1 00:00:00,012 --> 00:00:05,716 So, this video represents the combination of our exploration of signals in the 2 00:00:05,716 --> 00:00:09,320 frequency domain. Today, we're going to talk about 3 00:00:09,320 --> 00:00:13,729 non-periodic signals. These are the ones where we finally need 4 00:00:13,729 --> 00:00:17,149 to build up to. After today, we could handle both 5 00:00:17,149 --> 00:00:20,992 periodic and periodic signals with equal facility. 6 00:00:20,992 --> 00:00:26,474 The mathematical tool we're going to use is known as the Fourier transform. 7 00:00:26,474 --> 00:00:31,519 And so, we need to explore those properties and I think there will be no 8 00:00:31,519 --> 00:00:35,463 surprises there, we'll, we have seen a lot of them already. 9 00:00:35,463 --> 00:00:40,970 And finally, we will show you how you can figure out what happens when you filter a 10 00:00:40,970 --> 00:00:44,762 signal, be it periodic or non-periodic. 11 00:00:44,762 --> 00:00:48,554 Okay. So, first of all, we have to derive what 12 00:00:48,554 --> 00:00:54,436 the Fourier transform is. And it turns out what I'm going to do is 13 00:00:54,436 --> 00:01:00,462 I'm going to play with the period T. So, I'm going to be explicit here about 14 00:01:00,462 --> 00:01:06,938 what the period of the signal is. And as we have had and seen, the Fourier 15 00:01:06,938 --> 00:01:12,282 series coefficient is given by a formula that looks like this. 16 00:01:12,282 --> 00:01:17,335 And the one difference here are the range of integration. 17 00:01:17,335 --> 00:01:23,164 Turns out you can find the Fourier coefficient by integrating over any 18 00:01:23,164 --> 00:01:26,761 period, as long as you integrate over one period. 19 00:01:26,761 --> 00:01:32,808 So, for this derivation, I'm integrating from -T/2 to T/2 instead of zero to T. 20 00:01:32,808 --> 00:01:38,527 It doesn't really make any difference. I'm also throwing in over here the fact 21 00:01:38,527 --> 00:01:42,625 that the C sub k is are a function of what the period is. 22 00:01:42,625 --> 00:01:49,132 That doesn't really change anything. Now, I'm going to make a little simple 23 00:01:49,132 --> 00:01:54,162 definition. And that is that ST of f is just T * Ck 24 00:01:54,162 --> 00:01:58,757 of t. So, what I'm doing is just bringing that 25 00:01:58,757 --> 00:02:04,392 to the other side. And, of course, that's what you get. 26 00:02:04,392 --> 00:02:10,922 The other thing here is I'm defining f to be k/T, so that appears in the formula 27 00:02:10,922 --> 00:02:16,542 there and it appears over there. So, so far I'm just manipulating into 28 00:02:16,542 --> 00:02:20,892 finding things I haven't really done anything. 29 00:02:20,892 --> 00:02:27,786 Now, we know what the Fourier series expression is for the signal in terms of 30 00:02:27,786 --> 00:02:33,347 the ST(f). Since the ST(f) is just this in order to 31 00:02:33,347 --> 00:02:39,550 have the Fourier series formula, I have to divide by T and f is k/T. 32 00:02:39,550 --> 00:02:44,667 So it's, nothing has really changed. So, here's the fun. 33 00:02:44,667 --> 00:02:51,437 Here's where we start the mathematics. What happens if we make the period go to 34 00:02:51,437 --> 00:02:57,938 infinity? So, if you think about, for example, the periodic pulse string. 35 00:02:57,938 --> 00:03:02,812 So, it looks like that. Drawing way out there. 36 00:03:02,812 --> 00:03:11,597 T, there's even one over here at -T. I'm sorry, -T over here, goes on and on. 37 00:03:11,597 --> 00:03:19,842 What happens if T goes to infinity? Well, what's going to happen is all of the 38 00:03:19,842 --> 00:03:23,459 signals, all the pulses here are going to go out 39 00:03:23,459 --> 00:03:26,985 there because they have to be shifted out. 40 00:03:26,985 --> 00:03:30,743 And this one, these are going to go to minus infinity. 41 00:03:30,743 --> 00:03:36,007 So, the only thing that's left is a single pulse around the origin. 42 00:03:36,007 --> 00:03:42,553 It doesn't shift. And the reason for that, just to recall, 43 00:03:42,553 --> 00:03:46,830 is that if this is p(t), this is p(t) - T. 44 00:03:46,830 --> 00:03:51,295 So, when we let T go to infinity it shifts out. 45 00:03:51,295 --> 00:03:59,252 Okay. So, we, that's the mathematics. We need to look and, at what happens when 46 00:03:59,252 --> 00:04:06,927 t goes to infinity. So just repeating what we just looked at, and now, what 47 00:04:06,927 --> 00:04:13,867 happens if t goes to infinity. Well, what happens is that this, you can 48 00:04:13,867 --> 00:04:18,942 consider as a remind sum. So, when T goes to infinity, 49 00:04:18,942 --> 00:04:24,422 1/T turns out to be the separation between any two frequency values. 50 00:04:24,422 --> 00:04:30,092 And so, that becomes df. What's happening here is that since f in 51 00:04:30,092 --> 00:04:34,222 here is k/T as you're letting t go to infinity, 52 00:04:34,222 --> 00:04:40,722 you have to change the harmonic numbers so to keep you at that same frequency. 53 00:04:40,722 --> 00:04:48,007 And, that's essentially a detail, but we're now done. And the sum which is 54 00:04:48,007 --> 00:04:55,479 represented by the Fourier series expression, becomes an integral as we let 55 00:04:55,479 --> 00:05:01,951 the period go to infinity. So, to show you that that's kind of thing 56 00:05:01,951 --> 00:05:06,268 that's happening, I show two examples here of the period 57 00:05:06,268 --> 00:05:11,889 pulse sequence, periodic pulse train, with two different periods. 58 00:05:11,889 --> 00:05:16,123 Here, the T is 1. Here, I made it 5 times bigger, and you 59 00:05:16,123 --> 00:05:19,469 can see how the spectral lines fill in here. 60 00:05:19,469 --> 00:05:25,032 They get very dense. So, the sum, which is, corresponds to this term 61 00:05:25,032 --> 00:05:32,377 becomes an integral because the lines get closer and closer and closer until it get 62 00:05:32,377 --> 00:05:37,992 into a continuum. Okay. So now, we have what's known as the 63 00:05:37,992 --> 00:05:44,384 Fourier Transform Pair. So, if you're given a signal, you can 64 00:05:44,384 --> 00:05:49,555 find its spectrum by just doing, doing this calculation. 65 00:05:49,555 --> 00:05:55,667 If you have the spectrum, you can find the signal in the time domain. 66 00:05:55,667 --> 00:05:59,539 Here's the signal in the frequency domain. 67 00:05:59,539 --> 00:06:06,789 Here's the signal in the time domain. So, just like before with the Fourier 68 00:06:06,789 --> 00:06:11,467 series, you can express a signal either in a 69 00:06:11,467 --> 00:06:19,468 frequency domain or in a time domain. And the Fourier transform is your way 70 00:06:19,468 --> 00:06:24,999 going between the two. The definition is, this slide is what's 71 00:06:24,999 --> 00:06:29,922 called the Fourier transform, and you've heard that terminology. 72 00:06:29,922 --> 00:06:33,849 That's what they mean going from time to frequency. 73 00:06:33,849 --> 00:06:39,013 And the inverse Fourier transform is going from frequency to time. 74 00:06:39,013 --> 00:06:41,823 Just a terminology, nothing else. 75 00:06:41,823 --> 00:06:48,295 But, we can now calculate either the, the spectrum from the time domain's function, 76 00:06:48,295 --> 00:06:53,963 functional version of the signal or the time domain version from a frequency 77 00:06:53,963 --> 00:07:00,402 domain function or version of this. we point out this as a very interesting 78 00:07:00,402 --> 00:07:08,348 symmetry to it or they look very similar. The only difference between these two 79 00:07:08,348 --> 00:07:15,263 formulas is the sign of that exponent. So, the Fourier transform gets the minus 80 00:07:15,263 --> 00:07:22,312 sign in the exponential and the inverse Fourier transform gets the plus sign. 81 00:07:22,312 --> 00:07:28,522 That's going to be a very handy thing to note when we explore it's properties. 82 00:07:28,522 --> 00:07:33,945 Let's do a little example first. Let's take the Forier transform of a 83 00:07:33,945 --> 00:07:37,207 pulse. So, if you remember, a pulse is a 84 00:07:37,207 --> 00:07:40,992 standard definition, we've seen it many times. 85 00:07:40,992 --> 00:07:44,322 It's a pulse of type one, and with delta. 86 00:07:44,322 --> 00:07:49,422 Since it is zero out here, that makes our life very simple. 87 00:07:49,422 --> 00:07:55,847 When we plug it into this formula, we don't need to integrate from minus 88 00:07:55,847 --> 00:08:02,322 infinity to infinity, we know the integral over these areas is going to be 89 00:08:02,322 --> 00:08:06,990 zero. So, we just integrate from zero to delta. 90 00:08:06,990 --> 00:08:10,681 So, we plug it in and we just integrate one 91 00:08:10,681 --> 00:08:15,114 from zero to delta times the complex exponential. 92 00:08:15,114 --> 00:08:21,899 And that integral is easily calculated. And this should look very familiar to 93 00:08:21,899 --> 00:08:25,842 you. We're going to pull our same math trick. 94 00:08:25,842 --> 00:08:31,528 And we're going to write this. Pull out half the phase and simplifying, 95 00:08:31,528 --> 00:08:37,540 you get our very familiar sinc function. So, this is the spectrum of a pulse. 96 00:08:37,540 --> 00:08:43,460 And now, there are no spectral lines. It's a continuous function of f, the 97 00:08:43,460 --> 00:08:47,442 magnitude of p of f. Looks like a sync. 98 00:08:47,442 --> 00:08:56,901 Okay? And then, the first zero which is always an interesting place, is going to 99 00:08:56,901 --> 00:09:05,462 be the frequency at which this is equal to one because the sign is zero at 100 00:09:05,462 --> 00:09:11,044 integer multiples of pi. So, the first one out here is when f 101 00:09:11,044 --> 00:09:15,620 delta equals one, and so this frequency is one over delta. 102 00:09:15,620 --> 00:09:23,684 So, if you will, most of the energy and power in the spectrum is, is between -1 103 00:09:23,684 --> 00:09:29,838 over delta and one over delta, and then it decays by one/f. 104 00:09:29,838 --> 00:09:35,319 Okay. Let me do another example for you. 105 00:09:35,319 --> 00:09:43,022 And, this one is a decaying exponential times a unit step. 106 00:09:43,022 --> 00:09:49,222 Okay. So, what does this thing look like? So 107 00:09:49,222 --> 00:09:59,447 since the unit step is zero, for t less than zero, how about here that product is 108 00:09:59,447 --> 00:10:04,841 zero? For t greater than zero, the unit step is one. 109 00:10:04,841 --> 00:10:10,387 So, out here, we just get the decaying exponential. 110 00:10:10,387 --> 00:10:18,623 Okay. So, the a is positive in this case. So, to plug this into our Fourier 111 00:10:18,623 --> 00:10:27,024 transform expression, again we're just going to ingrate the exponential from 112 00:10:27,024 --> 00:10:33,449 zero to infinity, alright? And here's what we get for an answer. 113 00:10:33,449 --> 00:10:40,892 And I'll show you how to get that. So, instead you get the integral from 114 00:10:40,892 --> 00:10:44,009 zero to infinity, with e^-at, e^-j2πft dt. 115 00:10:50,742 --> 00:11:09,921 And you combine the exponentials [SOUND] to get a very easy integral to do. And I 116 00:11:09,921 --> 00:11:16,336 think it's pretty easy to see is that this is what you're going to get for the 117 00:11:16,336 --> 00:11:20,524 result. there's an important little detail in 118 00:11:20,524 --> 00:11:27,243 here and that a is positive so that when you evaluate the exponent and the 119 00:11:27,243 --> 00:11:33,544 infinity you get zero because this will dominate. 120 00:11:33,544 --> 00:11:40,046 Okay. So, this should look like a very familiar quantity here. 121 00:11:40,046 --> 00:11:44,952 and we'll get ot that in just a second. Okay. 122 00:11:44,952 --> 00:11:49,533 So, here's some interesting properties of spectra. 123 00:11:49,533 --> 00:11:55,229 None of these should be a surprise. We have a Parseval's Theorem. 124 00:11:55,229 --> 00:12:01,519 So, signal power is equal, has roughly the same expression except for the 125 00:12:01,519 --> 00:12:07,349 magnitude in either the time domain or frequency domain, will look very, very 126 00:12:07,349 --> 00:12:12,250 simple. So, you can calculate power in the frequency domain or time domain, 127 00:12:12,250 --> 00:12:16,346 whichever is easier. And if you decide one integral looks very 128 00:12:16,346 --> 00:12:21,370 hard, you can use the Fourier transform to go into the other domain and see if 129 00:12:21,370 --> 00:12:27,433 the integral looks simpler over there. We play that kind of game all the time 130 00:12:27,433 --> 00:12:30,560 because of the Fourier transform's properties. 131 00:12:30,560 --> 00:12:36,291 And, we have conjugate symmetry for real signals. And again, if it's even, the 132 00:12:36,291 --> 00:12:39,953 spectrum is even. And if it's odd, the signal in the time 133 00:12:39,953 --> 00:12:43,112 domain is odd, it's odd in the frequency domain. 134 00:12:43,112 --> 00:12:48,311 And furthermore, because of the conjugate symmetry properties, it's either real or 135 00:12:48,311 --> 00:12:52,303 imaginary depending if the time domain version is even or odd. 136 00:12:52,303 --> 00:12:57,247 No real surprises there, you should be able to prove these yourself quite 137 00:12:57,247 --> 00:13:00,074 easily. Let's go into some more interesting 138 00:13:00,074 --> 00:13:02,787 Fourier Transform Properties, okay? 139 00:13:02,787 --> 00:13:06,992 First one is, one that you've seen before. 140 00:13:06,992 --> 00:13:14,624 And then, if I delay in the time domain, that corresponds to the multiplying in 141 00:13:14,624 --> 00:13:19,522 the frequency domain by e^-j2πf times the delay, 142 00:13:19,522 --> 00:13:22,970 okay? We've seen this with the with the Foutier 143 00:13:22,970 --> 00:13:28,955 series and I think it's pretty easy to see when you plug that same formula, use 144 00:13:28,955 --> 00:13:34,375 the same mathematics in this case, too. However, what's really interesting is 145 00:13:34,375 --> 00:13:40,092 what happens if you delay in frequency. You take this spectrum and shift it over 146 00:13:40,092 --> 00:13:43,062 by f0. Because these two formulas were so 147 00:13:43,062 --> 00:13:48,052 similar, you know that the answer has to be it's multiplied by complex 148 00:13:48,052 --> 00:13:53,602 exponential. However, because of this sign difference in this formula, it's 149 00:13:53,602 --> 00:13:59,102 either the plus here. So, delaying infrequency corresponds to multiplying in 150 00:13:59,102 --> 00:14:05,386 the time domain but either the +j2πft0t. Whereas, if your delay in time 151 00:14:05,386 --> 00:14:09,398 corresponds in multiplying by e^-j2πf tap. 152 00:14:09,398 --> 00:14:15,555 Okay? So, this is where the, you can really use to your advantage these 153 00:14:15,555 --> 00:14:20,792 symmetry properties, the fact that these look so similar. 154 00:14:20,792 --> 00:14:25,159 You can mathematically, if you can drive one property, one 155 00:14:25,159 --> 00:14:30,506 domain, you can use it to see what happens in kind of the reverse sense, 156 00:14:30,506 --> 00:14:33,777 okay? Let's explore another property here. 157 00:14:33,777 --> 00:14:39,946 And this one is really easy. And that the signal at the origin in the time domain 158 00:14:39,946 --> 00:14:46,257 and it's value at the origin has to be the integral of the spectrum over all 159 00:14:46,257 --> 00:14:51,397 frequency. And the way to see that is to take this 160 00:14:51,397 --> 00:14:59,057 formula and just replace this by zero. Well, the complex exponential is of zero 161 00:14:59,057 --> 00:15:04,092 is just one, and so you just left with the integral. 162 00:15:04,092 --> 00:15:09,417 And that's pretty easy to see. And, of course, use the same thing over 163 00:15:09,417 --> 00:15:14,367 here and just just f at zero. So, the value of the spectrum at the 164 00:15:14,367 --> 00:15:18,037 origin is equal to the integral of the signal. 165 00:15:18,037 --> 00:15:21,642 And here's one, and it's a bit more exciting. 166 00:15:21,642 --> 00:15:29,807 The spectrum of the derivative of a signal is j2πf times the spectrum of the 167 00:15:29,807 --> 00:15:36,503 original signal S, okay? And how do you derive that? And the 168 00:15:36,503 --> 00:15:46,716 answer is, I'm simply going to take the derivative of this formula. 169 00:15:46,716 --> 00:15:52,953 So, ds(t)/dt is equal to, 170 00:15:59,648 --> 00:16:05,782 [SOUND] 171 00:16:05,782 --> 00:16:09,555 okay. So, what I'm going to do is I'm going to 172 00:16:09,555 --> 00:16:14,575 take this derivative and move it inside the integral. 173 00:16:14,575 --> 00:16:21,285 The only place that t occurs is right there in that complex exponential. 174 00:16:21,285 --> 00:16:31,979 Well, the derivative of complex exponential is just j2πf times the 175 00:16:31,979 --> 00:16:48,425 complex exponential times e^+j2πf t/dt. Well, this is the inverse Fourier 176 00:16:48,425 --> 00:16:54,679 transform formula. However, what it has in it, what you're 177 00:16:54,679 --> 00:17:02,182 taking inverse transform of is that. So that means these are related to each 178 00:17:02,182 --> 00:17:07,402 other. So, really pretty cool and that if you 179 00:17:07,402 --> 00:17:15,772 take a spectrum and you multiply it by j2πf, that corresponds in the time domain 180 00:17:15,772 --> 00:17:20,292 to taking a derivative. Very interesting. 181 00:17:20,292 --> 00:17:26,397 Well, if multiplying by j2πf corresponds to taking the derivative, 182 00:17:26,397 --> 00:17:30,402 guess what? Dividing by j2πf corresponds, too. Well, 183 00:17:30,402 --> 00:17:36,857 that's got to correspond to the equation. And again, you can show that by plugging 184 00:17:36,857 --> 00:17:40,702 this into the formula over here. I'll let you do that. 185 00:17:40,702 --> 00:17:45,880 It's pretty easy. and you can easily see that this is the 186 00:17:45,880 --> 00:17:49,633 way it works. I have a little question for you. 187 00:17:49,633 --> 00:17:55,861 What happens, what is the spectrum of this second derivative of a signal? 188 00:17:55,861 --> 00:18:08,428 What's the spectrum the second derivative of a signal? So, d^2s(t)/dt, well that's 189 00:18:08,428 --> 00:18:26,002 just a derivative of the derivative, right? So, the spectrum of this thing is 190 00:18:26,002 --> 00:18:32,771 j2 path. The spectrum corresponding to this is to 191 00:18:32,771 --> 00:18:36,553 multiply by a minus, another, another j2πf. 192 00:18:36,553 --> 00:18:41,812 And so, what you get is -4π^2f^2 *. s(f), 193 00:18:41,812 --> 00:18:48,498 okay? The minus sign of course is j^2. So, you can also talk about the, the 194 00:18:48,498 --> 00:18:55,816 double integral, the signal if you will, integrating it twice and all that. 195 00:18:55,816 --> 00:19:00,582 That, of course, when it's dividing twice by j2πf. 196 00:19:00,582 --> 00:19:06,186 Really interesting point. So, here's the summary of all those 197 00:19:06,186 --> 00:19:11,867 examples and properties that have done in the previous slides. 198 00:19:11,867 --> 00:19:18,820 And, they are very important for us. And we will use these properties over and 199 00:19:18,820 --> 00:19:24,669 over and over again with an example that's coming up is going to be 200 00:19:24,669 --> 00:19:33,087 particularly important to remember the integral, and to remember this. You'll 201 00:19:33,087 --> 00:19:38,358 see that in just a second. And I guess, that one's going to be 202 00:19:38,358 --> 00:19:42,194 important. First, I want to talk about a very 203 00:19:42,194 --> 00:19:49,072 interesting case that's very important. What is the spectrum of one plus the 204 00:19:49,072 --> 00:19:54,139 signal times the cosine? Okay. And s here is assumed to be a 205 00:19:54,139 --> 00:20:01,332 non-periodic signal, it has no special properties other than it's some signal. 206 00:20:01,332 --> 00:20:07,833 In fact, it could be something like a speech signal or something. 207 00:20:07,833 --> 00:20:17,183 So first of all, we note that if you just multiply everything through that we have 208 00:20:17,183 --> 00:20:21,511 a periodic part and we have a non-periodic part. 209 00:20:21,511 --> 00:20:28,192 The reason that this is non-periodic, the cosine might be periodic, but once you 210 00:20:28,192 --> 00:20:33,945 multiply by something that's not periodic, you get something that is 211 00:20:33,945 --> 00:20:37,322 clearly not going to be periodic. Okay. 212 00:20:37,322 --> 00:20:44,072 So, how do you find the spectrum of this? The way you do it, is you use the Fourier 213 00:20:44,072 --> 00:20:49,022 series for the periodic part. And now we know what to do. 214 00:20:49,022 --> 00:20:53,197 That gives you terms only at plus and minus fc, 215 00:20:53,197 --> 00:20:58,392 you'll see that in just a second. And we're going to use the Fourier 216 00:20:58,392 --> 00:21:04,102 transform on the non-periodic part. And then, because the Fourier transform is 217 00:21:04,102 --> 00:21:09,962 linear, the integral of a sum is the sum of the integral so it's easily seen to be 218 00:21:09,962 --> 00:21:13,307 linear. We're going to add the results together. 219 00:21:13,307 --> 00:21:16,092 Our only work is going to be on this second term. 220 00:21:16,092 --> 00:21:19,762 Okay. So, I don't want to, I don't like doing 221 00:21:19,762 --> 00:21:23,192 integral. I like using the properties. 222 00:21:23,192 --> 00:21:28,742 And, the way to get at those properties is through Euler's formula. 223 00:21:28,742 --> 00:21:33,937 Euler's formula comes up a lot and there's a reason for that. 224 00:21:33,937 --> 00:21:39,097 It's very very useful. So, cosine is easy plus, plus e to the 225 00:21:39,097 --> 00:21:42,268 minus. And put the s of t to the side. 226 00:21:42,268 --> 00:21:49,153 But from our properties we know that when you multiply in the time domain by a 227 00:21:49,153 --> 00:21:56,322 complex exponential, that corresponds to a delay or an advance in the frequency. 228 00:21:56,322 --> 00:22:03,245 And that's what each of these terms is. So, you write down the answer. 229 00:22:03,245 --> 00:22:10,763 It's easy to show that the Fourier transform of some signal times the cosine 230 00:22:10,763 --> 00:22:17,475 is going to be the spectrum of that signal, delayed, and advanced in 231 00:22:17,475 --> 00:22:21,957 frequency. And the amount of the frequency delayer 232 00:22:21,957 --> 00:22:27,233 advance is f sub c, okay? And the half is there because you 233 00:22:27,233 --> 00:22:33,400 get the half of Euler's form. So, and now we know what the Fourier 234 00:22:33,400 --> 00:22:37,232 series is and we now can add the spectrum. 235 00:22:37,232 --> 00:22:42,953 And we get the following plot. So, I like to use this example spectrum. 236 00:22:42,953 --> 00:22:48,685 it's called a rooftop spectrum because it sort of looks like a rooftop. 237 00:22:48,685 --> 00:22:54,209 And notice it has to be this, has to be symmetric, right? What happens in 238 00:22:54,209 --> 00:22:59,867 positive frequency has to occur in negative frequency and it has to be a 239 00:22:59,867 --> 00:23:06,967 conjugate symmetric which means for what I'm showing here, it's got to be an even 240 00:23:06,967 --> 00:23:10,517 function. We're going to use this a lot. 241 00:23:10,517 --> 00:23:17,292 So, the part that's due to the periodic part, that was of those lines, like we 242 00:23:17,292 --> 00:23:21,792 always do stem plots for things relative Fourier series. 243 00:23:21,792 --> 00:23:27,983 And the frequency at which they occur is -fc and +fc. 244 00:23:27,983 --> 00:23:38,322 For the non-periodic part, that's this, that's why you get this thing delayed and 245 00:23:38,322 --> 00:23:42,784 advanced in frequency. Let me add up. 246 00:23:42,784 --> 00:23:52,503 So you now, once you multiply by the cosine, you shift it up so the spectrum 247 00:23:52,503 --> 00:23:58,137 of the original signal now gets centered to f sub, f sub c. 248 00:23:58,137 --> 00:24:03,149 Okay. This is a very important example because 249 00:24:03,149 --> 00:24:08,917 this is amplitude modulation, and we're going to be very interested in how that, 250 00:24:08,917 --> 00:24:13,703 how radio signals work. but we have to do that a little bit 251 00:24:13,703 --> 00:24:17,246 later. Right now, it's just and example for us 252 00:24:17,246 --> 00:24:21,882 exploring Fourier transforms. Okay. Let's start filtering. 253 00:24:21,882 --> 00:24:32,558 I think it's pretty easy to see that the integral of a signal is just a sum. 254 00:24:32,558 --> 00:24:42,012 So, we're summing up complex exponentials to create our signal X(t). 255 00:24:42,012 --> 00:24:48,620 So, in terms of the spectra in the frequency domain, the Fourier transform 256 00:24:48,620 --> 00:24:54,571 of the output is just equal to the transfer function times the Fourier 257 00:24:54,571 --> 00:24:57,562 transform of the input. So, 258 00:24:57,562 --> 00:25:04,608 to figure out how to filter a signal, like on our friend the pulse, with our 259 00:25:04,608 --> 00:25:10,418 friend the RC low-pass, what we're going to do is all we have to 260 00:25:10,418 --> 00:25:16,857 do is find the Fourier transform of our pulse, which we've already done. 261 00:25:16,857 --> 00:25:21,604 Transfer functions are already in the frequency domain, 262 00:25:22,918 --> 00:25:29,640 in most cases. So, we're going to figure out what X(f) is going to be p, P(f), 263 00:25:29,640 --> 00:25:35,496 going to be that, multiply them together. And then, we have to find the inverse 264 00:25:35,496 --> 00:25:39,567 transform of of Y(f) to figure out what Y(t) is. 265 00:25:39,567 --> 00:25:44,059 That's the program. Okay. So, we know what the spectrum is. 266 00:25:44,059 --> 00:25:49,529 I'm not going to write it in terms of the sync function because there's a reason 267 00:25:49,529 --> 00:25:54,609 that you'll see in just a second, okay? So, we've already calculated the 268 00:25:54,609 --> 00:25:59,380 spectrum little bit earlier. And now, we're going to multiply them 269 00:25:59,380 --> 00:26:08,531 together and I think, and this is the answer for the H * X is this complicated 270 00:26:08,531 --> 00:26:13,375 thing. So, all we need to do is find it's 271 00:26:13,375 --> 00:26:20,875 inverse for a transform. Okay. And so, we just need to plug all of 272 00:26:20,875 --> 00:26:26,814 that, [LAUGH] into there. I don't know about you but I don't think 273 00:26:26,814 --> 00:26:30,631 that's going to be in an integral table anywhere. 274 00:26:30,631 --> 00:26:35,853 It's a little too complicated. And it's got lots of working, moving 275 00:26:35,853 --> 00:26:39,521 pieces. And again, I hate doing integrals, I'd 276 00:26:39,521 --> 00:26:44,402 rather use the properties if I can. This is what I normally do. 277 00:26:44,402 --> 00:26:49,488 I try to use the properties first. And if those fail, then I result to doing 278 00:26:49,488 --> 00:26:53,031 integral. You get a lot more incite of what's going 279 00:26:53,031 --> 00:26:57,788 on if you use the properties. In, I'm going to use this example to show 280 00:26:57,788 --> 00:27:01,684 you what I mean by that. So, I'm just going to look at this 281 00:27:01,684 --> 00:27:08,976 expression of the Y(f) in some detail. Well, some things ought to stand out for 282 00:27:08,976 --> 00:27:12,793 you. First of all, what's this do? So here we 283 00:27:12,793 --> 00:27:18,359 have some spectrum over here. We multiply it by one, we just get it 284 00:27:18,359 --> 00:27:22,004 back. What happens when we multiply it by 285 00:27:22,004 --> 00:27:29,928 e^-j2πf delta? Multiply the spectrum e^-j2πf delta, 286 00:27:29,928 --> 00:27:38,851 what happens in the time domain? So, if we call this S(t), that's this inverse 287 00:27:38,851 --> 00:27:48,397 Fourier transform, this means time delay. And so, this term in the time domain is 288 00:27:48,397 --> 00:27:56,267 going to be S(t) minus delta. So, all we have to figure out is what's 289 00:27:56,267 --> 00:28:01,557 the inverse transform of this term. Okay. 290 00:28:01,557 --> 00:28:11,159 To do that, I'm going to reorder things just a little change to make it a little 291 00:28:11,159 --> 00:28:17,490 bit clearer. And what this looks like to me is I have 292 00:28:17,490 --> 00:28:24,955 some spectrum multiplied by 1/j2πf. Well, that means integrate. 293 00:28:24,955 --> 00:28:30,817 What's the spectrum? Okay. I'm going to point out that one of the 294 00:28:30,817 --> 00:28:37,633 examples we did was this decaying exponential formula, and we found out 295 00:28:37,633 --> 00:28:41,892 that the Fourier transform of this was that. 296 00:28:41,892 --> 00:28:53,835 This is almost as the same form as that. And to give it in that form, I'm going to 297 00:28:53,835 --> 00:29:05,138 multiply and divide this by RC, and I think we can see now that a in the 298 00:29:05,138 --> 00:29:10,520 formula corresponds to R1/C. We have this one up top. 299 00:29:10,520 --> 00:29:18,197 I'm going to divide by multiply by a, and that multiplies this side by a. 300 00:29:18,197 --> 00:29:24,161 Again, because the Fourier transform formulas are linear. 301 00:29:24,161 --> 00:29:32,054 If you multiply one by a constant, the other one gets multiplied by a constant. 302 00:29:32,054 --> 00:29:40,099 And so, I know that the inverse Fourier transform of this term is this. 303 00:29:40,099 --> 00:29:46,172 And now, we're almost done. Now, I just have to, I know that to j, 304 00:29:46,172 --> 00:29:53,097 1/j2πf means integrate, so I just have to integrate this term again. 305 00:29:53,097 --> 00:29:56,722 This is an integral from minus infinity to T, 306 00:29:56,722 --> 00:30:02,297 it's always a definite integral, that's what integrate means. 307 00:30:02,297 --> 00:30:07,737 And so, that's what you get When you, perform the integral. 308 00:30:07,737 --> 00:30:14,639 And note that since this signal is zero for t less than zero, so is its integral. 309 00:30:14,639 --> 00:30:19,132 And everything in infinity is zero. Okay. 310 00:30:19,132 --> 00:30:24,404 And so, we're almost done. We know that this turnout here means 311 00:30:24,404 --> 00:30:27,433 delay. So, the final answer is that. 312 00:30:27,433 --> 00:30:33,211 It looks a little complicated, but actually we've seen this already. 313 00:30:33,211 --> 00:30:39,566 Remember, what happened when we took the periodic pulse train and sent it to our 314 00:30:39,566 --> 00:30:43,220 RC low-pass filter? I showed you some plots. 315 00:30:43,220 --> 00:30:50,020 I want to bring back one of those plots. So, the input here, if you recall, was 316 00:30:50,020 --> 00:30:55,882 the periodic pulse train. So, we have pulse there and the period 317 00:30:55,882 --> 00:30:59,564 was one millisecond. And it was like that. 318 00:30:59,564 --> 00:31:02,934 And you got that for an answer. Well, 319 00:31:02,934 --> 00:31:11,830 I can think about this periodic pulse train as being a superposition of pulses 320 00:31:11,830 --> 00:31:13,906 delayed, okay? 321 00:31:13,906 --> 00:31:23,730 So, that means I can think of the output as being the superposition of these 322 00:31:23,730 --> 00:31:29,335 outputs. And so, this formula should apply to that 323 00:31:29,335 --> 00:31:33,620 term. So, let's just not ignore this part for a 324 00:31:33,620 --> 00:31:39,935 second because that corresponds to this. And what we've done is put in a single 325 00:31:39,935 --> 00:31:45,508 pulse to figure out the answer. So now, I have a much clearer 326 00:31:45,508 --> 00:31:54,040 understanding of how this signal is constructed, but its structure is. 327 00:31:54,040 --> 00:31:57,647 Okay. So, let's go through it. 328 00:31:57,647 --> 00:32:07,102 The, this starts at zero, and what its going to do is go up and keep on going. 329 00:32:07,102 --> 00:32:15,882 And it has an asymptotic value of one, okay? The second term here, u(t) minus 330 00:32:15,882 --> 00:32:21,314 delta. Notice everything here is delay, delayed 331 00:32:21,314 --> 00:32:26,669 by delta, so this doesn't start until delta and 332 00:32:26,669 --> 00:32:34,999 it's going to be the negative of what we have over here, okay? So, at delta, you 333 00:32:34,999 --> 00:32:43,309 get the same thing headed toward -1. And when you add them up, you get this 334 00:32:43,309 --> 00:32:51,342 adds up and then it turns down, it goes down and it's going to go to zero since 335 00:32:51,342 --> 00:32:54,745 +1 and -1 will cancel when you add them up. 336 00:32:54,745 --> 00:32:58,424 And that's what is going on in this wave form. 337 00:32:58,424 --> 00:33:04,737 So, the shark fins that I referred to when we talked about the Fourier series, 338 00:33:04,737 --> 00:33:09,654 they actually consist of a rising exponential even, when minus, even -t of 339 00:33:09,654 --> 00:33:17,653 RC and then a decaying exponential, they are properties of this circuit because RC 340 00:33:17,653 --> 00:33:23,272 only comes from the circuit. I should point out that the formula for 341 00:33:23,272 --> 00:33:29,326 decaying exponential is frequently written like this, and it has a tail. 342 00:33:29,326 --> 00:33:36,316 And this is called the time constant of the exponential, just terminology. 343 00:33:36,316 --> 00:33:42,178 And so, the time constant of these exponentials here is RC. 344 00:33:42,178 --> 00:33:50,148 just a little fact, and R, so units of R * C by the way, are time, which is kind 345 00:33:50,148 --> 00:33:52,352 of interesting. Okay. 346 00:33:52,352 --> 00:33:57,708 So, this shows you how at least I calculate the output of a linear filter 347 00:33:57,708 --> 00:34:02,149 to any signal coming in. I find the Fourier transform of the 348 00:34:02,149 --> 00:34:07,539 signal, and then try to use the properties of the integrals, properties 349 00:34:07,539 --> 00:34:12,452 rather of the Fourier transform, to figure out what's going on. 350 00:34:12,452 --> 00:34:18,878 So we have finally finished our story of singnals in the frequency domain. 351 00:34:18,878 --> 00:34:25,256 So, if you have a periodic signal its spectrum is given by a Fourier series. 352 00:34:25,256 --> 00:34:31,657 If we have a non-periodic signal, it's the Fourier transform and its inverse 353 00:34:31,657 --> 00:34:38,101 Fourier transform that describe how you go back and forth. 354 00:34:38,101 --> 00:34:43,793 We now know how to filter anything in the frequency domain. 355 00:34:43,793 --> 00:34:50,148 we, if it's periodic, we use the Fourier series version of things. 356 00:34:50,148 --> 00:34:56,974 And if it's not periodic, we use the Fourier transform version of things. 357 00:34:56,974 --> 00:35:03,419 we either always decompose a signal into it's periodic, non-periodic parts, and 358 00:35:03,419 --> 00:35:08,105 then filter each separately and then add the results back in. 359 00:35:08,105 --> 00:35:13,948 Superposition, we use over, and over, and over again, it's very important. 360 00:35:13,948 --> 00:35:17,979 Well, we now know a lot, about how to think about signals. 361 00:35:17,979 --> 00:35:22,864 We can think about their structure in the time domain, we can think about their 362 00:35:22,864 --> 00:35:27,191 structure in the frequency domain. Our next video is going to talk about 363 00:35:27,191 --> 00:35:30,038 speech. I'm going to talk about the structure of 364 00:35:30,038 --> 00:35:34,252 speech and how it has both time domain and frequency domain parts. 365 00:35:34,252 --> 00:35:35,155 Really kind of interesting, in fact.