WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.059 Commons license. 00:00:04.059 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.350 To make a donation or view additional materials 00:00:13.350 --> 00:00:17.290 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.290 --> 00:00:18.294 at ocw.mit.edu. 00:00:26.652 --> 00:00:28.360 PROFESSOR: So today I'm going to continue 00:00:28.360 --> 00:00:30.820 with frequency response and filtering, 00:00:30.820 --> 00:00:36.260 but also begin the story of spectral content of signals. 00:00:36.260 --> 00:00:38.530 So our starting point is still something 00:00:38.530 --> 00:00:41.710 you've seen before, namely the statement that for an LTI 00:00:41.710 --> 00:00:43.990 system, a sinusoid into the system 00:00:43.990 --> 00:00:47.030 gives you a sinusoid out at the same frequency, 00:00:47.030 --> 00:00:51.970 but maybe shifted in phase and scaled in amplitude. 00:00:51.970 --> 00:00:57.220 So a bit of terminology here just for general interest, 00:00:57.220 --> 00:01:02.230 we refer to the exponential as an eigenfunction of the LTI 00:01:02.230 --> 00:01:04.239 system, because the only effect the LTI 00:01:04.239 --> 00:01:06.400 system has on it a scaling. 00:01:06.400 --> 00:01:10.330 So an input to some kind of a mapping, which comes out 00:01:10.330 --> 00:01:12.310 the same except for a scaling is referred 00:01:12.310 --> 00:01:14.440 to as an eigenfunction, or an eigenvector 00:01:14.440 --> 00:01:16.960 if you're talking about matrices. 00:01:16.960 --> 00:01:19.960 So we say that the exponential-- the complex exponential 00:01:19.960 --> 00:01:22.540 here is an eigenfunction of the LTI system. 00:01:22.540 --> 00:01:25.630 Because when it comes through, it's just the same exponential, 00:01:25.630 --> 00:01:28.480 but scaled by some number. 00:01:28.480 --> 00:01:31.450 And that number is what we refer to as a frequency response, 00:01:31.450 --> 00:01:32.020 right? 00:01:32.020 --> 00:01:35.133 And we've seen that there's a simple expression for it. 00:01:35.133 --> 00:01:36.550 And let me put that expression up, 00:01:36.550 --> 00:01:39.220 because we're going to use it repeatedly. 00:01:47.640 --> 00:01:49.270 The m here is irrelevant. 00:01:49.270 --> 00:01:52.140 It can be any dummy index, because we're 00:01:52.140 --> 00:01:53.393 summing over the m. 00:01:53.393 --> 00:01:54.810 You can call it anything you want. 00:01:58.597 --> 00:02:00.180 And I just should mention that there's 00:02:00.180 --> 00:02:03.120 other notation for this object. 00:02:07.280 --> 00:02:10.009 There are people who refer to it as-- 00:02:10.009 --> 00:02:15.800 well, it's often referred to as h of ej omega, 00:02:15.800 --> 00:02:19.610 because actually, the way omega enters is always in the term e 00:02:19.610 --> 00:02:20.640 to the j omega. 00:02:20.640 --> 00:02:23.840 So this is-- if you want to think of it that way, 00:02:23.840 --> 00:02:29.420 this is e to the j omega to the power minus m, right? 00:02:29.420 --> 00:02:35.176 Well, let me just write it as 1 over 1 over e to the j omega m. 00:02:37.870 --> 00:02:42.890 OK, so it's some function of e to the j omega. 00:02:42.890 --> 00:02:44.875 And people will often write it this way. 00:02:44.875 --> 00:02:46.250 And one of the advantages of this 00:02:46.250 --> 00:02:48.140 is the notation right away tells you 00:02:48.140 --> 00:02:51.080 that this object is periodic with period 2 pi. 00:02:51.080 --> 00:02:53.120 Because if you were to increase omega 00:02:53.120 --> 00:02:56.060 by an integer multiple of 2 pi in the numerator here, 00:02:56.060 --> 00:02:58.040 you'd get the same argument again. 00:02:58.040 --> 00:02:59.990 And therefore, h must be the same again. 00:02:59.990 --> 00:03:02.870 So this notation has the value that it 00:03:02.870 --> 00:03:05.820 keeps the periodicity front and center. 00:03:05.820 --> 00:03:08.750 It also makes sense when you're developing 00:03:08.750 --> 00:03:10.010 various other transforms. 00:03:10.010 --> 00:03:13.940 There's something called a z transform, which we 00:03:13.940 --> 00:03:15.380 won't deal with in this class. 00:03:15.380 --> 00:03:19.160 But it's used a lot when dealing with discrete-time systems. 00:03:19.160 --> 00:03:24.470 And the way that you get from the z transform 00:03:24.470 --> 00:03:28.130 to this object is by making the substitution z equals ej omega. 00:03:28.130 --> 00:03:31.320 So people will use this notation. 00:03:31.320 --> 00:03:34.580 So the z transform uses z exactly where 00:03:34.580 --> 00:03:36.650 we use e to the j omega. 00:03:36.650 --> 00:03:39.390 But for our purposes, this is a much simpler notation. 00:03:39.390 --> 00:03:42.230 It's just that we need you to remember when you see this 00:03:42.230 --> 00:03:43.790 that we're talking about something 00:03:43.790 --> 00:03:45.890 that's got period 2 pi. 00:03:45.890 --> 00:03:48.860 And if you look at the definition, that becomes clear. 00:03:48.860 --> 00:03:51.110 If you increase big omega here by any integer 00:03:51.110 --> 00:03:53.990 multiple of 2 pi, you're going to get the same thing 00:03:53.990 --> 00:03:56.610 back again. 00:03:56.610 --> 00:03:59.060 There's another bit of notational confusion that 00:03:59.060 --> 00:04:07.650 can arise, which is that people will sometimes 00:04:07.650 --> 00:04:11.040 write little omega instead of big omega. 00:04:15.390 --> 00:04:18.160 So that's also used. 00:04:18.160 --> 00:04:20.760 So this is other notation, and it's notation 00:04:20.760 --> 00:04:22.470 that we will try not to use, but you 00:04:22.470 --> 00:04:25.560 might see vestiges of this when you look through old problems. 00:04:25.560 --> 00:04:27.000 Because in some terms, we may have 00:04:27.000 --> 00:04:29.440 used this notation, and some terms, 00:04:29.440 --> 00:04:32.700 we may have used a little omega instead of a big omega. 00:04:32.700 --> 00:04:35.120 But for our purposes, we'll stick to this. 00:04:35.120 --> 00:04:37.890 OK, so when we say big omega, we're 00:04:37.890 --> 00:04:41.145 thinking of it as an angle around the unit circle. 00:04:43.960 --> 00:04:49.590 So if you've got the complex number here 00:04:49.590 --> 00:04:52.790 at an angle big omega, this complex number 00:04:52.790 --> 00:04:54.930 is e to the j omega, right? 00:04:54.930 --> 00:04:57.390 So we're thinking of big omega as an angle, something 00:04:57.390 --> 00:05:00.420 measured in radians, and it's different from little omega. 00:05:05.100 --> 00:05:08.220 You can write the expression for the frequency response 00:05:08.220 --> 00:05:09.150 in various ways. 00:05:09.150 --> 00:05:11.340 So here, I've just used Euler's identity 00:05:11.340 --> 00:05:14.130 to split that into a cosine and a sine, 00:05:14.130 --> 00:05:17.430 and that's straightforward enough. 00:05:17.430 --> 00:05:20.130 The sums are over infinite intervals. 00:05:20.130 --> 00:05:23.670 And we talked last time about how stability of the system-- 00:05:23.670 --> 00:05:25.950 bounded input, bounded output stability of the system 00:05:25.950 --> 00:05:30.830 will guarantee that those summations are well defined. 00:05:30.830 --> 00:05:34.700 OK, now there's another name for this formula. 00:05:34.700 --> 00:05:38.940 Basically, we've called it the frequency response. 00:05:38.940 --> 00:05:42.950 But when you compute an h of omega using this formula, 00:05:42.950 --> 00:05:44.450 another way to say what you're doing 00:05:44.450 --> 00:05:50.270 is to say that you're taking the discrete-time Fourier transform 00:05:50.270 --> 00:05:54.735 of the sequence h dot, OK? 00:05:54.735 --> 00:05:56.610 So it's the discrete-times Fourier transform. 00:05:56.610 --> 00:05:58.560 Again, that's just terminology for now. 00:05:58.560 --> 00:06:00.670 We'll come to expand our view of it later. 00:06:00.670 --> 00:06:04.170 But we've called it the frequency response so far, 00:06:04.170 --> 00:06:08.400 because it describes how sinusoids or exponentials here 00:06:08.400 --> 00:06:11.280 get to the output, but it's also referred to 00:06:11.280 --> 00:06:14.400 as the discrete-time Fourier transform of the unit sample 00:06:14.400 --> 00:06:15.310 response. 00:06:15.310 --> 00:06:17.280 So you've got some time signal-- 00:06:17.280 --> 00:06:20.100 happens to be a unit sample response. 00:06:20.100 --> 00:06:22.080 You compute an object through this formula 00:06:22.080 --> 00:06:24.090 to get an h of omega. 00:06:24.090 --> 00:06:28.330 That's the DTFT, OK? 00:06:28.330 --> 00:06:29.980 Another thing we've already seen is 00:06:29.980 --> 00:06:35.060 that knowing that you have an LTI system, 00:06:35.060 --> 00:06:37.300 and that a cosine is a superposition 00:06:37.300 --> 00:06:40.300 of complex exponentials, you can use the result 00:06:40.300 --> 00:06:43.300 that we had so far to just describe 00:06:43.300 --> 00:06:46.490 what happens to a cosine when it goes through the system. 00:06:46.490 --> 00:06:49.060 So it's no longer a complex exponential. 00:06:49.060 --> 00:06:51.160 It's a real signal of the kind that we're 00:06:51.160 --> 00:06:53.282 more likely to work with. 00:06:53.282 --> 00:06:55.240 And we've seen that the only thing that happens 00:06:55.240 --> 00:06:58.180 is the cosine that went in gets scaled 00:06:58.180 --> 00:07:01.280 in amplitude by an extra factor, which 00:07:01.280 --> 00:07:03.830 is the magnitude of the frequency response. 00:07:03.830 --> 00:07:06.270 And whatever phase it had, you get an extra phase, 00:07:06.270 --> 00:07:09.410 which is the angle of the frequency response. 00:07:09.410 --> 00:07:11.650 So actually, if you had an LTI system, 00:07:11.650 --> 00:07:14.770 this is a good way to measure the frequency response 00:07:14.770 --> 00:07:15.580 in the lab. 00:07:15.580 --> 00:07:17.770 What you do is you take your system there, 00:07:17.770 --> 00:07:20.640 excite it with a sinusoid. 00:07:20.640 --> 00:07:23.620 In continuous-time, we know we can do that with an oscillator. 00:07:23.620 --> 00:07:26.960 In discrete-time, you generate a sequence like this. 00:07:26.960 --> 00:07:30.100 And then look to see what comes out of the system 00:07:30.100 --> 00:07:34.100 and express it in this form, and you'll label the scale factor 00:07:34.100 --> 00:07:36.100 there as the magnitude of the frequency response 00:07:36.100 --> 00:07:39.550 and the extra phase angle as the phase angle of the frequency 00:07:39.550 --> 00:07:40.840 response. 00:07:40.840 --> 00:07:43.180 So it makes for a very systematic way 00:07:43.180 --> 00:07:48.820 to probe a system and get at the frequency response. 00:07:48.820 --> 00:07:51.710 Again, of a point that I've made before, 00:07:51.710 --> 00:07:54.580 which is that when you do this probing, 00:07:54.580 --> 00:08:02.380 you only need to vary big omega over the range minus pi to pi. 00:08:02.380 --> 00:08:09.580 So when we write a frequency response, because h of omega 00:08:09.580 --> 00:08:16.180 is periodic with period 2 pi, we only need to probe h of omega-- 00:08:16.180 --> 00:08:20.990 either the magnitude, that would be one plot, and the angle 00:08:20.990 --> 00:08:21.950 would be another plot. 00:08:26.210 --> 00:08:28.460 Both of these would be plotted from minus pi to pi. 00:08:32.240 --> 00:08:33.799 Because outside of that range-- well, 00:08:33.799 --> 00:08:35.960 you can see it already with the cosine. 00:08:35.960 --> 00:08:40.610 If I added an integer multiple of 2 pi to omega 0, 00:08:40.610 --> 00:08:43.280 I'm going to get an integer multiple of 2 pi added 00:08:43.280 --> 00:08:45.110 into the argument of a cosine. 00:08:45.110 --> 00:08:47.635 And I'm getting the same cosine back again. 00:08:47.635 --> 00:08:49.010 And the reason that's the case is 00:08:49.010 --> 00:08:51.960 because the n that's multiplying it here is an integer. 00:08:51.960 --> 00:08:53.720 So in continuous-time, it doesn't 00:08:53.720 --> 00:08:55.250 work quite the same way. 00:08:55.250 --> 00:08:58.760 If I had a little omega 0t and I added multiple of 2 pi, 00:08:58.760 --> 00:09:00.800 I wouldn't get the same argument back again. 00:09:00.800 --> 00:09:04.700 So what's different here is that if I increase omega 00:09:04.700 --> 00:09:07.310 0 by an integer multiple of 2 pi, 00:09:07.310 --> 00:09:09.410 because I've got an integer, and outside that, 00:09:09.410 --> 00:09:13.020 again, I end up adding an integer multiple of 2 pi, 00:09:13.020 --> 00:09:14.510 and I'm back at the same cosine. 00:09:14.510 --> 00:09:17.600 So frequency response for a discrete-time system 00:09:17.600 --> 00:09:20.443 is always in the interval minus pi to pi. 00:09:20.443 --> 00:09:22.610 It repeats periodically outside of that if you chose 00:09:22.610 --> 00:09:24.590 to look at some other omega. 00:09:27.390 --> 00:09:30.000 And I've said that already. 00:09:30.000 --> 00:09:33.120 You actually heard the term frequency response 00:09:33.120 --> 00:09:35.250 in all sorts of settings, I'm sure. 00:09:35.250 --> 00:09:38.505 One setting in which it's used a lot is in describing, 00:09:38.505 --> 00:09:40.380 for instance, the performance characteristics 00:09:40.380 --> 00:09:41.670 of a loudspeaker. 00:09:41.670 --> 00:09:45.120 So people will tell you how good their loudspeaker 00:09:45.120 --> 00:09:47.915 is by showing you the frequency response of the speaker. 00:09:47.915 --> 00:09:49.290 And what they're doing is they're 00:09:49.290 --> 00:09:52.470 applying a sinusoidal voltage to the input 00:09:52.470 --> 00:09:56.670 and looking at the sound pressure that comes out. 00:09:56.670 --> 00:10:01.440 SPL here is sound pressure level. 00:10:01.440 --> 00:10:05.000 This is measured in dB, so it's actually 00:10:05.000 --> 00:10:06.750 a measurement of the ratio of the pressure 00:10:06.750 --> 00:10:10.050 that you hear under certain standardized conditions 00:10:10.050 --> 00:10:15.900 to a pressure which is taken as the lowest 00:10:15.900 --> 00:10:18.060 audible pressure on the ear. 00:10:18.060 --> 00:10:20.340 So there's a particular ratio there. 00:10:20.340 --> 00:10:22.860 So what they'll do is they'll feed the loudspeaker 00:10:22.860 --> 00:10:27.690 with 1 watt at 1,000 Hertz, so just a steady tone. 00:10:27.690 --> 00:10:31.770 And then, a meter away from the speaker in an anechoic chamber, 00:10:31.770 --> 00:10:34.410 they'll look to see what sound pressure they pick up 00:10:34.410 --> 00:10:36.140 on a specialized sensor-- 00:10:36.140 --> 00:10:39.120 a detector, a microphone basically-- 00:10:39.120 --> 00:10:42.670 and that number in dB is what they'll represent. 00:10:42.670 --> 00:10:45.390 And so typical speakers are-- 00:10:45.390 --> 00:10:47.830 have values in that kind of range. 00:10:47.830 --> 00:10:49.920 Now, if you probe it at different frequencies 00:10:49.920 --> 00:10:54.840 applying the same input voltage and looking at pressure, 00:10:54.840 --> 00:10:58.440 you'll get varying pressure depending on the frequency 00:10:58.440 --> 00:10:59.250 that you probe at. 00:10:59.250 --> 00:11:01.790 So this is the frequency response of the speaker, 00:11:01.790 --> 00:11:04.237 and if you go too low in frequency, 00:11:04.237 --> 00:11:05.820 then you don't get much of a response. 00:11:05.820 --> 00:11:07.278 If you go to high in frequency, you 00:11:07.278 --> 00:11:08.925 don't get much of a response. 00:11:12.540 --> 00:11:16.740 Now, of course, when you use the speaker, 00:11:16.740 --> 00:11:19.133 you're not going to probe it with sines and cosines. 00:11:19.133 --> 00:11:21.300 You're actually going to put more complicated sounds 00:11:21.300 --> 00:11:22.120 in there. 00:11:22.120 --> 00:11:24.240 So what you're really interested in 00:11:24.240 --> 00:11:27.030 is how does the speaker behave to signals that 00:11:27.030 --> 00:11:29.670 are combinations of cosines? 00:11:29.670 --> 00:11:33.540 And again, we're using our model of the speaker as an LTI 00:11:33.540 --> 00:11:34.110 system. 00:11:34.110 --> 00:11:38.730 All bets are off if you drive your speaker so hard that you 00:11:38.730 --> 00:11:42.990 get distortion and exercise all the nonlinearities there 00:11:42.990 --> 00:11:44.310 or burn it out. 00:11:44.310 --> 00:11:48.180 But if you're in a normal range, the speaker is acting linearly, 00:11:48.180 --> 00:11:49.980 you can talk about its frequency response. 00:11:49.980 --> 00:11:51.210 And what you're really interested in 00:11:51.210 --> 00:11:53.640 is how does the speaker respond to linear combinations 00:11:53.640 --> 00:11:55.110 of cosines? 00:11:55.110 --> 00:12:01.770 And all of these various signals can be thought of as-- 00:12:01.770 --> 00:12:05.550 at least over reasonable time intervals-- 00:12:05.550 --> 00:12:09.850 as combinations of cosines appropriately chosen. 00:12:09.850 --> 00:12:12.250 So if you hit a particular key on the piano, 00:12:12.250 --> 00:12:15.570 you get a dominant note, but you'll get harmonics of that. 00:12:15.570 --> 00:12:17.700 And that's what's going into your speaker. 00:12:17.700 --> 00:12:22.170 So knowing how an LTI system responds to cosines 00:12:22.170 --> 00:12:24.750 then puts you in a position to say 00:12:24.750 --> 00:12:27.180 how it responds to combinations of cosines, 00:12:27.180 --> 00:12:29.490 or signals that are combinations of cosines. 00:12:29.490 --> 00:12:32.280 So the other part of the story that we're going to get 00:12:32.280 --> 00:12:35.070 to-- and maybe even by the end of this lecture-- 00:12:35.070 --> 00:12:37.800 is we need a way to take a general signal 00:12:37.800 --> 00:12:40.310 and represent it as a combination of cosines. 00:12:40.310 --> 00:12:42.690 And that's what we refer to as the spectral content 00:12:42.690 --> 00:12:43.750 of the signal. 00:12:43.750 --> 00:12:47.620 So when we talk of exposing the spectral content of a signal, 00:12:47.620 --> 00:12:49.800 as over here, what we're saying is 00:12:49.800 --> 00:12:52.830 we're going to show you what combination of cosines it 00:12:52.830 --> 00:12:54.750 takes to make up that signal. 00:12:54.750 --> 00:12:57.180 And once you figure that out, and you have the frequency 00:12:57.180 --> 00:12:59.400 response of your LTI system, you can say how 00:12:59.400 --> 00:13:01.180 your system responds to that. 00:13:01.180 --> 00:13:05.310 OK, so this theme runs through every stage 00:13:05.310 --> 00:13:09.950 of what happens, actually, in communication. 00:13:09.950 --> 00:13:11.950 Now, the example I've given you here 00:13:11.950 --> 00:13:14.840 is one that you would typically probe 00:13:14.840 --> 00:13:18.603 with a continuous-time oscillator in the lab. 00:13:18.603 --> 00:13:20.270 And so there's some connections that you 00:13:20.270 --> 00:13:24.990 might want to make between probing 00:13:24.990 --> 00:13:27.630 with a continuous-time signal and probing 00:13:27.630 --> 00:13:29.310 with a discrete-time sequence that comes 00:13:29.310 --> 00:13:30.900 from sampling that signal. 00:13:30.900 --> 00:13:33.450 But I'm going to leave you to look at that later or leave 00:13:33.450 --> 00:13:36.090 your recitation instructors to pull that back, 00:13:36.090 --> 00:13:39.030 or leave you to draw this up if you have a homework 00:13:39.030 --> 00:13:41.820 problem that needs you to think about how continuous-time maps 00:13:41.820 --> 00:13:42.900 to discrete-time. 00:13:42.900 --> 00:13:46.980 But the basic point is the actual, physical speaker 00:13:46.980 --> 00:13:51.390 you might probe with a cosine in continuous-time, 00:13:51.390 --> 00:13:53.970 if you're generating that signal from a computer, what you'd 00:13:53.970 --> 00:13:58.140 actually be sending to your amplifier 00:13:58.140 --> 00:13:59.950 is a sequence of numbers. 00:13:59.950 --> 00:14:03.810 And the frequency of the numbers that you would send, 00:14:03.810 --> 00:14:06.007 this frequency is related to the frequency 00:14:06.007 --> 00:14:07.590 of the continuous-time cosine that you 00:14:07.590 --> 00:14:09.923 want in a very particular way. 00:14:09.923 --> 00:14:11.340 So I'll leave you to chew on that. 00:14:11.340 --> 00:14:15.920 But I don't want to spend time on that now. 00:14:15.920 --> 00:14:18.210 OK, so let's spend a little time talking 00:14:18.210 --> 00:14:20.220 about the properties of frequency response now 00:14:20.220 --> 00:14:22.860 that we know why we would use it. 00:14:22.860 --> 00:14:24.060 And this I've already said. 00:14:26.990 --> 00:14:30.710 The value of the frequency response at-- 00:14:30.710 --> 00:14:33.380 some of this, by the way, you may have seen in recitation. 00:14:33.380 --> 00:14:35.840 But it doesn't hurt to repeat. 00:14:35.840 --> 00:14:41.420 The frequency response at frequency 0-- 00:14:41.420 --> 00:14:47.360 well, we've said if you've got e to the j omega sub 00:14:47.360 --> 00:14:53.770 0 and some frequency omega sub zero going into a system h 00:14:53.770 --> 00:14:57.790 omega, a system with frequency response h omega, 00:14:57.790 --> 00:15:00.860 an LTI system with frequency response h omega-- all right, 00:15:00.860 --> 00:15:02.632 I'm leaving out lots of words. 00:15:02.632 --> 00:15:04.340 But frequency response doesn't make sense 00:15:04.340 --> 00:15:07.000 unless you have an LTI system. 00:15:12.100 --> 00:15:14.940 OK, so for what kind of input signal 00:15:14.940 --> 00:15:17.270 would you be looking at omega equals 0? 00:15:21.380 --> 00:15:23.203 DC, right-- a constant signal. 00:15:23.203 --> 00:15:24.620 It's what the electrical engineers 00:15:24.620 --> 00:15:32.030 call DC, which used to stand for direct current 00:15:32.030 --> 00:15:35.090 but has now come to mean constant. 00:15:38.110 --> 00:15:41.900 When we say a DC input, we just mean a constant input. 00:15:41.900 --> 00:15:46.900 So if I pick omega sub 0 to be 0, then e to the j 0n, 00:15:46.900 --> 00:15:49.010 well, that's just one for all time. 00:15:49.010 --> 00:15:51.640 And so I'm feeding the system with a constant. 00:15:51.640 --> 00:15:54.130 That's the slowest possible input that you can find. 00:15:54.130 --> 00:15:56.620 It's a 0 frequency input. 00:15:56.620 --> 00:15:58.300 And the amount that it's scaled by 00:15:58.300 --> 00:16:00.580 is the number that you're going to plot here. 00:16:00.580 --> 00:16:04.030 So whatever value you get is going 00:16:04.030 --> 00:16:07.000 to end up being plotted there at omega equals 0. 00:16:09.550 --> 00:16:12.880 And let's see, do we believe this other statement-- 00:16:12.880 --> 00:16:15.290 h0? 00:16:15.290 --> 00:16:17.000 It's just a substitution in here. 00:16:17.000 --> 00:16:21.077 If I put omega equals 0, it's a summation of all the hm's. 00:16:21.077 --> 00:16:22.910 But there's another way to think of it also. 00:16:22.910 --> 00:16:25.560 If you want to think of it in the time domain-- 00:16:25.560 --> 00:16:26.060 let's see. 00:16:29.130 --> 00:16:32.350 I have an LTI system. 00:16:32.350 --> 00:16:33.955 It's got some unit sample response. 00:16:38.470 --> 00:16:42.255 And I'm feeding it with an input that's constant for all time. 00:16:42.255 --> 00:16:44.380 It's actually constant at the value 1 for all time. 00:16:50.150 --> 00:16:54.080 If you're thinking in terms of convolution, the flip slide 00:16:54.080 --> 00:16:59.910 and dot product picture, what is the output at any time here? 00:16:59.910 --> 00:17:03.580 You're going to draw out your unit sample response. 00:17:03.580 --> 00:17:07.329 You're going to draw out your input, which is 1 for all time, 00:17:07.329 --> 00:17:09.700 take one of them and flip it over, slide 00:17:09.700 --> 00:17:11.690 it the appropriate amount over the other, 00:17:11.690 --> 00:17:13.599 and then take the dot product. 00:17:13.599 --> 00:17:18.579 Well, for every shift of this flipped and shifted input, 00:17:18.579 --> 00:17:21.240 you're going to pick up all of the unit sample response. 00:17:21.240 --> 00:17:24.040 So every time, you're going to get summation hm outside. 00:17:29.940 --> 00:17:32.355 So if you fed an input that was DC at the value 1, 00:17:32.355 --> 00:17:34.230 this is what the output will be at all times. 00:17:34.230 --> 00:17:36.520 You can see that from the convolution picture. 00:17:36.520 --> 00:17:39.660 So what's the frequency response at frequency 0? 00:17:39.660 --> 00:17:42.300 What's the ratio of the output to the input-- 00:17:42.300 --> 00:17:44.460 the output amplitude to the input? 00:17:44.460 --> 00:17:45.390 This is for all time. 00:17:47.950 --> 00:17:50.580 So the input amplitude was 1 at each time. 00:17:50.580 --> 00:17:52.920 The output amplitude was that. 00:17:52.920 --> 00:17:54.823 And so that's the DC gain-- 00:17:54.823 --> 00:17:57.240 the DC gain of the system, or the frequency response at 0. 00:18:06.100 --> 00:18:10.840 Let's say, so h0 is what's referred to as the DC gain. 00:18:24.210 --> 00:18:26.280 What about high frequency? 00:18:26.280 --> 00:18:28.590 So what's the highest frequency variation 00:18:28.590 --> 00:18:31.800 that you can have with a discrete time sequence? 00:18:31.800 --> 00:18:33.780 I've got a sequence here at the input. 00:18:38.580 --> 00:18:40.940 We've seen what the slowest variation possible is. 00:18:40.940 --> 00:18:42.750 It's something that's constant. 00:18:42.750 --> 00:18:46.020 If you're talking about a discrete-time signal that 00:18:46.020 --> 00:18:48.690 can only take values at integer times, 00:18:48.690 --> 00:18:53.110 what's the highest frequency variation that you can get? 00:18:53.110 --> 00:18:55.148 Just something that alternates in sine, right? 00:18:55.148 --> 00:18:56.190 So you're going to have-- 00:19:03.260 --> 00:19:07.660 OK, so is this of the form e to the j omega 00:19:07.660 --> 00:19:11.590 0n for some omega 0? 00:19:11.590 --> 00:19:13.390 Is that a signal of exponential form? 00:19:16.228 --> 00:19:17.174 Yes? 00:19:17.174 --> 00:19:18.120 AUDIENCE: [INAUDIBLE] 00:19:18.120 --> 00:19:21.510 PROFESSOR: Yeah, if you take omega not equal to pi, 00:19:21.510 --> 00:19:25.620 this is just e to the j pi n. 00:19:25.620 --> 00:19:27.390 In fact, you can take plus or minus pi. 00:19:30.490 --> 00:19:33.277 So when you probe the system with an input 00:19:33.277 --> 00:19:35.110 of this type, which is the highest frequency 00:19:35.110 --> 00:19:37.450 input that you can probe with, what you're really 00:19:37.450 --> 00:19:40.960 probing is what's the frequency response at this point? 00:19:40.960 --> 00:19:43.510 You get the same value at minus pi or pi. 00:19:47.400 --> 00:19:48.650 So these are the two extremes. 00:19:48.650 --> 00:19:50.608 And then the frequency response, the rest of it 00:19:50.608 --> 00:19:52.730 lies in between for other sorts of inputs. 00:19:55.400 --> 00:19:59.660 Now, do you believe this other identity that I have up there? 00:19:59.660 --> 00:20:04.730 Well, you can go back to the definition, set 00:20:04.730 --> 00:20:06.830 big omega equal to pi or minus pi, 00:20:06.830 --> 00:20:10.430 and you'll get an alternating sequence of 1's and minus 00:20:10.430 --> 00:20:11.600 1's here. 00:20:11.600 --> 00:20:14.210 And so that verifies that identity. 00:20:14.210 --> 00:20:17.000 Or you can think in terms of convolution. 00:20:17.000 --> 00:20:22.760 If I convolve a sequence like this with a system 00:20:22.760 --> 00:20:27.470 with this unit sample response, what comes out at every time 00:20:27.470 --> 00:20:30.680 is an alternating sum of the hm's, except the sine flips 00:20:30.680 --> 00:20:32.480 from one time to the next. 00:20:32.480 --> 00:20:37.520 And so, again, you can verify in the time domain 00:20:37.520 --> 00:20:40.530 that that's actually the high frequency gain of the system, 00:20:40.530 --> 00:20:41.030 OK? 00:20:45.600 --> 00:20:49.260 Now, there's a bunch of other symmetry properties 00:20:49.260 --> 00:20:51.720 of the frequency response that I think 00:20:51.720 --> 00:20:55.470 in-- at least in some of the recitations you've done. 00:20:55.470 --> 00:20:58.350 And the easiest way to see these symmetry properties 00:20:58.350 --> 00:21:01.710 is to actually go back to the rewriting 00:21:01.710 --> 00:21:05.700 I did of the frequency response in terms of sines and cosines. 00:21:05.700 --> 00:21:09.570 This first term here I'm calling C of omega. 00:21:09.570 --> 00:21:11.250 The second term here, the summation, 00:21:11.250 --> 00:21:14.660 I'm calling S of omega. 00:21:14.660 --> 00:21:17.720 So where would a statement like this come from? 00:21:17.720 --> 00:21:18.560 Let's see. 00:21:18.560 --> 00:21:21.260 For real h of n, that's the only kind of h of n 00:21:21.260 --> 00:21:22.850 we're going to worry about in general. 00:21:22.850 --> 00:21:25.520 We're going to talk about systems with real unit sample 00:21:25.520 --> 00:21:27.110 responses. 00:21:27.110 --> 00:21:29.810 if h is real, why would it be true 00:21:29.810 --> 00:21:32.060 that the real part of the frequency response 00:21:32.060 --> 00:21:34.370 is an even function of frequency? 00:21:36.930 --> 00:21:40.020 Well, the real part of the frequency response 00:21:40.020 --> 00:21:44.500 is this term, because the other term is the imaginary part. 00:21:44.500 --> 00:21:46.540 So the real part of the frequency response 00:21:46.540 --> 00:21:48.940 is this term. 00:21:48.940 --> 00:21:51.110 And if I change big omega to minus omega, 00:21:51.110 --> 00:21:52.800 the cosine doesn't change. 00:21:52.800 --> 00:21:53.850 It's the same. 00:21:53.850 --> 00:21:56.090 And therefore, the real part is even, OK? 00:21:56.090 --> 00:21:58.330 So the real part of the frequency response 00:21:58.330 --> 00:22:00.110 is an even function of omega. 00:22:00.110 --> 00:22:03.910 The imaginary part, which is the minus S omega, 00:22:03.910 --> 00:22:07.100 well if I change omega to minus omega, I flip the sine. 00:22:07.100 --> 00:22:10.460 So that's an odd function of omega, and so on. 00:22:10.460 --> 00:22:12.350 So you can go through these properties. 00:22:12.350 --> 00:22:14.600 Whenever you're stuck trying to figure out a property, 00:22:14.600 --> 00:22:16.142 this is the expression to go back to. 00:22:16.142 --> 00:22:19.670 So rewrite the basic definition in this form, 00:22:19.670 --> 00:22:22.930 and you'll understand a lot of this. 00:22:22.930 --> 00:22:24.890 And again, you'll get practice in recitation 00:22:24.890 --> 00:22:27.680 if you haven't done that already. 00:22:30.680 --> 00:22:34.980 Another important property of-- 00:22:34.980 --> 00:22:37.520 that you encounter when you go from the time domain 00:22:37.520 --> 00:22:40.040 to the frequency domain-- 00:22:40.040 --> 00:22:41.690 so remember, in the time domain, we 00:22:41.690 --> 00:22:45.410 said that if you have an input here, you 00:22:45.410 --> 00:22:49.310 convolve that input with h1 to get the output 00:22:49.310 --> 00:22:50.760 of the intermediate point? 00:22:55.350 --> 00:22:58.330 OK, so if I call the output of the intermediate point-- 00:22:58.330 --> 00:22:59.980 I should have done it there. 00:22:59.980 --> 00:23:00.930 But here's h1. 00:23:03.480 --> 00:23:09.160 If I call this w, this is x. 00:23:09.160 --> 00:23:16.100 And then I go into a second system, h2. 00:23:16.100 --> 00:23:16.890 And here's y. 00:23:19.850 --> 00:23:26.210 OK, well w is equal to h1 convolved with x. 00:23:26.210 --> 00:23:33.680 And y equals h2 convolved with w. 00:23:33.680 --> 00:23:37.770 So that's this. 00:23:37.770 --> 00:23:40.780 But I can put the parentheses any way I like for convolution, 00:23:40.780 --> 00:23:41.280 right? 00:23:41.280 --> 00:23:42.947 We've already established that property. 00:23:48.240 --> 00:23:50.930 So the net effect of the cascade of systems 00:23:50.930 --> 00:23:55.840 is the effect you'd get by having a single system LTI 00:23:55.840 --> 00:23:59.930 with this unit sample response. 00:23:59.930 --> 00:24:03.430 Now, if I think the frequency domain-- 00:24:03.430 --> 00:24:12.490 if I put e to the j omega n here, then 00:24:12.490 --> 00:24:14.670 what comes out at the intermediate point? 00:24:14.670 --> 00:24:22.870 At the intermediate point, I get h1 omega ej omega n. 00:24:22.870 --> 00:24:24.220 All right, so that's wn. 00:24:27.710 --> 00:24:31.140 But this is, again, an input of exponential form. 00:24:31.140 --> 00:24:33.170 So what comes out of the second system when 00:24:33.170 --> 00:24:35.190 I put this input into it? 00:24:35.190 --> 00:24:37.640 So what's w of n-- sorry, what's y of n going to be? 00:24:41.060 --> 00:24:41.560 Yeah? 00:24:41.560 --> 00:24:43.860 AUDIENCE: [INAUDIBLE] 00:24:43.860 --> 00:24:46.110 PROFESSOR: Yeah, so it's basically the second system's 00:24:46.110 --> 00:24:50.370 frequency response scaling the exponential that 00:24:50.370 --> 00:24:52.470 went into the second system, which is this. 00:24:56.260 --> 00:25:00.040 So the net effect when I put ej omega in at the first spot 00:25:00.040 --> 00:25:03.400 is at the output, I get the same ej omega n, 00:25:03.400 --> 00:25:08.850 but scaled by the product of the two frequency responses. 00:25:08.850 --> 00:25:10.460 So the nice thing here is that when 00:25:10.460 --> 00:25:13.740 I'm describing a cascade of two systems, 00:25:13.740 --> 00:25:16.100 if I describe the net effect in the time domain, 00:25:16.100 --> 00:25:19.220 I've got to do a convolution of these two units' sample 00:25:19.220 --> 00:25:20.300 responses. 00:25:20.300 --> 00:25:22.700 If I think of it in the frequency domain, 00:25:22.700 --> 00:25:25.370 I just have to take the product of the individual frequency 00:25:25.370 --> 00:25:26.810 responses. 00:25:26.810 --> 00:25:31.490 So the key observation here is that convolution 00:25:31.490 --> 00:25:37.760 in the time domain maps to multiplication in the frequency 00:25:37.760 --> 00:25:39.390 domain. 00:25:39.390 --> 00:25:43.190 So if I wanted the DTFT of this-- 00:25:43.190 --> 00:25:44.780 if I wanted the discrete-time Fourier 00:25:44.780 --> 00:25:48.800 transform of this result of a convolution, 00:25:48.800 --> 00:25:50.660 I can find it by just multiplying 00:25:50.660 --> 00:25:55.490 the individual DTFTs, all right? 00:25:55.490 --> 00:26:04.670 So convolution in time maps to multiplication in frequency. 00:26:09.130 --> 00:26:14.530 And this actually makes design much more easy, 00:26:14.530 --> 00:26:17.590 because we're often cascading systems in this form. 00:26:17.590 --> 00:26:20.020 And if you think in terms of frequency, 00:26:20.020 --> 00:26:22.190 you can track a frequency component 00:26:22.190 --> 00:26:24.130 through a cascade of such systems 00:26:24.130 --> 00:26:26.950 just focusing on the frequency response of each system 00:26:26.950 --> 00:26:29.060 as you go. 00:26:29.060 --> 00:26:29.935 So here's an example. 00:26:33.112 --> 00:26:34.195 Suppose we have a channel. 00:26:39.840 --> 00:26:43.095 Let's say that it's a channel with an echo, so when I put-- 00:26:46.200 --> 00:26:49.880 let me actually draw it out here. 00:26:49.880 --> 00:26:59.020 So I've got a channel here which I'm modeling as LTI. 00:26:59.020 --> 00:27:02.020 And if I put in a unit sample function 00:27:02.020 --> 00:27:05.740 here, so this has the value 1 at time 1, 00:27:05.740 --> 00:27:09.710 suppose the channel is one that has some echoing in it. 00:27:09.710 --> 00:27:13.260 So what I actually get out for this input 00:27:13.260 --> 00:27:21.300 is the same delta of n plus 0.8 delta of n minus 1. 00:27:21.300 --> 00:27:27.270 So there is a later arrival scaled by something 00:27:27.270 --> 00:27:29.970 which corresponds to the echo. 00:27:29.970 --> 00:27:35.100 So this must be the unit sample response of the channel. 00:27:38.980 --> 00:27:42.750 What's the frequency response of the channel? 00:27:42.750 --> 00:27:48.640 So if I call this h1 of n, what is h1 big of omega-- 00:27:48.640 --> 00:27:51.940 big omega-- h1 of big omega? 00:27:57.968 --> 00:27:59.260 I don't have it up there, do I? 00:27:59.260 --> 00:28:01.400 No. 00:28:01.400 --> 00:28:03.210 Anyone? 00:28:03.210 --> 00:28:04.748 Just from the definition. 00:28:14.483 --> 00:28:16.150 Is the problem here that you don't quite 00:28:16.150 --> 00:28:20.920 see what h1, 0 is, h1, 1, h1, 2, and so on? 00:28:20.920 --> 00:28:24.490 If I asked you to plot this out, how would you plot it? 00:28:34.801 --> 00:28:35.783 Yeah? 00:28:35.783 --> 00:28:38.880 AUDIENCE: [INAUDIBLE] 00:28:38.880 --> 00:28:39.800 PROFESSOR: Is it 1.8? 00:28:39.800 --> 00:28:40.300 Where? 00:28:40.300 --> 00:28:42.340 Where would you put the 1-- 00:28:42.340 --> 00:28:43.060 just over there? 00:28:43.060 --> 00:28:45.060 Oh, you're talking about the frequency response. 00:28:45.060 --> 00:28:47.110 Let's get the unit sample response first. 00:28:47.110 --> 00:28:47.890 Let's sketch this. 00:28:47.890 --> 00:28:49.282 What's your sketch of that? 00:28:49.282 --> 00:28:50.430 AUDIENCE: At 0B1 00:28:50.430 --> 00:28:52.372 PROFESSOR: At 0B1? 00:28:52.372 --> 00:28:57.730 AUDIENCE: [INAUDIBLE] 00:28:57.730 --> 00:29:00.870 PROFESSOR: OK, on 0 everywhere else-- 00:29:00.870 --> 00:29:03.440 that's the unit sample response. 00:29:03.440 --> 00:29:05.070 OK, so what's the frequency response? 00:29:05.070 --> 00:29:06.820 Well, we just plug it into the definition. 00:29:06.820 --> 00:29:12.810 All the h's except the ones that argue in 0 on 1 are equal to 0. 00:29:12.810 --> 00:29:19.573 So this is going to be 1 plus 0.8 to the minus j omega. 00:29:19.573 --> 00:29:20.490 Is that what you said? 00:29:20.490 --> 00:29:23.700 It was not quite what you said, right? 00:29:23.700 --> 00:29:25.740 What you said was the number I'd get at omega 00:29:25.740 --> 00:29:28.500 equals 0-- the DC gain of the system. 00:29:28.500 --> 00:29:30.000 But the frequency response is that. 00:29:40.270 --> 00:29:43.240 Let's just work backwards here. 00:29:52.660 --> 00:29:54.130 So the frequency response is that. 00:29:54.130 --> 00:29:56.890 Or if I wanted to write it-- 00:29:56.890 --> 00:29:58.760 we're going from that board to here-- 00:29:58.760 --> 00:30:02.530 h1 of omega, I can write it as a real plus imaginary part. 00:30:02.530 --> 00:30:06.900 So it would be 1 plus 0.8 cosine omega. 00:30:06.900 --> 00:30:08.742 This would be the real part. 00:30:08.742 --> 00:30:12.140 Then I have a minus j-- 00:30:12.140 --> 00:30:14.140 sorry, 0.8 sine omega. 00:30:18.730 --> 00:30:21.850 OK, so that's the frequency response-- 00:30:21.850 --> 00:30:24.780 some complex number with a real part and an imaginary part. 00:30:31.770 --> 00:30:35.800 OK, and if I asked you to give it to me in magnitude and angle 00:30:35.800 --> 00:30:37.180 form, you could do that. 00:30:37.180 --> 00:30:40.480 It's just rearranging things. 00:30:40.480 --> 00:30:44.170 So you'd-- the magnitude would be the square root of the sum 00:30:44.170 --> 00:30:47.230 of squares of these two pieces. 00:30:47.230 --> 00:30:50.330 And the angle would be the arctan of the ratio. 00:30:50.330 --> 00:30:54.065 So I assume that you know how to do all of that. 00:30:54.065 --> 00:30:55.440 And what you find-- actually, you 00:30:55.440 --> 00:30:57.750 can see it in these expressions already. 00:30:57.750 --> 00:31:00.270 Just as-- well, I didn't quite claim this earlier. 00:31:00.270 --> 00:31:03.210 But the magnitude of the frequency response 00:31:03.210 --> 00:31:06.300 will always be a real function of frequency-- sorry, 00:31:06.300 --> 00:31:08.110 an even function of frequency. 00:31:08.110 --> 00:31:12.160 And the phase will always be an odd function of frequency. 00:31:12.160 --> 00:31:16.620 So if you're drawing the results of a computation like this 00:31:16.620 --> 00:31:21.990 and you find that you don't have an even function 00:31:21.990 --> 00:31:24.610 for the magnitude, then you know you've done something wrong. 00:31:24.610 --> 00:31:27.160 So I'm not-- I'm going to sketch something here 00:31:27.160 --> 00:31:29.340 which I'm not pretending is the magnitude of that. 00:31:29.340 --> 00:31:32.760 I just want you to get the idea of what I mean by even. 00:31:32.760 --> 00:31:41.350 It's going to be something that's symmetric in omega. 00:31:41.350 --> 00:31:42.990 This is the magnitude. 00:31:42.990 --> 00:31:45.630 And then, if I did the phase, the phase 00:31:45.630 --> 00:31:48.630 is always going to be something that's 00:31:48.630 --> 00:31:50.489 an odd function of frequency. 00:31:55.553 --> 00:31:57.220 So if it's an odd function of frequency, 00:31:57.220 --> 00:31:59.470 what's the value at 0 of the phase? 00:32:03.430 --> 00:32:05.690 It's got to go through 0, right? 00:32:05.690 --> 00:32:08.117 And so I might get-- 00:32:08.117 --> 00:32:09.450 well, what would it actually be? 00:32:09.450 --> 00:32:12.710 It would be some shape. 00:32:12.710 --> 00:32:14.810 I'm not pretending I have the right shape here. 00:32:14.810 --> 00:32:18.980 But it's going to have an odd symmetry. 00:32:18.980 --> 00:32:22.830 I'll leave you to figure out what it actually looks like. 00:32:22.830 --> 00:32:28.350 So that's the frequency response of this echo channel. 00:32:28.350 --> 00:32:30.930 So here's what I want you to do now. 00:32:30.930 --> 00:32:33.960 At your receiver, build for me a filter that's 00:32:33.960 --> 00:32:38.350 going to undo the distortion that the echo has produced. 00:32:38.350 --> 00:32:41.165 So what I'd like is, I'd like an output, 00:32:41.165 --> 00:32:42.790 after you've done your filtering, to be 00:32:42.790 --> 00:32:45.560 exactly equal to the input. 00:32:45.560 --> 00:32:47.350 So my question is, what should-- 00:32:47.350 --> 00:32:51.070 and my claim is you can do that with an LTI filter. 00:32:51.070 --> 00:32:53.170 How would you describe that LTI filter? 00:32:53.170 --> 00:32:56.182 What should that LTI filter be? 00:32:56.182 --> 00:32:57.170 Yeah? 00:32:57.170 --> 00:33:02.110 AUDIENCE: [INAUDIBLE] 00:33:02.110 --> 00:33:06.750 PROFESSOR: Right, OK, so if you wanted the output 00:33:06.750 --> 00:33:08.460 to be exactly equal to the input, 00:33:08.460 --> 00:33:13.890 no matter what input was, you want a frequency response of 1 00:33:13.890 --> 00:33:15.380 overall. 00:33:15.380 --> 00:33:17.220 And the overall frequency response we know 00:33:17.220 --> 00:33:19.680 is the product of the two individual ones. 00:33:19.680 --> 00:33:27.420 And so we want h2 omega times h1 omega to be equal to 1. 00:33:27.420 --> 00:33:29.790 And therefore, h2 should be 1 over h1. 00:33:29.790 --> 00:33:33.120 So you can see here how things get a lot easier when you 00:33:33.120 --> 00:33:34.650 think in the frequency domain. 00:33:34.650 --> 00:33:38.310 If I had to do this in the time domain, 00:33:38.310 --> 00:33:40.910 I would have had to say h2 convolved 00:33:40.910 --> 00:33:44.940 with h1 has got to give me the unit sample function. 00:33:44.940 --> 00:33:47.490 And I'll give you h1, now you've got to figure h2. 00:33:47.490 --> 00:33:50.850 Well, you've got to go and work the convolution picture 00:33:50.850 --> 00:33:54.480 backwards, which is doable for simple cases. 00:33:54.480 --> 00:33:56.320 But this is much simpler. 00:33:56.320 --> 00:34:03.150 So this shows that h2 should be 1 over h1. 00:34:09.020 --> 00:34:12.050 Seems like a reasonable way to go. 00:34:12.050 --> 00:34:16.500 And you can actually work the whole thing through. 00:34:16.500 --> 00:34:18.197 But there's a problem with this. 00:34:18.197 --> 00:34:19.739 And we've seen this in other settings 00:34:19.739 --> 00:34:23.190 as well, which is something that works 00:34:23.190 --> 00:34:27.090 fine in the noise-free case doesn't 00:34:27.090 --> 00:34:29.260 work so well when you've got noise in your system. 00:34:29.260 --> 00:34:33.219 So look at what this receiver filter is doing. 00:34:33.219 --> 00:34:38.040 The receiver filter-- let's see, what is its magnitude? 00:34:38.040 --> 00:34:40.770 How does the magnitude of the receiver filter 00:34:40.770 --> 00:34:44.370 relate to the magnitude of the channel filter-- 00:34:46.929 --> 00:34:49.820 of the channel frequency response? 00:34:49.820 --> 00:34:52.519 So this magnitude is a magnitude of 1 over h1. 00:34:52.519 --> 00:34:58.712 Is that the same as 1 over magnitude of h1? 00:34:58.712 --> 00:35:00.340 Is that how complex numbers work? 00:35:03.020 --> 00:35:05.800 OK, right? 00:35:05.800 --> 00:35:06.730 So look what happens. 00:35:06.730 --> 00:35:10.210 Where the channel has a very low frequency response-- 00:35:10.210 --> 00:35:12.820 in other words, where the channel output is very 00:35:12.820 --> 00:35:16.350 low for a sinusoidal input at that frequency, 00:35:16.350 --> 00:35:19.400 the receiver filter is going to have a very high magnitude. 00:35:19.400 --> 00:35:21.820 So the receiver filter is trying to boost up 00:35:21.820 --> 00:35:24.730 whatever signal it sees in a frequency range 00:35:24.730 --> 00:35:28.220 where the channel actually has very little output. 00:35:28.220 --> 00:35:30.880 So what happens if I come and have a bit of noise 00:35:30.880 --> 00:35:33.595 here where I'm receiving the signal? 00:35:33.595 --> 00:35:36.070 Well, it's going to be very badly exaggerated 00:35:36.070 --> 00:35:40.580 by the inverse filter. 00:35:40.580 --> 00:35:43.430 So a little bit of noise here will 00:35:43.430 --> 00:35:46.700 get accentuated at frequencies where 00:35:46.700 --> 00:35:50.000 the frequency response of the receiver filter is large. 00:35:50.000 --> 00:35:52.280 But that's precisely where the channel had 00:35:52.280 --> 00:35:53.810 a very low frequency response. 00:35:53.810 --> 00:35:55.935 And it's precisely where the output-- the channel-- 00:35:55.935 --> 00:35:57.360 has nothing interesting for me. 00:35:57.360 --> 00:36:00.420 So my receiver filter ends up accentuating the noise. 00:36:00.420 --> 00:36:02.960 OK, so yet again, we see that these sorts 00:36:02.960 --> 00:36:05.508 of inversion operations may look nice on paper. 00:36:05.508 --> 00:36:07.550 But if you don't take account of what noise does, 00:36:07.550 --> 00:36:09.200 then you can run into trouble. 00:36:09.200 --> 00:36:10.670 And the picture is very transparent 00:36:10.670 --> 00:36:14.660 when you think in the frequency domain. 00:36:14.660 --> 00:36:23.550 OK, some more practice with filters and cascade-- 00:36:23.550 --> 00:36:26.340 I think I'm going to leave you to work through this 00:36:26.340 --> 00:36:27.550 in recitation, perhaps. 00:36:27.550 --> 00:36:30.180 So I'll leave it on the slides. 00:36:30.180 --> 00:36:36.185 But let's go to design of filters. 00:36:40.190 --> 00:36:43.730 So now, we've seen one example of trying to design a filter-- 00:36:43.730 --> 00:36:47.090 the receiver filter-- to undo the distortion of the channel. 00:36:47.090 --> 00:36:50.320 Here's another-- actually, I want that. 00:36:50.320 --> 00:36:54.020 Here is another design problem that you 00:36:54.020 --> 00:37:02.960 run into all the time, which is that you see a signal that's 00:37:02.960 --> 00:37:05.270 got a whole bunch of frequencies mixed up in it, 00:37:05.270 --> 00:37:07.260 and you want to exclude some of them. 00:37:07.260 --> 00:37:09.560 So maybe you're looking for an audio signal. 00:37:09.560 --> 00:37:12.410 You know that the combinations of sinusoids 00:37:12.410 --> 00:37:16.010 that make up an audio signal are unlikely to go above-- 00:37:16.010 --> 00:37:16.760 whatever you want. 00:37:16.760 --> 00:37:19.620 Pick your number-- 10 kilohertz, 20 kilohertz. 00:37:19.620 --> 00:37:21.590 And so you want to exclude frequencies outside 00:37:21.590 --> 00:37:22.710 of that range. 00:37:22.710 --> 00:37:25.010 So you're very often in the position 00:37:25.010 --> 00:37:29.940 of trying to build what's called an ideal low pass filter. 00:37:29.940 --> 00:37:31.550 So here's an ideal low pass filter. 00:37:34.370 --> 00:37:37.280 I'd like you to build for me a filter that 00:37:37.280 --> 00:37:44.120 passes all frequencies in some range without distortion, 00:37:44.120 --> 00:37:46.445 and that completely kills everything outside. 00:37:49.590 --> 00:37:53.100 So let me call this the cutoff frequency. 00:38:00.150 --> 00:38:02.905 So that's the h of omega I want. 00:38:02.905 --> 00:38:05.530 And now my question is, how are you going to build this filter? 00:38:05.530 --> 00:38:07.690 I want you to give me the unit sample 00:38:07.690 --> 00:38:09.300 response that goes with it. 00:38:09.300 --> 00:38:11.140 And you see a hint over here. 00:38:11.140 --> 00:38:13.570 But can you tell me how you might go about that? 00:38:19.610 --> 00:38:20.630 Not so obvious, right? 00:38:20.630 --> 00:38:25.010 Because we've specified the filter characteristic 00:38:25.010 --> 00:38:27.740 in the frequency domain, and now we want to find the h's 00:38:27.740 --> 00:38:29.550 that go with it. 00:38:29.550 --> 00:38:32.270 So what we're really looking for is a formula 00:38:32.270 --> 00:38:36.500 that will give us the time domain signal in terms 00:38:36.500 --> 00:38:37.500 of the frequency domain. 00:38:37.500 --> 00:38:39.422 So we want to invert this somehow. 00:38:39.422 --> 00:38:40.880 So what we're looking for is really 00:38:40.880 --> 00:38:43.655 what's called the inverse DTFT. 00:39:00.627 --> 00:39:02.460 And actually, if you've done Fourier series, 00:39:02.460 --> 00:39:04.500 you've seen this trick before. 00:39:04.500 --> 00:39:06.837 Because really, we're not far from Fourier series here. 00:39:06.837 --> 00:39:08.920 It's just that the domains are a little different, 00:39:08.920 --> 00:39:10.590 so maybe you don't recognize it. 00:39:10.590 --> 00:39:13.920 Here, we've got a periodic something expressed 00:39:13.920 --> 00:39:16.350 as a combination of sines and cosines, 00:39:16.350 --> 00:39:19.320 or as a combination of exponentials. 00:39:19.320 --> 00:39:22.650 And now, we want to invert that, OK? 00:39:22.650 --> 00:39:25.860 If you thought of these as Fourier coefficients 00:39:25.860 --> 00:39:28.320 for some periodic signal, and then went and looked up 00:39:28.320 --> 00:39:30.540 whatever book you use for Fourier series, 00:39:30.540 --> 00:39:32.350 you'd get the formula. 00:39:32.350 --> 00:39:34.350 Because we're just trying to extract the Fourier 00:39:34.350 --> 00:39:38.120 coefficients for this periodic signal. 00:39:38.120 --> 00:39:39.790 But you can actually do it from scratch. 00:39:39.790 --> 00:39:44.880 So if you think of multiplying both sides of this by, 00:39:44.880 --> 00:39:46.440 let's say, e to the j omega n-- 00:39:49.085 --> 00:39:52.240 OK, so I'm going to multiply both sides. 00:39:52.240 --> 00:40:03.310 So I've got e to the minus j omega m minus n now, right? 00:40:06.110 --> 00:40:10.450 And I'm going to then integrate both sides over an interval 00:40:10.450 --> 00:40:11.440 of length 2 pi-- 00:40:15.180 --> 00:40:17.130 any contiguous interval of length 2 pi. 00:40:17.130 --> 00:40:21.593 It actually does matter because of the periodicity. 00:40:21.593 --> 00:40:23.260 So I'll take any interval of length 2 pi 00:40:23.260 --> 00:40:24.430 and I integrate both sides. 00:40:28.680 --> 00:40:31.020 And I'll assume that I can hop this integral in there. 00:40:31.020 --> 00:40:34.210 I'll assume my signal is well-behaved enough for that. 00:40:34.210 --> 00:40:44.130 So here's what I end up getting. 00:40:44.130 --> 00:40:54.080 On this right hand side, I get summation integral hm. 00:40:54.080 --> 00:40:55.670 Oh, I should put a d omega there. 00:40:55.670 --> 00:40:57.360 Sorry. 00:40:57.360 --> 00:40:59.497 I've gotten casual with my integration. 00:41:11.460 --> 00:41:13.720 So on this side, I have this integral. 00:41:13.720 --> 00:41:16.820 On this side, I have that integral. 00:41:16.820 --> 00:41:19.870 And if you work through this, out 00:41:19.870 --> 00:41:21.550 of all this infinity of terms, there's 00:41:21.550 --> 00:41:23.770 only one term that survives. 00:41:23.770 --> 00:41:27.370 Because any term in which m is different from n 00:41:27.370 --> 00:41:30.490 will have this exponential still sitting here. 00:41:30.490 --> 00:41:33.190 This exponential is like a cosine plus a j sine, 00:41:33.190 --> 00:41:35.650 or a cosine minus a j sine. 00:41:35.650 --> 00:41:39.710 You're integrating it over an interval of 2 pi. 00:41:39.710 --> 00:41:43.210 So any term here that has the exponential, 00:41:43.210 --> 00:41:45.833 or has the sine or cosine in it, will disappear 00:41:45.833 --> 00:41:46.750 under the integration. 00:41:46.750 --> 00:41:50.830 The only term that survives is the one where m equals n. 00:41:50.830 --> 00:41:54.160 And so what you discover is that this is 00:41:54.160 --> 00:41:57.915 2 pi hn when you're all done. 00:41:57.915 --> 00:41:59.540 I'm not going through the details here. 00:42:02.320 --> 00:42:05.455 So here is the formula we wanted for the inverse DTFT. 00:42:19.890 --> 00:42:26.430 Here's the inverse DTFT, OK? 00:42:26.430 --> 00:42:30.750 I've forgotten my colored chalk today, but that'll do. 00:42:34.030 --> 00:42:37.910 So if I gave you a filter characteristic like this 00:42:37.910 --> 00:42:40.430 and asked you to find the unit sample response of the filter 00:42:40.430 --> 00:42:44.480 that went with it, you would just have to plug 00:42:44.480 --> 00:42:47.750 in the frequency response characteristic that I gave you 00:42:47.750 --> 00:42:48.950 and solve for the h's. 00:42:52.080 --> 00:42:55.793 I think I have a bunch of this on the slides. 00:42:55.793 --> 00:42:57.210 This is what we just went through. 00:43:01.100 --> 00:43:05.543 So let's do this now for the ideal low pass filter. 00:43:05.543 --> 00:43:06.460 What is it that we do? 00:43:09.540 --> 00:43:13.440 I've got the formula that I just derived for you there. 00:43:13.440 --> 00:43:18.090 h is equal to 1 in the pass band of the filter, 00:43:18.090 --> 00:43:21.070 and it's 0 outside of that. 00:43:21.070 --> 00:43:24.380 So I set h equal to 1 in the pass band of the filter, which 00:43:24.380 --> 00:43:27.710 is from minus omega C to plus omega C, and the rest of it 00:43:27.710 --> 00:43:29.420 doesn't contribute anything. 00:43:29.420 --> 00:43:31.920 And then, I just work out this integral. 00:43:31.920 --> 00:43:34.320 And I've actually got to do it in two pieces. 00:43:34.320 --> 00:43:37.010 For n not equal to 0, this is what I get. 00:43:37.010 --> 00:43:40.790 For n equals 0, this is what I get. 00:43:40.790 --> 00:43:42.560 If n was continuous, actually, you'd 00:43:42.560 --> 00:43:46.580 say that this is the same expression as here, 00:43:46.580 --> 00:43:48.560 because you just use L'Hopital's rule 00:43:48.560 --> 00:43:50.660 and you'll get from here to here. 00:43:50.660 --> 00:43:52.730 But since n is an integer, we've got 00:43:52.730 --> 00:43:54.620 to be a little careful how we write it, OK? 00:43:54.620 --> 00:43:57.440 So you can't really say you're going to use L'Hopital's rule 00:43:57.440 --> 00:43:59.810 to see what this is in the limit of n going to 0, 00:43:59.810 --> 00:44:01.680 because n takes integer values. 00:44:01.680 --> 00:44:04.323 But if you work it out from scratch for n equals 0, 00:44:04.323 --> 00:44:05.990 you'll see that you get a formula that's 00:44:05.990 --> 00:44:09.110 consistent with using L'Hopital's rule. 00:44:09.110 --> 00:44:13.530 OK, so this is a function that we'll see again and again when 00:44:13.530 --> 00:44:15.960 we do filtering of this type, and it's 00:44:15.960 --> 00:44:18.100 referred to as a sinc function. 00:44:18.100 --> 00:44:22.380 So it's not S-I-N, but S-I-N-C. And if you plot it out, 00:44:22.380 --> 00:44:23.250 this is what it is. 00:44:26.410 --> 00:44:31.180 So it's got the oscilation that comes from the sine, 00:44:31.180 --> 00:44:33.970 but it's got a reduction in amplitude that comes from the 1 00:44:33.970 --> 00:44:34.720 over n. 00:44:34.720 --> 00:44:36.400 So it's a signal that falls off as 1 00:44:36.400 --> 00:44:42.642 over n with this kind of a characteristic. 00:44:47.290 --> 00:44:50.400 Do you think it's a bounded input, bounded output 00:44:50.400 --> 00:44:52.020 stable system? 00:44:52.020 --> 00:44:52.830 What's your hunch? 00:44:56.740 --> 00:44:59.030 Remember what it takes for a system to be stable? 00:44:59.030 --> 00:45:02.753 The unit sample response has to be absolutely summable. 00:45:02.753 --> 00:45:04.420 So if you take the absolute values here, 00:45:04.420 --> 00:45:06.370 and sum from minus infinity to infinity, 00:45:06.370 --> 00:45:10.890 you want to get something finite to call this stable. 00:45:10.890 --> 00:45:12.900 Well, since this only falls off as 1 over n, 00:45:12.900 --> 00:45:14.480 it turns out to not be stable. 00:45:14.480 --> 00:45:16.920 So it's actually an extreme idealization 00:45:16.920 --> 00:45:20.460 that is not bounded-- input bounded, output stable, 00:45:20.460 --> 00:45:21.150 but it's close. 00:45:24.870 --> 00:45:29.730 Just to go back when I showed you this filter characteristic 00:45:29.730 --> 00:45:35.790 here, to give the cheap version of a low pass filter, what 00:45:35.790 --> 00:45:41.670 we actually did was take the sinc function 00:45:41.670 --> 00:45:43.973 and truncated to a finite interval. 00:45:43.973 --> 00:45:46.140 And so what happens when you truncate it to a finite 00:45:46.140 --> 00:45:50.083 interval is that instead of the sharp box-like shape 00:45:50.083 --> 00:45:52.500 for the frequency response, you get a closer approximation 00:45:52.500 --> 00:45:53.160 to it-- 00:45:53.160 --> 00:45:56.678 not exactly the ideal low pass filter, but maybe good enough. 00:45:56.678 --> 00:45:58.220 The other thing that you might notice 00:45:58.220 --> 00:46:01.610 if you're looking carefully is that I had a sinc that 00:46:01.610 --> 00:46:04.130 was centered around 0 and even. 00:46:04.130 --> 00:46:07.530 And now, I seem to have a causal version of the filter. 00:46:07.530 --> 00:46:10.700 And I think I'll leave you in recitation to figure out 00:46:10.700 --> 00:46:13.910 how you can go from the centered, non-causal filter 00:46:13.910 --> 00:46:15.770 to a causal filter, and what that 00:46:15.770 --> 00:46:19.787 does to phase and to frequency response magnitude. 00:46:25.620 --> 00:46:29.370 So basically, I'll leave you to go through the details here. 00:46:29.370 --> 00:46:33.505 But the key idea here is the inverse DTFT. 00:46:37.690 --> 00:46:40.190 So now, I want to just take a slightly different perspective 00:46:40.190 --> 00:46:42.710 on this formula that we derived. 00:46:42.710 --> 00:46:45.080 We said we've got a frequency response, which we're 00:46:45.080 --> 00:46:49.910 calling the DTFT of the signal h of n-- the unit sample 00:46:49.910 --> 00:46:51.470 response. 00:46:51.470 --> 00:46:53.060 We've got an inverse formula that 00:46:53.060 --> 00:46:57.680 allows us to get the time signal from the frequency response. 00:46:57.680 --> 00:46:59.825 But here's yet another way of looking at what 00:46:59.825 --> 00:47:00.950 this formula is telling us. 00:47:00.950 --> 00:47:04.730 This formula is saying, I can think of h of n 00:47:04.730 --> 00:47:08.368 as being made up of a whole bunch of complex exponentials. 00:47:08.368 --> 00:47:10.410 So you see that this is what we were looking for. 00:47:10.410 --> 00:47:13.100 We were looking for a way to take a signal 00:47:13.100 --> 00:47:15.260 and figure out its spectral content. 00:47:15.260 --> 00:47:18.350 We want to know what complex exponentials, 00:47:18.350 --> 00:47:21.440 or what sinusoids does it take to make that signal? 00:47:21.440 --> 00:47:23.510 Well, we have a hint of that in this expression, 00:47:23.510 --> 00:47:26.420 because this is saying, take the time domain signal. 00:47:26.420 --> 00:47:28.610 I can think of it as being a combination. 00:47:28.610 --> 00:47:32.240 Now, this is not a finite combination, it's a continuum. 00:47:32.240 --> 00:47:35.720 But it is a combination of exponentials of the type 00:47:35.720 --> 00:47:37.730 that we know to work with. 00:47:37.730 --> 00:47:41.270 So this is actually giving us a spectral decomposition 00:47:41.270 --> 00:47:43.730 of the unit sample response, where 00:47:43.730 --> 00:47:47.240 the amount of e to the j omega n that it 00:47:47.240 --> 00:47:51.140 takes to make up the signal is told to me by h of omega. 00:47:51.140 --> 00:47:53.870 So the h of omegas are sort of the weights 00:47:53.870 --> 00:47:57.720 that we use to combine these exponentials to get the signal. 00:47:57.720 --> 00:48:01.340 So the idea for a spectral decomposition, 00:48:01.340 --> 00:48:03.560 or for describing the spectral nature of a signal 00:48:03.560 --> 00:48:05.480 is actually sitting there. 00:48:05.480 --> 00:48:08.360 All we have to do is say, we'll use the same formulas, 00:48:08.360 --> 00:48:12.050 but let's no longer restrict it to the unit sample 00:48:12.050 --> 00:48:14.750 response of a system and the frequency 00:48:14.750 --> 00:48:15.900 response of that system. 00:48:15.900 --> 00:48:17.850 Let's use it for any signal-- 00:48:17.850 --> 00:48:21.446 so the same formulas, but now for any signal xn. 00:48:21.446 --> 00:48:24.930 Give me any signal xn, I'll compute for you 00:48:24.930 --> 00:48:30.000 this object, which is the DTFT of that signal, just 00:48:30.000 --> 00:48:31.910 the same way I did for a frequency response. 00:48:31.910 --> 00:48:34.560 So I'll compute the x of omega for you. 00:48:34.560 --> 00:48:37.170 What's the significance of x of omega? 00:48:37.170 --> 00:48:39.240 Well, it tells me in what combination 00:48:39.240 --> 00:48:42.540 I have to wait the e to the j omega n's 00:48:42.540 --> 00:48:44.110 to construct for you the signal. 00:48:44.110 --> 00:48:49.650 So the x of big omega, the DTFT tells me 00:48:49.650 --> 00:48:51.720 what the spectral content of the signal is. 00:48:51.720 --> 00:48:53.640 If I plot that as a function of frequency, 00:48:53.640 --> 00:48:55.800 it tells me how to assemble the signal out 00:48:55.800 --> 00:48:59.160 of sums of sines and cosines. 00:48:59.160 --> 00:49:00.150 So let's see here. 00:49:05.880 --> 00:49:12.180 More specifically, what I would say is that the DTFT at omega 00:49:12.180 --> 00:49:12.830 0-- 00:49:12.830 --> 00:49:16.470 omega sub 0 times d omega is the spectral content 00:49:16.470 --> 00:49:19.800 of the signal in that particular interval. 00:49:19.800 --> 00:49:23.310 And if I add up all those components over all frequencies 00:49:23.310 --> 00:49:26.040 in this 2 pi range, I'll get the original signal 00:49:26.040 --> 00:49:27.580 that I'm interested in. 00:49:27.580 --> 00:49:31.080 So what we'll do next time is work with this idea 00:49:31.080 --> 00:49:34.290 to see how it lets us think about signals through systems, 00:49:34.290 --> 00:49:38.280 and how it enables us to do filtering in a systematic way. 00:49:38.280 --> 00:49:40.880 All right, let's leave it at that for now.