1 00:00:00,008 --> 00:00:04,480 In module 2.2 we're going to study the complex exponential. 2 00:00:04,480 --> 00:00:08,760 We had announced this sequence as being a fundamental character in our play. 3 00:00:08,760 --> 00:00:11,730 And it is time to meet it more in detail. 4 00:00:11,730 --> 00:00:13,790 It is not a completely simple object, but 5 00:00:13,790 --> 00:00:17,560 it has many features that will be useful later on in this class. 6 00:00:17,560 --> 00:00:22,410 In particular, there are periodic complex sequence that we are going to see, but 7 00:00:22,410 --> 00:00:25,940 there is also a notion of frequency which is a discrete time frequency 8 00:00:25,940 --> 00:00:27,480 that is quite subtle. 9 00:00:27,480 --> 00:00:29,730 In continuous time, frequency is very natural. 10 00:00:29,730 --> 00:00:33,870 Think of frequency for example in music, in musical instruments, and so on. 11 00:00:33,870 --> 00:00:37,440 In discrete time, the notion is more subtle, because there is 12 00:00:37,440 --> 00:00:42,330 an increasing frequency that after a while looks actually like a smaller frequency. 13 00:00:42,330 --> 00:00:45,940 And this phenomenon, which is well known if you watch old movies when you have 14 00:00:45,940 --> 00:00:47,720 wagon wheels that turn backwards, 15 00:00:47,720 --> 00:00:51,060 even though the wagon is actually moving forward. 16 00:00:51,060 --> 00:00:53,761 Is an effect called aliasing that we need to understand, 17 00:00:53,761 --> 00:00:57,042 because it's very particular to discrete time signal processing. 18 00:00:57,042 --> 00:01:00,500 Then we will finish the module 2.2 by seeing the relationship 19 00:01:00,500 --> 00:01:02,890 between a discrete time frequency and 20 00:01:02,890 --> 00:01:07,120 an equivalent continuous time frequency, which is very important, for example, 21 00:01:07,120 --> 00:01:12,410 if you want to synthesize a sound from the computer into an acoustic system. 22 00:01:12,410 --> 00:01:17,700 Or of course in the reverse, when you acquire a real-time signal from 23 00:01:17,700 --> 00:01:25,870 the continuous time domain and you want create a sequence. 24 00:01:25,870 --> 00:01:27,153 The overview is the following. 25 00:01:27,153 --> 00:01:32,180 We are going to introduce a complex exponential, see periodicity, 26 00:01:32,180 --> 00:01:36,030 then a special effect which is called the wagonwheel effect, and 27 00:01:36,030 --> 00:01:40,970 the maximum speed of discrete complex exponentials. 28 00:01:40,970 --> 00:01:46,414 Then we are also going to contrast digital and real-world frequency notions. 29 00:01:48,797 --> 00:01:52,598 As we have said in the previous sub-module, oscillations are everywhere. 30 00:01:52,598 --> 00:01:55,220 The cardiac beat, 31 00:01:55,220 --> 00:01:59,720 the whistle on the train, the waves on the ocean, the violin strings. 32 00:02:02,350 --> 00:02:05,700 The oscillation needs the following ingredients. 33 00:02:05,700 --> 00:02:07,710 First, we need a frequency. 34 00:02:07,710 --> 00:02:09,940 The unit will be radians. 35 00:02:09,940 --> 00:02:15,710 So, we will always look at the unit circle and the full circle is in radians 2 pi. 36 00:02:15,710 --> 00:02:19,500 There will be an initial phase, phi. 37 00:02:19,500 --> 00:02:22,850 In the same unit of course, radians. 38 00:02:22,850 --> 00:02:25,150 There'll be an amplitude, 39 00:02:25,150 --> 00:02:28,510 this amplitude will depend how we measure the physical phenomenon. 40 00:02:30,040 --> 00:02:35,520 And we therefore get the trigonometric function which is the sequence, 41 00:02:35,520 --> 00:02:40,505 for example, x[n], which is given by A, the amplitude, 42 00:02:40,505 --> 00:02:45,070 times cos(omega n + phi). 43 00:02:45,070 --> 00:02:52,030 So the phase term phi, the frequency omega, the time index n, okay? 44 00:02:53,280 --> 00:02:58,210 So this is the basic ingredient of a trigonometric 45 00:02:58,210 --> 00:03:02,650 sequence here is that thiis going to be periodic and has initial phase phi. 46 00:03:05,750 --> 00:03:08,660 Now, we always use complex exponentials. 47 00:03:08,660 --> 00:03:13,100 Now, some of you might not like complex numbers, but complex numbers 48 00:03:13,100 --> 00:03:18,560 are our friends here because they make trigonometry much simpler. 49 00:03:18,560 --> 00:03:25,340 So a trigonometric function that is expressed as a complex exponential 50 00:03:25,340 --> 00:03:31,000 of course can be reduced to sines and cosines using Euler's formulas. 51 00:03:31,000 --> 00:03:34,580 Okay, so we have a sequence x[n], just like previously. 52 00:03:34,580 --> 00:03:41,670 We have let's say A is the amplitude, e to the j(omega n + phi). 53 00:03:41,670 --> 00:03:46,517 And by Euler's formulas, this is the amplitude cos(omega n + phi), 54 00:03:46,517 --> 00:03:48,470 that's the real part. 55 00:03:48,470 --> 00:03:52,100 j times sin (omega n+ phi), 56 00:03:52,100 --> 00:03:55,980 which is the imaginary part, and together we have a complex exponential. 57 00:03:58,900 --> 00:04:04,400 So, we see that sines and cosines always come together. 58 00:04:04,400 --> 00:04:07,630 The reason why we prefer exponentials to sines and 59 00:04:07,630 --> 00:04:10,860 cosines is that the mathematics are simpler. 60 00:04:10,860 --> 00:04:13,870 Instead of doing trigonometry, we do algebra. 61 00:04:15,280 --> 00:04:17,550 And, last but not least, 62 00:04:17,550 --> 00:04:21,380 we actually have complex numbers in all digital systems that we use. 63 00:04:21,380 --> 00:04:24,352 So on your computer, you have real numbers, but 64 00:04:24,352 --> 00:04:28,495 you have also complex numbers, and you can do complex arithmetic. 65 00:04:30,839 --> 00:04:36,260 Okay, so let's look at an example of the advantages of using complex exponentials. 66 00:04:36,260 --> 00:04:39,440 So you want to do the change of phase of a pure cosine. 67 00:04:39,440 --> 00:04:44,440 So you start with the cos(omega n + phi). 68 00:04:44,440 --> 00:04:50,011 This will be written as a cos(omega n) + b sin(omega n), 69 00:04:50,011 --> 00:04:52,810 where a and b are given here. 70 00:04:52,810 --> 00:04:57,220 So if you know your trigonometry by heart, you know this is a very natural thing. 71 00:04:58,591 --> 00:05:02,519 But if you don't remember all the trigonometric formulas, 72 00:05:02,519 --> 00:05:05,870 you can always look them up on Wikipedia. 73 00:05:05,870 --> 00:05:09,950 But it is a little a cumbersome way to actually 74 00:05:09,950 --> 00:05:13,104 deal with such simple operations, okay? 75 00:05:13,104 --> 00:05:19,880 So instead, what we do is we look at the change of phase of a pure 76 00:05:19,880 --> 00:05:26,000 cosine by stating that the cosine is the real part of a complex exponential, 77 00:05:26,000 --> 00:05:30,030 the change of phase up here is an addition in the exponent. 78 00:05:30,030 --> 00:05:34,190 And the result is that we simply have a multiplication here 79 00:05:34,190 --> 00:05:36,050 in the complex exponential. 80 00:05:36,050 --> 00:05:40,408 So the phase that was up there, of course by the fact that 81 00:05:40,408 --> 00:05:44,910 it's in the exponent it will end up just being a multiplication here by the phase. 82 00:05:44,910 --> 00:05:49,950 So the phase change is extremely simple in the complex exponential domain. 83 00:05:49,950 --> 00:05:53,270 At the end, we simply take the real part, we get the cosine, and 84 00:05:53,270 --> 00:05:54,980 we're all done, okay? 85 00:05:56,570 --> 00:06:00,350 Notation is simpler, phase shift is a simple multiplication. 86 00:06:00,350 --> 00:06:06,080 So, these two effects are why we prefer complex exponentials. 87 00:06:08,300 --> 00:06:10,680 Okay, so what is a complex exponential? 88 00:06:10,680 --> 00:06:15,270 It is Euler's formula again, e to the j alpha 89 00:06:15,270 --> 00:06:20,730 is equal to cos alpha + j sin alpha, okay? 90 00:06:20,730 --> 00:06:21,820 I'm repeating myself, but 91 00:06:21,820 --> 00:06:28,320 it's such a beautiful formula that we can see it a couple of times, it won't hurt. 92 00:06:28,320 --> 00:06:33,290 Okay, so let's plot this in the complex plane, so 93 00:06:33,290 --> 00:06:38,690 the complex plane has a real axis, and an imaginary axis. 94 00:06:38,690 --> 00:06:47,730 So e to the j alpha, given by cos alpha + j sin alpha lives on a unit circle here. 95 00:06:47,730 --> 00:06:50,190 This is nicely the circle here. 96 00:06:50,190 --> 00:06:56,882 And the complex number e to the j alpha is, well, cos alpha, 97 00:06:56,882 --> 00:07:03,310 which would be here, + j sin alpha, which is here. 98 00:07:03,310 --> 00:07:09,390 And it is a vector of length 1 that reaches the unit circle and 99 00:07:09,390 --> 00:07:14,470 has an angle alpha here with the real axis, okay? 100 00:07:14,470 --> 00:07:19,680 So if you're not very familiar with this picture, please look at it, 101 00:07:19,680 --> 00:07:23,450 play with it, play with Euler's formula, go back and 102 00:07:23,450 --> 00:07:28,060 forth between real numbers and imaginary numbers, and so on. 103 00:07:28,060 --> 00:07:33,340 Because this is the alpha and omega of what we will be doing here in this class. 104 00:07:36,530 --> 00:07:41,068 Okay, so if we have a point on the complex plane, z, 105 00:07:41,068 --> 00:07:47,030 In this location, and we want to rotate it, 106 00:07:47,030 --> 00:07:52,040 then the argument is that we can simply multiply it by a complex 107 00:07:53,180 --> 00:07:57,660 number here, e to the j alpha, that's the previous character, and 108 00:07:57,660 --> 00:08:02,830 it will exactly move z by an angle alpha, okay? 109 00:08:02,830 --> 00:08:07,530 So rotation amounts to multiplying with the complex 110 00:08:07,530 --> 00:08:12,100 number which is of unit norm. 111 00:08:12,100 --> 00:08:15,200 So, a complex number of the form e to the j alpha. 112 00:08:16,720 --> 00:08:20,766 So you can see that a relatively complicated operation on 113 00:08:20,766 --> 00:08:24,560 the complex plane, which is moving from here to z', 114 00:08:24,560 --> 00:08:30,020 is actually very simple in this algebra using complex exponentials. 115 00:08:30,020 --> 00:08:34,148 Let's look at what we will call the complex exponential generating machine. 116 00:08:34,148 --> 00:08:38,255 x[n] = e to the j omega n. 117 00:08:38,255 --> 00:08:44,180 x[n+1] is simply e to the j times x[n]. 118 00:08:44,180 --> 00:08:47,800 And so, recursively we're going to generate successive 119 00:08:47,800 --> 00:08:51,650 samples of this discrete time complex exponential. 120 00:08:51,650 --> 00:08:56,887 So let's start at x[0], it's straight on the real line, 121 00:08:56,887 --> 00:09:03,450 as I will move by omega, that gives x[1[, by 2 omega, is x[2], etc. 122 00:09:03,450 --> 00:09:08,490 It will keep going x[4], 5, 6, 7, and so 123 00:09:08,490 --> 00:09:14,400 on, and in this particular case, x[12] happens to be back at the origin, 124 00:09:14,400 --> 00:09:20,102 and therefore, it will be a periodic discrete complex exponential. 125 00:09:20,102 --> 00:09:24,720 Now, if there is an initial phase, we don't start from the real line, but 126 00:09:24,720 --> 00:09:27,360 we'll start with an angle theta. 127 00:09:27,360 --> 00:09:31,395 But then the same story goes on, and as we go, 128 00:09:31,395 --> 00:09:35,000 x[6], 7, etc., at x[12], 129 00:09:35,000 --> 00:09:40,340 we will be back at the same location as x[0]. 130 00:09:40,340 --> 00:09:43,160 Now, one has to be careful, 131 00:09:43,160 --> 00:09:48,440 because not every sinusoid is periodic in discrete time, and we have an example 132 00:09:48,440 --> 00:09:53,339 here where omega is chosen such that the result will actually not be periodic. 133 00:09:54,450 --> 00:09:58,645 So x[1], move by omega on the unit circle, 134 00:09:58,645 --> 00:10:02,731 x[2], x[3], 4, 5, 6, 7, 8, 135 00:10:02,731 --> 00:10:08,190 is not back to the origin, and we can keep going like this. 136 00:10:08,190 --> 00:10:11,257 And it's not hard to see that this will not be periodic. 137 00:10:14,635 --> 00:10:18,830 What does periodicity mean for a discrete time exponential? 138 00:10:18,830 --> 00:10:22,740 So e to the j omega n is periodic if and 139 00:10:22,740 --> 00:10:28,320 only if omega is equal to a rational factor of 2 pi. 140 00:10:28,320 --> 00:10:31,939 So, M over N times 2 pi, where M and N are integers. 141 00:10:33,250 --> 00:10:37,100 And in this case, we can actually the fact, of course, 142 00:10:37,100 --> 00:10:43,290 that if we add a multiple of 2 pi to the angle of a complex exponential, 143 00:10:43,290 --> 00:10:45,280 we are back to e to the j omega. 144 00:10:46,660 --> 00:10:51,650 The quiz is about if a single e to the jn is periodic. 145 00:10:51,650 --> 00:10:56,590 On the complex plane, a single point can have many different names, and this 146 00:10:57,760 --> 00:11:02,490 uncertainty, in some sense, is one of the reasons why complex exponentials and 147 00:11:02,490 --> 00:11:09,130 discrete time are interesting but sometimes difficult objects to deal with. 148 00:11:09,130 --> 00:11:14,390 If I write the red dot here, e to the j alpha, I could have also 149 00:11:14,390 --> 00:11:19,340 said that it's actually the same, except that I had gone alpha plus 2 pi. 150 00:11:20,360 --> 00:11:23,020 Or I could have gone alpha plus 6 pi. 151 00:11:23,020 --> 00:11:25,700 Or any multiple of 2 pi. 152 00:11:25,700 --> 00:11:31,030 I could also have said that the angle was actually a negative angle of alpha 153 00:11:31,030 --> 00:11:36,570 minus 2 pi, and this will give me the exact same point, e to the j alpha. 154 00:11:36,570 --> 00:11:38,770 So the question is, how fast can we go? 155 00:11:38,770 --> 00:11:42,020 And here we are going to watch a movie, and 156 00:11:42,020 --> 00:11:46,860 the movie is a depiction of this famous wagon wheel effect that you probably have 157 00:11:46,860 --> 00:11:52,530 seen, watching western movies, old western movies on television. 158 00:11:52,530 --> 00:11:54,110 Let us do an experiment. 159 00:11:54,110 --> 00:11:56,410 Here we have the wheel of a bicycle. 160 00:11:56,410 --> 00:12:01,390 It has four spokes, so it has fourfold symmetry, and between two frames, 161 00:12:01,390 --> 00:12:06,100 we see it advances by the number of degrees from the upper left corner. 162 00:12:06,100 --> 00:12:09,180 So we see it goes faster and faster. 163 00:12:09,180 --> 00:12:11,930 Now, because of fourfold symmetry, 164 00:12:11,930 --> 00:12:16,610 everything we have seen before as complex exponentials will be divided by 4. 165 00:12:16,610 --> 00:12:22,560 So when we approach 45 degrees, we start to have the feeling the wheel goes 166 00:12:23,660 --> 00:12:27,639 backwards, at 45 it has a funny effect, and it starts to go backwards. 167 00:12:28,700 --> 00:12:30,900 And it will slow down. 168 00:12:32,020 --> 00:12:34,960 Again, the wheel has a fourfold symmetry. 169 00:12:34,960 --> 00:12:39,970 So we are approaching 90 degrees when it will stop. 170 00:12:39,970 --> 00:12:44,220 80, it gets slower and slower. 171 00:12:46,430 --> 00:12:47,270 Even slower. 172 00:12:48,300 --> 00:12:54,290 And at 90, we are soon there, it will actually stop. 173 00:12:54,290 --> 00:12:56,040 So the wheel is actually turning. 174 00:12:56,040 --> 00:13:00,680 It turns by a quarter of the circle every frame. 175 00:13:00,680 --> 00:13:04,170 But of course, because of fourfold symmetry, it actually stops. 176 00:13:05,440 --> 00:13:07,449 Okay, that's an illustration of aliasing. 177 00:13:09,400 --> 00:13:12,880 To understand this, we are going to look at frequencies and 178 00:13:12,880 --> 00:13:13,940 we increase the frequency. 179 00:13:13,940 --> 00:13:16,820 So the first frequency omega = 2 pi/12. 180 00:13:16,820 --> 00:13:19,620 So it is a very small frequency. 181 00:13:19,620 --> 00:13:24,840 It's one we have seen before, so x[0], x[1], etc., turns around and 182 00:13:24,840 --> 00:13:29,950 sure enough, x[12] is back to the real line. 183 00:13:29,950 --> 00:13:34,080 If we take omega is equal to 2 pi/6, so 184 00:13:34,080 --> 00:13:39,580 that's a much larger frequency here, but it's still a divisor of 2 pi, 185 00:13:39,580 --> 00:13:44,550 and so we have x[1], 2, 3, 4, 5, and x[6] is equal to x[0]. 186 00:13:44,550 --> 00:13:47,730 Again, a periodic signal. 187 00:13:47,730 --> 00:13:56,110 If we go up in frequency, omega is now equal to 2 pi/5, it is a periodic signal, 188 00:13:56,110 --> 00:14:00,270 sure enough, but you can see that its frequency now is very fast. 189 00:14:01,350 --> 00:14:07,950 2 pi/4, we simply see the four quadrants here, x[0], x[1], x[2], and x[3]. 190 00:14:07,950 --> 00:14:12,910 2 pi/3 divides nicely the unit circle into three pieces. 191 00:14:12,910 --> 00:14:17,032 2 pi/2 = pi, that's an interesting signal, 192 00:14:17,032 --> 00:14:22,260 because it simply alternates between +1 and -1. 193 00:14:22,260 --> 00:14:23,720 And it'll alternate like this, 194 00:14:23,720 --> 00:14:28,950 and so this is the maximum frequency discrete time complex exponential. 195 00:14:28,950 --> 00:14:35,430 Now, if the frequency is between pi and 2 pi, then 196 00:14:35,430 --> 00:14:40,460 there is a solution that we can easily think of it as a positive frequency, 197 00:14:40,460 --> 00:14:45,750 slightly larger than pi, or a negative frequency, slightly smaller than pi. 198 00:14:46,760 --> 00:14:52,290 Now, when omega = 2 pi- alpha, and alpha is small, then of 199 00:14:52,290 --> 00:14:57,190 course we are going to most likely think that it's actually a negative frequency. 200 00:14:57,190 --> 00:15:03,546 So here we have an example of x[1], which is 2 pi- alpha away, 201 00:15:03,546 --> 00:15:08,350 but is also minus alpha from the real line. 202 00:15:08,350 --> 00:15:13,510 And if we do this, we see now that we have the sense of a negative frequency. 203 00:15:13,510 --> 00:15:18,550 Let's now think about the difference between digital and physical frequency. 204 00:15:18,550 --> 00:15:23,450 In discrete time, n has no physical dimensions, it's just a counter. 205 00:15:23,450 --> 00:15:28,590 And periodicity is how many samples before the pattern repeats. 206 00:15:28,590 --> 00:15:33,840 In the real world, periodicity is how many seconds before the pattern repeats, 207 00:15:33,840 --> 00:15:38,780 and the frequency is measured in Hz, or s-1, 208 00:15:38,780 --> 00:15:43,370 which is 1 over the number of seconds until the pattern repeats. 209 00:15:43,370 --> 00:15:48,980 Now, if you have a PC that plays sound, you have to map digital frequency or 210 00:15:48,980 --> 00:15:54,180 a discrete time signal x[n] through a sound card into a loudspeaker. 211 00:15:54,180 --> 00:15:58,220 And so there has to be a relationship between the clock frequency on your 212 00:15:58,220 --> 00:16:03,530 computer that maps into a physical frequency out in the real world. 213 00:16:03,530 --> 00:16:05,160 This is given by the system clock. 214 00:16:05,160 --> 00:16:10,300 So, the sound card has a clock Ts and every Ts second it will 215 00:16:10,300 --> 00:16:14,990 generate a sample that will be interpolated and fed to the loudspeaker. 216 00:16:14,990 --> 00:16:18,740 We can set the time Ts between samples. 217 00:16:18,740 --> 00:16:25,680 So periodicity of M samples means a physical periodicity of M times T seconds. 218 00:16:25,680 --> 00:16:33,827 So the real world frequency is going to be f = 1/MTs.