1 00:00:06,770 --> 00:00:42,070 [MUSIC PLAYING] 2 00:00:42,070 --> 00:00:46,020 PROFESSOR: Last time, we began the discussion of 3 00:00:46,020 --> 00:00:50,510 discreet-time processing of continuous-time signals. 4 00:00:50,510 --> 00:00:56,170 And, as a reminder, let me review the basic notion. 5 00:00:56,170 --> 00:01:03,880 The idea was that we convert from a continuous-time signal 6 00:01:03,880 --> 00:01:09,900 to a sequence through an operation which I represent as 7 00:01:09,900 --> 00:01:12,890 a continuous to discrete time converter. 8 00:01:12,890 --> 00:01:22,740 And then that sequence is used as the input to an appropriate 9 00:01:22,740 --> 00:01:25,000 discreet-time system. 10 00:01:25,000 --> 00:01:29,250 And after appropriate discreet-time processing, that 11 00:01:29,250 --> 00:01:35,820 sequence is converted back to a continuous-time signal 12 00:01:35,820 --> 00:01:40,960 through an operation which I label as a discrete to 13 00:01:40,960 --> 00:01:44,180 continuous time converter. 14 00:01:44,180 --> 00:01:49,630 Now, in the lecture last time, we carried out some analysis 15 00:01:49,630 --> 00:01:54,910 which related for us the spectra in the first step of 16 00:01:54,910 --> 00:01:56,250 this operation. 17 00:01:56,250 --> 00:02:00,730 Namely in the transformation from a continuous-time time 18 00:02:00,730 --> 00:02:03,520 signal to a sequence. 19 00:02:03,520 --> 00:02:07,290 And let me, by the way, draw your attention to the fact 20 00:02:07,290 --> 00:02:12,970 that, in the real world, this operation is essentially 21 00:02:12,970 --> 00:02:17,540 implemented by what you would typically label as an analog 22 00:02:17,540 --> 00:02:21,430 to digital converter, if in fact the discreet-time 23 00:02:21,430 --> 00:02:23,990 processing is being done digitally. 24 00:02:23,990 --> 00:02:26,290 Now, it's important to emphasize that it's not 25 00:02:26,290 --> 00:02:30,290 exactly what an analog to digital converter does, but, 26 00:02:30,290 --> 00:02:34,060 in some sense at least, you should think of this mapping 27 00:02:34,060 --> 00:02:37,180 from continuous-time to discreet-time in very much the 28 00:02:37,180 --> 00:02:39,070 same way that one would think of an 29 00:02:39,070 --> 00:02:40,760 analog to digital converter. 30 00:02:40,760 --> 00:02:47,790 And the mapping back then corresponds, in some sense, to 31 00:02:47,790 --> 00:02:52,330 what would happen with a digital to analog converter. 32 00:02:52,330 --> 00:02:56,620 Well, let me review what is involved in the mapping from 33 00:02:56,620 --> 00:02:59,460 the continuous-time signal to the sequence. 34 00:02:59,460 --> 00:03:04,470 And, let me stress again that, this operation is basically-- 35 00:03:04,470 --> 00:03:07,400 in the continuous to discreet-time conversion-- 36 00:03:07,400 --> 00:03:10,170 a two-step process. 37 00:03:10,170 --> 00:03:13,940 In the first part of the process, the continuous-time 38 00:03:13,940 --> 00:03:20,740 signal is modulated with impulse train, where the 39 00:03:20,740 --> 00:03:25,510 period of the impulse train is capital T. And so we have a 40 00:03:25,510 --> 00:03:29,980 continuous-time time impulse train signal which captures 41 00:03:29,980 --> 00:03:35,470 the samples of the original continuous-time signal. 42 00:03:35,470 --> 00:03:42,280 That impulse train is then put through an operation which, 43 00:03:42,280 --> 00:03:49,100 essentially, re-labels the samples so that the sample 44 00:03:49,100 --> 00:03:55,400 values, the impulse areas, are re-labeled as sequence values. 45 00:03:55,400 --> 00:03:59,520 And the result of that conversion is then the 46 00:03:59,520 --> 00:04:02,020 sequence x of n. 47 00:04:02,020 --> 00:04:06,930 So the overall process, then, is a sampling process, 48 00:04:06,930 --> 00:04:10,610 followed by what is simply, in this box, 49 00:04:10,610 --> 00:04:12,510 a re-labeling process. 50 00:04:12,510 --> 00:04:18,110 And, although as I indicated just a minute ago, that is 51 00:04:18,110 --> 00:04:21,100 essentially what an analog to digital converter does. 52 00:04:21,100 --> 00:04:24,770 An analog to digital converter doesn't necessarily carry it 53 00:04:24,770 --> 00:04:30,280 out in those two steps, but particularly, in terms of 54 00:04:30,280 --> 00:04:33,610 carrying through an analysis, thinking of it as a two-step 55 00:04:33,610 --> 00:04:36,890 process is particularly convenient. 56 00:04:36,890 --> 00:04:43,200 Now we talked last time about what this mapping from 57 00:04:43,200 --> 00:04:46,370 continuous-time to discreet-time means, both in 58 00:04:46,370 --> 00:04:49,210 the time domain, and terms of the spectra. 59 00:04:49,210 --> 00:04:53,530 And in particular, in the time domain we begin with the 60 00:04:53,530 --> 00:04:58,550 continuous-time signal, which is then sampled with an 61 00:04:58,550 --> 00:05:04,010 impulse train and converted to a sequence by simply 62 00:05:04,010 --> 00:05:08,020 generating a sequence whose values are the 63 00:05:08,020 --> 00:05:10,490 areas of the impulses. 64 00:05:10,490 --> 00:05:14,420 And I stress the fact that what this corresponds to, 65 00:05:14,420 --> 00:05:17,620 essentially, is a normalization of the time 66 00:05:17,620 --> 00:05:23,950 axis, essentially, by dividing the time axis by capital T. 67 00:05:23,950 --> 00:05:29,430 In the frequency domain, then, we had the spectrum of the 68 00:05:29,430 --> 00:05:33,150 original signal, which, because of the sampling 69 00:05:33,150 --> 00:05:40,210 process, is replicated at integer multiples of the 70 00:05:40,210 --> 00:05:45,780 sampling frequency omega sub s, or 2 pi over capital T. And 71 00:05:45,780 --> 00:05:51,480 then, in converting the impulses to a sequence, we are 72 00:05:51,480 --> 00:05:57,150 essentially normalizing the frequency axis, so that the 73 00:05:57,150 --> 00:06:01,970 frequency 2 pi over capital T gets re-labeled as 2 pi. 74 00:06:01,970 --> 00:06:05,130 And the resulting discreet-time spectrum looks 75 00:06:05,130 --> 00:06:07,110 like I indicate here. 76 00:06:07,110 --> 00:06:12,310 Which really is nothing more than a frequency scaling 77 00:06:12,310 --> 00:06:15,140 corresponding to the associated time scaling. 78 00:06:15,140 --> 00:06:19,070 So the mapping from the impulse train spectrum to the 79 00:06:19,070 --> 00:06:23,450 discreet-time spectrum corresponds to a mapping 80 00:06:23,450 --> 00:06:27,620 specified by capital omega equal to small omega times 81 00:06:27,620 --> 00:06:29,800 capital T. 82 00:06:29,800 --> 00:06:35,470 And equivalently, it's the frequency 2 pi over capital T, 83 00:06:35,470 --> 00:06:39,670 which is, of course, the sampling frequency which gets 84 00:06:39,670 --> 00:06:44,730 normalized to the frequency 2 pi. 85 00:06:44,730 --> 00:06:49,070 And so, in the frequency domain, there is a frequency 86 00:06:49,070 --> 00:06:55,620 normalization associated with the fact that corresponding to 87 00:06:55,620 --> 00:07:01,290 this spectrum is a time sequence or discreet-time 88 00:07:01,290 --> 00:07:05,030 sequence, as I showed previously, and the 89 00:07:05,030 --> 00:07:10,900 discreet-time sequence is related to the original 90 00:07:10,900 --> 00:07:16,600 continuous-time signal through a time normalization. 91 00:07:16,600 --> 00:07:22,840 In other words, these sequence values are simply samples of 92 00:07:22,840 --> 00:07:26,510 the continuous-time signal with the time axis 93 00:07:26,510 --> 00:07:27,760 renormalized. 94 00:07:30,090 --> 00:07:31,910 Now, what we want to consider-- 95 00:07:31,910 --> 00:07:34,070 this is the conversion from continuous-time to 96 00:07:34,070 --> 00:07:35,260 discreet-time-- 97 00:07:35,260 --> 00:07:39,960 what we want to consider now is the overall system which 98 00:07:39,960 --> 00:07:42,960 implements not just the conversion, but filtering, and 99 00:07:42,960 --> 00:07:45,360 then coming back out of the conversion back to 100 00:07:45,360 --> 00:07:47,630 continuous-time. 101 00:07:47,630 --> 00:07:51,400 So let's look at the overall system. 102 00:07:51,400 --> 00:07:55,610 And the overall system, of course, as I've stressed 103 00:07:55,610 --> 00:08:01,080 several times in the past, consists of first, the 104 00:08:01,080 --> 00:08:09,980 sampling process, conversion to an impulse train, and the 105 00:08:09,980 --> 00:08:13,530 impulse train converted to a sequence. 106 00:08:13,530 --> 00:08:18,430 That sequence is then processed through our 107 00:08:18,430 --> 00:08:21,180 discreet-time filter. 108 00:08:21,180 --> 00:08:25,770 And after the discreet-time time processing, the result of 109 00:08:25,770 --> 00:08:30,700 that is converted back to an impulse train. 110 00:08:30,700 --> 00:08:35,710 So this resulting process sequence is then converted 111 00:08:35,710 --> 00:08:37,980 back to an impulse train. 112 00:08:37,980 --> 00:08:43,700 And then, finally, we carry out the desampling process by 113 00:08:43,700 --> 00:08:49,900 simply using a low-pass filter with a cutoff associated with 114 00:08:49,900 --> 00:08:53,170 the sampling frequency that we used. 115 00:08:53,170 --> 00:08:57,620 Now, typically in a system like that-- which implements 116 00:08:57,620 --> 00:09:01,690 discreet-time processing of continuous-time time signals-- 117 00:09:01,690 --> 00:09:07,030 we need to ensure in one way or another that the bandwidth 118 00:09:07,030 --> 00:09:10,100 of the input is sufficiently limited, so 119 00:09:10,100 --> 00:09:12,010 that we avoid aliasing. 120 00:09:12,010 --> 00:09:16,300 One way to do that is to force it one way or another, or 121 00:09:16,300 --> 00:09:19,190 simply know that our signal satisfies the bandwidth 122 00:09:19,190 --> 00:09:20,160 constraint. 123 00:09:20,160 --> 00:09:24,190 Although, a fairly typical thing to do in addition to 124 00:09:24,190 --> 00:09:30,770 this sampling process is to include what is referred to an 125 00:09:30,770 --> 00:09:33,440 anti-aliasing filter. 126 00:09:33,440 --> 00:09:37,580 In other words, this is a filter that would band-limit 127 00:09:37,580 --> 00:09:43,090 the input at at least half the sampling frequency, so that we 128 00:09:43,090 --> 00:09:47,850 are guaranteed, then, that there is no aliasing that's 129 00:09:47,850 --> 00:09:50,400 carried out in this process. 130 00:09:50,400 --> 00:09:55,000 And it's important to stress that, in this kind of 131 00:09:55,000 --> 00:09:55,610 processing-- 132 00:09:55,610 --> 00:09:59,190 discreet-time processing of continuous-time signals-- 133 00:09:59,190 --> 00:10:02,400 except in certain special situations, it's very 134 00:10:02,400 --> 00:10:06,750 important to avoid aliasing because we're going to want to 135 00:10:06,750 --> 00:10:09,660 do a reconstruction after we do the sampling and 136 00:10:09,660 --> 00:10:10,910 processing. 137 00:10:13,260 --> 00:10:17,070 OK, now, this is the sequence of steps in the time domain. 138 00:10:17,070 --> 00:10:20,610 Let's examine what happens as a consequence of this in the 139 00:10:20,610 --> 00:10:23,240 frequency domain. 140 00:10:23,240 --> 00:10:26,730 Well, let's choose some type of simple 141 00:10:26,730 --> 00:10:28,500 representative spectrum. 142 00:10:28,500 --> 00:10:32,320 And, of course, what's important about it is that the 143 00:10:32,320 --> 00:10:35,620 spectrum we choose is band-limited, or that there's 144 00:10:35,620 --> 00:10:37,470 an anti-aliasing filter. 145 00:10:37,470 --> 00:10:41,080 And it's not the shape, of course, that is critical. 146 00:10:41,080 --> 00:10:45,140 And as we work our way through the system, this is the 147 00:10:45,140 --> 00:10:47,460 continuous-time spectrum. 148 00:10:47,460 --> 00:10:53,530 After sampling, that spectrum is replicated at multiples of 149 00:10:53,530 --> 00:10:54,690 the sampling frequency-- 150 00:10:54,690 --> 00:10:56,840 integer multiples of the sampling frequency-- 151 00:10:56,840 --> 00:10:59,560 and so there would be another one over here, and another one 152 00:10:59,560 --> 00:11:01,380 over here, et cetera. 153 00:11:01,380 --> 00:11:05,370 And then, in converting to a discreet-time sequence, there 154 00:11:05,370 --> 00:11:10,500 is the associated frequency normalization, so that the 155 00:11:10,500 --> 00:11:14,360 sampling frequency gets normalized to a 156 00:11:14,360 --> 00:11:17,230 frequency of 2 pi. 157 00:11:17,230 --> 00:11:26,670 OK, now, at that point, where we are in the system is at 158 00:11:26,670 --> 00:11:30,750 this point, so that we've converted to a sequence. 159 00:11:30,750 --> 00:11:34,350 We now want to carry out some filtering, and then, after 160 00:11:34,350 --> 00:11:40,060 that filtering, convert back to a continuous-time signal. 161 00:11:40,060 --> 00:11:45,790 All right, so, here we are at the spectrum associated with 162 00:11:45,790 --> 00:11:49,040 the sequence. 163 00:11:49,040 --> 00:11:53,410 And now, the processing that we're carrying out is linear 164 00:11:53,410 --> 00:11:57,380 time and variant filtering in the discreet-time domain. 165 00:11:57,380 --> 00:12:01,560 And what that corresponds to, then, is multiplying this 166 00:12:01,560 --> 00:12:05,080 spectrum by the filter frequency response. 167 00:12:05,080 --> 00:12:08,250 And I've chosen a particular shape. 168 00:12:08,250 --> 00:12:10,430 And again, it's not the shape that's important to the 169 00:12:10,430 --> 00:12:14,310 discussion, but the fact, for example, that it has a 170 00:12:14,310 --> 00:12:18,070 particular cutoff frequency, which we will track as we work 171 00:12:18,070 --> 00:12:19,790 through this. 172 00:12:19,790 --> 00:12:24,550 And so now, the spectrum of y of n, the output of the 173 00:12:24,550 --> 00:12:30,120 digital filter, is the product of this spectrum, and the 174 00:12:30,120 --> 00:12:32,940 Fourier transform, or frequency response, of the 175 00:12:32,940 --> 00:12:34,190 digital filter. 176 00:12:36,320 --> 00:12:39,060 Now, in working our way through, we're going to take 177 00:12:39,060 --> 00:12:43,880 the output of the filter and undo the two-step process. 178 00:12:43,880 --> 00:12:46,700 So we now want to take that sequence, convert it to an 179 00:12:46,700 --> 00:12:50,600 impulse train, and then take that impulse train and 180 00:12:50,600 --> 00:12:53,460 desample through a low-pass filter. 181 00:12:53,460 --> 00:12:59,220 So, here we are now at the output of the digital filter. 182 00:12:59,220 --> 00:13:02,340 We then convert that to an impulse train. 183 00:13:02,340 --> 00:13:05,730 Well that's really undoing the original time normalization. 184 00:13:05,730 --> 00:13:10,520 And so, what that means, is that we are undoing the 185 00:13:10,520 --> 00:13:12,080 frequency normalization. 186 00:13:12,080 --> 00:13:17,320 In particular, we're dividing the frequency axis by capital 187 00:13:17,320 --> 00:13:24,610 T. Whereas, this point in y of omega was 2 pi, now it's 2 pi 188 00:13:24,610 --> 00:13:29,540 over capital T. What that means is that, equivalently, 189 00:13:29,540 --> 00:13:33,550 we're multiplying this spectrum by the frequency 190 00:13:33,550 --> 00:13:36,270 response of the digital filter, but now 191 00:13:36,270 --> 00:13:40,225 linearly-scaled in frequency, so that what was a cutoff 192 00:13:40,225 --> 00:13:44,830 frequency of omega sub c is now cutoff frequency of omega 193 00:13:44,830 --> 00:13:48,590 sub c, divided by capital T. 194 00:13:48,590 --> 00:13:53,760 So now, the next step in the process is the reconstructing 195 00:13:53,760 --> 00:13:55,290 low-pass filter. 196 00:13:55,290 --> 00:13:59,090 And what that extracts is simply the portion of this 197 00:13:59,090 --> 00:14:02,020 periodic spectrum around the origin. 198 00:14:02,020 --> 00:14:06,550 And so finally, then, the spectrum of the output of the 199 00:14:06,550 --> 00:14:11,920 overall system will be the spectrum of the input 200 00:14:11,920 --> 00:14:19,680 multiplied by a frequency response, which is the digital 201 00:14:19,680 --> 00:14:25,040 filter frequency response frequency scaled by dividing 202 00:14:25,040 --> 00:14:30,360 that digital filter frequency axis by capital T. 203 00:14:30,360 --> 00:14:35,580 OK, now, what we can ask is, we've got this processing-- 204 00:14:35,580 --> 00:14:37,390 we've converted to discreet-time, and we've gone 205 00:14:37,390 --> 00:14:41,590 back to continuous-time, and one can ask now what 206 00:14:41,590 --> 00:14:45,010 equivalent, overall continuous-time system does 207 00:14:45,010 --> 00:14:46,970 that correspond to? 208 00:14:46,970 --> 00:14:48,550 In other words, if we-- 209 00:14:48,550 --> 00:14:50,730 that, of course, is a continuous-time system, it's a 210 00:14:50,730 --> 00:14:53,790 continuous-time input and continuous-time time output-- 211 00:14:53,790 --> 00:14:58,980 and the overall system, then, would be one that would give 212 00:14:58,980 --> 00:15:02,540 us exactly the same output spectrum as we're getting. 213 00:15:02,540 --> 00:15:04,200 Well, what is that? 214 00:15:04,200 --> 00:15:08,480 What we have is an output spectrum, which is the product 215 00:15:08,480 --> 00:15:12,790 of the input spectrum and the digital filter frequency 216 00:15:12,790 --> 00:15:16,150 characteristic frequency-scaled. 217 00:15:16,150 --> 00:15:23,420 And so, in fact, the resulting continuous-time time filter is 218 00:15:23,420 --> 00:15:27,210 simply the digital filter with an 219 00:15:27,210 --> 00:15:29,120 appropriate frequency scaling. 220 00:15:29,120 --> 00:15:31,710 In other words, with the frequency axis divided by 221 00:15:31,710 --> 00:15:42,200 capital T. So said another way, if we show here the 222 00:15:42,200 --> 00:15:51,090 frequency response of the original digital filter, then 223 00:15:51,090 --> 00:15:56,150 the corresponding continuous-time filter would 224 00:15:56,150 --> 00:15:59,470 be this, frequency-scaled. 225 00:15:59,470 --> 00:16:02,930 And then, because of the associated low-pass filtering 226 00:16:02,930 --> 00:16:06,810 and the reconstruction, we would select out just one of 227 00:16:06,810 --> 00:16:10,070 these periods-- in particular, the portion around the origin. 228 00:16:10,070 --> 00:16:13,270 And the essential consequence of that is that the 229 00:16:13,270 --> 00:16:15,900 corresponding continuous-time filter, 230 00:16:15,900 --> 00:16:18,790 then, is given by this. 231 00:16:18,790 --> 00:16:24,150 And these two are related simply by a linear scaling of 232 00:16:24,150 --> 00:16:25,840 the frequency axis. 233 00:16:25,840 --> 00:16:28,830 And note that, where the digital filter has a cutoff 234 00:16:28,830 --> 00:16:33,100 frequency of omega sub c, the continuous-time filter has a 235 00:16:33,100 --> 00:16:38,560 cutoff frequency of omega sub c divided by capital T. 236 00:16:38,560 --> 00:16:41,750 So that's the linear frequency scaling. 237 00:16:41,750 --> 00:16:45,690 And, by the way, plant away for now-- and we'll return to 238 00:16:45,690 --> 00:16:48,170 this point later-- 239 00:16:48,170 --> 00:16:52,960 the observation that even if the digital filter frequency 240 00:16:52,960 --> 00:16:57,720 response is fixed, which we would assume it is, by 241 00:16:57,720 --> 00:17:00,740 changing the sampling frequency, in fact, what we're 242 00:17:00,740 --> 00:17:05,510 able to do is affect a linear scaling all of the equivalent 243 00:17:05,510 --> 00:17:09,170 continuous-time filter. 244 00:17:09,170 --> 00:17:12,960 OK, well, this is pretty much the process and the analysis, 245 00:17:12,960 --> 00:17:17,390 but to highlight a number of the issues and emphasize these 246 00:17:17,390 --> 00:17:24,530 points, what I'd like to do is illustrate some of this with a 247 00:17:24,530 --> 00:17:29,840 videotape demonstration that, in fact, was made originally 248 00:17:29,840 --> 00:17:35,390 as part of another course-- a course devoted entirely to 249 00:17:35,390 --> 00:17:38,120 digital signal processing, which essentially is 250 00:17:38,120 --> 00:17:41,200 discreet-time time processing, whether or not it's related to 251 00:17:41,200 --> 00:17:43,060 continuous-time signals. 252 00:17:43,060 --> 00:17:48,240 And what I'd like to now focus on are some of the details of 253 00:17:48,240 --> 00:17:50,980 that demonstration. 254 00:17:50,980 --> 00:17:57,450 In the demonstration, the specific impulse response that 255 00:17:57,450 --> 00:18:01,600 is used for the digital filter, or discreet-time 256 00:18:01,600 --> 00:18:06,160 filter, is the one that I show here. 257 00:18:06,160 --> 00:18:12,360 And the associated frequency response is the frequency 258 00:18:12,360 --> 00:18:16,460 response of a discreet-time, low-pass filter, 259 00:18:16,460 --> 00:18:18,800 as I indicate below. 260 00:18:18,800 --> 00:18:21,520 And the cutoff frequency of that filter-- 261 00:18:21,520 --> 00:18:24,890 as I indicate, the filter was designed as a discreet-time 262 00:18:24,890 --> 00:18:28,680 filter with a cutoff frequency of pi over 5. 263 00:18:28,680 --> 00:18:32,540 And let me just draw your attention to the fact that pi 264 00:18:32,540 --> 00:18:39,200 over 5 is also a 10th of 2 pi. 265 00:18:39,200 --> 00:18:44,140 And so in fact the digital or discreet-time filter cutoff 266 00:18:44,140 --> 00:18:47,250 frequency is a 10th of 2 pi. 267 00:18:47,250 --> 00:18:51,230 And as I'll stress again shortly, remember that, in the 268 00:18:51,230 --> 00:18:55,670 frequency normalization or unnormalization, 2 pi 269 00:18:55,670 --> 00:18:58,940 represents, in effect, the sampling frequency. 270 00:18:58,940 --> 00:19:03,080 And so the consequence of that is that the cutoff frequency 271 00:19:03,080 --> 00:19:06,750 really, is going to be associated with a 10th of the 272 00:19:06,750 --> 00:19:07,760 sampling frequency. 273 00:19:07,760 --> 00:19:09,720 But, for now, keep in mind that it's just 274 00:19:09,720 --> 00:19:13,040 simply a 10th of 2 pi. 275 00:19:13,040 --> 00:19:19,040 Now, the equivalent continuous-time system, in 276 00:19:19,040 --> 00:19:23,600 terms of the impulse response, is, of course, a band-limited 277 00:19:23,600 --> 00:19:31,350 interpolation of the impulse response associated with the 278 00:19:31,350 --> 00:19:33,380 discreet-time filter. 279 00:19:33,380 --> 00:19:37,080 And in the frequency domain, the frequency response is 280 00:19:37,080 --> 00:19:44,380 correspondingly a time-scaled, frequency scaled version of 281 00:19:44,380 --> 00:19:46,290 the frequency response. 282 00:19:46,290 --> 00:19:50,900 So, in fact, in the frequency domain and in the time domain 283 00:19:50,900 --> 00:19:57,320 related to the continuous-time signal, the associated impulse 284 00:19:57,320 --> 00:20:00,540 response is what I indicate here-- 285 00:20:00,540 --> 00:20:04,950 a band-limited interpolation of the discreet-time impulse 286 00:20:04,950 --> 00:20:09,340 response and time scale, in fact. 287 00:20:09,340 --> 00:20:11,000 And the frequency response-- 288 00:20:11,000 --> 00:20:12,680 following the discussion that we've 289 00:20:12,680 --> 00:20:14,410 previously gone through-- 290 00:20:14,410 --> 00:20:18,390 is a frequency-scaled version of the one associated with the 291 00:20:18,390 --> 00:20:20,630 digital filter. 292 00:20:20,630 --> 00:20:24,190 Well, the first thing that I'll want to look at is the 293 00:20:24,190 --> 00:20:25,980 impulse response. 294 00:20:25,980 --> 00:20:32,370 And when we do, let me just indicate that in the actual 295 00:20:32,370 --> 00:20:36,350 implementation things are slightly different than they 296 00:20:36,350 --> 00:20:39,720 are associated with the ideal analysis. 297 00:20:39,720 --> 00:20:46,920 In particular, in converting from a discreet-time sequence 298 00:20:46,920 --> 00:20:52,070 to the continuous-time signal, whereas this way of looking at 299 00:20:52,070 --> 00:20:56,530 it is convenient in the context of the analysis, in 300 00:20:56,530 --> 00:21:02,830 fact, the way it's done is using a more or less standard 301 00:21:02,830 --> 00:21:04,840 digital to analog converter. 302 00:21:04,840 --> 00:21:08,530 And what a digital to analog converter does, as I indicated 303 00:21:08,530 --> 00:21:13,900 in the previous lecture, is to convert the sequence not to an 304 00:21:13,900 --> 00:21:17,400 impulse train but, in fact, to go directly through a 305 00:21:17,400 --> 00:21:18,930 zero-order hold. 306 00:21:18,930 --> 00:21:22,140 And so, usually what comes out of a digital to analog 307 00:21:22,140 --> 00:21:27,590 converter is a staircase type of signal associated with a 308 00:21:27,590 --> 00:21:29,150 zero-order hold. 309 00:21:29,150 --> 00:21:33,260 And then, the result of that is low-pass filtered to do the 310 00:21:33,260 --> 00:21:34,590 reconstruction. 311 00:21:34,590 --> 00:21:39,410 So what we'll want to look at, then, is that reconstruction, 312 00:21:39,410 --> 00:21:42,170 first with just an impulse input. 313 00:21:42,170 --> 00:21:47,420 And so, what we'll see after the low-pass filter, for the 314 00:21:47,420 --> 00:21:52,900 impulse response, is a smooth curve like this. 315 00:21:52,900 --> 00:21:57,320 But also, as part of the demonstration, what I'll do, 316 00:21:57,320 --> 00:22:01,300 just to show the zero-order hold, is to take the low-pass 317 00:22:01,300 --> 00:22:05,610 filter out temporarily and then put it back in. 318 00:22:05,610 --> 00:22:11,010 So first, let's just look at the filter impulse response. 319 00:22:11,010 --> 00:22:14,790 What we see here is the impulse response of the 320 00:22:14,790 --> 00:22:17,040 overall system. 321 00:22:17,040 --> 00:22:20,600 And we observe, for one thing, that it's a symmetrical 322 00:22:20,600 --> 00:22:21,620 impulse response. 323 00:22:21,620 --> 00:22:25,170 In other words, corresponds to a linear phase filter. 324 00:22:25,170 --> 00:22:28,190 We could also look at the impulse response before the 325 00:22:28,190 --> 00:22:31,340 desampling low-pass filter-- let's take out the desampling 326 00:22:31,340 --> 00:22:34,050 low-pass filter slowly-- 327 00:22:34,050 --> 00:22:38,700 and what we observe is, basically, the output of the 328 00:22:38,700 --> 00:22:40,570 digital to analog converter. 329 00:22:40,570 --> 00:22:43,440 Which, of course, is a staircase, or boxcar, 330 00:22:43,440 --> 00:22:45,425 function, not an impulse train. 331 00:22:45,425 --> 00:22:48,640 In the real world, the output of a D to A converter, 332 00:22:48,640 --> 00:22:51,590 generally, is a boxcar type of function. 333 00:22:51,590 --> 00:22:55,990 We can put the desampling filter back in now and notice 334 00:22:55,990 --> 00:22:58,330 that the effect of the desampling filter is, 335 00:22:58,330 --> 00:23:04,210 basically, to smooth out the rough edges in the boxcar 336 00:23:04,210 --> 00:23:05,605 output from the D to A converter. 337 00:23:08,570 --> 00:23:12,210 OK, so, that's the impulse response of the system. 338 00:23:12,210 --> 00:23:15,910 Now, what I'd like to show is the frequency 339 00:23:15,910 --> 00:23:17,650 response of the system. 340 00:23:17,650 --> 00:23:20,390 And to measure the frequency response, of course, what we 341 00:23:20,390 --> 00:23:28,260 can do is put a sine wave into the system and look at the 342 00:23:28,260 --> 00:23:29,780 sinusoidal output. 343 00:23:29,780 --> 00:23:33,590 So, in particular now, what will happen is that, with the 344 00:23:33,590 --> 00:23:38,540 system, we will put in a continuous-time sinusoid, 345 00:23:38,540 --> 00:23:43,350 which is sampled, converted to a sequence. 346 00:23:43,350 --> 00:23:46,630 The sampled continuous-time sinusoid is a 347 00:23:46,630 --> 00:23:48,670 discreet-time sinusoid. 348 00:23:48,670 --> 00:23:52,650 That goes through the digital filter and gets attenuated, or 349 00:23:52,650 --> 00:23:55,110 amplified appropriately. 350 00:23:55,110 --> 00:23:59,400 And then the output of that is converted back-- and that's, 351 00:23:59,400 --> 00:24:01,450 again, a sinusoidal output-- 352 00:24:01,450 --> 00:24:08,000 that gets converted back to a continuous-time sinusoid. 353 00:24:08,000 --> 00:24:11,110 Theoretically, as I indicate here-- but again, as we just 354 00:24:11,110 --> 00:24:15,520 saw, really, represented by a zero-order hold, followed by a 355 00:24:15,520 --> 00:24:17,680 low-pass filter. 356 00:24:17,680 --> 00:24:22,120 So, that's the overall operation, with one 357 00:24:22,120 --> 00:24:25,310 modification from the diagram that we have here. 358 00:24:25,310 --> 00:24:30,430 In this particular diagram I've included and 359 00:24:30,430 --> 00:24:32,540 anti-aliasing filter. 360 00:24:32,540 --> 00:24:35,540 In fact, in the demonstration there is no 361 00:24:35,540 --> 00:24:37,510 anti-aliasing filter. 362 00:24:37,510 --> 00:24:43,750 And so, in fact, the input is a sinusoidal input which is 363 00:24:43,750 --> 00:24:48,990 not band-limited by virtue of an anti-aliasing filter. 364 00:24:48,990 --> 00:24:53,610 It's only, of course, band-limited appropriately if 365 00:24:53,610 --> 00:24:57,530 we choose the sinusoidal frequency that way. 366 00:24:57,530 --> 00:24:59,900 So, there is no anti-aliasing filter, 367 00:24:59,900 --> 00:25:02,070 and this is the system. 368 00:25:02,070 --> 00:25:09,020 And one consequence of that is that, in fact, if we sweep the 369 00:25:09,020 --> 00:25:13,900 input sinusoid only up to half the sampling frequency, we'll 370 00:25:13,900 --> 00:25:15,490 see no aliasing. 371 00:25:15,490 --> 00:25:17,360 But if we let it sweep past that, we're 372 00:25:17,360 --> 00:25:19,100 going to get aliasing. 373 00:25:19,100 --> 00:25:23,560 Now, in the demonstration, the sampling rate that's picked 374 00:25:23,560 --> 00:25:25,950 for this part of the demonstration is a 20 375 00:25:25,950 --> 00:25:28,790 kilohertz sampling rate. 376 00:25:28,790 --> 00:25:31,950 That means, based on the sampling theorem, that as long 377 00:25:31,950 --> 00:25:34,940 as the input frequency is below 10 378 00:25:34,940 --> 00:25:39,300 kilohertz we get no aliasing. 379 00:25:39,300 --> 00:25:43,900 When the input frequency goes beyond 10 kilohertz , that 380 00:25:43,900 --> 00:25:46,760 higher frequency is going to get aliased down 381 00:25:46,760 --> 00:25:48,670 into a lower frequency. 382 00:25:48,670 --> 00:25:51,590 A consequence of that, then, is that as we go through the 383 00:25:51,590 --> 00:25:55,910 processing, and we demonstrate the frequency response of the 384 00:25:55,910 --> 00:26:02,310 system, what we'll see in the output is no aliasing when the 385 00:26:02,310 --> 00:26:04,660 input is below 10 kilohertz. 386 00:26:04,660 --> 00:26:08,310 As the input sweeps past 10 kilohertz-- 387 00:26:08,310 --> 00:26:10,450 when we let it, which we will eventually in the 388 00:26:10,450 --> 00:26:11,880 demonstration-- 389 00:26:11,880 --> 00:26:16,460 then, in fact, that frequency, as it finally shows up here, 390 00:26:16,460 --> 00:26:21,870 will begin to be aliased down into a lower frequency. 391 00:26:21,870 --> 00:26:26,640 Another way of thinking about that is that, when we watch 392 00:26:26,640 --> 00:26:33,090 the frequency response of the system, as we look at the 393 00:26:33,090 --> 00:26:37,750 digital filter frequency response, what we're sweeping 394 00:26:37,750 --> 00:26:45,180 as we go from 0 up to 10 kilohertz in the input 395 00:26:45,180 --> 00:26:49,010 frequency is this portion of the frequency response. 396 00:26:49,010 --> 00:26:53,170 As we sweep from 10 kilohertz out to 20 kilohertz, what 397 00:26:53,170 --> 00:26:57,170 we'll see is this portion of the frequency response. 398 00:26:57,170 --> 00:27:00,880 In other words, we'll see it periodically replicated. 399 00:27:00,880 --> 00:27:06,520 Or, if we look at the corresponding continuous-time 400 00:27:06,520 --> 00:27:10,350 frequency response, what it means, really, is that 401 00:27:10,350 --> 00:27:14,510 sweeping from 0 to 10 kilohertz is 402 00:27:14,510 --> 00:27:16,190 moving up this way. 403 00:27:16,190 --> 00:27:20,360 And then sweeping from 10 kilohertz to 20 kilohertz on 404 00:27:20,360 --> 00:27:26,330 the input, really because of the aliasing, reflects itself 405 00:27:26,330 --> 00:27:30,460 in the digital filter by looking back toward lower 406 00:27:30,460 --> 00:27:31,590 frequencies. 407 00:27:31,590 --> 00:27:36,310 And so the continuous-time filter sweeps back down from 408 00:27:36,310 --> 00:27:40,520 10 kilohertz back to 0. 409 00:27:40,520 --> 00:27:44,020 OK, so, that's what we'll see, and we'll see it in several 410 00:27:44,020 --> 00:27:46,560 different ways as explained in the demonstration. 411 00:27:46,560 --> 00:27:49,130 So now let's look at the frequency 412 00:27:49,130 --> 00:27:51,930 response of the filter. 413 00:27:51,930 --> 00:27:54,440 Now what we'd like to illustrate is the frequency 414 00:27:54,440 --> 00:27:58,610 response of the equivalent continuous-time filter. 415 00:27:58,610 --> 00:28:01,010 And we can do that by sweeping the filter 416 00:28:01,010 --> 00:28:02,880 with sinusoidal input. 417 00:28:02,880 --> 00:28:06,890 So, what we'll see in this demonstration is, on the upper 418 00:28:06,890 --> 00:28:10,180 trace, the input sinusoid, on the lower trace, the output 419 00:28:10,180 --> 00:28:14,740 sinusoid, using a 20 kilohertz sampling rate, and a sweep 420 00:28:14,740 --> 00:28:16,420 from 0 to 10 kilohertz. 421 00:28:16,420 --> 00:28:21,080 In other words, a sweep from 0 to, effectively, pi, in terms 422 00:28:21,080 --> 00:28:23,320 of the digital filter. 423 00:28:23,320 --> 00:28:28,430 So what we'll observe as the input frequency increases, is 424 00:28:28,430 --> 00:28:31,450 that the output sinusoid will have, essentially, constant 425 00:28:31,450 --> 00:28:35,020 amplitude up to the cutoff frequency of the filter, and 426 00:28:35,020 --> 00:28:38,170 then approximately zero amplitude past. 427 00:28:38,170 --> 00:28:41,525 So let's now sweep the filter frequency response. 428 00:28:46,580 --> 00:28:50,410 And there is the filter cutoff frequency. 429 00:28:55,300 --> 00:28:59,310 Now, we can also observe the filter frequency response in 430 00:28:59,310 --> 00:29:00,840 several other ways. 431 00:29:00,840 --> 00:29:04,180 One way in which we can observe it is by looking, 432 00:29:04,180 --> 00:29:09,880 also, at the amplitude of the output sinusoid as a function 433 00:29:09,880 --> 00:29:12,980 of frequency, rather than as a function of time. 434 00:29:12,980 --> 00:29:16,530 And so we'll observe that on the left-hand scope. 435 00:29:16,530 --> 00:29:19,280 While on the right-hand scope, we'll have the same trace the 436 00:29:19,280 --> 00:29:21,760 we just saw, namely two traces-- 437 00:29:21,760 --> 00:29:24,990 the upper trace is the inputs sinusoid, the lower trace is 438 00:29:24,990 --> 00:29:26,800 the output sinusoid. 439 00:29:26,800 --> 00:29:30,730 And, in addition to observing the frequency response, let's 440 00:29:30,730 --> 00:29:35,430 also listen to the output sinusoid and observe the 441 00:29:35,430 --> 00:29:38,730 attenuation in the output as we go from the filter passband 442 00:29:38,730 --> 00:29:40,520 to the filter stopband. 443 00:29:40,520 --> 00:29:44,550 Again, a 20 kilohertz sampling rate and a sweep range from 0 444 00:29:44,550 --> 00:29:45,800 to 10 kilohertz. 445 00:29:53,290 --> 00:29:54,850 Now, of course, we're in the filter stopband. 446 00:29:58,360 --> 00:30:04,310 Now, if we increase the sweep range from 10 kilohertz the 20 447 00:30:04,310 --> 00:30:07,880 kilohertz, so that the sweep range is equal to the sampling 448 00:30:07,880 --> 00:30:11,310 frequency, in essence, that corresponds to sweeping out 449 00:30:11,310 --> 00:30:15,110 the digital filter from 0 to 2 pi. 450 00:30:15,110 --> 00:30:18,810 And, in that case, we'll begin to see some of the periodicity 451 00:30:18,810 --> 00:30:21,210 in the digital filter frequency response. 452 00:30:21,210 --> 00:30:25,890 So let's do that now with a 20 kilohertz sampling rate and a 453 00:30:25,890 --> 00:30:27,870 sweep range of 0 to 20 kilohertz. 454 00:30:32,790 --> 00:30:41,140 Now as we come near 2 pi, we get back the past-band. 455 00:30:41,140 --> 00:30:45,780 And, finally, back to a 0 to 10 kilohertz sweep, so that 456 00:30:45,780 --> 00:30:50,130 we're again sweeping only from 0 to pi with regard to the 457 00:30:50,130 --> 00:30:51,380 digital filter. 458 00:31:05,070 --> 00:31:10,560 Now, an important observation is that, with the digital or 459 00:31:10,560 --> 00:31:15,040 discreet-time filter cutoff frequency fixed as I've 460 00:31:15,040 --> 00:31:16,130 indicated here-- 461 00:31:16,130 --> 00:31:20,110 and I remind you that what the cutoff frequency is, 462 00:31:20,110 --> 00:31:22,370 is a 10th of 2 pi-- 463 00:31:22,370 --> 00:31:26,060 with that cutoff frequency fixed, because of the 464 00:31:26,060 --> 00:31:30,470 normalization that we get as we come back to a 465 00:31:30,470 --> 00:31:35,335 continuous-time filter, in fact, what we have is a cutoff 466 00:31:35,335 --> 00:31:41,140 frequency that is dependent on the sampling frequency or on 467 00:31:41,140 --> 00:31:42,620 the sampling period. 468 00:31:42,620 --> 00:31:46,580 And, more specifically, since the discreet-time, or digital, 469 00:31:46,580 --> 00:31:50,010 filter or has a cutoff frequency which is a 10th of 2 470 00:31:50,010 --> 00:31:56,630 pi, the normalization, as you recall, is that 2 pi, in 471 00:31:56,630 --> 00:32:00,780 discreet-time frequency, corresponds to omega sub s, 472 00:32:00,780 --> 00:32:02,750 the sampling frequency, in terms of 473 00:32:02,750 --> 00:32:05,180 continuous-time frequency. 474 00:32:05,180 --> 00:32:09,540 The consequence is that this cutoff frequency, in 475 00:32:09,540 --> 00:32:13,970 fact, is 1/10 of-- 476 00:32:13,970 --> 00:32:16,680 not 2 pi now because of the normalization-- 477 00:32:16,680 --> 00:32:20,180 it's 1/10 of the sampling frequency. 478 00:32:20,180 --> 00:32:25,980 So, consequently, as we change the sampling frequency, what 479 00:32:25,980 --> 00:32:29,010 will happen is that, even with the discreet-time filter 480 00:32:29,010 --> 00:32:33,870 cutoff fixed, the cutoff frequency of the equivalent 481 00:32:33,870 --> 00:32:37,810 continuous-time filter will change. 482 00:32:37,810 --> 00:32:40,570 Now, that's what I want to demonstrate. 483 00:32:40,570 --> 00:32:44,790 But let me again stress and ask you to keep in mind that 484 00:32:44,790 --> 00:32:47,010 this demonstration is done without an 485 00:32:47,010 --> 00:32:49,480 anti-aliasing filter in. 486 00:32:49,480 --> 00:32:53,590 And we are going to be changing the sampling 487 00:32:53,590 --> 00:33:00,650 frequency and, so keep in mind that, as we look at this, as 488 00:33:00,650 --> 00:33:04,940 the input frequency sweeps past half the sampling 489 00:33:04,940 --> 00:33:05,560 frequency-- 490 00:33:05,560 --> 00:33:08,680 whatever sampling frequency we happen to be looking at-- 491 00:33:08,680 --> 00:33:11,550 then, because of the fact that there's no anti-aliasing 492 00:33:11,550 --> 00:33:14,360 filter we'll get aliasing. 493 00:33:14,360 --> 00:33:17,230 In other words, the frequency and the digital filter, or 494 00:33:17,230 --> 00:33:21,040 discreet-time filter, as we sweep the input frequency up, 495 00:33:21,040 --> 00:33:25,280 will move up in frequency until we get past half the 496 00:33:25,280 --> 00:33:28,130 sampling frequency and then essentially will move back 497 00:33:28,130 --> 00:33:31,170 down in frequency. 498 00:33:31,170 --> 00:33:33,570 Consequently, what we'll get, then-- 499 00:33:33,570 --> 00:33:36,340 or what we'll see-- are periodic replications of the 500 00:33:36,340 --> 00:33:39,370 frequency response when we swept past half 501 00:33:39,370 --> 00:33:40,910 the sampling frequency. 502 00:33:40,910 --> 00:33:45,440 All right, so now, let's look at the same digital filter, 503 00:33:45,440 --> 00:33:50,040 but the frequency response, as we change 504 00:33:50,040 --> 00:33:52,980 the sampling frequency. 505 00:33:52,980 --> 00:33:56,070 Now, what we would like to demonstrate is the effect of 506 00:33:56,070 --> 00:33:58,450 changing the sampling frequency. 507 00:33:58,450 --> 00:34:03,930 And we know that the effective filter cutoff frequency is 508 00:34:03,930 --> 00:34:08,139 tied to the sampling frequency and, for this particular 509 00:34:08,139 --> 00:34:12,719 filter, corresponds to a 10th of the sampling frequency. 510 00:34:12,719 --> 00:34:15,690 Consequently, if we double the sampling frequency, we should 511 00:34:15,690 --> 00:34:20,690 double the effective filter passband width, or double the 512 00:34:20,690 --> 00:34:22,730 filter cutoff frequency. 513 00:34:22,730 --> 00:34:25,100 And, so, let's do that now. 514 00:34:25,100 --> 00:34:29,070 Again a 0 to 10 kilohertz sweep range, but a 40 515 00:34:29,070 --> 00:34:30,355 kilohertz sampling frequency. 516 00:34:34,690 --> 00:34:38,560 And we should observe that the filter cutoff frequency has 517 00:34:38,560 --> 00:34:40,574 now doubled out to four kilohertz. 518 00:34:43,469 --> 00:34:47,889 Now, let's begin to decrease the filter sampling frequency. 519 00:34:47,889 --> 00:34:50,110 So from 40, let's change the sampling 520 00:34:50,110 --> 00:34:52,730 frequency to 20 kilohertz. 521 00:34:52,730 --> 00:34:55,489 We should see the cutoff frequency cut in half. 522 00:35:05,970 --> 00:35:07,710 Now, we can go even further. 523 00:35:07,710 --> 00:35:08,810 We can cut the sampling 524 00:35:08,810 --> 00:35:10,650 frequency down to 10 kilohertz. 525 00:35:10,650 --> 00:35:13,800 And remember that the sweep range is 0 to 10 kilohertz. 526 00:35:13,800 --> 00:35:16,080 So now we'll be sweeping from 0 to 2 pi. 527 00:35:23,830 --> 00:35:29,240 So as we get close to 2 pi, we'll see the passband again. 528 00:35:29,240 --> 00:35:33,040 And, now, let's cut down the sampling frequency even 529 00:35:33,040 --> 00:35:34,965 further, to 5 kilohertz. 530 00:35:40,990 --> 00:35:42,240 Here we are at 2 pi. 531 00:35:45,020 --> 00:35:46,830 And then at 4pi. 532 00:35:49,390 --> 00:35:52,110 All right, so, that illustrates the effect of 533 00:35:52,110 --> 00:35:54,610 changing the sampling frequency. 534 00:35:54,610 --> 00:35:57,920 Now let's conclude this demonstration of the effect of 535 00:35:57,920 --> 00:36:01,180 the sampling frequency on the filter cutoff frequency by 536 00:36:01,180 --> 00:36:04,810 carrying out some filtering on some live audio. 537 00:36:04,810 --> 00:36:09,870 What we'll watch, in this case, is the output audio 538 00:36:09,870 --> 00:36:15,220 waveform as a function of time on the single tray scope, and 539 00:36:15,220 --> 00:36:18,220 also we'll listen to the output. 540 00:36:18,220 --> 00:36:21,870 We'll begin it with a 40 kilohertz sampling rate, then 541 00:36:21,870 --> 00:36:23,870 reduce that to 20 kilohertz, 10 542 00:36:23,870 --> 00:36:26,100 kilohertz, and then 5 kilohertz. 543 00:36:26,100 --> 00:36:29,040 And in each of those cases, the effective filter cutoff 544 00:36:29,040 --> 00:36:33,130 frequency, then, is cut in half from 4 kilohertz, to 2 545 00:36:33,130 --> 00:36:37,080 kilohertz, to 1 kilohertz, and then to 500 cycles. 546 00:36:37,080 --> 00:36:40,120 So let's begin with a 40 kilohertz sampling frequency, 547 00:36:40,120 --> 00:36:43,375 or an effective filter cutoff frequency of 4 kilohertz. 548 00:36:49,950 --> 00:36:54,090 Now, let's reduce that a 20 kilohertz sampling frequency, 549 00:36:54,090 --> 00:36:55,340 or a 2 kilohertz filter. 550 00:37:00,090 --> 00:37:02,655 Then a 10 kilohertz sampling frequency. 551 00:37:07,950 --> 00:37:11,130 And, finally, a 5 kilohertz sampling frequency 552 00:37:11,130 --> 00:37:15,615 corresponding to a 500 cycle equivalent analog filter. 553 00:37:20,280 --> 00:37:24,760 Alright, now, let's finally conclude by returning to a 554 00:37:24,760 --> 00:37:27,916 little higher quality ragtime by changing the sampling 555 00:37:27,916 --> 00:37:30,406 frequency back to 40 kilohertz. 556 00:37:37,880 --> 00:37:40,700 Alright, well, hopefully what you've seen in the 557 00:37:40,700 --> 00:37:46,775 demonstration and in this lecture gives you a sense and 558 00:37:46,775 --> 00:37:52,780 a feeling for the analysis and the use of discreet-time 559 00:37:52,780 --> 00:37:56,160 filters for processing continuous-time signals. 560 00:37:56,160 --> 00:38:00,480 And as you may be aware, and as I've tried to indicate 561 00:38:00,480 --> 00:38:04,020 previously in the past, this, in fact, is one very 562 00:38:04,020 --> 00:38:08,670 important-- but not the only-- but one very important context 563 00:38:08,670 --> 00:38:12,000 in which discreet-time filtering is used. 564 00:38:12,000 --> 00:38:16,480 And this, in fact, is an area that is developing rapidly 565 00:38:16,480 --> 00:38:20,800 because of the fact that microprocessors, digital 566 00:38:20,800 --> 00:38:25,100 technology, computers, et cetera afford considerable 567 00:38:25,100 --> 00:38:30,710 flexibility in carrying out digital processing of signals. 568 00:38:30,710 --> 00:38:37,540 And when digital processing is used, that naturally 569 00:38:37,540 --> 00:38:44,190 corresponds to implementing the processing and analyzing 570 00:38:44,190 --> 00:38:45,440 it in discreet-time. 571 00:38:48,100 --> 00:38:52,120 Now, in the next lecture we'll be 572 00:38:52,120 --> 00:38:54,980 continuing on another aspect-- 573 00:38:54,980 --> 00:38:56,460 developing another aspect-- 574 00:38:56,460 --> 00:38:58,400 of sampling. 575 00:38:58,400 --> 00:39:02,550 And, in particular, what we'll be talking about is sampling 576 00:39:02,550 --> 00:39:04,760 of discreet-time signals. 577 00:39:04,760 --> 00:39:08,440 As I'll indicate there, one of the contexts in which 578 00:39:08,440 --> 00:39:12,890 discreet-time sampling, in fact, plays an important role 579 00:39:12,890 --> 00:39:18,180 is in the context in which we are processing continuous-time 580 00:39:18,180 --> 00:39:21,170 signals using discreet-time processing. 581 00:39:21,170 --> 00:39:24,510 Where, in fact, one step that we might want to take-- in 582 00:39:24,510 --> 00:39:27,130 addition to the steps so we've talked about here-- 583 00:39:27,130 --> 00:39:31,470 is an additional sampling process following whatever 584 00:39:31,470 --> 00:39:33,770 kinds of filtering that we do. 585 00:39:33,770 --> 00:39:36,650 Well, that's a discussion and a topic that we'll be going 586 00:39:36,650 --> 00:39:37,930 into in the next lecture. 587 00:39:37,930 --> 00:39:39,180 Thank you.