 2pi over omega 0. 389 00:20:51,690 --> 00:20:54,800 So I'm saying that's approximately 18. 390 00:20:54,800 --> 00:20:57,110 So the frequency that I expect to see, 391 00:20:57,110 --> 00:21:04,190 the fastest frequency there, is 2pi over 18 or pi over 9. 392 00:21:07,430 --> 00:21:10,370 Is that roughly consistent with what we're seeing there? 393 00:21:10,370 --> 00:21:13,370 Does the spectral content drop off? 394 00:21:13,370 --> 00:21:15,420 Well, here is pi over 4. 395 00:21:15,420 --> 00:21:17,630 Here's pi over 8. 396 00:21:17,630 --> 00:21:19,760 Somewhere on pi over 9, we've run out 397 00:21:19,760 --> 00:21:22,850 of underlying components. 398 00:21:22,850 --> 00:21:24,920 It's because the frequency content 399 00:21:24,920 --> 00:21:28,820 is limited to that range that the associated signal doesn't 400 00:21:28,820 --> 00:21:30,200 wiggle any faster than this. 401 00:21:32,990 --> 00:21:35,450 Here's another example-- a signal 402 00:21:35,450 --> 00:21:36,990 that actually-- well, in this case, 403 00:21:36,990 --> 00:21:40,880 it seems to have some fairly regular periodicity to it, 404 00:21:40,880 --> 00:21:42,710 and then it damps out. 405 00:21:42,710 --> 00:21:44,935 By the way, in all these cases, I'm assuming-- 406 00:21:44,935 --> 00:21:46,310 actually, in all these cases, I'm 407 00:21:46,310 --> 00:21:48,800 assuming the signal's identically 0 outside-- 408 00:21:48,800 --> 00:21:52,250 outside of what I've shown you here. 409 00:21:52,250 --> 00:21:55,250 If you take the DTFT of this, you 410 00:21:55,250 --> 00:21:58,550 find that the spectral content is what-- low frequency, 411 00:21:58,550 --> 00:22:01,978 mid frequency, high frequency? 412 00:22:01,978 --> 00:22:02,882 STUDENT: Mid? 413 00:22:02,882 --> 00:22:04,630 PROFESSOR 1: Mid frequency right? 414 00:22:04,630 --> 00:22:06,730 Because this is low frequency. 415 00:22:06,730 --> 00:22:07,780 Here is 0. 416 00:22:07,780 --> 00:22:09,340 This is high frequency. 417 00:22:09,340 --> 00:22:11,440 This is just a reflection on the left side. 418 00:22:11,440 --> 00:22:14,390 So at some intermediate frequency, 419 00:22:14,390 --> 00:22:16,630 there's a peak in the spectral content. 420 00:22:16,630 --> 00:22:19,090 And again, you can go through the rough calculation 421 00:22:19,090 --> 00:22:20,150 I just made. 422 00:22:20,150 --> 00:22:22,730 So we see some oscillation here. 423 00:22:22,730 --> 00:22:23,230 Let's see. 424 00:22:23,230 --> 00:22:24,313 Let's estimate the period. 425 00:22:24,313 --> 00:22:27,070 That's 1, 2, 3, 4, 5. 426 00:22:27,070 --> 00:22:32,110 Let's say it's about a period of 5 for those oscillations. 427 00:22:32,110 --> 00:22:37,800 So 2 pi over omega 0 is approximately 5. 428 00:22:37,800 --> 00:22:42,370 So omega 0 is approximately 2 over 5 pi. 429 00:22:46,780 --> 00:22:52,660 So we expect to see a spectral peak somewhere around 0.4 pi. 430 00:22:52,660 --> 00:22:54,100 Here's 0.25 pi. 431 00:22:54,100 --> 00:22:55,960 Here's 0.5 pi. 432 00:22:55,960 --> 00:22:57,910 We're about right. 433 00:22:57,910 --> 00:22:59,240 So make these sorts of checks. 434 00:22:59,240 --> 00:23:00,291 Yeah, question? 435 00:23:00,291 --> 00:23:01,874 STUDENT: So for that first calculation 436 00:23:01,874 --> 00:23:03,362 with the [INAUDIBLE]. 437 00:23:08,102 --> 00:23:09,060 PROFESSOR 1: Oh, sorry. 438 00:23:09,060 --> 00:23:12,630 Yeah, I should have done twice this, right? 439 00:23:12,630 --> 00:23:14,790 What I did was estimate-- thanks for catching that. 440 00:23:14,790 --> 00:23:17,920 I estimated at about 18 is the half period. 441 00:23:17,920 --> 00:23:19,920 And so the period I should have had-- 442 00:23:19,920 --> 00:23:20,830 36 here. 443 00:23:20,830 --> 00:23:22,170 Good. 444 00:23:22,170 --> 00:23:27,040 This is all ballpark, of course, but no reason 445 00:23:27,040 --> 00:23:29,740 to make it worse than it has to be. 446 00:23:33,810 --> 00:23:35,130 Vibrating, right? 447 00:23:37,660 --> 00:23:41,340 And that's not exactly where it sits down there, 448 00:23:41,340 --> 00:23:43,650 but it's in the right region. 449 00:23:52,530 --> 00:23:55,200 Now here's the part that we've already seen on the board. 450 00:23:55,200 --> 00:23:57,750 Once you know the spectral content of the input, 451 00:23:57,750 --> 00:24:00,300 you can assemble-- or you can think of the time domain signal 452 00:24:00,300 --> 00:24:02,880 as being made up of those components. 453 00:24:02,880 --> 00:24:04,980 Correspondingly, that's what the output is. 454 00:24:04,980 --> 00:24:06,750 And all we're doing is invoking the fact 455 00:24:06,750 --> 00:24:09,480 that this is an LTI system for which we 456 00:24:09,480 --> 00:24:11,370 know the frequency response gives us 457 00:24:11,370 --> 00:24:14,460 the output for exponential inputs. 458 00:24:14,460 --> 00:24:16,710 And then we compare that with what 459 00:24:16,710 --> 00:24:20,100 we expect to be seeing for the DTFT of y, 460 00:24:20,100 --> 00:24:23,760 and we make this conclusion. 461 00:24:23,760 --> 00:24:27,330 So this is exactly what we had earlier. 462 00:24:30,070 --> 00:24:35,505 One thing to keep in mind, the DTFT, the frequency response, 463 00:24:35,505 --> 00:24:36,420 the DTFT-- 464 00:24:36,420 --> 00:24:40,150 these are all complex functions of omega in general. 465 00:24:40,150 --> 00:24:42,960 Each of them will have a magnitude and an angle. 466 00:24:42,960 --> 00:24:46,620 So make sure you understand why the magnitude of y 467 00:24:46,620 --> 00:24:49,920 is the product of the magnitudes of h and x 468 00:24:49,920 --> 00:24:53,740 and why the angle of y is the sum of the individual angles, 469 00:24:53,740 --> 00:24:54,240 all right? 470 00:24:57,280 --> 00:25:04,940 It's basically the fact that for a complex number c, 471 00:25:04,940 --> 00:25:06,950 you can write it as the magnitude times e 472 00:25:06,950 --> 00:25:09,610 to the j angle. 473 00:25:09,610 --> 00:25:11,560 So that's really what's being invoked there. 474 00:25:18,680 --> 00:25:23,000 So really, what the story is about-- we've done 475 00:25:23,000 --> 00:25:24,320 a lot of math along the way. 476 00:25:24,320 --> 00:25:26,690 But this is really the story. 477 00:25:26,690 --> 00:25:29,540 And I've only exposed a little part of it 478 00:25:29,540 --> 00:25:31,820 for you because I've only dealt with DT signals. 479 00:25:31,820 --> 00:25:34,490 But the same thing holds for continuous time signals. 480 00:25:34,490 --> 00:25:36,560 A huge class of such signals can be 481 00:25:36,560 --> 00:25:39,920 written as linear combinations of sinusoids. 482 00:25:39,920 --> 00:25:41,540 And when I say "linear combination," 483 00:25:41,540 --> 00:25:45,890 it could be a combination of a discrete set, finite 484 00:25:45,890 --> 00:25:46,790 or infinite. 485 00:25:46,790 --> 00:25:49,520 Or it could be a continuous combination of exponentials 486 00:25:49,520 --> 00:25:52,730 or sinusoids under an integral sign, but the idea is the same. 487 00:25:52,730 --> 00:25:57,560 If you've done 1803, you've seen this kind of thing happening, 488 00:25:57,560 --> 00:26:00,500 at least for periodic signals. 489 00:26:00,500 --> 00:26:02,480 And then the other piece of what we rely on 490 00:26:02,480 --> 00:26:07,010 is that LTI systems are very easy to understand in terms 491 00:26:07,010 --> 00:26:08,660 of their action on sinusoids. 492 00:26:08,660 --> 00:26:11,000 So once you put these two pieces together, 493 00:26:11,000 --> 00:26:14,000 you've got a very powerful way to analyze LTI systems. 494 00:26:19,080 --> 00:26:21,510 So just to go back to the kind of example 495 00:26:21,510 --> 00:26:23,280 I had last time, in, which you'll be-- 496 00:26:23,280 --> 00:26:26,640 or you're already dealing with in the lab-- 497 00:26:26,640 --> 00:26:29,920 we're talking about an audio channel, for instance. 498 00:26:29,920 --> 00:26:34,030 The frequency response, in this case, is the magnitude. 499 00:26:34,030 --> 00:26:36,650 Some characteristic here-- this is a bit of a cartoon. 500 00:26:36,650 --> 00:26:41,650 But let me show you more typical experimental plots 501 00:26:41,650 --> 00:26:42,640 of frequency response. 502 00:26:42,640 --> 00:26:45,040 This is frequency response magnitude. 503 00:26:45,040 --> 00:26:48,250 In most of these plots, people don't show you the phase. 504 00:26:48,250 --> 00:26:50,500 Part of the reason is, or maybe the major reason 505 00:26:50,500 --> 00:26:52,870 is, that for audio, the ear is not 506 00:26:52,870 --> 00:26:54,632 all that sensitive to phase. 507 00:26:54,632 --> 00:26:56,590 If we were doing the analogous thing for video, 508 00:26:56,590 --> 00:26:58,660 then you'd be very concerned about phase. 509 00:26:58,660 --> 00:27:00,220 But in audio characteristics, people 510 00:27:00,220 --> 00:27:02,530 will typically only show you the magnitude 511 00:27:02,530 --> 00:27:04,180 because the phase distortions aren't 512 00:27:04,180 --> 00:27:07,600 picked up quite that readily. 513 00:27:07,600 --> 00:27:09,160 So here is three speakers. 514 00:27:09,160 --> 00:27:13,450 If you look on the site there, you'll see many more tested. 515 00:27:13,450 --> 00:27:16,510 This is the frequency range. 516 00:27:16,510 --> 00:27:19,570 Now, I should make some comments about that. 517 00:27:19,570 --> 00:27:24,040 We're talking about doing minus pi to pi. 518 00:27:29,100 --> 00:27:34,710 We've been talking about filters with frequency responses 519 00:27:34,710 --> 00:27:38,490 that we show from minus pi to pi. 520 00:27:38,490 --> 00:27:41,020 So for instance, if I had a band pass filter, 521 00:27:41,020 --> 00:27:43,350 it would be something with-- 522 00:27:43,350 --> 00:27:45,830 in the ideal case, something like this, right? 523 00:27:48,530 --> 00:27:50,450 This is because we're writing things 524 00:27:50,450 --> 00:27:54,590 in terms of big omega for a discrete time filter. 525 00:27:54,590 --> 00:27:57,350 These are actually written-- 526 00:27:57,350 --> 00:27:58,910 the scale here is hertz. 527 00:27:58,910 --> 00:28:00,410 And they're talking about the action 528 00:28:00,410 --> 00:28:02,610 on an underlying continuous time signal. 529 00:28:02,610 --> 00:28:06,140 So you actually need a way to go from an underlying continuous 530 00:28:06,140 --> 00:28:12,360 time signal that sampled at a particular sampling rate-- 531 00:28:12,360 --> 00:28:17,075 let's say f of s samples per second-- 532 00:28:21,230 --> 00:28:25,910 to a corresponding omega for the underlying discrete time 533 00:28:25,910 --> 00:28:27,530 sequence. 534 00:28:27,530 --> 00:28:30,170 So the question is, how does fs map to omega? 535 00:28:30,170 --> 00:28:31,910 And I had a slide last time. 536 00:28:31,910 --> 00:28:35,030 I haven't gone through it in detail. 537 00:28:35,030 --> 00:28:36,530 Maybe we'll have you work through it 538 00:28:36,530 --> 00:28:38,250 on a homework problem. 539 00:28:38,250 --> 00:28:42,020 But this is actually the mapping. 540 00:28:42,020 --> 00:28:43,880 If you have a sequence that comes 541 00:28:43,880 --> 00:28:47,450 from an underlying continuous time sinusoid 542 00:28:47,450 --> 00:28:51,420 by sampling at fs samples per second, 543 00:28:51,420 --> 00:28:53,520 and you're doing all your calculations 544 00:28:53,520 --> 00:28:55,470 in the discrete time domain, if you 545 00:28:55,470 --> 00:28:57,720 want to think about what that means for the underlying 546 00:28:57,720 --> 00:29:02,910 continuous time domain, you want to map pi to fs over 2. 547 00:29:02,910 --> 00:29:04,560 It's not the omega that maps to that. 548 00:29:04,560 --> 00:29:05,160 It's the pi. 549 00:29:11,950 --> 00:29:14,020 So for instance, in the lab, I think 550 00:29:14,020 --> 00:29:17,350 you're using 48 kilohertz, for instance, 551 00:29:17,350 --> 00:29:20,860 at least for some part of it, as a sampling rate. 552 00:29:20,860 --> 00:29:22,880 You get a discrete time sequence out of that. 553 00:29:22,880 --> 00:29:26,830 You do various DTFT-type computations-- 554 00:29:26,830 --> 00:29:29,560 spectral content or frequency response. 555 00:29:29,560 --> 00:29:31,930 Then you plot them on this kind of a scale. 556 00:29:31,930 --> 00:29:35,090 If you want to think about what the underlying continuous line 557 00:29:35,090 --> 00:29:41,290 frequency is, well, that's 24 kilohertz in this case, 558 00:29:41,290 --> 00:29:44,980 and minus 24 kilohertz, it's at this end. 559 00:29:47,570 --> 00:29:49,520 So when you're trying to visualize 560 00:29:49,520 --> 00:29:52,910 what this characteristic is telling you about what you're 561 00:29:52,910 --> 00:29:54,470 seeing with a discrete time sequence, 562 00:29:54,470 --> 00:29:55,738 that's really the mapping. 563 00:29:55,738 --> 00:29:57,530 The other thing about these characteristics 564 00:29:57,530 --> 00:30:01,650 is that people only plot the positive frequency part. 565 00:30:01,650 --> 00:30:05,000 So they ignore the negative frequency because 566 00:30:05,000 --> 00:30:08,025 of the symmetries there. 567 00:30:08,025 --> 00:30:10,400 In applications, when they give you a frequency response, 568 00:30:10,400 --> 00:30:12,800 they will typically just give you the positive frequency 569 00:30:12,800 --> 00:30:15,890 part of that. 570 00:30:15,890 --> 00:30:19,340 All right, so this is what we're seeing here. 571 00:30:21,940 --> 00:30:24,070 This is the characteristic of the LTI system 572 00:30:24,070 --> 00:30:27,190 that you're going to be sending signals into. 573 00:30:27,190 --> 00:30:29,357 And then you've got to characterize the signals 574 00:30:29,357 --> 00:30:30,940 that you're going to send through it-- 575 00:30:30,940 --> 00:30:33,190 voice or music or whatever. 576 00:30:33,190 --> 00:30:35,840 And this is a figure I showed you last time. 577 00:30:35,840 --> 00:30:38,350 But basically, you're looking at the spectral content 578 00:30:38,350 --> 00:30:41,170 of the signal of interest and seeing how it matches up 579 00:30:41,170 --> 00:30:44,030 with the channel that you have. 580 00:30:44,030 --> 00:30:45,970 And if you compare with-- 581 00:30:45,970 --> 00:30:48,670 well, let's actually look at the previous case 582 00:30:48,670 --> 00:30:52,280 and get a few landmarks, here. 583 00:30:52,280 --> 00:30:57,770 So let's take the Sony speaker, for instance, down here. 584 00:30:57,770 --> 00:31:00,470 OK, so it's got a fairly flat frequency response 585 00:31:00,470 --> 00:31:01,730 for a range of frequencies. 586 00:31:01,730 --> 00:31:05,570 But you've got to get fairly high up before you get there. 587 00:31:05,570 --> 00:31:11,390 For frequencies lower than about 100 hertz, 588 00:31:11,390 --> 00:31:15,290 this is not doing a very good job of propagating the sound. 589 00:31:15,290 --> 00:31:16,790 The frequency response is measured 590 00:31:16,790 --> 00:31:19,350 by having a microphone at a fixed distance from the speaker 591 00:31:19,350 --> 00:31:21,922 in an anechoic chamber. 592 00:31:21,922 --> 00:31:23,630 And you can see that this one is actually 593 00:31:23,630 --> 00:31:25,047 perhaps the poorest of the-- it is 594 00:31:25,047 --> 00:31:28,010 the poorest of the speakers in that it does it very poorly 595 00:31:28,010 --> 00:31:31,580 with the low-frequency sound. 596 00:31:31,580 --> 00:31:33,260 So for this particular one, you would 597 00:31:33,260 --> 00:31:35,210 hope that the spectral content of what you're 598 00:31:35,210 --> 00:31:37,010 trying to send through the channel 599 00:31:37,010 --> 00:31:40,970 lives somewhere in maybe 300 to-- 600 00:31:40,970 --> 00:31:44,030 300 hertz to 10 kilohertz-- if you want to get it across 601 00:31:44,030 --> 00:31:45,710 the channel-- 602 00:31:45,710 --> 00:31:47,840 from the speaker with high fidelity. 603 00:31:47,840 --> 00:31:50,413 But if you've got low-frequency signal 604 00:31:50,413 --> 00:31:52,580 that you're trying to send, and you use the speaker, 605 00:31:52,580 --> 00:31:54,122 well, you're going to be out of luck. 606 00:31:54,122 --> 00:31:57,830 It's not going to propagate it very well. 607 00:31:57,830 --> 00:31:59,960 So thinking in terms of frequency response 608 00:31:59,960 --> 00:32:02,960 and spectral content is really key to making 609 00:32:02,960 --> 00:32:04,310 sense of a lot of this. 610 00:32:07,130 --> 00:32:09,750 All right, let's get a little more practice with this. 611 00:32:09,750 --> 00:32:13,190 And I just want to show you that once you've 612 00:32:13,190 --> 00:32:15,050 learned how to deal with frequency response, 613 00:32:15,050 --> 00:32:17,570 there's not new stuff that you're going to do 614 00:32:17,570 --> 00:32:19,040 to deal with spectral content. 615 00:32:19,040 --> 00:32:21,470 It's just a change of perspective. 616 00:32:21,470 --> 00:32:22,130 So let's see. 617 00:32:22,130 --> 00:32:25,370 If I asked you for a signal that had its frequency 618 00:32:25,370 --> 00:32:29,122 content uniformly distributed in some finite range, 619 00:32:29,122 --> 00:32:31,580 can someone tell me what that signal is going to look like? 620 00:32:35,930 --> 00:32:39,460 I'm asking you for a signal whose spectral 621 00:32:39,460 --> 00:32:48,460 content is uniformly in some range minus omega 622 00:32:48,460 --> 00:32:49,435 c to plus omega c. 623 00:33:01,210 --> 00:33:03,430 Have you met such a signal before? 624 00:33:06,040 --> 00:33:07,975 Anyone? 625 00:33:07,975 --> 00:33:11,370 STUDENT: [INAUDIBLE] 626 00:33:11,370 --> 00:33:12,171 PROFESSOR 1: Sorry? 627 00:33:12,171 --> 00:33:14,280 STUDENT: [INAUDIBLE] 628 00:33:14,280 --> 00:33:15,540 PROFESSOR 1: It was the unit-- 629 00:33:15,540 --> 00:33:17,790 we've seen the same kind of thing with the unit sample 630 00:33:17,790 --> 00:33:19,910 response to the low pass filter, right? 631 00:33:19,910 --> 00:33:22,590 So if this was a frequency response, 632 00:33:22,590 --> 00:33:24,540 then the associated unit sample response 633 00:33:24,540 --> 00:33:26,350 would be the signal we're talking about. 634 00:33:26,350 --> 00:33:29,880 So remember what that was called? 635 00:33:29,880 --> 00:33:33,480 We called it a sinc function-- 636 00:33:33,480 --> 00:33:36,130 sinc function in time. 637 00:33:36,130 --> 00:33:39,160 So if the spectral content is this in frequency, 638 00:33:39,160 --> 00:33:41,338 then the signal that you're talking about 639 00:33:41,338 --> 00:33:42,630 is going to be a sinc function. 640 00:33:42,630 --> 00:33:46,090 Now you can actually work that out. 641 00:33:46,090 --> 00:33:49,170 You don't have to take my word for it. 642 00:33:49,170 --> 00:33:52,020 You want flat spectral content in the range minus omega 643 00:33:52,020 --> 00:33:53,910 c to plus omega c. 644 00:33:53,910 --> 00:33:58,330 So the signal that you're going to get as a result 645 00:33:58,330 --> 00:34:00,610 you can extract from this computation. 646 00:34:00,610 --> 00:34:03,900 And this is exactly the same function that we saw last time. 647 00:34:08,380 --> 00:34:13,090 So the DTFT does what your eye may not do very well. 648 00:34:13,090 --> 00:34:15,370 If I had just given you the signal 649 00:34:15,370 --> 00:34:17,230 and asked you to take a guess as to what 650 00:34:17,230 --> 00:34:20,980 the spectral content of that is, you're 651 00:34:20,980 --> 00:34:23,960 not very likely to have ended up deducing 652 00:34:23,960 --> 00:34:27,100 that the spectral content is flat in some range and 0 653 00:34:27,100 --> 00:34:28,340 outside of that. 654 00:34:28,340 --> 00:34:32,739 So the DTFT is valuable in actually 655 00:34:32,739 --> 00:34:34,179 doing this analysis for you. 656 00:34:38,710 --> 00:34:43,659 So there is a signal that has flat spectral content. 657 00:34:43,659 --> 00:34:46,389 More examples of a similar type-- 658 00:34:46,389 --> 00:34:48,370 and again, we first encountered these 659 00:34:48,370 --> 00:34:51,949 in the context of unit sample responses and frequency 660 00:34:51,949 --> 00:34:52,580 responses. 661 00:34:52,580 --> 00:34:54,747 But now, I just want to change perspective and think 662 00:34:54,747 --> 00:34:59,215 in terms of time domain signals and their associated DTFTs. 663 00:35:01,810 --> 00:35:03,490 So if we look at the top one there, 664 00:35:03,490 --> 00:35:05,920 this is the case we just saw, except I've 665 00:35:05,920 --> 00:35:08,900 truncated the sinc function. 666 00:35:08,900 --> 00:35:12,657 And so what I get is not the perfectly uniform distribution 667 00:35:12,657 --> 00:35:13,990 of frequencies in some interval. 668 00:35:13,990 --> 00:35:15,950 There's a little bit of a wiggle to it. 669 00:35:15,950 --> 00:35:18,160 But this is essentially the sinc function 670 00:35:18,160 --> 00:35:21,570 and its spectral content. 671 00:35:21,570 --> 00:35:23,610 Here's another signal whose spectral content 672 00:35:23,610 --> 00:35:26,010 is at high frequencies and essentially 673 00:35:26,010 --> 00:35:30,710 0 in the low-frequency range. 674 00:35:30,710 --> 00:35:32,460 What does it look like in the time domain? 675 00:35:32,460 --> 00:35:35,220 Well, you can actually work it out. 676 00:35:35,220 --> 00:35:39,330 And here's what you see-- that this has actually more wiggle 677 00:35:39,330 --> 00:35:42,150 to it than the sinc does. 678 00:35:42,150 --> 00:35:45,270 Alternate samples seem to take opposite signs, at least 679 00:35:45,270 --> 00:35:48,420 of the dominant ones over here, reflecting 680 00:35:48,420 --> 00:35:51,600 the high-frequency content of that signal. 681 00:35:51,600 --> 00:35:54,660 Here's something that's intermediate. 682 00:35:54,660 --> 00:35:56,700 This also has the oscillation in sign, 683 00:35:56,700 --> 00:35:59,220 but it's not necessarily in alternate samples. 684 00:35:59,220 --> 00:36:02,390 It's a little bit more leisurely. 685 00:36:02,390 --> 00:36:05,490 Here's something that has low frequency and high frequency, 686 00:36:05,490 --> 00:36:06,960 but not intermediate frequency. 687 00:36:06,960 --> 00:36:10,470 So you see a component that's rapid wiggling, 688 00:36:10,470 --> 00:36:13,030 but you also see this lower-frequency content 689 00:36:13,030 --> 00:36:13,530 in there. 690 00:36:16,260 --> 00:36:18,150 So this is what the DTFT does for us. 691 00:36:21,090 --> 00:36:23,280 Now there's an issue of how you compute these, 692 00:36:23,280 --> 00:36:56,240 because if you look at the formula for the DTFT, 693 00:36:56,240 --> 00:37:00,590 you could certainly do analytical things 694 00:37:00,590 --> 00:37:02,120 with that expression. 695 00:37:02,120 --> 00:37:04,430 And that's the case that-- 696 00:37:04,430 --> 00:37:07,070 we've treated cases of that type, where you write down 697 00:37:07,070 --> 00:37:10,160 an analytical formula for the DTFT. 698 00:37:10,160 --> 00:37:13,730 And then you do things with that, like plotting. 699 00:37:13,730 --> 00:37:18,170 But if I gave you some numerical sequence here, 700 00:37:18,170 --> 00:37:20,510 there's certain simplifications. 701 00:37:20,510 --> 00:37:23,200 For one thing, you really aren't going 702 00:37:23,200 --> 00:37:27,705 to expect to compute this at a continuum of values of omega. 703 00:37:27,705 --> 00:37:29,080 You're not really going to expect 704 00:37:29,080 --> 00:37:35,440 to construct the values from minus pi to pi 705 00:37:35,440 --> 00:37:38,840 at every real number omega in that interval, right? 706 00:37:38,840 --> 00:37:40,310 That would take you a long time. 707 00:37:40,310 --> 00:37:41,740 So what you're likely to be doing 708 00:37:41,740 --> 00:37:48,520 is asking for what the DTFT is at some grid of points. 709 00:37:53,120 --> 00:37:54,650 So you'll form a little grid. 710 00:37:54,650 --> 00:37:58,198 And it's on that grid of points that you want the DTFT. 711 00:37:58,198 --> 00:37:59,990 That's the only practical thing you can do. 712 00:37:59,990 --> 00:38:03,110 You're not going to compute it at all omega outside 713 00:38:03,110 --> 00:38:05,400 of toy examples like that. 714 00:38:05,400 --> 00:38:09,150 So if you had a numerical sequence collected in the lab, 715 00:38:09,150 --> 00:38:12,170 for instance, this is what you'd be aiming to do. 716 00:38:12,170 --> 00:38:14,990 What's the other thing that's likely to be the case if you've 717 00:38:14,990 --> 00:38:17,180 got a numerical sequence collected in the lab? 718 00:38:21,560 --> 00:38:22,310 Any thoughts here? 719 00:38:26,020 --> 00:38:27,610 It's unlikely that my summation is 720 00:38:27,610 --> 00:38:29,470 going to go from minus infinity to infinity 721 00:38:29,470 --> 00:38:31,345 because I'd be waiting a long time to collect 722 00:38:31,345 --> 00:38:32,470 that signal, right? 723 00:38:32,470 --> 00:38:35,350 So in practice, what we're dealing with 724 00:38:35,350 --> 00:38:38,560 are signals of finite duration, typically assumed 725 00:38:38,560 --> 00:38:40,120 to be 0 outside of that interval, 726 00:38:40,120 --> 00:38:43,660 though you might have reasons in some context for assuming 727 00:38:43,660 --> 00:38:44,888 otherwise. 728 00:38:44,888 --> 00:38:46,930 We're always going to take finite length signals. 729 00:38:46,930 --> 00:38:49,870 So the summation will be over a finite interval. 730 00:38:49,870 --> 00:38:53,050 And we're going to want to compute the DTFT 731 00:38:53,050 --> 00:38:55,090 on a finite group of points. 732 00:38:55,090 --> 00:38:58,840 And that makes for some simplifications. 733 00:38:58,840 --> 00:39:01,630 So let's see here. 734 00:39:01,630 --> 00:39:03,950 You've probably heard people talk of FFT, 735 00:39:03,950 --> 00:39:07,690 or the fast Fourier transform. 736 00:39:07,690 --> 00:39:10,540 The fast Fourier transform is not a new kind of transform, 737 00:39:10,540 --> 00:39:12,530 so the name is a little bit misleading. 738 00:39:12,530 --> 00:39:16,570 It's a good way of computing samples of a DTFT. 739 00:39:16,570 --> 00:39:18,970 So you don't have to learn a new transform. 740 00:39:18,970 --> 00:39:22,270 We're still talking about this object, the DTFT. 741 00:39:22,270 --> 00:39:25,810 The FFT is an efficient way of computing 742 00:39:25,810 --> 00:39:29,170 the DTFT on a grid of points, given 743 00:39:29,170 --> 00:39:30,838 a signal of finite duration. 744 00:39:30,838 --> 00:39:33,130 Now, I've got a lot on this slide, and I hope all of it 745 00:39:33,130 --> 00:39:33,630 is right. 746 00:39:33,630 --> 00:39:38,450 But let me talk you through the basic idea, here. 747 00:39:38,450 --> 00:39:44,600 So we're going to compute the DTFT 748 00:39:44,600 --> 00:39:47,810 on a finite grid of points. 749 00:39:47,810 --> 00:39:50,840 So that's the omega k's that I've shown you over there. 750 00:39:50,840 --> 00:39:52,820 We've only got a finite duration signal. 751 00:39:52,820 --> 00:39:57,320 Let me say that it exists only from 0 to p minus 1. 752 00:39:57,320 --> 00:40:00,000 So the signal is 0 outside of that interval. 753 00:40:00,000 --> 00:40:03,350 And therefore, all the other terms drop out of this. 754 00:40:03,350 --> 00:40:05,660 So there's nothing new in this formula. 755 00:40:05,660 --> 00:40:07,550 This is just acknowledging that I only 756 00:40:07,550 --> 00:40:10,250 want to compute the DTFT at a grid point, 757 00:40:10,250 --> 00:40:13,350 and I only have a finite duration signal. 758 00:40:13,350 --> 00:40:18,480 Now the interesting thing is that if your signal is 759 00:40:18,480 --> 00:40:21,540 0 outside of this interval, well, 760 00:40:21,540 --> 00:40:23,220 that means your signal is completely 761 00:40:23,220 --> 00:40:35,660 specified if xn is known to be non-zero only on the interval, 762 00:40:35,660 --> 00:40:37,370 let's say, 0 to p minus 1. 763 00:40:37,370 --> 00:40:40,470 So that's p values. 764 00:40:40,470 --> 00:40:45,210 Then you would hope that just having p samples of the DTFT 765 00:40:45,210 --> 00:40:47,820 will allow you to go the other way. 766 00:40:47,820 --> 00:40:49,800 We know for sure that if I gave you 767 00:40:49,800 --> 00:40:56,800 the entire DTFT, if I gave you the entire DTFT, 768 00:40:56,800 --> 00:41:07,130 that you could go the other way, because we 769 00:41:07,130 --> 00:41:08,560 have this expression. 770 00:41:12,060 --> 00:41:14,010 If I gave you the entire DTFT, you 771 00:41:14,010 --> 00:41:15,360 would just plug it into here. 772 00:41:21,740 --> 00:41:23,990 And you'd get the time domain signal. 773 00:41:23,990 --> 00:41:26,450 What's interesting, though, as it turns out-- 774 00:41:26,450 --> 00:41:28,040 and you might expect this. 775 00:41:28,040 --> 00:41:31,620 Since your signal takes non-zero values only at p points, 776 00:41:31,620 --> 00:41:34,760 you only need p samples of the DTFT 777 00:41:34,760 --> 00:41:36,680 to get an exact reconstruction. 778 00:41:36,680 --> 00:41:37,980 And here is the formula. 779 00:41:37,980 --> 00:41:40,550 And the derivation is not hard. 780 00:41:40,550 --> 00:41:41,540 I've omitted it here. 781 00:41:41,540 --> 00:41:46,280 It's the same kind of idea that we used in the full case. 782 00:41:46,280 --> 00:41:47,930 But you can actually-- 783 00:41:47,930 --> 00:41:52,250 using the values of the DTFT at these grid points, 784 00:41:52,250 --> 00:41:57,200 you can reconstruct the signal x sub n. 785 00:41:57,200 --> 00:41:59,900 So with these simplifications, you actually 786 00:41:59,900 --> 00:42:03,950 have a simple pair that gets you through the numerics. 787 00:42:03,950 --> 00:42:07,430 If you followed these formulas exactly as they're written, 788 00:42:07,430 --> 00:42:10,310 you'd end up doing work on the order of p squared, 789 00:42:10,310 --> 00:42:12,200 because you see each of these summations 790 00:42:12,200 --> 00:42:14,467 involves taking p products. 791 00:42:14,467 --> 00:42:15,800 And then you've got to sum them. 792 00:42:15,800 --> 00:42:18,537 But you've got to do it at p different frequencies. 793 00:42:18,537 --> 00:42:20,120 And the same thing on the other side-- 794 00:42:20,120 --> 00:42:24,290 you've got to do p products, but you've got to do it p times. 795 00:42:24,290 --> 00:42:26,820 So it's order p squared computation. 796 00:42:26,820 --> 00:42:28,910 The fast Fourier transform is actually 797 00:42:28,910 --> 00:42:32,060 a clever way of using the symmetries associated 798 00:42:32,060 --> 00:42:35,650 with these exponentials to group the computations 799 00:42:35,650 --> 00:42:36,650 and make it much faster. 800 00:42:36,650 --> 00:42:40,310 And you can actually reduce it to order p log p. 801 00:42:40,310 --> 00:42:42,170 So it's a huge simplification. 802 00:42:42,170 --> 00:42:44,960 I've got some illustrative numbers down there. 803 00:42:44,960 --> 00:42:50,750 So the FFT actually is a major reason 804 00:42:50,750 --> 00:42:53,150 for advances in numerical computations, 805 00:42:53,150 --> 00:42:56,103 including signal processing of various kinds-- 806 00:42:56,103 --> 00:42:57,770 the fact that you can get this reduction 807 00:42:57,770 --> 00:43:01,708 from p squared to p log p. 808 00:43:01,708 --> 00:43:03,500 All right, I don't think I need to say much 809 00:43:03,500 --> 00:43:04,550 about the grid of points. 810 00:43:04,550 --> 00:43:10,880 But let's move on to thinking about spectral content 811 00:43:10,880 --> 00:43:12,602 of signals going through channels. 812 00:43:12,602 --> 00:43:14,810 All right, so we'll get closer to signals of the type 813 00:43:14,810 --> 00:43:16,268 that we're interested in, which are 814 00:43:16,268 --> 00:43:20,930 these signals in this case for on-off keying 815 00:43:20,930 --> 00:43:22,853 that are signaling 1's and 0's and that we're 816 00:43:22,853 --> 00:43:24,770 trying to get across a channel-- for instance, 817 00:43:24,770 --> 00:43:26,960 the audio channel. 818 00:43:26,960 --> 00:43:30,590 So this might be a typical finite length sequence. 819 00:43:30,590 --> 00:43:33,410 This particular case, we had chosen 7 samples per bit. 820 00:43:33,410 --> 00:43:35,120 That's why the shortest interval you see 821 00:43:35,120 --> 00:43:38,240 has 7 non-zero bits, there. 822 00:43:38,240 --> 00:43:42,020 Here's the spectral content of the signal. 823 00:43:42,020 --> 00:43:45,830 And actually, I've taken this figure 824 00:43:45,830 --> 00:43:48,080 from an earlier version of the course, where 825 00:43:48,080 --> 00:43:50,520 we talked about the discrete time Fourier series, 826 00:43:50,520 --> 00:43:52,710 not the discrete time Fourier transform. 827 00:43:52,710 --> 00:43:54,770 The discrete time Fourier series turns out 828 00:43:54,770 --> 00:43:59,090 to be something very similar to the formulas 829 00:43:59,090 --> 00:44:01,130 I showed you for the FFT. 830 00:44:01,130 --> 00:44:06,380 So the discrete time Fourier series, 831 00:44:06,380 --> 00:44:11,780 apart from a scale factor, is essentially a story 832 00:44:11,780 --> 00:44:15,650 built around this relationship. 833 00:44:15,650 --> 00:44:18,320 And that's developed in some detail in Section 13.2, 834 00:44:18,320 --> 00:44:20,330 but we're actually bypassing about this term 835 00:44:20,330 --> 00:44:23,870 to try and keep the story simpler. 836 00:44:23,870 --> 00:44:26,240 So when you see these plots, you'll 837 00:44:26,240 --> 00:44:29,600 see on a, at magnitude a of k, that's 838 00:44:29,600 --> 00:44:32,570 a symbol associated with the discrete time Fourier series. 839 00:44:32,570 --> 00:44:34,940 These are actually Fourier coefficients 840 00:44:34,940 --> 00:44:38,810 associated with the periodic replication of this signal 841 00:44:38,810 --> 00:44:39,702 outside. 842 00:44:39,702 --> 00:44:41,910 So it's a discrete time version of the Fourier series 843 00:44:41,910 --> 00:44:43,910 you may have seen in 1803. 844 00:44:43,910 --> 00:44:45,980 All you have to do when you see a plot like this 845 00:44:45,980 --> 00:44:50,600 is think of it as a scaled version of the DTFT samples. 846 00:44:50,600 --> 00:44:54,740 So we're just talking about samples of the DTFT taken 847 00:44:54,740 --> 00:44:56,120 at a grid of points. 848 00:44:56,120 --> 00:44:59,030 The scale factor may be off by a factor of p-- 849 00:44:59,030 --> 00:45:01,070 the length of the signal. 850 00:45:01,070 --> 00:45:04,160 But the shape is entirely told to you here. 851 00:45:04,160 --> 00:45:07,220 So think of this as samples of a DTFT. 852 00:45:07,220 --> 00:45:09,800 We're going from minus pi to plus pi. 853 00:45:12,870 --> 00:45:15,210 What's down at the bottom here is 854 00:45:15,210 --> 00:45:18,990 imagining that you've sent this signal over a channel that 855 00:45:18,990 --> 00:45:22,110 could absorb the entire spectral content of the signal. 856 00:45:22,110 --> 00:45:26,556 So suppose you had a channel whose bandwidth-- 857 00:45:26,556 --> 00:45:38,670 suppose I had a low pass channel whose bandwidth 858 00:45:38,670 --> 00:45:41,610 has got some cutoff. 859 00:45:41,610 --> 00:45:42,540 This is low pass. 860 00:45:46,640 --> 00:45:48,920 Suppose this bandwidth could absorb 861 00:45:48,920 --> 00:45:50,880 the entire spectral content of the signal. 862 00:45:50,880 --> 00:45:56,540 In other words, what I mean is that all these DTFT numbers 863 00:45:56,540 --> 00:45:59,550 that are significant actually fit in under this. 864 00:45:59,550 --> 00:46:02,930 So suppose your channel was such that it didn't attenuate 865 00:46:02,930 --> 00:46:05,520 the DTFT coefficients. 866 00:46:05,520 --> 00:46:08,600 So the spectral content is unmodified when 867 00:46:08,600 --> 00:46:09,920 it gets through the channel. 868 00:46:09,920 --> 00:46:14,370 And so if you resynthesize the signal using this formula 869 00:46:14,370 --> 00:46:17,660 at the receiving end, you'd get back the same thing again, 870 00:46:17,660 --> 00:46:20,525 because the channel's not induced any distortion. 871 00:46:24,250 --> 00:46:26,740 Let me go past this and actually show you 872 00:46:26,740 --> 00:46:35,160 what happens when you start to distort the-- 873 00:46:35,160 --> 00:46:36,360 what goes across. 874 00:46:36,360 --> 00:46:39,180 So here, what we're doing in this succession of experiments 875 00:46:39,180 --> 00:46:43,710 is sending that same signal through a channel 876 00:46:43,710 --> 00:46:47,040 with successively smaller bandwidth. 877 00:46:47,040 --> 00:46:49,800 So in the first case, everything goes through. 878 00:46:49,800 --> 00:46:50,970 There's no distortion. 879 00:46:50,970 --> 00:46:53,040 You get the same thing back again. 880 00:46:53,040 --> 00:46:55,200 In this case, the channel actually 881 00:46:55,200 --> 00:46:57,510 has a cutoff that ends up zeroing out 882 00:46:57,510 --> 00:47:01,500 all spectral content outside of some frequency range. 883 00:47:01,500 --> 00:47:04,105 So it's a low pass channel whose bandwidth 884 00:47:04,105 --> 00:47:06,480 is not enough to take the spectral content of what you're 885 00:47:06,480 --> 00:47:08,110 feeding across. 886 00:47:08,110 --> 00:47:10,420 So what do you expect to happen? 887 00:47:10,420 --> 00:47:13,080 Well, the higher-frequency components of the signal 888 00:47:13,080 --> 00:47:15,160 have been zeroed out. 889 00:47:15,160 --> 00:47:17,500 So what should happen? 890 00:47:17,500 --> 00:47:20,410 You expect the signal to be more rounded because it can't 891 00:47:20,410 --> 00:47:22,520 make these sharp transitions. 892 00:47:22,520 --> 00:47:27,070 It takes high-frequency content to make sharp transitions. 893 00:47:27,070 --> 00:47:31,043 So what happens is you trim the spectral content by sending it 894 00:47:31,043 --> 00:47:32,710 through a channel that's not wide enough 895 00:47:32,710 --> 00:47:35,500 to contain all of the signal is that you 896 00:47:35,500 --> 00:47:38,680 get a more rounded signal at the other end. 897 00:47:38,680 --> 00:47:40,180 So you sent this. 898 00:47:40,180 --> 00:47:41,950 This is what you're receiving. 899 00:47:41,950 --> 00:47:43,990 This is the distortion that the channel 900 00:47:43,990 --> 00:47:45,850 has imposed on your signal. 901 00:47:45,850 --> 00:47:48,760 And you can imagine if you tried to find a place to sample this, 902 00:47:48,760 --> 00:47:51,650 you might run into some trouble. 903 00:47:51,650 --> 00:47:53,570 If you go even more extreme, here 904 00:47:53,570 --> 00:47:56,360 is an even narrower channel. 905 00:47:56,360 --> 00:47:59,060 What comes out is even more rounded than what we had there 906 00:47:59,060 --> 00:48:01,550 because you've taken away more high-frequency components. 907 00:48:01,550 --> 00:48:03,830 The signal just can't wiggle that fast, 908 00:48:03,830 --> 00:48:10,130 so it takes its leisurely time going through its paces here. 909 00:48:10,130 --> 00:48:13,190 And you can imagine that you can be thrown off 910 00:48:13,190 --> 00:48:15,770 when you try and take samples. 911 00:48:15,770 --> 00:48:19,670 This is actually even more evident on the eye diagram, 912 00:48:19,670 --> 00:48:21,440 here. 913 00:48:21,440 --> 00:48:26,030 So these are eye diagrams-- again, the same kind of thing. 914 00:48:26,030 --> 00:48:31,190 As you successively transmit fewer and fewer-- 915 00:48:31,190 --> 00:48:33,500 let's say less and less of the high-frequency content 916 00:48:33,500 --> 00:48:37,100 of the signal, what gets picked up-- 917 00:48:37,100 --> 00:48:39,200 what gets received is a more rounded version 918 00:48:39,200 --> 00:48:40,610 of what was sent in. 919 00:48:40,610 --> 00:48:42,230 And the corresponding eye diagrams 920 00:48:42,230 --> 00:48:44,330 that you construct-- well, at a certain point, 921 00:48:44,330 --> 00:48:45,830 I guess somewhere around here, you'd 922 00:48:45,830 --> 00:48:48,710 be a little nervous about trying to find a place to threshold 923 00:48:48,710 --> 00:48:50,720 and decide on what signal you have. 924 00:48:50,720 --> 00:48:52,100 So this is not a noise issue. 925 00:48:52,100 --> 00:48:53,630 This is a distortion issue. 926 00:48:53,630 --> 00:48:56,300 It's distortion induced by the channel. 927 00:48:56,300 --> 00:48:58,100 And it can all be understood in terms 928 00:48:58,100 --> 00:49:01,040 of what the channel is doing to the spectral content 929 00:49:01,040 --> 00:49:02,420 of the input. 930 00:49:02,420 --> 00:49:06,350 I think we'll continue next time to get more insight into this 931 00:49:06,350 --> 00:49:10,330 and start on the topic of modulation.