WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:55.941 --> 00:00:58.330 ALAN V. OPPENHEIM: Last time, we introduced the notion of 00:00:58.330 --> 00:01:01.120 mapping continuous time filters to discrete time 00:01:01.120 --> 00:01:06.150 filters, and we developed impulse invariance as one 00:01:06.150 --> 00:01:10.470 useful technique for carrying out that type of mapping. 00:01:10.470 --> 00:01:15.520 What I'd like to do in this lecture is illustrate impulse 00:01:15.520 --> 00:01:19.600 invariance as a design procedure in the context of 00:01:19.600 --> 00:01:24.900 one specific class of filters, namely Butterworth filters. 00:01:24.900 --> 00:01:29.790 And then following that, we'll proceed on to discuss another 00:01:29.790 --> 00:01:33.610 very important and useful mapping or design procedure 00:01:33.610 --> 00:01:36.810 referred to as the bilinear transformation. 00:01:36.810 --> 00:01:42.020 So to begin, let me just discuss briefly what the class 00:01:42.020 --> 00:01:44.580 of Butterworth filters is. 00:01:44.580 --> 00:01:49.720 And specifically, the Butterworth filters are 00:01:49.720 --> 00:01:52.710 defined through their frequency 00:01:52.710 --> 00:01:54.390 response or transfer function. 00:01:54.390 --> 00:01:58.530 And I'm using capital B to denote a Butterworth filter. 00:01:58.530 --> 00:02:01.080 And by definition, the magnitude squared of the 00:02:01.080 --> 00:02:04.020 frequency response of a Butterworth filter is given by 00:02:04.020 --> 00:02:05.310 this expression. 00:02:05.310 --> 00:02:09.750 And for example, if N were equal to 1, then this would 00:02:09.750 --> 00:02:13.350 simply correspond to the magnitude squared of the 00:02:13.350 --> 00:02:18.130 frequency response for a first order differential equation. 00:02:18.130 --> 00:02:23.790 Now if you look at the frequency response, B of omega 00:02:23.790 --> 00:02:25.930 for this class of filters-- 00:02:25.930 --> 00:02:28.610 I've illustrated that below-- 00:02:28.610 --> 00:02:35.370 and what we see is that the frequency response starts at 00:02:35.370 --> 00:02:39.110 unity, because that's the way that it's normalized, and it 00:02:39.110 --> 00:02:42.730 has a monotonic characteristic in the pass band 00:02:42.730 --> 00:02:44.820 and in the stop band. 00:02:44.820 --> 00:02:49.340 At a frequency equal to the parameter omega sub c, up 00:02:49.340 --> 00:02:52.800 here, which is referred to as the cutoff frequency, the 00:02:52.800 --> 00:02:56.480 Butterworth filter frequency response always goes through 00:02:56.480 --> 00:03:00.100 the same point, namely 0.707. 00:03:00.100 --> 00:03:04.130 And as the order of the Butterworth filter, capital N, 00:03:04.130 --> 00:03:11.220 increases, the transition from the pass band to the stop band 00:03:11.220 --> 00:03:12.780 becomes sharper and sharper. 00:03:12.780 --> 00:03:16.320 So for higher order filters, then, the frequency response 00:03:16.320 --> 00:03:20.810 is flatter in the pass band and drops off more quickly, 00:03:20.810 --> 00:03:24.950 and attenuates more in the stop band. 00:03:24.950 --> 00:03:29.520 Now in designing Butterworth filters, what we want to look 00:03:29.520 --> 00:03:36.060 at is the location of the poles of the system function. 00:03:36.060 --> 00:03:40.780 And we can infer those from our definition of what the 00:03:40.780 --> 00:03:44.050 frequency response for the Butterworth filter is. 00:03:44.050 --> 00:03:48.760 In particular, we have this expression for the magnitude 00:03:48.760 --> 00:03:51.190 squared of the frequency response. 00:03:51.190 --> 00:03:58.170 And we recognize that, of course, as B of j omega times 00:03:58.170 --> 00:04:03.240 B of minus j omega, that's what the magnitude squared 00:04:03.240 --> 00:04:04.910 will be equal to. 00:04:04.910 --> 00:04:10.110 And in order to convert this to an expression describing 00:04:10.110 --> 00:04:13.110 the system function, or in terms of the more general 00:04:13.110 --> 00:04:18.000 Laplace transform variable s, what we recognize is that j 00:04:18.000 --> 00:04:23.005 omega, in the more general setting, simply corresponds to 00:04:23.005 --> 00:04:25.860 the Laplace transform variable s. 00:04:25.860 --> 00:04:32.440 So this product, in fact, is the Laplace transform for s 00:04:32.440 --> 00:04:34.070 equal to j omega. 00:04:34.070 --> 00:04:38.970 More generally, then, this is the result of evaluating B of 00:04:38.970 --> 00:04:42.830 s times B of minus s at s equals j omega. 00:04:42.830 --> 00:04:48.060 Consequently, comparing this expression with this statement 00:04:48.060 --> 00:04:52.490 leads us to the conclusion that the transfer function B 00:04:52.490 --> 00:04:57.730 of s of the Butterworth filter, times B of minus s, is 00:04:57.730 --> 00:05:01.250 given by the expression that I indicate here, simply 00:05:01.250 --> 00:05:05.490 replacing j omega by s. 00:05:05.490 --> 00:05:10.540 Now what we want to look at are the poles of B of s. 00:05:10.540 --> 00:05:14.200 That's what we'll want to get as we design 00:05:14.200 --> 00:05:15.870 a Butterworth filter. 00:05:15.870 --> 00:05:21.520 And we can recognize the poles of this product simply by 00:05:21.520 --> 00:05:24.470 looking at the roots of the denominative polynomial. 00:05:24.470 --> 00:05:26.740 And those roots are-- 00:05:26.740 --> 00:05:29.220 just taking account of this factor 2N-- 00:05:29.220 --> 00:05:36.000 those roots are at j omega sub c times the 2N 00:05:36.000 --> 00:05:39.450 roots of minus 1. 00:05:39.450 --> 00:05:43.740 Those roots, in fact, all lie on a circle, and the 00:05:43.740 --> 00:05:53.140 consequence of that is that the poles of this expression 00:05:53.140 --> 00:05:55.400 are on a circle. 00:05:55.400 --> 00:05:59.770 The circle is of radius omega sub c, and the poles are 00:05:59.770 --> 00:06:02.550 distributed around the circle. 00:06:02.550 --> 00:06:07.130 So here I've illustrated the poles of B of s times B of 00:06:07.130 --> 00:06:09.920 minus s, for this specific case where capital 00:06:09.920 --> 00:06:11.320 N is equal to 3. 00:06:11.320 --> 00:06:15.970 So there are a total of six poles around this circle. 00:06:15.970 --> 00:06:20.800 And then for this specific case, the poles are spaced by 00:06:20.800 --> 00:06:22.850 60 degrees. 00:06:22.850 --> 00:06:26.020 Now we have B of s times B of minus s. 00:06:26.020 --> 00:06:29.020 To get the system function for the Butterworth filter, we'd 00:06:29.020 --> 00:06:33.320 like to get B of s, and the question now is 00:06:33.320 --> 00:06:35.140 how do we get that? 00:06:35.140 --> 00:06:40.250 Well, the thing to recognize is that wherever this factor 00:06:40.250 --> 00:06:45.590 has a root, this factor has to have a root at 00:06:45.590 --> 00:06:47.420 the negative location. 00:06:47.420 --> 00:06:51.600 So in fact, when we look at these poles, we compare them 00:06:51.600 --> 00:06:55.180 with this, for example, associated with B of s, this 00:06:55.180 --> 00:06:57.560 associated with B of minus s. 00:06:57.560 --> 00:07:01.200 And likewise, we compare these two together, likewise we 00:07:01.200 --> 00:07:03.010 compare these two together. 00:07:03.010 --> 00:07:07.190 And so we can extract B of s from this product, simply by 00:07:07.190 --> 00:07:11.360 taking one pole out of each of those pairs. 00:07:11.360 --> 00:07:15.110 Now a question, of course, is out of each pair, which one do 00:07:15.110 --> 00:07:17.860 we associate with B of s, and which do we associate with B 00:07:17.860 --> 00:07:19.470 of minus s? 00:07:19.470 --> 00:07:24.040 And the answer drops out fairly simply if we recognize 00:07:24.040 --> 00:07:30.510 that if we want to design filters that are stable, then 00:07:30.510 --> 00:07:33.570 B of s, the transfer function that we're designing, must 00:07:33.570 --> 00:07:36.440 have all its poles in the left half of the s plane. 00:07:36.440 --> 00:07:41.760 So in fact, out of each of these pairs we would associate 00:07:41.760 --> 00:07:47.900 the left half plane pole with B of s, and so the transfer 00:07:47.900 --> 00:07:52.170 function for the Butterworth filter for this particular 00:07:52.170 --> 00:07:57.020 case, where the this designates the parameter omega 00:07:57.020 --> 00:07:59.950 sub c and capital N is 3, namely a third order 00:07:59.950 --> 00:08:03.850 Butterworth filter, is this set of pole locations. 00:08:03.850 --> 00:08:08.020 Given those, of course, we can figure out, simply through 00:08:08.020 --> 00:08:13.360 Algebraic means, what the transfer function B of s is. 00:08:13.360 --> 00:08:16.700 All right, so that's what Butterworth filters are. 00:08:16.700 --> 00:08:22.430 And now what I'd like to do is talk about the design of a 00:08:22.430 --> 00:08:27.460 digital Butterworth filter using the design technique 00:08:27.460 --> 00:08:31.680 that we introduced last time, namely impulse invariance. 00:08:31.680 --> 00:08:37.360 And the context in which I will phrase the design is the 00:08:37.360 --> 00:08:42.809 context of mapping a continuous time signal to a 00:08:42.809 --> 00:08:47.390 discrete time signal, carrying out filtering using the 00:08:47.390 --> 00:08:50.360 discrete time filter that we're designing, and then 00:08:50.360 --> 00:08:51.150 mapping back. 00:08:51.150 --> 00:08:56.980 So we're talking about now a discrete time filter that we 00:08:56.980 --> 00:09:02.240 want to design through impulse invariance from continuous 00:09:02.240 --> 00:09:05.950 time Butterworth filters, and we're going to get our design 00:09:05.950 --> 00:09:11.990 specifications in the context of having considered discrete 00:09:11.990 --> 00:09:16.660 time processing of continuous time signals, where we will 00:09:16.660 --> 00:09:23.020 map from a continuous time signal to a sequence, carry 00:09:23.020 --> 00:09:26.580 out the filtering with the discrete time filter that 00:09:26.580 --> 00:09:28.440 we're going to design. 00:09:28.440 --> 00:09:35.860 And then we will take the resulting filtered output and 00:09:35.860 --> 00:09:39.920 map it back to the continuous time signal. 00:09:39.920 --> 00:09:44.460 But this discrete time filter is the one that we're talking 00:09:44.460 --> 00:09:45.820 about designing. 00:09:45.820 --> 00:09:49.780 And for a choice of parameter, there's a sampling frequency, 00:09:49.780 --> 00:09:51.910 of course, involved in this process. 00:09:51.910 --> 00:09:56.120 And the value that I'll pick for the sampling frequency is 00:09:56.120 --> 00:09:58.840 10 kilohertz. 00:09:58.840 --> 00:10:02.030 OK, all fairly straightforward so far. 00:10:02.030 --> 00:10:08.470 And so since we have a sampling rate of 10 kilohertz, 00:10:08.470 --> 00:10:16.140 we want to first look at our specifications on the desired 00:10:16.140 --> 00:10:21.400 continuous time filter and then map those to appropriate 00:10:21.400 --> 00:10:24.540 specifications on the discrete time filter. 00:10:24.540 --> 00:10:29.740 And what I'll pick for the desired specifications on the 00:10:29.740 --> 00:10:39.470 continuous time filter is that at 1 kilohertz, I will ask 00:10:39.470 --> 00:10:44.070 that the continuous time frequency response be down by 00:10:44.070 --> 00:10:50.610 no more than 1 db in comparison with its value at 00:10:50.610 --> 00:10:53.130 omega equals 0. 00:10:53.130 --> 00:10:57.580 So that, in effect, specifies the behavior in the pass band, 00:10:57.580 --> 00:11:00.480 or the specifications on the pass band. 00:11:00.480 --> 00:11:06.070 And for the stop band, I'll specify that the filter is 00:11:06.070 --> 00:11:12.830 down by 15 db by the time we've gotten to 1.5 kilohertz. 00:11:12.830 --> 00:11:16.560 So we have, essentially, the beginning of the transition 00:11:16.560 --> 00:11:19.610 band-- the end of the pass band--at 1 kilohertz, and the 00:11:19.610 --> 00:11:23.850 beginning of the stop band at 1.5 kilohertz. 00:11:23.850 --> 00:11:27.670 And since we're talking about designing a Butterworth 00:11:27.670 --> 00:11:31.420 filter, we know that the Butterworth filter is 00:11:31.420 --> 00:11:36.190 monotonic in the pass band and stop band, and so we'll have a 00:11:36.190 --> 00:11:40.560 filter specification, something as I show here. 00:11:40.560 --> 00:11:44.940 This represents the allowable pass band tolerance. 00:11:44.940 --> 00:11:48.130 This is the allowable stop band tolerance. 00:11:48.130 --> 00:11:53.460 And if I can draw this without getting myself into trouble, 00:11:53.460 --> 00:11:57.220 essentially we're looking for a filter, then, that always 00:11:57.220 --> 00:12:04.580 stays between the specified boundaries here. 00:12:07.580 --> 00:12:10.930 Now what we have to figure out is what the corresponding 00:12:10.930 --> 00:12:13.820 specifications are for the digital filter. 00:12:13.820 --> 00:12:19.640 And the strategy, let me emphasize, is that we have a 00:12:19.640 --> 00:12:23.990 situation where we're doing discrete time processing of 00:12:23.990 --> 00:12:29.350 continuous time signals, and we have a set of 00:12:29.350 --> 00:12:31.850 specifications associated with that. 00:12:31.850 --> 00:12:36.270 That imposes specifications on our discrete time filter, and 00:12:36.270 --> 00:12:40.460 then we want to design the discrete time filter using 00:12:40.460 --> 00:12:43.940 impulse invariance, and that's the discrete time filter that 00:12:43.940 --> 00:12:47.360 we'll use in the overall system. 00:12:47.360 --> 00:12:52.240 All right, now we want specifications on the discrete 00:12:52.240 --> 00:12:58.430 time filter, and we want the continuous overall equivalent 00:12:58.430 --> 00:13:04.560 system to meet certain specifications at certain 00:13:04.560 --> 00:13:09.010 frequencies related to continuous time frequencies. 00:13:09.010 --> 00:13:14.550 Recall that when we sample a continuous time signal, 00:13:14.550 --> 00:13:17.990 there's a very specific mapping from the continuous 00:13:17.990 --> 00:13:21.590 time frequency axis to the discrete time frequency axis. 00:13:21.590 --> 00:13:29.120 In particular, the sampling frequency gets mapped to 2 pi. 00:13:29.120 --> 00:13:32.550 Well, that means that our other critical frequencies get 00:13:32.550 --> 00:13:34.500 mapped in proportion to that. 00:13:34.500 --> 00:13:39.000 So 1 kilohertz, which is a 1/10 of the sampling 00:13:39.000 --> 00:13:42.590 frequency, would then convert to a discrete time 00:13:42.590 --> 00:13:45.860 frequency of 0.2 pi. 00:13:45.860 --> 00:13:51.030 And 1.5 kilohertz will convert to a discrete time 00:13:51.030 --> 00:13:54.620 frequency of 0.3 pi. 00:13:54.620 --> 00:14:00.230 So what this says is that for the discrete time filter, we 00:14:00.230 --> 00:14:05.860 would like the same behavior, but at frequencies-- 00:14:05.860 --> 00:14:10.980 or the same specifications, but at frequencies normalized 00:14:10.980 --> 00:14:14.360 to the discrete time frequency axis. 00:14:14.360 --> 00:14:20.120 That means that we want the discrete time frequency 00:14:20.120 --> 00:14:23.250 response magnitude to be greater than or equal to minus 00:14:23.250 --> 00:14:26.770 1 db at 2/10 pi-- 00:14:26.770 --> 00:14:30.060 corresponding to the 1 kilohertz in continuous time-- 00:14:30.060 --> 00:14:36.340 and for the beginning of the stop band, that would occur at 00:14:36.340 --> 00:14:38.970 0.3 pi, at which point we want this less than or 00:14:38.970 --> 00:14:42.230 equal to minus 15. 00:14:42.230 --> 00:14:45.680 So those are our discrete time specifications. 00:14:45.680 --> 00:14:51.960 And we now want to design the discrete time filter using 00:14:51.960 --> 00:14:54.260 impulse invariance. 00:14:54.260 --> 00:15:02.670 Now in impulse invariance, as you recall, it corresponds to 00:15:02.670 --> 00:15:06.920 generating an impulse response which is a sampled version of 00:15:06.920 --> 00:15:11.180 the continuous time impulse response, and there is a 00:15:11.180 --> 00:15:16.050 temptation naturally to think all of this parameter, capital 00:15:16.050 --> 00:15:25.190 T, as necessarily identical to the sampling in the system in 00:15:25.190 --> 00:15:27.710 which the filter is going to be used. 00:15:27.710 --> 00:15:31.200 Now this is a fairly subtle, complicated, tongue twisting 00:15:31.200 --> 00:15:36.800 issue, but the bottom line on it, the essential point, is 00:15:36.800 --> 00:15:41.140 that the parameter capital T that we use in impulse 00:15:41.140 --> 00:15:46.060 invariant design is a totally different, unrelated, and in 00:15:46.060 --> 00:15:50.660 fact, as it turns out, arbitrary parameter, which is 00:15:50.660 --> 00:15:56.480 not necessarily pegged to the sampling frequency. 00:15:56.480 --> 00:15:58.720 And I think it would be difficult for me to totally 00:15:58.720 --> 00:16:03.180 clarify that during the lecture. 00:16:03.180 --> 00:16:05.710 It's discussed more in the book, and certainly, you 00:16:05.710 --> 00:16:08.850 should take plenty of time to reflect on it. 00:16:08.850 --> 00:16:14.510 All right, but let's now look, then, at where we are in our 00:16:14.510 --> 00:16:18.570 design procedure, and specifically, what it is that 00:16:18.570 --> 00:16:21.720 we need to do in order to design the 00:16:21.720 --> 00:16:26.000 digital Butterworth filter. 00:16:26.000 --> 00:16:30.900 Now we have a set of specifications that we've 00:16:30.900 --> 00:16:35.000 generated relating, essentially, to how we want 00:16:35.000 --> 00:16:38.950 the pass band of the digital filter and the stop band of 00:16:38.950 --> 00:16:41.050 the digital filter to behave. 00:16:41.050 --> 00:16:46.170 Of course, since this isn't an ideal filter, it has some 00:16:46.170 --> 00:16:50.510 transition from pass band to stop band, and as we discussed 00:16:50.510 --> 00:16:54.430 last time, there is aliasing, which we need to at 00:16:54.430 --> 00:16:57.140 least be aware of. 00:16:57.140 --> 00:17:01.500 We specified certain frequencies along this axis 00:17:01.500 --> 00:17:07.420 which are easily converted by relating to the two axes 00:17:07.420 --> 00:17:13.920 through this mapping, are easily related back to the 00:17:13.920 --> 00:17:21.040 continuous time frequency axis, as we have here. 00:17:21.040 --> 00:17:26.079 And in particular, now if we were to simply pick that 00:17:26.079 --> 00:17:29.800 parameter in the impulse invariant design, capital T, 00:17:29.800 --> 00:17:34.160 as equal to unity, and I indicated just a minute ago 00:17:34.160 --> 00:17:37.850 that we can pick it arbitrarily, if I pick it as 00:17:37.850 --> 00:17:43.580 unity, then the procedure would consist of designing the 00:17:43.580 --> 00:17:47.800 continuous time Butterworth filter with meeting or 00:17:47.800 --> 00:17:50.870 exceeding the appropriate specifications and then going 00:17:50.870 --> 00:17:55.090 through the impulse invariant procedure. 00:17:55.090 --> 00:17:58.250 All right, so let's do that then. 00:18:01.070 --> 00:18:05.290 We want the discrete time impulse response to be the 00:18:05.290 --> 00:18:09.750 continuous time impulse response sampled, and for 00:18:09.750 --> 00:18:11.700 convenience, I'm going to pick this parameter 00:18:11.700 --> 00:18:14.560 capital T equal to 1. 00:18:14.560 --> 00:18:20.080 That means that the frequency normalization between the 00:18:20.080 --> 00:18:22.730 discrete time frequency axis and the continuous time 00:18:22.730 --> 00:18:26.640 frequency axis in fact is-- 00:18:26.640 --> 00:18:30.500 those axes are scaled identically, because capital T 00:18:30.500 --> 00:18:32.000 is equal to 1. 00:18:32.000 --> 00:18:37.590 And so now we want the analog, or continuous time, 00:18:37.590 --> 00:18:39.550 specifications. 00:18:39.550 --> 00:18:42.690 And so what we need to do then is design 00:18:42.690 --> 00:18:44.000 a Butterworth filter. 00:18:44.000 --> 00:18:47.370 I'm using capital B here again to denote the frequency 00:18:47.370 --> 00:18:49.400 response of the Butterworth filter. 00:18:49.400 --> 00:18:53.300 The Butterworth filter to have a magnitude which is greater 00:18:53.300 --> 00:18:59.590 than or equal to minus 1 db prior to the frequency 0.2 pi, 00:18:59.590 --> 00:19:02.520 and less than or equal to minus 15 at a 00:19:02.520 --> 00:19:07.770 frequency beyond 0.3 pi. 00:19:07.770 --> 00:19:14.820 And so now what we need to do is determine capital N and 00:19:14.820 --> 00:19:17.910 omega sub c in order to meet or exceed those 00:19:17.910 --> 00:19:19.160 specifications. 00:19:21.210 --> 00:19:24.300 Now if you go through the associated algebra in doing 00:19:24.300 --> 00:19:27.360 that, let's say that you decide that you want to pick 00:19:27.360 --> 00:19:31.040 capital N and omega sub c to exactly meet those 00:19:31.040 --> 00:19:35.860 inequalities at the two frequencies, 0.2 pi and 0.3 00:19:35.860 --> 00:19:40.320 pi, what you'll find after going through the algebra is 00:19:40.320 --> 00:19:46.170 that they're exactly met if capital N is 5.88 and omega 00:19:46.170 --> 00:19:50.320 sub c is 0.7047. 00:19:50.320 --> 00:19:55.560 And this obviously isn't satisfactory as parameters for 00:19:55.560 --> 00:19:56.820 the Butterworth filter. 00:19:56.820 --> 00:19:58.720 Why is that? 00:19:58.720 --> 00:20:03.480 The reason is that capital N isn't an integer, and in the 00:20:03.480 --> 00:20:06.320 class of Butterworth filters, to generate a rational 00:20:06.320 --> 00:20:08.960 transfer function, capital N, this 00:20:08.960 --> 00:20:11.420 parameter, must be an integer. 00:20:11.420 --> 00:20:18.480 So since it can't be 5.88, the natural thing to do is to move 00:20:18.480 --> 00:20:22.510 it up to the next closest integer, namely 6. 00:20:22.510 --> 00:20:26.330 And that means that we'll end up with a filter that does 00:20:26.330 --> 00:20:30.100 even better than the specifications. 00:20:30.100 --> 00:20:33.320 On the other hand, there's something kind of underneath 00:20:33.320 --> 00:20:36.950 the surface that is inherent in the impulse invariant 00:20:36.950 --> 00:20:41.350 design procedure, namely the fact that there will always be 00:20:41.350 --> 00:20:42.960 some aliasing. 00:20:42.960 --> 00:20:46.960 So one strategy, and a natural one, often, in impulse 00:20:46.960 --> 00:20:53.780 invariant design, is to choose N as the next highest integer 00:20:53.780 --> 00:20:59.960 as I've done, and then choose the parameter omega sub c so 00:20:59.960 --> 00:21:04.810 that the pass band specifications are exactly 00:21:04.810 --> 00:21:07.940 met, and the stop band specifications are then 00:21:07.940 --> 00:21:10.390 slightly exceeded, and that will leave 00:21:10.390 --> 00:21:13.860 some margin for aliasing. 00:21:13.860 --> 00:21:18.660 All right, now continuing this example, then, how would we 00:21:18.660 --> 00:21:21.910 complete the design? 00:21:21.910 --> 00:21:28.840 Well, we know what our two parameters, capital N and 00:21:28.840 --> 00:21:31.680 omega sub c are. 00:21:31.680 --> 00:21:34.160 That means that we can determine B of s 00:21:34.160 --> 00:21:35.970 times B of minus s. 00:21:35.970 --> 00:21:42.190 Those poles are located on a circle in the complex s plane. 00:21:42.190 --> 00:21:47.420 And the poles on that circle are paired, some being 00:21:47.420 --> 00:21:50.770 associated with B of s and some with B of minus s. 00:21:50.770 --> 00:21:55.540 And in particular, to determine B of s, we would 00:21:55.540 --> 00:22:00.780 simply take the poles on the portion of the circle that's 00:22:00.780 --> 00:22:02.180 in the left half of the s plane. 00:22:04.720 --> 00:22:11.280 Now what that gives us is a Butterworth continuous time 00:22:11.280 --> 00:22:16.720 filter, which we are then mapping through impulse 00:22:16.720 --> 00:22:20.960 invariance with capital T equal to 1 to a 00:22:20.960 --> 00:22:22.960 discrete time filter. 00:22:22.960 --> 00:22:28.800 That discrete time filter to be used in a system which has 00:22:28.800 --> 00:22:29.990 an associated sampling 00:22:29.990 --> 00:22:34.620 frequency which is 10 kilohertz. 00:22:34.620 --> 00:22:38.220 So to get the discrete time filter, then, we first 00:22:38.220 --> 00:22:41.900 determine B of s, as we just did-- 00:22:41.900 --> 00:22:45.360 or at least I indicated how to do it. 00:22:45.360 --> 00:22:48.940 We would then expand that out in a partial fraction 00:22:48.940 --> 00:22:54.580 expansion, and then apply the impulse invariant procedure, 00:22:54.580 --> 00:23:00.660 which consists of mapping the poles in the s plane to poles 00:23:00.660 --> 00:23:05.230 in the z plane at locations e to the s of k, capital T, 00:23:05.230 --> 00:23:07.310 capital T is equal to 1. 00:23:07.310 --> 00:23:10.190 And retaining the residue, or the 00:23:10.190 --> 00:23:12.860 coefficient, in the expansion. 00:23:12.860 --> 00:23:17.050 And this, then, will give us the transfer function for the 00:23:17.050 --> 00:23:20.370 discrete time filter. 00:23:20.370 --> 00:23:28.130 So if in fact we did that, then the resulting frequency 00:23:28.130 --> 00:23:32.050 response that we would get is what I've indicated here. 00:23:32.050 --> 00:23:37.310 And I indicated first on a magnitude scale, linear 00:23:37.310 --> 00:23:42.580 magnitude scale, and second on a logarithmic scale. 00:23:42.580 --> 00:23:47.120 And as we had originally specified, the digital filter 00:23:47.120 --> 00:23:52.430 is supposed to be greater than or equal to minus 1 db at 00:23:52.430 --> 00:23:55.680 point 2 pi, and less than or equal to minus 00:23:55.680 --> 00:23:59.540 15 db at 0.3 pi. 00:23:59.540 --> 00:24:03.740 And in fact, this slightly exceeds those specifications 00:24:03.740 --> 00:24:08.385 since we had purposely allowed some margin in the stop band. 00:24:14.090 --> 00:24:17.540 Now this is an illustration of the impulse invariant 00:24:17.540 --> 00:24:22.360 procedure, and it has a number of very nice properties, one 00:24:22.360 --> 00:24:26.100 of which is that it takes a continuous time filter 00:24:26.100 --> 00:24:31.630 frequency response and it converts it to a discrete time 00:24:31.630 --> 00:24:35.270 frequency response, which in the absence of aliasing, looks 00:24:35.270 --> 00:24:39.500 identical, except for a linear frequency scale change. 00:24:39.500 --> 00:24:41.680 And in fact, if we picked capital T equal to 1, then 00:24:41.680 --> 00:24:43.520 there is no scale change. 00:24:43.520 --> 00:24:49.950 It has the major disadvantage that there is always aliasing, 00:24:49.950 --> 00:24:54.610 and for some problems, for example, if the filter that 00:24:54.610 --> 00:24:59.940 we're trying to design is not band limited or low pass, then 00:24:59.940 --> 00:25:04.820 the aliasing will naturally become intolerable. 00:25:04.820 --> 00:25:07.340 Well, there's another design procedure which I now want to 00:25:07.340 --> 00:25:11.300 introduce, which totally avoids the problems of 00:25:11.300 --> 00:25:15.040 aliasing, but, obviously, then has its own costs 00:25:15.040 --> 00:25:16.460 associated with it. 00:25:16.460 --> 00:25:20.640 And that's procedure is referred to as the bilinear 00:25:20.640 --> 00:25:21.890 transformation. 00:25:24.430 --> 00:25:28.660 The bilinear transformation, which I won't try to derive 00:25:28.660 --> 00:25:34.770 here in any detail, is a mapping of continuous time 00:25:34.770 --> 00:25:40.910 filters to discrete time filters, corresponding to 00:25:40.910 --> 00:25:45.110 taking the Laplace transform variable s in the continuous 00:25:45.110 --> 00:25:50.220 time filter and replacing it by what is referred to as a 00:25:50.220 --> 00:25:53.560 bilinear function of z. 00:25:53.560 --> 00:25:58.530 And so if I substitute this in here, that will give me the 00:25:58.530 --> 00:26:02.370 discrete time frequency response. 00:26:02.370 --> 00:26:05.670 Again in this procedure, there's a parameter capital T, 00:26:05.670 --> 00:26:08.500 which again is totally irrelevant given the approach 00:26:08.500 --> 00:26:11.610 that we're taking, and which we will generally tend to 00:26:11.610 --> 00:26:13.570 normalize out to unity. 00:26:13.570 --> 00:26:19.130 And let me just say quickly and in passing that although 00:26:19.130 --> 00:26:23.080 we won't go through this, the notion that the bilinear 00:26:23.080 --> 00:26:29.370 transformation can be tied to the concept of taking the 00:26:29.370 --> 00:26:32.730 differential equation for the continuous time filter, 00:26:32.730 --> 00:26:36.560 converting it to an integral equation by integrating enough 00:26:36.560 --> 00:26:42.700 times on both sides, and then converting that to a 00:26:42.700 --> 00:26:47.600 difference equation by approximating the integrals 00:26:47.600 --> 00:26:50.200 with the trapezoidal rule. 00:26:50.200 --> 00:26:54.380 And that, in effect, will correspond to mapping the 00:26:54.380 --> 00:26:57.600 continuous time filter to a discrete time filter with the 00:26:57.600 --> 00:26:59.930 bilinear transformation. 00:26:59.930 --> 00:27:04.400 Well, we'll just focus on the properties of the mapping. 00:27:04.400 --> 00:27:09.190 And in particular, if we were to substitute into the 00:27:09.190 --> 00:27:14.370 expression z equal to e to the j omega corresponding to the 00:27:14.370 --> 00:27:20.020 unit circle, we would find that the unit circle in 00:27:20.020 --> 00:27:26.510 discrete time corresponds to mapping the j omega axis in 00:27:26.510 --> 00:27:30.970 continuous time, which is exactly what we want. 00:27:30.970 --> 00:27:37.290 Now the mapping between the continuous time frequency and 00:27:37.290 --> 00:27:43.260 the discrete time frequency is a nonlinear mapping, which is 00:27:43.260 --> 00:27:44.940 given by the algebraic expression 00:27:44.940 --> 00:27:46.530 that I indicate here. 00:27:46.530 --> 00:27:56.170 And if we plot this mapping, what we have is this curve. 00:27:56.170 --> 00:28:03.510 And what this corresponds to, then, is a mapping of the j 00:28:03.510 --> 00:28:07.410 omega axis, or continuous time frequency, to 00:28:07.410 --> 00:28:10.270 discrete time frequency. 00:28:10.270 --> 00:28:14.390 And if we think more generally of the mapping represented by 00:28:14.390 --> 00:28:18.730 the bilinear transformation in the context of the s-plane and 00:28:18.730 --> 00:28:25.100 the z-plane, it corresponds to mapping the j omega axis in 00:28:25.100 --> 00:28:31.170 the s-plane to once around the unit circle in the z-plane. 00:28:31.170 --> 00:28:35.260 And you could also convince yourself that the left half of 00:28:35.260 --> 00:28:39.850 the s-plane maps to the inside of the unit circle. 00:28:39.850 --> 00:28:45.250 And so that means that stable continuous time filters will 00:28:45.250 --> 00:28:48.690 always map to stable discrete time filters, which is exactly 00:28:48.690 --> 00:28:51.310 what we desire. 00:28:51.310 --> 00:28:54.990 Now notice in this that there's no issue of aliasing. 00:28:54.990 --> 00:28:58.320 What's happened is that we've replaced s by a function of z, 00:28:58.320 --> 00:29:01.850 and it corresponds to mapping the s-plane to the z-plane. 00:29:01.850 --> 00:29:05.970 In fact, the whole j omega axis has mapped to once around 00:29:05.970 --> 00:29:13.150 the unit circle, which obviously requires some type 00:29:13.150 --> 00:29:17.280 of nonlinear mapping, because, look, the j omega axis is 00:29:17.280 --> 00:29:18.640 infinitely long. 00:29:18.640 --> 00:29:21.960 The unit circle has a finite radius. 00:29:21.960 --> 00:29:25.720 And so essentially, what has to happen is if you think of 00:29:25.720 --> 00:29:29.720 walking along the continuous time frequency axis, and 00:29:29.720 --> 00:29:33.840 simultaneously walking around the unit circle, if you walk 00:29:33.840 --> 00:29:37.580 at a constant rate around the unit circle and you're 00:29:37.580 --> 00:29:41.330 simultaneously walking up the j omega axis, if you want to 00:29:41.330 --> 00:29:45.810 get around to pi by the time, here, you've gotten to 00:29:45.810 --> 00:29:50.530 infinity along the continuous time frequency axis, you 00:29:50.530 --> 00:29:54.760 better start walking faster, and faster, and faster, 00:29:54.760 --> 00:29:58.640 because you've got an infinite distance to cover while you 00:29:58.640 --> 00:30:01.410 just go over a finite distance here. 00:30:01.410 --> 00:30:07.420 Well, what all that says, really, is that that's why, in 00:30:07.420 --> 00:30:10.200 fact, we're taking the entire-- 00:30:10.200 --> 00:30:14.850 or we're able to take the entire omega axis and just map 00:30:14.850 --> 00:30:19.450 it into an interval of length pi. 00:30:22.544 --> 00:30:27.640 All right, now that means that there is a nonlinear 00:30:27.640 --> 00:30:31.170 distortion of the frequency axis if we were to take a 00:30:31.170 --> 00:30:35.320 continuous time filter and convert it to a discrete time 00:30:35.320 --> 00:30:37.860 filter with the bilinear transformation. 00:30:37.860 --> 00:30:42.120 How do we account for that, or for that matter, when can we 00:30:42.120 --> 00:30:44.230 really use it? 00:30:44.230 --> 00:30:49.050 And we can see how to both take account of it, and what 00:30:49.050 --> 00:30:54.300 its limitations are, by recognizing the following. 00:30:54.300 --> 00:31:02.690 Suppose that I want to design a discrete time filter and 00:31:02.690 --> 00:31:06.220 what is going to happen is that it will be mapped to a 00:31:06.220 --> 00:31:09.340 continuous time filter, or the relationship between the two 00:31:09.340 --> 00:31:14.300 frequency axes will be given by this curve. 00:31:14.300 --> 00:31:19.190 So let's suppose that the continuous time frequency 00:31:19.190 --> 00:31:23.510 response looks as I've shown here, with a pass band cutoff 00:31:23.510 --> 00:31:27.900 frequency and a stop band cutoff frequency. 00:31:27.900 --> 00:31:34.270 If this were mapped through the bilinear transformation to 00:31:34.270 --> 00:31:38.950 a discrete time filter, then this cutoff frequency would 00:31:38.950 --> 00:31:44.410 fall over here, which is related through this curve, 00:31:44.410 --> 00:31:49.950 and this cutoff frequency, the stop band edge would be here. 00:31:49.950 --> 00:31:56.930 Again, it's these frequencies reflected through this curve. 00:31:56.930 --> 00:32:02.620 So let's suppose now that what I'd like to do is design a 00:32:02.620 --> 00:32:07.810 discrete time filter, where the discrete time filter has 00:32:07.810 --> 00:32:11.650 certain frequency specifications, which is what 00:32:11.650 --> 00:32:14.320 our previous example has. 00:32:14.320 --> 00:32:18.600 It has, let's say, a specified stop band edge and a specified 00:32:18.600 --> 00:32:23.070 pass band edge, and a specified stop band edge. 00:32:23.070 --> 00:32:29.100 The design procedure would then correspond to mapping 00:32:29.100 --> 00:32:34.230 those frequencies to the corresponding continuous time 00:32:34.230 --> 00:32:40.790 frequencies, designing the continuous time filter to meet 00:32:40.790 --> 00:32:46.480 the specifications based on those critical frequencies, 00:32:46.480 --> 00:32:51.300 then taking the continuous time design and mapping it 00:32:51.300 --> 00:32:54.820 back to a discrete time filter through the bilinear 00:32:54.820 --> 00:32:57.101 transformation. 00:32:57.101 --> 00:33:02.780 All right, now what we want to do is again map a Butterworth 00:33:02.780 --> 00:33:05.460 filter, continuous time Butterworth filter to a 00:33:05.460 --> 00:33:08.510 digital filter in such a way that the digital filter 00:33:08.510 --> 00:33:11.050 approximately meets the 00:33:11.050 --> 00:33:13.720 specifications that we had before. 00:33:13.720 --> 00:33:16.720 And let me just remind you of where we were. 00:33:16.720 --> 00:33:24.600 The critical frequencies were at 0.2 pi and 0.3 pi. 00:33:24.600 --> 00:33:30.260 And what we had asked for is that the frequency response be 00:33:30.260 --> 00:33:37.370 down by no more than 1 db up to 0.2 pi, and down by at 00:33:37.370 --> 00:33:41.800 least 15 db at 0.3 pi. 00:33:41.800 --> 00:33:47.630 So we want to design the same discrete time filter as we did 00:33:47.630 --> 00:33:49.780 before with impulse invariance. 00:33:49.780 --> 00:33:52.900 We now want to do it with the bilinear transformation 00:33:52.900 --> 00:33:56.160 applied to an appropriate Butterworth design. 00:33:59.289 --> 00:34:05.970 OK, well, let's see what kind of specifications we have. 00:34:05.970 --> 00:34:09.590 We know that the frequency, the critical frequencies, are 00:34:09.590 --> 00:34:13.530 mapped through this curve, or this equation. 00:34:13.530 --> 00:34:18.489 And I indicated again that this parameter capital T is 00:34:18.489 --> 00:34:21.770 arbitrary in the design procedure. 00:34:21.770 --> 00:34:26.120 That may seem confusing, initially, but there's some 00:34:26.120 --> 00:34:28.719 further discussion of it in the book, and it's true and 00:34:28.719 --> 00:34:31.000 important to sort out. 00:34:31.000 --> 00:34:35.070 So we're going to pick capital T equal to unity. 00:34:35.070 --> 00:34:40.280 And that means then that the corresponding critical 00:34:40.280 --> 00:34:46.650 frequencies of the continuous time filter are at twice the 00:34:46.650 --> 00:34:51.690 tangent of 0.2 pi over 2, and twice the tangent 00:34:51.690 --> 00:34:54.920 of 0.3 pi over 2. 00:34:54.920 --> 00:35:00.340 So that means then that the specifications on our 00:35:00.340 --> 00:35:05.790 continuous time filter are given by this, essentially 00:35:05.790 --> 00:35:09.380 pass band and stop band edges warped through 00:35:09.380 --> 00:35:10.870 that nonlinear curve-- 00:35:10.870 --> 00:35:12.620 this nonlinear curve. 00:35:12.620 --> 00:35:18.550 And I'm using capital G here to denote the system function 00:35:18.550 --> 00:35:24.350 for the resulting continuous time filter. 00:35:24.350 --> 00:35:28.760 And so now we could think of designing a Butterworth filter 00:35:28.760 --> 00:35:34.430 that, let's say, exactly meets these specifications. 00:35:34.430 --> 00:35:38.760 Well, if you do that, what you'll find is that you get 00:35:38.760 --> 00:35:49.770 exact equality here if you pick capital N equal to 5.3. 00:35:49.770 --> 00:35:55.240 And again, we have the issue that if we want to meet or 00:35:55.240 --> 00:35:58.520 exceed the specifications, we can't make the filter order 00:35:58.520 --> 00:36:00.440 lower, we have to make it higher. 00:36:00.440 --> 00:36:03.740 And so we would make it, instead of 5.3, we would make 00:36:03.740 --> 00:36:07.010 the filter order equal to 6. 00:36:07.010 --> 00:36:10.260 And now, again, we have several 00:36:10.260 --> 00:36:11.430 options and trade offs. 00:36:11.430 --> 00:36:15.600 Before, with impulse invariance, we essentially 00:36:15.600 --> 00:36:20.930 decided to meet the pass band specifications and exceed the 00:36:20.930 --> 00:36:23.530 stop band specifications to provide 00:36:23.530 --> 00:36:26.940 some margin for aliasing. 00:36:26.940 --> 00:36:31.650 Here, we don't have any aliasing, and we can trade 00:36:31.650 --> 00:36:33.470 things off anyway we'd like. 00:36:33.470 --> 00:36:40.390 The way that I've chosen to do it is to exactly meet the stop 00:36:40.390 --> 00:36:46.910 band cut off, and exceed the pass band specifications. 00:36:46.910 --> 00:36:52.490 And the result of doing that is to choose a filter order 00:36:52.490 --> 00:36:57.580 capital N equal to 6, and the parameter omega sub c in the 00:36:57.580 --> 00:37:04.350 Butterworth filter is given by 0.76622. 00:37:04.350 --> 00:37:12.090 All right, so now we have the parameters for the continuous 00:37:12.090 --> 00:37:18.920 time Butterworth filter, which when mapped to the discrete 00:37:18.920 --> 00:37:23.690 time filter through the bilinear transformation, will 00:37:23.690 --> 00:37:27.640 exceed our requirements in the pass band and just meet the 00:37:27.640 --> 00:37:29.450 stop band cutoff. 00:37:29.450 --> 00:37:32.950 And so now we want to complete the design procedure. 00:37:32.950 --> 00:37:41.240 And given the parameters of the Butterworth filter, we can 00:37:41.240 --> 00:37:45.330 draw our appropriate circle in the s-plane, which happens to 00:37:45.330 --> 00:37:49.580 be the same circle as before, but with a different radius. 00:37:49.580 --> 00:37:56.800 And then associate poles on this circle with B of s and B 00:37:56.800 --> 00:37:57.950 of minus s. 00:37:57.950 --> 00:37:59.760 In particular, take the ones in the 00:37:59.760 --> 00:38:02.170 left half of the s-plane-- 00:38:02.170 --> 00:38:05.070 same ones we had before except we have a different value for 00:38:05.070 --> 00:38:07.380 omega sub c-- 00:38:07.380 --> 00:38:11.880 and those, then, represent B of s. 00:38:11.880 --> 00:38:16.810 So we can now determine B of s. 00:38:16.810 --> 00:38:22.800 And once we have the transfer function B of s, we then map 00:38:22.800 --> 00:38:28.060 that to a discrete time filter by mapping through the 00:38:28.060 --> 00:38:30.760 bilinear transformation. 00:38:30.760 --> 00:38:35.790 And that then is the design procedure which we would 00:38:35.790 --> 00:38:37.660 follow this example. 00:38:37.660 --> 00:38:41.880 All right, well, let's just see what the result looks like 00:38:41.880 --> 00:38:44.260 when we're done. 00:38:44.260 --> 00:38:50.940 Here I show, again, on both a linear magnitude scale and on 00:38:50.940 --> 00:38:55.010 a logarithmic or DB scale, the frequency response of the 00:38:55.010 --> 00:38:57.440 resulting filter. 00:38:57.440 --> 00:39:00.310 And recall that the specifications were that at 00:39:00.310 --> 00:39:05.730 0.2 pi, we should be down by no more than 1 db, and clearly 00:39:05.730 --> 00:39:08.120 we've exceeded that considerably. 00:39:08.120 --> 00:39:13.870 And at 0.3 pi, we should be down by at least 15 db, and we 00:39:13.870 --> 00:39:17.750 chose the design so that we would exactly meet that edge. 00:39:17.750 --> 00:39:21.910 So we've met this point and exceeded this point, and this 00:39:21.910 --> 00:39:24.070 is our design. 00:39:24.070 --> 00:39:29.220 Now let's just compare this with the impulse invariant 00:39:29.220 --> 00:39:31.870 design that we saw a few minutes ago. 00:39:35.050 --> 00:39:41.046 The impulse invariant design is what I indicate here. 00:39:41.046 --> 00:39:44.930 The bilinear transformation design is now 00:39:44.930 --> 00:39:47.860 overlaid on top of it. 00:39:47.860 --> 00:39:51.420 It's hard to see much difference on a linear 00:39:51.420 --> 00:39:54.820 magnitude scale, but let's look on a logarithmic scale. 00:39:54.820 --> 00:39:58.750 And this one that I'm lifting is the bilinear 00:39:58.750 --> 00:40:03.870 transformation, and this is impulse invariance. 00:40:03.870 --> 00:40:08.410 And notice, in fact, something very interesting, which is 00:40:08.410 --> 00:40:16.670 that the filter that we obtained by mapping the 00:40:16.670 --> 00:40:19.655 Butterworth filter to a digital filter through the 00:40:19.655 --> 00:40:24.740 bilinear transformation in fact falls off in frequency 00:40:24.740 --> 00:40:28.900 much more rapidly than the one that we got through impulse 00:40:28.900 --> 00:40:31.690 invariance. 00:40:31.690 --> 00:40:34.190 The question is why? 00:40:34.190 --> 00:40:36.600 Now one thought that might come to mind is, well, impulse 00:40:36.600 --> 00:40:39.160 invariance has aliasing, the bilinear transformation 00:40:39.160 --> 00:40:41.870 doesn't have aliasing, that must be a 00:40:41.870 --> 00:40:44.990 consequence of aliasing. 00:40:44.990 --> 00:40:47.850 In fact, that's not the reason. 00:40:47.850 --> 00:40:51.370 Aliasing, as it turns out in this particular design, was 00:40:51.370 --> 00:40:54.460 relatively minimal in the impulse invariant design. 00:40:54.460 --> 00:41:00.070 The reason has to do with this nonlinear mapping in the 00:41:00.070 --> 00:41:03.710 bilinear transformation from the continuous time frequency 00:41:03.710 --> 00:41:06.920 to the discrete time frequency. 00:41:06.920 --> 00:41:12.070 Keep in mind that through that mapping, as you start walking 00:41:12.070 --> 00:41:15.900 around the unit circle and moving up to j omega axis, as 00:41:15.900 --> 00:41:18.700 you move up the j omega axis, you have to move faster, and 00:41:18.700 --> 00:41:21.450 faster, and faster, and faster. 00:41:21.450 --> 00:41:28.580 And what's in continuous time frequency at infinity is what 00:41:28.580 --> 00:41:31.370 you get to in the discrete time frequency by the time you 00:41:31.370 --> 00:41:32.920 get around to pi. 00:41:32.920 --> 00:41:38.870 So in fact, what we're looking at as we look out along this 00:41:38.870 --> 00:41:45.340 frequency axis is we're seeing higher, and higher, and higher 00:41:45.340 --> 00:41:48.520 frequencies in the continuous time filter. 00:41:48.520 --> 00:41:53.550 By the time we get to pi, we should in fact be in the 00:41:53.550 --> 00:41:57.560 continuous time filter equivalently off to infinity, 00:41:57.560 --> 00:42:00.540 which sounds like a pretty uncomfortable place to be. 00:42:04.380 --> 00:42:10.920 OK, now this was a fairly rapid trip through a number of 00:42:10.920 --> 00:42:14.200 issues, in particular, some of the issues associated with the 00:42:14.200 --> 00:42:20.980 bilinear transformation, and also this issue of how you 00:42:20.980 --> 00:42:24.330 pick this parameter capital T, and how it might be associated 00:42:24.330 --> 00:42:27.190 with a sampling frequency if you're doing discrete time 00:42:27.190 --> 00:42:30.130 processing of continuous time signals. 00:42:30.130 --> 00:42:35.020 And we don't have time to explore some of those issues 00:42:35.020 --> 00:42:37.880 more fully in this lecture. 00:42:37.880 --> 00:42:41.220 But I'd like to conclude by making a couple of comments. 00:42:41.220 --> 00:42:45.160 One comment is that the two techniques that we've talked 00:42:45.160 --> 00:42:49.510 about, impulse invariance and the bilinear transformation, 00:42:49.510 --> 00:42:54.610 are the two techniques that are principally used when one 00:42:54.610 --> 00:42:58.900 thinks of mapping continuous time filters to discrete time 00:42:58.900 --> 00:43:00.830 filters for whatever application. 00:43:00.830 --> 00:43:05.890 And I stress again that you may want to do that mapping 00:43:05.890 --> 00:43:10.080 whether or not the discrete time filter is eventually 00:43:10.080 --> 00:43:11.910 going to be used for processing 00:43:11.910 --> 00:43:13.160 continuous time signals. 00:43:15.970 --> 00:43:24.020 Now impulse invariance had the very nice characteristic that 00:43:24.020 --> 00:43:26.830 it corresponds to a linear mapping between the two 00:43:26.830 --> 00:43:32.560 frequency axes, except for the issue of aliasing, and that's 00:43:32.560 --> 00:43:36.020 a problem with it, and in particular, limits its 00:43:36.020 --> 00:43:41.280 usefulness to filter designs, or for mapping continuous time 00:43:41.280 --> 00:43:42.970 filters that are band limited. 00:43:46.610 --> 00:43:50.840 On the other hand, we have the bilinear transformation as a 00:43:50.840 --> 00:43:55.830 design procedure, which totally avoids aliasing, but 00:43:55.830 --> 00:44:00.210 has the disadvantage or difficulty that it represents 00:44:00.210 --> 00:44:03.510 a nonlinear mapping from the continuous time filter to the 00:44:03.510 --> 00:44:05.420 discrete time filter. 00:44:05.420 --> 00:44:10.230 Now this nonlinear distortion is perfectly acceptable if 00:44:10.230 --> 00:44:13.210 we're designing or attempting to design filters that have 00:44:13.210 --> 00:44:16.400 flat frequency characteristics. 00:44:16.400 --> 00:44:19.740 It's not acceptable if, for example, we had a linear 00:44:19.740 --> 00:44:23.650 frequency characteristic that we wanted to map to a discrete 00:44:23.650 --> 00:44:25.750 time filter and end up with a linear frequency 00:44:25.750 --> 00:44:26.800 characteristic. 00:44:26.800 --> 00:44:31.480 It won't come out to be linear because of this nonlinear 00:44:31.480 --> 00:44:34.320 mapping of the frequency axis. 00:44:34.320 --> 00:44:39.890 Now there are also a number of other design procedures, which 00:44:39.890 --> 00:44:42.620 we won't go into, for designing discrete time 00:44:42.620 --> 00:44:47.560 filters, and among them are a variety of techniques, 00:44:47.560 --> 00:44:51.440 including, for example, computer aided design 00:44:51.440 --> 00:44:52.700 procedures. 00:44:52.700 --> 00:44:56.460 And I invite you, if you're interested and want to dig 00:44:56.460 --> 00:45:01.330 into that in more detail and more deeply, to explore that 00:45:01.330 --> 00:45:08.060 topic by making reference to various of the books listed in 00:45:08.060 --> 00:45:10.390 the bibliography in the text. 00:45:10.390 --> 00:45:11.640 Thank you.