1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,540 ocw.mit.edu. 8 00:00:35,720 --> 00:00:55,260 [MUSIC PLAYING] 9 00:00:55,260 --> 00:00:58,560 PROFESSOR: Last time we began the development of the 10 00:00:58,560 --> 00:01:01,430 discrete-time Fourier transform. 11 00:01:01,430 --> 00:01:06,350 And just as with the continuous-time case, we first 12 00:01:06,350 --> 00:01:08,710 treated the notion of periodic signals. 13 00:01:08,710 --> 00:01:11,060 This led to the Fourier series. 14 00:01:11,060 --> 00:01:15,350 And then we generalized that to the Fourier transform, and 15 00:01:15,350 --> 00:01:18,080 finally incorporated within the framework of the Fourier 16 00:01:18,080 --> 00:01:23,070 transform both aperiodic and periodic signals. 17 00:01:23,070 --> 00:01:27,820 In today's lecture, what I'd like to do is expand on some 18 00:01:27,820 --> 00:01:30,940 of the properties of the Fourier transform, and 19 00:01:30,940 --> 00:01:32,875 indicate how those properties are used for 20 00:01:32,875 --> 00:01:34,770 a variety of things. 21 00:01:34,770 --> 00:01:39,470 Well, let's begin by reviewing the Fourier transform as we 22 00:01:39,470 --> 00:01:41,310 developed it last time. 23 00:01:41,310 --> 00:01:45,410 It, of course, involves a synthesis equation and an 24 00:01:45,410 --> 00:01:47,040 analysis equation. 25 00:01:47,040 --> 00:01:51,340 The synthesis equation expressing x of n, the 26 00:01:51,340 --> 00:01:55,090 sequence, in terms of the Fourier transform, and the 27 00:01:55,090 --> 00:01:58,880 analysis equation telling us how to obtain the Fourier 28 00:01:58,880 --> 00:02:02,890 transform from the original sequence. 29 00:02:02,890 --> 00:02:07,480 And I draw your attention again to the basic point that 30 00:02:07,480 --> 00:02:11,650 the synthesis equation essentially corresponds to 31 00:02:11,650 --> 00:02:17,660 decomposing the sequence as a linear combination of complex 32 00:02:17,660 --> 00:02:21,590 exponentials with amplitudes that are, in effect, 33 00:02:21,590 --> 00:02:25,820 proportional to the Fourier transform. 34 00:02:25,820 --> 00:02:30,290 Now, the discrete-time Fourier transform, just as the 35 00:02:30,290 --> 00:02:34,380 continuous-time Fourier transform, has a number of 36 00:02:34,380 --> 00:02:37,050 important and useful properties. 37 00:02:37,050 --> 00:02:41,600 Of course, as I stressed last time, it's a function of a 38 00:02:41,600 --> 00:02:44,140 continuous variable. 39 00:02:44,140 --> 00:02:48,870 And it's also a complex-valued function, which means that 40 00:02:48,870 --> 00:02:51,680 when we represent it in general it requires a 41 00:02:51,680 --> 00:02:56,860 representation in terms of its real part and imaginary part, 42 00:02:56,860 --> 00:03:00,540 or in terms of magnitude and angle. 43 00:03:00,540 --> 00:03:06,360 Also, as I indicated last time, the Fourier transform is 44 00:03:06,360 --> 00:03:12,600 a periodic function of frequency, and the periodicity 45 00:03:12,600 --> 00:03:16,030 is with a period of 2 pi. 46 00:03:16,030 --> 00:03:20,750 And so it says, in effect, that the Fourier transform, if 47 00:03:20,750 --> 00:03:24,110 we replace the frequency variable by an integer 48 00:03:24,110 --> 00:03:28,580 multiple of 2 pi, the function repeats. 49 00:03:28,580 --> 00:03:34,680 And I stress again that the underlying basis for this 50 00:03:34,680 --> 00:03:39,080 periodicity property is the fact that it's the set of 51 00:03:39,080 --> 00:03:41,560 complex exponentials that are 52 00:03:41,560 --> 00:03:44,190 inherently periodic in frequency. 53 00:03:44,190 --> 00:03:47,040 And so, of course, any representation using them 54 00:03:47,040 --> 00:03:52,660 would, in effect, generate a periodicity with 55 00:03:52,660 --> 00:03:53,950 this period of 2 pi. 56 00:03:56,780 --> 00:04:01,320 Just as in continuous time, the Fourier transform has 57 00:04:01,320 --> 00:04:03,560 important symmetry properties. 58 00:04:03,560 --> 00:04:08,060 And in particular, if the sequence x sub n is 59 00:04:08,060 --> 00:04:12,130 real-valued, then the Fourier transform 60 00:04:12,130 --> 00:04:14,680 is conjugate symmetric. 61 00:04:14,680 --> 00:04:19,420 In other words, if we replace omega by minus omega, that's 62 00:04:19,420 --> 00:04:22,930 equivalent to applying complex conjugation 63 00:04:22,930 --> 00:04:25,000 to the Fourier transform. 64 00:04:25,000 --> 00:04:30,780 And as a consequence of this conjugate symmetry, this 65 00:04:30,780 --> 00:04:35,880 results in a symmetry in the real part that is an even 66 00:04:35,880 --> 00:04:41,430 symmetry, or the magnitude has an even symmetry, whereas the 67 00:04:41,430 --> 00:04:47,590 imaginary part or the phase angle are both odd symmetric. 68 00:04:47,590 --> 00:04:50,850 And these are symmetry properties, again, that are 69 00:04:50,850 --> 00:04:53,480 identical to the symmetry properties that we saw in 70 00:04:53,480 --> 00:04:55,230 continuous time. 71 00:04:55,230 --> 00:04:58,820 Well, let's see this in the context of an example that we 72 00:04:58,820 --> 00:05:01,940 worked last time and that we'll want to draw attention 73 00:05:01,940 --> 00:05:04,960 to in reference to several issues as this 74 00:05:04,960 --> 00:05:06,730 lecture goes along. 75 00:05:06,730 --> 00:05:11,890 And that is the Fourier transform of a real damped 76 00:05:11,890 --> 00:05:13,330 exponential. 77 00:05:13,330 --> 00:05:19,330 So the sequence that we are talking about is a to the n u 78 00:05:19,330 --> 00:05:23,590 of n, and let's consider a to be positive. 79 00:05:23,590 --> 00:05:27,060 We saw last time that the Fourier transform for this 80 00:05:27,060 --> 00:05:31,490 sequence algebraically is of this form. 81 00:05:31,490 --> 00:05:35,860 And if we look at its magnitude and angle, the 82 00:05:35,860 --> 00:05:40,110 magnitude I show here. 83 00:05:40,110 --> 00:05:43,120 And the magnitude, as we see, has the 84 00:05:43,120 --> 00:05:45,130 properties that we indicated. 85 00:05:45,130 --> 00:05:48,400 It is an even function of frequency. 86 00:05:48,400 --> 00:05:52,150 Of course, it's a function of a continuous variable. 87 00:05:52,150 --> 00:05:55,400 And it, in addition, is periodic with a 88 00:05:55,400 --> 00:05:58,300 period of two pi. 89 00:05:58,300 --> 00:06:02,050 On the other hand, if we look at the phase angle below it, 90 00:06:02,050 --> 00:06:06,310 the phase angle has a symmetry which is odd symmetric. 91 00:06:06,310 --> 00:06:09,640 And that's indicated clearly in this picture. 92 00:06:09,640 --> 00:06:12,100 And of course, in addition to being odd symmetric, it 93 00:06:12,100 --> 00:06:16,700 naturally has to be, again, a periodic function of frequency 94 00:06:16,700 --> 00:06:20,080 with a period of 2 pi. 95 00:06:20,080 --> 00:06:22,630 OK, so we have some symmetry properties. 96 00:06:22,630 --> 00:06:25,220 We have this inherent periodicity in the Fourier 97 00:06:25,220 --> 00:06:31,260 transform, which I'm stressing very heavily because it forms 98 00:06:31,260 --> 00:06:34,060 the basic difference between continuous time 99 00:06:34,060 --> 00:06:36,170 and discrete time. 100 00:06:36,170 --> 00:06:39,420 In addition to these properties of the Fourier 101 00:06:39,420 --> 00:06:44,940 transform, there are a number of other properties that are 102 00:06:44,940 --> 00:06:48,820 particularly useful in the manipulation of the Fourier 103 00:06:48,820 --> 00:06:53,050 transform, and, in fact, in using the Fourier transform 104 00:06:53,050 --> 00:06:57,550 to, for example, analyze systems represented by linear 105 00:06:57,550 --> 00:07:00,660 constant coefficient difference equations. 106 00:07:00,660 --> 00:07:05,410 There in the text is a longer list of properties, but let me 107 00:07:05,410 --> 00:07:09,490 just draw your attention to several of them. 108 00:07:09,490 --> 00:07:13,210 One is the time shifting property. 109 00:07:13,210 --> 00:07:19,620 And the time shifting property tells us that if x of omega is 110 00:07:19,620 --> 00:07:24,870 the Fourier transform of x of n, then the Fourier transform 111 00:07:24,870 --> 00:07:30,090 of x of n shifted in time is that same Fourier transform 112 00:07:30,090 --> 00:07:34,730 multiplied by this factor, which is a 113 00:07:34,730 --> 00:07:36,690 linear phase factor. 114 00:07:36,690 --> 00:07:41,670 So time shifting introduces a linear phase term. 115 00:07:41,670 --> 00:07:46,410 And, by the way, recall that in the continuous-time case we 116 00:07:46,410 --> 00:07:49,530 had a similar situation, namely that a time shift 117 00:07:49,530 --> 00:07:53,050 corresponded to a linear phase. 118 00:07:53,050 --> 00:07:57,260 There also is a dual to the time shifting property, which 119 00:07:57,260 --> 00:08:01,000 is referred to as the frequency shifting property, 120 00:08:01,000 --> 00:08:06,520 which tells us that if we multiply a time function by a 121 00:08:06,520 --> 00:08:09,600 complex exponential, that, in effect, 122 00:08:09,600 --> 00:08:13,030 generates a frequency shift. 123 00:08:13,030 --> 00:08:17,000 And we'll see this frequency shifting property surface in a 124 00:08:17,000 --> 00:08:20,310 slightly different way shortly, when we talk about 125 00:08:20,310 --> 00:08:24,840 the modulation property in the discrete-time case. 126 00:08:24,840 --> 00:08:28,080 Another important property that we'll want to make use of 127 00:08:28,080 --> 00:08:31,390 shortly is linearity, which follows in a very 128 00:08:31,390 --> 00:08:36,860 straightforward way from the Fourier transform definition. 129 00:08:36,860 --> 00:08:42,220 And the linearity property says simply that the Fourier 130 00:08:42,220 --> 00:08:46,510 transform of a sum, or linear combination, is the same 131 00:08:46,510 --> 00:08:49,020 linear combination of the Fourier transforms. 132 00:08:49,020 --> 00:08:52,750 Again, that's a property that we saw in continuous time. 133 00:08:52,750 --> 00:08:58,570 And, also, among other properties there is a 134 00:08:58,570 --> 00:09:01,790 Parseval's relation for the discrete-time case that in 135 00:09:01,790 --> 00:09:05,110 effect says something similar to continuous time, 136 00:09:05,110 --> 00:09:10,750 specifically that the energy in the sequence is 137 00:09:10,750 --> 00:09:15,340 proportional to the energy in the Fourier transform, the 138 00:09:15,340 --> 00:09:17,370 energy over one period. 139 00:09:17,370 --> 00:09:20,820 Or, said another way, in fact, or another way that it can be 140 00:09:20,820 --> 00:09:23,760 said, is that the energy in the time domain is 141 00:09:23,760 --> 00:09:26,070 proportional to the power in this 142 00:09:26,070 --> 00:09:30,060 periodic Fourier transform. 143 00:09:30,060 --> 00:09:33,510 OK, so these are some of the properties. 144 00:09:33,510 --> 00:09:37,750 And, as I indicated, parallel somewhat properties that we 145 00:09:37,750 --> 00:09:40,140 saw in continuous time. 146 00:09:40,140 --> 00:09:44,730 Two additional properties that will play important roles in 147 00:09:44,730 --> 00:09:48,530 discrete time just as they did in continuous time are the 148 00:09:48,530 --> 00:09:53,730 convolution property and the modulation property. 149 00:09:53,730 --> 00:09:57,760 The convolution property is the property that tells us how 150 00:09:57,760 --> 00:10:03,100 to relate the Fourier transform of the convolution 151 00:10:03,100 --> 00:10:06,990 of two sequences to the Fourier transforms of the 152 00:10:06,990 --> 00:10:08,940 individual sequences. 153 00:10:08,940 --> 00:10:12,240 And, not surprisingly, what happens-- 154 00:10:12,240 --> 00:10:15,530 and this can be demonstrated algebraically-- 155 00:10:15,530 --> 00:10:20,390 not surprisingly, the Fourier transform of the convolution 156 00:10:20,390 --> 00:10:24,960 is simply the product of the Fourier transforms. 157 00:10:24,960 --> 00:10:30,460 So, Fourier transform maps convolution in the time domain 158 00:10:30,460 --> 00:10:33,650 to multiplication in the frequency domain. 159 00:10:33,650 --> 00:10:37,300 Now convolution, of course, arises in the context of 160 00:10:37,300 --> 00:10:39,610 linear time-invariant systems. 161 00:10:39,610 --> 00:10:42,120 In particular, if we have a system with an impulse 162 00:10:42,120 --> 00:10:47,710 response h of n, input x of n, the output is the convolution. 163 00:10:47,710 --> 00:10:51,860 The convolution property then tells us that in the frequency 164 00:10:51,860 --> 00:10:57,780 domain, the Fourier transform is the product of the Fourier 165 00:10:57,780 --> 00:11:01,420 transform of the impulse response and the Fourier 166 00:11:01,420 --> 00:11:05,120 transform of the input. 167 00:11:05,120 --> 00:11:10,560 Now we also saw and have talked about a relationship 168 00:11:10,560 --> 00:11:14,950 between the Fourier transform, the impulse response, and what 169 00:11:14,950 --> 00:11:22,250 we call the frequency response in the context of the response 170 00:11:22,250 --> 00:11:25,840 of a system to a complex exponential. 171 00:11:25,840 --> 00:11:29,030 Specifically, complex exponentials are 172 00:11:29,030 --> 00:11:33,210 eigenfunctions of linear time-invariant systems. 173 00:11:33,210 --> 00:11:37,810 One of these into the system gives us, as an output, a 174 00:11:37,810 --> 00:11:42,070 complex exponential with the same complex frequency 175 00:11:42,070 --> 00:11:45,920 multiplied by what we refer to as the eigenvalue. 176 00:11:45,920 --> 00:11:52,030 And as you saw in the video course manual, this 177 00:11:52,030 --> 00:11:56,010 eigenvalue, this constant, multiplier on the exponential 178 00:11:56,010 --> 00:12:00,530 is, in fact, the Fourier transform of the impulse 179 00:12:00,530 --> 00:12:05,520 response evaluated at that frequency. 180 00:12:05,520 --> 00:12:10,700 Now, we saw exactly the same statement in continuous time. 181 00:12:10,700 --> 00:12:13,280 And, in fact, we used that statement-- 182 00:12:13,280 --> 00:12:17,860 the frequency response interpretation of the Fourier 183 00:12:17,860 --> 00:12:20,040 transform, the impulse response-- 184 00:12:20,040 --> 00:12:24,710 we use that to motivate an intuitive interpretation of 185 00:12:24,710 --> 00:12:26,550 the convolution property. 186 00:12:26,550 --> 00:12:30,460 Now, formally the convolution property can be developed by 187 00:12:30,460 --> 00:12:33,840 taking the convolution sum, applying the Fourier transform 188 00:12:33,840 --> 00:12:38,240 sum to it, doing the appropriate substitution of 189 00:12:38,240 --> 00:12:40,810 variables, interchanging order of summations, et cetera, and 190 00:12:40,810 --> 00:12:44,800 all the algebra works out to show that it's a product. 191 00:12:44,800 --> 00:12:48,610 But as I stressed when we discussed this with continuous 192 00:12:48,610 --> 00:12:50,550 time, the interpretation-- 193 00:12:50,550 --> 00:12:52,160 the underlying interpretation-- 194 00:12:52,160 --> 00:12:55,040 is particularly important to understand. 195 00:12:55,040 --> 00:12:58,160 So let me review it again in the discrete-time case, and 196 00:12:58,160 --> 00:13:01,080 it's exactly the same for discrete time or for 197 00:13:01,080 --> 00:13:03,330 continuous time. 198 00:13:03,330 --> 00:13:10,100 Specifically, the argument was that the Fourier transform of 199 00:13:10,100 --> 00:13:14,760 a sequence or signal corresponds to decomposing it 200 00:13:14,760 --> 00:13:18,330 into a linear combination of complex exponentials. 201 00:13:18,330 --> 00:13:21,310 What's the amplitude of those complex exponentials? 202 00:13:21,310 --> 00:13:25,990 It's basically proportional to the Fourier transform. 203 00:13:25,990 --> 00:13:29,290 If we think of pushing through the system that linear 204 00:13:29,290 --> 00:13:35,220 combination, then each of those complex exponentials 205 00:13:35,220 --> 00:13:40,540 gets the amplitude modified, or multiplied, by the Fourier 206 00:13:40,540 --> 00:13:42,500 transform of-- 207 00:13:42,500 --> 00:13:44,110 by the frequency response-- 208 00:13:44,110 --> 00:13:46,670 which we saw is the Fourier transform 209 00:13:46,670 --> 00:13:49,130 of the impulse response. 210 00:13:49,130 --> 00:13:54,170 So the amplitudes of the output complex exponentials is 211 00:13:54,170 --> 00:13:58,240 then the amplitudes of the input complex exponentials 212 00:13:58,240 --> 00:14:01,470 multiplied by the frequency response. 213 00:14:01,470 --> 00:14:06,240 And the Fourier transform of the output, in effect, is an 214 00:14:06,240 --> 00:14:11,450 expression expressing the summation, or integration, of 215 00:14:11,450 --> 00:14:14,790 the output as a linear combination of all of these 216 00:14:14,790 --> 00:14:17,400 exponentials with the appropriate complex 217 00:14:17,400 --> 00:14:19,580 amplitudes. 218 00:14:19,580 --> 00:14:23,720 So, it's important, in thinking about the convolution 219 00:14:23,720 --> 00:14:29,880 property, to think about it in terms of nothing more than the 220 00:14:29,880 --> 00:14:34,740 fact that we've decomposed the input, and we're now modifying 221 00:14:34,740 --> 00:14:38,410 separately through multiplication, through 222 00:14:38,410 --> 00:14:43,010 scaling, the amplitudes of each of the complex 223 00:14:43,010 --> 00:14:44,260 exponential components. 224 00:14:47,230 --> 00:14:50,780 Now what we saw in continuous time is that this 225 00:14:50,780 --> 00:14:56,390 interpretation and the convolution property led to an 226 00:14:56,390 --> 00:15:00,640 important concept, namely the concept of filtering. 227 00:15:00,640 --> 00:15:04,140 Kind of the idea that if we decompose the input as a 228 00:15:04,140 --> 00:15:08,990 linear combination of complex exponentials, we can 229 00:15:08,990 --> 00:15:12,320 separately attenuate or amplify each of those 230 00:15:12,320 --> 00:15:13,910 components. 231 00:15:13,910 --> 00:15:19,190 And, in fact, we could exactly pass some set of frequencies 232 00:15:19,190 --> 00:15:23,160 and totally eliminate other set of frequencies. 233 00:15:23,160 --> 00:15:29,430 So, again, just as in continuous time, we can talk 234 00:15:29,430 --> 00:15:32,130 about an ideal filter. 235 00:15:32,130 --> 00:15:37,820 And what I show here is the frequency response of an ideal 236 00:15:37,820 --> 00:15:39,990 lowpass filter. 237 00:15:39,990 --> 00:15:45,050 The ideal lowpass filter, of course, passes exactly, with a 238 00:15:45,050 --> 00:15:51,800 gain of 1, frequencies around 0, and eliminates totally 239 00:15:51,800 --> 00:15:53,050 other frequencies. 240 00:15:55,030 --> 00:15:58,480 However, an important distinction here between 241 00:15:58,480 --> 00:16:02,530 continuous time and discrete time is the fact that, whereas 242 00:16:02,530 --> 00:16:06,540 in continuous time when we talked about an ideal filter, 243 00:16:06,540 --> 00:16:09,110 we passed a band of frequencies and totally 244 00:16:09,110 --> 00:16:12,070 eliminated everything else out to infinity. 245 00:16:12,070 --> 00:16:15,070 In the discrete time case, the 246 00:16:15,070 --> 00:16:18,210 frequency response is periodic. 247 00:16:18,210 --> 00:16:22,400 So, obviously, the frequency response must periodically 248 00:16:22,400 --> 00:16:24,430 repeat for the lowpass filter. 249 00:16:24,430 --> 00:16:26,700 And in fact we see that here. 250 00:16:26,700 --> 00:16:31,570 If we look at the lowpass filter, then we've eliminated 251 00:16:31,570 --> 00:16:33,240 some frequencies. 252 00:16:33,240 --> 00:16:39,260 But then we pass, of course, frequencies around 2 pi, and 253 00:16:39,260 --> 00:16:43,160 also frequencies around minus 2 pi, and for that matter 254 00:16:43,160 --> 00:16:46,030 around any multiple of 2 pi. 255 00:16:46,030 --> 00:16:49,770 Although it's important to recognize that because of the 256 00:16:49,770 --> 00:16:53,310 inherent periodicity of the complex exponentials, these 257 00:16:53,310 --> 00:16:58,590 frequencies are exactly the same frequencies as these 258 00:16:58,590 --> 00:16:59,670 frequencies. 259 00:16:59,670 --> 00:17:03,720 So it's lowpass filtering interpreted in terms of 260 00:17:03,720 --> 00:17:09,180 frequencies over a range from minus pi to pi. 261 00:17:09,180 --> 00:17:12,670 Well, just as we talk about a lowpass filter, we can also 262 00:17:12,670 --> 00:17:15,609 talk about a highpass filter. 263 00:17:15,609 --> 00:17:19,500 And a highpass filter, of course, would pass high 264 00:17:19,500 --> 00:17:21,790 frequencies. 265 00:17:21,790 --> 00:17:24,460 In a continuous-time case, high frequencies meant 266 00:17:24,460 --> 00:17:28,050 frequencies that go out to infinity. 267 00:17:28,050 --> 00:17:30,720 In the discrete-time case, of course, the highest 268 00:17:30,720 --> 00:17:34,800 frequencies we can generate are frequencies up to pi. 269 00:17:34,800 --> 00:17:39,760 And once our complex exponentials go past pi, then, 270 00:17:39,760 --> 00:17:42,570 in fact, we start seeing the lower frequencies again. 271 00:17:42,570 --> 00:17:45,400 Let me indicate what I mean. 272 00:17:45,400 --> 00:17:49,560 If we think in the context of the lowpass filter, these are 273 00:17:49,560 --> 00:17:51,300 low frequencies. 274 00:17:51,300 --> 00:17:54,030 As we move along the frequency axis, these become high 275 00:17:54,030 --> 00:17:55,350 frequencies. 276 00:17:55,350 --> 00:17:59,490 And as we move further along the frequency axis, what we'll 277 00:17:59,490 --> 00:18:03,330 see when we get to, for example, a frequency of 2 pi 278 00:18:03,330 --> 00:18:09,030 are the same low frequencies that we see around 0. 279 00:18:09,030 --> 00:18:12,900 In particular then, an ideal highpass filter in the 280 00:18:12,900 --> 00:18:16,790 discrete-time case would be a filter that eliminates these 281 00:18:16,790 --> 00:18:20,660 frequencies and passes frequencies around pi. 282 00:18:23,620 --> 00:18:28,860 OK, so we've seen the convolution property and its 283 00:18:28,860 --> 00:18:31,760 interpretation in terms of filtering. 284 00:18:31,760 --> 00:18:36,040 More broadly, the convolution property in combination with a 285 00:18:36,040 --> 00:18:38,470 number of the other properties that I introduced, in 286 00:18:38,470 --> 00:18:42,040 particular the time shifting and linearity property, allows 287 00:18:42,040 --> 00:18:49,290 us to generate or analyze systems that are described by 288 00:18:49,290 --> 00:18:52,100 linear constant coefficient difference equations. 289 00:18:52,100 --> 00:18:56,330 And this, again, parallels very strongly the discussion 290 00:18:56,330 --> 00:19:00,650 we carried out in the continuous-time case. 291 00:19:00,650 --> 00:19:08,510 In particular, let's think of a discrete-time system that is 292 00:19:08,510 --> 00:19:09,900 described by a linear constant 293 00:19:09,900 --> 00:19:11,910 coefficient difference equation. 294 00:19:11,910 --> 00:19:15,470 And we'll restrict the initial conditions on the equation 295 00:19:15,470 --> 00:19:19,340 such that it corresponds to a linear time-invariant system. 296 00:19:19,340 --> 00:19:23,360 And recall that, in fact, in our discussion of linear 297 00:19:23,360 --> 00:19:26,810 constant coefficient difference equations, it is 298 00:19:26,810 --> 00:19:31,230 the condition of initial rest that-- on the equation-- 299 00:19:31,230 --> 00:19:35,240 that guarantees for us that the system will be causal, 300 00:19:35,240 --> 00:19:38,720 linear, and time invariant. 301 00:19:38,720 --> 00:19:41,790 OK, now let's consider a first-order difference 302 00:19:41,790 --> 00:19:43,340 equation, a system described by a 303 00:19:43,340 --> 00:19:44,740 first-order difference equation. 304 00:19:44,740 --> 00:19:46,260 And we've talked about the solution of 305 00:19:46,260 --> 00:19:48,160 this equation before. 306 00:19:48,160 --> 00:19:51,140 Essentially, we run the solution recursively. 307 00:19:51,140 --> 00:19:55,370 Let's now consider generating the solution by taking 308 00:19:55,370 --> 00:19:59,270 advantage of the properties of the Fourier transform. 309 00:19:59,270 --> 00:20:02,360 Well, just as we did in continuous time, we can 310 00:20:02,360 --> 00:20:05,870 consider applying the Fourier transform to both sides of 311 00:20:05,870 --> 00:20:07,210 this equation. 312 00:20:07,210 --> 00:20:10,520 And the Fourier transform of y of n, of 313 00:20:10,520 --> 00:20:12,680 course, is Y of omega. 314 00:20:12,680 --> 00:20:16,090 And then using the shifting property, the time shifting 315 00:20:16,090 --> 00:20:21,270 property, the Fourier transform of y of n minus 1 is 316 00:20:21,270 --> 00:20:25,520 Y of omega multiplied by e to the minus j omega. 317 00:20:25,520 --> 00:20:30,180 And so we have this, using a linearity property we can 318 00:20:30,180 --> 00:20:32,840 carry down the scale factor, and add these two together as 319 00:20:32,840 --> 00:20:34,290 they're added here. 320 00:20:34,290 --> 00:20:39,020 And the Fourier transform of x of n is X of omega. 321 00:20:39,020 --> 00:20:43,150 Well, we can solve this equation for the Fourier 322 00:20:43,150 --> 00:20:47,630 transform of the output in terms of the Fourier transform 323 00:20:47,630 --> 00:20:51,650 of the input and an appropriate 324 00:20:51,650 --> 00:20:53,220 complex scale factor. 325 00:20:53,220 --> 00:20:58,170 And simply solving this for Y of omega yields 326 00:20:58,170 --> 00:21:00,800 what we have here. 327 00:21:00,800 --> 00:21:04,740 Now what we've used in going from this point to this point 328 00:21:04,740 --> 00:21:11,750 is both the shifting property and we've also used the 329 00:21:11,750 --> 00:21:13,000 linearity property. 330 00:21:15,860 --> 00:21:19,940 At this point, we can recognize that here the 331 00:21:19,940 --> 00:21:22,810 Fourier transform of the output is the product of the 332 00:21:22,810 --> 00:21:28,860 Fourier transform of the input and some complex function. 333 00:21:28,860 --> 00:21:33,300 And from the convolution property, then, that complex 334 00:21:33,300 --> 00:21:38,480 function must in fact correspond to the frequency 335 00:21:38,480 --> 00:21:43,440 response, or equivalently, the Fourier transform of the 336 00:21:43,440 --> 00:21:45,620 impulse response. 337 00:21:45,620 --> 00:21:50,460 So if we want to determine the Fourier transform of the-- 338 00:21:50,460 --> 00:21:55,170 or the impulse response of the system, let's say for example, 339 00:21:55,170 --> 00:21:59,980 then it becomes a matter of having identified the Fourier 340 00:21:59,980 --> 00:22:02,080 transform of the impulse response, which is the 341 00:22:02,080 --> 00:22:03,500 frequency response. 342 00:22:03,500 --> 00:22:05,800 We now want to inverse transform to 343 00:22:05,800 --> 00:22:08,600 get the impulse response. 344 00:22:08,600 --> 00:22:10,730 Well, how do we inverse transform? 345 00:22:10,730 --> 00:22:15,420 Of course, we could do it by attempting to go through the 346 00:22:15,420 --> 00:22:18,880 synthesis equation for the Fourier transform. 347 00:22:18,880 --> 00:22:22,520 Or we can do as we did in the continuous-time case which is 348 00:22:22,520 --> 00:22:25,362 to take advantage of what we know. 349 00:22:25,362 --> 00:22:30,450 And in particular, we know that from an example that we 350 00:22:30,450 --> 00:22:36,910 worked before, this is in fact the Fourier transform of a 351 00:22:36,910 --> 00:22:43,960 sequence which is a to the n times u of n. 352 00:22:43,960 --> 00:22:46,860 And so, in essence, by inspection-- 353 00:22:46,860 --> 00:22:51,770 very similar to what has gone on in continuous time-- 354 00:22:51,770 --> 00:22:55,900 essentially by inspection, we can then solve for the impulse 355 00:22:55,900 --> 00:22:58,400 response to the system. 356 00:22:58,400 --> 00:23:02,940 OK, so that procedure follows very much the kind of 357 00:23:02,940 --> 00:23:06,630 procedure that we've carried out in continuous time. 358 00:23:06,630 --> 00:23:08,630 And this, of course, is discussed in more 359 00:23:08,630 --> 00:23:11,480 detail in the text. 360 00:23:11,480 --> 00:23:16,380 Well, let's look at that example then. 361 00:23:16,380 --> 00:23:21,530 Here we have the impulse response for that, associated 362 00:23:21,530 --> 00:23:22,960 with the system described by that 363 00:23:22,960 --> 00:23:26,340 particular difference equation. 364 00:23:26,340 --> 00:23:29,630 And to the right of that, we have the 365 00:23:29,630 --> 00:23:32,140 associated frequency response. 366 00:23:32,140 --> 00:23:34,150 And one of the things that we notice-- 367 00:23:34,150 --> 00:23:39,080 and this is drawn for a positive between 0 and 1-- 368 00:23:39,080 --> 00:23:44,650 what we notice, in fact, is that it is an approximation to 369 00:23:44,650 --> 00:23:48,670 a lowpass filter, because it tends to attenuate the high 370 00:23:48,670 --> 00:23:52,380 frequencies and retain and, in fact, amplify the low 371 00:23:52,380 --> 00:23:54,520 frequencies. 372 00:23:54,520 --> 00:23:59,840 Now if instead, actually, the impulse response was such that 373 00:23:59,840 --> 00:24:04,540 we picked a to be negative between minus 1 and 0, then 374 00:24:04,540 --> 00:24:08,110 the impulse response in the time domain looks like this. 375 00:24:08,110 --> 00:24:10,380 And the corresponding frequency 376 00:24:10,380 --> 00:24:12,980 response looks like this. 377 00:24:12,980 --> 00:24:15,920 And that becomes an approximation 378 00:24:15,920 --> 00:24:19,790 to a highpass filter. 379 00:24:19,790 --> 00:24:24,330 So, in fact, a first-order difference equation, as we 380 00:24:24,330 --> 00:24:28,250 see, has a frequency response, depending on the value of a, 381 00:24:28,250 --> 00:24:31,930 that either looks approximately like a lowpass 382 00:24:31,930 --> 00:24:34,540 filter for a positive or a highpass filter for a 383 00:24:34,540 --> 00:24:38,360 negative, very much like the first-order differential 384 00:24:38,360 --> 00:24:41,560 equation looked like a lowpass filter in the 385 00:24:41,560 --> 00:24:43,710 continuous-time case. 386 00:24:43,710 --> 00:24:48,950 And, in fact, what I'd like to illustrate is the filtering 387 00:24:48,950 --> 00:24:50,340 characteristics-- 388 00:24:50,340 --> 00:24:52,680 or an example of filtering-- 389 00:24:52,680 --> 00:24:55,600 using a first-order difference equation. 390 00:24:55,600 --> 00:25:00,400 And the example that I'll illustrate is a filtering of a 391 00:25:00,400 --> 00:25:04,190 sequence that in fact is filtered very often for very 392 00:25:04,190 --> 00:25:07,880 practical reasons, namely a sequence which represents the 393 00:25:07,880 --> 00:25:12,030 Dow Jones Industrial Average over a fairly long period. 394 00:25:12,030 --> 00:25:17,040 And we'll process the Dow Jones Industrial Average first 395 00:25:17,040 --> 00:25:19,610 through a first-order difference equation, where, if 396 00:25:19,610 --> 00:25:23,460 we begin with a equals 0, then, referring to the 397 00:25:23,460 --> 00:25:27,760 frequency response that we have here, a equals 0 would 398 00:25:27,760 --> 00:25:30,960 simply be passing all frequencies. 399 00:25:30,960 --> 00:25:37,140 As a is positive we start to retain mostly low frequencies, 400 00:25:37,140 --> 00:25:44,370 and the larger a gets, but still less than 1, the more it 401 00:25:44,370 --> 00:25:46,770 attenuates high frequencies at the expense of low 402 00:25:46,770 --> 00:25:48,270 frequencies. 403 00:25:48,270 --> 00:25:53,490 So let's watch the filtering, first with a positive and 404 00:25:53,490 --> 00:25:56,500 we'll see it behave as a lowpass filter, and then with 405 00:25:56,500 --> 00:26:02,270 a negative and we'll see the difference equation behaving 406 00:26:02,270 --> 00:26:05,020 as a highpass filter. 407 00:26:05,020 --> 00:26:09,520 What we see here is the Dow Jones Industrial Average over 408 00:26:09,520 --> 00:26:15,100 roughly a five-year period from 1927 to 1932. 409 00:26:15,100 --> 00:26:19,470 And, in fact, that big dip in the middle is the famous stock 410 00:26:19,470 --> 00:26:22,010 market crash of 1929. 411 00:26:22,010 --> 00:26:25,190 And we can see that following that, in fact, the market 412 00:26:25,190 --> 00:26:29,030 continued a very long downward trend. 413 00:26:29,030 --> 00:26:34,600 And what we now want to do is process this through a 414 00:26:34,600 --> 00:26:36,720 difference equation. 415 00:26:36,720 --> 00:26:40,950 Above the Dow Jones average we show the impulse response of 416 00:26:40,950 --> 00:26:42,020 the difference equation. 417 00:26:42,020 --> 00:26:46,080 Here we've chosen the parameter a equal to 0. 418 00:26:46,080 --> 00:26:50,570 And the impulse response will be displayed on an expanded 419 00:26:50,570 --> 00:26:57,700 scale in relation to the scale of the input and, for that 420 00:26:57,700 --> 00:27:00,300 matter, the scale of the output. 421 00:27:00,300 --> 00:27:03,900 Now with the impulse response shown here which is just an 422 00:27:03,900 --> 00:27:09,160 impulse, in fact, the output shown on the bottom trace is 423 00:27:09,160 --> 00:27:11,880 exactly identical to the input. 424 00:27:11,880 --> 00:27:16,730 And what we'll want to do now is increase, first, the 425 00:27:16,730 --> 00:27:22,000 parameter a, and the impulse response will begin to look 426 00:27:22,000 --> 00:27:27,140 like an exponential with a duration that's longer and 427 00:27:27,140 --> 00:27:30,670 longer as a moves from 0 to 1. 428 00:27:33,180 --> 00:27:35,850 Correspondingly we'll get more and more lowpass filtering as 429 00:27:35,850 --> 00:27:39,300 the coefficient a increases from 0 toward 1. 430 00:27:39,300 --> 00:27:43,340 So now we are increasing the parameter a. 431 00:27:43,340 --> 00:27:47,300 We see that the bottom trace in relation to the middle 432 00:27:47,300 --> 00:27:52,480 trace in fact is looking more and more smoothed or 433 00:27:52,480 --> 00:27:53,700 lowpass-filtered. 434 00:27:53,700 --> 00:27:57,440 And here now we have a fair amount of smoothing, to the 435 00:27:57,440 --> 00:28:02,480 point where the stock market crash of 1929 is totally lost. 436 00:28:02,480 --> 00:28:05,470 And in fact I'm sure there are many people who wish that 437 00:28:05,470 --> 00:28:09,090 through filtering we could, in fact, have avoided the stock 438 00:28:09,090 --> 00:28:11,940 market crash altogether. 439 00:28:11,940 --> 00:28:19,490 Now, let's decrease a from 1 back towards 0. 440 00:28:19,490 --> 00:28:22,410 And as we do that, we will be taking 441 00:28:22,410 --> 00:28:25,930 out the lowpass filtering. 442 00:28:25,930 --> 00:28:29,840 And when a finally reaches 0, the impulse response of the 443 00:28:29,840 --> 00:28:34,180 filter will again be an impulse, and so the output 444 00:28:34,180 --> 00:28:38,650 will be once again identical to the input. 445 00:28:38,650 --> 00:28:42,000 And that's where we are now. 446 00:28:42,000 --> 00:28:46,930 All right now we want to continue to decrease a so that 447 00:28:46,930 --> 00:28:51,160 it becomes negative, moving from 0 toward minus 1. 448 00:28:51,160 --> 00:28:55,890 And what we will see in that case is more and more highpass 449 00:28:55,890 --> 00:29:00,550 filtering on the output in relation to the input. 450 00:29:00,550 --> 00:29:04,500 And this will be particularly evident in, again, the region 451 00:29:04,500 --> 00:29:07,090 of high frequencies represented by sharp 452 00:29:07,090 --> 00:29:10,160 transitions which, of course, the market crash 453 00:29:10,160 --> 00:29:12,870 of 1929 would represent. 454 00:29:12,870 --> 00:29:18,740 So here, now, a is decreasing toward minus 1. 455 00:29:18,740 --> 00:29:22,140 We see that the high frequencies, or rapid 456 00:29:22,140 --> 00:29:29,230 variations are emphasized., And finally, let's move from 457 00:29:29,230 --> 00:29:35,620 minus 1 back towards 0, taking out the highpass filtering and 458 00:29:35,620 --> 00:29:40,390 ending up with a equal to 0, corresponding to an impulse 459 00:29:40,390 --> 00:29:42,290 response which is an impulse, in other 460 00:29:42,290 --> 00:29:44,010 words, an identity system. 461 00:29:44,010 --> 00:29:47,050 And let me stress once again that the time scale on which 462 00:29:47,050 --> 00:29:51,030 we displayed the impulse response is an expanded time 463 00:29:51,030 --> 00:29:55,320 scale in relation to the time scale on which we displayed 464 00:29:55,320 --> 00:29:56,820 the input and the output. 465 00:29:59,650 --> 00:30:04,090 OK, so we see that, in fact, a first-order difference 466 00:30:04,090 --> 00:30:07,370 equation is a filter. 467 00:30:07,370 --> 00:30:10,270 And, in fact, it's a very important class of filters, 468 00:30:10,270 --> 00:30:13,320 and it's used very often to do approximate lowpass and 469 00:30:13,320 --> 00:30:14,570 highpass filtering. 470 00:30:17,200 --> 00:30:24,310 Now, in addition to the convolution property, another 471 00:30:24,310 --> 00:30:27,420 important property that we had in continuous time, and that 472 00:30:27,420 --> 00:30:31,500 we have in discrete time, is the modulation property. 473 00:30:31,500 --> 00:30:35,080 The modulation property tells us what happens in the 474 00:30:35,080 --> 00:30:38,730 frequency domain when you multiply 475 00:30:38,730 --> 00:30:41,700 signals in the time domain. 476 00:30:41,700 --> 00:30:45,100 In continuous time, the modulation property 477 00:30:45,100 --> 00:30:48,760 corresponded to the statement that if we multiply the time 478 00:30:48,760 --> 00:30:54,400 domain, we convolve the Fourier transforms in the 479 00:30:54,400 --> 00:30:56,470 frequency domain. 480 00:30:56,470 --> 00:31:02,730 And in discrete time we have very much the same kind of 481 00:31:02,730 --> 00:31:05,060 relationship. 482 00:31:05,060 --> 00:31:11,090 The only real distinction between these is that in the 483 00:31:11,090 --> 00:31:15,160 discrete-time case, in carrying out the convolution, 484 00:31:15,160 --> 00:31:18,870 it's an integration only over a 2 pi interval. 485 00:31:18,870 --> 00:31:25,120 And what that corresponds to is what's referred to as a 486 00:31:25,120 --> 00:31:31,550 periodic convolution, as opposed to the continuous-time 487 00:31:31,550 --> 00:31:37,800 case where what we have is a convolution that is an 488 00:31:37,800 --> 00:31:41,050 aperiodic convolution. 489 00:31:41,050 --> 00:31:44,930 So, again, we have a convolution property in 490 00:31:44,930 --> 00:31:48,680 discrete time that is very much like the convolution 491 00:31:48,680 --> 00:31:51,060 property in continuous time. 492 00:31:51,060 --> 00:31:54,570 The only real difference is that here we're convolving 493 00:31:54,570 --> 00:31:56,300 periodic functions. 494 00:31:56,300 --> 00:32:01,050 And so it's a periodic convolution which involves an 495 00:32:01,050 --> 00:32:04,510 integration only over a 2 pi interval, rather than an 496 00:32:04,510 --> 00:32:08,570 integration from minus infinity to plus infinity. 497 00:32:08,570 --> 00:32:14,650 Well, let's take a look at an example of the modulation 498 00:32:14,650 --> 00:32:20,070 property, which will then lead to one particular application, 499 00:32:20,070 --> 00:32:24,110 and a very useful application, of the modulation property in 500 00:32:24,110 --> 00:32:26,180 discrete time. 501 00:32:26,180 --> 00:32:31,580 The example that I want to pick is an example in which we 502 00:32:31,580 --> 00:32:35,590 consider modulating a signal with-- 503 00:32:35,590 --> 00:32:40,150 a signal with another signal, x of n, or x1 of n as I 504 00:32:40,150 --> 00:32:44,170 indicated here, which is minus 1 to the n. 505 00:32:44,170 --> 00:32:48,050 Essentially what that says is that any signal which I 506 00:32:48,050 --> 00:32:51,640 modulate with this in effect corresponds to taking the 507 00:32:51,640 --> 00:32:56,110 original signal and then going through that signal 508 00:32:56,110 --> 00:33:00,690 alternating the algebraic signs. 509 00:33:00,690 --> 00:33:03,200 Now we-- 510 00:33:03,200 --> 00:33:06,430 in applying the modulation property, of course, what we 511 00:33:06,430 --> 00:33:09,390 need to do is develop the Fourier 512 00:33:09,390 --> 00:33:12,410 transform of this signal. 513 00:33:12,410 --> 00:33:14,820 This signal which I rewrite-- 514 00:33:14,820 --> 00:33:17,450 I can write either as minus 1 to the n or rewrite as e to 515 00:33:17,450 --> 00:33:22,520 the j pi n since e to the j pi is equal to minus 1-- 516 00:33:22,520 --> 00:33:25,210 is a periodic signal. 517 00:33:25,210 --> 00:33:28,700 And it's the periodic signal that I show here. 518 00:33:28,700 --> 00:33:32,860 And recall that to get the Fourier transform of a 519 00:33:32,860 --> 00:33:39,020 periodic signal, one way to do it is to generate the Fourier 520 00:33:39,020 --> 00:33:42,360 series coefficients for the periodic signal, and then 521 00:33:42,360 --> 00:33:47,450 identify the Fourier transform as an impulse train where the 522 00:33:47,450 --> 00:33:49,900 heights of the impulses in the impulse train are 523 00:33:49,900 --> 00:33:52,510 proportional, with a proportionality factor of 2 524 00:33:52,510 --> 00:33:58,200 pi, proportional to the Fourier series coefficients. 525 00:33:58,200 --> 00:34:01,380 So let's first work out what the Fourier series is and for 526 00:34:01,380 --> 00:34:04,580 this example, in fact, it's fairly easy. 527 00:34:04,580 --> 00:34:08,886 Here is the general synthesis equation 528 00:34:08,886 --> 00:34:12,429 for the Fourier series. 529 00:34:12,429 --> 00:34:18,969 And if we take our particular example where, if we look back 530 00:34:18,969 --> 00:34:25,130 at the curve above, what we recognize is that the period 531 00:34:25,130 --> 00:34:29,139 is equal to 2, namely it repeats after 2 points. 532 00:34:29,139 --> 00:34:34,389 Then capital N is equal to 2, and so we can just write this 533 00:34:34,389 --> 00:34:36,389 out with the two terms. 534 00:34:36,389 --> 00:34:40,900 And the two terms involved are x1 of n is a0, the 0-th 535 00:34:40,900 --> 00:34:46,520 coefficient, that's with k equals 0, and a1, and this is 536 00:34:46,520 --> 00:34:49,719 with k equals 1, and we substituted in 537 00:34:49,719 --> 00:34:52,520 capital N equal to 2. 538 00:34:52,520 --> 00:34:55,290 All right, well, we can do a little bit of algebra here, 539 00:34:55,290 --> 00:34:58,800 obviously cross off the factors of 2. 540 00:34:58,800 --> 00:35:04,370 And what we recognize, if we compare this expression with 541 00:35:04,370 --> 00:35:08,840 the original signal which is e to the j pi n, then we can 542 00:35:08,840 --> 00:35:13,780 simply identify the fact that a0, the 0-th coefficient is 0, 543 00:35:13,780 --> 00:35:15,510 that's the DC term. 544 00:35:15,510 --> 00:35:19,760 And the coefficient a1 is equal to 1. 545 00:35:19,760 --> 00:35:25,160 So we've done it simply by essentially inspecting the 546 00:35:25,160 --> 00:35:30,950 Fourier series synthesis equation. 547 00:35:30,950 --> 00:35:35,380 OK, now, if we want to get the Fourier transform for this, we 548 00:35:35,380 --> 00:35:40,470 take those coefficients and essentially generate an 549 00:35:40,470 --> 00:35:46,830 impulse train where we choose as values for the impulses 2 550 00:35:46,830 --> 00:35:50,680 pi times the Fourier series coefficients. 551 00:35:50,680 --> 00:35:54,010 So, the Fourier series coefficients are a0 is equal 552 00:35:54,010 --> 00:35:56,630 to 0 and a1 is equal to 1. 553 00:35:56,630 --> 00:36:03,370 So, notice that in the plot that I've shown here of the 554 00:36:03,370 --> 00:36:09,460 Fourier transform of x1 of n, we have the 0-th coefficient, 555 00:36:09,460 --> 00:36:14,320 which happens to be 0, and so I have it indicated, an 556 00:36:14,320 --> 00:36:16,440 impulse there. 557 00:36:16,440 --> 00:36:23,360 We have the coefficient a1, and the coefficient a1 occurs 558 00:36:23,360 --> 00:36:28,540 at a frequency which is omega 0, and omega 0 in fact is 559 00:36:28,540 --> 00:36:33,100 equal to pi because the signal is e to the j pi n. 560 00:36:33,100 --> 00:36:37,730 Well, what's this impulse over here? 561 00:36:37,730 --> 00:36:41,150 Well, that impulse is a-- 562 00:36:41,150 --> 00:36:42,940 corresponds to the Fourier series 563 00:36:42,940 --> 00:36:45,560 coefficient a sub minus 1. 564 00:36:45,560 --> 00:36:49,300 And, of course, if we drew this out over a longer 565 00:36:49,300 --> 00:36:52,850 frequency axis, we would see lots of other impulses because 566 00:36:52,850 --> 00:36:56,900 of the fact that the Fourier transform periodically repeats 567 00:36:56,900 --> 00:36:59,660 or, equivalently, the Fourier series coefficients 568 00:36:59,660 --> 00:37:01,790 periodically repeat. 569 00:37:01,790 --> 00:37:08,580 So this is the coefficient a0, This is the coefficient a1 570 00:37:08,580 --> 00:37:12,180 with a factor of 2 pi, this is 2 pi times a0 571 00:37:12,180 --> 00:37:15,680 and 2 pi times a1. 572 00:37:15,680 --> 00:37:20,260 And then this is simply an indication that it's 573 00:37:20,260 --> 00:37:21,510 periodically repeated. 574 00:37:24,220 --> 00:37:24,670 All right. 575 00:37:24,670 --> 00:37:28,830 Now, let's consider what happens if we take a signal 576 00:37:28,830 --> 00:37:34,470 and multiply it, modulate it, by minus 1 to the n. 577 00:37:34,470 --> 00:37:37,540 Well in the frequency domain that corresponds to a 578 00:37:37,540 --> 00:37:39,800 convolution. 579 00:37:39,800 --> 00:37:43,820 Let's consider a signal x2 of n which has a Fourier 580 00:37:43,820 --> 00:37:47,300 transform as I've indicated here. 581 00:37:47,300 --> 00:37:52,460 Then the Fourier transform of the product of x1 of n and x2 582 00:37:52,460 --> 00:37:57,270 of n is the convolution of these two spectra. 583 00:37:57,270 --> 00:38:01,850 And recall that if you could convolve something with an 584 00:38:01,850 --> 00:38:05,800 impulse train, as this is, that simply corresponds to 585 00:38:05,800 --> 00:38:10,920 taking the something and placing it at the positions of 586 00:38:10,920 --> 00:38:12,830 each of the impulses. 587 00:38:12,830 --> 00:38:17,850 So, in fact, the result of the convolution of this with this 588 00:38:17,850 --> 00:38:23,320 would then be the spectrum that I indicate here, namely 589 00:38:23,320 --> 00:38:28,920 this spectrum shifted up to pi and of course to minus pi. 590 00:38:28,920 --> 00:38:34,550 And then of course to not only pi but 3 pi and 5 pi, et 591 00:38:34,550 --> 00:38:36,180 cetera, et cetera. 592 00:38:36,180 --> 00:38:42,020 And so this spectrum, finally, corresponds to the Fourier 593 00:38:42,020 --> 00:38:48,010 transform of minus 1 to the n times x2 of n where x2 of n is 594 00:38:48,010 --> 00:38:53,780 the sequence whose spectrum was X2 of omega. 595 00:38:53,780 --> 00:38:57,750 OK, now, this is in fact an important, useful, and 596 00:38:57,750 --> 00:38:58,570 interesting point. 597 00:38:58,570 --> 00:39:03,210 What it says is if I have a signal with a certain spectrum 598 00:39:03,210 --> 00:39:05,230 and if I modulate-- 599 00:39:05,230 --> 00:39:06,180 multiply-- 600 00:39:06,180 --> 00:39:10,130 that signal by minus 1 to the n, meaning that I alternate 601 00:39:10,130 --> 00:39:14,470 the signs, then it takes the low frequencies-- 602 00:39:14,470 --> 00:39:17,160 in effect, it shifts the spectrum by pi. 603 00:39:17,160 --> 00:39:19,120 So it takes the low frequencies and moves them up 604 00:39:19,120 --> 00:39:22,470 to high frequencies, and will incidentally take the high 605 00:39:22,470 --> 00:39:26,440 frequencies and move them to low frequencies. 606 00:39:26,440 --> 00:39:30,400 So in fact we, in essence, saw this when we took-- 607 00:39:30,400 --> 00:39:33,920 or when I talked about the example of a sequence which 608 00:39:33,920 --> 00:39:36,120 was a to the n times u of n. 609 00:39:36,120 --> 00:39:36,730 Notice-- 610 00:39:36,730 --> 00:39:40,670 let me draw your attention to the fact that when a is 611 00:39:40,670 --> 00:39:46,460 positive, we have this sequence and its Fourier 612 00:39:46,460 --> 00:39:52,220 transform is as I show on the right. 613 00:39:52,220 --> 00:40:00,820 For a negative, the sequence is identical to a positive but 614 00:40:00,820 --> 00:40:03,030 with alternating sines. 615 00:40:03,030 --> 00:40:06,730 And the Fourier transform of that you can now see, and 616 00:40:06,730 --> 00:40:12,900 verify also algebraically if you'd like, is identical to 617 00:40:12,900 --> 00:40:17,640 this spectrum, simply shifted by pi. 618 00:40:17,640 --> 00:40:21,280 So it says in fact that multiplying that impulse 619 00:40:21,280 --> 00:40:25,530 response, or if we think of a positive and a negative, that 620 00:40:25,530 --> 00:40:29,260 is algebraically similar to multiplying the impulse 621 00:40:29,260 --> 00:40:31,650 response by minus 1 to the n. 622 00:40:31,650 --> 00:40:35,150 And in the frequency domain, the effect of that, 623 00:40:35,150 --> 00:40:38,250 essentially, is shifting the spectrum by pi. 624 00:40:38,250 --> 00:40:40,440 And we can interpret that in the context of 625 00:40:40,440 --> 00:40:43,060 the modulation property. 626 00:40:43,060 --> 00:40:50,260 Now it's interesting that what that says is that if we have a 627 00:40:50,260 --> 00:40:58,070 system which corresponds to a lowpass filter, as I indicate 628 00:40:58,070 --> 00:41:03,320 here, with an impulse response h of n. 629 00:41:03,320 --> 00:41:06,530 And it can be any approximation to a lowpass 630 00:41:06,530 --> 00:41:09,860 filter and even an ideal lowpass filter. 631 00:41:09,860 --> 00:41:14,900 If we want to convert that to a highpass filter, we can do 632 00:41:14,900 --> 00:41:19,980 that by generating a new system whose impulse response 633 00:41:19,980 --> 00:41:24,010 is minus 1 to the n times the impulse response of the 634 00:41:24,010 --> 00:41:25,490 lowpass filter. 635 00:41:25,490 --> 00:41:31,920 And this modulation by minus 1 to the n will take the 636 00:41:31,920 --> 00:41:36,980 frequency response of this system and shift it by pi so 637 00:41:36,980 --> 00:41:40,500 that what's going on here at low frequencies will now go on 638 00:41:40,500 --> 00:41:41,750 here at high frequencies. 639 00:41:44,600 --> 00:41:53,010 This also says, incidentally, that if we look at an ideal 640 00:41:53,010 --> 00:41:58,610 lowpass filter and an ideal highpass filter, and we choose 641 00:41:58,610 --> 00:42:02,270 the cutoff frequencies for comparison, or the bandwidth 642 00:42:02,270 --> 00:42:04,650 of the filter to be equal. 643 00:42:04,650 --> 00:42:10,190 Since this ideal highpass filter is this ideal lowpass 644 00:42:10,190 --> 00:42:16,580 filter with the frequency response shifted by pi, the 645 00:42:16,580 --> 00:42:20,640 modulation property tells us that in the time domain, what 646 00:42:20,640 --> 00:42:25,520 that corresponds to is an impulse response multiplied by 647 00:42:25,520 --> 00:42:26,950 minus 1 to the n. 648 00:42:26,950 --> 00:42:32,500 So it says that the impulse response of the highpass 649 00:42:32,500 --> 00:42:36,230 filter, or equivalently the inverse Fourier transform of 650 00:42:36,230 --> 00:42:40,720 the highpass filter frequency response, is minus 1 to the n 651 00:42:40,720 --> 00:42:43,780 times the impulse response for the lowpass filter. 652 00:42:43,780 --> 00:42:47,860 That all follows from the modulation property. 653 00:42:47,860 --> 00:42:51,710 Now there's another way, an interesting and useful way, 654 00:42:51,710 --> 00:42:57,200 that modulation can be used to implement or convert from 655 00:42:57,200 --> 00:43:00,710 lowpass filtering to highpass filtering. 656 00:43:00,710 --> 00:43:04,300 The modulation property tells us about multiplying the time 657 00:43:04,300 --> 00:43:07,170 domain is shifting in the frequency domain. 658 00:43:07,170 --> 00:43:09,810 And in the example that we happened to pick said if you 659 00:43:09,810 --> 00:43:14,120 multiply or modulate by minus 1 to the n, that takes low 660 00:43:14,120 --> 00:43:17,580 frequencies and shifts them to high frequencies. 661 00:43:17,580 --> 00:43:23,280 What that tells us, as a practical and useful notion, 662 00:43:23,280 --> 00:43:24,520 is the following. 663 00:43:24,520 --> 00:43:28,540 Suppose we have a system that we know is a lowpass filter, 664 00:43:28,540 --> 00:43:31,410 and it's a good lowpass filter. 665 00:43:31,410 --> 00:43:34,800 How might we use it as a highpass filter? 666 00:43:34,800 --> 00:43:38,080 Well, one way to do it, instead of shifting its 667 00:43:38,080 --> 00:43:43,810 frequency response, is to take the original signal, shift its 668 00:43:43,810 --> 00:43:46,080 low frequencies to high frequencies and its high 669 00:43:46,080 --> 00:43:49,500 frequencies to low frequencies by multiplying the input 670 00:43:49,500 --> 00:43:54,160 signal, the original signal, by minus 1 to the n, process 671 00:43:54,160 --> 00:43:57,070 that with a lowpass filter where now what's sitting at 672 00:43:57,070 --> 00:44:00,430 the low frequencies were the high frequencies. 673 00:44:00,430 --> 00:44:05,320 And then unscramble it all at the output so that we put the 674 00:44:05,320 --> 00:44:07,760 frequencies back where they belong. 675 00:44:07,760 --> 00:44:10,060 And I summarize that here. 676 00:44:10,060 --> 00:44:13,860 Let's suppose, for example, that this system was a lowpass 677 00:44:13,860 --> 00:44:17,840 filter, and so it lowpass-filters 678 00:44:17,840 --> 00:44:20,120 whatever comes into it. 679 00:44:20,120 --> 00:44:24,560 Down below, I indicate taking the input and first 680 00:44:24,560 --> 00:44:28,830 interchanging the high and low frequencies through modulation 681 00:44:28,830 --> 00:44:32,460 with minus 1 to the n. 682 00:44:32,460 --> 00:44:35,220 Doing the lowpass filtering, which-- 683 00:44:35,220 --> 00:44:37,660 and what's sitting at the low frequencies here were the high 684 00:44:37,660 --> 00:44:39,990 frequencies of this signal. 685 00:44:39,990 --> 00:44:42,830 And then after the lowpass filtering, moving the 686 00:44:42,830 --> 00:44:46,860 frequencies back where they belong by again modulating 687 00:44:46,860 --> 00:44:48,560 with minus 1 to the n. 688 00:44:48,560 --> 00:44:56,150 And that, in fact, turns out to be a very useful notion for 689 00:44:56,150 --> 00:45:01,240 applying a fixed lowpass filter to do highpass 690 00:45:01,240 --> 00:45:02,550 filtering and vice versa. 691 00:45:06,740 --> 00:45:15,470 OK, now, what we've seen and what we've talked about are 692 00:45:15,470 --> 00:45:20,740 the Fourier representation for discrete-time signals, and 693 00:45:20,740 --> 00:45:24,080 prior to that, continuous-time signals. 694 00:45:24,080 --> 00:45:27,250 And we've seen some very important similarities and 695 00:45:27,250 --> 00:45:28,470 differences. 696 00:45:28,470 --> 00:45:34,660 And what I'd like to do is conclude this lecture by 697 00:45:34,660 --> 00:45:38,910 summarizing those various relationships kind of all in 698 00:45:38,910 --> 00:45:43,550 one package, and in fact drawing your attention to both 699 00:45:43,550 --> 00:45:46,250 the similarities and differences and comparisons 700 00:45:46,250 --> 00:45:48,930 between them. 701 00:45:48,930 --> 00:45:54,710 Well, let's begin this summary by first looking at the 702 00:45:54,710 --> 00:45:57,240 continuous-time Fourier series. 703 00:45:57,240 --> 00:46:02,280 In the continuous-time Fourier series, we have a periodic 704 00:46:02,280 --> 00:46:06,940 time function expanded as a linear combination of 705 00:46:06,940 --> 00:46:10,000 harmonically-related complex exponentials. 706 00:46:10,000 --> 00:46:12,420 And there are an infinite number of these that are 707 00:46:12,420 --> 00:46:15,710 required to do the decomposition. 708 00:46:15,710 --> 00:46:20,230 And we saw an analysis equation which tells us how to 709 00:46:20,230 --> 00:46:24,380 get these Fourier series coefficients through an 710 00:46:24,380 --> 00:46:28,330 integration on the original time function. 711 00:46:28,330 --> 00:46:32,260 And notice in this that what we have is a continuous 712 00:46:32,260 --> 00:46:34,250 periodic time function. 713 00:46:34,250 --> 00:46:38,990 What we end up with in the frequency domain is a sequence 714 00:46:38,990 --> 00:46:42,670 of Fourier series coefficients which in fact is an infinite 715 00:46:42,670 --> 00:46:46,700 sequence, namely, requires all values of k in general. 716 00:46:49,380 --> 00:46:53,650 We had then generalized that to the continuous-time Fourier 717 00:46:53,650 --> 00:46:57,750 transform, and, in effect, in doing that what happened is 718 00:46:57,750 --> 00:47:07,580 that the synthesis equation in the Fourier series became an 719 00:47:07,580 --> 00:47:11,830 integral relationship in the Fourier transform. 720 00:47:11,830 --> 00:47:17,360 And we now have a continuous-time function which 721 00:47:17,360 --> 00:47:21,290 is no longer periodic, this was for the aperiodic case, 722 00:47:21,290 --> 00:47:25,080 represented as a linear combination of infinitesimally 723 00:47:25,080 --> 00:47:29,870 close-in-frequency complex exponentials with complex 724 00:47:29,870 --> 00:47:35,850 amplitudes given by X of omega d omega divided by 2 pi. 725 00:47:35,850 --> 00:47:39,210 And we had of course the corresponding analysis 726 00:47:39,210 --> 00:47:43,360 equation that told us how to get X of omega. 727 00:47:43,360 --> 00:47:48,060 Here we have a continuous-time function which is aperiodic, 728 00:47:48,060 --> 00:47:53,485 and a continuous function of frequency which is aperiodic. 729 00:47:56,990 --> 00:48:01,780 The conceptual strategy in the discrete-time case was very 730 00:48:01,780 --> 00:48:08,640 similar, with some differences resulting in the relationships 731 00:48:08,640 --> 00:48:11,920 because of some inherent differences between continuous 732 00:48:11,920 --> 00:48:15,110 time and discrete time. 733 00:48:15,110 --> 00:48:19,560 We began with the discrete-time Fourier series, 734 00:48:19,560 --> 00:48:24,360 corresponding to representing a periodic sequence through a 735 00:48:24,360 --> 00:48:29,780 set of complex exponentials, where now we only required a 736 00:48:29,780 --> 00:48:33,990 finite number of these because of the fact that, in fact, 737 00:48:33,990 --> 00:48:36,830 there are only a finite number of harmonically-related 738 00:48:36,830 --> 00:48:38,580 complex exponentials. 739 00:48:38,580 --> 00:48:41,710 That's an inherent property of discrete-time complex 740 00:48:41,710 --> 00:48:43,070 exponentials. 741 00:48:43,070 --> 00:48:49,640 And so we have a discrete, periodic time function. 742 00:48:49,640 --> 00:48:53,270 And we ended up with a set of Fourier series coefficients, 743 00:48:53,270 --> 00:48:56,670 which of course are discrete, as Fourier series coefficients 744 00:48:56,670 --> 00:49:01,960 are, and which periodically repeat because of the fact 745 00:49:01,960 --> 00:49:04,290 that the associated complex exponentials 746 00:49:04,290 --> 00:49:05,540 periodically repeat. 747 00:49:07,960 --> 00:49:11,030 We then used an argument similar to the continuous-time 748 00:49:11,030 --> 00:49:14,840 case for going from periodic time functions to aperiodic 749 00:49:14,840 --> 00:49:16,290 time functions. 750 00:49:16,290 --> 00:49:20,190 And we ended up with a relationship describing a 751 00:49:20,190 --> 00:49:25,710 representation for aperiodic discrete-time signals in which 752 00:49:25,710 --> 00:49:30,460 now the synthesis equation went from a summation to an 753 00:49:30,460 --> 00:49:32,620 integration, since the frequencies are now 754 00:49:32,620 --> 00:49:36,770 infinitesimally close, involving frequencies only 755 00:49:36,770 --> 00:49:40,500 over a 2 pi interval, and for which the 756 00:49:40,500 --> 00:49:42,800 amplitude factor X of omega-- 757 00:49:42,800 --> 00:49:45,140 well, the amplitude factor is X of omega d omega 758 00:49:45,140 --> 00:49:47,190 divided by 2 pi. 759 00:49:47,190 --> 00:49:51,060 And this term, X of omega, which is the Fourier 760 00:49:51,060 --> 00:49:57,120 transform, is given by this summation, and of course 761 00:49:57,120 --> 00:50:01,500 involves all of the values of x of n. 762 00:50:01,500 --> 00:50:05,420 And so the important difference between the 763 00:50:05,420 --> 00:50:08,390 continuous-time and discrete-time case kind of 764 00:50:08,390 --> 00:50:11,460 arose, in part, out of the fact that discrete time is 765 00:50:11,460 --> 00:50:14,830 discrete time, continuous time is continuous time, and the 766 00:50:14,830 --> 00:50:18,770 fact that complex exponentials are periodic in discrete time. 767 00:50:18,770 --> 00:50:21,910 The harmonically-related ones periodically repeat whereas 768 00:50:21,910 --> 00:50:25,470 they don't in continuous time. 769 00:50:25,470 --> 00:50:28,830 Now this, among other things, has an important consequence 770 00:50:28,830 --> 00:50:30,270 for duality. 771 00:50:30,270 --> 00:50:34,540 And let's go back again and look at this equation, this 772 00:50:34,540 --> 00:50:35,660 pair of equations. 773 00:50:35,660 --> 00:50:38,800 And clearly there is no duality 774 00:50:38,800 --> 00:50:41,040 between these two equations. 775 00:50:41,040 --> 00:50:44,230 This involves a summation, this involves an integration. 776 00:50:44,230 --> 00:50:50,810 And so, in fact, if we make reference to duality, there 777 00:50:50,810 --> 00:50:52,810 isn't duality in the 778 00:50:52,810 --> 00:50:55,900 continuous-time Fourier series. 779 00:50:55,900 --> 00:50:58,910 However, for the continuous-time Fourier 780 00:50:58,910 --> 00:51:03,910 transform, we're talking about aperiodic time functions and 781 00:51:03,910 --> 00:51:06,310 aperiodic frequency functions. 782 00:51:06,310 --> 00:51:09,350 And, in fact, when we look at these two equations, we see 783 00:51:09,350 --> 00:51:11,550 very definitely a duality. 784 00:51:11,550 --> 00:51:15,530 In other words, the time function effectively is the 785 00:51:15,530 --> 00:51:18,340 Fourier transform of the Fourier transform. 786 00:51:18,340 --> 00:51:20,660 There's a little time reversal in there, but basically that's 787 00:51:20,660 --> 00:51:21,780 the result. 788 00:51:21,780 --> 00:51:26,890 And, in fact, we had exploited that duality property when we 789 00:51:26,890 --> 00:51:27,620 talked about the 790 00:51:27,620 --> 00:51:32,020 continuous-time Fourier transform. 791 00:51:32,020 --> 00:51:40,370 With the discrete-time Fourier series, we have a duality 792 00:51:40,370 --> 00:51:44,590 indicated by the fact that we have a periodic time function 793 00:51:44,590 --> 00:51:48,910 and a sequence which is periodic in 794 00:51:48,910 --> 00:51:50,170 the frequency domain. 795 00:51:50,170 --> 00:51:53,150 And in fact, if you look at these two expressions, you see 796 00:51:53,150 --> 00:51:55,200 the duality very clearly. 797 00:51:55,200 --> 00:51:58,630 And so it's the discrete-time Fourier 798 00:51:58,630 --> 00:52:02,630 series that has a duality. 799 00:52:02,630 --> 00:52:07,080 And finally the discrete-time Fourier transform loses the 800 00:52:07,080 --> 00:52:11,150 duality because of the fact, among other things, that in 801 00:52:11,150 --> 00:52:15,040 the time domain things are inherently discrete whereas in 802 00:52:15,040 --> 00:52:18,580 the frequency domain they're inherently continuous. 803 00:52:18,580 --> 00:52:21,395 So, in fact, here there is no duality. 804 00:52:25,690 --> 00:52:30,280 OK, now that says that there's a difference in the duality, 805 00:52:30,280 --> 00:52:32,510 continuous time and discrete time. 806 00:52:32,510 --> 00:52:37,780 And there's one more very important piece to the duality 807 00:52:37,780 --> 00:52:39,610 relationships. 808 00:52:39,610 --> 00:52:44,400 And we can see that first algebraically by comparing the 809 00:52:44,400 --> 00:52:49,280 continuous-time Fourier series and the 810 00:52:49,280 --> 00:52:51,435 discrete-time Fourier transform. 811 00:52:54,240 --> 00:52:59,210 The continuous-time Fourier series in the time domain is a 812 00:52:59,210 --> 00:53:03,580 periodic continuous function, in the frequency domain is an 813 00:53:03,580 --> 00:53:06,930 aperiodic sequence. 814 00:53:06,930 --> 00:53:13,680 In the discrete-time case, in the time domain we have an 815 00:53:13,680 --> 00:53:19,850 aperiodic sequence, and in the frequency domain we have a 816 00:53:19,850 --> 00:53:21,990 function of a continuous variable 817 00:53:21,990 --> 00:53:24,480 which we know is periodic. 818 00:53:24,480 --> 00:53:28,100 And so in fact we have, in the time domain 819 00:53:28,100 --> 00:53:30,370 here, aperiodic sequence. 820 00:53:30,370 --> 00:53:32,300 In the frequency domain we have a 821 00:53:32,300 --> 00:53:34,990 continuous periodic function. 822 00:53:34,990 --> 00:53:38,600 And in fact, if you look at the relationship between these 823 00:53:38,600 --> 00:53:48,390 two, then what we see in fact is a duality between the 824 00:53:48,390 --> 00:53:53,370 continuous-time Fourier series and the 825 00:53:53,370 --> 00:53:56,500 discrete-time Fourier transform. 826 00:53:56,500 --> 00:54:00,170 One way of thinking of that is to kind of think, and this is 827 00:54:00,170 --> 00:54:02,820 a little bit of a tongue twister which you might want 828 00:54:02,820 --> 00:54:08,240 to get straightened out slowly, but the Fourier 829 00:54:08,240 --> 00:54:10,000 transform in discrete time is a 830 00:54:10,000 --> 00:54:12,270 periodic function of frequency. 831 00:54:12,270 --> 00:54:17,190 That periodic function has a Fourier series representation. 832 00:54:17,190 --> 00:54:19,540 What is this Fourier series? 833 00:54:19,540 --> 00:54:21,170 What are the Fourier series coefficients of 834 00:54:21,170 --> 00:54:22,730 that periodic function? 835 00:54:22,730 --> 00:54:26,030 Well in fact, except for an issue of time reversal, what 836 00:54:26,030 --> 00:54:29,920 it is the original sequence for which 837 00:54:29,920 --> 00:54:31,670 that's the Fourier transform. 838 00:54:31,670 --> 00:54:35,610 And that is the duality that I'm trying to emphasize here. 839 00:54:38,600 --> 00:54:42,910 OK, well, so what we see is that these four sets of 840 00:54:42,910 --> 00:54:47,110 relationships all tie together in a whole variety of ways. 841 00:54:47,110 --> 00:54:51,070 And we will be exploiting as the discussion goes on the 842 00:54:51,070 --> 00:54:52,690 inner-connections and relationships 843 00:54:52,690 --> 00:54:55,000 that I've talked about. 844 00:54:55,000 --> 00:54:58,730 Also, as we've talked about the Fourier transform, both 845 00:54:58,730 --> 00:55:02,750 continuous time and discrete time, two important properties 846 00:55:02,750 --> 00:55:06,570 that we focused on, among many of the properties, are the 847 00:55:06,570 --> 00:55:09,950 convolution property and the modulation property. 848 00:55:09,950 --> 00:55:14,670 We've also shown that the convolution property leads to 849 00:55:14,670 --> 00:55:19,020 a very important concept, namely filtering. 850 00:55:19,020 --> 00:55:22,700 The modulation property leads to an important concept, 851 00:55:22,700 --> 00:55:24,380 namely modulation. 852 00:55:24,380 --> 00:55:31,200 We've also very briefly indicated how these properties 853 00:55:31,200 --> 00:55:35,000 and how these concepts have practical implications. 854 00:55:35,000 --> 00:55:37,700 In the next several lectures, we'll focus in more 855 00:55:37,700 --> 00:55:41,100 specifically first on filtering, and then on 856 00:55:41,100 --> 00:55:42,570 modulation. 857 00:55:42,570 --> 00:55:49,000 And as we'll see the filtering and modulation concepts form 858 00:55:49,000 --> 00:55:51,750 really the cornerstone of many, many 859 00:55:51,750 --> 00:55:53,530 signal processing ideas. 860 00:55:53,530 --> 00:55:54,780 Thank you.