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 --> 00:00:17.400 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.400 --> 00:00:18.650 ocw.mit.edu. 00:00:55.550 --> 00:00:58.110 PROFESSOR: In the last lecture, I introduced and 00:00:58.110 --> 00:01:01.610 illustrated the kinds of signals and systems that we'll 00:01:01.610 --> 00:01:05.010 be dealing with throughout this course. 00:01:05.010 --> 00:01:08.820 In today's lecture I'd like to be a little more specific, and 00:01:08.820 --> 00:01:13.080 in particular, talk about some of the basic signals, both 00:01:13.080 --> 00:01:16.430 continuous-time and discrete-time that will form 00:01:16.430 --> 00:01:21.020 important building blocks as the course progresses. 00:01:21.020 --> 00:01:25.420 Let's begin with one signal, the continuous-time sinusoidal 00:01:25.420 --> 00:01:27.550 signal, which perhaps you're already 00:01:27.550 --> 00:01:29.650 somewhat familiar with. 00:01:29.650 --> 00:01:32.790 Mathematically, the continuous-time sinusoidal 00:01:32.790 --> 00:01:36.500 signal is expressed as I've indicated here. 00:01:36.500 --> 00:01:41.750 There are three parameters, A, omega_0 and phi. 00:01:41.750 --> 00:01:46.760 The parameter A is referred to as the amplitude, the 00:01:46.760 --> 00:01:51.150 parameter omega 0 as the frequency, and the parameter 00:01:51.150 --> 00:01:54.950 phi as the phase. 00:01:54.950 --> 00:02:00.010 And graphically, the continuous-time sinusoidal 00:02:00.010 --> 00:02:04.440 signal has the form shown here. 00:02:04.440 --> 00:02:08.759 Now, the sinusoidal signal has a number of important 00:02:08.759 --> 00:02:13.160 properties that we'll find it convenient to exploit as the 00:02:13.160 --> 00:02:18.800 course goes along, one of which is the fact that the 00:02:18.800 --> 00:02:24.250 sinusoidal signal is what is referred to as periodic. 00:02:24.250 --> 00:02:28.920 What I mean by periodic is that under an appropriate time 00:02:28.920 --> 00:02:34.380 shift, which I indicate here as T_0, the signal replicates 00:02:34.380 --> 00:02:36.110 or repeats itself. 00:02:36.110 --> 00:02:40.820 Or said another way, if we shift the time origin by an 00:02:40.820 --> 00:02:44.690 appropriate amount T_0, the smallest value T_0 being 00:02:44.690 --> 00:02:48.190 referred to as the period, then x(t) is 00:02:48.190 --> 00:02:51.420 equal to itself, shifted. 00:02:51.420 --> 00:02:57.630 And we can demonstrate it mathematically by simply 00:02:57.630 --> 00:03:00.700 substituting into the mathematical expression for 00:03:00.700 --> 00:03:07.630 the sinusoidal signal t + T_0, in place of t. 00:03:07.630 --> 00:03:11.310 When we carry out the expansion we then have, for 00:03:11.310 --> 00:03:20.040 the argument of the sinusoid, omega_0 t + omega_0 T_0 + phi. 00:03:20.040 --> 00:03:23.350 Now, one of the things that we know about sinusoidal 00:03:23.350 --> 00:03:27.080 functions is that if you change the argument by any 00:03:27.080 --> 00:03:30.630 integer multiple of 2 pi, then the function 00:03:30.630 --> 00:03:32.320 has the same value. 00:03:32.320 --> 00:03:36.110 And so we can exploit that here, in particular with 00:03:36.110 --> 00:03:41.400 omega_0 T_0 and integer multiple of 2 pi. 00:03:41.400 --> 00:03:44.700 Then the right-hand side of this equation is equal to the 00:03:44.700 --> 00:03:47.040 left-hand side of the equation. 00:03:47.040 --> 00:03:54.880 So with omega_0 T_0 equal to 2 pi times an integer, or T_0 00:03:54.880 --> 00:03:57.420 equal to 2 pi times an integer divided by 00:03:57.420 --> 00:04:00.280 omega_0, the signal repeats. 00:04:00.280 --> 00:04:03.700 The period is defined as the smallest value of T_0. 00:04:03.700 --> 00:04:08.750 And so the period is 2 pi divided by omega_0. 00:04:08.750 --> 00:04:11.190 And going back to our sinusoidal 00:04:11.190 --> 00:04:16.160 signal, we can see that-- 00:04:16.160 --> 00:04:23.100 and I've indicated here, then, the period as 2 pi / omega_0. 00:04:23.100 --> 00:04:26.990 And that's the value under which the signal repeats. 00:04:30.160 --> 00:04:35.710 Now in addition, a useful property of the sinusoidal 00:04:35.710 --> 00:04:42.070 signal is the fact that a time shift of a sinusoid is 00:04:42.070 --> 00:04:44.290 equivalent to a phase change. 00:04:44.290 --> 00:04:48.670 And we can demonstrate that again mathematically, in 00:04:48.670 --> 00:04:52.610 particular if we put the sinusoidal signal 00:04:52.610 --> 00:04:54.540 under a time shift-- 00:04:54.540 --> 00:04:57.710 I've indicated the time shift that I'm 00:04:57.710 --> 00:05:00.680 talking about by t_0-- 00:05:00.680 --> 00:05:06.570 and expand this out, then we see that that is equivalent to 00:05:06.570 --> 00:05:08.760 a change in phase. 00:05:08.760 --> 00:05:12.640 And an important thing to recognize about this statement 00:05:12.640 --> 00:05:18.260 is that not only is a time shift generating a phase 00:05:18.260 --> 00:05:24.890 change, but, in fact, if we inserted a phase change, there 00:05:24.890 --> 00:05:31.570 is always a value of t_0 which would correspond to an 00:05:31.570 --> 00:05:33.420 equivalent time shift. 00:05:33.420 --> 00:05:39.630 Said another way, if we take omega_0 t_0 and think of that 00:05:39.630 --> 00:05:45.190 as our change in phase, for any change in phase, we can 00:05:45.190 --> 00:05:48.900 solve this equation for a time shift, or conversely for any 00:05:48.900 --> 00:05:51.810 value of time shift, that represents 00:05:51.810 --> 00:05:54.740 an appropriate phase. 00:05:54.740 --> 00:05:58.820 So a time shift corresponds to a phase change, and a phase 00:05:58.820 --> 00:06:02.970 change, likewise, corresponds to time shift. 00:06:02.970 --> 00:06:11.160 And so for example, if we look at the general sinusoidal 00:06:11.160 --> 00:06:17.530 signal that we saw previously, in effect, changing the phase 00:06:17.530 --> 00:06:21.090 corresponds to moving this signal in time 00:06:21.090 --> 00:06:22.990 one way or the other. 00:06:22.990 --> 00:06:27.110 For example, if we look at the sinusoidal signal with a phase 00:06:27.110 --> 00:06:31.820 equal to 0 that corresponds to locating the time 00:06:31.820 --> 00:06:33.950 origin at this peak. 00:06:33.950 --> 00:06:41.820 And I've indicated that on the following graph. 00:06:41.820 --> 00:06:46.980 So here we have illustrated a sinusoid with 0 phase, or a 00:06:46.980 --> 00:06:50.650 cosine with 0 phase, corresponding to taking our 00:06:50.650 --> 00:06:52.890 general picture and shifting it. 00:06:52.890 --> 00:06:56.730 Shifting it appropriately as I've indicated here. 00:06:56.730 --> 00:07:00.370 This, of course, still has the property that it's a periodic 00:07:00.370 --> 00:07:05.520 function, since we simply displaced it in time. 00:07:05.520 --> 00:07:09.580 And by looking at the graph, what we see is that it has 00:07:09.580 --> 00:07:12.540 another very important property, a property referred 00:07:12.540 --> 00:07:14.070 to as even. 00:07:14.070 --> 00:07:18.180 And that's a property that we'll find useful, in general, 00:07:18.180 --> 00:07:20.970 to refer to in relation to signals. 00:07:20.970 --> 00:07:24.860 A signal is said to be even if, when we reflect it about 00:07:24.860 --> 00:07:27.170 the origin, it looks exactly the same. 00:07:27.170 --> 00:07:30.600 So it's symmetric about the origin. 00:07:30.600 --> 00:07:34.670 And looking at this sinusoid, that, in 00:07:34.670 --> 00:07:37.010 fact, has that property. 00:07:37.010 --> 00:07:41.550 And mathematically, the statement that it's even is 00:07:41.550 --> 00:07:45.570 equivalent to the statement that if we replace the time 00:07:45.570 --> 00:07:48.630 argument by its negative, the function 00:07:48.630 --> 00:07:50.070 itself doesn't change. 00:07:53.290 --> 00:07:56.660 Now this corresponded to a phase shift of 0 in our 00:07:56.660 --> 00:07:58.990 original cosine expression. 00:07:58.990 --> 00:08:03.350 If instead, we had chosen a phase shift of, let's say, 00:08:03.350 --> 00:08:08.275 -pi/2, then instead of a cosinusoidal signal, what we 00:08:08.275 --> 00:08:14.770 would regenerate is a sinusoid with the appropriate phase. 00:08:14.770 --> 00:08:22.260 Or, said another way, if we take our original cosine and 00:08:22.260 --> 00:08:27.120 substitute in for the phase -pi/2, then of course we have 00:08:27.120 --> 00:08:29.540 this mathematical expression. 00:08:29.540 --> 00:08:32.900 Using just straightforward trigonometric identities, we 00:08:32.900 --> 00:08:37.799 can express that alternately as sin(omega_0*t). 00:08:37.799 --> 00:08:40.990 The frequency and amplitude, of course, haven't changed. 00:08:40.990 --> 00:08:45.750 And that, you can convince yourself, also is equivalent 00:08:45.750 --> 00:08:50.500 to shifting the cosine by an amount in time that I've 00:08:50.500 --> 00:08:54.600 indicated here, namely a quarter of a period. 00:08:54.600 --> 00:09:00.160 So illustrated below is the graph now, when we have a 00:09:00.160 --> 00:09:07.290 phase of -pi/2 in our cosine, which is a sinusoidal signal. 00:09:07.290 --> 00:09:09.270 Of course, it's still periodic. 00:09:09.270 --> 00:09:15.220 It's periodic with a period of 2 pi / omega_0 again, because 00:09:15.220 --> 00:09:19.690 all that we've done by introducing a phase change is 00:09:19.690 --> 00:09:20.940 introduced the time shift. 00:09:23.780 --> 00:09:27.420 Now, when we look at the sinusoid in comparison with 00:09:27.420 --> 00:09:30.730 the cosine, namely with this particular choice of phase, 00:09:30.730 --> 00:09:34.960 this has a different symmetry, and that symmetry 00:09:34.960 --> 00:09:37.020 is referred to odd. 00:09:37.020 --> 00:09:41.590 What odd symmetry means, graphically, is that when we 00:09:41.590 --> 00:09:50.230 flip the signal about the time origin, we also multiply it by 00:09:50.230 --> 00:09:51.410 a minus sign. 00:09:51.410 --> 00:09:55.110 So that's, in effect, anti-symmetric. 00:09:55.110 --> 00:09:56.880 It's not the mirror image, but it's the mirror 00:09:56.880 --> 00:09:58.390 image flipped over. 00:09:58.390 --> 00:10:02.420 And we'll find many occasions, not only to refer to signals 00:10:02.420 --> 00:10:06.610 more general than sinusoidal signals, as even in some cases 00:10:06.610 --> 00:10:08.950 and odd in other cases. 00:10:08.950 --> 00:10:14.240 And in general, mathematically, an odd signal 00:10:14.240 --> 00:10:22.190 is one which satisfies the algebraic expression, x(t). 00:10:22.190 --> 00:10:29.660 When you replace t by its negative, is equal to -x(-t). 00:10:29.660 --> 00:10:35.640 So replacing the argument by its negative corresponds to an 00:10:35.640 --> 00:10:36.910 algebraic sign reversal. 00:10:39.570 --> 00:10:39.900 OK. 00:10:39.900 --> 00:10:43.610 So this is the class of continuous-time sinusoids. 00:10:43.610 --> 00:10:46.470 We'll have a little more to say about it later. 00:10:46.470 --> 00:10:51.720 But I'd now like to turn to discrete-time sinusoids. 00:10:51.720 --> 00:10:54.790 What we'll see is that discrete-time sinusoids are 00:10:54.790 --> 00:10:59.070 very much like continuous-time ones, but also with some very 00:10:59.070 --> 00:11:00.650 important differences. 00:11:00.650 --> 00:11:03.920 And we want to focus, not only on the similarities, but also 00:11:03.920 --> 00:11:06.040 on the differences. 00:11:06.040 --> 00:11:10.530 Well, let's begin with the mathematical expression. 00:11:10.530 --> 00:11:16.740 A discrete-time sinusoidal signal, mathematically, is as 00:11:16.740 --> 00:11:22.040 I've indicated here, A cos(omega_0 n + phi). 00:11:22.040 --> 00:11:26.050 And just as in the continuous-time case, the 00:11:26.050 --> 00:11:31.360 parameter A is what we'll refer to as the amplitude, 00:11:31.360 --> 00:11:39.820 omega_0 as the frequency, and phi as the phase. 00:11:39.820 --> 00:11:46.420 And I've illustrated here several discrete-time 00:11:46.420 --> 00:11:48.480 sinusoidal signals. 00:11:48.480 --> 00:11:51.000 And they kind of look similar. 00:11:51.000 --> 00:11:53.720 In fact, if you track what you might think of as the 00:11:53.720 --> 00:11:57.670 envelope, it looks very much like what a continuous-time 00:11:57.670 --> 00:12:00.090 sinusoid might look like. 00:12:00.090 --> 00:12:03.720 But keep in mind that the independent variable, in this 00:12:03.720 --> 00:12:06.890 case, is an integer variable. 00:12:06.890 --> 00:12:13.050 And so the sequence only takes on values at integer values of 00:12:13.050 --> 00:12:13.810 the argument. 00:12:13.810 --> 00:12:19.540 And we'll see that has a very important implication, and 00:12:19.540 --> 00:12:22.000 we'll see that shortly. 00:12:22.000 --> 00:12:24.950 Now, one of the issues that we addressed in the 00:12:24.950 --> 00:12:27.500 continuous-time case was periodicity. 00:12:27.500 --> 00:12:33.150 And I want to return to that shortly, because that is one 00:12:33.150 --> 00:12:35.085 of the areas where there is an important distinction. 00:12:37.770 --> 00:12:42.880 Let's first, though, examine the statement similar to the 00:12:42.880 --> 00:12:46.170 one that we examined for continuous time, namely the 00:12:46.170 --> 00:12:51.730 relationship between a time shift and a phase change. 00:12:51.730 --> 00:12:56.490 Now, in continuous time, of course, we saw that a time 00:12:56.490 --> 00:13:00.360 shift corresponds to a phase change, and vice versa. 00:13:00.360 --> 00:13:04.460 Let's first look at the relationship between shifting 00:13:04.460 --> 00:13:07.470 time and generating a change in phase. 00:13:07.470 --> 00:13:12.400 In particular for discrete time, if I implement a time 00:13:12.400 --> 00:13:16.480 shift that generates a phase change-- 00:13:16.480 --> 00:13:21.160 and we can see that easily by simply inserting a 00:13:21.160 --> 00:13:24.300 time shift, n + n_0. 00:13:24.300 --> 00:13:29.860 And if we expand out this argument, we have omega_0 n + 00:13:29.860 --> 00:13:31.670 omega_0 n_0. 00:13:31.670 --> 00:13:34.300 And so I've done that on the right-hand side of the 00:13:34.300 --> 00:13:36.900 equation here. 00:13:36.900 --> 00:13:41.300 And the omega_0 n_0, then, simply corresponds 00:13:41.300 --> 00:13:44.360 to a change in phase. 00:13:44.360 --> 00:13:51.390 So clearly, a shift in time generates a change in phase. 00:13:51.390 --> 00:13:57.650 And for example, if we take a particular sinusoidal signal, 00:13:57.650 --> 00:14:01.050 let's say we take the cosine signal at a particular 00:14:01.050 --> 00:14:06.830 frequency, and with a phase equal to 0, a sequence that we 00:14:06.830 --> 00:14:12.370 might generate is one that I've illustrated here. 00:14:12.370 --> 00:14:16.510 So what I'm illustrating here is the cosine 00:14:16.510 --> 00:14:20.230 signal with 0 phase. 00:14:20.230 --> 00:14:24.570 And it has a particular behavior to it, which will 00:14:24.570 --> 00:14:27.600 depend somewhat on the frequency. 00:14:27.600 --> 00:14:34.310 If I now take this same sequence and shift it so that 00:14:34.310 --> 00:14:39.230 the time origin is shifted a quarter of a period away, then 00:14:39.230 --> 00:14:41.140 you can convince yourself-- 00:14:41.140 --> 00:14:43.780 and it's straightforward to work out-- that that time 00:14:43.780 --> 00:14:49.030 shift corresponds to a phase shift of pi/2. 00:14:49.030 --> 00:14:57.580 So in that case, with the cosine with a phase of -pi/2, 00:14:57.580 --> 00:15:05.410 that will correspond to the expression that I have here. 00:15:05.410 --> 00:15:07.900 We could alternately write that, using again a 00:15:07.900 --> 00:15:12.080 trigonometric identity, as a sine function. 00:15:12.080 --> 00:15:17.790 And that, I've stated, is equivalent to a time shift. 00:15:17.790 --> 00:15:23.360 Namely, this shift of pi/2 is equal to a certain time shift, 00:15:23.360 --> 00:15:27.470 and the time shift for this particular example is a 00:15:27.470 --> 00:15:29.560 quarter of a period. 00:15:29.560 --> 00:15:35.210 So here, we have the sinusoid. 00:15:35.210 --> 00:15:37.700 Previously we had the cosine. 00:15:37.700 --> 00:15:40.590 The cosine was exactly the same sequence, but with the 00:15:40.590 --> 00:15:42.520 origin located here. 00:15:42.520 --> 00:15:45.310 And in fact, that's exactly the way we drew this graph. 00:15:45.310 --> 00:15:49.120 Namely, we just simply took the same values and changed 00:15:49.120 --> 00:15:51.330 the time origin. 00:15:51.330 --> 00:15:56.070 Now, looking at this sequence, which is the sinusoidal 00:15:56.070 --> 00:16:01.120 sequence, the phase of -pi/2, that has a certain symmetry. 00:16:01.120 --> 00:16:05.440 And in fact, what we see is that it has an odd symmetry, 00:16:05.440 --> 00:16:07.790 just as in the continuous-time case. 00:16:07.790 --> 00:16:11.190 Namely, if we take that sequence, flip it about the 00:16:11.190 --> 00:16:15.080 axis, and flip it over in sign, that we get the same 00:16:15.080 --> 00:16:17.380 sequence back again. 00:16:17.380 --> 00:16:22.050 Whereas with 0 phase corresponding to the cosine 00:16:22.050 --> 00:16:26.330 that I showed previously, that has an even symmetry. 00:16:26.330 --> 00:16:31.070 Namely, if I flip it about the time origin and don't do a 00:16:31.070 --> 00:16:34.650 sign reversal, then the sequence is maintained. 00:16:34.650 --> 00:16:40.650 So here, we have an odd symmetry, expressed 00:16:40.650 --> 00:16:43.840 mathematically as I've indicated. 00:16:43.840 --> 00:16:46.980 Namely, replacing the independent variable by its 00:16:46.980 --> 00:16:52.370 negative attaches a negative sign to the whole sequence. 00:16:52.370 --> 00:17:00.660 Whereas in the previous case, what we have is 0 phase and an 00:17:00.660 --> 00:17:02.230 even symmetry. 00:17:02.230 --> 00:17:06.240 And that's expressed mathematically as x[n] 00:17:06.240 --> 00:17:07.490 = x[-n]. 00:17:11.839 --> 00:17:14.900 Now, one of the things I've said so far about 00:17:14.900 --> 00:17:19.849 discrete-time sinusoids is that a time shift corresponds 00:17:19.849 --> 00:17:22.140 to a phase change. 00:17:22.140 --> 00:17:26.960 And we can then ask whether the reverse statement is also 00:17:26.960 --> 00:17:29.950 true, and we knew that the reverse statement was true in 00:17:29.950 --> 00:17:32.160 continuous time. 00:17:32.160 --> 00:17:38.920 Specifically, is it true that a phase change always 00:17:38.920 --> 00:17:40.580 corresponds to a time shift? 00:17:40.580 --> 00:17:43.100 Now, we know that that is true, namely, that this 00:17:43.100 --> 00:17:47.160 statement works both ways in continuous time. 00:17:47.160 --> 00:17:49.300 Does it in discrete time? 00:17:49.300 --> 00:17:54.040 Well, the answer, somewhat interestingly or surprisingly 00:17:54.040 --> 00:17:58.480 until you sit down and think about it, is no. 00:17:58.480 --> 00:18:02.460 It is not necessarily true in discrete time that any phase 00:18:02.460 --> 00:18:06.830 change can be interpreted as a simple time 00:18:06.830 --> 00:18:08.430 shift of the sequence. 00:18:08.430 --> 00:18:12.210 And let me just indicate what the problem is. 00:18:12.210 --> 00:18:17.880 If we look at the relationship between the left side and the 00:18:17.880 --> 00:18:21.660 right side of this equation, expanding this out as we did 00:18:21.660 --> 00:18:27.950 previously, we have that omega_0 n + omega_0 n_0 must 00:18:27.950 --> 00:18:31.370 correspond to omega_0 n + phi. 00:18:31.370 --> 00:18:35.200 And so omega_0 n_0 must correspond 00:18:35.200 --> 00:18:37.630 to the phase change. 00:18:37.630 --> 00:18:43.200 Now, what you can see pretty clearly is that depending on 00:18:43.200 --> 00:18:48.730 the relationship between phi and omega_0, n_0 may or may 00:18:48.730 --> 00:18:51.950 not come out to be an integer. 00:18:51.950 --> 00:18:56.400 Now, in continuous time, the amount of time shift did not 00:18:56.400 --> 00:18:58.350 have to be an integer amount. 00:18:58.350 --> 00:19:01.520 In discrete time, when we talk about a time shift, the amount 00:19:01.520 --> 00:19:03.760 of time shift-- obviously, because of the nature of 00:19:03.760 --> 00:19:05.270 discrete time signals-- 00:19:05.270 --> 00:19:06.640 must be an integer. 00:19:06.640 --> 00:19:12.570 So the phase changes related to time shifts must satisfy 00:19:12.570 --> 00:19:14.580 this particular relationship. 00:19:14.580 --> 00:19:19.380 Namely, that omega_0 n_0, where n_0 is an integer, is 00:19:19.380 --> 00:19:21.145 equal to the change in phase. 00:19:24.940 --> 00:19:25.480 OK. 00:19:25.480 --> 00:19:29.350 Now, that's one distinction between continuous time and 00:19:29.350 --> 00:19:30.290 discrete time. 00:19:30.290 --> 00:19:34.530 Let's now focus on another one, namely the issue of 00:19:34.530 --> 00:19:36.190 periodicity. 00:19:36.190 --> 00:19:39.930 And what we'll see is that again, whereas in continuous 00:19:39.930 --> 00:19:44.930 time, all continuous-time sinusoids are periodic, in the 00:19:44.930 --> 00:19:48.220 discrete-time case that is not necessarily true. 00:19:51.480 --> 00:19:56.960 To explore that a little more carefully, let's look at the 00:19:56.960 --> 00:20:00.750 expression, again, for a general sinusoidal signal with 00:20:00.750 --> 00:20:05.520 an arbitrary amplitude, frequency, and phase. 00:20:05.520 --> 00:20:10.590 And for this to be periodic, what we require is that there 00:20:10.590 --> 00:20:15.850 be some value, N, under which, when we shift the sequence by 00:20:15.850 --> 00:20:19.150 that amount, we get the same sequence back again. 00:20:19.150 --> 00:20:22.000 And the smallest-value N is what we've 00:20:22.000 --> 00:20:24.050 defined as the period. 00:20:24.050 --> 00:20:27.870 Now, when we try that on a sinusoid, we of course 00:20:27.870 --> 00:20:35.460 substitute in for n, n + N. And when we expand out the 00:20:35.460 --> 00:20:41.910 argument here, we'll get the argument that I have on the 00:20:41.910 --> 00:20:44.210 right-hand side. 00:20:44.210 --> 00:20:48.960 And in order for this to repeat, in other words, in 00:20:48.960 --> 00:20:55.630 order for us to discard this term, omega_0 N, where N is 00:20:55.630 --> 00:21:00.100 the period, must be an integer multiple of 2 pi. 00:21:00.100 --> 00:21:06.720 And in that case, it's periodic as long as omega_0 N, 00:21:06.720 --> 00:21:11.550 N being the period, is 2 pi times an integer. 00:21:11.550 --> 00:21:15.550 Just simply dividing this out, we have N, the period, is 2 pi 00:21:15.550 --> 00:21:20.900 m / omega_0. 00:21:20.900 --> 00:21:23.880 Well, you could say, OK what's the big deal? 00:21:23.880 --> 00:21:26.690 Whatever N happens to come out to be when we do that little 00:21:26.690 --> 00:21:29.010 bit of algebra, that's the period. 00:21:29.010 --> 00:21:35.910 But in fact, N, or 2 pi m / omega_0, may not ever come out 00:21:35.910 --> 00:21:37.270 to be an integer. 00:21:37.270 --> 00:21:38.970 Or it may not come out to be the one that 00:21:38.970 --> 00:21:39.850 you thought it might. 00:21:39.850 --> 00:21:43.340 For example, let's look at some 00:21:43.340 --> 00:21:45.895 particular sinusoidal signals. 00:21:49.050 --> 00:21:49.430 Let's see. 00:21:49.430 --> 00:21:53.160 We have the first one here, which is a 00:21:53.160 --> 00:21:55.880 sinusoid, as I've shown. 00:21:55.880 --> 00:21:59.110 And it has a frequency, what I've referred to as the 00:21:59.110 --> 00:22:05.130 frequency, omega_0 = 2 pi / 12. 00:22:05.130 --> 00:22:11.720 And what we'd like to look at is 2 pi / omega_0, then find 00:22:11.720 --> 00:22:14.070 an integer to multiply that by in order 00:22:14.070 --> 00:22:15.740 to get another integer. 00:22:15.740 --> 00:22:17.110 Let's just try that here. 00:22:17.110 --> 00:22:24.760 If we look at 2 pi / omega_0, 2 pi / omega_0, for this case, 00:22:24.760 --> 00:22:27.130 is equal to 12. 00:22:27.130 --> 00:22:28.530 Well, that's fine. 00:22:28.530 --> 00:22:29.890 12 is an integer. 00:22:29.890 --> 00:22:35.030 So what that says is that this sinusoidal signal is periodic. 00:22:35.030 --> 00:22:39.990 And in fact, it's periodic with a period of 12. 00:22:39.990 --> 00:22:41.820 Let's look at the next one. 00:22:41.820 --> 00:22:46.810 The next one, we would have 2 pi / omega_0 again. 00:22:46.810 --> 00:22:48.060 And that's equal to 31/4. 00:22:50.560 --> 00:22:57.080 So what that says is that the period is 31/4. 00:22:57.080 --> 00:22:58.440 But wait a minute. 00:22:58.440 --> 00:23:00.730 31/4 isn't an integer. 00:23:00.730 --> 00:23:02.820 We have to multiply that by an integer 00:23:02.820 --> 00:23:04.730 to get another integer. 00:23:04.730 --> 00:23:09.680 Well, we'd multiply that by 4, so (2 pi / omega_0) times 4 is 00:23:09.680 --> 00:23:11.790 31, 31 is an integer. 00:23:11.790 --> 00:23:16.290 And so what that says is this is periodic, not with a period 00:23:16.290 --> 00:23:22.130 of 2 pi / omega_0, but with a period of (2 pi / omega_0) 00:23:22.130 --> 00:23:29.950 times 4, namely with a period of 31. 00:23:29.950 --> 00:23:34.310 Finally, let's take the example where omega_0 is equal 00:23:34.310 --> 00:23:37.030 to 1/6, as I've shown here. 00:23:37.030 --> 00:23:40.330 That actually looks, if you track it with your eye, like 00:23:40.330 --> 00:23:41.970 it's periodic. 00:23:41.970 --> 00:23:47.660 2 pi / omega_0, in that case, is equal to 12 pi. 00:23:47.660 --> 00:23:53.070 Well, what integer can I multiply 12 pi by and get 00:23:53.070 --> 00:23:54.400 another integer? 00:23:54.400 --> 00:23:58.350 The answer is none, because pi is an irrational number. 00:23:58.350 --> 00:24:03.860 So in fact, what that says is that if you look at this 00:24:03.860 --> 00:24:09.380 sinusoidal signal, it's not periodic at all, even though 00:24:09.380 --> 00:24:13.120 you might fool yourself into thinking it is simply because 00:24:13.120 --> 00:24:15.830 the envelope looks periodic. 00:24:15.830 --> 00:24:19.510 Namely, the continuous-time equivalent of this is 00:24:19.510 --> 00:24:22.580 periodic, the discrete-time sequence is not. 00:24:26.750 --> 00:24:27.180 OK. 00:24:27.180 --> 00:24:31.860 Well, we've seen, then, some important distinctions between 00:24:31.860 --> 00:24:34.780 continuous-time sinusoidal signals and discrete-time 00:24:34.780 --> 00:24:36.730 sinusoidal signals. 00:24:36.730 --> 00:24:41.900 The first one is the fact that in the continuous-time case, a 00:24:41.900 --> 00:24:45.830 time shift and phase change are always equivalent. 00:24:45.830 --> 00:24:49.210 Whereas in the discrete-time case, in effect, it works one 00:24:49.210 --> 00:24:52.400 way but not the other way. 00:24:52.400 --> 00:24:58.060 We've also seen that for a continuous-time signal, the 00:24:58.060 --> 00:25:01.990 continuous-time signal is always periodic, whereas the 00:25:01.990 --> 00:25:04.850 discrete-time signal is not necessarily. 00:25:04.850 --> 00:25:08.640 In particular, for the continuous-time case, if we 00:25:08.640 --> 00:25:12.320 have a general expression for the sinusoidal signal that 00:25:12.320 --> 00:25:15.650 I've indicated here, that's periodic for 00:25:15.650 --> 00:25:18.440 any choice of omega_0. 00:25:18.440 --> 00:25:30.880 Whereas in the discrete-time case, it's periodic only if 2 00:25:30.880 --> 00:25:34.740 pi / omega_0 can be multiplied by an integer 00:25:34.740 --> 00:25:37.020 to get another integer. 00:25:37.020 --> 00:25:41.470 Now, another important and, as it turns out, useful 00:25:41.470 --> 00:25:44.940 distinction between the continuous-time and 00:25:44.940 --> 00:25:49.370 discrete-time case is the fact that in the discrete-time 00:25:49.370 --> 00:25:56.550 case, as we vary what I've called the frequency omega_0, 00:25:56.550 --> 00:26:02.390 we only see distinct signals as omega_0 00:26:02.390 --> 00:26:04.900 varies over a 2 pi interval. 00:26:04.900 --> 00:26:09.940 And if we let omega_0 vary outside the range of, let's 00:26:09.940 --> 00:26:15.130 say, -pi to pi, or 0 to 2 pi, we'll see the same sequences 00:26:15.130 --> 00:26:18.870 all over again, even though at first glance, the mathematical 00:26:18.870 --> 00:26:20.870 expression might look different. 00:26:20.870 --> 00:26:26.980 So in the discrete-time case, this class of signals is 00:26:26.980 --> 00:26:33.340 identical for values of omega_0 separated by 2 pi, 00:26:33.340 --> 00:26:37.890 whereas in the continuous-time case, that is not true. 00:26:37.890 --> 00:26:41.020 In particular, if I consider these sinusoidal 00:26:41.020 --> 00:26:48.260 continuous-time signals, as I vary omega_0, what will happen 00:26:48.260 --> 00:26:53.200 is that I will always see different sinusoidal signals. 00:26:53.200 --> 00:26:55.210 Namely, these won't be equal. 00:26:55.210 --> 00:27:00.970 And in effect, we can justify that statement algebraically. 00:27:00.970 --> 00:27:04.410 And I won't take the time to do it carefully. 00:27:04.410 --> 00:27:09.670 But let's look, first of all, at the discrete-time case. 00:27:09.670 --> 00:27:15.130 And the statement that I'm making is that if I have two 00:27:15.130 --> 00:27:18.700 discrete-time sinusoidal signals at two different 00:27:18.700 --> 00:27:24.980 frequencies, and if these frequencies are separated by 00:27:24.980 --> 00:27:30.170 an integer multiple of 2 pi-- namely if omega_2 is equal to 00:27:30.170 --> 00:27:35.650 omega_1 + 2 pi times an integer m-- 00:27:35.650 --> 00:27:40.070 when I substitute this into this expression, because of 00:27:40.070 --> 00:27:46.330 the fact that n is also an integer, I'll have m * n as an 00:27:46.330 --> 00:27:48.270 integer multiple of 2 pi. 00:27:48.270 --> 00:27:50.520 And that term, of course, will disappear because of the 00:27:50.520 --> 00:27:53.370 periodicity of the sinusoid, and these two 00:27:53.370 --> 00:27:56.350 sequences will be equal. 00:27:56.350 --> 00:28:03.450 On the other hand in the continuous-time case, since t 00:28:03.450 --> 00:28:06.560 is not restricted to be an integer variable, for 00:28:06.560 --> 00:28:11.230 different values of omega_1 and omega_2, these sinusoidal 00:28:11.230 --> 00:28:13.770 signals will always be different. 00:28:17.280 --> 00:28:17.650 OK. 00:28:17.650 --> 00:28:21.540 Now, many of the issues that I've raised so far, in 00:28:21.540 --> 00:28:24.550 relation to sinusoidal signals, are elaborated on in 00:28:24.550 --> 00:28:26.240 more detail in the text. 00:28:26.240 --> 00:28:30.890 And of course, you'll have an opportunity to exercise some 00:28:30.890 --> 00:28:36.150 of this as you work through the video course manual. 00:28:36.150 --> 00:28:40.550 Let me stress that sinusoidal signals will play an extremely 00:28:40.550 --> 00:28:44.260 important role for us as building blocks for general 00:28:44.260 --> 00:28:47.770 signals and descriptions of systems, and leads to the 00:28:47.770 --> 00:28:51.870 whole concept Fourier analysis, which is very 00:28:51.870 --> 00:28:53.755 heavily exploited throughout the course. 00:28:56.760 --> 00:29:02.870 What I'd now like to turn to is another class of important 00:29:02.870 --> 00:29:03.780 building blocks. 00:29:03.780 --> 00:29:06.820 And in fact, we'll see that under certain conditions, 00:29:06.820 --> 00:29:10.970 these relate strongly to sinusoidal signals, namely the 00:29:10.970 --> 00:29:14.580 class of real and complex exponentials. 00:29:14.580 --> 00:29:19.810 Let me begin, first of all, with the real exponential, and 00:29:19.810 --> 00:29:23.420 in particular, in the continuous-time case. 00:29:23.420 --> 00:29:26.980 A real continuous-time exponential is mathematically 00:29:26.980 --> 00:29:36.000 expressed, as I indicate here, x(t) = C e ^ (a t), where for 00:29:36.000 --> 00:29:40.320 the real exponential, C and a are real numbers. 00:29:40.320 --> 00:29:43.900 And that's what we mean by the real exponential. 00:29:43.900 --> 00:29:47.010 Shortly, we'll also consider complex exponentials, where 00:29:47.010 --> 00:29:50.240 these numbers can then become complex. 00:29:50.240 --> 00:29:54.120 So this is an exponential function. 00:29:54.120 --> 00:29:59.220 And for example, if the parameter a is positive, that 00:29:59.220 --> 00:30:03.320 means that we have a growing exponential function. 00:30:03.320 --> 00:30:08.410 If the parameter a is negative, then that means that 00:30:08.410 --> 00:30:13.760 we have a decaying exponential function. 00:30:13.760 --> 00:30:17.750 Now, somewhat as an aside, it's kind of interesting to 00:30:17.750 --> 00:30:23.160 note that for exponentials, a time shift corresponds to a 00:30:23.160 --> 00:30:26.950 scale change, which is somewhat different than what 00:30:26.950 --> 00:30:29.700 happens with sinusoids. 00:30:29.700 --> 00:30:33.380 In the sinusoidal case, we saw that a time shift corresponded 00:30:33.380 --> 00:30:34.940 to a phase change. 00:30:34.940 --> 00:30:37.850 With the real exponential, a time shift, as it turns out, 00:30:37.850 --> 00:30:41.070 corresponds to simply changing the scale. 00:30:41.070 --> 00:30:45.730 There's nothing particularly crucial or 00:30:45.730 --> 00:30:47.300 exciting about that. 00:30:47.300 --> 00:30:51.050 And in fact, perhaps stressing it is a little misleading. 00:30:51.050 --> 00:30:53.620 For general functions, of course, about all that you can 00:30:53.620 --> 00:30:58.430 say about what happens when you implement a time shift is 00:30:58.430 --> 00:31:02.720 that it implements a time shift. 00:31:02.720 --> 00:31:03.060 OK. 00:31:03.060 --> 00:31:05.890 So here's the real exponential. 00:31:05.890 --> 00:31:08.600 Just C e ^ (a t). 00:31:08.600 --> 00:31:12.160 Let's look at the real exponential, now, in the 00:31:12.160 --> 00:31:14.480 discrete-time case. 00:31:14.480 --> 00:31:21.550 And in the discrete-time case, we have several alternate ways 00:31:21.550 --> 00:31:23.910 of expressing it. 00:31:23.910 --> 00:31:29.020 We can express the real exponential in the form C e ^ 00:31:29.020 --> 00:31:35.120 (beta n), or as we'll find more convenient, in part for a 00:31:35.120 --> 00:31:38.810 reason at I'll indicate shortly, we can rewrite this 00:31:38.810 --> 00:31:44.820 as C alpha ^ n, where of course, alpha = e ^ beta. 00:31:44.820 --> 00:31:47.610 More typically in the discrete-time case, we'll 00:31:47.610 --> 00:31:53.720 express the exponential as C alpha ^ n. 00:31:53.720 --> 00:31:57.690 So for example, this becomes, essentially, a geometric 00:31:57.690 --> 00:32:01.920 series or progression as n continues for 00:32:01.920 --> 00:32:03.740 certain values of alpha. 00:32:03.740 --> 00:32:11.070 Here for example, we have for alpha greater than 0, first of 00:32:11.070 --> 00:32:15.010 all on the top, the case where the magnitude of alpha is 00:32:15.010 --> 00:32:21.430 greater than 1, so that the sequence is exponentially or 00:32:21.430 --> 00:32:23.420 geometrically growing. 00:32:23.420 --> 00:32:27.770 On the bottom, again with alpha positive, but now with 00:32:27.770 --> 00:32:32.517 its magnitude less than 1, we have a geometric progression 00:32:32.517 --> 00:32:38.360 that is exponentially or geometrically decaying. 00:32:38.360 --> 00:32:38.650 OK. 00:32:38.650 --> 00:32:42.350 So this, in both of these cases, is with alpha 00:32:42.350 --> 00:32:45.070 greater than 0. 00:32:45.070 --> 00:32:48.480 Now the function that we're talking about is alpha ^ n. 00:32:48.480 --> 00:32:51.310 And of course, what you can see is that if alpha is 00:32:51.310 --> 00:32:57.910 negative instead of positive, then when n is even, that 00:32:57.910 --> 00:32:59.620 minus sign is going to disappear. 00:32:59.620 --> 00:33:02.320 When n is odd, there will be a minus sign. 00:33:02.320 --> 00:33:06.610 And so for alpha negative, the sequence is going to alternate 00:33:06.610 --> 00:33:08.500 positive and negative values. 00:33:08.500 --> 00:33:15.870 So for example, here we have alpha negative, with its 00:33:15.870 --> 00:33:17.230 magnitude less than 1. 00:33:17.230 --> 00:33:21.130 And you can see that, again, its envelope decays 00:33:21.130 --> 00:33:25.320 geometrically, and the values alternate in sign. 00:33:25.320 --> 00:33:29.310 And here we have the magnitude of alpha greater than 1, with 00:33:29.310 --> 00:33:30.600 alpha negative. 00:33:30.600 --> 00:33:33.360 Again, they alternate in sign, and of course it's growing 00:33:33.360 --> 00:33:35.740 geometrically. 00:33:35.740 --> 00:33:42.910 Now, if you think about alpha positive and go back to the 00:33:42.910 --> 00:33:48.640 expression that I have at the top, namely C alpha ^ n. 00:33:48.640 --> 00:33:54.670 With alpha positive, you can see a straightforward 00:33:54.670 --> 00:33:58.440 relationship between alpha and beta. 00:33:58.440 --> 00:34:01.830 Namely, beta is the natural logarithm of alpha. 00:34:04.520 --> 00:34:06.830 Something to think about is what 00:34:06.830 --> 00:34:09.070 happens if alpha is negative? 00:34:09.070 --> 00:34:13.230 Which is, of course, a very important and useful class of 00:34:13.230 --> 00:34:16.190 real discrete-time exponentials also. 00:34:16.190 --> 00:34:19.560 Well, it turns out that with alpha negative, if you try to 00:34:19.560 --> 00:34:25.020 express it as C e ^ (beta n), then beta comes out to be an 00:34:25.020 --> 00:34:26.659 imaginary number. 00:34:26.659 --> 00:34:30.889 And that is one, but not the only reason why, in the 00:34:30.889 --> 00:34:36.389 discrete-time case, it's often most convenient to phrase real 00:34:36.389 --> 00:34:42.420 exponentials in the form alpha ^ n, rather than e ^ (beta n). 00:34:42.420 --> 00:34:47.650 In other words, to express them in this form rather than 00:34:47.650 --> 00:34:48.900 in this form. 00:34:54.520 --> 00:34:56.699 Those are real exponentials, continuous-time and 00:34:56.699 --> 00:34:57.960 discrete-time. 00:34:57.960 --> 00:35:01.710 Now let's look at the continuous-time complex 00:35:01.710 --> 00:35:03.130 exponential. 00:35:03.130 --> 00:35:09.910 And what I mean by a complex exponential, again, is an 00:35:09.910 --> 00:35:15.750 exponential of the form C e ^ (a t). 00:35:15.750 --> 00:35:21.050 But in this case, we allow the parameters C and a to be 00:35:21.050 --> 00:35:22.970 complex numbers. 00:35:22.970 --> 00:35:25.850 And let's just track this through algebraically. 00:35:25.850 --> 00:35:28.820 If C and a are complex numbers, let's write C in 00:35:28.820 --> 00:35:33.350 polar form, so it has a magnitude and an angle. 00:35:33.350 --> 00:35:36.800 Let's write a in rectangular form, so it has a real part 00:35:36.800 --> 00:35:38.950 and an imaginary part. 00:35:38.950 --> 00:35:43.060 And when we substitute these two in here, combine some 00:35:43.060 --> 00:35:44.680 things together-- 00:35:44.680 --> 00:35:46.100 well actually, I haven't combined yet. 00:35:46.100 --> 00:35:52.160 I have this for the amplitude factor, and this for the 00:35:52.160 --> 00:35:53.960 exponential factor. 00:35:53.960 --> 00:35:58.597 I can now pull out of this the term corresponding to e ^ (r 00:35:58.597 --> 00:36:02.320 t), and combine the imaginary parts together. 00:36:02.320 --> 00:36:05.760 And I come down to the expression that I have here. 00:36:05.760 --> 00:36:12.800 So following this further, an exponential of this form, e ^ 00:36:12.800 --> 00:36:19.640 (j omega) or e ^ (j phi), using Euler's relation, can be 00:36:19.640 --> 00:36:28.150 expressed as the sum of a cosine plus j times a sine. 00:36:28.150 --> 00:36:31.560 And so that corresponds to this factor. 00:36:31.560 --> 00:36:35.290 And then there is this time-varying amplitude factor 00:36:35.290 --> 00:36:36.420 on top of it. 00:36:36.420 --> 00:36:39.390 Finally putting those together, we end up with the 00:36:39.390 --> 00:36:42.580 expression that I show on the bottom. 00:36:42.580 --> 00:36:49.010 And what this corresponds to are two sinusoidal signals, 90 00:36:49.010 --> 00:36:51.733 degrees out of phase, as indicated by the fact that 00:36:51.733 --> 00:36:54.220 there's a cosine and a sine. 00:36:54.220 --> 00:36:57.560 So there's a real part and an imaginary part, with 00:36:57.560 --> 00:37:03.210 sinusoidal components 90 degrees out of phase, and a 00:37:03.210 --> 00:37:07.140 time-varying amplitude factor, which is a real exponential. 00:37:07.140 --> 00:37:11.670 So it's a sinusoid multiplied by a real exponential in both 00:37:11.670 --> 00:37:14.870 the real part and the imaginary part. 00:37:14.870 --> 00:37:19.070 And let's just see what one of those terms might look like. 00:37:19.070 --> 00:37:29.290 What I've indicated at the top is a sinusoidal signal with a 00:37:29.290 --> 00:37:32.330 time-varying exponential envelope, or an envelope which 00:37:32.330 --> 00:37:37.780 is a real exponential, and in particular which is growing, 00:37:37.780 --> 00:37:40.960 namely with r greater than 0. 00:37:40.960 --> 00:37:44.430 And on the bottom, I've indicated the same thing with 00:37:44.430 --> 00:37:46.240 r less than 0. 00:37:46.240 --> 00:37:51.300 And this kind of sinusoidal signal, by the way, is 00:37:51.300 --> 00:37:54.730 typically referred to as a damped sinusoid. 00:37:54.730 --> 00:37:58.110 So with r negative, what we have in the real and imaginary 00:37:58.110 --> 00:38:01.130 parts are damped sinusoids. 00:38:01.130 --> 00:38:07.270 And the sinusoidal components of that are 90 degrees out of 00:38:07.270 --> 00:38:12.260 phase, in the real part and in the imaginary part. 00:38:12.260 --> 00:38:12.560 OK. 00:38:12.560 --> 00:38:17.040 Now, in the discrete-time case, we have more or less the 00:38:17.040 --> 00:38:19.350 same kind of outcome. 00:38:19.350 --> 00:38:26.220 In particular we'll make reference to our complex 00:38:26.220 --> 00:38:28.860 exponentials in the discrete-time case. 00:38:28.860 --> 00:38:33.200 The expression for the complex exponential looks very much 00:38:33.200 --> 00:38:36.430 like the expression for the real exponential, except that 00:38:36.430 --> 00:38:40.460 now we have complex factors. 00:38:40.460 --> 00:38:46.040 So C and alpha are complex numbers. 00:38:46.040 --> 00:38:53.050 And again, if we track through the algebra, and get to a 00:38:53.050 --> 00:38:57.740 point where we have a real exponential multiplied by a 00:38:57.740 --> 00:39:03.930 factor which is a purely imaginary exponential, apply 00:39:03.930 --> 00:39:10.430 Euler's relationship to this, we then finally come down to a 00:39:10.430 --> 00:39:16.980 sequence, which has a real exponential amplitude 00:39:16.980 --> 00:39:21.480 multiplying one sinusoid in the real part. 00:39:21.480 --> 00:39:25.120 And in the imaginary part, exactly the same kind of 00:39:25.120 --> 00:39:30.560 exponential multiplying a sinusoid that's 90 degrees out 00:39:30.560 --> 00:39:33.150 of phase from that. 00:39:33.150 --> 00:39:37.750 And so if we look at what one of these factors might look 00:39:37.750 --> 00:39:42.510 like, it's what we would expect given the analogy with 00:39:42.510 --> 00:39:44.580 the continuous-time case. 00:39:44.580 --> 00:39:53.140 Namely, it's a sinusoidal sequence with a real 00:39:53.140 --> 00:39:55.060 exponential envelope. 00:39:55.060 --> 00:39:58.500 In the case where alpha is positive, then 00:39:58.500 --> 00:40:00.250 it's a growing envelope. 00:40:00.250 --> 00:40:03.050 In the case where alpha is negative-- 00:40:03.050 --> 00:40:03.990 I'm sorry-- 00:40:03.990 --> 00:40:08.200 where the magnitude of alpha is greater than 1, it's a 00:40:08.200 --> 00:40:10.070 growing exponential envelope. 00:40:10.070 --> 00:40:14.700 Where the magnitude of alpha is less than 1, it's a 00:40:14.700 --> 00:40:17.570 decaying exponential envelope. 00:40:17.570 --> 00:40:20.370 And so I've illustrated that here. 00:40:20.370 --> 00:40:24.050 Here we have the magnitude of alpha greater than 1. 00:40:24.050 --> 00:40:27.510 And here we have the magnitude of alpha less than 1. 00:40:27.510 --> 00:40:30.960 In both cases, sinusoidal sequences underneath the 00:40:30.960 --> 00:40:35.920 envelope, and then an envelope that is dictated by what the 00:40:35.920 --> 00:40:39.710 magnitude of alpha is. 00:40:39.710 --> 00:40:40.200 OK. 00:40:40.200 --> 00:40:44.900 Now, in the discrete-time case, then, we have results 00:40:44.900 --> 00:40:47.080 similar to the continuous-time case. 00:40:47.080 --> 00:40:53.170 Namely, components in a real and imaginary part that have a 00:40:53.170 --> 00:40:57.370 real exponential factor times a sinusoid. 00:40:57.370 --> 00:41:07.240 Of course, if the magnitude of alpha is equal to 1, then this 00:41:07.240 --> 00:41:10.010 factor disappears, or is equal to 1. 00:41:10.010 --> 00:41:12.350 And this factor is equal to 1. 00:41:12.350 --> 00:41:15.250 And so we have sinusoids in both the real 00:41:15.250 --> 00:41:17.620 and imaginary parts. 00:41:17.620 --> 00:41:22.660 Now, one can ask whether, in general, the complex 00:41:22.660 --> 00:41:26.580 exponential with the magnitude of alpha equal to 1 is 00:41:26.580 --> 00:41:28.640 periodic or not periodic. 00:41:28.640 --> 00:41:34.450 And the clue to that can be inferred by examining this 00:41:34.450 --> 00:41:36.680 expression. 00:41:36.680 --> 00:41:39.600 In particular, in the discrete-time case with the 00:41:39.600 --> 00:41:44.600 magnitude of alpha equal to 1, we have pure sinusoids in the 00:41:44.600 --> 00:41:48.250 real part and the imaginary part. 00:41:48.250 --> 00:41:51.240 And in fact, in a continuous-time case with r 00:41:51.240 --> 00:41:56.910 equal to 0, we have sinusoids in the real part and the 00:41:56.910 --> 00:41:59.570 imaginary part. 00:41:59.570 --> 00:42:02.520 In a continuous-time case when we have a pure complex 00:42:02.520 --> 00:42:07.580 exponential, so that the terms aren't exponentially growing 00:42:07.580 --> 00:42:09.410 or decaying, those 00:42:09.410 --> 00:42:13.420 exponentials are always periodic. 00:42:13.420 --> 00:42:17.630 Because, of course, the real and imaginary sinusoidal 00:42:17.630 --> 00:42:19.640 components are periodic. 00:42:19.640 --> 00:42:23.100 In the discrete-time case, we know that the sinusoids may or 00:42:23.100 --> 00:42:28.150 may not be periodic, depending on the value of omega_0. 00:42:28.150 --> 00:42:31.390 And so in fact, in the discrete-time case, the 00:42:31.390 --> 00:42:38.630 exponential e ^ (j omega_0 n), that I've indicated here, may 00:42:38.630 --> 00:42:42.780 or may not be periodic depending on what the value of 00:42:42.780 --> 00:42:44.030 omega_0 is. 00:42:47.140 --> 00:42:47.660 OK. 00:42:47.660 --> 00:42:53.410 Now, to summarize, in this lecture I've introduced and 00:42:53.410 --> 00:42:58.000 discussed a number of important basic signals. 00:42:58.000 --> 00:43:04.710 In particular, sinusoids and real and complex exponentials. 00:43:04.710 --> 00:43:07.660 One of the important outcomes of the discussion, emphasized 00:43:07.660 --> 00:43:11.190 further in the text, is that there are some very important 00:43:11.190 --> 00:43:13.060 similarities between them. 00:43:13.060 --> 00:43:16.700 But there are also some very important differences. 00:43:16.700 --> 00:43:20.770 And these differences will surface when we exploit 00:43:20.770 --> 00:43:26.220 sinusoids and complex exponentials as basic building 00:43:26.220 --> 00:43:29.350 blocks for more general continuous-time and 00:43:29.350 --> 00:43:32.160 discrete-time signals. 00:43:32.160 --> 00:43:35.900 In the next lecture, what I'll discuss are some other very 00:43:35.900 --> 00:43:40.070 important building blocks, namely, what are referred to 00:43:40.070 --> 00:43:43.230 as step signals and impulse signals. 00:43:43.230 --> 00:43:47.970 And those, together with the sinusoidal signals and 00:43:47.970 --> 00:43:52.160 exponentials as we've talked about today, will really form 00:43:52.160 --> 00:43:57.300 the cornerstone for, essentially, all of the signal 00:43:57.300 --> 00:44:00.270 and system analysis that we'll be dealing with for the 00:44:00.270 --> 00:44:01.780 remainder of course. 00:44:01.780 --> 00:44:03.030 Thank you.