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 00:00:56,140 --> 00:00:59,190 PROFESSOR: In this lecture we begin a discussion of the 9 00:00:59,190 --> 00:01:03,860 topic of modulation, which is, among other things, a very 10 00:01:03,860 --> 00:01:06,160 important topic in practical terms. 11 00:01:06,160 --> 00:01:10,400 For example, it forms the cornerstone for many 12 00:01:10,400 --> 00:01:12,280 communication systems. 13 00:01:12,280 --> 00:01:16,290 And also, as we'll see as these lectures go along, a 14 00:01:16,290 --> 00:01:20,380 particular form of modulation referred to as pulse amplitude 15 00:01:20,380 --> 00:01:24,350 modulation, and eventually impulse modulation or impulse 16 00:01:24,350 --> 00:01:28,930 train modulation, forms a very important bridge between 17 00:01:28,930 --> 00:01:33,670 continuous time signals and discrete time signals. 18 00:01:33,670 --> 00:01:37,590 Now in general terms what we mean when we refer to 19 00:01:37,590 --> 00:01:43,280 modulation is the notion of using one signal to vary a 20 00:01:43,280 --> 00:01:46,110 parameter of another signal. 21 00:01:46,110 --> 00:01:51,510 For example, a sinusoidal signal has three parameters, 22 00:01:51,510 --> 00:01:54,520 amplitude, frequency, and phase. 23 00:01:54,520 --> 00:01:59,920 And we could think, for example, of using one signal 24 00:01:59,920 --> 00:02:03,080 to vary, let's say, the amplitude of 25 00:02:03,080 --> 00:02:04,730 a sinusoidal signal. 26 00:02:04,730 --> 00:02:08,259 And what that leads to is a notion, which we'll develop in 27 00:02:08,259 --> 00:02:12,300 some detail, referred to as sinusoidal amplitude 28 00:02:12,300 --> 00:02:17,780 modulation, and would correspond to a sinusoidal 29 00:02:17,780 --> 00:02:22,670 signal, referred to as the carrier, and it's amplitude 30 00:02:22,670 --> 00:02:27,700 being varied on the basis of another signal. 31 00:02:27,700 --> 00:02:31,800 Now alternatively we could think of varying either the 32 00:02:31,800 --> 00:02:36,790 frequency or the phase of a sinusoidal signal, again with 33 00:02:36,790 --> 00:02:38,200 another signal. 34 00:02:38,200 --> 00:02:41,850 And what that leads to is another very important notion, 35 00:02:41,850 --> 00:02:45,480 which is referred to sinusoidal frequency 36 00:02:45,480 --> 00:02:49,850 modulation, where essentially it's the frequency of a 37 00:02:49,850 --> 00:02:54,850 sinusoid that's changing depending on the signal that's 38 00:02:54,850 --> 00:02:59,460 we're using to modulate the sinusoid. 39 00:02:59,460 --> 00:03:03,200 Now sinusoidal amplitude, frequency, and phase 40 00:03:03,200 --> 00:03:07,950 modulation are extremely important topics and ideas in 41 00:03:07,950 --> 00:03:11,570 the context of communication systems. 42 00:03:11,570 --> 00:03:14,800 One of the reasons is that if you want to transmit a signal, 43 00:03:14,800 --> 00:03:17,880 let's say for example a voice signal, the voice signal that 44 00:03:17,880 --> 00:03:19,980 you're listening to now. 45 00:03:19,980 --> 00:03:23,320 If you try to transmit that over long distances, because 46 00:03:23,320 --> 00:03:27,740 of the frequencies involved the medium that you use to 47 00:03:27,740 --> 00:03:31,530 transmit it won't carry it long distances. 48 00:03:31,530 --> 00:03:35,030 The idea then is to essentially take that signal, 49 00:03:35,030 --> 00:03:38,760 like a voice signal, use it to modulate a much 50 00:03:38,760 --> 00:03:41,980 higher-frequency signal, and then transmit that 51 00:03:41,980 --> 00:03:46,510 higher-frequency signal over a medium that essentially can 52 00:03:46,510 --> 00:03:49,060 support long-distance transmission at those 53 00:03:49,060 --> 00:03:50,420 frequencies. 54 00:03:50,420 --> 00:03:53,240 Then at the other end of course, the voice information, 55 00:03:53,240 --> 00:03:57,480 or whatever else the information is, is taken off. 56 00:03:57,480 --> 00:04:00,370 Now also, a notion that that leads to, and we'll be 57 00:04:00,370 --> 00:04:06,140 developing in some detail, is the idea that you can 58 00:04:06,140 --> 00:04:11,940 simultaneously transmit more than one signal by in essence 59 00:04:11,940 --> 00:04:16,000 taking several voice signals or other signals, using them 60 00:04:16,000 --> 00:04:20,279 to modulate either the frequency or amplitude of 61 00:04:20,279 --> 00:04:23,750 sinusoidal signals at different frequencies, adding 62 00:04:23,750 --> 00:04:24,940 all those together-- 63 00:04:24,940 --> 00:04:27,330 that's a process called multiplexing-- 64 00:04:27,330 --> 00:04:31,040 and then at the other end of the transmission system, 65 00:04:31,040 --> 00:04:33,570 taking those sinusoidal signals apart. 66 00:04:33,570 --> 00:04:37,610 And then extracting the envelope or frequency 67 00:04:37,610 --> 00:04:41,910 modulation information to get back to the voice signal or 68 00:04:41,910 --> 00:04:44,630 other information-carrying signal. 69 00:04:44,630 --> 00:04:50,420 So that's one of the very important ways in which 70 00:04:50,420 --> 00:04:54,670 sinusoidal modulation is used in communication systems. 71 00:04:54,670 --> 00:04:59,100 And what we'll see, in particular as we go through 72 00:04:59,100 --> 00:05:04,980 today's lecture, is that sinusoidal amplitude 73 00:05:04,980 --> 00:05:10,630 modulation, follows in a fairly straightforward way 74 00:05:10,630 --> 00:05:14,250 from the properties of the Fourier transform that we've 75 00:05:14,250 --> 00:05:16,920 developed in some of the earlier lectures. 76 00:05:16,920 --> 00:05:22,520 So our focus in today's lecture will be on sinusoidal 77 00:05:22,520 --> 00:05:26,580 amplitude modulation in continuous time. 78 00:05:26,580 --> 00:05:29,740 In the next lecture we'll consider the same set of 79 00:05:29,740 --> 00:05:32,950 notions related to discrete time, and also a concept 80 00:05:32,950 --> 00:05:36,040 referred to as pulse amplitude modulation. 81 00:05:36,040 --> 00:05:39,690 And all of these follow, in a very straightforward way, from 82 00:05:39,690 --> 00:05:43,470 the modulation property for the Fourier transform. 83 00:05:43,470 --> 00:05:47,460 Issues of frequency and phase modulation are a little more 84 00:05:47,460 --> 00:05:48,760 difficult to analyze. 85 00:05:48,760 --> 00:05:52,570 But many of the techniques that we've developed in the 86 00:05:52,570 --> 00:05:57,190 previous lectures also provide important insights into 87 00:05:57,190 --> 00:05:58,930 frequency and phase modulation. 88 00:05:58,930 --> 00:06:04,060 And some of this is developed in more detail in the book. 89 00:06:04,060 --> 00:06:09,050 So what I'd like to do is focus, for now, on the concept 90 00:06:09,050 --> 00:06:12,520 of amplitude modulation. 91 00:06:12,520 --> 00:06:18,270 And as I indicated, there are several kinds of carrier 92 00:06:18,270 --> 00:06:22,010 signals on which the modulation can be 93 00:06:22,010 --> 00:06:23,460 superimposed. 94 00:06:23,460 --> 00:06:27,230 The basic structure for an amplitude modulation system is 95 00:06:27,230 --> 00:06:30,870 one in which there is the modulating signal, let's say 96 00:06:30,870 --> 00:06:34,400 for example, voice, and a carrier signal-- what's 97 00:06:34,400 --> 00:06:36,560 referred to as the carrier. 98 00:06:36,560 --> 00:06:39,550 And then of course the resulting output is the 99 00:06:39,550 --> 00:06:41,640 modulated output. 100 00:06:41,640 --> 00:06:45,890 Now to analyze this, since we have multiplication in the 101 00:06:45,890 --> 00:06:49,350 time domain, we know from the property of the Fourier 102 00:06:49,350 --> 00:06:51,920 transform that we've developed previously-- 103 00:06:51,920 --> 00:06:53,750 the modulation property-- 104 00:06:53,750 --> 00:06:57,900 that multiplication in the time domain corresponds to 105 00:06:57,900 --> 00:07:00,550 convolution in the frequency domain. 106 00:07:00,550 --> 00:07:04,890 And it's this basic property or equation that lets us 107 00:07:04,890 --> 00:07:09,420 analyze, in some detail in fact, the notions of amplitude 108 00:07:09,420 --> 00:07:11,590 modulation. 109 00:07:11,590 --> 00:07:15,240 As we go through this lecture and the next lecture, we'll be 110 00:07:15,240 --> 00:07:18,920 talking, as I indicated, about several different types of 111 00:07:18,920 --> 00:07:20,700 carrier signals. 112 00:07:20,700 --> 00:07:24,910 One is what's referred to as pulse carriers. 113 00:07:24,910 --> 00:07:28,390 And that leads to, among other things, the concept of pulse 114 00:07:28,390 --> 00:07:30,220 amplitude modulation. 115 00:07:30,220 --> 00:07:34,050 That will be deferred until the next lecture. 116 00:07:34,050 --> 00:07:39,720 In today's lecture what I'll focus on is first, the case of 117 00:07:39,720 --> 00:07:43,240 a complex exponential carrier, second, the case 118 00:07:43,240 --> 00:07:45,210 of sinusoidal carrier. 119 00:07:45,210 --> 00:07:48,490 And in fact the complex exponential carrier and 120 00:07:48,490 --> 00:07:51,770 sinusoidal carrier are obviously very closely 121 00:07:51,770 --> 00:07:56,710 related, since the complex exponential carrier is, in 122 00:07:56,710 --> 00:08:00,350 effect, two sinusoidal carriers. 123 00:08:00,350 --> 00:08:04,900 One for the real part and one for the imaginary part. 124 00:08:04,900 --> 00:08:08,700 So let's first begin the discussion of amplitude 125 00:08:08,700 --> 00:08:12,840 modulation by considering a complex exponential carrier, 126 00:08:12,840 --> 00:08:14,900 and then moving on to a discussion of 127 00:08:14,900 --> 00:08:18,490 a sinusoidal carrier. 128 00:08:18,490 --> 00:08:26,930 So the issue then is that we have a signal, x of t. 129 00:08:26,930 --> 00:08:29,060 It's multiplied by a carrier. 130 00:08:29,060 --> 00:08:31,860 And the carrier that we're considering is a carrier 131 00:08:31,860 --> 00:08:39,919 signal, c of t, of the form e to the j omega c t 132 00:08:39,919 --> 00:08:42,030 plus theta sub c. 133 00:08:42,030 --> 00:08:45,430 That's the form of our carrier signal. 134 00:08:45,430 --> 00:08:52,110 And what we can first analyze is what the resulting signal 135 00:08:52,110 --> 00:08:55,870 or spectrum is at the output of the modulator. 136 00:08:55,870 --> 00:08:59,870 Well, we can do that by concentrating on the 137 00:08:59,870 --> 00:09:01,650 modulation property. 138 00:09:01,650 --> 00:09:06,820 And let's consider, just as a general form for a spectrum, 139 00:09:06,820 --> 00:09:14,220 what I've indicated here for the Fourier transform of the 140 00:09:14,220 --> 00:09:18,260 input signal or modulating signal, X of omega. 141 00:09:18,260 --> 00:09:22,650 And so this is intended to represent the 142 00:09:22,650 --> 00:09:25,760 spectrum of x of t. 143 00:09:25,760 --> 00:09:29,140 And then the carrier signal, since it's a single complex 144 00:09:29,140 --> 00:09:35,340 exponential, has a Fourier transform which is an impulse 145 00:09:35,340 --> 00:09:37,300 in the frequency domain. 146 00:09:37,300 --> 00:09:42,450 And the amplitude of the impulse is 2 pi e to the j 147 00:09:42,450 --> 00:09:46,310 theta sub c, where we notice that the complex amplitude 148 00:09:46,310 --> 00:09:50,110 incorporates the phase information. 149 00:09:50,110 --> 00:09:54,690 So now if we multiply in the time domain, we convolve in 150 00:09:54,690 --> 00:09:55,920 the frequency domain. 151 00:09:55,920 --> 00:09:59,850 And as you know, convolving a signal with an impulse just 152 00:09:59,850 --> 00:10:03,380 shifts that signal to the location of the impulse. 153 00:10:03,380 --> 00:10:07,400 And so as a consequence of taking care of various 154 00:10:07,400 --> 00:10:13,250 factors, what we end up with is a spectrum that is centered 155 00:10:13,250 --> 00:10:18,660 at the carrier frequency omega sub c. 156 00:10:18,660 --> 00:10:25,330 So what this says is that if we have a signal, x of t, and 157 00:10:25,330 --> 00:10:31,040 we use it to modulate a complex exponential carrier in 158 00:10:31,040 --> 00:10:34,570 the frequency domain, what we've simply done is to take 159 00:10:34,570 --> 00:10:39,760 the original spectrum and shift it in frequency. 160 00:10:39,760 --> 00:10:43,340 So that what was originally at zero frequency is now centered 161 00:10:43,340 --> 00:10:44,590 around the carrier frequency. 162 00:10:46,960 --> 00:10:50,160 We've now modulated, in effect, to a higher frequency. 163 00:10:50,160 --> 00:10:53,180 Things are happening in a higher-frequency band. 164 00:10:53,180 --> 00:10:57,200 And the next question is, how do we demodulate, or in other 165 00:10:57,200 --> 00:11:02,540 words, how do we get the original signal back? 166 00:11:02,540 --> 00:11:05,210 Of course one way that we can think of doing it, 167 00:11:05,210 --> 00:11:09,790 particularly in the context of this specific carrier, if we 168 00:11:09,790 --> 00:11:14,660 look back at the top equation we have, as the result of the 169 00:11:14,660 --> 00:11:19,470 modulation, x of t times c of t, where c of t is this. 170 00:11:19,470 --> 00:11:24,370 And we could consider, for example, just simply dividing 171 00:11:24,370 --> 00:11:26,530 the modulator output by this. 172 00:11:26,530 --> 00:11:29,790 Or equivalently, taking the modulated output and 173 00:11:29,790 --> 00:11:37,170 multiplying by e to the minus j omega c t plus theta c. 174 00:11:37,170 --> 00:11:40,790 Let's track that through in terms of the spectra. 175 00:11:40,790 --> 00:11:46,020 We have, again, the spectrum of the output of the 176 00:11:46,020 --> 00:11:49,130 modulator, which is the original spectrum shifted up 177 00:11:49,130 --> 00:11:51,530 to the carrier frequency. 178 00:11:51,530 --> 00:11:58,130 We have, below that, the spectrum of e to the minus j 179 00:11:58,130 --> 00:12:01,050 omega c t plus theta c. 180 00:12:01,050 --> 00:12:06,440 And if we now convolve this with this, that results in 181 00:12:06,440 --> 00:12:09,300 simply shifting this spectrum-- except for an issue 182 00:12:09,300 --> 00:12:10,750 of a scale factor-- 183 00:12:10,750 --> 00:12:14,430 shifting this spectrum back down to the origin. 184 00:12:14,430 --> 00:12:17,560 So convolving these two together, the spectrum that we 185 00:12:17,560 --> 00:12:20,740 end up with is that. 186 00:12:20,740 --> 00:12:25,370 So we can track this through in the frequency domain. 187 00:12:25,370 --> 00:12:28,900 In the frequency domain it says shift the spectrum up. 188 00:12:28,900 --> 00:12:32,430 When you want to demodulate, shift the spectrum back down. 189 00:12:32,430 --> 00:12:35,780 And alternatively, we can look at it algebraically in the 190 00:12:35,780 --> 00:12:36,320 time domain. 191 00:12:36,320 --> 00:12:40,010 And what it says is, if you multiply by e to the plus j 192 00:12:40,010 --> 00:12:44,150 omega c t, then when you want to get back, multiply by e to 193 00:12:44,150 --> 00:12:46,920 the minus j omega c t. 194 00:12:46,920 --> 00:12:51,280 Now one question that you could conceivably be asking 195 00:12:51,280 --> 00:12:55,730 is, if we're talking about practical systems and not 196 00:12:55,730 --> 00:13:01,890 simply mathematics, does it make sense in the real world 197 00:13:01,890 --> 00:13:06,530 to consider using a complex exponential carrier? 198 00:13:06,530 --> 00:13:11,300 And the answer to that, in fact, is yes. 199 00:13:11,300 --> 00:13:16,460 That very often in practical systems one considers using a 200 00:13:16,460 --> 00:13:20,640 carrier which in fact is a complex exponential. 201 00:13:20,640 --> 00:13:22,890 Well, a complex exponential is complex. 202 00:13:22,890 --> 00:13:25,440 There's a square root of minus one in there. 203 00:13:25,440 --> 00:13:27,590 And you could ask well, how do we get a square 204 00:13:27,590 --> 00:13:29,170 root of minus one? 205 00:13:29,170 --> 00:13:31,970 And the answer is fairly simple. 206 00:13:31,970 --> 00:13:37,030 Let's look again at the modulator, which we have here. 207 00:13:37,030 --> 00:13:40,980 And in effect, what that says is we want to multiply a 208 00:13:40,980 --> 00:13:45,840 real-valued signal by e to the j omega c t plus theta c. 209 00:13:45,840 --> 00:13:51,030 Now, we can equivalently use Euler's relationship to break 210 00:13:51,030 --> 00:13:54,650 this down into a cosine and sine term. 211 00:13:54,650 --> 00:13:57,660 And so what that means in terms of an implementation, 212 00:13:57,660 --> 00:14:07,140 equivalently, is modulating x of t onto a cosine carrier. 213 00:14:07,140 --> 00:14:10,850 And that then gives us the real part 214 00:14:10,850 --> 00:14:12,830 of the complex output. 215 00:14:12,830 --> 00:14:17,780 And modulating it onto a sinusoidal carrier-- 216 00:14:17,780 --> 00:14:20,340 these two being 90 degrees out of phase-- 217 00:14:20,340 --> 00:14:23,500 and that gives us the imaginary part. 218 00:14:23,500 --> 00:14:27,010 And so in effect, this is the complex signal. 219 00:14:27,010 --> 00:14:30,440 If we just simply think of hanging a tag on here that 220 00:14:30,440 --> 00:14:34,500 says square root of minus 1, or j, and we appropriately 221 00:14:34,500 --> 00:14:38,210 combine complex signals following the rules of complex 222 00:14:38,210 --> 00:14:39,110 arithmetic. 223 00:14:39,110 --> 00:14:42,020 And indeed, that's exactly the way things are done in the 224 00:14:42,020 --> 00:14:43,010 real world. 225 00:14:43,010 --> 00:14:48,660 A complex signal is simply a set of two real signals. 226 00:14:48,660 --> 00:14:54,220 And of course, if we look at the spectra involved, we have 227 00:14:54,220 --> 00:14:56,690 here the real part and the imaginary part 228 00:14:56,690 --> 00:14:59,990 of the complex output. 229 00:14:59,990 --> 00:15:05,330 If we again refer back to the original spectrum, X of omega, 230 00:15:05,330 --> 00:15:10,580 and the modulated spectrum which I show down here, the 231 00:15:10,580 --> 00:15:16,030 original spectrum shifted up to the carrier frequency. 232 00:15:16,030 --> 00:15:20,610 In effect we're building this out of two lines. 233 00:15:20,610 --> 00:15:25,870 One line representing the real part of that. 234 00:15:25,870 --> 00:15:29,610 And the real part in the time domain corresponds to the even 235 00:15:29,610 --> 00:15:31,290 part in the frequency domain. 236 00:15:31,290 --> 00:15:37,680 And so with the output of the cosine modulator, we have a 237 00:15:37,680 --> 00:15:40,750 spectrum that looks like this. 238 00:15:40,750 --> 00:15:47,120 And the output along the imaginary branch has a 239 00:15:47,120 --> 00:15:50,310 spectrum that looks like this. 240 00:15:50,310 --> 00:15:54,690 Recall in the top branch that this, for positive frequencies 241 00:15:54,690 --> 00:15:57,210 was positive, and was positive here. 242 00:15:57,210 --> 00:16:00,260 And so in effect when you add them, this portion of the 243 00:16:00,260 --> 00:16:02,490 spectrum will cancel out. 244 00:16:02,490 --> 00:16:10,360 So in effect, what we're doing is building the complex signal 245 00:16:10,360 --> 00:16:11,850 out of two real signals. 246 00:16:11,850 --> 00:16:16,710 Or we're building the spectrum of the complex signal out of 247 00:16:16,710 --> 00:16:20,745 separate lines that represent the even and the odd parts. 248 00:16:23,540 --> 00:16:29,640 Now, there are lots of applications of amplitude 249 00:16:29,640 --> 00:16:30,910 modulation. 250 00:16:30,910 --> 00:16:33,390 And we'll be seeing a number of these as we go through the 251 00:16:33,390 --> 00:16:34,420 discussion. 252 00:16:34,420 --> 00:16:39,660 What I'd like to do is just indicate briefly one now, 253 00:16:39,660 --> 00:16:44,560 which is an application that in fact surfaces fairly often 254 00:16:44,560 --> 00:16:49,540 in the context of a complex exponential carrier. 255 00:16:49,540 --> 00:16:58,440 And that is the notion of using modulation to permit the 256 00:16:58,440 --> 00:17:03,550 application of a very well designed and implemented 257 00:17:03,550 --> 00:17:08,240 low-pass filter to be used as a band-pass filter and in 258 00:17:08,240 --> 00:17:10,520 fact, as a set of band-pass filters. 259 00:17:10,520 --> 00:17:11,680 And here's the idea. 260 00:17:11,680 --> 00:17:18,230 The idea is if we have a fixed filter-- 261 00:17:18,230 --> 00:17:19,990 let's say we have a signal. 262 00:17:19,990 --> 00:17:24,359 And we want to think of a filter, which we want to move 263 00:17:24,359 --> 00:17:28,640 along the signal, one way to do it is to somehow have 264 00:17:28,640 --> 00:17:30,950 filters that move along the signal. 265 00:17:30,950 --> 00:17:34,870 The other possibility is to keep the filter fixed and let 266 00:17:34,870 --> 00:17:39,250 the signal move in frequency in front of the filter. 267 00:17:39,250 --> 00:17:42,540 Let me be a little more specific. 268 00:17:42,540 --> 00:17:47,860 Suppose that we have a signal, x of t. 269 00:17:47,860 --> 00:17:52,500 And we modulate it with a complex exponential carrier 270 00:17:52,500 --> 00:17:56,490 with a carrier frequency, omega c. 271 00:17:56,490 --> 00:18:01,020 And the output of that is then processed with a low-pass 272 00:18:01,020 --> 00:18:06,910 filter and then we demodulate the result. 273 00:18:06,910 --> 00:18:12,770 Then what we've done is to take the spectrum of the input 274 00:18:12,770 --> 00:18:18,440 signal, shift it, pull out what is now around low 275 00:18:18,440 --> 00:18:22,200 frequencies, and then shift that part of the spectrum back 276 00:18:22,200 --> 00:18:24,190 to where it belongs. 277 00:18:24,190 --> 00:18:28,900 So if we look at that in terms of actually tracking through 278 00:18:28,900 --> 00:18:34,180 the spectra, we would have initially a spectrum for the 279 00:18:34,180 --> 00:18:40,050 original signal, which I show at the top as X of omega. 280 00:18:40,050 --> 00:18:44,750 After modulating or shifting that spectrum up to a center 281 00:18:44,750 --> 00:18:49,530 frequency of omega c, we then have what I indicate here. 282 00:18:49,530 --> 00:18:53,490 And the dotted line corresponds to the pass band 283 00:18:53,490 --> 00:18:55,430 of the low-pass filter. 284 00:18:55,430 --> 00:19:00,590 Well, the result of low-pass filtering rejects all the 285 00:19:00,590 --> 00:19:04,970 spectrum except the part around low frequencies. 286 00:19:04,970 --> 00:19:10,260 And the next step is then to demodulate this. 287 00:19:10,260 --> 00:19:14,640 And so in effect, demodulating will shift this spectrum back 288 00:19:14,640 --> 00:19:17,490 to where it originally came from. 289 00:19:17,490 --> 00:19:22,710 And so that result will be what I show in the final 290 00:19:22,710 --> 00:19:25,680 result, which is here. 291 00:19:25,680 --> 00:19:31,140 And what we can see is that this is equivalent. 292 00:19:31,140 --> 00:19:36,170 If we can look back at the top spectrum, this is equivalent 293 00:19:36,170 --> 00:19:41,070 to having extracted, with a band-pass filter, a section 294 00:19:41,070 --> 00:19:43,860 out of this part of the spectrum. 295 00:19:43,860 --> 00:19:47,840 So in terms of tracking through the spectrum and 296 00:19:47,840 --> 00:19:54,240 looking at the equivalent filtering operation, then what 297 00:19:54,240 --> 00:20:00,470 we accomplished was to pull out this part of the spectrum 298 00:20:00,470 --> 00:20:03,950 using a low-pass filter and modulation. 299 00:20:03,950 --> 00:20:07,770 But equivalently what we implemented was a band-pass 300 00:20:07,770 --> 00:20:10,750 filter as I indicated here. 301 00:20:10,750 --> 00:20:14,030 Now of course, a signal with this spectrum, since the 302 00:20:14,030 --> 00:20:17,030 spectrum is not conjugate symmetric, we know that this 303 00:20:17,030 --> 00:20:20,620 signal does not correspond to a real-valued signal. 304 00:20:20,620 --> 00:20:24,400 Equivalently this filter doesn't correspond to a filter 305 00:20:24,400 --> 00:20:26,990 whose impulse response is real. 306 00:20:26,990 --> 00:20:32,140 If we add another step to this, which is to take the 307 00:20:32,140 --> 00:20:36,290 real part of the output, then by taking the real part of the 308 00:20:36,290 --> 00:20:40,550 output we would be taking the even part of the spectrum 309 00:20:40,550 --> 00:20:43,080 associated with that complex signal. 310 00:20:43,080 --> 00:20:47,860 And the equivalent filter that we would end up with then is 311 00:20:47,860 --> 00:20:52,250 the filter that I indicate at the bottom, which is a 312 00:20:52,250 --> 00:20:54,950 band-pass filter. 313 00:20:54,950 --> 00:21:00,750 Now just to reiterate a point that I made earlier. 314 00:21:00,750 --> 00:21:03,860 A question, of course, is why would you go to this trouble? 315 00:21:03,860 --> 00:21:06,520 Why not just build a band-pass filter? 316 00:21:06,520 --> 00:21:13,300 And one of the reasons is that it's often much easier to 317 00:21:13,300 --> 00:21:17,920 build a fixed filter, a filter with a fixed-center frequency, 318 00:21:17,920 --> 00:21:21,680 for example a low-pass filter, than it is to build a filter 319 00:21:21,680 --> 00:21:25,030 that has variable components in it so that when you vary 320 00:21:25,030 --> 00:21:28,920 them the filter's center frequency shifts around. 321 00:21:28,920 --> 00:21:31,880 Now, if you want to look at the energy in a signal in 322 00:21:31,880 --> 00:21:35,450 different frequency bands, then you'd like to look at it 323 00:21:35,450 --> 00:21:37,660 through different filters. 324 00:21:37,660 --> 00:21:41,390 And so the idea here, which is really the basis for many 325 00:21:41,390 --> 00:21:45,810 spectrum analyzers, is to build a really good quality 326 00:21:45,810 --> 00:21:50,320 low-pass filter and then use modulation, which is often 327 00:21:50,320 --> 00:21:51,260 easier to implement. 328 00:21:51,260 --> 00:21:56,470 Use modulation to shift the signal essentially in front of 329 00:21:56,470 --> 00:21:57,720 the filter. 330 00:22:00,600 --> 00:22:05,600 So we've worked our way through modulation with a 331 00:22:05,600 --> 00:22:08,350 complex exponential carrier. 332 00:22:08,350 --> 00:22:13,090 And what we saw, among other things with a complex 333 00:22:13,090 --> 00:22:17,340 exponential carrier, is that what it 334 00:22:17,340 --> 00:22:23,260 corresponds to is two branches. 335 00:22:23,260 --> 00:22:28,130 One being modulation with a cosine, and the other, 336 00:22:28,130 --> 00:22:29,570 modulation with a sine. 337 00:22:29,570 --> 00:22:34,340 And so in the real world, or in a practical system, 338 00:22:34,340 --> 00:22:37,610 modulation of the complex exponential carrier really 339 00:22:37,610 --> 00:22:41,930 would be accomplished with modulation with a sinusoidal 340 00:22:41,930 --> 00:22:47,330 carrier, and in particular with sinusoidal carriers that 341 00:22:47,330 --> 00:22:50,660 are in quadrature, as it's referred to, or equivalently 342 00:22:50,660 --> 00:22:54,170 90 degrees out of phase. 343 00:22:54,170 --> 00:23:00,750 Well, in fact sinusoidal modulation, in other words, 344 00:23:00,750 --> 00:23:04,970 modulation using only a sinusoidal carrier, very often 345 00:23:04,970 --> 00:23:10,020 is used in its own right not only for generating a complex 346 00:23:10,020 --> 00:23:17,060 exponential carrier, but as a carrier by itself. 347 00:23:17,060 --> 00:23:21,370 Let's look at what the consequences of modulation 348 00:23:21,370 --> 00:23:23,500 with a sinusoidal carrier are. 349 00:23:23,500 --> 00:23:26,830 And in particular work through, again, what the 350 00:23:26,830 --> 00:23:32,410 spectra are and how we get the original signal back again. 351 00:23:32,410 --> 00:23:39,150 So we are talking about a carrier signal which is simply 352 00:23:39,150 --> 00:23:42,920 a sinusoidal signal with some phase. 353 00:23:42,920 --> 00:23:48,650 And of course we can write that as the sum of two complex 354 00:23:48,650 --> 00:23:51,460 exponential signals. 355 00:23:51,460 --> 00:23:56,500 And so now, when we apply the modulation property we have 356 00:23:56,500 --> 00:24:01,190 the original spectrum, which I show here, X of omega. 357 00:24:01,190 --> 00:24:05,340 And that's convolved with the spectrum of the carrier. 358 00:24:05,340 --> 00:24:07,310 And the spectrum of the carrier, in this 359 00:24:07,310 --> 00:24:10,520 case, is two impulses. 360 00:24:10,520 --> 00:24:16,270 One at plus omega c, and one at minus omega c. 361 00:24:16,270 --> 00:24:19,510 And the amplitudes of these incorporate the phase. 362 00:24:19,510 --> 00:24:25,620 And later on in the lecture, and in subsequent lectures, 363 00:24:25,620 --> 00:24:30,500 I'll have a tendency to drop the theta sub c, just to keep 364 00:24:30,500 --> 00:24:33,820 the notation and algebra a little cleaner, but for now 365 00:24:33,820 --> 00:24:35,340 I've incorporated it. 366 00:24:35,340 --> 00:24:41,040 And so now when we apply the modulation property, what we 367 00:24:41,040 --> 00:24:46,270 will do is convolve this spectrum with this spectrum, 368 00:24:46,270 --> 00:24:50,060 and the result is that the spectrum of the original 369 00:24:50,060 --> 00:24:55,690 signal gets replicated at both omega sub c and at 370 00:24:55,690 --> 00:24:57,460 minus omega sub c. 371 00:24:57,460 --> 00:25:00,800 And the resulting spectrum at the output of the modulator, 372 00:25:00,800 --> 00:25:03,540 then, is the spectrum that I show here. 373 00:25:06,940 --> 00:25:09,930 Now the question, of course, is-- 374 00:25:09,930 --> 00:25:12,990 so now what's happened is that with a sinusoidal carrier, 375 00:25:12,990 --> 00:25:16,710 we've moved the spectrum to both plus omega c 376 00:25:16,710 --> 00:25:18,870 and minus omega c. 377 00:25:18,870 --> 00:25:22,570 And now if we want to get the original signal back again, 378 00:25:22,570 --> 00:25:26,480 what we would like to do somehow is move that spectrum 379 00:25:26,480 --> 00:25:29,500 back down to the origin. 380 00:25:29,500 --> 00:25:32,050 Now in the case of a complex exponential, 381 00:25:32,050 --> 00:25:33,020 that was easy to do. 382 00:25:33,020 --> 00:25:35,910 We'd shifted one up, we'd just shift it back down. 383 00:25:35,910 --> 00:25:41,320 Let's see what happens if we attempt to demodulate by again 384 00:25:41,320 --> 00:25:45,960 multiplying by the same sinusoidal carrier. 385 00:25:45,960 --> 00:25:55,690 So let's examine what happens if we now take our modulated 386 00:25:55,690 --> 00:26:03,600 signal and, again, modulate it onto the same sinusoidal 387 00:26:03,600 --> 00:26:08,010 carrier to generate the output w of t. 388 00:26:08,010 --> 00:26:14,340 If we look at the spectra, we have the modulated spectrum 389 00:26:14,340 --> 00:26:16,940 which we had initially. 390 00:26:16,940 --> 00:26:21,300 And we now want to convolve that, again, with the spectrum 391 00:26:21,300 --> 00:26:23,470 of the carrier signal. 392 00:26:23,470 --> 00:26:26,840 The spectrum of the carrier signal, I indicate here. 393 00:26:26,840 --> 00:26:30,350 And if you track through the convolution, which is fairly 394 00:26:30,350 --> 00:26:34,870 straightforward, then what happens as you convolve this 395 00:26:34,870 --> 00:26:43,270 with this is you end up with a composite spectrum, which is 396 00:26:43,270 --> 00:26:48,730 what I've indicated on the bottom curve, and has the 397 00:26:48,730 --> 00:26:52,990 spectrum of the original signal, x of t, replicated in 398 00:26:52,990 --> 00:26:54,220 three places. 399 00:26:54,220 --> 00:26:58,920 One is at minus 2 omega sub c. 400 00:26:58,920 --> 00:27:00,960 One is around the origin. 401 00:27:00,960 --> 00:27:05,980 And one is shifted up to twice the carrier frequency. 402 00:27:05,980 --> 00:27:08,150 Well it's this piece that we want. 403 00:27:08,150 --> 00:27:12,480 If we could eliminate everything else and keep this, 404 00:27:12,480 --> 00:27:18,360 then that would correspond to the spectrum of the original 405 00:27:18,360 --> 00:27:20,020 signal, x of t. 406 00:27:20,020 --> 00:27:21,410 How do we do that? 407 00:27:21,410 --> 00:27:26,870 Well, we know how to eliminate part of the spectrum and keep 408 00:27:26,870 --> 00:27:27,990 another part of the spectrum. 409 00:27:27,990 --> 00:27:29,320 That's called filtering. 410 00:27:29,320 --> 00:27:35,440 So what we would do is put the result of this through a 411 00:27:35,440 --> 00:27:37,035 low-pass filter. 412 00:27:37,035 --> 00:27:39,480 The low-pass filter route would retain the part of the 413 00:27:39,480 --> 00:27:44,720 spectrum around DC and eliminate the remaining part 414 00:27:44,720 --> 00:27:45,330 of the spectrum. 415 00:27:45,330 --> 00:27:49,090 So we would keep this part and eliminate the part of the 416 00:27:49,090 --> 00:27:51,410 spectrum that we have over here. 417 00:27:51,410 --> 00:27:54,490 And let me just draw your attention to the fact that, 418 00:27:54,490 --> 00:27:59,190 because of the way the algebra works out, the amplitude of 419 00:27:59,190 --> 00:28:01,950 this replication of the spectrum is half what the 420 00:28:01,950 --> 00:28:04,320 original spectrum was. 421 00:28:04,320 --> 00:28:09,660 And that means that ideally, to keep scale factors correct, 422 00:28:09,660 --> 00:28:14,890 we would choose the amplitude of this to be 2, to scale this 423 00:28:14,890 --> 00:28:18,150 back up to 1. 424 00:28:18,150 --> 00:28:23,140 So what we have is the modulator and demodulator. 425 00:28:23,140 --> 00:28:30,010 And just to summarize, for the case of a sinusoidal carrier 426 00:28:30,010 --> 00:28:33,860 as opposed to a complex exponential carrier, the 427 00:28:33,860 --> 00:28:39,060 modulator is just as it is in the complex exponential case. 428 00:28:39,060 --> 00:28:44,560 It's multiplication with the sinusoidal carrier, with 429 00:28:44,560 --> 00:28:47,190 frequency, omega c, and phase, theta sub c. 430 00:28:51,680 --> 00:28:57,740 In the demodulator we would take the modulated signal, 431 00:28:57,740 --> 00:29:00,820 modulate it again with the same carrier signal-- 432 00:29:00,820 --> 00:29:03,740 and as we'll see later, it's important to keep the same 433 00:29:03,740 --> 00:29:06,100 phase relationship. 434 00:29:06,100 --> 00:29:11,870 This result is not yet quite the demodulated signal. 435 00:29:11,870 --> 00:29:17,090 We need to process that with a low-pass filter that extracts 436 00:29:17,090 --> 00:29:21,550 the part of the spectrum around DC and throws away the 437 00:29:21,550 --> 00:29:24,560 upper part of the spectrum that gets generated in the 438 00:29:24,560 --> 00:29:26,770 second modulation process. 439 00:29:26,770 --> 00:29:32,380 And the resulting output is the original signal, x of t. 440 00:29:35,020 --> 00:29:37,490 What we've done then is we've taken x of t. 441 00:29:37,490 --> 00:29:41,120 We've modulated it onto a carrier. 442 00:29:41,120 --> 00:29:43,380 And then we've taken that modulated signal and we've 443 00:29:43,380 --> 00:29:45,360 figured out how to get back x of t. 444 00:29:45,360 --> 00:29:49,545 And of course one could ask, well, if you start with x of t 445 00:29:49,545 --> 00:29:52,470 and you want to get x of t back again, why bother going 446 00:29:52,470 --> 00:29:53,160 through all that? 447 00:29:53,160 --> 00:29:55,980 Why not just use x of t at the beginning and at the end? 448 00:29:55,980 --> 00:29:59,300 And obviously there are lots of reasons 449 00:29:59,300 --> 00:30:01,880 as I indicated before. 450 00:30:01,880 --> 00:30:06,060 And just to reiterate what they are. 451 00:30:06,060 --> 00:30:10,540 The notion, often, is that what you'd like to do is shift 452 00:30:10,540 --> 00:30:14,330 the signal into a different frequency band for 453 00:30:14,330 --> 00:30:19,370 transmission over some medium that is more matched to that 454 00:30:19,370 --> 00:30:22,090 frequency band than the frequency range of the 455 00:30:22,090 --> 00:30:23,440 original signal. 456 00:30:23,440 --> 00:30:27,280 Also, as I alluded to, is the notion that you can take lots 457 00:30:27,280 --> 00:30:29,460 of signals and transmit them 458 00:30:29,460 --> 00:30:32,650 simultaneously over one channel-- 459 00:30:32,650 --> 00:30:35,400 whether the channel is a wire, a microwave link, a satellite 460 00:30:35,400 --> 00:30:36,650 link, or whatever-- 461 00:30:36,650 --> 00:30:40,430 again, using the idea of modulation. 462 00:30:40,430 --> 00:30:45,460 And what that process is referred to as is 463 00:30:45,460 --> 00:30:46,680 multiplexing. 464 00:30:46,680 --> 00:30:49,270 And let me just quickly indicate what that 465 00:30:49,270 --> 00:30:53,280 multiplexing process corresponds to. 466 00:30:53,280 --> 00:31:00,560 We could think, for example, of taking one signal and 467 00:31:00,560 --> 00:31:06,760 modulating in it onto one carrier with one carrier 468 00:31:06,760 --> 00:31:12,280 frequency, taking a second signal, modulating it onto a 469 00:31:12,280 --> 00:31:16,690 different carrier frequency, taking a third signal and 470 00:31:16,690 --> 00:31:20,940 modulating it onto a third carrier frequency, et cetera. 471 00:31:20,940 --> 00:31:26,520 And if we choose these carrier frequencies appropriately, 472 00:31:26,520 --> 00:31:30,380 then we can add all those together-- 473 00:31:30,380 --> 00:31:33,540 and do it in such a way that the spectra don't overlap-- 474 00:31:33,540 --> 00:31:38,920 and end up with one broader band signal that incorporates 475 00:31:38,920 --> 00:31:45,070 the information simultaneously in all of those signals. 476 00:31:45,070 --> 00:31:49,690 So just to illustrate that in the frequency domain. 477 00:31:49,690 --> 00:31:56,890 What we have are our three spectra, Xa, Xb, and Xc. 478 00:31:56,890 --> 00:32:01,020 And we would, for example, take this spectrum and 479 00:32:01,020 --> 00:32:05,780 modulate it to a carrier frequency, omega sub a. 480 00:32:05,780 --> 00:32:09,140 We can take this spectrum and modulate it to a carrier 481 00:32:09,140 --> 00:32:14,600 frequency, omega sub b, where omega sub b is chosen so that 482 00:32:14,600 --> 00:32:18,140 when we add these two together they don't overlap, so that 483 00:32:18,140 --> 00:32:20,640 they can eventually be separated out. 484 00:32:20,640 --> 00:32:24,380 And then we can do the same thing with the third signal, 485 00:32:24,380 --> 00:32:27,990 and put that in a frequency range over here, being careful 486 00:32:27,990 --> 00:32:30,330 that none of those overlap. 487 00:32:30,330 --> 00:32:32,720 And when we add all those together, the composite 488 00:32:32,720 --> 00:32:36,890 spectrum is what I show here. 489 00:32:36,890 --> 00:32:43,430 And as you can see, essentially, by doing 490 00:32:43,430 --> 00:32:46,280 appropriate band-pass filtering we can pull out 491 00:32:46,280 --> 00:32:50,710 whatever part of the spectrum we choose to, and then 492 00:32:50,710 --> 00:32:53,020 demodulate that in the appropriate way. 493 00:32:53,020 --> 00:32:55,740 And of course we can do this, not just with three signals, 494 00:32:55,740 --> 00:32:59,140 but perhaps with tens or hundreds of signals. 495 00:32:59,140 --> 00:33:03,460 So that's a process that is typically referred to as 496 00:33:03,460 --> 00:33:04,380 multiplexing. 497 00:33:04,380 --> 00:33:08,130 And as I've described it here, it's referred to as 498 00:33:08,130 --> 00:33:10,460 frequency-division multiplexing. 499 00:33:10,460 --> 00:33:14,030 That is, dividing the frequency band into cells and 500 00:33:14,030 --> 00:33:19,560 plunking different signals into each one of those. 501 00:33:19,560 --> 00:33:23,650 And so if we want now to recover one of those channels 502 00:33:23,650 --> 00:33:27,250 in a frequency-division multiplex system, as I 503 00:33:27,250 --> 00:33:31,060 indicated, we would first demultiplex. 504 00:33:31,060 --> 00:33:35,590 Demultiplexing corresponding to pulling out the appropriate 505 00:33:35,590 --> 00:33:40,020 channel with a band-pass filter. 506 00:33:40,020 --> 00:33:45,190 And after demultiplexing, we would then demodulate. 507 00:33:45,190 --> 00:33:50,510 And we would demodulate with the carrier appropriate to the 508 00:33:50,510 --> 00:33:52,340 channel that we've pulled out. 509 00:33:52,340 --> 00:33:56,800 And the demodulation, of course, involves multiplying 510 00:33:56,800 --> 00:34:00,790 by the carrier and doing appropriate low-pass filtering 511 00:34:00,790 --> 00:34:03,510 to finally get the signal back. 512 00:34:03,510 --> 00:34:13,850 And frequency-division multiplexing is the type of 513 00:34:13,850 --> 00:34:18,679 multiplexing that's used, for example, in typical broadcast 514 00:34:18,679 --> 00:34:22,090 AM radio systems, where all the channels are 515 00:34:22,090 --> 00:34:23,780 superimposed together. 516 00:34:23,780 --> 00:34:26,520 And it's your home radio receiver that does the 517 00:34:26,520 --> 00:34:29,760 appropriate demultiplexing and demodulating. 518 00:34:29,760 --> 00:34:32,800 And of course, you can see that not only is modulation an 519 00:34:32,800 --> 00:34:36,830 important part of that, but as I alluded to in the last 520 00:34:36,830 --> 00:34:40,750 lecture, filtering also becomes important part of 521 00:34:40,750 --> 00:34:42,000 these practical systems. 522 00:34:46,020 --> 00:34:51,070 Now, the kind of amplitude modulation that I've talked 523 00:34:51,070 --> 00:34:57,310 about so far is what's referred to as synchronous 524 00:34:57,310 --> 00:34:58,880 modulation. 525 00:34:58,880 --> 00:35:04,060 And the reason for the term synchronous is that what's 526 00:35:04,060 --> 00:35:10,520 implied in these systems is a synchronization between the 527 00:35:10,520 --> 00:35:12,550 transmitter and receiver. 528 00:35:12,550 --> 00:35:19,200 In particular, in the system as we've talked about it, the 529 00:35:19,200 --> 00:35:29,010 modulator and the demodulator have a synchronization in both 530 00:35:29,010 --> 00:35:30,790 frequency and phase. 531 00:35:30,790 --> 00:35:33,630 The phase here is indicated as theta sub c. 532 00:35:33,630 --> 00:35:39,860 And if we take a look at the demodulator, the demodulator 533 00:35:39,860 --> 00:35:44,980 has phase of theta sub c. 534 00:35:44,980 --> 00:35:50,620 And in general, there's the issue of whether we can 535 00:35:50,620 --> 00:35:53,480 maintain that synchronization between the modulator and 536 00:35:53,480 --> 00:35:54,830 demodulator. 537 00:35:54,830 --> 00:36:00,160 And so what we want to examine now, more generally, is what 538 00:36:00,160 --> 00:36:03,750 the consequence might be, and the solution to the resulting 539 00:36:03,750 --> 00:36:09,300 problems, if we don't have synchronization between the 540 00:36:09,300 --> 00:36:11,380 modulator and demodulator. 541 00:36:11,380 --> 00:36:14,070 Synchronization in terms of phase. 542 00:36:14,070 --> 00:36:18,260 And there also is another problem, which is the issue of 543 00:36:18,260 --> 00:36:21,230 synchronization in frequency. 544 00:36:21,230 --> 00:36:23,840 That's examined more in the text. 545 00:36:23,840 --> 00:36:26,330 And what I'll focus on here is just the issue of 546 00:36:26,330 --> 00:36:28,900 synchronization in phase, to give you some sense of what 547 00:36:28,900 --> 00:36:31,920 the issue is. 548 00:36:31,920 --> 00:36:35,240 So now what we want to look at is what happens if we have a 549 00:36:35,240 --> 00:36:43,580 modulator with phase, theta sub c, and a demodulator where 550 00:36:43,580 --> 00:36:47,440 the phase, instead of being theta sub c, is some other 551 00:36:47,440 --> 00:36:51,280 phase, phi sub c. 552 00:36:51,280 --> 00:36:56,260 And if you track through the details and the algebra, then 553 00:36:56,260 --> 00:37:02,040 what you'll find is that the output of the low-pass filter, 554 00:37:02,040 --> 00:37:07,650 rather than being x of t, the signal that we want, is x of t 555 00:37:07,650 --> 00:37:10,660 multiplied by a scale factor. 556 00:37:10,660 --> 00:37:13,340 And the scale factor is the cosine of the phase 557 00:37:13,340 --> 00:37:14,790 difference. 558 00:37:14,790 --> 00:37:18,910 Now one could ask, OK well, what's the big deal about 559 00:37:18,910 --> 00:37:19,790 scale factor? 560 00:37:19,790 --> 00:37:21,930 If it's too small we'll make it big, it it's too big we'll 561 00:37:21,930 --> 00:37:23,680 make it small. 562 00:37:23,680 --> 00:37:25,000 But there are several points. 563 00:37:25,000 --> 00:37:28,600 One is, notice, for example, that if the phase difference 564 00:37:28,600 --> 00:37:32,750 between the modulator and demodulator is 90 degrees, 565 00:37:32,750 --> 00:37:36,920 then the output of the demodulator is zero. 566 00:37:36,920 --> 00:37:39,970 Or if it isn't quite 90 degrees, the 567 00:37:39,970 --> 00:37:41,580 amplitude might be small. 568 00:37:41,580 --> 00:37:44,300 And the implication would be that if there's other noise it 569 00:37:44,300 --> 00:37:46,980 gets injected in the system, the signal-to-noise 570 00:37:46,980 --> 00:37:49,120 ratio is very low. 571 00:37:49,120 --> 00:37:55,430 Now even worse is the issue that if there's a phase 572 00:37:55,430 --> 00:37:58,220 difference, but the exact phase difference isn't 573 00:37:58,220 --> 00:38:01,990 maintained, so that the modulator and demodulator kind 574 00:38:01,990 --> 00:38:05,990 of fade in and out of phase, then the output of the 575 00:38:05,990 --> 00:38:14,240 demodulator is x of t multiplied by a time-varying 576 00:38:14,240 --> 00:38:16,740 fading term, which is the cosine of the phase 577 00:38:16,740 --> 00:38:17,500 difference. 578 00:38:17,500 --> 00:38:21,080 Well what that means, essentially, is that if you 579 00:38:21,080 --> 00:38:26,470 use this kind of system to do the demodulation, then what 580 00:38:26,470 --> 00:38:31,860 you need to be careful about is maintaining synchronization 581 00:38:31,860 --> 00:38:35,560 in phase, and also in frequency, between the 582 00:38:35,560 --> 00:38:39,470 modulator and the demodulator. 583 00:38:39,470 --> 00:38:44,200 Now there are alternatives to this. 584 00:38:44,200 --> 00:38:48,590 And the alternative is what's referred to as asynchronous 585 00:38:48,590 --> 00:38:50,420 demodulation. 586 00:38:50,420 --> 00:38:54,530 And let me indicate what the idea behind asynchronous 587 00:38:54,530 --> 00:38:57,760 demodulation is. 588 00:38:57,760 --> 00:39:01,190 Now, recall that what we've done in amplitude modulation 589 00:39:01,190 --> 00:39:06,040 is to take the carrier signal and vary its amplitude with 590 00:39:06,040 --> 00:39:09,070 the signal that eventually we want to get back. 591 00:39:09,070 --> 00:39:12,530 So if we look at the amplitude-modulated waveform, 592 00:39:12,530 --> 00:39:16,880 it might typically look as I indicate here. 593 00:39:16,880 --> 00:39:22,910 And we're trying to get back the envelope. 594 00:39:22,910 --> 00:39:26,060 Well, one could imagine building a circuit, or 595 00:39:26,060 --> 00:39:28,450 designing a device, which in some sense 596 00:39:28,450 --> 00:39:30,490 will track the envelope. 597 00:39:30,490 --> 00:39:36,550 And a common circuit to do that is a fairly simple 598 00:39:36,550 --> 00:39:41,750 circuit consisting of a diode and a resistor and capacitor 599 00:39:41,750 --> 00:39:43,230 in parallel. 600 00:39:43,230 --> 00:39:48,580 The idea being that the capacitor charges up as this 601 00:39:48,580 --> 00:39:50,920 waveform moves up to its peak. 602 00:39:50,920 --> 00:39:55,690 And then as the waveform drops down, the capacitor discharges 603 00:39:55,690 --> 00:39:56,760 through the resistor. 604 00:39:56,760 --> 00:39:59,520 And it kind of tracks the envelope. 605 00:39:59,520 --> 00:40:03,740 In fact, the kind of output that we would get is the type 606 00:40:03,740 --> 00:40:07,150 of behavior that I've indicated here. 607 00:40:07,150 --> 00:40:13,090 And then that is a type of demodulation. 608 00:40:13,090 --> 00:40:15,570 It's a demodulation that doesn't require 609 00:40:15,570 --> 00:40:19,470 synchronization between the modulator and demodulator. 610 00:40:19,470 --> 00:40:25,400 And it's fairly inexpensive to build. 611 00:40:25,400 --> 00:40:30,660 But it has, obviously, some tradeoffs associated with it. 612 00:40:30,660 --> 00:40:34,580 Well, to indicate where the tradeoff comes from, or where 613 00:40:34,580 --> 00:40:39,210 the issue surfaces, notice that what we're doing is 614 00:40:39,210 --> 00:40:43,420 tracking the envelope of the sinusoidal signal. 615 00:40:43,420 --> 00:40:46,110 And we're calling that, or we're assuming that that is 616 00:40:46,110 --> 00:40:50,010 our original signal, x of t. 617 00:40:50,010 --> 00:40:56,300 Well, suppose that x of t, the original signal, is sometimes 618 00:40:56,300 --> 00:40:59,090 positive and sometimes negative. 619 00:40:59,090 --> 00:41:03,490 What might we see as we look at the output of the 620 00:41:03,490 --> 00:41:04,590 demodulator? 621 00:41:04,590 --> 00:41:07,950 Well, the output of the demodulator would follow the 622 00:41:07,950 --> 00:41:11,570 envelope down, and then it would follow the 623 00:41:11,570 --> 00:41:13,940 envelope back up again. 624 00:41:13,940 --> 00:41:17,090 In other words, what it would tend to generate is a 625 00:41:17,090 --> 00:41:20,680 full-wave rectified version of the signal that you were 626 00:41:20,680 --> 00:41:22,610 really trying to get back. 627 00:41:22,610 --> 00:41:24,970 Now, there's a simple solution to this. 628 00:41:24,970 --> 00:41:30,290 The simple solution is to make sure that the signal that is 629 00:41:30,290 --> 00:41:33,510 the modulating signal, x of t, never goes negative. 630 00:41:33,510 --> 00:41:38,460 So if it happens to-- a voice signal tends to go negative. 631 00:41:38,460 --> 00:41:42,570 If it happens to, we can simply add a constant to it, 632 00:41:42,570 --> 00:41:44,540 add a large enough constant, so that it 633 00:41:44,540 --> 00:41:46,670 always stays positive. 634 00:41:46,670 --> 00:41:49,840 Well let's look at that. 635 00:41:49,840 --> 00:41:52,845 What we want to do then, if we're considering asynchronous 636 00:41:52,845 --> 00:42:03,090 demodulation, is to take our original signal, x of t, and 637 00:42:03,090 --> 00:42:07,960 add to it a constant, where the constant is made large 638 00:42:07,960 --> 00:42:13,300 enough so that we're sure that this is a positive signal. 639 00:42:13,300 --> 00:42:15,410 And incidentally, let me just draw your attention to the 640 00:42:15,410 --> 00:42:19,870 fact that I'm now suppressing the phase on the carrier 641 00:42:19,870 --> 00:42:24,280 signal, since the phase is not important to the argument and 642 00:42:24,280 --> 00:42:28,550 it's just some additional notation to carry around. 643 00:42:28,550 --> 00:42:32,040 So the idea then, is add a constant to x of t. 644 00:42:32,040 --> 00:42:36,530 Notice that if we just take this term and expand it out 645 00:42:36,530 --> 00:42:41,940 into two terms, x of t cosine omega c t plus a times cosine 646 00:42:41,940 --> 00:42:47,690 omega c t, then in block diagram terms we can represent 647 00:42:47,690 --> 00:42:50,430 that as I've shown here. 648 00:42:50,430 --> 00:42:56,090 And so it would correspond to modulating the signal, x of t, 649 00:42:56,090 --> 00:43:01,430 onto the carrier, omega sub c t, and also injecting some 650 00:43:01,430 --> 00:43:05,500 carrier with an amplitude, A. And the output of the 651 00:43:05,500 --> 00:43:08,920 modulator is then the sum of those two. 652 00:43:08,920 --> 00:43:14,610 And depending on exactly what this value A is will influence 653 00:43:14,610 --> 00:43:16,610 what the envelope will look like. 654 00:43:16,610 --> 00:43:21,140 And I indicate below, two possibilities. 655 00:43:21,140 --> 00:43:26,110 One is where I've made A fairly large, and one is where 656 00:43:26,110 --> 00:43:29,350 I've made A significantly smaller. 657 00:43:29,350 --> 00:43:34,010 And there are both positive and negative issues associated 658 00:43:34,010 --> 00:43:36,860 with whether A is too large or A is too small. 659 00:43:36,860 --> 00:43:49,300 For example, if A is large in relation to the amplitude of 660 00:43:49,300 --> 00:43:53,360 the signal, then this envelope tends to be very flat. 661 00:43:53,360 --> 00:43:56,510 And it tends to be easy to track it with that simple 662 00:43:56,510 --> 00:44:03,450 diode RC circuit, as compared with the case down here. 663 00:44:03,450 --> 00:44:09,060 On the other hand, there is a price that you pay for this 664 00:44:09,060 --> 00:44:10,660 kind of envelope. 665 00:44:10,660 --> 00:44:13,350 And the price that you pay is perhaps best seen in the 666 00:44:13,350 --> 00:44:15,720 frequency domain. 667 00:44:15,720 --> 00:44:19,650 If we look in the frequency domain, here is 668 00:44:19,650 --> 00:44:22,620 our original spectrum. 669 00:44:22,620 --> 00:44:27,450 Here is the spectrum at the output of the modulator. 670 00:44:27,450 --> 00:44:31,030 And the impulse that occurs here corresponds to the 671 00:44:31,030 --> 00:44:33,370 carrier that's injected. 672 00:44:33,370 --> 00:44:37,510 The larger A is, the more carrier is injected. 673 00:44:37,510 --> 00:44:41,900 The more carrier that's injected, the easier it is for 674 00:44:41,900 --> 00:44:45,960 the envelope detector to demodulate. 675 00:44:45,960 --> 00:44:49,360 So one can ask, why not just put a lot in? 676 00:44:49,360 --> 00:44:54,670 Well, the obvious answer is that it's not an 677 00:44:54,670 --> 00:44:58,130 information-carrying part of the signal. 678 00:44:58,130 --> 00:45:01,670 And so in some sense it represents an inefficiency in 679 00:45:01,670 --> 00:45:04,890 transmission, because what you're transmitting is power, 680 00:45:04,890 --> 00:45:08,820 energy, that doesn't have any information 681 00:45:08,820 --> 00:45:09,900 associated with it. 682 00:45:09,900 --> 00:45:13,120 It's simply the injection of a carrier to make the 683 00:45:13,120 --> 00:45:18,770 demodulation for an asynchronous demodulator-- 684 00:45:18,770 --> 00:45:23,630 to make the demodulation easier. 685 00:45:23,630 --> 00:45:25,340 And so there's this tradeoff. 686 00:45:25,340 --> 00:45:29,590 And in fact, one represents the tradeoff and the 687 00:45:29,590 --> 00:45:32,230 associated parameters very often in terms of percent 688 00:45:32,230 --> 00:45:40,240 modulation, where the percent modulation is essentially the 689 00:45:40,240 --> 00:45:45,280 ratio of the maximum signal level to the amplitude of the 690 00:45:45,280 --> 00:45:47,380 injected carrier. 691 00:45:47,380 --> 00:45:52,750 And depending on whether the modulation's very high or very 692 00:45:52,750 --> 00:45:57,670 low, the tradeoff is that the transmission is more 693 00:45:57,670 --> 00:45:59,800 inefficient and it takes more energy, but the 694 00:45:59,800 --> 00:46:01,190 demodulator is simpler. 695 00:46:01,190 --> 00:46:04,910 Or the demodulator is more complicated but the 696 00:46:04,910 --> 00:46:06,840 transmission is simpler. 697 00:46:06,840 --> 00:46:09,780 Now, there are situations where you might very well want 698 00:46:09,780 --> 00:46:10,720 to use one or the other. 699 00:46:10,720 --> 00:46:16,670 For example, in home radio you're often willing to 700 00:46:16,670 --> 00:46:21,610 transmit a lot of power so that you can have inexpensive 701 00:46:21,610 --> 00:46:24,100 consumer-oriented receivers. 702 00:46:24,100 --> 00:46:26,790 On the other hand, in satellite communication you're 703 00:46:26,790 --> 00:46:31,070 willing to pay a very high price for the modulators and 704 00:46:31,070 --> 00:46:34,620 demodulators, but it's the amount of power that's 705 00:46:34,620 --> 00:46:37,010 transmitted that's at a premium. 706 00:46:37,010 --> 00:46:41,120 And so in one case, satellite communication, you would use 707 00:46:41,120 --> 00:46:43,890 synchronous modulation and demodulation. 708 00:46:43,890 --> 00:46:46,090 Whereas in typical consumer-oriented 709 00:46:46,090 --> 00:46:54,000 broadcasting, you would use an asynchronous system and 710 00:46:54,000 --> 00:46:56,980 transmit more power, even if it's inefficient, so that the 711 00:46:56,980 --> 00:47:00,470 demodulator can be simpler. 712 00:47:00,470 --> 00:47:06,400 Now, in the asynchronous system, as we've indicated, 713 00:47:06,400 --> 00:47:09,970 there's one source of inefficiency, which is this 714 00:47:09,970 --> 00:47:11,900 injection of the carrier. 715 00:47:11,900 --> 00:47:16,930 There also is a somewhat different issue, related to 716 00:47:16,930 --> 00:47:21,120 inefficiency in sinusoidal amplitude modulation. 717 00:47:21,120 --> 00:47:26,710 And it's an inefficiency that is separate from the issue of 718 00:47:26,710 --> 00:47:28,850 synchronous versus asynchronous systems. 719 00:47:28,850 --> 00:47:31,760 In other words, it's not associated with the injection 720 00:47:31,760 --> 00:47:35,410 of the carrier, it's a very different issue. 721 00:47:35,410 --> 00:47:38,140 Let me indicate what that is. 722 00:47:38,140 --> 00:47:44,050 Let's look again at the spectrum of x of t, which I've 723 00:47:44,050 --> 00:47:45,960 indicated here. 724 00:47:45,960 --> 00:47:54,600 And in a sinusoidal amplitude modulation system, we would 725 00:47:54,600 --> 00:47:58,650 center it around plus and minus the carrier frequency. 726 00:47:58,650 --> 00:48:02,500 Now, notice that in the original system we occupy a 727 00:48:02,500 --> 00:48:07,490 frequency spectrum that's 2 times omega sub M. By the time 728 00:48:07,490 --> 00:48:11,450 we've shifted it, thinking of positive and negative 729 00:48:11,450 --> 00:48:15,660 frequencies, we've used up twice as much of 730 00:48:15,660 --> 00:48:17,950 the frequency spectrum. 731 00:48:17,950 --> 00:48:22,300 Well you could say, OK, let's just shift this up this way 732 00:48:22,300 --> 00:48:24,470 and get rid of this part. 733 00:48:24,470 --> 00:48:25,860 That's of course what the complex 734 00:48:25,860 --> 00:48:27,860 exponential carrier did. 735 00:48:27,860 --> 00:48:31,360 And the issue there is that now you've got to transmit 736 00:48:31,360 --> 00:48:35,110 both a real part and an imaginary part. 737 00:48:35,110 --> 00:48:39,190 So what you can think about, and ask, is if you still want 738 00:48:39,190 --> 00:48:43,760 to transmit a real-valued signal, how can you somehow 739 00:48:43,760 --> 00:48:47,540 remove the inefficiency or redundancy in the spectrum? 740 00:48:47,540 --> 00:48:52,750 Well, notice that what we have is this spectrum moved here, 741 00:48:52,750 --> 00:48:55,090 and moved here. 742 00:48:55,090 --> 00:49:02,180 And we could imagine building real-valued signal by 743 00:49:02,180 --> 00:49:06,770 eliminating what I refer to here as the lower sideband out 744 00:49:06,770 --> 00:49:11,210 of the positive frequencies, and the lower sideband out of 745 00:49:11,210 --> 00:49:13,710 the negative frequencies. 746 00:49:13,710 --> 00:49:18,650 And in effect, what we've done is taken just the positive 747 00:49:18,650 --> 00:49:22,510 frequencies here, shifted them there, the negative 748 00:49:22,510 --> 00:49:24,460 frequencies here, and shifted them here. 749 00:49:24,460 --> 00:49:29,000 And the resulting spectrum is what I indicate below. 750 00:49:29,000 --> 00:49:32,840 Well, this is what is often done. 751 00:49:32,840 --> 00:49:37,020 And what it's referred to as is single sideband. 752 00:49:37,020 --> 00:49:40,790 What we've done is kept the upper sideband, in this 753 00:49:40,790 --> 00:49:42,050 particular case. 754 00:49:42,050 --> 00:49:46,390 We could alternatively think of putting this system 755 00:49:46,390 --> 00:49:52,760 together where we retain the lower sideband instead of the 756 00:49:52,760 --> 00:49:54,250 upper sideband. 757 00:49:54,250 --> 00:49:59,870 And in either case, what we've removed is an inefficiency in 758 00:49:59,870 --> 00:50:01,760 transmission of the signal. 759 00:50:01,760 --> 00:50:05,380 Namely we have a real-valued signal, but it only requires 760 00:50:05,380 --> 00:50:10,710 as much total bandwidth, in terms of the frequencies in 761 00:50:10,710 --> 00:50:14,480 which there's energy present, as the original signal. 762 00:50:14,480 --> 00:50:16,190 Well how do we do this? 763 00:50:16,190 --> 00:50:18,160 There are a variety of ways. 764 00:50:18,160 --> 00:50:22,810 And there's one procedure that is discussed in more detail in 765 00:50:22,810 --> 00:50:26,160 the text, which uses what's referred to as a 90 degree 766 00:50:26,160 --> 00:50:27,690 phase splitter. 767 00:50:27,690 --> 00:50:30,720 The simplest way, at least conceptually, is to think 768 00:50:30,720 --> 00:50:33,420 about doing it with filtering. 769 00:50:33,420 --> 00:50:39,810 And the idea simply is that if we have our modulated signal-- 770 00:50:39,810 --> 00:50:41,810 here's the spectrum of the modulated signal. 771 00:50:41,810 --> 00:50:44,780 And if that modulated signal is simply put through a 772 00:50:44,780 --> 00:50:51,740 high-pass filter, then the result will be to eliminate 773 00:50:51,740 --> 00:50:56,240 the lower sideband, if we choose the high-pass filter to 774 00:50:56,240 --> 00:50:59,530 have a characteristic as I indicate here. 775 00:50:59,530 --> 00:51:03,820 So this, conceptually, is a very sharp cutoff filter. 776 00:51:03,820 --> 00:51:06,810 And what it eliminates are the lower sidebands. 777 00:51:06,810 --> 00:51:11,770 And the resulting spectrum is what we have below. 778 00:51:11,770 --> 00:51:17,870 And this in fact is really the basic idea behind 779 00:51:17,870 --> 00:51:20,130 single-sideband transmission. 780 00:51:20,130 --> 00:51:22,180 Again, there's a tradeoff. 781 00:51:22,180 --> 00:51:26,280 It's clearly more efficient than double-sideband 782 00:51:26,280 --> 00:51:32,770 transmission, but also has the complication, or additional 783 00:51:32,770 --> 00:51:35,760 issue, that the modulator becomes a little more 784 00:51:35,760 --> 00:51:39,240 complicated because you need this filtering operation, or 785 00:51:39,240 --> 00:51:41,770 some equivalent operation, to get rid of 786 00:51:41,770 --> 00:51:43,020 the unwanted sideband. 787 00:51:46,730 --> 00:51:52,050 Well, this is a fairly quick tour through a variety of 788 00:51:52,050 --> 00:51:53,990 issues related to modulation. 789 00:51:53,990 --> 00:51:58,180 And it really is just the tip of the iceberg, obviously. 790 00:51:58,180 --> 00:52:02,180 Modulation in the context of sinusoidal modulation, as 791 00:52:02,180 --> 00:52:06,710 we've talked about, has a lot of detailed issues 792 00:52:06,710 --> 00:52:09,030 associated with it. 793 00:52:09,030 --> 00:52:14,220 It's important to recognize, and to be somewhat pleased by 794 00:52:14,220 --> 00:52:19,030 the fact, that not only with the mathematical foundations 795 00:52:19,030 --> 00:52:24,130 that we've developed can we understand the basics of 796 00:52:24,130 --> 00:52:25,610 sinusoidal amplitude modulation. 797 00:52:25,610 --> 00:52:32,190 But what you'll find if you dig into this somewhat deeper 798 00:52:32,190 --> 00:52:35,940 that the basic background that we built up so far-- 799 00:52:35,940 --> 00:52:38,190 the mathematical tools-- 800 00:52:38,190 --> 00:52:42,080 are really pretty much what you need for a much deeper 801 00:52:42,080 --> 00:52:44,505 understanding of all of the issues involved. 802 00:52:47,180 --> 00:52:51,800 So from what might have seemed like a fairly abstract 803 00:52:51,800 --> 00:52:54,920 mathematical property associated with the Fourier 804 00:52:54,920 --> 00:53:00,460 transform, we've begun to develop what should give you 805 00:53:00,460 --> 00:53:05,000 the sense of some important practical considerations. 806 00:53:05,000 --> 00:53:09,770 And as we'll see the next lecture, very much the same 807 00:53:09,770 --> 00:53:15,350 kinds of notions apply for discrete time, sinusoidal, and 808 00:53:15,350 --> 00:53:19,000 complex exponential amplitude modulation. 809 00:53:19,000 --> 00:53:23,060 And also as I indicated at the beginning of the lecture, in 810 00:53:23,060 --> 00:53:27,120 the next lecture we'll also talk about what's referred to 811 00:53:27,120 --> 00:53:28,770 as pulse amplitude modulation. 812 00:53:28,770 --> 00:53:30,770 It's a different kind of carrier. 813 00:53:30,770 --> 00:53:36,260 And what that will lead to, among other things, is a very 814 00:53:36,260 --> 00:53:43,540 important bridge between the notions of continuous time and 815 00:53:43,540 --> 00:53:45,080 the notions of discrete time. 816 00:53:45,080 --> 00:53:46,330 Thank you.