WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:56.140 --> 00:00:59.190 PROFESSOR: In this lecture we begin a discussion of the 00:00:59.190 --> 00:01:03.860 topic of modulation, which is, among other things, a very 00:01:03.860 --> 00:01:06.160 important topic in practical terms. 00:01:06.160 --> 00:01:10.400 For example, it forms the cornerstone for many 00:01:10.400 --> 00:01:12.280 communication systems. 00:01:12.280 --> 00:01:16.290 And also, as we'll see as these lectures go along, a 00:01:16.290 --> 00:01:20.380 particular form of modulation referred to as pulse amplitude 00:01:20.380 --> 00:01:24.350 modulation, and eventually impulse modulation or impulse 00:01:24.350 --> 00:01:28.930 train modulation, forms a very important bridge between 00:01:28.930 --> 00:01:33.670 continuous time signals and discrete time signals. 00:01:33.670 --> 00:01:37.590 Now in general terms what we mean when we refer to 00:01:37.590 --> 00:01:43.280 modulation is the notion of using one signal to vary a 00:01:43.280 --> 00:01:46.110 parameter of another signal. 00:01:46.110 --> 00:01:51.510 For example, a sinusoidal signal has three parameters, 00:01:51.510 --> 00:01:54.520 amplitude, frequency, and phase. 00:01:54.520 --> 00:01:59.920 And we could think, for example, of using one signal 00:01:59.920 --> 00:02:03.080 to vary, let's say, the amplitude of 00:02:03.080 --> 00:02:04.730 a sinusoidal signal. 00:02:04.730 --> 00:02:08.259 And what that leads to is a notion, which we'll develop in 00:02:08.259 --> 00:02:12.300 some detail, referred to as sinusoidal amplitude 00:02:12.300 --> 00:02:17.780 modulation, and would correspond to a sinusoidal 00:02:17.780 --> 00:02:22.670 signal, referred to as the carrier, and it's amplitude 00:02:22.670 --> 00:02:27.700 being varied on the basis of another signal. 00:02:27.700 --> 00:02:31.800 Now alternatively we could think of varying either the 00:02:31.800 --> 00:02:36.790 frequency or the phase of a sinusoidal signal, again with 00:02:36.790 --> 00:02:38.200 another signal. 00:02:38.200 --> 00:02:41.850 And what that leads to is another very important notion, 00:02:41.850 --> 00:02:45.480 which is referred to sinusoidal frequency 00:02:45.480 --> 00:02:49.850 modulation, where essentially it's the frequency of a 00:02:49.850 --> 00:02:54.850 sinusoid that's changing depending on the signal that's 00:02:54.850 --> 00:02:59.460 we're using to modulate the sinusoid. 00:02:59.460 --> 00:03:03.200 Now sinusoidal amplitude, frequency, and phase 00:03:03.200 --> 00:03:07.950 modulation are extremely important topics and ideas in 00:03:07.950 --> 00:03:11.570 the context of communication systems. 00:03:11.570 --> 00:03:14.800 One of the reasons is that if you want to transmit a signal, 00:03:14.800 --> 00:03:17.880 let's say for example a voice signal, the voice signal that 00:03:17.880 --> 00:03:19.980 you're listening to now. 00:03:19.980 --> 00:03:23.320 If you try to transmit that over long distances, because 00:03:23.320 --> 00:03:27.740 of the frequencies involved the medium that you use to 00:03:27.740 --> 00:03:31.530 transmit it won't carry it long distances. 00:03:31.530 --> 00:03:35.030 The idea then is to essentially take that signal, 00:03:35.030 --> 00:03:38.760 like a voice signal, use it to modulate a much 00:03:38.760 --> 00:03:41.980 higher-frequency signal, and then transmit that 00:03:41.980 --> 00:03:46.510 higher-frequency signal over a medium that essentially can 00:03:46.510 --> 00:03:49.060 support long-distance transmission at those 00:03:49.060 --> 00:03:50.420 frequencies. 00:03:50.420 --> 00:03:53.240 Then at the other end of course, the voice information, 00:03:53.240 --> 00:03:57.480 or whatever else the information is, is taken off. 00:03:57.480 --> 00:04:00.370 Now also, a notion that that leads to, and we'll be 00:04:00.370 --> 00:04:06.140 developing in some detail, is the idea that you can 00:04:06.140 --> 00:04:11.940 simultaneously transmit more than one signal by in essence 00:04:11.940 --> 00:04:16.000 taking several voice signals or other signals, using them 00:04:16.000 --> 00:04:20.279 to modulate either the frequency or amplitude of 00:04:20.279 --> 00:04:23.750 sinusoidal signals at different frequencies, adding 00:04:23.750 --> 00:04:24.940 all those together-- 00:04:24.940 --> 00:04:27.330 that's a process called multiplexing-- 00:04:27.330 --> 00:04:31.040 and then at the other end of the transmission system, 00:04:31.040 --> 00:04:33.570 taking those sinusoidal signals apart. 00:04:33.570 --> 00:04:37.610 And then extracting the envelope or frequency 00:04:37.610 --> 00:04:41.910 modulation information to get back to the voice signal or 00:04:41.910 --> 00:04:44.630 other information-carrying signal. 00:04:44.630 --> 00:04:50.420 So that's one of the very important ways in which 00:04:50.420 --> 00:04:54.670 sinusoidal modulation is used in communication systems. 00:04:54.670 --> 00:04:59.100 And what we'll see, in particular as we go through 00:04:59.100 --> 00:05:04.980 today's lecture, is that sinusoidal amplitude 00:05:04.980 --> 00:05:10.630 modulation, follows in a fairly straightforward way 00:05:10.630 --> 00:05:14.250 from the properties of the Fourier transform that we've 00:05:14.250 --> 00:05:16.920 developed in some of the earlier lectures. 00:05:16.920 --> 00:05:22.520 So our focus in today's lecture will be on sinusoidal 00:05:22.520 --> 00:05:26.580 amplitude modulation in continuous time. 00:05:26.580 --> 00:05:29.740 In the next lecture we'll consider the same set of 00:05:29.740 --> 00:05:32.950 notions related to discrete time, and also a concept 00:05:32.950 --> 00:05:36.040 referred to as pulse amplitude modulation. 00:05:36.040 --> 00:05:39.690 And all of these follow, in a very straightforward way, from 00:05:39.690 --> 00:05:43.470 the modulation property for the Fourier transform. 00:05:43.470 --> 00:05:47.460 Issues of frequency and phase modulation are a little more 00:05:47.460 --> 00:05:48.760 difficult to analyze. 00:05:48.760 --> 00:05:52.570 But many of the techniques that we've developed in the 00:05:52.570 --> 00:05:57.190 previous lectures also provide important insights into 00:05:57.190 --> 00:05:58.930 frequency and phase modulation. 00:05:58.930 --> 00:06:04.060 And some of this is developed in more detail in the book. 00:06:04.060 --> 00:06:09.050 So what I'd like to do is focus, for now, on the concept 00:06:09.050 --> 00:06:12.520 of amplitude modulation. 00:06:12.520 --> 00:06:18.270 And as I indicated, there are several kinds of carrier 00:06:18.270 --> 00:06:22.010 signals on which the modulation can be 00:06:22.010 --> 00:06:23.460 superimposed. 00:06:23.460 --> 00:06:27.230 The basic structure for an amplitude modulation system is 00:06:27.230 --> 00:06:30.870 one in which there is the modulating signal, let's say 00:06:30.870 --> 00:06:34.400 for example, voice, and a carrier signal-- what's 00:06:34.400 --> 00:06:36.560 referred to as the carrier. 00:06:36.560 --> 00:06:39.550 And then of course the resulting output is the 00:06:39.550 --> 00:06:41.640 modulated output. 00:06:41.640 --> 00:06:45.890 Now to analyze this, since we have multiplication in the 00:06:45.890 --> 00:06:49.350 time domain, we know from the property of the Fourier 00:06:49.350 --> 00:06:51.920 transform that we've developed previously-- 00:06:51.920 --> 00:06:53.750 the modulation property-- 00:06:53.750 --> 00:06:57.900 that multiplication in the time domain corresponds to 00:06:57.900 --> 00:07:00.550 convolution in the frequency domain. 00:07:00.550 --> 00:07:04.890 And it's this basic property or equation that lets us 00:07:04.890 --> 00:07:09.420 analyze, in some detail in fact, the notions of amplitude 00:07:09.420 --> 00:07:11.590 modulation. 00:07:11.590 --> 00:07:15.240 As we go through this lecture and the next lecture, we'll be 00:07:15.240 --> 00:07:18.920 talking, as I indicated, about several different types of 00:07:18.920 --> 00:07:20.700 carrier signals. 00:07:20.700 --> 00:07:24.910 One is what's referred to as pulse carriers. 00:07:24.910 --> 00:07:28.390 And that leads to, among other things, the concept of pulse 00:07:28.390 --> 00:07:30.220 amplitude modulation. 00:07:30.220 --> 00:07:34.050 That will be deferred until the next lecture. 00:07:34.050 --> 00:07:39.720 In today's lecture what I'll focus on is first, the case of 00:07:39.720 --> 00:07:43.240 a complex exponential carrier, second, the case 00:07:43.240 --> 00:07:45.210 of sinusoidal carrier. 00:07:45.210 --> 00:07:48.490 And in fact the complex exponential carrier and 00:07:48.490 --> 00:07:51.770 sinusoidal carrier are obviously very closely 00:07:51.770 --> 00:07:56.710 related, since the complex exponential carrier is, in 00:07:56.710 --> 00:08:00.350 effect, two sinusoidal carriers. 00:08:00.350 --> 00:08:04.900 One for the real part and one for the imaginary part. 00:08:04.900 --> 00:08:08.700 So let's first begin the discussion of amplitude 00:08:08.700 --> 00:08:12.840 modulation by considering a complex exponential carrier, 00:08:12.840 --> 00:08:14.900 and then moving on to a discussion of 00:08:14.900 --> 00:08:18.490 a sinusoidal carrier. 00:08:18.490 --> 00:08:26.930 So the issue then is that we have a signal, x of t. 00:08:26.930 --> 00:08:29.060 It's multiplied by a carrier. 00:08:29.060 --> 00:08:31.860 And the carrier that we're considering is a carrier 00:08:31.860 --> 00:08:39.919 signal, c of t, of the form e to the j omega c t 00:08:39.919 --> 00:08:42.030 plus theta sub c. 00:08:42.030 --> 00:08:45.430 That's the form of our carrier signal. 00:08:45.430 --> 00:08:52.110 And what we can first analyze is what the resulting signal 00:08:52.110 --> 00:08:55.870 or spectrum is at the output of the modulator. 00:08:55.870 --> 00:08:59.870 Well, we can do that by concentrating on the 00:08:59.870 --> 00:09:01.650 modulation property. 00:09:01.650 --> 00:09:06.820 And let's consider, just as a general form for a spectrum, 00:09:06.820 --> 00:09:14.220 what I've indicated here for the Fourier transform of the 00:09:14.220 --> 00:09:18.260 input signal or modulating signal, X of omega. 00:09:18.260 --> 00:09:22.650 And so this is intended to represent the 00:09:22.650 --> 00:09:25.760 spectrum of x of t. 00:09:25.760 --> 00:09:29.140 And then the carrier signal, since it's a single complex 00:09:29.140 --> 00:09:35.340 exponential, has a Fourier transform which is an impulse 00:09:35.340 --> 00:09:37.300 in the frequency domain. 00:09:37.300 --> 00:09:42.450 And the amplitude of the impulse is 2 pi e to the j 00:09:42.450 --> 00:09:46.310 theta sub c, where we notice that the complex amplitude 00:09:46.310 --> 00:09:50.110 incorporates the phase information. 00:09:50.110 --> 00:09:54.690 So now if we multiply in the time domain, we convolve in 00:09:54.690 --> 00:09:55.920 the frequency domain. 00:09:55.920 --> 00:09:59.850 And as you know, convolving a signal with an impulse just 00:09:59.850 --> 00:10:03.380 shifts that signal to the location of the impulse. 00:10:03.380 --> 00:10:07.400 And so as a consequence of taking care of various 00:10:07.400 --> 00:10:13.250 factors, what we end up with is a spectrum that is centered 00:10:13.250 --> 00:10:18.660 at the carrier frequency omega sub c. 00:10:18.660 --> 00:10:25.330 So what this says is that if we have a signal, x of t, and 00:10:25.330 --> 00:10:31.040 we use it to modulate a complex exponential carrier in 00:10:31.040 --> 00:10:34.570 the frequency domain, what we've simply done is to take 00:10:34.570 --> 00:10:39.760 the original spectrum and shift it in frequency. 00:10:39.760 --> 00:10:43.340 So that what was originally at zero frequency is now centered 00:10:43.340 --> 00:10:44.590 around the carrier frequency. 00:10:46.960 --> 00:10:50.160 We've now modulated, in effect, to a higher frequency. 00:10:50.160 --> 00:10:53.180 Things are happening in a higher-frequency band. 00:10:53.180 --> 00:10:57.200 And the next question is, how do we demodulate, or in other 00:10:57.200 --> 00:11:02.540 words, how do we get the original signal back? 00:11:02.540 --> 00:11:05.210 Of course one way that we can think of doing it, 00:11:05.210 --> 00:11:09.790 particularly in the context of this specific carrier, if we 00:11:09.790 --> 00:11:14.660 look back at the top equation we have, as the result of the 00:11:14.660 --> 00:11:19.470 modulation, x of t times c of t, where c of t is this. 00:11:19.470 --> 00:11:24.370 And we could consider, for example, just simply dividing 00:11:24.370 --> 00:11:26.530 the modulator output by this. 00:11:26.530 --> 00:11:29.790 Or equivalently, taking the modulated output and 00:11:29.790 --> 00:11:37.170 multiplying by e to the minus j omega c t plus theta c. 00:11:37.170 --> 00:11:40.790 Let's track that through in terms of the spectra. 00:11:40.790 --> 00:11:46.020 We have, again, the spectrum of the output of the 00:11:46.020 --> 00:11:49.130 modulator, which is the original spectrum shifted up 00:11:49.130 --> 00:11:51.530 to the carrier frequency. 00:11:51.530 --> 00:11:58.130 We have, below that, the spectrum of e to the minus j 00:11:58.130 --> 00:12:01.050 omega c t plus theta c. 00:12:01.050 --> 00:12:06.440 And if we now convolve this with this, that results in 00:12:06.440 --> 00:12:09.300 simply shifting this spectrum-- except for an issue 00:12:09.300 --> 00:12:10.750 of a scale factor-- 00:12:10.750 --> 00:12:14.430 shifting this spectrum back down to the origin. 00:12:14.430 --> 00:12:17.560 So convolving these two together, the spectrum that we 00:12:17.560 --> 00:12:20.740 end up with is that. 00:12:20.740 --> 00:12:25.370 So we can track this through in the frequency domain. 00:12:25.370 --> 00:12:28.900 In the frequency domain it says shift the spectrum up. 00:12:28.900 --> 00:12:32.430 When you want to demodulate, shift the spectrum back down. 00:12:32.430 --> 00:12:35.780 And alternatively, we can look at it algebraically in the 00:12:35.780 --> 00:12:36.320 time domain. 00:12:36.320 --> 00:12:40.010 And what it says is, if you multiply by e to the plus j 00:12:40.010 --> 00:12:44.150 omega c t, then when you want to get back, multiply by e to 00:12:44.150 --> 00:12:46.920 the minus j omega c t. 00:12:46.920 --> 00:12:51.280 Now one question that you could conceivably be asking 00:12:51.280 --> 00:12:55.730 is, if we're talking about practical systems and not 00:12:55.730 --> 00:13:01.890 simply mathematics, does it make sense in the real world 00:13:01.890 --> 00:13:06.530 to consider using a complex exponential carrier? 00:13:06.530 --> 00:13:11.300 And the answer to that, in fact, is yes. 00:13:11.300 --> 00:13:16.460 That very often in practical systems one considers using a 00:13:16.460 --> 00:13:20.640 carrier which in fact is a complex exponential. 00:13:20.640 --> 00:13:22.890 Well, a complex exponential is complex. 00:13:22.890 --> 00:13:25.440 There's a square root of minus one in there. 00:13:25.440 --> 00:13:27.590 And you could ask well, how do we get a square 00:13:27.590 --> 00:13:29.170 root of minus one? 00:13:29.170 --> 00:13:31.970 And the answer is fairly simple. 00:13:31.970 --> 00:13:37.030 Let's look again at the modulator, which we have here. 00:13:37.030 --> 00:13:40.980 And in effect, what that says is we want to multiply a 00:13:40.980 --> 00:13:45.840 real-valued signal by e to the j omega c t plus theta c. 00:13:45.840 --> 00:13:51.030 Now, we can equivalently use Euler's relationship to break 00:13:51.030 --> 00:13:54.650 this down into a cosine and sine term. 00:13:54.650 --> 00:13:57.660 And so what that means in terms of an implementation, 00:13:57.660 --> 00:14:07.140 equivalently, is modulating x of t onto a cosine carrier. 00:14:07.140 --> 00:14:10.850 And that then gives us the real part 00:14:10.850 --> 00:14:12.830 of the complex output. 00:14:12.830 --> 00:14:17.780 And modulating it onto a sinusoidal carrier-- 00:14:17.780 --> 00:14:20.340 these two being 90 degrees out of phase-- 00:14:20.340 --> 00:14:23.500 and that gives us the imaginary part. 00:14:23.500 --> 00:14:27.010 And so in effect, this is the complex signal. 00:14:27.010 --> 00:14:30.440 If we just simply think of hanging a tag on here that 00:14:30.440 --> 00:14:34.500 says square root of minus 1, or j, and we appropriately 00:14:34.500 --> 00:14:38.210 combine complex signals following the rules of complex 00:14:38.210 --> 00:14:39.110 arithmetic. 00:14:39.110 --> 00:14:42.020 And indeed, that's exactly the way things are done in the 00:14:42.020 --> 00:14:43.010 real world. 00:14:43.010 --> 00:14:48.660 A complex signal is simply a set of two real signals. 00:14:48.660 --> 00:14:54.220 And of course, if we look at the spectra involved, we have 00:14:54.220 --> 00:14:56.690 here the real part and the imaginary part 00:14:56.690 --> 00:14:59.990 of the complex output. 00:14:59.990 --> 00:15:05.330 If we again refer back to the original spectrum, X of omega, 00:15:05.330 --> 00:15:10.580 and the modulated spectrum which I show down here, the 00:15:10.580 --> 00:15:16.030 original spectrum shifted up to the carrier frequency. 00:15:16.030 --> 00:15:20.610 In effect we're building this out of two lines. 00:15:20.610 --> 00:15:25.870 One line representing the real part of that. 00:15:25.870 --> 00:15:29.610 And the real part in the time domain corresponds to the even 00:15:29.610 --> 00:15:31.290 part in the frequency domain. 00:15:31.290 --> 00:15:37.680 And so with the output of the cosine modulator, we have a 00:15:37.680 --> 00:15:40.750 spectrum that looks like this. 00:15:40.750 --> 00:15:47.120 And the output along the imaginary branch has a 00:15:47.120 --> 00:15:50.310 spectrum that looks like this. 00:15:50.310 --> 00:15:54.690 Recall in the top branch that this, for positive frequencies 00:15:54.690 --> 00:15:57.210 was positive, and was positive here. 00:15:57.210 --> 00:16:00.260 And so in effect when you add them, this portion of the 00:16:00.260 --> 00:16:02.490 spectrum will cancel out. 00:16:02.490 --> 00:16:10.360 So in effect, what we're doing is building the complex signal 00:16:10.360 --> 00:16:11.850 out of two real signals. 00:16:11.850 --> 00:16:16.710 Or we're building the spectrum of the complex signal out of 00:16:16.710 --> 00:16:20.745 separate lines that represent the even and the odd parts. 00:16:23.540 --> 00:16:29.640 Now, there are lots of applications of amplitude 00:16:29.640 --> 00:16:30.910 modulation. 00:16:30.910 --> 00:16:33.390 And we'll be seeing a number of these as we go through the 00:16:33.390 --> 00:16:34.420 discussion. 00:16:34.420 --> 00:16:39.660 What I'd like to do is just indicate briefly one now, 00:16:39.660 --> 00:16:44.560 which is an application that in fact surfaces fairly often 00:16:44.560 --> 00:16:49.540 in the context of a complex exponential carrier. 00:16:49.540 --> 00:16:58.440 And that is the notion of using modulation to permit the 00:16:58.440 --> 00:17:03.550 application of a very well designed and implemented 00:17:03.550 --> 00:17:08.240 low-pass filter to be used as a band-pass filter and in 00:17:08.240 --> 00:17:10.520 fact, as a set of band-pass filters. 00:17:10.520 --> 00:17:11.680 And here's the idea. 00:17:11.680 --> 00:17:18.230 The idea is if we have a fixed filter-- 00:17:18.230 --> 00:17:19.990 let's say we have a signal. 00:17:19.990 --> 00:17:24.359 And we want to think of a filter, which we want to move 00:17:24.359 --> 00:17:28.640 along the signal, one way to do it is to somehow have 00:17:28.640 --> 00:17:30.950 filters that move along the signal. 00:17:30.950 --> 00:17:34.870 The other possibility is to keep the filter fixed and let 00:17:34.870 --> 00:17:39.250 the signal move in frequency in front of the filter. 00:17:39.250 --> 00:17:42.540 Let me be a little more specific. 00:17:42.540 --> 00:17:47.860 Suppose that we have a signal, x of t. 00:17:47.860 --> 00:17:52.500 And we modulate it with a complex exponential carrier 00:17:52.500 --> 00:17:56.490 with a carrier frequency, omega c. 00:17:56.490 --> 00:18:01.020 And the output of that is then processed with a low-pass 00:18:01.020 --> 00:18:06.910 filter and then we demodulate the result. 00:18:06.910 --> 00:18:12.770 Then what we've done is to take the spectrum of the input 00:18:12.770 --> 00:18:18.440 signal, shift it, pull out what is now around low 00:18:18.440 --> 00:18:22.200 frequencies, and then shift that part of the spectrum back 00:18:22.200 --> 00:18:24.190 to where it belongs. 00:18:24.190 --> 00:18:28.900 So if we look at that in terms of actually tracking through 00:18:28.900 --> 00:18:34.180 the spectra, we would have initially a spectrum for the 00:18:34.180 --> 00:18:40.050 original signal, which I show at the top as X of omega. 00:18:40.050 --> 00:18:44.750 After modulating or shifting that spectrum up to a center 00:18:44.750 --> 00:18:49.530 frequency of omega c, we then have what I indicate here. 00:18:49.530 --> 00:18:53.490 And the dotted line corresponds to the pass band 00:18:53.490 --> 00:18:55.430 of the low-pass filter. 00:18:55.430 --> 00:19:00.590 Well, the result of low-pass filtering rejects all the 00:19:00.590 --> 00:19:04.970 spectrum except the part around low frequencies. 00:19:04.970 --> 00:19:10.260 And the next step is then to demodulate this. 00:19:10.260 --> 00:19:14.640 And so in effect, demodulating will shift this spectrum back 00:19:14.640 --> 00:19:17.490 to where it originally came from. 00:19:17.490 --> 00:19:22.710 And so that result will be what I show in the final 00:19:22.710 --> 00:19:25.680 result, which is here. 00:19:25.680 --> 00:19:31.140 And what we can see is that this is equivalent. 00:19:31.140 --> 00:19:36.170 If we can look back at the top spectrum, this is equivalent 00:19:36.170 --> 00:19:41.070 to having extracted, with a band-pass filter, a section 00:19:41.070 --> 00:19:43.860 out of this part of the spectrum. 00:19:43.860 --> 00:19:47.840 So in terms of tracking through the spectrum and 00:19:47.840 --> 00:19:54.240 looking at the equivalent filtering operation, then what 00:19:54.240 --> 00:20:00.470 we accomplished was to pull out this part of the spectrum 00:20:00.470 --> 00:20:03.950 using a low-pass filter and modulation. 00:20:03.950 --> 00:20:07.770 But equivalently what we implemented was a band-pass 00:20:07.770 --> 00:20:10.750 filter as I indicated here. 00:20:10.750 --> 00:20:14.030 Now of course, a signal with this spectrum, since the 00:20:14.030 --> 00:20:17.030 spectrum is not conjugate symmetric, we know that this 00:20:17.030 --> 00:20:20.620 signal does not correspond to a real-valued signal. 00:20:20.620 --> 00:20:24.400 Equivalently this filter doesn't correspond to a filter 00:20:24.400 --> 00:20:26.990 whose impulse response is real. 00:20:26.990 --> 00:20:32.140 If we add another step to this, which is to take the 00:20:32.140 --> 00:20:36.290 real part of the output, then by taking the real part of the 00:20:36.290 --> 00:20:40.550 output we would be taking the even part of the spectrum 00:20:40.550 --> 00:20:43.080 associated with that complex signal. 00:20:43.080 --> 00:20:47.860 And the equivalent filter that we would end up with then is 00:20:47.860 --> 00:20:52.250 the filter that I indicate at the bottom, which is a 00:20:52.250 --> 00:20:54.950 band-pass filter. 00:20:54.950 --> 00:21:00.750 Now just to reiterate a point that I made earlier. 00:21:00.750 --> 00:21:03.860 A question, of course, is why would you go to this trouble? 00:21:03.860 --> 00:21:06.520 Why not just build a band-pass filter? 00:21:06.520 --> 00:21:13.300 And one of the reasons is that it's often much easier to 00:21:13.300 --> 00:21:17.920 build a fixed filter, a filter with a fixed-center frequency, 00:21:17.920 --> 00:21:21.680 for example a low-pass filter, than it is to build a filter 00:21:21.680 --> 00:21:25.030 that has variable components in it so that when you vary 00:21:25.030 --> 00:21:28.920 them the filter's center frequency shifts around. 00:21:28.920 --> 00:21:31.880 Now, if you want to look at the energy in a signal in 00:21:31.880 --> 00:21:35.450 different frequency bands, then you'd like to look at it 00:21:35.450 --> 00:21:37.660 through different filters. 00:21:37.660 --> 00:21:41.390 And so the idea here, which is really the basis for many 00:21:41.390 --> 00:21:45.810 spectrum analyzers, is to build a really good quality 00:21:45.810 --> 00:21:50.320 low-pass filter and then use modulation, which is often 00:21:50.320 --> 00:21:51.260 easier to implement. 00:21:51.260 --> 00:21:56.470 Use modulation to shift the signal essentially in front of 00:21:56.470 --> 00:21:57.720 the filter. 00:22:00.600 --> 00:22:05.600 So we've worked our way through modulation with a 00:22:05.600 --> 00:22:08.350 complex exponential carrier. 00:22:08.350 --> 00:22:13.090 And what we saw, among other things with a complex 00:22:13.090 --> 00:22:17.340 exponential carrier, is that what it 00:22:17.340 --> 00:22:23.260 corresponds to is two branches. 00:22:23.260 --> 00:22:28.130 One being modulation with a cosine, and the other, 00:22:28.130 --> 00:22:29.570 modulation with a sine. 00:22:29.570 --> 00:22:34.340 And so in the real world, or in a practical system, 00:22:34.340 --> 00:22:37.610 modulation of the complex exponential carrier really 00:22:37.610 --> 00:22:41.930 would be accomplished with modulation with a sinusoidal 00:22:41.930 --> 00:22:47.330 carrier, and in particular with sinusoidal carriers that 00:22:47.330 --> 00:22:50.660 are in quadrature, as it's referred to, or equivalently 00:22:50.660 --> 00:22:54.170 90 degrees out of phase. 00:22:54.170 --> 00:23:00.750 Well, in fact sinusoidal modulation, in other words, 00:23:00.750 --> 00:23:04.970 modulation using only a sinusoidal carrier, very often 00:23:04.970 --> 00:23:10.020 is used in its own right not only for generating a complex 00:23:10.020 --> 00:23:17.060 exponential carrier, but as a carrier by itself. 00:23:17.060 --> 00:23:21.370 Let's look at what the consequences of modulation 00:23:21.370 --> 00:23:23.500 with a sinusoidal carrier are. 00:23:23.500 --> 00:23:26.830 And in particular work through, again, what the 00:23:26.830 --> 00:23:32.410 spectra are and how we get the original signal back again. 00:23:32.410 --> 00:23:39.150 So we are talking about a carrier signal which is simply 00:23:39.150 --> 00:23:42.920 a sinusoidal signal with some phase. 00:23:42.920 --> 00:23:48.650 And of course we can write that as the sum of two complex 00:23:48.650 --> 00:23:51.460 exponential signals. 00:23:51.460 --> 00:23:56.500 And so now, when we apply the modulation property we have 00:23:56.500 --> 00:24:01.190 the original spectrum, which I show here, X of omega. 00:24:01.190 --> 00:24:05.340 And that's convolved with the spectrum of the carrier. 00:24:05.340 --> 00:24:07.310 And the spectrum of the carrier, in this 00:24:07.310 --> 00:24:10.520 case, is two impulses. 00:24:10.520 --> 00:24:16.270 One at plus omega c, and one at minus omega c. 00:24:16.270 --> 00:24:19.510 And the amplitudes of these incorporate the phase. 00:24:19.510 --> 00:24:25.620 And later on in the lecture, and in subsequent lectures, 00:24:25.620 --> 00:24:30.500 I'll have a tendency to drop the theta sub c, just to keep 00:24:30.500 --> 00:24:33.820 the notation and algebra a little cleaner, but for now 00:24:33.820 --> 00:24:35.340 I've incorporated it. 00:24:35.340 --> 00:24:41.040 And so now when we apply the modulation property, what we 00:24:41.040 --> 00:24:46.270 will do is convolve this spectrum with this spectrum, 00:24:46.270 --> 00:24:50.060 and the result is that the spectrum of the original 00:24:50.060 --> 00:24:55.690 signal gets replicated at both omega sub c and at 00:24:55.690 --> 00:24:57.460 minus omega sub c. 00:24:57.460 --> 00:25:00.800 And the resulting spectrum at the output of the modulator, 00:25:00.800 --> 00:25:03.540 then, is the spectrum that I show here. 00:25:06.940 --> 00:25:09.930 Now the question, of course, is-- 00:25:09.930 --> 00:25:12.990 so now what's happened is that with a sinusoidal carrier, 00:25:12.990 --> 00:25:16.710 we've moved the spectrum to both plus omega c 00:25:16.710 --> 00:25:18.870 and minus omega c. 00:25:18.870 --> 00:25:22.570 And now if we want to get the original signal back again, 00:25:22.570 --> 00:25:26.480 what we would like to do somehow is move that spectrum 00:25:26.480 --> 00:25:29.500 back down to the origin. 00:25:29.500 --> 00:25:32.050 Now in the case of a complex exponential, 00:25:32.050 --> 00:25:33.020 that was easy to do. 00:25:33.020 --> 00:25:35.910 We'd shifted one up, we'd just shift it back down. 00:25:35.910 --> 00:25:41.320 Let's see what happens if we attempt to demodulate by again 00:25:41.320 --> 00:25:45.960 multiplying by the same sinusoidal carrier. 00:25:45.960 --> 00:25:55.690 So let's examine what happens if we now take our modulated 00:25:55.690 --> 00:26:03.600 signal and, again, modulate it onto the same sinusoidal 00:26:03.600 --> 00:26:08.010 carrier to generate the output w of t. 00:26:08.010 --> 00:26:14.340 If we look at the spectra, we have the modulated spectrum 00:26:14.340 --> 00:26:16.940 which we had initially. 00:26:16.940 --> 00:26:21.300 And we now want to convolve that, again, with the spectrum 00:26:21.300 --> 00:26:23.470 of the carrier signal. 00:26:23.470 --> 00:26:26.840 The spectrum of the carrier signal, I indicate here. 00:26:26.840 --> 00:26:30.350 And if you track through the convolution, which is fairly 00:26:30.350 --> 00:26:34.870 straightforward, then what happens as you convolve this 00:26:34.870 --> 00:26:43.270 with this is you end up with a composite spectrum, which is 00:26:43.270 --> 00:26:48.730 what I've indicated on the bottom curve, and has the 00:26:48.730 --> 00:26:52.990 spectrum of the original signal, x of t, replicated in 00:26:52.990 --> 00:26:54.220 three places. 00:26:54.220 --> 00:26:58.920 One is at minus 2 omega sub c. 00:26:58.920 --> 00:27:00.960 One is around the origin. 00:27:00.960 --> 00:27:05.980 And one is shifted up to twice the carrier frequency. 00:27:05.980 --> 00:27:08.150 Well it's this piece that we want. 00:27:08.150 --> 00:27:12.480 If we could eliminate everything else and keep this, 00:27:12.480 --> 00:27:18.360 then that would correspond to the spectrum of the original 00:27:18.360 --> 00:27:20.020 signal, x of t. 00:27:20.020 --> 00:27:21.410 How do we do that? 00:27:21.410 --> 00:27:26.870 Well, we know how to eliminate part of the spectrum and keep 00:27:26.870 --> 00:27:27.990 another part of the spectrum. 00:27:27.990 --> 00:27:29.320 That's called filtering. 00:27:29.320 --> 00:27:35.440 So what we would do is put the result of this through a 00:27:35.440 --> 00:27:37.035 low-pass filter. 00:27:37.035 --> 00:27:39.480 The low-pass filter route would retain the part of the 00:27:39.480 --> 00:27:44.720 spectrum around DC and eliminate the remaining part 00:27:44.720 --> 00:27:45.330 of the spectrum. 00:27:45.330 --> 00:27:49.090 So we would keep this part and eliminate the part of the 00:27:49.090 --> 00:27:51.410 spectrum that we have over here. 00:27:51.410 --> 00:27:54.490 And let me just draw your attention to the fact that, 00:27:54.490 --> 00:27:59.190 because of the way the algebra works out, the amplitude of 00:27:59.190 --> 00:28:01.950 this replication of the spectrum is half what the 00:28:01.950 --> 00:28:04.320 original spectrum was. 00:28:04.320 --> 00:28:09.660 And that means that ideally, to keep scale factors correct, 00:28:09.660 --> 00:28:14.890 we would choose the amplitude of this to be 2, to scale this 00:28:14.890 --> 00:28:18.150 back up to 1. 00:28:18.150 --> 00:28:23.140 So what we have is the modulator and demodulator. 00:28:23.140 --> 00:28:30.010 And just to summarize, for the case of a sinusoidal carrier 00:28:30.010 --> 00:28:33.860 as opposed to a complex exponential carrier, the 00:28:33.860 --> 00:28:39.060 modulator is just as it is in the complex exponential case. 00:28:39.060 --> 00:28:44.560 It's multiplication with the sinusoidal carrier, with 00:28:44.560 --> 00:28:47.190 frequency, omega c, and phase, theta sub c. 00:28:51.680 --> 00:28:57.740 In the demodulator we would take the modulated signal, 00:28:57.740 --> 00:29:00.820 modulate it again with the same carrier signal-- 00:29:00.820 --> 00:29:03.740 and as we'll see later, it's important to keep the same 00:29:03.740 --> 00:29:06.100 phase relationship. 00:29:06.100 --> 00:29:11.870 This result is not yet quite the demodulated signal. 00:29:11.870 --> 00:29:17.090 We need to process that with a low-pass filter that extracts 00:29:17.090 --> 00:29:21.550 the part of the spectrum around DC and throws away the 00:29:21.550 --> 00:29:24.560 upper part of the spectrum that gets generated in the 00:29:24.560 --> 00:29:26.770 second modulation process. 00:29:26.770 --> 00:29:32.380 And the resulting output is the original signal, x of t. 00:29:35.020 --> 00:29:37.490 What we've done then is we've taken x of t. 00:29:37.490 --> 00:29:41.120 We've modulated it onto a carrier. 00:29:41.120 --> 00:29:43.380 And then we've taken that modulated signal and we've 00:29:43.380 --> 00:29:45.360 figured out how to get back x of t. 00:29:45.360 --> 00:29:49.545 And of course one could ask, well, if you start with x of t 00:29:49.545 --> 00:29:52.470 and you want to get x of t back again, why bother going 00:29:52.470 --> 00:29:53.160 through all that? 00:29:53.160 --> 00:29:55.980 Why not just use x of t at the beginning and at the end? 00:29:55.980 --> 00:29:59.300 And obviously there are lots of reasons 00:29:59.300 --> 00:30:01.880 as I indicated before. 00:30:01.880 --> 00:30:06.060 And just to reiterate what they are. 00:30:06.060 --> 00:30:10.540 The notion, often, is that what you'd like to do is shift 00:30:10.540 --> 00:30:14.330 the signal into a different frequency band for 00:30:14.330 --> 00:30:19.370 transmission over some medium that is more matched to that 00:30:19.370 --> 00:30:22.090 frequency band than the frequency range of the 00:30:22.090 --> 00:30:23.440 original signal. 00:30:23.440 --> 00:30:27.280 Also, as I alluded to, is the notion that you can take lots 00:30:27.280 --> 00:30:29.460 of signals and transmit them 00:30:29.460 --> 00:30:32.650 simultaneously over one channel-- 00:30:32.650 --> 00:30:35.400 whether the channel is a wire, a microwave link, a satellite 00:30:35.400 --> 00:30:36.650 link, or whatever-- 00:30:36.650 --> 00:30:40.430 again, using the idea of modulation. 00:30:40.430 --> 00:30:45.460 And what that process is referred to as is 00:30:45.460 --> 00:30:46.680 multiplexing. 00:30:46.680 --> 00:30:49.270 And let me just quickly indicate what that 00:30:49.270 --> 00:30:53.280 multiplexing process corresponds to. 00:30:53.280 --> 00:31:00.560 We could think, for example, of taking one signal and 00:31:00.560 --> 00:31:06.760 modulating in it onto one carrier with one carrier 00:31:06.760 --> 00:31:12.280 frequency, taking a second signal, modulating it onto a 00:31:12.280 --> 00:31:16.690 different carrier frequency, taking a third signal and 00:31:16.690 --> 00:31:20.940 modulating it onto a third carrier frequency, et cetera. 00:31:20.940 --> 00:31:26.520 And if we choose these carrier frequencies appropriately, 00:31:26.520 --> 00:31:30.380 then we can add all those together-- 00:31:30.380 --> 00:31:33.540 and do it in such a way that the spectra don't overlap-- 00:31:33.540 --> 00:31:38.920 and end up with one broader band signal that incorporates 00:31:38.920 --> 00:31:45.070 the information simultaneously in all of those signals. 00:31:45.070 --> 00:31:49.690 So just to illustrate that in the frequency domain. 00:31:49.690 --> 00:31:56.890 What we have are our three spectra, Xa, Xb, and Xc. 00:31:56.890 --> 00:32:01.020 And we would, for example, take this spectrum and 00:32:01.020 --> 00:32:05.780 modulate it to a carrier frequency, omega sub a. 00:32:05.780 --> 00:32:09.140 We can take this spectrum and modulate it to a carrier 00:32:09.140 --> 00:32:14.600 frequency, omega sub b, where omega sub b is chosen so that 00:32:14.600 --> 00:32:18.140 when we add these two together they don't overlap, so that 00:32:18.140 --> 00:32:20.640 they can eventually be separated out. 00:32:20.640 --> 00:32:24.380 And then we can do the same thing with the third signal, 00:32:24.380 --> 00:32:27.990 and put that in a frequency range over here, being careful 00:32:27.990 --> 00:32:30.330 that none of those overlap. 00:32:30.330 --> 00:32:32.720 And when we add all those together, the composite 00:32:32.720 --> 00:32:36.890 spectrum is what I show here. 00:32:36.890 --> 00:32:43.430 And as you can see, essentially, by doing 00:32:43.430 --> 00:32:46.280 appropriate band-pass filtering we can pull out 00:32:46.280 --> 00:32:50.710 whatever part of the spectrum we choose to, and then 00:32:50.710 --> 00:32:53.020 demodulate that in the appropriate way. 00:32:53.020 --> 00:32:55.740 And of course we can do this, not just with three signals, 00:32:55.740 --> 00:32:59.140 but perhaps with tens or hundreds of signals. 00:32:59.140 --> 00:33:03.460 So that's a process that is typically referred to as 00:33:03.460 --> 00:33:04.380 multiplexing. 00:33:04.380 --> 00:33:08.130 And as I've described it here, it's referred to as 00:33:08.130 --> 00:33:10.460 frequency-division multiplexing. 00:33:10.460 --> 00:33:14.030 That is, dividing the frequency band into cells and 00:33:14.030 --> 00:33:19.560 plunking different signals into each one of those. 00:33:19.560 --> 00:33:23.650 And so if we want now to recover one of those channels 00:33:23.650 --> 00:33:27.250 in a frequency-division multiplex system, as I 00:33:27.250 --> 00:33:31.060 indicated, we would first demultiplex. 00:33:31.060 --> 00:33:35.590 Demultiplexing corresponding to pulling out the appropriate 00:33:35.590 --> 00:33:40.020 channel with a band-pass filter. 00:33:40.020 --> 00:33:45.190 And after demultiplexing, we would then demodulate. 00:33:45.190 --> 00:33:50.510 And we would demodulate with the carrier appropriate to the 00:33:50.510 --> 00:33:52.340 channel that we've pulled out. 00:33:52.340 --> 00:33:56.800 And the demodulation, of course, involves multiplying 00:33:56.800 --> 00:34:00.790 by the carrier and doing appropriate low-pass filtering 00:34:00.790 --> 00:34:03.510 to finally get the signal back. 00:34:03.510 --> 00:34:13.850 And frequency-division multiplexing is the type of 00:34:13.850 --> 00:34:18.679 multiplexing that's used, for example, in typical broadcast 00:34:18.679 --> 00:34:22.090 AM radio systems, where all the channels are 00:34:22.090 --> 00:34:23.780 superimposed together. 00:34:23.780 --> 00:34:26.520 And it's your home radio receiver that does the 00:34:26.520 --> 00:34:29.760 appropriate demultiplexing and demodulating. 00:34:29.760 --> 00:34:32.800 And of course, you can see that not only is modulation an 00:34:32.800 --> 00:34:36.830 important part of that, but as I alluded to in the last 00:34:36.830 --> 00:34:40.750 lecture, filtering also becomes important part of 00:34:40.750 --> 00:34:42.000 these practical systems. 00:34:46.020 --> 00:34:51.070 Now, the kind of amplitude modulation that I've talked 00:34:51.070 --> 00:34:57.310 about so far is what's referred to as synchronous 00:34:57.310 --> 00:34:58.880 modulation. 00:34:58.880 --> 00:35:04.060 And the reason for the term synchronous is that what's 00:35:04.060 --> 00:35:10.520 implied in these systems is a synchronization between the 00:35:10.520 --> 00:35:12.550 transmitter and receiver. 00:35:12.550 --> 00:35:19.200 In particular, in the system as we've talked about it, the 00:35:19.200 --> 00:35:29.010 modulator and the demodulator have a synchronization in both 00:35:29.010 --> 00:35:30.790 frequency and phase. 00:35:30.790 --> 00:35:33.630 The phase here is indicated as theta sub c. 00:35:33.630 --> 00:35:39.860 And if we take a look at the demodulator, the demodulator 00:35:39.860 --> 00:35:44.980 has phase of theta sub c. 00:35:44.980 --> 00:35:50.620 And in general, there's the issue of whether we can 00:35:50.620 --> 00:35:53.480 maintain that synchronization between the modulator and 00:35:53.480 --> 00:35:54.830 demodulator. 00:35:54.830 --> 00:36:00.160 And so what we want to examine now, more generally, is what 00:36:00.160 --> 00:36:03.750 the consequence might be, and the solution to the resulting 00:36:03.750 --> 00:36:09.300 problems, if we don't have synchronization between the 00:36:09.300 --> 00:36:11.380 modulator and demodulator. 00:36:11.380 --> 00:36:14.070 Synchronization in terms of phase. 00:36:14.070 --> 00:36:18.260 And there also is another problem, which is the issue of 00:36:18.260 --> 00:36:21.230 synchronization in frequency. 00:36:21.230 --> 00:36:23.840 That's examined more in the text. 00:36:23.840 --> 00:36:26.330 And what I'll focus on here is just the issue of 00:36:26.330 --> 00:36:28.900 synchronization in phase, to give you some sense of what 00:36:28.900 --> 00:36:31.920 the issue is. 00:36:31.920 --> 00:36:35.240 So now what we want to look at is what happens if we have a 00:36:35.240 --> 00:36:43.580 modulator with phase, theta sub c, and a demodulator where 00:36:43.580 --> 00:36:47.440 the phase, instead of being theta sub c, is some other 00:36:47.440 --> 00:36:51.280 phase, phi sub c. 00:36:51.280 --> 00:36:56.260 And if you track through the details and the algebra, then 00:36:56.260 --> 00:37:02.040 what you'll find is that the output of the low-pass filter, 00:37:02.040 --> 00:37:07.650 rather than being x of t, the signal that we want, is x of t 00:37:07.650 --> 00:37:10.660 multiplied by a scale factor. 00:37:10.660 --> 00:37:13.340 And the scale factor is the cosine of the phase 00:37:13.340 --> 00:37:14.790 difference. 00:37:14.790 --> 00:37:18.910 Now one could ask, OK well, what's the big deal about 00:37:18.910 --> 00:37:19.790 scale factor? 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 00:37:21.930 --> 00:37:23.680 make it small. 00:37:23.680 --> 00:37:25.000 But there are several points. 00:37:25.000 --> 00:37:28.600 One is, notice, for example, that if the phase difference 00:37:28.600 --> 00:37:32.750 between the modulator and demodulator is 90 degrees, 00:37:32.750 --> 00:37:36.920 then the output of the demodulator is zero. 00:37:36.920 --> 00:37:39.970 Or if it isn't quite 90 degrees, the 00:37:39.970 --> 00:37:41.580 amplitude might be small. 00:37:41.580 --> 00:37:44.300 And the implication would be that if there's other noise it 00:37:44.300 --> 00:37:46.980 gets injected in the system, the signal-to-noise 00:37:46.980 --> 00:37:49.120 ratio is very low. 00:37:49.120 --> 00:37:55.430 Now even worse is the issue that if there's a phase 00:37:55.430 --> 00:37:58.220 difference, but the exact phase difference isn't 00:37:58.220 --> 00:38:01.990 maintained, so that the modulator and demodulator kind 00:38:01.990 --> 00:38:05.990 of fade in and out of phase, then the output of the 00:38:05.990 --> 00:38:14.240 demodulator is x of t multiplied by a time-varying 00:38:14.240 --> 00:38:16.740 fading term, which is the cosine of the phase 00:38:16.740 --> 00:38:17.500 difference. 00:38:17.500 --> 00:38:21.080 Well what that means, essentially, is that if you 00:38:21.080 --> 00:38:26.470 use this kind of system to do the demodulation, then what 00:38:26.470 --> 00:38:31.860 you need to be careful about is maintaining synchronization 00:38:31.860 --> 00:38:35.560 in phase, and also in frequency, between the 00:38:35.560 --> 00:38:39.470 modulator and the demodulator. 00:38:39.470 --> 00:38:44.200 Now there are alternatives to this. 00:38:44.200 --> 00:38:48.590 And the alternative is what's referred to as asynchronous 00:38:48.590 --> 00:38:50.420 demodulation. 00:38:50.420 --> 00:38:54.530 And let me indicate what the idea behind asynchronous 00:38:54.530 --> 00:38:57.760 demodulation is. 00:38:57.760 --> 00:39:01.190 Now, recall that what we've done in amplitude modulation 00:39:01.190 --> 00:39:06.040 is to take the carrier signal and vary its amplitude with 00:39:06.040 --> 00:39:09.070 the signal that eventually we want to get back. 00:39:09.070 --> 00:39:12.530 So if we look at the amplitude-modulated waveform, 00:39:12.530 --> 00:39:16.880 it might typically look as I indicate here. 00:39:16.880 --> 00:39:22.910 And we're trying to get back the envelope. 00:39:22.910 --> 00:39:26.060 Well, one could imagine building a circuit, or 00:39:26.060 --> 00:39:28.450 designing a device, which in some sense 00:39:28.450 --> 00:39:30.490 will track the envelope. 00:39:30.490 --> 00:39:36.550 And a common circuit to do that is a fairly simple 00:39:36.550 --> 00:39:41.750 circuit consisting of a diode and a resistor and capacitor 00:39:41.750 --> 00:39:43.230 in parallel. 00:39:43.230 --> 00:39:48.580 The idea being that the capacitor charges up as this 00:39:48.580 --> 00:39:50.920 waveform moves up to its peak. 00:39:50.920 --> 00:39:55.690 And then as the waveform drops down, the capacitor discharges 00:39:55.690 --> 00:39:56.760 through the resistor. 00:39:56.760 --> 00:39:59.520 And it kind of tracks the envelope. 00:39:59.520 --> 00:40:03.740 In fact, the kind of output that we would get is the type 00:40:03.740 --> 00:40:07.150 of behavior that I've indicated here. 00:40:07.150 --> 00:40:13.090 And then that is a type of demodulation. 00:40:13.090 --> 00:40:15.570 It's a demodulation that doesn't require 00:40:15.570 --> 00:40:19.470 synchronization between the modulator and demodulator. 00:40:19.470 --> 00:40:25.400 And it's fairly inexpensive to build. 00:40:25.400 --> 00:40:30.660 But it has, obviously, some tradeoffs associated with it. 00:40:30.660 --> 00:40:34.580 Well, to indicate where the tradeoff comes from, or where 00:40:34.580 --> 00:40:39.210 the issue surfaces, notice that what we're doing is 00:40:39.210 --> 00:40:43.420 tracking the envelope of the sinusoidal signal. 00:40:43.420 --> 00:40:46.110 And we're calling that, or we're assuming that that is 00:40:46.110 --> 00:40:50.010 our original signal, x of t. 00:40:50.010 --> 00:40:56.300 Well, suppose that x of t, the original signal, is sometimes 00:40:56.300 --> 00:40:59.090 positive and sometimes negative. 00:40:59.090 --> 00:41:03.490 What might we see as we look at the output of the 00:41:03.490 --> 00:41:04.590 demodulator? 00:41:04.590 --> 00:41:07.950 Well, the output of the demodulator would follow the 00:41:07.950 --> 00:41:11.570 envelope down, and then it would follow the 00:41:11.570 --> 00:41:13.940 envelope back up again. 00:41:13.940 --> 00:41:17.090 In other words, what it would tend to generate is a 00:41:17.090 --> 00:41:20.680 full-wave rectified version of the signal that you were 00:41:20.680 --> 00:41:22.610 really trying to get back. 00:41:22.610 --> 00:41:24.970 Now, there's a simple solution to this. 00:41:24.970 --> 00:41:30.290 The simple solution is to make sure that the signal that is 00:41:30.290 --> 00:41:33.510 the modulating signal, x of t, never goes negative. 00:41:33.510 --> 00:41:38.460 So if it happens to-- a voice signal tends to go negative. 00:41:38.460 --> 00:41:42.570 If it happens to, we can simply add a constant to it, 00:41:42.570 --> 00:41:44.540 add a large enough constant, so that it 00:41:44.540 --> 00:41:46.670 always stays positive. 00:41:46.670 --> 00:41:49.840 Well let's look at that. 00:41:49.840 --> 00:41:52.845 What we want to do then, if we're considering asynchronous 00:41:52.845 --> 00:42:03.090 demodulation, is to take our original signal, x of t, and 00:42:03.090 --> 00:42:07.960 add to it a constant, where the constant is made large 00:42:07.960 --> 00:42:13.300 enough so that we're sure that this is a positive signal. 00:42:13.300 --> 00:42:15.410 And incidentally, let me just draw your attention to the 00:42:15.410 --> 00:42:19.870 fact that I'm now suppressing the phase on the carrier 00:42:19.870 --> 00:42:24.280 signal, since the phase is not important to the argument and 00:42:24.280 --> 00:42:28.550 it's just some additional notation to carry around. 00:42:28.550 --> 00:42:32.040 So the idea then, is add a constant to x of t. 00:42:32.040 --> 00:42:36.530 Notice that if we just take this term and expand it out 00:42:36.530 --> 00:42:41.940 into two terms, x of t cosine omega c t plus a times cosine 00:42:41.940 --> 00:42:47.690 omega c t, then in block diagram terms we can represent 00:42:47.690 --> 00:42:50.430 that as I've shown here. 00:42:50.430 --> 00:42:56.090 And so it would correspond to modulating the signal, x of t, 00:42:56.090 --> 00:43:01.430 onto the carrier, omega sub c t, and also injecting some 00:43:01.430 --> 00:43:05.500 carrier with an amplitude, A. And the output of the 00:43:05.500 --> 00:43:08.920 modulator is then the sum of those two. 00:43:08.920 --> 00:43:14.610 And depending on exactly what this value A is will influence 00:43:14.610 --> 00:43:16.610 what the envelope will look like. 00:43:16.610 --> 00:43:21.140 And I indicate below, two possibilities. 00:43:21.140 --> 00:43:26.110 One is where I've made A fairly large, and one is where 00:43:26.110 --> 00:43:29.350 I've made A significantly smaller. 00:43:29.350 --> 00:43:34.010 And there are both positive and negative issues associated 00:43:34.010 --> 00:43:36.860 with whether A is too large or A is too small. 00:43:36.860 --> 00:43:49.300 For example, if A is large in relation to the amplitude of 00:43:49.300 --> 00:43:53.360 the signal, then this envelope tends to be very flat. 00:43:53.360 --> 00:43:56.510 And it tends to be easy to track it with that simple 00:43:56.510 --> 00:44:03.450 diode RC circuit, as compared with the case down here. 00:44:03.450 --> 00:44:09.060 On the other hand, there is a price that you pay for this 00:44:09.060 --> 00:44:10.660 kind of envelope. 00:44:10.660 --> 00:44:13.350 And the price that you pay is perhaps best seen in the 00:44:13.350 --> 00:44:15.720 frequency domain. 00:44:15.720 --> 00:44:19.650 If we look in the frequency domain, here is 00:44:19.650 --> 00:44:22.620 our original spectrum. 00:44:22.620 --> 00:44:27.450 Here is the spectrum at the output of the modulator. 00:44:27.450 --> 00:44:31.030 And the impulse that occurs here corresponds to the 00:44:31.030 --> 00:44:33.370 carrier that's injected. 00:44:33.370 --> 00:44:37.510 The larger A is, the more carrier is injected. 00:44:37.510 --> 00:44:41.900 The more carrier that's injected, the easier it is for 00:44:41.900 --> 00:44:45.960 the envelope detector to demodulate. 00:44:45.960 --> 00:44:49.360 So one can ask, why not just put a lot in? 00:44:49.360 --> 00:44:54.670 Well, the obvious answer is that it's not an 00:44:54.670 --> 00:44:58.130 information-carrying part of the signal. 00:44:58.130 --> 00:45:01.670 And so in some sense it represents an inefficiency in 00:45:01.670 --> 00:45:04.890 transmission, because what you're transmitting is power, 00:45:04.890 --> 00:45:08.820 energy, that doesn't have any information 00:45:08.820 --> 00:45:09.900 associated with it. 00:45:09.900 --> 00:45:13.120 It's simply the injection of a carrier to make the 00:45:13.120 --> 00:45:18.770 demodulation for an asynchronous demodulator-- 00:45:18.770 --> 00:45:23.630 to make the demodulation easier. 00:45:23.630 --> 00:45:25.340 And so there's this tradeoff. 00:45:25.340 --> 00:45:29.590 And in fact, one represents the tradeoff and the 00:45:29.590 --> 00:45:32.230 associated parameters very often in terms of percent 00:45:32.230 --> 00:45:40.240 modulation, where the percent modulation is essentially the 00:45:40.240 --> 00:45:45.280 ratio of the maximum signal level to the amplitude of the 00:45:45.280 --> 00:45:47.380 injected carrier. 00:45:47.380 --> 00:45:52.750 And depending on whether the modulation's very high or very 00:45:52.750 --> 00:45:57.670 low, the tradeoff is that the transmission is more 00:45:57.670 --> 00:45:59.800 inefficient and it takes more energy, but the 00:45:59.800 --> 00:46:01.190 demodulator is simpler. 00:46:01.190 --> 00:46:04.910 Or the demodulator is more complicated but the 00:46:04.910 --> 00:46:06.840 transmission is simpler. 00:46:06.840 --> 00:46:09.780 Now, there are situations where you might very well want 00:46:09.780 --> 00:46:10.720 to use one or the other. 00:46:10.720 --> 00:46:16.670 For example, in home radio you're often willing to 00:46:16.670 --> 00:46:21.610 transmit a lot of power so that you can have inexpensive 00:46:21.610 --> 00:46:24.100 consumer-oriented receivers. 00:46:24.100 --> 00:46:26.790 On the other hand, in satellite communication you're 00:46:26.790 --> 00:46:31.070 willing to pay a very high price for the modulators and 00:46:31.070 --> 00:46:34.620 demodulators, but it's the amount of power that's 00:46:34.620 --> 00:46:37.010 transmitted that's at a premium. 00:46:37.010 --> 00:46:41.120 And so in one case, satellite communication, you would use 00:46:41.120 --> 00:46:43.890 synchronous modulation and demodulation. 00:46:43.890 --> 00:46:46.090 Whereas in typical consumer-oriented 00:46:46.090 --> 00:46:54.000 broadcasting, you would use an asynchronous system and 00:46:54.000 --> 00:46:56.980 transmit more power, even if it's inefficient, so that the 00:46:56.980 --> 00:47:00.470 demodulator can be simpler. 00:47:00.470 --> 00:47:06.400 Now, in the asynchronous system, as we've indicated, 00:47:06.400 --> 00:47:09.970 there's one source of inefficiency, which is this 00:47:09.970 --> 00:47:11.900 injection of the carrier. 00:47:11.900 --> 00:47:16.930 There also is a somewhat different issue, related to 00:47:16.930 --> 00:47:21.120 inefficiency in sinusoidal amplitude modulation. 00:47:21.120 --> 00:47:26.710 And it's an inefficiency that is separate from the issue of 00:47:26.710 --> 00:47:28.850 synchronous versus asynchronous systems. 00:47:28.850 --> 00:47:31.760 In other words, it's not associated with the injection 00:47:31.760 --> 00:47:35.410 of the carrier, it's a very different issue. 00:47:35.410 --> 00:47:38.140 Let me indicate what that is. 00:47:38.140 --> 00:47:44.050 Let's look again at the spectrum of x of t, which I've 00:47:44.050 --> 00:47:45.960 indicated here. 00:47:45.960 --> 00:47:54.600 And in a sinusoidal amplitude modulation system, we would 00:47:54.600 --> 00:47:58.650 center it around plus and minus the carrier frequency. 00:47:58.650 --> 00:48:02.500 Now, notice that in the original system we occupy a 00:48:02.500 --> 00:48:07.490 frequency spectrum that's 2 times omega sub M. By the time 00:48:07.490 --> 00:48:11.450 we've shifted it, thinking of positive and negative 00:48:11.450 --> 00:48:15.660 frequencies, we've used up twice as much of 00:48:15.660 --> 00:48:17.950 the frequency spectrum. 00:48:17.950 --> 00:48:22.300 Well you could say, OK, let's just shift this up this way 00:48:22.300 --> 00:48:24.470 and get rid of this part. 00:48:24.470 --> 00:48:25.860 That's of course what the complex 00:48:25.860 --> 00:48:27.860 exponential carrier did. 00:48:27.860 --> 00:48:31.360 And the issue there is that now you've got to transmit 00:48:31.360 --> 00:48:35.110 both a real part and an imaginary part. 00:48:35.110 --> 00:48:39.190 So what you can think about, and ask, is if you still want 00:48:39.190 --> 00:48:43.760 to transmit a real-valued signal, how can you somehow 00:48:43.760 --> 00:48:47.540 remove the inefficiency or redundancy in the spectrum? 00:48:47.540 --> 00:48:52.750 Well, notice that what we have is this spectrum moved here, 00:48:52.750 --> 00:48:55.090 and moved here. 00:48:55.090 --> 00:49:02.180 And we could imagine building real-valued signal by 00:49:02.180 --> 00:49:06.770 eliminating what I refer to here as the lower sideband out 00:49:06.770 --> 00:49:11.210 of the positive frequencies, and the lower sideband out of 00:49:11.210 --> 00:49:13.710 the negative frequencies. 00:49:13.710 --> 00:49:18.650 And in effect, what we've done is taken just the positive 00:49:18.650 --> 00:49:22.510 frequencies here, shifted them there, the negative 00:49:22.510 --> 00:49:24.460 frequencies here, and shifted them here. 00:49:24.460 --> 00:49:29.000 And the resulting spectrum is what I indicate below. 00:49:29.000 --> 00:49:32.840 Well, this is what is often done. 00:49:32.840 --> 00:49:37.020 And what it's referred to as is single sideband. 00:49:37.020 --> 00:49:40.790 What we've done is kept the upper sideband, in this 00:49:40.790 --> 00:49:42.050 particular case. 00:49:42.050 --> 00:49:46.390 We could alternatively think of putting this system 00:49:46.390 --> 00:49:52.760 together where we retain the lower sideband instead of the 00:49:52.760 --> 00:49:54.250 upper sideband. 00:49:54.250 --> 00:49:59.870 And in either case, what we've removed is an inefficiency in 00:49:59.870 --> 00:50:01.760 transmission of the signal. 00:50:01.760 --> 00:50:05.380 Namely we have a real-valued signal, but it only requires 00:50:05.380 --> 00:50:10.710 as much total bandwidth, in terms of the frequencies in 00:50:10.710 --> 00:50:14.480 which there's energy present, as the original signal. 00:50:14.480 --> 00:50:16.190 Well how do we do this? 00:50:16.190 --> 00:50:18.160 There are a variety of ways. 00:50:18.160 --> 00:50:22.810 And there's one procedure that is discussed in more detail in 00:50:22.810 --> 00:50:26.160 the text, which uses what's referred to as a 90 degree 00:50:26.160 --> 00:50:27.690 phase splitter. 00:50:27.690 --> 00:50:30.720 The simplest way, at least conceptually, is to think 00:50:30.720 --> 00:50:33.420 about doing it with filtering. 00:50:33.420 --> 00:50:39.810 And the idea simply is that if we have our modulated signal-- 00:50:39.810 --> 00:50:41.810 here's the spectrum of the modulated signal. 00:50:41.810 --> 00:50:44.780 And if that modulated signal is simply put through a 00:50:44.780 --> 00:50:51.740 high-pass filter, then the result will be to eliminate 00:50:51.740 --> 00:50:56.240 the lower sideband, if we choose the high-pass filter to 00:50:56.240 --> 00:50:59.530 have a characteristic as I indicate here. 00:50:59.530 --> 00:51:03.820 So this, conceptually, is a very sharp cutoff filter. 00:51:03.820 --> 00:51:06.810 And what it eliminates are the lower sidebands. 00:51:06.810 --> 00:51:11.770 And the resulting spectrum is what we have below. 00:51:11.770 --> 00:51:17.870 And this in fact is really the basic idea behind 00:51:17.870 --> 00:51:20.130 single-sideband transmission. 00:51:20.130 --> 00:51:22.180 Again, there's a tradeoff. 00:51:22.180 --> 00:51:26.280 It's clearly more efficient than double-sideband 00:51:26.280 --> 00:51:32.770 transmission, but also has the complication, or additional 00:51:32.770 --> 00:51:35.760 issue, that the modulator becomes a little more 00:51:35.760 --> 00:51:39.240 complicated because you need this filtering operation, or 00:51:39.240 --> 00:51:41.770 some equivalent operation, to get rid of 00:51:41.770 --> 00:51:43.020 the unwanted sideband. 00:51:46.730 --> 00:51:52.050 Well, this is a fairly quick tour through a variety of 00:51:52.050 --> 00:51:53.990 issues related to modulation. 00:51:53.990 --> 00:51:58.180 And it really is just the tip of the iceberg, obviously. 00:51:58.180 --> 00:52:02.180 Modulation in the context of sinusoidal modulation, as 00:52:02.180 --> 00:52:06.710 we've talked about, has a lot of detailed issues 00:52:06.710 --> 00:52:09.030 associated with it. 00:52:09.030 --> 00:52:14.220 It's important to recognize, and to be somewhat pleased by 00:52:14.220 --> 00:52:19.030 the fact, that not only with the mathematical foundations 00:52:19.030 --> 00:52:24.130 that we've developed can we understand the basics of 00:52:24.130 --> 00:52:25.610 sinusoidal amplitude modulation. 00:52:25.610 --> 00:52:32.190 But what you'll find if you dig into this somewhat deeper 00:52:32.190 --> 00:52:35.940 that the basic background that we built up so far-- 00:52:35.940 --> 00:52:38.190 the mathematical tools-- 00:52:38.190 --> 00:52:42.080 are really pretty much what you need for a much deeper 00:52:42.080 --> 00:52:44.505 understanding of all of the issues involved. 00:52:47.180 --> 00:52:51.800 So from what might have seemed like a fairly abstract 00:52:51.800 --> 00:52:54.920 mathematical property associated with the Fourier 00:52:54.920 --> 00:53:00.460 transform, we've begun to develop what should give you 00:53:00.460 --> 00:53:05.000 the sense of some important practical considerations. 00:53:05.000 --> 00:53:09.770 And as we'll see the next lecture, very much the same 00:53:09.770 --> 00:53:15.350 kinds of notions apply for discrete time, sinusoidal, and 00:53:15.350 --> 00:53:19.000 complex exponential amplitude modulation. 00:53:19.000 --> 00:53:23.060 And also as I indicated at the beginning of the lecture, in 00:53:23.060 --> 00:53:27.120 the next lecture we'll also talk about what's referred to 00:53:27.120 --> 00:53:28.770 as pulse amplitude modulation. 00:53:28.770 --> 00:53:30.770 It's a different kind of carrier. 00:53:30.770 --> 00:53:36.260 And what that will lead to, among other things, is a very 00:53:36.260 --> 00:53:43.540 important bridge between the notions of continuous time and 00:53:43.540 --> 00:53:45.080 the notions of discrete time. 00:53:45.080 --> 00:53:46.330 Thank you.