1 00:00:00,620 --> 00:00:04,170 Welcome to module 2 of Digital Signal Processing. 2 00:00:04,170 --> 00:00:08,580 In this module, we are going to see the basics of Digital Signal Processing and 3 00:00:08,580 --> 00:00:11,970 we'll start by considering discrete time signals and 4 00:00:11,970 --> 00:00:15,168 operators of discrete time signals in module 2.1. 5 00:00:15,168 --> 00:00:20,160 In module 2.2, we are going to consider complex exponentials. 6 00:00:20,160 --> 00:00:25,180 These are the most elementary and fundamental discrete time signals. 7 00:00:25,180 --> 00:00:25,570 Finally, 8 00:00:25,570 --> 00:00:28,190 in module 2.3, we are going to take 9 00:00:28,190 --> 00:00:32,330 building blocks, elementary ones from signal processing to see 10 00:00:32,330 --> 00:00:35,800 some elementary operations on signals, like moving averages 11 00:00:35,800 --> 00:00:38,620 and recursive filters, and finally build a simple synthesizer. 12 00:00:38,620 --> 00:00:43,112 Let's now get started and see what discrete time signals really are. 13 00:00:43,112 --> 00:00:45,649 [BLANK_AUDIO] 14 00:00:45,649 --> 00:00:49,720 Module 2.1, Discrete-time signals. The Overview is the following. 15 00:00:49,720 --> 00:00:54,180 We will definie discrete-time signals to know what we are 16 00:00:54,180 --> 00:00:56,860 talking about here as a main component in the class. 17 00:00:58,380 --> 00:01:01,744 We are going to look at different types of signals, give examples. 18 00:01:01,744 --> 00:01:07,880 And then go onto operators, which are elementary blocks, 19 00:01:07,880 --> 00:01:10,660 building blocks for more complex systems that will be 20 00:01:10,660 --> 00:01:13,690 used in the discrete-time signal processing. 21 00:01:13,690 --> 00:01:20,050 And finally finish with two concepts which will come also over and over in 22 00:01:20,050 --> 00:01:23,010 the class which is the energy of a signal and the power of a signal. 23 00:01:26,070 --> 00:01:29,660 What is the earliest discreet time signals we can think of? 24 00:01:29,660 --> 00:01:35,840 Well, probably the earliest one that has been recorded are the floods of the Nile. 25 00:01:35,840 --> 00:01:40,270 So the Nile of course is extremely important in the agriculture of Egypt and 26 00:01:40,270 --> 00:01:46,000 so from very early on, people recorded how high the Nile would come in 27 00:01:46,000 --> 00:01:51,972 any given year, and this representation has been caught here on this beautiful 28 00:01:51,972 --> 00:01:58,224 historical record that is about 4 and a half thousand 29 00:01:58,224 --> 00:02:01,290 years old, and these are representation of the flood data. 30 00:02:01,290 --> 00:02:04,830 Unfortunately, we don't have digital version of this data, 31 00:02:04,830 --> 00:02:07,940 so we can really do signal processing on this. 32 00:02:07,940 --> 00:02:09,490 Maybe somebody wants to pick up a 33 00:02:09,490 --> 00:02:13,390 research project and actually, do this transformation. 34 00:02:13,390 --> 00:02:16,930 In the meantime, we have access to more recent data. 35 00:02:16,930 --> 00:02:24,060 So this is an example of the, flood in the Nile in cubic meters per second. 36 00:02:24,060 --> 00:02:27,920 Yearly measurements over the last 100 years or so. 37 00:02:29,562 --> 00:02:34,310 you can see the representation is these lollipop diagrams. 38 00:02:34,310 --> 00:02:40,150 So as a lollipop is, at any given year, we represent one data point. 39 00:02:40,150 --> 00:02:40,960 So we have here, 40 00:02:42,040 --> 00:02:48,660 almost a bit more than 100. It's a very busy signal, but overall, even 41 00:02:48,660 --> 00:02:52,760 if you just look at it approximately, you can sort of see there is a trend, okay? 42 00:02:52,760 --> 00:02:57,720 And the trend is not a good one, means there is less and less water in the Nile. 43 00:03:00,190 --> 00:03:04,030 The second example of, signal is daily temperature. 44 00:03:04,030 --> 00:03:06,640 It's probably your first scientific experiment. 45 00:03:06,640 --> 00:03:11,730 Every day, at 8 am, you look at the temperature outside. 46 00:03:11,730 --> 00:03:15,490 You do this every day for a year, for a couple of years and so on. 47 00:03:15,490 --> 00:03:18,000 And you record something that could also be 48 00:03:18,000 --> 00:03:20,820 a lollipop diagram, like in the previous slide. 49 00:03:20,820 --> 00:03:24,980 Except there are so many points here, that we decided to actually join them. 50 00:03:24,980 --> 00:03:25,270 So it 51 00:03:25,270 --> 00:03:28,020 looks almost as a continuous time function even though 52 00:03:28,020 --> 00:03:33,350 it's one measurement every day, or a few thousand days. 53 00:03:33,350 --> 00:03:35,120 Now, when you look at this, you, of 54 00:03:35,120 --> 00:03:38,810 course, immediately see that there are seasons happening. 55 00:03:38,810 --> 00:03:42,810 So the temperature here in Centigrades, you know, goes up and down 56 00:03:42,810 --> 00:03:47,530 on a, with a yearly variation, and this we can actually show. 57 00:03:47,530 --> 00:03:50,900 We can fit the sinusoid, here. It's a blue curve 58 00:03:50,900 --> 00:03:54,948 that has been added. And this is simply the trends that these 59 00:03:54,948 --> 00:03:55,957 [INAUDIBLE] 60 00:03:55,957 --> 00:03:59,540 seasonal changing, daily temperature. 61 00:03:59,540 --> 00:04:02,660 Now, this blue sinusoid is actually one of the 62 00:04:02,660 --> 00:04:04,350 things we're going to do here in the class. 63 00:04:04,350 --> 00:04:08,100 We're going to take a data set like the original data set and 64 00:04:08,100 --> 00:04:12,690 going to fit a sinusoid instead of our algorithmic methods to actually do this. 65 00:04:15,810 --> 00:04:20,490 The next signal is, so called solar activity. 66 00:04:20,490 --> 00:04:25,720 So as you probably know, the sun has different intensity due to 67 00:04:25,720 --> 00:04:31,300 solar spots, and this has been measured over several hundred years now, and 68 00:04:31,300 --> 00:04:37,130 we again see this this curve here, which has a lot of variation. 69 00:04:37,130 --> 00:04:41,400 It's not seasonal variation. It's a different variation and 70 00:04:41,400 --> 00:04:44,870 we shall also analyze this variation later on in the class. 71 00:04:48,100 --> 00:04:51,290 Now all the signals so far had to do with physics. 72 00:04:51,290 --> 00:04:53,490 Now we can see a man made signal 73 00:04:53,490 --> 00:04:57,580 that's a world population, from the beginning of current 74 00:04:57,580 --> 00:05:01,310 times until now, and you see that it's a 75 00:05:01,310 --> 00:05:05,740 slowly evolving curve up to the industrial revolution here. 76 00:05:05,740 --> 00:05:10,206 This is the industrial revolution, and then for various reasons which 77 00:05:10,206 --> 00:05:13,500 are not the topics of this class, there was a population explosion. 78 00:05:13,500 --> 00:05:16,430 By now we are close to 8 billion, and 79 00:05:16,430 --> 00:05:18,756 it's not completely clear what's going to happen next. 80 00:05:18,756 --> 00:05:26,130 Another man-made signal is, the value of the stock market. 81 00:05:26,130 --> 00:05:29,490 Here, it's the Dow Jones and again it's something that looks a 82 00:05:29,490 --> 00:05:33,680 little bit like the world population in the beginning, so very small. 83 00:05:33,680 --> 00:05:38,840 Then there is an exponential increase, then there is the internet bubble, 84 00:05:38,840 --> 00:05:43,630 and the financial crisis, etc. Okay, so it's a good question mark. 85 00:05:43,630 --> 00:05:45,780 What is going to happen next? 86 00:05:45,780 --> 00:05:47,990 We could claim that thanks to signal processing, 87 00:05:47,990 --> 00:05:51,740 we can predict the future of the stock market. 88 00:05:51,740 --> 00:05:55,420 But I will be honest, this is a very hard problem. 89 00:05:55,420 --> 00:06:01,340 If we knew how to do this, we probably wouldn't be, teaching free online course. 90 00:06:01,340 --> 00:06:04,070 We would be do, doing something else for a living. 91 00:06:04,070 --> 00:06:06,690 But anyway, it's a typical discrete time signal. 92 00:06:06,690 --> 00:06:11,020 You can measure the Dow Jones every day and record this as we did here. 93 00:06:11,020 --> 00:06:15,910 Again, it would be a lollipop a stick diagram. 94 00:06:15,910 --> 00:06:17,720 But here, we have joined the points because there 95 00:06:17,720 --> 00:06:20,310 are so many points over this period of time. 96 00:06:22,600 --> 00:06:26,520 So let's be a little bit more formal. We saw several examples. 97 00:06:26,520 --> 00:06:28,920 All of these examples where a number per 98 00:06:28,920 --> 00:06:31,650 day, a number per second, a number per year. 99 00:06:32,850 --> 00:06:36,300 in the channel case this number can be a complex number, so 100 00:06:36,300 --> 00:06:40,980 for us a digi, discrete-time signal is a sequence of complex numbers. 101 00:06:42,090 --> 00:06:44,920 We will look at mostly one dimensional 102 00:06:44,920 --> 00:06:48,390 signals and the dimension typically is time. 103 00:06:48,390 --> 00:06:53,430 The notation, that's very important to note is that x is the name of the signal, 104 00:06:53,430 --> 00:06:58,080 then we have a square bracket and finally we have the index m which is an 105 00:06:58,080 --> 00:07:02,724 integer. We typically have two sided 106 00:07:02,724 --> 00:07:08,034 sequences, so they start from minus infinity, to, 107 00:07:08,034 --> 00:07:13,580 let's say x minus 1, x at 0, x at 1, etc and goes of to 108 00:07:13,580 --> 00:07:19,850 plus infinity. And the index is dimension-less. 109 00:07:19,850 --> 00:07:23,530 So it could be seconds, years, micro seconds or whatever, 110 00:07:23,530 --> 00:07:28,350 but we just index it as a dimensionless integer index. 111 00:07:30,120 --> 00:07:34,060 The analysis of a signal means we take periodic measurements. 112 00:07:34,060 --> 00:07:39,190 This is a topic that we'll study in detail under the name of sampling. 113 00:07:39,190 --> 00:07:43,080 So for example, in the case of the Nile, we took a sample every year. 114 00:07:43,080 --> 00:07:46,230 In the case of the stock market, every day. 115 00:07:46,230 --> 00:07:51,380 And synthesis is when we generate a sequence of samples. 116 00:07:51,380 --> 00:07:54,870 We will see this when we talk about the Karplus-Strong Algorithm later in 117 00:07:54,870 --> 00:07:57,610 this module, where we indeed generate a 118 00:07:57,610 --> 00:08:01,200 musical signal by generating discrete time samples. 119 00:08:04,480 --> 00:08:09,770 So among the formal signals, or elementary signals, we will be use in this class. 120 00:08:09,770 --> 00:08:12,470 We have for example the delta signal. 121 00:08:12,470 --> 00:08:17,810 So delta signal denoted by delta of n here, is very simple. 122 00:08:17,810 --> 00:08:19,180 It's the simplest possible signal. 123 00:08:19,180 --> 00:08:23,570 It's 0 everywhere except at the origin, where it's equal to 1. 124 00:08:23,570 --> 00:08:25,170 Okay. 125 00:08:25,170 --> 00:08:30,710 Now do such signals exist in reality or is it a pure mathematical abstraction? 126 00:08:30,710 --> 00:08:32,130 It is an abstraction. 127 00:08:32,130 --> 00:08:34,730 Most signals will not be that simple, but let 128 00:08:34,730 --> 00:08:38,300 me show you one signal which comes close to that. 129 00:08:38,300 --> 00:08:41,690 Let us look at natural device called the clappers that 130 00:08:41,690 --> 00:08:46,160 is used in Hollywood studios to synchronize audio and video. 131 00:08:46,160 --> 00:08:48,220 Why do they need this? 132 00:08:48,220 --> 00:08:51,780 Well, sometimes the audio is recorded on one machine, the video, 133 00:08:51,780 --> 00:08:56,150 on another machine, and later, you have to splice them together. 134 00:08:56,150 --> 00:08:58,600 And so the way it is done is that you use this clapper. 135 00:08:58,600 --> 00:09:05,360 You write down what scene is shot by what director in what movie. 136 00:09:05,360 --> 00:09:09,940 And what you do is that you have the clapper that is open. 137 00:09:09,940 --> 00:09:14,670 You put it in front of the camera then you close it and that gives a click sound. 138 00:09:15,730 --> 00:09:18,620 And we see the click sound here at the bottom. 139 00:09:18,620 --> 00:09:20,380 And it''s not exactly a d rack. 140 00:09:20,380 --> 00:09:21,940 It's more complicated. 141 00:09:21,940 --> 00:09:24,320 But it is really an impulse. 142 00:09:24,320 --> 00:09:27,910 An impulse of sound when you listen to it it sounds like a d rack. 143 00:09:27,910 --> 00:09:30,490 When you look at it in detail it has oscillations 144 00:09:30,490 --> 00:09:34,130 because it's a piece of wood that is going to vibrate. 145 00:09:34,130 --> 00:09:40,300 But to a first approximation, this is as close as it gets to a d rack. 146 00:09:40,300 --> 00:09:43,965 The next simple signal is a so called unit step. 147 00:09:43,965 --> 00:09:46,968 It is sequence that is 0 from minus 148 00:09:46,968 --> 00:09:49,270 infinity till the origin. 149 00:09:49,270 --> 00:09:55,140 At the origin it flips to 1 and is 1 everywhere after that. 150 00:09:55,140 --> 00:09:59,570 Okay. Does such a signal exist in reality? 151 00:09:59,570 --> 00:10:01,980 A physical implementation would be a switch 152 00:10:01,980 --> 00:10:04,970 like this blade switch that we see here. 153 00:10:04,970 --> 00:10:08,655 We call this the Frankenstein switch, if you have seen the movie. 154 00:10:08,655 --> 00:10:12,420 Dr. Frankenstein, at some point, gives life to the monster. 155 00:10:12,420 --> 00:10:16,420 So, he flips a switch, a blade switch and 156 00:10:16,420 --> 00:10:20,420 then some current flows into the monster that comes alive. 157 00:10:20,420 --> 00:10:25,300 So, that's a beautiful picture of one of these blade switches. 158 00:10:25,300 --> 00:10:31,260 The third, signal that is very typical is a so called exponential decay sequence. 159 00:10:32,330 --> 00:10:37,800 So it's a combination of two things. Okay, it has, a unit step, 160 00:10:37,800 --> 00:10:43,358 so the signal is 0 before the origin and then it is positive. 161 00:10:43,358 --> 00:10:46,770 And that's at point we apply an exponential here 162 00:10:46,770 --> 00:10:50,090 with a root a that is smaller than 1. 163 00:10:50,090 --> 00:10:51,930 Taken to the nth power. 164 00:10:51,930 --> 00:10:54,950 And so it nicely comes down as an exponential. 165 00:10:54,950 --> 00:10:58,820 Okay, if we did not apply the unit step, 166 00:10:58,820 --> 00:11:03,110 then of course the exponential would grow up here exponentially, 167 00:11:03,110 --> 00:11:04,580 and it would be unbounded. 168 00:11:04,580 --> 00:11:09,007 So that would be, not signal you would encounter in practice. 169 00:11:09,007 --> 00:11:12,090 So we see here a signal that has been specifically engineered that 170 00:11:12,090 --> 00:11:16,290 it looks actually like a signal that you would encounter in real life. 171 00:11:17,770 --> 00:11:22,630 And real life is coffee. So the way coffee will lose its 172 00:11:22,630 --> 00:11:28,710 temperature is governed by partial differential equation, that we see here 173 00:11:28,710 --> 00:11:32,650 goes actually back to Newton, one of the greatest scientist of all time. 174 00:11:32,650 --> 00:11:38,610 So, the law of cooling says that the rate of cooling is proportional to 175 00:11:38,610 --> 00:11:43,540 the difference of temperature between the cup of coffee and the environment, okay. 176 00:11:43,540 --> 00:11:45,578 And there is a constant c here that 177 00:11:45,578 --> 00:11:49,010 says how fast actually the temperature will decay. 178 00:11:49,010 --> 00:11:53,910 So negative sign means the temperature will go down when the coffee is warm and 179 00:11:53,910 --> 00:11:55,580 the environment is cold. 180 00:11:55,580 --> 00:11:57,840 And the solutions of the differential equation is 181 00:11:57,840 --> 00:12:00,520 given here and it has, indeed, an exponential term. 182 00:12:01,920 --> 00:12:04,680 There are some practical issues. 183 00:12:04,680 --> 00:12:09,300 you have only convection at work and there has to be sufficient conductivity, 184 00:12:09,300 --> 00:12:13,920 and then you will see a decay, that looks a little bit like this. 185 00:12:13,920 --> 00:12:14,038 Okay? 186 00:12:14,038 --> 00:12:18,370 And here, let's say the ambient temperature is 0. 187 00:12:18,370 --> 00:12:19,110 We start with 188 00:12:19,110 --> 00:12:24,550 the coffee at 100 degrees, boiling water. And it will start to decay like this. 189 00:12:24,550 --> 00:12:26,090 And the speed of decay will depend 190 00:12:26,090 --> 00:12:30,550 how much convection there is, how much conductivity. 191 00:12:30,550 --> 00:12:31,080 Okay. 192 00:12:31,080 --> 00:12:33,780 So this was all to show you a little bit a sense 193 00:12:33,780 --> 00:12:39,060 of you have physics at work and you see signals in real life. 194 00:12:39,060 --> 00:12:41,500 A discreet time version of these signals 195 00:12:41,500 --> 00:12:44,200 will be typically sequences we'll be interested 196 00:12:44,200 --> 00:12:45,220 in in this class. 197 00:12:47,140 --> 00:12:50,520 The next character in the play is a sinusoid. 198 00:12:50,520 --> 00:12:57,580 So here is the sinusoid indicated as sine of frequency omega naught 199 00:12:57,580 --> 00:13:02,980 times n, the index, plus a phase, theta. Okay. 200 00:13:02,980 --> 00:13:07,600 In the drawing, we see theta is equal to 0. 201 00:13:07,600 --> 00:13:08,680 actually, it's not equal to zero. 202 00:13:08,680 --> 00:13:09,916 I'm sorry. 203 00:13:09,916 --> 00:13:11,960 theta is equal to pi over 2. 204 00:13:11,960 --> 00:13:16,600 So, at the origin it's maximum and then we have the sinusoidal variation. 205 00:13:19,060 --> 00:13:25,450 Now sinusoidal signals are everywhere. The heartbeat is an example. 206 00:13:25,450 --> 00:13:26,990 The whistle on 207 00:13:29,600 --> 00:13:31,470 the whistle used on a train that we 208 00:13:31,470 --> 00:13:35,570 will also analyze later is, has sinusoidal components. 209 00:13:35,570 --> 00:13:41,810 Waves that are showing up on the shore are another example of sinusoidal components. 210 00:13:41,810 --> 00:13:46,640 And, of course, musical instruments, really leave off, strings 211 00:13:46,640 --> 00:13:50,510 that have sinusoidal models, something we'll also study in detail. 212 00:13:53,200 --> 00:13:55,560 There are really four classes of signals. 213 00:13:55,560 --> 00:14:00,420 And, each one of them is a natural example 214 00:14:00,420 --> 00:14:03,090 of a signal and is used in different cases. 215 00:14:03,090 --> 00:14:08,030 The first one, the simplest one is, the finite length signal, then the infinite 216 00:14:08,030 --> 00:14:14,040 length sequence, periodic signals, and finite support sequences. 217 00:14:15,060 --> 00:14:17,830 Let's look at each of them in a little bit more detail. 218 00:14:19,820 --> 00:14:24,060 So finite length signal can be either written in the sequence notations. 219 00:14:24,060 --> 00:14:28,440 So the sequence notation we have seen before x of square bracket n. 220 00:14:28,440 --> 00:14:33,190 But n is only going from 0 to n minus 1. 221 00:14:33,190 --> 00:14:36,140 So it's a length's capital n signal. 222 00:14:36,140 --> 00:14:41,150 The vector notation puts this into a vector, and in this class vectors are 223 00:14:41,150 --> 00:14:45,000 column vectors, so we have the entry of the vectors at x0, x1 up 224 00:14:45,000 --> 00:14:49,132 to xn minus 1, a transpose to get a column vector. 225 00:14:49,132 --> 00:14:54,260 Finite length signals or vectors are very useful, for 226 00:14:54,260 --> 00:14:57,445 example, in numerical packages like Matlab and so on. 227 00:14:57,445 --> 00:15:02,450 Infinite length signals. 228 00:15:02,450 --> 00:15:06,090 We again have the sequence notations that we had seen before. 229 00:15:06,090 --> 00:15:10,240 And the index and n now is from the integer 230 00:15:10,240 --> 00:15:14,700 set, meaning from minus infinity to plus infinity. 231 00:15:14,700 --> 00:15:19,210 This is very good for abstraction, for theorems, for analysis, using 232 00:15:19,210 --> 00:15:24,090 Furey analysis, for example, z transforms, that we will see later. 233 00:15:24,090 --> 00:15:26,550 But, of course, in real life, and fortunately, 234 00:15:26,550 --> 00:15:30,330 nothing lives from minus infinity to plus infinity. 235 00:15:30,330 --> 00:15:32,960 That depends a little bit on your beliefs, but 236 00:15:32,960 --> 00:15:35,310 if you believe in big bang, then it started at 237 00:15:35,310 --> 00:15:36,720 some point. 238 00:15:36,720 --> 00:15:42,360 And if you are not certain about reincarnation, it will stop at some point. 239 00:15:42,360 --> 00:15:45,910 So the sequence notation is really a mathematical abstraction. 240 00:15:48,340 --> 00:15:51,940 Periodic signals we told earlier about sinusoids. 241 00:15:51,940 --> 00:15:56,150 So periodic sequences do exist also in real life. 242 00:15:56,150 --> 00:16:00,590 They are denoted by x tilde, okay. 243 00:16:00,590 --> 00:16:06,700 So the notation here tilde means there is a periodicity and it means that x 244 00:16:06,700 --> 00:16:13,740 n is equal to x n plus k times the period. So, capital N, is the period, 245 00:16:13,740 --> 00:16:16,260 k is an integer, it means how many periods 246 00:16:16,260 --> 00:16:19,330 we shift and n is the usual index for time. 247 00:16:21,330 --> 00:16:25,620 So, one of these periods of lengths n has all the information needed. 248 00:16:25,620 --> 00:16:28,210 The rest is just a repetition. 249 00:16:28,210 --> 00:16:29,050 Okay. 250 00:16:29,050 --> 00:16:31,070 And very often, periodic signals are a 251 00:16:31,070 --> 00:16:34,660 bridge between finite and infinite length signals because 252 00:16:34,660 --> 00:16:39,180 you start with a, with a finite-length signal and maybe you let the period go 253 00:16:39,180 --> 00:16:41,890 to infinity, or you take a periodic signal and you 254 00:16:41,890 --> 00:16:45,870 let the N, the length of the period, go to infinity. 255 00:16:47,740 --> 00:16:54,720 A finite-support sequence is defined by x over bar, given here and 256 00:16:54,720 --> 00:17:01,950 it's a sequence x n that is only different from 0 for n between 0 and N minus 1. 257 00:17:01,950 --> 00:17:04,310 It is 0 otherwise, and it has the same 258 00:17:04,310 --> 00:17:08,560 information as a finite-lengths signal of lengths N, and it 259 00:17:08,560 --> 00:17:12,760 is another possible bridge between finite and infinite length signals. 260 00:17:16,510 --> 00:17:20,770 Now that we have seen elementary signals, let's talk a little bit about operators. 261 00:17:22,110 --> 00:17:25,010 So if we have a sequence, we can 262 00:17:25,010 --> 00:17:29,820 scale the sequence by multiplying it with scalar alpha. 263 00:17:29,820 --> 00:17:30,040 Okay. 264 00:17:30,040 --> 00:17:33,900 So the natural thing is you have a sequence like this. 265 00:17:33,900 --> 00:17:37,390 You multiply it by 2. It's going to look the same. 266 00:17:37,390 --> 00:17:40,670 It's simply going to have amplitudes twice as large. 267 00:17:41,820 --> 00:17:46,630 The sum of two signals is obvious. Take two sequences and the result will be 268 00:17:46,630 --> 00:17:51,760 the sum of the two. The product of two sequences, same story, 269 00:17:51,760 --> 00:17:57,300 you take two elementary sequences x n and z n, you multiply them term by term. 270 00:17:58,370 --> 00:18:03,760 And last but not least, very important, the shift by k. 271 00:18:03,760 --> 00:18:05,430 Or the delay by k. 272 00:18:05,430 --> 00:18:08,060 So k is a positive integer. 273 00:18:08,060 --> 00:18:13,180 So when we write y n is equal to y n minus k. 274 00:18:13,180 --> 00:18:19,180 Then y n is a delayed version here of the sequence x mainly by k terms. 275 00:18:19,180 --> 00:18:23,150 Right, and to look at this in detail, it's sometimes a little bit confusing, 276 00:18:23,150 --> 00:18:28,650 but if you want to think about it very quickly, is that y of 0. 277 00:18:28,650 --> 00:18:33,170 So the output sequence at 0 is equal to x of 278 00:18:33,170 --> 00:18:38,070 0 minus k, so x of minus k. This is in the past so it 279 00:18:38,070 --> 00:18:43,200 is a single x from index minus k. 280 00:18:49,290 --> 00:18:52,160 Let's look specifically at these shifts on 281 00:18:54,210 --> 00:18:56,860 the various types of sequences we have seen. 282 00:18:56,860 --> 00:19:03,465 So the shift of finite-lengths sequence. 283 00:19:03,465 --> 00:19:08,750 Well, we have on the left side the original sequence which is the 284 00:19:08,750 --> 00:19:13,900 decaying set here, of samples, and we are going to 285 00:19:13,900 --> 00:19:19,510 shift it by extending this sequence with 0's left and right. 286 00:19:19,510 --> 00:19:19,650 Okay. 287 00:19:19,650 --> 00:19:23,392 So this, the only thing we did is that we added 0's 288 00:19:23,392 --> 00:19:27,950 before the 7 as the eight samples and after the eight samples. 289 00:19:29,130 --> 00:19:30,650 And if we shift it. 290 00:19:31,920 --> 00:19:36,510 Well, we delayed by 1, so it has shifted right. 291 00:19:36,510 --> 00:19:39,130 And what has happened is that 0 showed up 292 00:19:39,130 --> 00:19:43,580 here and the x7 fell off on the right side. 293 00:19:43,580 --> 00:19:44,890 We see this here 294 00:19:44,890 --> 00:19:50,160 at the bottom, that we can follow what happens as we shift further and further. 295 00:19:50,160 --> 00:19:51,670 Okay, shift by 2. 296 00:19:52,790 --> 00:19:54,880 Again it is delayed by 2. 297 00:19:54,880 --> 00:19:59,720 So we have two 0's at the beginning and two samples that fall off. 298 00:19:59,720 --> 00:20:03,580 And we see the result at the bottom right side. 299 00:20:03,580 --> 00:20:09,120 And shift by 3, shift by 4 and you see the idea. 300 00:20:11,070 --> 00:20:14,490 We can also have a model of the world where the signal 301 00:20:14,490 --> 00:20:19,250 is periodically extended and then the shift operator will have another effect. 302 00:20:19,250 --> 00:20:20,660 Let's look at this. 303 00:20:20,660 --> 00:20:24,926 So we start with the sequence eight samples from x0 to x7. 304 00:20:24,926 --> 00:20:30,270 We extend it periodically. Okay, so here we have a second 305 00:20:30,270 --> 00:20:35,470 period, here we have the previous period. And now when we shift, 306 00:20:36,490 --> 00:20:41,524 what will happen is that, we have a shift pile to the right okay? 307 00:20:41,524 --> 00:20:43,520 x tilde n minus one. 308 00:20:43,520 --> 00:20:50,750 So, x7 enters on the left side, and x7 leaves at the right side, okay. 309 00:20:50,750 --> 00:20:56,964 So, the new signal, given here at the bottom right, has x7 entering 310 00:20:56,964 --> 00:21:01,610 here, and having left on the other side. Now this of course 311 00:21:01,610 --> 00:21:04,600 is the same as doing a circular shift, okay. 312 00:21:04,600 --> 00:21:06,550 Let's do a few more. 313 00:21:06,550 --> 00:21:10,850 So x tilde n minus 2, we have now 314 00:21:10,850 --> 00:21:14,220 two new samples entering and two that fell off there. 315 00:21:14,220 --> 00:21:16,920 And at the bottom right, we see the picture again. 316 00:21:19,830 --> 00:21:23,050 X tilde n minus 3, same story. 317 00:21:23,050 --> 00:21:28,870 X tilde n minus 4, one more step and you start to see what is happening here. 318 00:21:33,770 --> 00:21:38,120 We have now seen the shift operator, which is one of the most fundamental 319 00:21:38,120 --> 00:21:43,110 ones we are going to use over and over again, and study in great detail. 320 00:21:43,110 --> 00:21:47,350 Let's talk about two conceptual characteristics of signals. 321 00:21:47,350 --> 00:21:50,470 The first one is something we call the energy. 322 00:21:50,470 --> 00:21:54,040 The energy is a sum of squares, of the samples. 323 00:21:54,040 --> 00:21:56,560 Okay. It is given in the formula here. 324 00:21:56,560 --> 00:21:58,980 So you take a sequence. You take the absolute 325 00:21:58,980 --> 00:22:00,960 value of the samples. You square it. 326 00:22:00,960 --> 00:22:03,640 And you sum from minus infinity to plus infinity. 327 00:22:05,480 --> 00:22:07,370 This may or may not be finite. 328 00:22:07,370 --> 00:22:12,710 If you take a sinsusoid it's, signals that starts at minus infinity, goes to plus 329 00:22:12,710 --> 00:22:17,960 infinity, has amplitudes that vary, let's say between minus 1 and 1. 330 00:22:17,960 --> 00:22:20,220 And of course if you square it and you sum 331 00:22:20,220 --> 00:22:24,790 from minus infinity to plus infinity, It will blow up, okay. 332 00:22:24,790 --> 00:22:29,300 So, a notion that can still be defined even for such 333 00:22:29,300 --> 00:22:33,750 signals that might have infinite, energy is a notion of power. 334 00:22:33,750 --> 00:22:37,870 The notion of power is that, you look at the energy over 335 00:22:37,870 --> 00:22:43,270 a window, the window is of size, 2 n plus 1, right? 336 00:22:43,270 --> 00:22:48,030 Because we sum from minus n here to plus n, the sum of squares. 337 00:22:48,030 --> 00:22:50,070 But we normalize this by 338 00:22:50,070 --> 00:22:55,420 the length of the window. So we put the 1 over 2 n plus 1, and 339 00:22:55,420 --> 00:23:00,909 this is the power, because it's like the instantaneous energy for the sequence. 340 00:23:04,760 --> 00:23:06,570 So for periodic signals as I have all 341 00:23:06,570 --> 00:23:09,580 ready alluded to, the energy will be infinite. 342 00:23:10,680 --> 00:23:14,640 And the power is simply the energy of 343 00:23:14,640 --> 00:23:17,130 one period divided by the length of the period. 344 00:23:17,130 --> 00:23:19,310 So this is given down here. 345 00:23:19,310 --> 00:23:21,380 So, this part is the energy. 346 00:23:21,380 --> 00:23:24,290 1 over gives the power, and this typically will 347 00:23:24,290 --> 00:23:28,240 be actually finite, even when this guy is infinite. 348 00:23:30,500 --> 00:23:35,974 End of module 2.1 349 00:23:35,974 --> 00:23:38,249 [SOUND].