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 --> 00:00:55.940 ocw.mit.edu 00:00:55.940 --> 00:00:58.420 PROFESSOR: In the last lecture, we began the 00:00:58.420 --> 00:01:00.360 discussion of modulation. 00:01:00.360 --> 00:01:03.930 And in particular, what we focused on, was the 00:01:03.930 --> 00:01:05.710 continuous time case. 00:01:05.710 --> 00:01:08.420 And in talking about continuous time modulation, 00:01:08.420 --> 00:01:09.950 we've covered a number of topics. 00:01:09.950 --> 00:01:15.330 We talked about the properties and analysis of modulation 00:01:15.330 --> 00:01:20.160 when we had a complex exponential carrier signal. 00:01:20.160 --> 00:01:23.960 We talked about the properties and analysis in the case of a 00:01:23.960 --> 00:01:26.130 sinusoidal carrier. 00:01:26.130 --> 00:01:31.180 And in that context and related to the application 00:01:31.180 --> 00:01:33.900 associated with communications, we talked 00:01:33.900 --> 00:01:39.660 about synchronous modulation, asynchronous modulation and 00:01:39.660 --> 00:01:45.360 also the notion of single side band modulation. 00:01:45.360 --> 00:01:49.960 In the lecture today, there are two issues that I'd like 00:01:49.960 --> 00:01:51.920 to address, broad topics. 00:01:51.920 --> 00:01:57.270 One is a parallel discussion, particularly, as associated 00:01:57.270 --> 00:02:00.940 with complex exponential and sinusoidal modulation for 00:02:00.940 --> 00:02:03.260 discrete time signals. 00:02:03.260 --> 00:02:07.940 And the second is the introduction and analysis of 00:02:07.940 --> 00:02:11.710 another kind of carriers, specifically a pulse kind of 00:02:11.710 --> 00:02:15.680 carrier in continuous time leading to the notions of 00:02:15.680 --> 00:02:19.150 pulse amplitude modulation and, eventually, a very 00:02:19.150 --> 00:02:22.990 powerful theorem and result called the sampling theorem. 00:02:22.990 --> 00:02:27.100 Well, let me begin the lecture, though, focusing on 00:02:27.100 --> 00:02:31.380 the discrete time modulation to essentially draw your 00:02:31.380 --> 00:02:36.090 attention to the fact that the analysis in discrete time very 00:02:36.090 --> 00:02:40.200 much parallels the analysis in continuous time. 00:02:40.200 --> 00:02:44.330 Well, let's consider, in the discrete time case, just as we 00:02:44.330 --> 00:02:49.810 had in continuous time, a signal modulating a carrier 00:02:49.810 --> 00:02:56.170 signal and the resulting modulated signal is y of n. 00:02:56.170 --> 00:03:01.510 And it was in continuous time the modulation property 00:03:01.510 --> 00:03:06.460 associated with the Fourier Transform that provided the 00:03:06.460 --> 00:03:08.340 basis for the analysis. 00:03:08.340 --> 00:03:10.610 And exactly the same thing is true in the 00:03:10.610 --> 00:03:12.120 discrete time case. 00:03:12.120 --> 00:03:16.200 In particular, what we have in discrete time, is the 00:03:16.200 --> 00:03:19.050 modulation property as it relates to the Fourier 00:03:19.050 --> 00:03:23.820 Transform, which tells us that the Fourier Transform of the 00:03:23.820 --> 00:03:30.080 modulated signal is the convolution of the Fourier 00:03:30.080 --> 00:03:34.840 Transform of the carrier and the Fourier Transform of the 00:03:34.840 --> 00:03:36.460 modulated signal. 00:03:36.460 --> 00:03:41.070 And the only real difference at issue here is that, in the 00:03:41.070 --> 00:03:45.590 discrete time case, what we're talking about is a periodic 00:03:45.590 --> 00:03:47.970 convolution because the specter, 00:03:47.970 --> 00:03:49.490 of course is periodic. 00:03:49.490 --> 00:03:53.080 Whereas, in the continuous time case, it was an aperiodic 00:03:53.080 --> 00:03:55.010 convolution. 00:03:55.010 --> 00:03:58.860 So let's parallel the discussion, and in particular, 00:03:58.860 --> 00:04:05.300 what we'll focus on is, first, a complex exponential carrier 00:04:05.300 --> 00:04:08.820 and second a sinusoidal carrier. 00:04:08.820 --> 00:04:11.470 And we'll see how this parallels our discussion in 00:04:11.470 --> 00:04:16.029 continuous time, and we'll make fairly brief reference as 00:04:16.029 --> 00:04:20.350 we introduce the pulse carrier for continuous time. 00:04:20.350 --> 00:04:22.900 We'll make very brief reference to the pulse carrier 00:04:22.900 --> 00:04:26.790 for discrete time indicating that, again, the analysis and 00:04:26.790 --> 00:04:30.280 discrete time and continuous time is very parallel. 00:04:30.280 --> 00:04:33.990 So let's, first, consider complex exponential and 00:04:33.990 --> 00:04:37.650 sinusoidal carriers for the discrete time case, 00:04:37.650 --> 00:04:41.960 emphasizing the very strong parallel and similarity 00:04:41.960 --> 00:04:46.170 between discrete time and continuous time. 00:04:46.170 --> 00:04:50.740 Well we have, once, again the modulation property. 00:04:50.740 --> 00:04:55.000 And the modulation property tells us that the spectrum of 00:04:55.000 --> 00:05:00.280 the modulated signal is the periodic convolution of the 00:05:00.280 --> 00:05:02.120 two spectra. 00:05:02.120 --> 00:05:06.200 And let's consider, for example, an input, or 00:05:06.200 --> 00:05:09.480 modulating spectrum, as I've indicated here. 00:05:09.480 --> 00:05:14.530 And since we want to consider, first of all, a complex 00:05:14.530 --> 00:05:19.670 exponential carrier, we'll consider the case of c of n 00:05:19.670 --> 00:05:23.940 equal to e to the j omega sub cn. 00:05:23.940 --> 00:05:29.040 And let me stress, by the way, as I did in the continuous 00:05:29.040 --> 00:05:33.510 time case, that I'll tend to suppress the phase angle 00:05:33.510 --> 00:05:35.260 which, of course, can be associated 00:05:35.260 --> 00:05:37.610 with the carrier also. 00:05:37.610 --> 00:05:41.260 All right, so we have, then, the spectrum of 00:05:41.260 --> 00:05:43.320 the modulated signal. 00:05:43.320 --> 00:05:46.130 The spectra, the carrier signal, if this is the 00:05:46.130 --> 00:05:50.160 carrier, then it's spectrum is an impulse train, and that 00:05:50.160 --> 00:05:54.690 impulse train, I've indicated here. 00:05:54.690 --> 00:05:59.110 And let me stress, also, that in the discrete time case, of 00:05:59.110 --> 00:06:03.820 course, these spectra and all of the spectra involved, are 00:06:03.820 --> 00:06:06.500 periodic with a period of 2 pi. 00:06:06.500 --> 00:06:10.000 So this then is the spectrum of the carrier signal. 00:06:10.000 --> 00:06:12.440 This is the spectrum of the input signal. 00:06:12.440 --> 00:06:15.730 The periodic convolution of these two is the spectrum the 00:06:15.730 --> 00:06:17.360 modulated signal. 00:06:17.360 --> 00:06:22.940 And the result is, then, this spectrum shifted to a center 00:06:22.940 --> 00:06:28.170 frequency, which is the carrier frequency omega sub c. 00:06:28.170 --> 00:06:34.120 So the result of modulation with a complex exponential is 00:06:34.120 --> 00:06:39.940 a straightforward shift of the spectrum so that what occurred 00:06:39.940 --> 00:06:43.820 around zero frequency now occurs around the frequency 00:06:43.820 --> 00:06:47.040 omega sub c. 00:06:47.040 --> 00:06:51.670 Now, in the continuous time case, we demodulated, when we 00:06:51.670 --> 00:06:55.600 had a complex exponential carrier, we demodulated by, 00:06:55.600 --> 00:06:58.930 essentially, just shifting the spectrum back. 00:06:58.930 --> 00:07:01.950 And in fact, in the discrete time case, were able to do 00:07:01.950 --> 00:07:04.180 exactly the same thing. 00:07:04.180 --> 00:07:11.660 So if we were to replace c of n which is either j omega sub 00:07:11.660 --> 00:07:17.490 cn by c of n equals e to the minus j omega sub cn, the 00:07:17.490 --> 00:07:23.920 resulting spectra would be an impulse train, as I indicate 00:07:23.920 --> 00:07:32.340 here, and the result of multiply y of n by that new 00:07:32.340 --> 00:07:36.240 carrier, in the frequency domain as a convolution of 00:07:36.240 --> 00:07:40.300 these two, and it's relatively straightforward to verify that 00:07:40.300 --> 00:07:44.160 if you convolve these with a periodic convolution, then 00:07:44.160 --> 00:07:47.300 that will get us back to the original spectrum that we 00:07:47.300 --> 00:07:49.460 started with. 00:07:49.460 --> 00:07:52.270 So what's happened in the discrete time case, with the 00:07:52.270 --> 00:07:55.970 complex exponential, is exactly the same as in 00:07:55.970 --> 00:07:57.420 continuous time. 00:07:57.420 --> 00:08:00.040 Namely, we modulate that corresponds to 00:08:00.040 --> 00:08:01.470 shifting the spectrum. 00:08:01.470 --> 00:08:05.960 We demodulate by multiplying by the complex conjugate of 00:08:05.960 --> 00:08:10.710 the original modulated carrier and that shifts the spectrum 00:08:10.710 --> 00:08:14.290 back to where it was originally. 00:08:14.290 --> 00:08:14.760 OK. 00:08:14.760 --> 00:08:18.540 Now let's consider the case of a sinusoidal carrier in 00:08:18.540 --> 00:08:19.910 discrete time. 00:08:19.910 --> 00:08:24.390 And again, things very much parallel what we saw in 00:08:24.390 --> 00:08:26.190 continuous time. 00:08:26.190 --> 00:08:31.460 And again, as we look at the spectra, I will choose a phase 00:08:31.460 --> 00:08:35.770 angle of zero, mainly for notational and analytical 00:08:35.770 --> 00:08:37.460 convenience. 00:08:37.460 --> 00:08:42.190 So in this case, now, rather than a carrier signal, which 00:08:42.190 --> 00:08:45.740 is a single complex exponential, it's now a 00:08:45.740 --> 00:08:49.470 sinusoidal carrier and the sinusoidal carrier is the sum 00:08:49.470 --> 00:08:52.320 of two complex exponential. 00:08:52.320 --> 00:08:56.960 And so if we consider a modulated spectrum, that is 00:08:56.960 --> 00:09:01.510 the spectrum of x of n, something of the type that I 00:09:01.510 --> 00:09:09.430 indicate here, and the spectrum of the carrier, now, 00:09:09.430 --> 00:09:12.460 since the carrier is sinusoidal rather than a 00:09:12.460 --> 00:09:16.750 complex exponential consists of two impulses, one at plus 00:09:16.750 --> 00:09:22.070 omega sub c and one at minus omega sub c, convolving this 00:09:22.070 --> 00:09:26.440 spectrum with this spectrum gives us a replication of x of 00:09:26.440 --> 00:09:32.450 omega around plus and minus omega sub c. 00:09:32.450 --> 00:09:36.800 And incidentally, with an amplitude change of a half. 00:09:36.800 --> 00:09:41.540 So again, things have worked as they did 00:09:41.540 --> 00:09:43.640 in continuous time. 00:09:43.640 --> 00:09:47.690 In continuous time or in discrete time, modulating with 00:09:47.690 --> 00:09:52.590 a sinusoidal carrier would correspond to a replication of 00:09:52.590 --> 00:09:56.800 the spectrum around, plus the carrier frequency and a 00:09:56.800 --> 00:09:59.450 replication of the spectrum around minus the carrier 00:09:59.450 --> 00:10:05.350 frequency, in both cases, as long as the carrier frequency 00:10:05.350 --> 00:10:09.060 is large enough compared with the bandwidth of the signal so 00:10:09.060 --> 00:10:13.920 that those two replication don't overlap, then it's 00:10:13.920 --> 00:10:17.850 reasonable to suppose that we should be able to recover the 00:10:17.850 --> 00:10:19.330 original signal. 00:10:19.330 --> 00:10:25.470 Well, in fact, to demodulate in the discrete time case, we 00:10:25.470 --> 00:10:30.170 would again follow very much the strategy that we did in 00:10:30.170 --> 00:10:32.080 continuous time. 00:10:32.080 --> 00:10:37.180 In particular, let's consider demodulating by taking the 00:10:37.180 --> 00:10:40.040 modulated signal and, again, putting that through a 00:10:40.040 --> 00:10:43.220 modulator, again, with the carrier which is 00:10:43.220 --> 00:10:45.690 cosine omega sub cn. 00:10:45.690 --> 00:10:50.700 If we do that, we have a demodulator or what will turn 00:10:50.700 --> 00:10:53.010 out to be part of a demodulator, as I indicate 00:10:53.010 --> 00:11:00.320 here, the spectrum of the input signal is, as I had just 00:11:00.320 --> 00:11:04.040 developed, a replication of the original spectrum around 00:11:04.040 --> 00:11:09.760 plus and minus omega sub c with an amplitude of a half. 00:11:09.760 --> 00:11:14.410 When this is, again, convolved with the spectrum of the 00:11:14.410 --> 00:11:18.860 carrier, then we get a replication of the original 00:11:18.860 --> 00:11:25.650 spectrum, first around zero frequency, as I indicate here, 00:11:25.650 --> 00:11:30.680 and then around twice the carrier frequency and minus 00:11:30.680 --> 00:11:33.010 twice the carrier frequency. 00:11:33.010 --> 00:11:37.030 And as long as the carrier frequency is large enough 00:11:37.030 --> 00:11:42.610 compared with the width of the original signal, then, as you 00:11:42.610 --> 00:11:48.360 can see, by extracting this part of the spectrum with a 00:11:48.360 --> 00:11:51.780 low pass filter, we can, in principle, recover the 00:11:51.780 --> 00:11:54.810 spectrum associated with the original signal. 00:11:54.810 --> 00:11:58.060 And again, just as in continuous time, because this 00:11:58.060 --> 00:12:02.860 amplitude is a half, we would want to choose, for scaling 00:12:02.860 --> 00:12:08.850 purposes, a low pass filter amplitude which is 2 to 00:12:08.850 --> 00:12:12.630 compensate for this factor of a half. 00:12:12.630 --> 00:12:18.400 So once again things work out basically the same way as they 00:12:18.400 --> 00:12:20.350 had in continuous time. 00:12:20.350 --> 00:12:26.250 We have sinusoidal modulation which consists of using a 00:12:26.250 --> 00:12:28.010 sinusoidal carrier. 00:12:28.010 --> 00:12:33.100 And we have the demodulator which consists of taking a 00:12:33.100 --> 00:12:38.260 modulated signal, multiplying by the carrier, and then 00:12:38.260 --> 00:12:44.650 processing that with a low pass filter to extract the 00:12:44.650 --> 00:12:49.070 portion of the spectrum, which is around zero frequency, as I 00:12:49.070 --> 00:12:55.220 indicate in the spectrum below and the result, then, that 00:12:55.220 --> 00:12:59.730 this low pass filter having a gain of 2 is that we've 00:12:59.730 --> 00:13:06.020 recovered the original spectrum, x of omega, which is 00:13:06.020 --> 00:13:09.390 the spectrum that we started with. 00:13:09.390 --> 00:13:12.060 Now several other things to stress. 00:13:12.060 --> 00:13:17.310 This is a fairly quick tour through sinusoidal modulation 00:13:17.310 --> 00:13:19.000 for discrete time. 00:13:19.000 --> 00:13:22.720 There are very similar issues that arise in the discrete 00:13:22.720 --> 00:13:27.040 time case in terms of having phase synchronization and 00:13:27.040 --> 00:13:29.820 frequency synchronization between the modulator and 00:13:29.820 --> 00:13:31.110 demodulator. 00:13:31.110 --> 00:13:34.120 And we had discussed that in a fair amount of detail for the 00:13:34.120 --> 00:13:36.900 continuous time case. 00:13:36.900 --> 00:13:40.680 In some sense, in practical terms, that becomes much more 00:13:40.680 --> 00:13:45.860 of an issue in continuous time than it does in discrete time, 00:13:45.860 --> 00:13:50.550 in part, because synchronization between a 00:13:50.550 --> 00:13:55.050 modulator and demodulator is often much harder in a 00:13:55.050 --> 00:13:59.570 continuous time system, which is essentially an analog 00:13:59.570 --> 00:14:03.490 system as compared with a digital system. 00:14:03.490 --> 00:14:06.520 Another very important reason and it's important to stress 00:14:06.520 --> 00:14:11.550 this at the outset is that, whereas the theory involving 00:14:11.550 --> 00:14:15.750 the use of complex exponential and sinusoidal modulation 00:14:15.750 --> 00:14:19.480 parallels very strongly in the continuous time and the 00:14:19.480 --> 00:14:21.240 discrete time case. 00:14:21.240 --> 00:14:25.350 In practical terms, it has much more significance in 00:14:25.350 --> 00:14:28.810 continuous time than it does in discrete time. 00:14:28.810 --> 00:14:33.140 That is, the notion called sinusidal modulation, in the 00:14:33.140 --> 00:14:36.410 context of communication systems, is extremely 00:14:36.410 --> 00:14:40.740 important for continuous time systems, and less so in 00:14:40.740 --> 00:14:42.720 discrete time systems. 00:14:42.720 --> 00:14:47.690 Now as a preview of a point to be raised later on, I should 00:14:47.690 --> 00:14:52.040 modify that slightly with the statement that one very 00:14:52.040 --> 00:14:56.700 important place in which sinusoidal modulation in a 00:14:56.700 --> 00:15:01.420 discrete time context arises, is in a class of systems 00:15:01.420 --> 00:15:06.410 called transmultiplexers or transmodulation systems. 00:15:06.410 --> 00:15:11.190 And this surface is basically because so many communication 00:15:11.190 --> 00:15:16.080 systems are now becoming digital and, specifically, 00:15:16.080 --> 00:15:18.400 discrete time, although the actual transmission is 00:15:18.400 --> 00:15:22.510 continuous time, the signal processing manipulation and 00:15:22.510 --> 00:15:25.080 switching is discrete time. 00:15:25.080 --> 00:15:28.510 And so, in fact, it turns out to be very important and 00:15:28.510 --> 00:15:35.200 useful to take a discrete time representation of the analog 00:15:35.200 --> 00:15:39.010 signals or continuous time signals and, in a discrete 00:15:39.010 --> 00:15:42.380 time, or digital representation, to convert 00:15:42.380 --> 00:15:46.240 them from one modulation scheme or one multiplexing 00:15:46.240 --> 00:15:48.070 scheme to another. 00:15:48.070 --> 00:15:53.280 And although I said a lot there that really requires 00:15:53.280 --> 00:16:00.900 much more detail to develop in any sense at all, you should 00:16:00.900 --> 00:16:06.530 get the notion that discrete time modulation systems become 00:16:06.530 --> 00:16:09.560 very important, in part, because of 00:16:09.560 --> 00:16:13.110 implementational issues. 00:16:13.110 --> 00:16:17.720 OK, now, there is another application that we have 00:16:17.720 --> 00:16:20.820 discussed for both continuous time and actually, previously, 00:16:20.820 --> 00:16:25.510 for discrete time, amplitude modulation with sinusoidal 00:16:25.510 --> 00:16:27.630 complex exponential carriers. 00:16:27.630 --> 00:16:30.800 And let me just remind you of that, because, in fact, it 00:16:30.800 --> 00:16:34.930 becomes a very important one in the case of 00:16:34.930 --> 00:16:36.980 discrete time systems. 00:16:36.980 --> 00:16:42.360 And that is the notion of using modulation together with 00:16:42.360 --> 00:16:48.170 fixed filtering to implement a filter, which either has a 00:16:48.170 --> 00:16:52.940 variable cut off or converts, let's say, a low pass filter 00:16:52.940 --> 00:16:54.580 to a high pass filter. 00:16:54.580 --> 00:16:57.240 We had originally talked about this when we introduce the 00:16:57.240 --> 00:17:02.260 modulation property in the context of converting a low 00:17:02.260 --> 00:17:05.579 pass filter to a high pass filter. 00:17:05.579 --> 00:17:12.180 And the notion was that, if we modulate the signal with a 00:17:12.180 --> 00:17:15.520 carrier which is minus 1 to the n, and that's just simply 00:17:15.520 --> 00:17:19.540 a complex exponential or sinusoidal carrier with a 00:17:19.540 --> 00:17:25.240 carrier frequency of pi, then that, in effect, interchanges 00:17:25.240 --> 00:17:28.000 the low frequencies and the high frequencies. 00:17:28.000 --> 00:17:34.450 And if, after modulation, that is processed with a low pass 00:17:34.450 --> 00:17:42.150 filter, and then the result is demodulated, using exactly the 00:17:42.150 --> 00:17:47.840 same carrier, namely a carrier which is minus 1 to the n, 00:17:47.840 --> 00:17:52.900 then the effect of that is equivalent to high pass 00:17:52.900 --> 00:17:56.170 filtering on the original signal. 00:17:56.170 --> 00:18:00.210 And a generalization of that would involve, instead of this 00:18:00.210 --> 00:18:04.530 specific choice of minus 1 to the n, would involve a choice, 00:18:04.530 --> 00:18:08.830 in general, of e to the j omega sub cn, that is a more 00:18:08.830 --> 00:18:14.000 general carrier frequency, and a demodulator which is e to 00:18:14.000 --> 00:18:16.590 the minus j omega sub cn. 00:18:16.590 --> 00:18:20.360 And as I've represented it here, and as we had talked 00:18:20.360 --> 00:18:23.780 about it when we talked about the modulation property for 00:18:23.780 --> 00:18:27.020 discrete time signals, we had specifically chosen the 00:18:27.020 --> 00:18:30.060 conversion of a low pass to a high pass filter. 00:18:30.060 --> 00:18:34.530 Well, let me continue the review of that just by 00:18:34.530 --> 00:18:37.090 reminding you of the details of what happens with the 00:18:37.090 --> 00:18:43.390 spectra, and, specifically, the notion, if we take this 00:18:43.390 --> 00:18:48.200 particular case of omega sub c is equal to pi, or 00:18:48.200 --> 00:18:51.950 equivalently, a carrier signal which is minus 1 to the n, 00:18:51.950 --> 00:19:01.090 then if we have the original spectra and the spectrum of 00:19:01.090 --> 00:19:06.100 the carrier signal, the spectrum of the carrier signal 00:19:06.100 --> 00:19:10.690 convolved with this spectrum will then, in effect, shift 00:19:10.690 --> 00:19:13.900 this by pi. 00:19:13.900 --> 00:19:19.410 And so, after modulating, the result that we have is a shift 00:19:19.410 --> 00:19:23.260 of that spectrum so that what happened in low frequencies 00:19:23.260 --> 00:19:26.740 now happens at high frequencies, namely around pi, 00:19:26.740 --> 00:19:32.280 and what happened at high frequencies now happens at low 00:19:32.280 --> 00:19:33.900 frequencies. 00:19:33.900 --> 00:19:38.480 Well, if that's processed now, with a low pass filter, and 00:19:38.480 --> 00:19:42.690 this dashed line indicates the low pass filter, then the 00:19:42.690 --> 00:19:47.090 result that we get is shown here, where we've extracted 00:19:47.090 --> 00:19:50.630 the low frequency portion of the modulated signal. 00:19:50.630 --> 00:19:56.990 And now when we modulate or demodulate back, then this 00:19:56.990 --> 00:20:00.640 spectrum is shifted back to where it belongs. 00:20:00.640 --> 00:20:06.580 Namely, it's shifted back to be centered around minus pi 00:20:06.580 --> 00:20:08.780 and around plus pi. 00:20:08.780 --> 00:20:13.660 So if we just compare this spectrum with the original 00:20:13.660 --> 00:20:18.240 spectrum at the top, what we can see is that, in effect, 00:20:18.240 --> 00:20:23.690 what we've done is to extract a portion of the spectrum 00:20:23.690 --> 00:20:28.380 equivalent to processing with a high pass filter. 00:20:28.380 --> 00:20:32.350 And, again, this is very similar to what we did in 00:20:32.350 --> 00:20:38.090 continuous time and all of the analytical processes and 00:20:38.090 --> 00:20:41.180 convolution involved are very much the same. 00:20:41.180 --> 00:20:43.610 Really, the biggest difference between continuous time 00:20:43.610 --> 00:20:47.520 discrete time has to do, not so much with the details of 00:20:47.520 --> 00:20:53.670 the analysis, but perhaps has more to do with issues of 00:20:53.670 --> 00:20:54.920 practical applications. 00:20:59.130 --> 00:21:05.020 OK, well, so what we've done, so far, for continuous time 00:21:05.020 --> 00:21:10.250 and discrete time, is to talk about modulation, amplitude 00:21:10.250 --> 00:21:13.730 modulation with complex exponential 00:21:13.730 --> 00:21:15.540 and sinusoidal carriers. 00:21:15.540 --> 00:21:19.010 We saw that the analysis is very similar, although 00:21:19.010 --> 00:21:21.190 applications are slightly different. 00:21:21.190 --> 00:21:26.480 And now what I'd like to turn to is a different choice of 00:21:26.480 --> 00:21:27.650 carrier signal. 00:21:27.650 --> 00:21:31.400 And the carrier signal, in this particular case, is a 00:21:31.400 --> 00:21:36.450 pulse train rather than a sinusoidal signal. 00:21:36.450 --> 00:21:42.930 Now the idea is the following. 00:21:42.930 --> 00:21:47.700 In general, of course, the modulator consists of all of 00:21:47.700 --> 00:21:52.490 multiplying x of t by whatever the carrier signal is. 00:21:52.490 --> 00:21:55.960 And previously, we've talked about a carrier signal which 00:21:55.960 --> 00:21:58.435 is sinusoidal signal. 00:21:58.435 --> 00:22:00.740 The carrier signal that we want to consider now is a 00:22:00.740 --> 00:22:05.860 carrier signal which, in fact, is a pulse train. 00:22:05.860 --> 00:22:10.530 And so, in fact, what we want to do is multiply the input 00:22:10.530 --> 00:22:17.160 signal by a pulse train and, in effect, then, the modulated 00:22:17.160 --> 00:22:22.070 signal consists of the original signal, simply with 00:22:22.070 --> 00:22:25.250 time slices pulled out of it, as I've indicated in the 00:22:25.250 --> 00:22:27.030 bottom curve. 00:22:27.030 --> 00:22:33.930 So what we have now is a modulated signal that is a 00:22:33.930 --> 00:22:40.880 chopped or sliced version of the original waveform and that 00:22:40.880 --> 00:22:46.610 is what's referred to as pulse amplitude modulation. 00:22:46.610 --> 00:22:49.430 Now it seems like it's kind of a crazy idea. 00:22:49.430 --> 00:22:55.000 The idea is to chop out slices of the wave form and hope that 00:22:55.000 --> 00:22:57.680 you could put things back together again. 00:22:57.680 --> 00:23:01.480 And the amazing thing about it, as we'll see, is that, in 00:23:01.480 --> 00:23:06.140 fact, under fairly broad general and applicable 00:23:06.140 --> 00:23:10.800 conditions, you really can put the waveform back together 00:23:10.800 --> 00:23:14.720 again if you just have these time slices. 00:23:14.720 --> 00:23:20.350 Not only that, but that basic notion, as we'll see, is 00:23:20.350 --> 00:23:22.820 independent, in fact, of what the width of 00:23:22.820 --> 00:23:24.370 those time slices are. 00:23:24.370 --> 00:23:25.900 In fact the width can go to zero. 00:23:25.900 --> 00:23:29.530 And, in fact, we're going to make it go to zero, and really 00:23:29.530 --> 00:23:32.920 only dependent on what the frequency of 00:23:32.920 --> 00:23:35.700 the pulse train is. 00:23:35.700 --> 00:23:38.950 So let's explore that in some detail. 00:23:38.950 --> 00:23:45.010 And what we want to look at is the analysis, but let me, 00:23:45.010 --> 00:23:48.680 first, just comment, very briefly, that all of the 00:23:48.680 --> 00:23:51.520 analysis we go through, as has been true in the case of 00:23:51.520 --> 00:23:55.140 sinusoidal modulation, all of the analysis then we go 00:23:55.140 --> 00:24:00.660 through holds just as well with, essentially minor 00:24:00.660 --> 00:24:05.510 analytical modifications, to discrete time pulse amplitude 00:24:05.510 --> 00:24:09.190 modulation as it does to continuous time pulse 00:24:09.190 --> 00:24:10.920 amplitude modulation. 00:24:10.920 --> 00:24:14.390 And so we'll really only go through this in terms of 00:24:14.390 --> 00:24:16.900 tracking the wave forms and spectra for the 00:24:16.900 --> 00:24:18.320 continuous time case. 00:24:18.320 --> 00:24:22.370 But bear in mind that the results are basically similar 00:24:22.370 --> 00:24:25.940 for discrete time. 00:24:25.940 --> 00:24:30.350 OK, well, let's see how so we get the basic result that we 00:24:30.350 --> 00:24:33.110 want to get. 00:24:33.110 --> 00:24:38.860 What we have is modulated signal which is a pulse train, 00:24:38.860 --> 00:24:42.620 basically a square wave, and as we've seen in previous 00:24:42.620 --> 00:24:47.870 lectures, the spectra or Fourier transform associated 00:24:47.870 --> 00:24:50.420 with that is an impulse train. 00:24:50.420 --> 00:24:54.840 And the envelope of that impulse train is on the form 00:24:54.840 --> 00:24:57.430 of a sine x over x function. 00:24:57.430 --> 00:25:00.220 The Fourier transform is impulses. 00:25:00.220 --> 00:25:05.970 And the spacing of the impulses is associated with 00:25:05.970 --> 00:25:09.470 the fundamental frequency of the pulse train and that's 00:25:09.470 --> 00:25:10.790 omega sub p. 00:25:10.790 --> 00:25:13.630 So omega sub p is pi divided by the period 00:25:13.630 --> 00:25:15.570 of the pulse train. 00:25:15.570 --> 00:25:23.350 And the amplitude and shape of this envelope is dictated by 00:25:23.350 --> 00:25:26.570 the parameter delta, which has to do with how 00:25:26.570 --> 00:25:29.340 wide the pulses are. 00:25:29.340 --> 00:25:32.100 OK, so we have a time function. 00:25:32.100 --> 00:25:34.750 It's multiplied by this pulse train. 00:25:34.750 --> 00:25:37.780 Now we're talking continuous time. 00:25:37.780 --> 00:25:41.800 So, in the frequency domain, we have the Fourier transform 00:25:41.800 --> 00:25:45.350 of the time function convolved with this Fourier transform 00:25:45.350 --> 00:25:46.720 for the pulse train. 00:25:46.720 --> 00:25:48.550 And let's see what that looks like. 00:25:48.550 --> 00:25:52.830 If we were to consider, let's say, a Fourier transform, 00:25:52.830 --> 00:25:57.910 which I have chosen as more or less a general one, then in 00:25:57.910 --> 00:26:03.370 fact, when we convert all of this with this impulse train, 00:26:03.370 --> 00:26:11.240 what we end up with is a replication of this spectrum 00:26:11.240 --> 00:26:16.700 at the places in the frequency domain where the individual 00:26:16.700 --> 00:26:19.260 impulses occurred. 00:26:19.260 --> 00:26:23.200 So we can see that this spectrum is replicated at each 00:26:23.200 --> 00:26:25.530 of these locations. 00:26:25.530 --> 00:26:33.220 And as long as the frequency of the pulse train is large 00:26:33.220 --> 00:26:38.410 enough, compared with the maximum frequency in the 00:26:38.410 --> 00:26:42.750 original signal, x of t, so that there's no overlap 00:26:42.750 --> 00:26:47.570 between these triangles, then what you can see, in fact, 00:26:47.570 --> 00:26:52.130 somewhat amazingly is that, simply by low pass filtering 00:26:52.130 --> 00:26:57.100 the result, we can get back, except for amplitude factor, 00:26:57.100 --> 00:27:00.490 we can get back to the original signal. 00:27:00.490 --> 00:27:02.870 Now it's amazing. 00:27:02.870 --> 00:27:08.470 It really is amazing that all that this depends on is the 00:27:08.470 --> 00:27:12.530 original signal being band limited and the frequency of 00:27:12.530 --> 00:27:15.870 the pulse train being high enough so that when you 00:27:15.870 --> 00:27:19.700 replicate the spectrum the frequency domain, there's no 00:27:19.700 --> 00:27:22.480 overlap between these individual replications. 00:27:22.480 --> 00:27:26.360 And we'll have address that a little more in a few minutes. 00:27:26.360 --> 00:27:30.550 But let me, first of all, point out that this has a 00:27:30.550 --> 00:27:33.300 whole variety of very important implications. 00:27:33.300 --> 00:27:37.350 One is, in the context of communications, it leads to 00:27:37.350 --> 00:27:41.290 another very important multiplexing scheme for 00:27:41.290 --> 00:27:42.510 communications. 00:27:42.510 --> 00:27:45.530 We had talked last time about frequency division 00:27:45.530 --> 00:27:49.010 multiplexing, where individual signals were put into 00:27:49.010 --> 00:27:53.250 individual frequencies slots by choosing different carrier 00:27:53.250 --> 00:27:56.790 frequencies for a sinusoidal modulating signal. 00:27:56.790 --> 00:28:05.400 What this suggests is that what we can put different 00:28:05.400 --> 00:28:12.050 signals into, non-overlapping time slots and, in fact, be 00:28:12.050 --> 00:28:14.960 able to recover the original signals back again. 00:28:14.960 --> 00:28:19.460 So in particular, suppose that I had a signal which I 00:28:19.460 --> 00:28:25.070 modulated with a pulse train and I chose another signal, 00:28:25.070 --> 00:28:28.660 modulated with another pulse train, where the time slot was 00:28:28.660 --> 00:28:32.470 different, and I continued this process. 00:28:32.470 --> 00:28:37.330 And after I'd done this with some number of channels, 00:28:37.330 --> 00:28:43.910 simply added all those together as I indicate here. 00:28:43.910 --> 00:28:47.220 Then as long as I knew what time slots to associate with 00:28:47.220 --> 00:28:53.370 what signal, I could get the original modulated signals 00:28:53.370 --> 00:28:54.920 back again. 00:28:54.920 --> 00:28:59.590 And then as long as the frequency of the impulse train 00:28:59.590 --> 00:29:05.400 was such that I was able to do this reconstruction by simply 00:29:05.400 --> 00:29:10.100 low pass filtering, then I would be able to demodulate. 00:29:10.100 --> 00:29:12.770 So it's a very different very important modulation scheme 00:29:12.770 --> 00:29:16.890 called time division multiplexing in contrast to 00:29:16.890 --> 00:29:19.060 frequency division multiplexing as we had talked 00:29:19.060 --> 00:29:20.740 about last time. 00:29:20.740 --> 00:29:23.750 I had made reference earlier to the concept of 00:29:23.750 --> 00:29:24.570 trans-multiplexing. 00:29:24.570 --> 00:29:29.760 And in fact, what happens in many communication systems is 00:29:29.760 --> 00:29:33.530 that the signals are represented, in fact, in 00:29:33.530 --> 00:29:34.760 discrete time. 00:29:34.760 --> 00:29:37.380 The analog and continuous time signals are represented in 00:29:37.380 --> 00:29:39.750 discrete time. 00:29:39.750 --> 00:29:42.330 And very often the conversion from frequency division 00:29:42.330 --> 00:29:45.970 multiplexing to time division multiplexing and back is done, 00:29:45.970 --> 00:29:50.160 in fact, in the discrete time domain. 00:29:50.160 --> 00:30:01.800 OK, so what we have then, is the notion that we can 00:30:01.800 --> 00:30:08.670 multiply a time function by a pulse train, 00:30:08.670 --> 00:30:10.940 as I indicate here. 00:30:10.940 --> 00:30:17.020 And from the output I can, if the frequency of this pulse 00:30:17.020 --> 00:30:19.770 train is high enough in relation to this bandwidth, 00:30:19.770 --> 00:30:25.640 from the output, which consists of time slices, from 00:30:25.640 --> 00:30:31.450 those time slices I can recover the original signal. 00:30:31.450 --> 00:30:37.770 Stressing again the reason it relates to the spectra, and 00:30:37.770 --> 00:30:41.760 the reason is that the original spectra is simply 00:30:41.760 --> 00:30:48.170 replicated at multiples of the fundamental frequency of the 00:30:48.170 --> 00:30:49.420 pulse train. 00:30:51.960 --> 00:30:55.320 Now there's a very important thing to observe here, which 00:30:55.320 --> 00:31:01.620 is that the ability to do the reconstruction is associated 00:31:01.620 --> 00:31:03.690 with the notion of whether we can extract 00:31:03.690 --> 00:31:05.360 that central triangle. 00:31:05.360 --> 00:31:08.030 I happened to choose a triangular shape but obviously 00:31:08.030 --> 00:31:10.260 I could be talking about any shape, as long as it's band 00:31:10.260 --> 00:31:13.230 limited, the ability to extract that. 00:31:13.230 --> 00:31:19.270 And notice that, in this modulated output spectrum, the 00:31:19.270 --> 00:31:23.800 ability to recover this is totally independent called 00:31:23.800 --> 00:31:26.880 what the value of delta is. 00:31:26.880 --> 00:31:35.520 In other words, if we look back at the modulator, then, 00:31:35.520 --> 00:31:40.360 in fact, we can make delta, the width of these pulses, 00:31:40.360 --> 00:31:42.470 arbitrarily small. 00:31:42.470 --> 00:31:46.710 And, in theory, that doesn't affect our ability to do the 00:31:46.710 --> 00:31:48.150 reconstruction. 00:31:48.150 --> 00:31:50.590 Now in practical terms it might. 00:31:50.590 --> 00:31:55.360 Looking back once more at the spectrum of the output, notice 00:31:55.360 --> 00:31:59.060 that this amplitude is proportional to delta. 00:31:59.060 --> 00:32:03.470 And what that suggests is that, as we make delta smaller 00:32:03.470 --> 00:32:06.320 and smaller, which we might, in fact, want to do, if you 00:32:06.320 --> 00:32:10.370 want to time division multiplex lots of channels, in 00:32:10.370 --> 00:32:12.210 principle, in theory, you could make it an infinite 00:32:12.210 --> 00:32:13.680 number of channels just by making that 00:32:13.680 --> 00:32:15.780 infinitesimally small. 00:32:15.780 --> 00:32:19.030 The smaller it is, in some sense, the less 00:32:19.030 --> 00:32:20.140 energy there is. 00:32:20.140 --> 00:32:23.340 And again, in practical terms, this one of those things if 00:32:23.340 --> 00:32:27.310 you push down here pops up there, namely, you eventually 00:32:27.310 --> 00:32:30.480 run into issues such as noise problems. 00:32:30.480 --> 00:32:36.510 So, more typically what's done is to, in fact, eliminate this 00:32:36.510 --> 00:32:38.540 scale factor of delta. 00:32:38.540 --> 00:32:42.570 And the way that that's done is very simply. 00:32:42.570 --> 00:32:51.600 It's done by choosing the width of the pulses, and the 00:32:51.600 --> 00:32:56.240 height of the pulses, in such a way that the area is 00:32:56.240 --> 00:33:01.890 constant, even as we make delta get arbitrarily small. 00:33:01.890 --> 00:33:07.820 So we can just modify our argument so that what we're 00:33:07.820 --> 00:33:12.890 referring to is a modulated pulse train, which is a pulse 00:33:12.890 --> 00:33:15.360 train with pulses of width delta and 00:33:15.360 --> 00:33:18.900 height, 1 over delta. 00:33:18.900 --> 00:33:24.160 In that case, as delta gets arbitrarily small, then, in 00:33:24.160 --> 00:33:29.570 fact, what these rectangles become are impulses, in which 00:33:29.570 --> 00:33:36.290 case, what we're talking about is a carrier signal which, in 00:33:36.290 --> 00:33:39.550 fact, is an impulse train. 00:33:39.550 --> 00:33:45.200 And the resulting modulated signal is an impulse train for 00:33:45.200 --> 00:33:50.630 which the amplitudes of the impulses are proportional to 00:33:50.630 --> 00:33:56.290 the original input waveform at the times at which these 00:33:56.290 --> 00:33:59.050 impulses occur. 00:33:59.050 --> 00:34:03.510 OK well, let's look at the analysis of that. 00:34:03.510 --> 00:34:15.010 And so now, what we're talking about, is a spectrum that 00:34:15.010 --> 00:34:20.840 consists of the result of the spectrum we talked about 00:34:20.840 --> 00:34:24.900 before with the sine x over x envelope, except that, now, as 00:34:24.900 --> 00:34:28.260 delta goes to zero that becomes flat. 00:34:28.260 --> 00:34:32.000 In other words, the modulated signal is an impulse train. 00:34:32.000 --> 00:34:36.210 And so as we look at the spectrum of the modulated 00:34:36.210 --> 00:34:39.460 signal, that is, then, an impulse train in 00:34:39.460 --> 00:34:41.380 the frequency domain. 00:34:41.380 --> 00:34:46.389 The height is proportional to the frequency of the impulse 00:34:46.389 --> 00:34:50.659 train and omega sub s now denotes the frequency of the 00:34:50.659 --> 00:34:52.170 impulse train. 00:34:52.170 --> 00:34:59.090 And the resulting output of the modulator has a spectrum 00:34:59.090 --> 00:35:04.060 which is this original spectrum, again, replicated 00:35:04.060 --> 00:35:08.800 around each of these impulses, in other words, replicated in 00:35:08.800 --> 00:35:12.410 multiples of the sampling frequency 00:35:12.410 --> 00:35:16.410 Now this is very much identical to the 00:35:16.410 --> 00:35:17.460 more general case. 00:35:17.460 --> 00:35:21.130 We have this replication of the spectra. 00:35:21.130 --> 00:35:30.220 And as long as the frequency of the impulse train is large 00:35:30.220 --> 00:35:33.840 enough, compared with the bandwidth of the signal so 00:35:33.840 --> 00:35:40.670 that these triangles don't overlap, I can extract this 00:35:40.670 --> 00:35:45.070 portion of the spectrum by low pass filtering, in fact, would 00:35:45.070 --> 00:35:49.340 then give us back the original signal. 00:35:49.340 --> 00:35:54.380 Now if, instead, this frequency omega sub m is 00:35:54.380 --> 00:36:00.060 greater than omega sub s minus omega sub m, we would have a 00:36:00.060 --> 00:36:04.140 spectrum that looked something more like this. 00:36:04.140 --> 00:36:08.640 And what's happened, in this case, is that, because we have 00:36:08.640 --> 00:36:12.980 an overlap here, we've destroyed the ability to 00:36:12.980 --> 00:36:16.430 recover the original signal from the impulse train. 00:36:16.430 --> 00:36:20.750 And that would be true, also in a more general case, of 00:36:20.750 --> 00:36:26.820 pulse amplitude modulation with pulses of non-zero width. 00:36:26.820 --> 00:36:32.520 This effect by the way, is one that we'll be exploring in 00:36:32.520 --> 00:36:35.100 considerably more detail in the next lecture. 00:36:35.100 --> 00:36:38.440 And it's a phenomenon or distortion refer to as 00:36:38.440 --> 00:36:40.540 aliasing which, in fact, is an important 00:36:40.540 --> 00:36:42.090 and interesting topic. 00:36:42.090 --> 00:36:45.620 But going back to the case in which we've chosen the 00:36:45.620 --> 00:36:50.070 frequency of the impulse train high enough, then we would 00:36:50.070 --> 00:36:57.300 recover the original signal by processing it through a low 00:36:57.300 --> 00:37:00.050 pass filter. 00:37:00.050 --> 00:37:06.480 And in that case, what this says is, that if we have a 00:37:06.480 --> 00:37:13.030 signal, and we modulate it with an impulse train, if we 00:37:13.030 --> 00:37:16.960 then process that impulse train through an idea low pass 00:37:16.960 --> 00:37:20.950 filter, given the right conditions on the frequency of 00:37:20.950 --> 00:37:23.760 impulse train and the bandwidth of the signal, we 00:37:23.760 --> 00:37:27.050 can recover the original signal back again. 00:37:27.050 --> 00:37:32.770 Now let me stress, just going back to the picture in which 00:37:32.770 --> 00:37:37.560 we had done this modulation, that this process, where the 00:37:37.560 --> 00:37:40.880 modulation, where the carrier signal involves an impulse 00:37:40.880 --> 00:37:45.380 train, is often referred to as sampling. 00:37:45.380 --> 00:37:50.290 And what that means, specifically, is that, if we 00:37:50.290 --> 00:37:58.510 notice, this resulting impulse train is, in fact, a sequence 00:37:58.510 --> 00:38:04.190 of samples of the original continuous time signal. 00:38:04.190 --> 00:38:07.930 In other words, what we've done, in effect, is taken 00:38:07.930 --> 00:38:10.940 instantaneous sample of this wave form. 00:38:10.940 --> 00:38:14.560 And the implication is that, if we do that at a rapid 00:38:14.560 --> 00:38:21.190 enough rate in relation to the bandwidth of the signal, then 00:38:21.190 --> 00:38:25.280 we can, in fact, recover the original signal back again. 00:38:25.280 --> 00:38:31.960 And, finally to remind you of the argument once more, we 00:38:31.960 --> 00:38:37.910 have an original signal and we have its spectrum. 00:38:37.910 --> 00:38:40.800 When we've sampled it, and this is now the sampled 00:38:40.800 --> 00:38:45.400 signal, it's an impulse train whose instantaneous values are 00:38:45.400 --> 00:38:49.840 samples of the original waveform, the spectrum of that 00:38:49.840 --> 00:38:53.890 is the original one replicated. 00:38:53.890 --> 00:38:59.110 And when that is processed, through a low pass filter, to 00:38:59.110 --> 00:39:03.500 extract this part of the spectrum, then, after the low 00:39:03.500 --> 00:39:06.530 pass filter, we can recover the 00:39:06.530 --> 00:39:07.910 original signal back again. 00:39:11.390 --> 00:39:18.650 OK well, in fact, although if you follow through the spectra 00:39:18.650 --> 00:39:22.610 and the wave forms, this all seems fairly straightforward 00:39:22.610 --> 00:39:29.210 and, perhaps or perhaps not, obvious, it's really worth 00:39:29.210 --> 00:39:34.510 reflecting on how amazing the result really is. 00:39:34.510 --> 00:39:39.040 We began this discussion by talking about modulation. 00:39:39.040 --> 00:39:42.030 And in fact modulation and sinusoidal of modulation is 00:39:42.030 --> 00:39:44.410 important in its own right. 00:39:44.410 --> 00:39:49.890 We ended the discussion by talking about first pulse 00:39:49.890 --> 00:39:55.210 amplitude modulation, and then showing how, under the right 00:39:55.210 --> 00:40:01.320 set of conditions, you can, in fact, take a wave form and 00:40:01.320 --> 00:40:05.520 sample it with a set of instantaneous samples. 00:40:05.520 --> 00:40:08.820 And that set of instantaneous samples, in fact, are 00:40:08.820 --> 00:40:11.690 sufficient to totally represent and 00:40:11.690 --> 00:40:14.280 reconstruct the signal. 00:40:14.280 --> 00:40:19.070 What in fact, the formal statement that is, is refer to 00:40:19.070 --> 00:40:24.030 as the sampling theorem, a very powerful theorem that 00:40:24.030 --> 00:40:28.190 says, if we're given equally spaced samples of a time 00:40:28.190 --> 00:40:36.950 function, and if that time function is band limited, and 00:40:36.950 --> 00:40:45.110 if the bandwidth and if the sampling frequency is chosen 00:40:45.110 --> 00:40:54.100 in the right way, in relation to the bandwidth, then, in 00:40:54.100 --> 00:40:58.990 fact, the original time function is uniquely 00:40:58.990 --> 00:41:02.320 recoverable with a low pass filter. 00:41:05.150 --> 00:41:11.110 Now the sampling theorem is, I would say, a watershed or 00:41:11.110 --> 00:41:14.520 cornerstone of a lot of the discussion that we've been 00:41:14.520 --> 00:41:17.450 having for a whole variety of reasons. 00:41:17.450 --> 00:41:21.620 It, first of all, drops out almost as a straightforward 00:41:21.620 --> 00:41:22.920 obvious statement. 00:41:22.920 --> 00:41:28.700 But more importantly what it says is, if I have a 00:41:28.700 --> 00:41:32.160 continuous time signal which satisfies the right set of 00:41:32.160 --> 00:41:38.300 conditions, I could represent it by what it does at sampling 00:41:38.300 --> 00:41:42.200 instance or, equivalently, at discrete instance of time. 00:41:44.950 --> 00:41:49.920 Now what that leads to is a whole host of things. 00:41:49.920 --> 00:41:53.630 One of which is this statement that says, if we have a 00:41:53.630 --> 00:41:58.880 continuous time signal, I could in fact, represent it as 00:41:58.880 --> 00:42:01.090 a discrete time signal. 00:42:01.090 --> 00:42:04.890 And I could even think of processing a continuous time 00:42:04.890 --> 00:42:09.520 signal using discrete time concepts. 00:42:09.520 --> 00:42:13.010 And when I'm all done converting back, through the 00:42:13.010 --> 00:42:15.230 power of the sampling theorem, converting back to a 00:42:15.230 --> 00:42:16.870 continuous time signal. 00:42:16.870 --> 00:42:21.790 So the sampling theorem provides us with a very major 00:42:21.790 --> 00:42:25.240 important bridge between continuous time and discrete 00:42:25.240 --> 00:42:28.930 time implementations and ideas. 00:42:28.930 --> 00:42:31.950 In the next several lectures, we will be exploring some of 00:42:31.950 --> 00:42:34.140 this in considerable detail. 00:42:34.140 --> 00:42:38.860 First, to focus in more, next time, on some of the specific 00:42:38.860 --> 00:42:42.620 issues and distortions associated with sampling. 00:42:42.620 --> 00:42:47.520 And following that, a discussion of what is referred 00:42:47.520 --> 00:42:52.400 to discrete time processing of continuous time signals. 00:42:52.400 --> 00:42:53.650 Thank you.