WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:35.720 --> 00:00:55.260 [MUSIC PLAYING] 00:00:55.260 --> 00:00:58.560 PROFESSOR: Last time we began the development of the 00:00:58.560 --> 00:01:01.430 discrete-time Fourier transform. 00:01:01.430 --> 00:01:06.350 And just as with the continuous-time case, we first 00:01:06.350 --> 00:01:08.710 treated the notion of periodic signals. 00:01:08.710 --> 00:01:11.060 This led to the Fourier series. 00:01:11.060 --> 00:01:15.350 And then we generalized that to the Fourier transform, and 00:01:15.350 --> 00:01:18.080 finally incorporated within the framework of the Fourier 00:01:18.080 --> 00:01:23.070 transform both aperiodic and periodic signals. 00:01:23.070 --> 00:01:27.820 In today's lecture, what I'd like to do is expand on some 00:01:27.820 --> 00:01:30.940 of the properties of the Fourier transform, and 00:01:30.940 --> 00:01:32.875 indicate how those properties are used for 00:01:32.875 --> 00:01:34.770 a variety of things. 00:01:34.770 --> 00:01:39.470 Well, let's begin by reviewing the Fourier transform as we 00:01:39.470 --> 00:01:41.310 developed it last time. 00:01:41.310 --> 00:01:45.410 It, of course, involves a synthesis equation and an 00:01:45.410 --> 00:01:47.040 analysis equation. 00:01:47.040 --> 00:01:51.340 The synthesis equation expressing x of n, the 00:01:51.340 --> 00:01:55.090 sequence, in terms of the Fourier transform, and the 00:01:55.090 --> 00:01:58.880 analysis equation telling us how to obtain the Fourier 00:01:58.880 --> 00:02:02.890 transform from the original sequence. 00:02:02.890 --> 00:02:07.480 And I draw your attention again to the basic point that 00:02:07.480 --> 00:02:11.650 the synthesis equation essentially corresponds to 00:02:11.650 --> 00:02:17.660 decomposing the sequence as a linear combination of complex 00:02:17.660 --> 00:02:21.590 exponentials with amplitudes that are, in effect, 00:02:21.590 --> 00:02:25.820 proportional to the Fourier transform. 00:02:25.820 --> 00:02:30.290 Now, the discrete-time Fourier transform, just as the 00:02:30.290 --> 00:02:34.380 continuous-time Fourier transform, has a number of 00:02:34.380 --> 00:02:37.050 important and useful properties. 00:02:37.050 --> 00:02:41.600 Of course, as I stressed last time, it's a function of a 00:02:41.600 --> 00:02:44.140 continuous variable. 00:02:44.140 --> 00:02:48.870 And it's also a complex-valued function, which means that 00:02:48.870 --> 00:02:51.680 when we represent it in general it requires a 00:02:51.680 --> 00:02:56.860 representation in terms of its real part and imaginary part, 00:02:56.860 --> 00:03:00.540 or in terms of magnitude and angle. 00:03:00.540 --> 00:03:06.360 Also, as I indicated last time, the Fourier transform is 00:03:06.360 --> 00:03:12.600 a periodic function of frequency, and the periodicity 00:03:12.600 --> 00:03:16.030 is with a period of 2 pi. 00:03:16.030 --> 00:03:20.750 And so it says, in effect, that the Fourier transform, if 00:03:20.750 --> 00:03:24.110 we replace the frequency variable by an integer 00:03:24.110 --> 00:03:28.580 multiple of 2 pi, the function repeats. 00:03:28.580 --> 00:03:34.680 And I stress again that the underlying basis for this 00:03:34.680 --> 00:03:39.080 periodicity property is the fact that it's the set of 00:03:39.080 --> 00:03:41.560 complex exponentials that are 00:03:41.560 --> 00:03:44.190 inherently periodic in frequency. 00:03:44.190 --> 00:03:47.040 And so, of course, any representation using them 00:03:47.040 --> 00:03:52.660 would, in effect, generate a periodicity with 00:03:52.660 --> 00:03:53.950 this period of 2 pi. 00:03:56.780 --> 00:04:01.320 Just as in continuous time, the Fourier transform has 00:04:01.320 --> 00:04:03.560 important symmetry properties. 00:04:03.560 --> 00:04:08.060 And in particular, if the sequence x sub n is 00:04:08.060 --> 00:04:12.130 real-valued, then the Fourier transform 00:04:12.130 --> 00:04:14.680 is conjugate symmetric. 00:04:14.680 --> 00:04:19.420 In other words, if we replace omega by minus omega, that's 00:04:19.420 --> 00:04:22.930 equivalent to applying complex conjugation 00:04:22.930 --> 00:04:25.000 to the Fourier transform. 00:04:25.000 --> 00:04:30.780 And as a consequence of this conjugate symmetry, this 00:04:30.780 --> 00:04:35.880 results in a symmetry in the real part that is an even 00:04:35.880 --> 00:04:41.430 symmetry, or the magnitude has an even symmetry, whereas the 00:04:41.430 --> 00:04:47.590 imaginary part or the phase angle are both odd symmetric. 00:04:47.590 --> 00:04:50.850 And these are symmetry properties, again, that are 00:04:50.850 --> 00:04:53.480 identical to the symmetry properties that we saw in 00:04:53.480 --> 00:04:55.230 continuous time. 00:04:55.230 --> 00:04:58.820 Well, let's see this in the context of an example that we 00:04:58.820 --> 00:05:01.940 worked last time and that we'll want to draw attention 00:05:01.940 --> 00:05:04.960 to in reference to several issues as this 00:05:04.960 --> 00:05:06.730 lecture goes along. 00:05:06.730 --> 00:05:11.890 And that is the Fourier transform of a real damped 00:05:11.890 --> 00:05:13.330 exponential. 00:05:13.330 --> 00:05:19.330 So the sequence that we are talking about is a to the n u 00:05:19.330 --> 00:05:23.590 of n, and let's consider a to be positive. 00:05:23.590 --> 00:05:27.060 We saw last time that the Fourier transform for this 00:05:27.060 --> 00:05:31.490 sequence algebraically is of this form. 00:05:31.490 --> 00:05:35.860 And if we look at its magnitude and angle, the 00:05:35.860 --> 00:05:40.110 magnitude I show here. 00:05:40.110 --> 00:05:43.120 And the magnitude, as we see, has the 00:05:43.120 --> 00:05:45.130 properties that we indicated. 00:05:45.130 --> 00:05:48.400 It is an even function of frequency. 00:05:48.400 --> 00:05:52.150 Of course, it's a function of a continuous variable. 00:05:52.150 --> 00:05:55.400 And it, in addition, is periodic with a 00:05:55.400 --> 00:05:58.300 period of two pi. 00:05:58.300 --> 00:06:02.050 On the other hand, if we look at the phase angle below it, 00:06:02.050 --> 00:06:06.310 the phase angle has a symmetry which is odd symmetric. 00:06:06.310 --> 00:06:09.640 And that's indicated clearly in this picture. 00:06:09.640 --> 00:06:12.100 And of course, in addition to being odd symmetric, it 00:06:12.100 --> 00:06:16.700 naturally has to be, again, a periodic function of frequency 00:06:16.700 --> 00:06:20.080 with a period of 2 pi. 00:06:20.080 --> 00:06:22.630 OK, so we have some symmetry properties. 00:06:22.630 --> 00:06:25.220 We have this inherent periodicity in the Fourier 00:06:25.220 --> 00:06:31.260 transform, which I'm stressing very heavily because it forms 00:06:31.260 --> 00:06:34.060 the basic difference between continuous time 00:06:34.060 --> 00:06:36.170 and discrete time. 00:06:36.170 --> 00:06:39.420 In addition to these properties of the Fourier 00:06:39.420 --> 00:06:44.940 transform, there are a number of other properties that are 00:06:44.940 --> 00:06:48.820 particularly useful in the manipulation of the Fourier 00:06:48.820 --> 00:06:53.050 transform, and, in fact, in using the Fourier transform 00:06:53.050 --> 00:06:57.550 to, for example, analyze systems represented by linear 00:06:57.550 --> 00:07:00.660 constant coefficient difference equations. 00:07:00.660 --> 00:07:05.410 There in the text is a longer list of properties, but let me 00:07:05.410 --> 00:07:09.490 just draw your attention to several of them. 00:07:09.490 --> 00:07:13.210 One is the time shifting property. 00:07:13.210 --> 00:07:19.620 And the time shifting property tells us that if x of omega is 00:07:19.620 --> 00:07:24.870 the Fourier transform of x of n, then the Fourier transform 00:07:24.870 --> 00:07:30.090 of x of n shifted in time is that same Fourier transform 00:07:30.090 --> 00:07:34.730 multiplied by this factor, which is a 00:07:34.730 --> 00:07:36.690 linear phase factor. 00:07:36.690 --> 00:07:41.670 So time shifting introduces a linear phase term. 00:07:41.670 --> 00:07:46.410 And, by the way, recall that in the continuous-time case we 00:07:46.410 --> 00:07:49.530 had a similar situation, namely that a time shift 00:07:49.530 --> 00:07:53.050 corresponded to a linear phase. 00:07:53.050 --> 00:07:57.260 There also is a dual to the time shifting property, which 00:07:57.260 --> 00:08:01.000 is referred to as the frequency shifting property, 00:08:01.000 --> 00:08:06.520 which tells us that if we multiply a time function by a 00:08:06.520 --> 00:08:09.600 complex exponential, that, in effect, 00:08:09.600 --> 00:08:13.030 generates a frequency shift. 00:08:13.030 --> 00:08:17.000 And we'll see this frequency shifting property surface in a 00:08:17.000 --> 00:08:20.310 slightly different way shortly, when we talk about 00:08:20.310 --> 00:08:24.840 the modulation property in the discrete-time case. 00:08:24.840 --> 00:08:28.080 Another important property that we'll want to make use of 00:08:28.080 --> 00:08:31.390 shortly is linearity, which follows in a very 00:08:31.390 --> 00:08:36.860 straightforward way from the Fourier transform definition. 00:08:36.860 --> 00:08:42.220 And the linearity property says simply that the Fourier 00:08:42.220 --> 00:08:46.510 transform of a sum, or linear combination, is the same 00:08:46.510 --> 00:08:49.020 linear combination of the Fourier transforms. 00:08:49.020 --> 00:08:52.750 Again, that's a property that we saw in continuous time. 00:08:52.750 --> 00:08:58.570 And, also, among other properties there is a 00:08:58.570 --> 00:09:01.790 Parseval's relation for the discrete-time case that in 00:09:01.790 --> 00:09:05.110 effect says something similar to continuous time, 00:09:05.110 --> 00:09:10.750 specifically that the energy in the sequence is 00:09:10.750 --> 00:09:15.340 proportional to the energy in the Fourier transform, the 00:09:15.340 --> 00:09:17.370 energy over one period. 00:09:17.370 --> 00:09:20.820 Or, said another way, in fact, or another way that it can be 00:09:20.820 --> 00:09:23.760 said, is that the energy in the time domain is 00:09:23.760 --> 00:09:26.070 proportional to the power in this 00:09:26.070 --> 00:09:30.060 periodic Fourier transform. 00:09:30.060 --> 00:09:33.510 OK, so these are some of the properties. 00:09:33.510 --> 00:09:37.750 And, as I indicated, parallel somewhat properties that we 00:09:37.750 --> 00:09:40.140 saw in continuous time. 00:09:40.140 --> 00:09:44.730 Two additional properties that will play important roles in 00:09:44.730 --> 00:09:48.530 discrete time just as they did in continuous time are the 00:09:48.530 --> 00:09:53.730 convolution property and the modulation property. 00:09:53.730 --> 00:09:57.760 The convolution property is the property that tells us how 00:09:57.760 --> 00:10:03.100 to relate the Fourier transform of the convolution 00:10:03.100 --> 00:10:06.990 of two sequences to the Fourier transforms of the 00:10:06.990 --> 00:10:08.940 individual sequences. 00:10:08.940 --> 00:10:12.240 And, not surprisingly, what happens-- 00:10:12.240 --> 00:10:15.530 and this can be demonstrated algebraically-- 00:10:15.530 --> 00:10:20.390 not surprisingly, the Fourier transform of the convolution 00:10:20.390 --> 00:10:24.960 is simply the product of the Fourier transforms. 00:10:24.960 --> 00:10:30.460 So, Fourier transform maps convolution in the time domain 00:10:30.460 --> 00:10:33.650 to multiplication in the frequency domain. 00:10:33.650 --> 00:10:37.300 Now convolution, of course, arises in the context of 00:10:37.300 --> 00:10:39.610 linear time-invariant systems. 00:10:39.610 --> 00:10:42.120 In particular, if we have a system with an impulse 00:10:42.120 --> 00:10:47.710 response h of n, input x of n, the output is the convolution. 00:10:47.710 --> 00:10:51.860 The convolution property then tells us that in the frequency 00:10:51.860 --> 00:10:57.780 domain, the Fourier transform is the product of the Fourier 00:10:57.780 --> 00:11:01.420 transform of the impulse response and the Fourier 00:11:01.420 --> 00:11:05.120 transform of the input. 00:11:05.120 --> 00:11:10.560 Now we also saw and have talked about a relationship 00:11:10.560 --> 00:11:14.950 between the Fourier transform, the impulse response, and what 00:11:14.950 --> 00:11:22.250 we call the frequency response in the context of the response 00:11:22.250 --> 00:11:25.840 of a system to a complex exponential. 00:11:25.840 --> 00:11:29.030 Specifically, complex exponentials are 00:11:29.030 --> 00:11:33.210 eigenfunctions of linear time-invariant systems. 00:11:33.210 --> 00:11:37.810 One of these into the system gives us, as an output, a 00:11:37.810 --> 00:11:42.070 complex exponential with the same complex frequency 00:11:42.070 --> 00:11:45.920 multiplied by what we refer to as the eigenvalue. 00:11:45.920 --> 00:11:52.030 And as you saw in the video course manual, this 00:11:52.030 --> 00:11:56.010 eigenvalue, this constant, multiplier on the exponential 00:11:56.010 --> 00:12:00.530 is, in fact, the Fourier transform of the impulse 00:12:00.530 --> 00:12:05.520 response evaluated at that frequency. 00:12:05.520 --> 00:12:10.700 Now, we saw exactly the same statement in continuous time. 00:12:10.700 --> 00:12:13.280 And, in fact, we used that statement-- 00:12:13.280 --> 00:12:17.860 the frequency response interpretation of the Fourier 00:12:17.860 --> 00:12:20.040 transform, the impulse response-- 00:12:20.040 --> 00:12:24.710 we use that to motivate an intuitive interpretation of 00:12:24.710 --> 00:12:26.550 the convolution property. 00:12:26.550 --> 00:12:30.460 Now, formally the convolution property can be developed by 00:12:30.460 --> 00:12:33.840 taking the convolution sum, applying the Fourier transform 00:12:33.840 --> 00:12:38.240 sum to it, doing the appropriate substitution of 00:12:38.240 --> 00:12:40.810 variables, interchanging order of summations, et cetera, and 00:12:40.810 --> 00:12:44.800 all the algebra works out to show that it's a product. 00:12:44.800 --> 00:12:48.610 But as I stressed when we discussed this with continuous 00:12:48.610 --> 00:12:50.550 time, the interpretation-- 00:12:50.550 --> 00:12:52.160 the underlying interpretation-- 00:12:52.160 --> 00:12:55.040 is particularly important to understand. 00:12:55.040 --> 00:12:58.160 So let me review it again in the discrete-time case, and 00:12:58.160 --> 00:13:01.080 it's exactly the same for discrete time or for 00:13:01.080 --> 00:13:03.330 continuous time. 00:13:03.330 --> 00:13:10.100 Specifically, the argument was that the Fourier transform of 00:13:10.100 --> 00:13:14.760 a sequence or signal corresponds to decomposing it 00:13:14.760 --> 00:13:18.330 into a linear combination of complex exponentials. 00:13:18.330 --> 00:13:21.310 What's the amplitude of those complex exponentials? 00:13:21.310 --> 00:13:25.990 It's basically proportional to the Fourier transform. 00:13:25.990 --> 00:13:29.290 If we think of pushing through the system that linear 00:13:29.290 --> 00:13:35.220 combination, then each of those complex exponentials 00:13:35.220 --> 00:13:40.540 gets the amplitude modified, or multiplied, by the Fourier 00:13:40.540 --> 00:13:42.500 transform of-- 00:13:42.500 --> 00:13:44.110 by the frequency response-- 00:13:44.110 --> 00:13:46.670 which we saw is the Fourier transform 00:13:46.670 --> 00:13:49.130 of the impulse response. 00:13:49.130 --> 00:13:54.170 So the amplitudes of the output complex exponentials is 00:13:54.170 --> 00:13:58.240 then the amplitudes of the input complex exponentials 00:13:58.240 --> 00:14:01.470 multiplied by the frequency response. 00:14:01.470 --> 00:14:06.240 And the Fourier transform of the output, in effect, is an 00:14:06.240 --> 00:14:11.450 expression expressing the summation, or integration, of 00:14:11.450 --> 00:14:14.790 the output as a linear combination of all of these 00:14:14.790 --> 00:14:17.400 exponentials with the appropriate complex 00:14:17.400 --> 00:14:19.580 amplitudes. 00:14:19.580 --> 00:14:23.720 So, it's important, in thinking about the convolution 00:14:23.720 --> 00:14:29.880 property, to think about it in terms of nothing more than the 00:14:29.880 --> 00:14:34.740 fact that we've decomposed the input, and we're now modifying 00:14:34.740 --> 00:14:38.410 separately through multiplication, through 00:14:38.410 --> 00:14:43.010 scaling, the amplitudes of each of the complex 00:14:43.010 --> 00:14:44.260 exponential components. 00:14:47.230 --> 00:14:50.780 Now what we saw in continuous time is that this 00:14:50.780 --> 00:14:56.390 interpretation and the convolution property led to an 00:14:56.390 --> 00:15:00.640 important concept, namely the concept of filtering. 00:15:00.640 --> 00:15:04.140 Kind of the idea that if we decompose the input as a 00:15:04.140 --> 00:15:08.990 linear combination of complex exponentials, we can 00:15:08.990 --> 00:15:12.320 separately attenuate or amplify each of those 00:15:12.320 --> 00:15:13.910 components. 00:15:13.910 --> 00:15:19.190 And, in fact, we could exactly pass some set of frequencies 00:15:19.190 --> 00:15:23.160 and totally eliminate other set of frequencies. 00:15:23.160 --> 00:15:29.430 So, again, just as in continuous time, we can talk 00:15:29.430 --> 00:15:32.130 about an ideal filter. 00:15:32.130 --> 00:15:37.820 And what I show here is the frequency response of an ideal 00:15:37.820 --> 00:15:39.990 lowpass filter. 00:15:39.990 --> 00:15:45.050 The ideal lowpass filter, of course, passes exactly, with a 00:15:45.050 --> 00:15:51.800 gain of 1, frequencies around 0, and eliminates totally 00:15:51.800 --> 00:15:53.050 other frequencies. 00:15:55.030 --> 00:15:58.480 However, an important distinction here between 00:15:58.480 --> 00:16:02.530 continuous time and discrete time is the fact that, whereas 00:16:02.530 --> 00:16:06.540 in continuous time when we talked about an ideal filter, 00:16:06.540 --> 00:16:09.110 we passed a band of frequencies and totally 00:16:09.110 --> 00:16:12.070 eliminated everything else out to infinity. 00:16:12.070 --> 00:16:15.070 In the discrete time case, the 00:16:15.070 --> 00:16:18.210 frequency response is periodic. 00:16:18.210 --> 00:16:22.400 So, obviously, the frequency response must periodically 00:16:22.400 --> 00:16:24.430 repeat for the lowpass filter. 00:16:24.430 --> 00:16:26.700 And in fact we see that here. 00:16:26.700 --> 00:16:31.570 If we look at the lowpass filter, then we've eliminated 00:16:31.570 --> 00:16:33.240 some frequencies. 00:16:33.240 --> 00:16:39.260 But then we pass, of course, frequencies around 2 pi, and 00:16:39.260 --> 00:16:43.160 also frequencies around minus 2 pi, and for that matter 00:16:43.160 --> 00:16:46.030 around any multiple of 2 pi. 00:16:46.030 --> 00:16:49.770 Although it's important to recognize that because of the 00:16:49.770 --> 00:16:53.310 inherent periodicity of the complex exponentials, these 00:16:53.310 --> 00:16:58.590 frequencies are exactly the same frequencies as these 00:16:58.590 --> 00:16:59.670 frequencies. 00:16:59.670 --> 00:17:03.720 So it's lowpass filtering interpreted in terms of 00:17:03.720 --> 00:17:09.180 frequencies over a range from minus pi to pi. 00:17:09.180 --> 00:17:12.670 Well, just as we talk about a lowpass filter, we can also 00:17:12.670 --> 00:17:15.609 talk about a highpass filter. 00:17:15.609 --> 00:17:19.500 And a highpass filter, of course, would pass high 00:17:19.500 --> 00:17:21.790 frequencies. 00:17:21.790 --> 00:17:24.460 In a continuous-time case, high frequencies meant 00:17:24.460 --> 00:17:28.050 frequencies that go out to infinity. 00:17:28.050 --> 00:17:30.720 In the discrete-time case, of course, the highest 00:17:30.720 --> 00:17:34.800 frequencies we can generate are frequencies up to pi. 00:17:34.800 --> 00:17:39.760 And once our complex exponentials go past pi, then, 00:17:39.760 --> 00:17:42.570 in fact, we start seeing the lower frequencies again. 00:17:42.570 --> 00:17:45.400 Let me indicate what I mean. 00:17:45.400 --> 00:17:49.560 If we think in the context of the lowpass filter, these are 00:17:49.560 --> 00:17:51.300 low frequencies. 00:17:51.300 --> 00:17:54.030 As we move along the frequency axis, these become high 00:17:54.030 --> 00:17:55.350 frequencies. 00:17:55.350 --> 00:17:59.490 And as we move further along the frequency axis, what we'll 00:17:59.490 --> 00:18:03.330 see when we get to, for example, a frequency of 2 pi 00:18:03.330 --> 00:18:09.030 are the same low frequencies that we see around 0. 00:18:09.030 --> 00:18:12.900 In particular then, an ideal highpass filter in the 00:18:12.900 --> 00:18:16.790 discrete-time case would be a filter that eliminates these 00:18:16.790 --> 00:18:20.660 frequencies and passes frequencies around pi. 00:18:23.620 --> 00:18:28.860 OK, so we've seen the convolution property and its 00:18:28.860 --> 00:18:31.760 interpretation in terms of filtering. 00:18:31.760 --> 00:18:36.040 More broadly, the convolution property in combination with a 00:18:36.040 --> 00:18:38.470 number of the other properties that I introduced, in 00:18:38.470 --> 00:18:42.040 particular the time shifting and linearity property, allows 00:18:42.040 --> 00:18:49.290 us to generate or analyze systems that are described by 00:18:49.290 --> 00:18:52.100 linear constant coefficient difference equations. 00:18:52.100 --> 00:18:56.330 And this, again, parallels very strongly the discussion 00:18:56.330 --> 00:19:00.650 we carried out in the continuous-time case. 00:19:00.650 --> 00:19:08.510 In particular, let's think of a discrete-time system that is 00:19:08.510 --> 00:19:09.900 described by a linear constant 00:19:09.900 --> 00:19:11.910 coefficient difference equation. 00:19:11.910 --> 00:19:15.470 And we'll restrict the initial conditions on the equation 00:19:15.470 --> 00:19:19.340 such that it corresponds to a linear time-invariant system. 00:19:19.340 --> 00:19:23.360 And recall that, in fact, in our discussion of linear 00:19:23.360 --> 00:19:26.810 constant coefficient difference equations, it is 00:19:26.810 --> 00:19:31.230 the condition of initial rest that-- on the equation-- 00:19:31.230 --> 00:19:35.240 that guarantees for us that the system will be causal, 00:19:35.240 --> 00:19:38.720 linear, and time invariant. 00:19:38.720 --> 00:19:41.790 OK, now let's consider a first-order difference 00:19:41.790 --> 00:19:43.340 equation, a system described by a 00:19:43.340 --> 00:19:44.740 first-order difference equation. 00:19:44.740 --> 00:19:46.260 And we've talked about the solution of 00:19:46.260 --> 00:19:48.160 this equation before. 00:19:48.160 --> 00:19:51.140 Essentially, we run the solution recursively. 00:19:51.140 --> 00:19:55.370 Let's now consider generating the solution by taking 00:19:55.370 --> 00:19:59.270 advantage of the properties of the Fourier transform. 00:19:59.270 --> 00:20:02.360 Well, just as we did in continuous time, we can 00:20:02.360 --> 00:20:05.870 consider applying the Fourier transform to both sides of 00:20:05.870 --> 00:20:07.210 this equation. 00:20:07.210 --> 00:20:10.520 And the Fourier transform of y of n, of 00:20:10.520 --> 00:20:12.680 course, is Y of omega. 00:20:12.680 --> 00:20:16.090 And then using the shifting property, the time shifting 00:20:16.090 --> 00:20:21.270 property, the Fourier transform of y of n minus 1 is 00:20:21.270 --> 00:20:25.520 Y of omega multiplied by e to the minus j omega. 00:20:25.520 --> 00:20:30.180 And so we have this, using a linearity property we can 00:20:30.180 --> 00:20:32.840 carry down the scale factor, and add these two together as 00:20:32.840 --> 00:20:34.290 they're added here. 00:20:34.290 --> 00:20:39.020 And the Fourier transform of x of n is X of omega. 00:20:39.020 --> 00:20:43.150 Well, we can solve this equation for the Fourier 00:20:43.150 --> 00:20:47.630 transform of the output in terms of the Fourier transform 00:20:47.630 --> 00:20:51.650 of the input and an appropriate 00:20:51.650 --> 00:20:53.220 complex scale factor. 00:20:53.220 --> 00:20:58.170 And simply solving this for Y of omega yields 00:20:58.170 --> 00:21:00.800 what we have here. 00:21:00.800 --> 00:21:04.740 Now what we've used in going from this point to this point 00:21:04.740 --> 00:21:11.750 is both the shifting property and we've also used the 00:21:11.750 --> 00:21:13.000 linearity property. 00:21:15.860 --> 00:21:19.940 At this point, we can recognize that here the 00:21:19.940 --> 00:21:22.810 Fourier transform of the output is the product of the 00:21:22.810 --> 00:21:28.860 Fourier transform of the input and some complex function. 00:21:28.860 --> 00:21:33.300 And from the convolution property, then, that complex 00:21:33.300 --> 00:21:38.480 function must in fact correspond to the frequency 00:21:38.480 --> 00:21:43.440 response, or equivalently, the Fourier transform of the 00:21:43.440 --> 00:21:45.620 impulse response. 00:21:45.620 --> 00:21:50.460 So if we want to determine the Fourier transform of the-- 00:21:50.460 --> 00:21:55.170 or the impulse response of the system, let's say for example, 00:21:55.170 --> 00:21:59.980 then it becomes a matter of having identified the Fourier 00:21:59.980 --> 00:22:02.080 transform of the impulse response, which is the 00:22:02.080 --> 00:22:03.500 frequency response. 00:22:03.500 --> 00:22:05.800 We now want to inverse transform to 00:22:05.800 --> 00:22:08.600 get the impulse response. 00:22:08.600 --> 00:22:10.730 Well, how do we inverse transform? 00:22:10.730 --> 00:22:15.420 Of course, we could do it by attempting to go through the 00:22:15.420 --> 00:22:18.880 synthesis equation for the Fourier transform. 00:22:18.880 --> 00:22:22.520 Or we can do as we did in the continuous-time case which is 00:22:22.520 --> 00:22:25.362 to take advantage of what we know. 00:22:25.362 --> 00:22:30.450 And in particular, we know that from an example that we 00:22:30.450 --> 00:22:36.910 worked before, this is in fact the Fourier transform of a 00:22:36.910 --> 00:22:43.960 sequence which is a to the n times u of n. 00:22:43.960 --> 00:22:46.860 And so, in essence, by inspection-- 00:22:46.860 --> 00:22:51.770 very similar to what has gone on in continuous time-- 00:22:51.770 --> 00:22:55.900 essentially by inspection, we can then solve for the impulse 00:22:55.900 --> 00:22:58.400 response to the system. 00:22:58.400 --> 00:23:02.940 OK, so that procedure follows very much the kind of 00:23:02.940 --> 00:23:06.630 procedure that we've carried out in continuous time. 00:23:06.630 --> 00:23:08.630 And this, of course, is discussed in more 00:23:08.630 --> 00:23:11.480 detail in the text. 00:23:11.480 --> 00:23:16.380 Well, let's look at that example then. 00:23:16.380 --> 00:23:21.530 Here we have the impulse response for that, associated 00:23:21.530 --> 00:23:22.960 with the system described by that 00:23:22.960 --> 00:23:26.340 particular difference equation. 00:23:26.340 --> 00:23:29.630 And to the right of that, we have the 00:23:29.630 --> 00:23:32.140 associated frequency response. 00:23:32.140 --> 00:23:34.150 And one of the things that we notice-- 00:23:34.150 --> 00:23:39.080 and this is drawn for a positive between 0 and 1-- 00:23:39.080 --> 00:23:44.650 what we notice, in fact, is that it is an approximation to 00:23:44.650 --> 00:23:48.670 a lowpass filter, because it tends to attenuate the high 00:23:48.670 --> 00:23:52.380 frequencies and retain and, in fact, amplify the low 00:23:52.380 --> 00:23:54.520 frequencies. 00:23:54.520 --> 00:23:59.840 Now if instead, actually, the impulse response was such that 00:23:59.840 --> 00:24:04.540 we picked a to be negative between minus 1 and 0, then 00:24:04.540 --> 00:24:08.110 the impulse response in the time domain looks like this. 00:24:08.110 --> 00:24:10.380 And the corresponding frequency 00:24:10.380 --> 00:24:12.980 response looks like this. 00:24:12.980 --> 00:24:15.920 And that becomes an approximation 00:24:15.920 --> 00:24:19.790 to a highpass filter. 00:24:19.790 --> 00:24:24.330 So, in fact, a first-order difference equation, as we 00:24:24.330 --> 00:24:28.250 see, has a frequency response, depending on the value of a, 00:24:28.250 --> 00:24:31.930 that either looks approximately like a lowpass 00:24:31.930 --> 00:24:34.540 filter for a positive or a highpass filter for a 00:24:34.540 --> 00:24:38.360 negative, very much like the first-order differential 00:24:38.360 --> 00:24:41.560 equation looked like a lowpass filter in the 00:24:41.560 --> 00:24:43.710 continuous-time case. 00:24:43.710 --> 00:24:48.950 And, in fact, what I'd like to illustrate is the filtering 00:24:48.950 --> 00:24:50.340 characteristics-- 00:24:50.340 --> 00:24:52.680 or an example of filtering-- 00:24:52.680 --> 00:24:55.600 using a first-order difference equation. 00:24:55.600 --> 00:25:00.400 And the example that I'll illustrate is a filtering of a 00:25:00.400 --> 00:25:04.190 sequence that in fact is filtered very often for very 00:25:04.190 --> 00:25:07.880 practical reasons, namely a sequence which represents the 00:25:07.880 --> 00:25:12.030 Dow Jones Industrial Average over a fairly long period. 00:25:12.030 --> 00:25:17.040 And we'll process the Dow Jones Industrial Average first 00:25:17.040 --> 00:25:19.610 through a first-order difference equation, where, if 00:25:19.610 --> 00:25:23.460 we begin with a equals 0, then, referring to the 00:25:23.460 --> 00:25:27.760 frequency response that we have here, a equals 0 would 00:25:27.760 --> 00:25:30.960 simply be passing all frequencies. 00:25:30.960 --> 00:25:37.140 As a is positive we start to retain mostly low frequencies, 00:25:37.140 --> 00:25:44.370 and the larger a gets, but still less than 1, the more it 00:25:44.370 --> 00:25:46.770 attenuates high frequencies at the expense of low 00:25:46.770 --> 00:25:48.270 frequencies. 00:25:48.270 --> 00:25:53.490 So let's watch the filtering, first with a positive and 00:25:53.490 --> 00:25:56.500 we'll see it behave as a lowpass filter, and then with 00:25:56.500 --> 00:26:02.270 a negative and we'll see the difference equation behaving 00:26:02.270 --> 00:26:05.020 as a highpass filter. 00:26:05.020 --> 00:26:09.520 What we see here is the Dow Jones Industrial Average over 00:26:09.520 --> 00:26:15.100 roughly a five-year period from 1927 to 1932. 00:26:15.100 --> 00:26:19.470 And, in fact, that big dip in the middle is the famous stock 00:26:19.470 --> 00:26:22.010 market crash of 1929. 00:26:22.010 --> 00:26:25.190 And we can see that following that, in fact, the market 00:26:25.190 --> 00:26:29.030 continued a very long downward trend. 00:26:29.030 --> 00:26:34.600 And what we now want to do is process this through a 00:26:34.600 --> 00:26:36.720 difference equation. 00:26:36.720 --> 00:26:40.950 Above the Dow Jones average we show the impulse response of 00:26:40.950 --> 00:26:42.020 the difference equation. 00:26:42.020 --> 00:26:46.080 Here we've chosen the parameter a equal to 0. 00:26:46.080 --> 00:26:50.570 And the impulse response will be displayed on an expanded 00:26:50.570 --> 00:26:57.700 scale in relation to the scale of the input and, for that 00:26:57.700 --> 00:27:00.300 matter, the scale of the output. 00:27:00.300 --> 00:27:03.900 Now with the impulse response shown here which is just an 00:27:03.900 --> 00:27:09.160 impulse, in fact, the output shown on the bottom trace is 00:27:09.160 --> 00:27:11.880 exactly identical to the input. 00:27:11.880 --> 00:27:16.730 And what we'll want to do now is increase, first, the 00:27:16.730 --> 00:27:22.000 parameter a, and the impulse response will begin to look 00:27:22.000 --> 00:27:27.140 like an exponential with a duration that's longer and 00:27:27.140 --> 00:27:30.670 longer as a moves from 0 to 1. 00:27:33.180 --> 00:27:35.850 Correspondingly we'll get more and more lowpass filtering as 00:27:35.850 --> 00:27:39.300 the coefficient a increases from 0 toward 1. 00:27:39.300 --> 00:27:43.340 So now we are increasing the parameter a. 00:27:43.340 --> 00:27:47.300 We see that the bottom trace in relation to the middle 00:27:47.300 --> 00:27:52.480 trace in fact is looking more and more smoothed or 00:27:52.480 --> 00:27:53.700 lowpass-filtered. 00:27:53.700 --> 00:27:57.440 And here now we have a fair amount of smoothing, to the 00:27:57.440 --> 00:28:02.480 point where the stock market crash of 1929 is totally lost. 00:28:02.480 --> 00:28:05.470 And in fact I'm sure there are many people who wish that 00:28:05.470 --> 00:28:09.090 through filtering we could, in fact, have avoided the stock 00:28:09.090 --> 00:28:11.940 market crash altogether. 00:28:11.940 --> 00:28:19.490 Now, let's decrease a from 1 back towards 0. 00:28:19.490 --> 00:28:22.410 And as we do that, we will be taking 00:28:22.410 --> 00:28:25.930 out the lowpass filtering. 00:28:25.930 --> 00:28:29.840 And when a finally reaches 0, the impulse response of the 00:28:29.840 --> 00:28:34.180 filter will again be an impulse, and so the output 00:28:34.180 --> 00:28:38.650 will be once again identical to the input. 00:28:38.650 --> 00:28:42.000 And that's where we are now. 00:28:42.000 --> 00:28:46.930 All right now we want to continue to decrease a so that 00:28:46.930 --> 00:28:51.160 it becomes negative, moving from 0 toward minus 1. 00:28:51.160 --> 00:28:55.890 And what we will see in that case is more and more highpass 00:28:55.890 --> 00:29:00.550 filtering on the output in relation to the input. 00:29:00.550 --> 00:29:04.500 And this will be particularly evident in, again, the region 00:29:04.500 --> 00:29:07.090 of high frequencies represented by sharp 00:29:07.090 --> 00:29:10.160 transitions which, of course, the market crash 00:29:10.160 --> 00:29:12.870 of 1929 would represent. 00:29:12.870 --> 00:29:18.740 So here, now, a is decreasing toward minus 1. 00:29:18.740 --> 00:29:22.140 We see that the high frequencies, or rapid 00:29:22.140 --> 00:29:29.230 variations are emphasized., And finally, let's move from 00:29:29.230 --> 00:29:35.620 minus 1 back towards 0, taking out the highpass filtering and 00:29:35.620 --> 00:29:40.390 ending up with a equal to 0, corresponding to an impulse 00:29:40.390 --> 00:29:42.290 response which is an impulse, in other 00:29:42.290 --> 00:29:44.010 words, an identity system. 00:29:44.010 --> 00:29:47.050 And let me stress once again that the time scale on which 00:29:47.050 --> 00:29:51.030 we displayed the impulse response is an expanded time 00:29:51.030 --> 00:29:55.320 scale in relation to the time scale on which we displayed 00:29:55.320 --> 00:29:56.820 the input and the output. 00:29:59.650 --> 00:30:04.090 OK, so we see that, in fact, a first-order difference 00:30:04.090 --> 00:30:07.370 equation is a filter. 00:30:07.370 --> 00:30:10.270 And, in fact, it's a very important class of filters, 00:30:10.270 --> 00:30:13.320 and it's used very often to do approximate lowpass and 00:30:13.320 --> 00:30:14.570 highpass filtering. 00:30:17.200 --> 00:30:24.310 Now, in addition to the convolution property, another 00:30:24.310 --> 00:30:27.420 important property that we had in continuous time, and that 00:30:27.420 --> 00:30:31.500 we have in discrete time, is the modulation property. 00:30:31.500 --> 00:30:35.080 The modulation property tells us what happens in the 00:30:35.080 --> 00:30:38.730 frequency domain when you multiply 00:30:38.730 --> 00:30:41.700 signals in the time domain. 00:30:41.700 --> 00:30:45.100 In continuous time, the modulation property 00:30:45.100 --> 00:30:48.760 corresponded to the statement that if we multiply the time 00:30:48.760 --> 00:30:54.400 domain, we convolve the Fourier transforms in the 00:30:54.400 --> 00:30:56.470 frequency domain. 00:30:56.470 --> 00:31:02.730 And in discrete time we have very much the same kind of 00:31:02.730 --> 00:31:05.060 relationship. 00:31:05.060 --> 00:31:11.090 The only real distinction between these is that in the 00:31:11.090 --> 00:31:15.160 discrete-time case, in carrying out the convolution, 00:31:15.160 --> 00:31:18.870 it's an integration only over a 2 pi interval. 00:31:18.870 --> 00:31:25.120 And what that corresponds to is what's referred to as a 00:31:25.120 --> 00:31:31.550 periodic convolution, as opposed to the continuous-time 00:31:31.550 --> 00:31:37.800 case where what we have is a convolution that is an 00:31:37.800 --> 00:31:41.050 aperiodic convolution. 00:31:41.050 --> 00:31:44.930 So, again, we have a convolution property in 00:31:44.930 --> 00:31:48.680 discrete time that is very much like the convolution 00:31:48.680 --> 00:31:51.060 property in continuous time. 00:31:51.060 --> 00:31:54.570 The only real difference is that here we're convolving 00:31:54.570 --> 00:31:56.300 periodic functions. 00:31:56.300 --> 00:32:01.050 And so it's a periodic convolution which involves an 00:32:01.050 --> 00:32:04.510 integration only over a 2 pi interval, rather than an 00:32:04.510 --> 00:32:08.570 integration from minus infinity to plus infinity. 00:32:08.570 --> 00:32:14.650 Well, let's take a look at an example of the modulation 00:32:14.650 --> 00:32:20.070 property, which will then lead to one particular application, 00:32:20.070 --> 00:32:24.110 and a very useful application, of the modulation property in 00:32:24.110 --> 00:32:26.180 discrete time. 00:32:26.180 --> 00:32:31.580 The example that I want to pick is an example in which we 00:32:31.580 --> 00:32:35.590 consider modulating a signal with-- 00:32:35.590 --> 00:32:40.150 a signal with another signal, x of n, or x1 of n as I 00:32:40.150 --> 00:32:44.170 indicated here, which is minus 1 to the n. 00:32:44.170 --> 00:32:48.050 Essentially what that says is that any signal which I 00:32:48.050 --> 00:32:51.640 modulate with this in effect corresponds to taking the 00:32:51.640 --> 00:32:56.110 original signal and then going through that signal 00:32:56.110 --> 00:33:00.690 alternating the algebraic signs. 00:33:00.690 --> 00:33:03.200 Now we-- 00:33:03.200 --> 00:33:06.430 in applying the modulation property, of course, what we 00:33:06.430 --> 00:33:09.390 need to do is develop the Fourier 00:33:09.390 --> 00:33:12.410 transform of this signal. 00:33:12.410 --> 00:33:14.820 This signal which I rewrite-- 00:33:14.820 --> 00:33:17.450 I can write either as minus 1 to the n or rewrite as e to 00:33:17.450 --> 00:33:22.520 the j pi n since e to the j pi is equal to minus 1-- 00:33:22.520 --> 00:33:25.210 is a periodic signal. 00:33:25.210 --> 00:33:28.700 And it's the periodic signal that I show here. 00:33:28.700 --> 00:33:32.860 And recall that to get the Fourier transform of a 00:33:32.860 --> 00:33:39.020 periodic signal, one way to do it is to generate the Fourier 00:33:39.020 --> 00:33:42.360 series coefficients for the periodic signal, and then 00:33:42.360 --> 00:33:47.450 identify the Fourier transform as an impulse train where the 00:33:47.450 --> 00:33:49.900 heights of the impulses in the impulse train are 00:33:49.900 --> 00:33:52.510 proportional, with a proportionality factor of 2 00:33:52.510 --> 00:33:58.200 pi, proportional to the Fourier series coefficients. 00:33:58.200 --> 00:34:01.380 So let's first work out what the Fourier series is and for 00:34:01.380 --> 00:34:04.580 this example, in fact, it's fairly easy. 00:34:04.580 --> 00:34:08.886 Here is the general synthesis equation 00:34:08.886 --> 00:34:12.429 for the Fourier series. 00:34:12.429 --> 00:34:18.969 And if we take our particular example where, if we look back 00:34:18.969 --> 00:34:25.130 at the curve above, what we recognize is that the period 00:34:25.130 --> 00:34:29.139 is equal to 2, namely it repeats after 2 points. 00:34:29.139 --> 00:34:34.389 Then capital N is equal to 2, and so we can just write this 00:34:34.389 --> 00:34:36.389 out with the two terms. 00:34:36.389 --> 00:34:40.900 And the two terms involved are x1 of n is a0, the 0-th 00:34:40.900 --> 00:34:46.520 coefficient, that's with k equals 0, and a1, and this is 00:34:46.520 --> 00:34:49.719 with k equals 1, and we substituted in 00:34:49.719 --> 00:34:52.520 capital N equal to 2. 00:34:52.520 --> 00:34:55.290 All right, well, we can do a little bit of algebra here, 00:34:55.290 --> 00:34:58.800 obviously cross off the factors of 2. 00:34:58.800 --> 00:35:04.370 And what we recognize, if we compare this expression with 00:35:04.370 --> 00:35:08.840 the original signal which is e to the j pi n, then we can 00:35:08.840 --> 00:35:13.780 simply identify the fact that a0, the 0-th coefficient is 0, 00:35:13.780 --> 00:35:15.510 that's the DC term. 00:35:15.510 --> 00:35:19.760 And the coefficient a1 is equal to 1. 00:35:19.760 --> 00:35:25.160 So we've done it simply by essentially inspecting the 00:35:25.160 --> 00:35:30.950 Fourier series synthesis equation. 00:35:30.950 --> 00:35:35.380 OK, now, if we want to get the Fourier transform for this, we 00:35:35.380 --> 00:35:40.470 take those coefficients and essentially generate an 00:35:40.470 --> 00:35:46.830 impulse train where we choose as values for the impulses 2 00:35:46.830 --> 00:35:50.680 pi times the Fourier series coefficients. 00:35:50.680 --> 00:35:54.010 So, the Fourier series coefficients are a0 is equal 00:35:54.010 --> 00:35:56.630 to 0 and a1 is equal to 1. 00:35:56.630 --> 00:36:03.370 So, notice that in the plot that I've shown here of the 00:36:03.370 --> 00:36:09.460 Fourier transform of x1 of n, we have the 0-th coefficient, 00:36:09.460 --> 00:36:14.320 which happens to be 0, and so I have it indicated, an 00:36:14.320 --> 00:36:16.440 impulse there. 00:36:16.440 --> 00:36:23.360 We have the coefficient a1, and the coefficient a1 occurs 00:36:23.360 --> 00:36:28.540 at a frequency which is omega 0, and omega 0 in fact is 00:36:28.540 --> 00:36:33.100 equal to pi because the signal is e to the j pi n. 00:36:33.100 --> 00:36:37.730 Well, what's this impulse over here? 00:36:37.730 --> 00:36:41.150 Well, that impulse is a-- 00:36:41.150 --> 00:36:42.940 corresponds to the Fourier series 00:36:42.940 --> 00:36:45.560 coefficient a sub minus 1. 00:36:45.560 --> 00:36:49.300 And, of course, if we drew this out over a longer 00:36:49.300 --> 00:36:52.850 frequency axis, we would see lots of other impulses because 00:36:52.850 --> 00:36:56.900 of the fact that the Fourier transform periodically repeats 00:36:56.900 --> 00:36:59.660 or, equivalently, the Fourier series coefficients 00:36:59.660 --> 00:37:01.790 periodically repeat. 00:37:01.790 --> 00:37:08.580 So this is the coefficient a0, This is the coefficient a1 00:37:08.580 --> 00:37:12.180 with a factor of 2 pi, this is 2 pi times a0 00:37:12.180 --> 00:37:15.680 and 2 pi times a1. 00:37:15.680 --> 00:37:20.260 And then this is simply an indication that it's 00:37:20.260 --> 00:37:21.510 periodically repeated. 00:37:24.220 --> 00:37:24.670 All right. 00:37:24.670 --> 00:37:28.830 Now, let's consider what happens if we take a signal 00:37:28.830 --> 00:37:34.470 and multiply it, modulate it, by minus 1 to the n. 00:37:34.470 --> 00:37:37.540 Well in the frequency domain that corresponds to a 00:37:37.540 --> 00:37:39.800 convolution. 00:37:39.800 --> 00:37:43.820 Let's consider a signal x2 of n which has a Fourier 00:37:43.820 --> 00:37:47.300 transform as I've indicated here. 00:37:47.300 --> 00:37:52.460 Then the Fourier transform of the product of x1 of n and x2 00:37:52.460 --> 00:37:57.270 of n is the convolution of these two spectra. 00:37:57.270 --> 00:38:01.850 And recall that if you could convolve something with an 00:38:01.850 --> 00:38:05.800 impulse train, as this is, that simply corresponds to 00:38:05.800 --> 00:38:10.920 taking the something and placing it at the positions of 00:38:10.920 --> 00:38:12.830 each of the impulses. 00:38:12.830 --> 00:38:17.850 So, in fact, the result of the convolution of this with this 00:38:17.850 --> 00:38:23.320 would then be the spectrum that I indicate here, namely 00:38:23.320 --> 00:38:28.920 this spectrum shifted up to pi and of course to minus pi. 00:38:28.920 --> 00:38:34.550 And then of course to not only pi but 3 pi and 5 pi, et 00:38:34.550 --> 00:38:36.180 cetera, et cetera. 00:38:36.180 --> 00:38:42.020 And so this spectrum, finally, corresponds to the Fourier 00:38:42.020 --> 00:38:48.010 transform of minus 1 to the n times x2 of n where x2 of n is 00:38:48.010 --> 00:38:53.780 the sequence whose spectrum was X2 of omega. 00:38:53.780 --> 00:38:57.750 OK, now, this is in fact an important, useful, and 00:38:57.750 --> 00:38:58.570 interesting point. 00:38:58.570 --> 00:39:03.210 What it says is if I have a signal with a certain spectrum 00:39:03.210 --> 00:39:05.230 and if I modulate-- 00:39:05.230 --> 00:39:06.180 multiply-- 00:39:06.180 --> 00:39:10.130 that signal by minus 1 to the n, meaning that I alternate 00:39:10.130 --> 00:39:14.470 the signs, then it takes the low frequencies-- 00:39:14.470 --> 00:39:17.160 in effect, it shifts the spectrum by pi. 00:39:17.160 --> 00:39:19.120 So it takes the low frequencies and moves them up 00:39:19.120 --> 00:39:22.470 to high frequencies, and will incidentally take the high 00:39:22.470 --> 00:39:26.440 frequencies and move them to low frequencies. 00:39:26.440 --> 00:39:30.400 So in fact we, in essence, saw this when we took-- 00:39:30.400 --> 00:39:33.920 or when I talked about the example of a sequence which 00:39:33.920 --> 00:39:36.120 was a to the n times u of n. 00:39:36.120 --> 00:39:36.730 Notice-- 00:39:36.730 --> 00:39:40.670 let me draw your attention to the fact that when a is 00:39:40.670 --> 00:39:46.460 positive, we have this sequence and its Fourier 00:39:46.460 --> 00:39:52.220 transform is as I show on the right. 00:39:52.220 --> 00:40:00.820 For a negative, the sequence is identical to a positive but 00:40:00.820 --> 00:40:03.030 with alternating sines. 00:40:03.030 --> 00:40:06.730 And the Fourier transform of that you can now see, and 00:40:06.730 --> 00:40:12.900 verify also algebraically if you'd like, is identical to 00:40:12.900 --> 00:40:17.640 this spectrum, simply shifted by pi. 00:40:17.640 --> 00:40:21.280 So it says in fact that multiplying that impulse 00:40:21.280 --> 00:40:25.530 response, or if we think of a positive and a negative, that 00:40:25.530 --> 00:40:29.260 is algebraically similar to multiplying the impulse 00:40:29.260 --> 00:40:31.650 response by minus 1 to the n. 00:40:31.650 --> 00:40:35.150 And in the frequency domain, the effect of that, 00:40:35.150 --> 00:40:38.250 essentially, is shifting the spectrum by pi. 00:40:38.250 --> 00:40:40.440 And we can interpret that in the context of 00:40:40.440 --> 00:40:43.060 the modulation property. 00:40:43.060 --> 00:40:50.260 Now it's interesting that what that says is that if we have a 00:40:50.260 --> 00:40:58.070 system which corresponds to a lowpass filter, as I indicate 00:40:58.070 --> 00:41:03.320 here, with an impulse response h of n. 00:41:03.320 --> 00:41:06.530 And it can be any approximation to a lowpass 00:41:06.530 --> 00:41:09.860 filter and even an ideal lowpass filter. 00:41:09.860 --> 00:41:14.900 If we want to convert that to a highpass filter, we can do 00:41:14.900 --> 00:41:19.980 that by generating a new system whose impulse response 00:41:19.980 --> 00:41:24.010 is minus 1 to the n times the impulse response of the 00:41:24.010 --> 00:41:25.490 lowpass filter. 00:41:25.490 --> 00:41:31.920 And this modulation by minus 1 to the n will take the 00:41:31.920 --> 00:41:36.980 frequency response of this system and shift it by pi so 00:41:36.980 --> 00:41:40.500 that what's going on here at low frequencies will now go on 00:41:40.500 --> 00:41:41.750 here at high frequencies. 00:41:44.600 --> 00:41:53.010 This also says, incidentally, that if we look at an ideal 00:41:53.010 --> 00:41:58.610 lowpass filter and an ideal highpass filter, and we choose 00:41:58.610 --> 00:42:02.270 the cutoff frequencies for comparison, or the bandwidth 00:42:02.270 --> 00:42:04.650 of the filter to be equal. 00:42:04.650 --> 00:42:10.190 Since this ideal highpass filter is this ideal lowpass 00:42:10.190 --> 00:42:16.580 filter with the frequency response shifted by pi, the 00:42:16.580 --> 00:42:20.640 modulation property tells us that in the time domain, what 00:42:20.640 --> 00:42:25.520 that corresponds to is an impulse response multiplied by 00:42:25.520 --> 00:42:26.950 minus 1 to the n. 00:42:26.950 --> 00:42:32.500 So it says that the impulse response of the highpass 00:42:32.500 --> 00:42:36.230 filter, or equivalently the inverse Fourier transform of 00:42:36.230 --> 00:42:40.720 the highpass filter frequency response, is minus 1 to the n 00:42:40.720 --> 00:42:43.780 times the impulse response for the lowpass filter. 00:42:43.780 --> 00:42:47.860 That all follows from the modulation property. 00:42:47.860 --> 00:42:51.710 Now there's another way, an interesting and useful way, 00:42:51.710 --> 00:42:57.200 that modulation can be used to implement or convert from 00:42:57.200 --> 00:43:00.710 lowpass filtering to highpass filtering. 00:43:00.710 --> 00:43:04.300 The modulation property tells us about multiplying the time 00:43:04.300 --> 00:43:07.170 domain is shifting in the frequency domain. 00:43:07.170 --> 00:43:09.810 And in the example that we happened to pick said if you 00:43:09.810 --> 00:43:14.120 multiply or modulate by minus 1 to the n, that takes low 00:43:14.120 --> 00:43:17.580 frequencies and shifts them to high frequencies. 00:43:17.580 --> 00:43:23.280 What that tells us, as a practical and useful notion, 00:43:23.280 --> 00:43:24.520 is the following. 00:43:24.520 --> 00:43:28.540 Suppose we have a system that we know is a lowpass filter, 00:43:28.540 --> 00:43:31.410 and it's a good lowpass filter. 00:43:31.410 --> 00:43:34.800 How might we use it as a highpass filter? 00:43:34.800 --> 00:43:38.080 Well, one way to do it, instead of shifting its 00:43:38.080 --> 00:43:43.810 frequency response, is to take the original signal, shift its 00:43:43.810 --> 00:43:46.080 low frequencies to high frequencies and its high 00:43:46.080 --> 00:43:49.500 frequencies to low frequencies by multiplying the input 00:43:49.500 --> 00:43:54.160 signal, the original signal, by minus 1 to the n, process 00:43:54.160 --> 00:43:57.070 that with a lowpass filter where now what's sitting at 00:43:57.070 --> 00:44:00.430 the low frequencies were the high frequencies. 00:44:00.430 --> 00:44:05.320 And then unscramble it all at the output so that we put the 00:44:05.320 --> 00:44:07.760 frequencies back where they belong. 00:44:07.760 --> 00:44:10.060 And I summarize that here. 00:44:10.060 --> 00:44:13.860 Let's suppose, for example, that this system was a lowpass 00:44:13.860 --> 00:44:17.840 filter, and so it lowpass-filters 00:44:17.840 --> 00:44:20.120 whatever comes into it. 00:44:20.120 --> 00:44:24.560 Down below, I indicate taking the input and first 00:44:24.560 --> 00:44:28.830 interchanging the high and low frequencies through modulation 00:44:28.830 --> 00:44:32.460 with minus 1 to the n. 00:44:32.460 --> 00:44:35.220 Doing the lowpass filtering, which-- 00:44:35.220 --> 00:44:37.660 and what's sitting at the low frequencies here were the high 00:44:37.660 --> 00:44:39.990 frequencies of this signal. 00:44:39.990 --> 00:44:42.830 And then after the lowpass filtering, moving the 00:44:42.830 --> 00:44:46.860 frequencies back where they belong by again modulating 00:44:46.860 --> 00:44:48.560 with minus 1 to the n. 00:44:48.560 --> 00:44:56.150 And that, in fact, turns out to be a very useful notion for 00:44:56.150 --> 00:45:01.240 applying a fixed lowpass filter to do highpass 00:45:01.240 --> 00:45:02.550 filtering and vice versa. 00:45:06.740 --> 00:45:15.470 OK, now, what we've seen and what we've talked about are 00:45:15.470 --> 00:45:20.740 the Fourier representation for discrete-time signals, and 00:45:20.740 --> 00:45:24.080 prior to that, continuous-time signals. 00:45:24.080 --> 00:45:27.250 And we've seen some very important similarities and 00:45:27.250 --> 00:45:28.470 differences. 00:45:28.470 --> 00:45:34.660 And what I'd like to do is conclude this lecture by 00:45:34.660 --> 00:45:38.910 summarizing those various relationships kind of all in 00:45:38.910 --> 00:45:43.550 one package, and in fact drawing your attention to both 00:45:43.550 --> 00:45:46.250 the similarities and differences and comparisons 00:45:46.250 --> 00:45:48.930 between them. 00:45:48.930 --> 00:45:54.710 Well, let's begin this summary by first looking at the 00:45:54.710 --> 00:45:57.240 continuous-time Fourier series. 00:45:57.240 --> 00:46:02.280 In the continuous-time Fourier series, we have a periodic 00:46:02.280 --> 00:46:06.940 time function expanded as a linear combination of 00:46:06.940 --> 00:46:10.000 harmonically-related complex exponentials. 00:46:10.000 --> 00:46:12.420 And there are an infinite number of these that are 00:46:12.420 --> 00:46:15.710 required to do the decomposition. 00:46:15.710 --> 00:46:20.230 And we saw an analysis equation which tells us how to 00:46:20.230 --> 00:46:24.380 get these Fourier series coefficients through an 00:46:24.380 --> 00:46:28.330 integration on the original time function. 00:46:28.330 --> 00:46:32.260 And notice in this that what we have is a continuous 00:46:32.260 --> 00:46:34.250 periodic time function. 00:46:34.250 --> 00:46:38.990 What we end up with in the frequency domain is a sequence 00:46:38.990 --> 00:46:42.670 of Fourier series coefficients which in fact is an infinite 00:46:42.670 --> 00:46:46.700 sequence, namely, requires all values of k in general. 00:46:49.380 --> 00:46:53.650 We had then generalized that to the continuous-time Fourier 00:46:53.650 --> 00:46:57.750 transform, and, in effect, in doing that what happened is 00:46:57.750 --> 00:47:07.580 that the synthesis equation in the Fourier series became an 00:47:07.580 --> 00:47:11.830 integral relationship in the Fourier transform. 00:47:11.830 --> 00:47:17.360 And we now have a continuous-time function which 00:47:17.360 --> 00:47:21.290 is no longer periodic, this was for the aperiodic case, 00:47:21.290 --> 00:47:25.080 represented as a linear combination of infinitesimally 00:47:25.080 --> 00:47:29.870 close-in-frequency complex exponentials with complex 00:47:29.870 --> 00:47:35.850 amplitudes given by X of omega d omega divided by 2 pi. 00:47:35.850 --> 00:47:39.210 And we had of course the corresponding analysis 00:47:39.210 --> 00:47:43.360 equation that told us how to get X of omega. 00:47:43.360 --> 00:47:48.060 Here we have a continuous-time function which is aperiodic, 00:47:48.060 --> 00:47:53.485 and a continuous function of frequency which is aperiodic. 00:47:56.990 --> 00:48:01.780 The conceptual strategy in the discrete-time case was very 00:48:01.780 --> 00:48:08.640 similar, with some differences resulting in the relationships 00:48:08.640 --> 00:48:11.920 because of some inherent differences between continuous 00:48:11.920 --> 00:48:15.110 time and discrete time. 00:48:15.110 --> 00:48:19.560 We began with the discrete-time Fourier series, 00:48:19.560 --> 00:48:24.360 corresponding to representing a periodic sequence through a 00:48:24.360 --> 00:48:29.780 set of complex exponentials, where now we only required a 00:48:29.780 --> 00:48:33.990 finite number of these because of the fact that, in fact, 00:48:33.990 --> 00:48:36.830 there are only a finite number of harmonically-related 00:48:36.830 --> 00:48:38.580 complex exponentials. 00:48:38.580 --> 00:48:41.710 That's an inherent property of discrete-time complex 00:48:41.710 --> 00:48:43.070 exponentials. 00:48:43.070 --> 00:48:49.640 And so we have a discrete, periodic time function. 00:48:49.640 --> 00:48:53.270 And we ended up with a set of Fourier series coefficients, 00:48:53.270 --> 00:48:56.670 which of course are discrete, as Fourier series coefficients 00:48:56.670 --> 00:49:01.960 are, and which periodically repeat because of the fact 00:49:01.960 --> 00:49:04.290 that the associated complex exponentials 00:49:04.290 --> 00:49:05.540 periodically repeat. 00:49:07.960 --> 00:49:11.030 We then used an argument similar to the continuous-time 00:49:11.030 --> 00:49:14.840 case for going from periodic time functions to aperiodic 00:49:14.840 --> 00:49:16.290 time functions. 00:49:16.290 --> 00:49:20.190 And we ended up with a relationship describing a 00:49:20.190 --> 00:49:25.710 representation for aperiodic discrete-time signals in which 00:49:25.710 --> 00:49:30.460 now the synthesis equation went from a summation to an 00:49:30.460 --> 00:49:32.620 integration, since the frequencies are now 00:49:32.620 --> 00:49:36.770 infinitesimally close, involving frequencies only 00:49:36.770 --> 00:49:40.500 over a 2 pi interval, and for which the 00:49:40.500 --> 00:49:42.800 amplitude factor X of omega-- 00:49:42.800 --> 00:49:45.140 well, the amplitude factor is X of omega d omega 00:49:45.140 --> 00:49:47.190 divided by 2 pi. 00:49:47.190 --> 00:49:51.060 And this term, X of omega, which is the Fourier 00:49:51.060 --> 00:49:57.120 transform, is given by this summation, and of course 00:49:57.120 --> 00:50:01.500 involves all of the values of x of n. 00:50:01.500 --> 00:50:05.420 And so the important difference between the 00:50:05.420 --> 00:50:08.390 continuous-time and discrete-time case kind of 00:50:08.390 --> 00:50:11.460 arose, in part, out of the fact that discrete time is 00:50:11.460 --> 00:50:14.830 discrete time, continuous time is continuous time, and the 00:50:14.830 --> 00:50:18.770 fact that complex exponentials are periodic in discrete time. 00:50:18.770 --> 00:50:21.910 The harmonically-related ones periodically repeat whereas 00:50:21.910 --> 00:50:25.470 they don't in continuous time. 00:50:25.470 --> 00:50:28.830 Now this, among other things, has an important consequence 00:50:28.830 --> 00:50:30.270 for duality. 00:50:30.270 --> 00:50:34.540 And let's go back again and look at this equation, this 00:50:34.540 --> 00:50:35.660 pair of equations. 00:50:35.660 --> 00:50:38.800 And clearly there is no duality 00:50:38.800 --> 00:50:41.040 between these two equations. 00:50:41.040 --> 00:50:44.230 This involves a summation, this involves an integration. 00:50:44.230 --> 00:50:50.810 And so, in fact, if we make reference to duality, there 00:50:50.810 --> 00:50:52.810 isn't duality in the 00:50:52.810 --> 00:50:55.900 continuous-time Fourier series. 00:50:55.900 --> 00:50:58.910 However, for the continuous-time Fourier 00:50:58.910 --> 00:51:03.910 transform, we're talking about aperiodic time functions and 00:51:03.910 --> 00:51:06.310 aperiodic frequency functions. 00:51:06.310 --> 00:51:09.350 And, in fact, when we look at these two equations, we see 00:51:09.350 --> 00:51:11.550 very definitely a duality. 00:51:11.550 --> 00:51:15.530 In other words, the time function effectively is the 00:51:15.530 --> 00:51:18.340 Fourier transform of the Fourier transform. 00:51:18.340 --> 00:51:20.660 There's a little time reversal in there, but basically that's 00:51:20.660 --> 00:51:21.780 the result. 00:51:21.780 --> 00:51:26.890 And, in fact, we had exploited that duality property when we 00:51:26.890 --> 00:51:27.620 talked about the 00:51:27.620 --> 00:51:32.020 continuous-time Fourier transform. 00:51:32.020 --> 00:51:40.370 With the discrete-time Fourier series, we have a duality 00:51:40.370 --> 00:51:44.590 indicated by the fact that we have a periodic time function 00:51:44.590 --> 00:51:48.910 and a sequence which is periodic in 00:51:48.910 --> 00:51:50.170 the frequency domain. 00:51:50.170 --> 00:51:53.150 And in fact, if you look at these two expressions, you see 00:51:53.150 --> 00:51:55.200 the duality very clearly. 00:51:55.200 --> 00:51:58.630 And so it's the discrete-time Fourier 00:51:58.630 --> 00:52:02.630 series that has a duality. 00:52:02.630 --> 00:52:07.080 And finally the discrete-time Fourier transform loses the 00:52:07.080 --> 00:52:11.150 duality because of the fact, among other things, that in 00:52:11.150 --> 00:52:15.040 the time domain things are inherently discrete whereas in 00:52:15.040 --> 00:52:18.580 the frequency domain they're inherently continuous. 00:52:18.580 --> 00:52:21.395 So, in fact, here there is no duality. 00:52:25.690 --> 00:52:30.280 OK, now that says that there's a difference in the duality, 00:52:30.280 --> 00:52:32.510 continuous time and discrete time. 00:52:32.510 --> 00:52:37.780 And there's one more very important piece to the duality 00:52:37.780 --> 00:52:39.610 relationships. 00:52:39.610 --> 00:52:44.400 And we can see that first algebraically by comparing the 00:52:44.400 --> 00:52:49.280 continuous-time Fourier series and the 00:52:49.280 --> 00:52:51.435 discrete-time Fourier transform. 00:52:54.240 --> 00:52:59.210 The continuous-time Fourier series in the time domain is a 00:52:59.210 --> 00:53:03.580 periodic continuous function, in the frequency domain is an 00:53:03.580 --> 00:53:06.930 aperiodic sequence. 00:53:06.930 --> 00:53:13.680 In the discrete-time case, in the time domain we have an 00:53:13.680 --> 00:53:19.850 aperiodic sequence, and in the frequency domain we have a 00:53:19.850 --> 00:53:21.990 function of a continuous variable 00:53:21.990 --> 00:53:24.480 which we know is periodic. 00:53:24.480 --> 00:53:28.100 And so in fact we have, in the time domain 00:53:28.100 --> 00:53:30.370 here, aperiodic sequence. 00:53:30.370 --> 00:53:32.300 In the frequency domain we have a 00:53:32.300 --> 00:53:34.990 continuous periodic function. 00:53:34.990 --> 00:53:38.600 And in fact, if you look at the relationship between these 00:53:38.600 --> 00:53:48.390 two, then what we see in fact is a duality between the 00:53:48.390 --> 00:53:53.370 continuous-time Fourier series and the 00:53:53.370 --> 00:53:56.500 discrete-time Fourier transform. 00:53:56.500 --> 00:54:00.170 One way of thinking of that is to kind of think, and this is 00:54:00.170 --> 00:54:02.820 a little bit of a tongue twister which you might want 00:54:02.820 --> 00:54:08.240 to get straightened out slowly, but the Fourier 00:54:08.240 --> 00:54:10.000 transform in discrete time is a 00:54:10.000 --> 00:54:12.270 periodic function of frequency. 00:54:12.270 --> 00:54:17.190 That periodic function has a Fourier series representation. 00:54:17.190 --> 00:54:19.540 What is this Fourier series? 00:54:19.540 --> 00:54:21.170 What are the Fourier series coefficients of 00:54:21.170 --> 00:54:22.730 that periodic function? 00:54:22.730 --> 00:54:26.030 Well in fact, except for an issue of time reversal, what 00:54:26.030 --> 00:54:29.920 it is the original sequence for which 00:54:29.920 --> 00:54:31.670 that's the Fourier transform. 00:54:31.670 --> 00:54:35.610 And that is the duality that I'm trying to emphasize here. 00:54:38.600 --> 00:54:42.910 OK, well, so what we see is that these four sets of 00:54:42.910 --> 00:54:47.110 relationships all tie together in a whole variety of ways. 00:54:47.110 --> 00:54:51.070 And we will be exploiting as the discussion goes on the 00:54:51.070 --> 00:54:52.690 inner-connections and relationships 00:54:52.690 --> 00:54:55.000 that I've talked about. 00:54:55.000 --> 00:54:58.730 Also, as we've talked about the Fourier transform, both 00:54:58.730 --> 00:55:02.750 continuous time and discrete time, two important properties 00:55:02.750 --> 00:55:06.570 that we focused on, among many of the properties, are the 00:55:06.570 --> 00:55:09.950 convolution property and the modulation property. 00:55:09.950 --> 00:55:14.670 We've also shown that the convolution property leads to 00:55:14.670 --> 00:55:19.020 a very important concept, namely filtering. 00:55:19.020 --> 00:55:22.700 The modulation property leads to an important concept, 00:55:22.700 --> 00:55:24.380 namely modulation. 00:55:24.380 --> 00:55:31.200 We've also very briefly indicated how these properties 00:55:31.200 --> 00:55:35.000 and how these concepts have practical implications. 00:55:35.000 --> 00:55:37.700 In the next several lectures, we'll focus in more 00:55:37.700 --> 00:55:41.100 specifically first on filtering, and then on 00:55:41.100 --> 00:55:42.570 modulation. 00:55:42.570 --> 00:55:49.000 And as we'll see the filtering and modulation concepts form 00:55:49.000 --> 00:55:51.750 really the cornerstone of many, many 00:55:51.750 --> 00:55:53.530 signal processing ideas. 00:55:53.530 --> 00:55:54.780 Thank you.