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 --> 00:00:20.600 ocw.mit.edu. 00:00:20.600 --> 00:00:55.444 [MUSIC PLAYING] 00:00:55.444 --> 00:00:58.610 PROFESSOR: Last time, we began the discussion of the 00:00:58.610 --> 00:01:01.240 z-transform. 00:01:01.240 --> 00:01:04.110 As with the Laplace transform in continuous time, we 00:01:04.110 --> 00:01:06.190 developed it as a generalization 00:01:06.190 --> 00:01:09.000 of the Fourier transform. 00:01:09.000 --> 00:01:14.230 The expression that we got for the z-transform is the sum 00:01:14.230 --> 00:01:17.050 that I indicate here. 00:01:17.050 --> 00:01:20.750 We also briefly talked about an inverse z-transform 00:01:20.750 --> 00:01:25.320 integral and some other informal methods of computing 00:01:25.320 --> 00:01:27.660 the inverse z-transform. 00:01:27.660 --> 00:01:32.220 But we focused in particular on the relationship between 00:01:32.220 --> 00:01:36.090 the z-transform and the Fourier transform, pointing 00:01:36.090 --> 00:01:41.800 out first of all that the z-transform, when we choose 00:01:41.800 --> 00:01:44.590 the magnitude of z equal to 1-- 00:01:44.590 --> 00:01:48.440 so the magnitude of z of the form e to the j omega-- 00:01:48.440 --> 00:01:52.150 just simply reduces to the Fourier 00:01:52.150 --> 00:01:55.210 transform of the sequence. 00:01:55.210 --> 00:02:00.160 Then, in addition, we explored the z-transform for z, a more 00:02:00.160 --> 00:02:02.640 general complex number. 00:02:02.640 --> 00:02:06.390 In the discrete time z-transform case, we expressed 00:02:06.390 --> 00:02:12.130 that complex number in polar form as r e to the j omega, 00:02:12.130 --> 00:02:17.800 and recognize that the z-transform expression in fact 00:02:17.800 --> 00:02:22.900 corresponds to the Fourier transform of the sequence 00:02:22.900 --> 00:02:25.400 exponentially weighted. 00:02:25.400 --> 00:02:30.610 Because of the exponential weighting, the z-transform 00:02:30.610 --> 00:02:35.100 converges for some values of r corresponding to some 00:02:35.100 --> 00:02:39.320 exponential waiting, and perhaps not for others, and 00:02:39.320 --> 00:02:44.890 that led to a notion which corresponded to the region of 00:02:44.890 --> 00:02:49.180 convergence associated with the z-transform. 00:02:49.180 --> 00:02:51.680 We talked some about properties of the region of 00:02:51.680 --> 00:02:55.230 convergence, particularly in relation to 00:02:55.230 --> 00:02:57.900 the pole-zero pattern. 00:02:57.900 --> 00:03:03.090 Now, they z-transform has a number of important and useful 00:03:03.090 --> 00:03:06.780 properties, just as the Laplace transform does. 00:03:06.780 --> 00:03:10.580 As one part of this lecture, what we'll want to do is 00:03:10.580 --> 00:03:14.440 exploit some of these properties in the context of 00:03:14.440 --> 00:03:16.930 systems described by linear constant coefficient 00:03:16.930 --> 00:03:19.360 difference equations. 00:03:19.360 --> 00:03:23.660 These particular properties that play an important role in 00:03:23.660 --> 00:03:25.280 that context are the properties 00:03:25.280 --> 00:03:27.570 that I indicate here. 00:03:27.570 --> 00:03:29.500 In particular, there is-- 00:03:29.500 --> 00:03:31.780 as with continuous time-- 00:03:31.780 --> 00:03:36.500 a linearity property that tells us that the z-transform 00:03:36.500 --> 00:03:40.870 of a linear combination of sequences is the same linear 00:03:40.870 --> 00:03:45.140 combination of the z-transforms, a shifting 00:03:45.140 --> 00:03:50.580 property that indicates that the z-transform of x of n 00:03:50.580 --> 00:03:56.740 shifted is the z transform of x of n multiplied by a factor 00:03:56.740 --> 00:03:59.780 z to the minus n 0. 00:03:59.780 --> 00:04:06.120 Then the convolution property for which the z-transform of a 00:04:06.120 --> 00:04:11.500 convolution of sequences is the product of the associated 00:04:11.500 --> 00:04:14.250 z-transforms. 00:04:14.250 --> 00:04:17.769 With all of these properties, of course, there is again the 00:04:17.769 --> 00:04:21.760 issue of what the associated region of convergence is in 00:04:21.760 --> 00:04:23.840 comparison with the region of convergence of 00:04:23.840 --> 00:04:26.500 the original sequences. 00:04:26.500 --> 00:04:30.040 That is an issue that's addressed somewhat more in the 00:04:30.040 --> 00:04:31.730 text and let's not go into it here. 00:04:34.240 --> 00:04:40.890 With the convolution property, the convolution property as in 00:04:40.890 --> 00:04:45.220 continuous time, of course, provides a mechanism for 00:04:45.220 --> 00:04:48.750 dealing with linear time invariant systems. 00:04:48.750 --> 00:04:52.640 In particular, in the time domain a linear time invariant 00:04:52.640 --> 00:04:55.980 system is described through convolution-- 00:04:55.980 --> 00:05:00.610 namely, the output is the convolution of the input and 00:05:00.610 --> 00:05:02.770 the impulse response. 00:05:02.770 --> 00:05:06.810 Because of the convolution property associated with the 00:05:06.810 --> 00:05:11.590 z-transform, the z-transform of the output is the 00:05:11.590 --> 00:05:15.650 z-transform of the input times the z-transform 00:05:15.650 --> 00:05:17.580 of the impulse response. 00:05:17.580 --> 00:05:22.410 Again, very much the same as what we had in continuous time 00:05:22.410 --> 00:05:26.590 and also what we had in the context of the discussion with 00:05:26.590 --> 00:05:29.410 the Fourier transform. 00:05:29.410 --> 00:05:32.290 In fact, because of the relationship between the 00:05:32.290 --> 00:05:37.280 z-transform and the Fourier transform, the z-transform of 00:05:37.280 --> 00:05:41.690 the impulse response evaluated on the unit circle-- 00:05:41.690 --> 00:05:44.570 in other words, for the magnitude of z equal to 1-- 00:05:44.570 --> 00:05:47.620 in fact corresponds to the frequency 00:05:47.620 --> 00:05:50.060 response of the system. 00:05:50.060 --> 00:05:53.400 More generally, when we talk about the z-transform of the 00:05:53.400 --> 00:05:58.300 impulse response, we will refer to it as the system 00:05:58.300 --> 00:06:00.225 function associated with the system. 00:06:03.130 --> 00:06:06.030 Now, the convolution property and these other properties, as 00:06:06.030 --> 00:06:09.740 I indicated, we will find useful in talking about 00:06:09.740 --> 00:06:13.260 systems which are described by linear constant coefficient 00:06:13.260 --> 00:06:18.460 difference equations, and in fact, we'll do that shortly. 00:06:18.460 --> 00:06:25.320 But first what I'd like to do is to continue to focus on the 00:06:25.320 --> 00:06:29.270 system function for linear time invariant systems, and 00:06:29.270 --> 00:06:33.350 make a couple of comments that tie back to some things that 00:06:33.350 --> 00:06:38.280 we said in the last lecture relating to the relationship 00:06:38.280 --> 00:06:43.510 between the region of convergence of a system, or-- 00:06:43.510 --> 00:06:47.080 I'm sorry, the region of convergence of a z-transform-- 00:06:47.080 --> 00:06:54.040 and the issue of where that is in relation to the poles of 00:06:54.040 --> 00:06:55.290 the z-transform. 00:06:56.980 --> 00:07:00.260 In particular, we can draw some conclusions tying back to 00:07:00.260 --> 00:07:05.780 that discussion about the pole locations of the system 00:07:05.780 --> 00:07:11.810 function in relation to whether the system is stable 00:07:11.810 --> 00:07:15.510 and whether the system is causal. 00:07:15.510 --> 00:07:19.960 In particular, recall from one of the early lectures way back 00:07:19.960 --> 00:07:25.830 when that stability for a system corresponded to the 00:07:25.830 --> 00:07:28.500 statement that the impulse response 00:07:28.500 --> 00:07:31.450 is absolutely summable. 00:07:31.450 --> 00:07:35.640 Furthermore, when we talked about the Fourier transform, 00:07:35.640 --> 00:07:42.220 the Fourier transform of a sequence converges if the 00:07:42.220 --> 00:07:44.500 sequence is absolutely summable. 00:07:44.500 --> 00:07:50.630 So in fact, the condition for stability of a system and the 00:07:50.630 --> 00:07:53.540 condition for convergence of the Fourier transform of its 00:07:53.540 --> 00:07:58.820 impulse response are the same condition, namely absolute 00:07:58.820 --> 00:08:00.900 summability. 00:08:00.900 --> 00:08:02.750 Now what does this mean? 00:08:02.750 --> 00:08:07.870 What it means is that if the Fourier transform converges, 00:08:07.870 --> 00:08:11.140 that means that the z-transform converges on the 00:08:11.140 --> 00:08:13.640 unit circle. 00:08:13.640 --> 00:08:18.200 Consequently, if the system is stable, then the system 00:08:18.200 --> 00:08:22.500 function, the z-transform or the impulse response, must 00:08:22.500 --> 00:08:25.210 also converge on the unit circle. 00:08:25.210 --> 00:08:28.610 In other words, the impulse response must have a Fourier 00:08:28.610 --> 00:08:31.160 transform that converges. 00:08:31.160 --> 00:08:36.549 So for a stable system, then, the region of convergence of 00:08:36.549 --> 00:08:39.450 the system function must include the unit 00:08:39.450 --> 00:08:42.620 circle in the z-plane. 00:08:42.620 --> 00:08:50.080 So we see how stability relates to the location of the 00:08:50.080 --> 00:08:54.450 region of convergence, and we can also relate causality to 00:08:54.450 --> 00:08:55.670 the region of convergence. 00:08:55.670 --> 00:09:01.850 In particular, we know that if a system is causal, then the 00:09:01.850 --> 00:09:05.980 impulse response is right-sided. 00:09:05.980 --> 00:09:08.250 For a sequence that's right-sided, the region of 00:09:08.250 --> 00:09:12.060 convergence of its z-transform must be outside 00:09:12.060 --> 00:09:15.110 the outermost pole. 00:09:15.110 --> 00:09:18.800 So for causality, the region of convergence of the system 00:09:18.800 --> 00:09:21.710 function must be outside the outermost pole. 00:09:21.710 --> 00:09:25.350 For stability, the region of convergence must include the 00:09:25.350 --> 00:09:30.150 unit circle, and we can also then draw from that the 00:09:30.150 --> 00:09:34.840 conclusion that if we have a system that's causal unstable, 00:09:34.840 --> 00:09:39.430 then all poles must be inside the unit circle because of the 00:09:39.430 --> 00:09:41.130 fact that the poles must-- 00:09:41.130 --> 00:09:44.420 because of the fact that the region of convergence must be 00:09:44.420 --> 00:09:47.480 outside the outermost pole and has to also 00:09:47.480 --> 00:09:49.910 include the unit circle. 00:09:49.910 --> 00:09:55.780 So for example, if we had let's say a system with a 00:09:55.780 --> 00:10:05.840 system function as I indicate here with the algebraic 00:10:05.840 --> 00:10:09.930 expression for the system function being the expression 00:10:09.930 --> 00:10:13.810 that I indicate here with the pole at z equals a third and 00:10:13.810 --> 00:10:18.290 another pole at z equals 2 and a zero at the origin. 00:10:18.290 --> 00:10:24.210 If, in fact, the system was causal, corresponding to an 00:10:24.210 --> 00:10:28.050 impulse response that's right-sided, this then would 00:10:28.050 --> 00:10:34.050 be the region of convergence of the system function. 00:10:34.050 --> 00:10:38.220 Alternatively, if I knew that the system was stable, then I 00:10:38.220 --> 00:10:40.910 know that the region of convergence must include the 00:10:40.910 --> 00:10:45.080 unit circle, and so this would be then the region of 00:10:45.080 --> 00:10:46.640 convergence. 00:10:46.640 --> 00:10:53.690 And what you might now want to ask yourselves is if instead 00:10:53.690 --> 00:10:57.980 the region of convergence for the system function is this, 00:10:57.980 --> 00:11:00.570 then is the system causal? 00:11:00.570 --> 00:11:02.390 That's the first question. 00:11:02.390 --> 00:11:06.340 The second question is is the system stable? 00:11:06.340 --> 00:11:08.840 Remembering that for causality, the region of 00:11:08.840 --> 00:11:12.150 convergence must be outside the outermost pole, and for 00:11:12.150 --> 00:11:16.540 stability it must include the unit circle. 00:11:16.540 --> 00:11:25.200 Now what I'd like to do is look at the properties of the 00:11:25.200 --> 00:11:29.430 z-transform, and in particular exploit these properties in 00:11:29.430 --> 00:11:33.850 the context of systems that are described by linear 00:11:33.850 --> 00:11:36.930 constant coefficient difference equations. 00:11:36.930 --> 00:11:40.230 The three basic properties that play a key role in that 00:11:40.230 --> 00:11:44.980 discussion are the linearity property, the shifting 00:11:44.980 --> 00:11:49.860 property, and the convolution property. 00:11:49.860 --> 00:11:52.430 These, then, are the properties 00:11:52.430 --> 00:11:55.380 that I want to exploit. 00:11:55.380 --> 00:11:59.740 So let's do that by first looking at a first order 00:11:59.740 --> 00:12:02.480 difference equation. 00:12:02.480 --> 00:12:06.310 In the case of a first order difference equation, which 00:12:06.310 --> 00:12:09.170 I've written as I indicate here-- 00:12:09.170 --> 00:12:11.930 no terms on the right hand side, but we could have terms 00:12:11.930 --> 00:12:13.860 of course, in general-- 00:12:13.860 --> 00:12:17.900 y of n minus ay of n minus 1 equals x of n. 00:12:17.900 --> 00:12:24.630 We can use the linearity property, so that if we take 00:12:24.630 --> 00:12:28.570 the z-transform of both sides of this expression, that will 00:12:28.570 --> 00:12:32.110 then be the z-transform of this term plus the z-transform 00:12:32.110 --> 00:12:34.180 of this term. 00:12:34.180 --> 00:12:39.790 So using those properties and together with the shifting 00:12:39.790 --> 00:12:44.010 property, the property that tells us that the z-transform 00:12:44.010 --> 00:12:49.110 of y of n minus 1 is z to the minus 1 times the z-transform 00:12:49.110 --> 00:12:53.520 of y of n, we then convert the difference equation to an 00:12:53.520 --> 00:12:55.400 algebraic expression. 00:12:55.400 --> 00:12:58.940 And we can solve this algebraic expression for the 00:12:58.940 --> 00:13:02.600 z-transform of the output in terms of the 00:13:02.600 --> 00:13:06.100 z-transform of the input. 00:13:06.100 --> 00:13:11.820 Now what we know from the convolution property is that 00:13:11.820 --> 00:13:15.670 for a system, the z-transform of the output is the system 00:13:15.670 --> 00:13:18.440 function times the z-transform of the input. 00:13:18.440 --> 00:13:24.720 So this factor that we have, then, must correspond to the 00:13:24.720 --> 00:13:29.260 system function, or equivalently the z-transform 00:13:29.260 --> 00:13:32.360 of the impulse response of the system. 00:13:32.360 --> 00:13:36.200 In fact, then, if we have this z-transform, we could figure 00:13:36.200 --> 00:13:41.560 out what the impulse response of the system is by computing 00:13:41.560 --> 00:13:45.730 or determining what the inverse z-transform, except 00:13:45.730 --> 00:13:52.590 for the fact that expression is an algebraic expression and 00:13:52.590 --> 00:13:55.940 doesn't yet totally specify the z-transform because we 00:13:55.940 --> 00:14:00.210 don't yet know what the region of convergence is. 00:14:00.210 --> 00:14:02.670 How do we get the region of convergence? 00:14:02.670 --> 00:14:05.320 Well, we have the same issue here as we had with the 00:14:05.320 --> 00:14:06.980 Laplace transform-- 00:14:06.980 --> 00:14:13.440 namely, the point that the difference equation tells us, 00:14:13.440 --> 00:14:17.340 in essence, what the algebraic expression is for the system 00:14:17.340 --> 00:14:22.080 function, but doesn't specify the region of convergence. 00:14:22.080 --> 00:14:26.910 That is specified by either explicitly, because one way or 00:14:26.910 --> 00:14:30.590 another we know what the impulse response is, or 00:14:30.590 --> 00:14:33.940 implicitly, because we know certain properties of the 00:14:33.940 --> 00:14:40.130 system, such as causality and or stability. 00:14:40.130 --> 00:14:46.030 If I, let's say, imposed on this system, in addition to 00:14:46.030 --> 00:14:55.580 the difference equation, the condition of causality, then 00:14:55.580 --> 00:14:59.530 what that requires is that the impulse response be 00:14:59.530 --> 00:15:03.700 right-sided or the region of convergence be outside the 00:15:03.700 --> 00:15:05.330 outermost pole. 00:15:05.330 --> 00:15:09.350 So, for this example that would require, then, that the 00:15:09.350 --> 00:15:12.940 region of convergence correspond to the magnitude of 00:15:12.940 --> 00:15:18.190 z greater than the magnitude of a. 00:15:18.190 --> 00:15:21.670 What you might think about is whether I would get the same 00:15:21.670 --> 00:15:25.960 condition if I required instead that 00:15:25.960 --> 00:15:29.310 the system be stable. 00:15:29.310 --> 00:15:31.500 Furthermore, you could think about the question of whether 00:15:31.500 --> 00:15:35.480 I could specify or impose on this system that it be both 00:15:35.480 --> 00:15:37.090 stable and causal. 00:15:37.090 --> 00:15:39.760 The real issue-- and let me just kind of point to it-- 00:15:39.760 --> 00:15:43.970 is that the answer to those questions relate to whether 00:15:43.970 --> 00:15:46.730 the magnitude of a is less than 1 or the magnitude of a 00:15:46.730 --> 00:15:48.160 is greater than 1. 00:15:48.160 --> 00:15:50.810 If the magnitude of a is less than 1 and I specify 00:15:50.810 --> 00:15:57.320 causality, that will also mean that the system is stable. 00:15:57.320 --> 00:16:02.170 In any case, given this region of convergence, then the 00:16:02.170 --> 00:16:06.340 impulse response is the inverse transform of that 00:16:06.340 --> 00:16:08.780 z-transform, which is a to the n times u of n. 00:16:11.490 --> 00:16:15.680 Now let's look at a second order equation, and there's a 00:16:15.680 --> 00:16:17.470 very similar strategy. 00:16:17.470 --> 00:16:21.430 For the second order equation, the one that I've picked is of 00:16:21.430 --> 00:16:24.700 this particular form, and I've written the coefficients 00:16:24.700 --> 00:16:28.270 parametrically as I indicate here for a specific reason, 00:16:28.270 --> 00:16:29.880 which we'll see shortly. 00:16:29.880 --> 00:16:32.870 Again, I can apply the z-transform to this 00:16:32.870 --> 00:16:38.580 expression, and I've skipped an algebraic step or two here. 00:16:38.580 --> 00:16:42.970 When I do this, then, again I use the linearity property and 00:16:42.970 --> 00:16:46.790 the shifting property, and I end up with this algebraic 00:16:46.790 --> 00:16:47.890 expression. 00:16:47.890 --> 00:16:54.240 If I now solve that to express y of z in terms of x of z and 00:16:54.240 --> 00:16:58.790 a function of z, then this is what I get 00:16:58.790 --> 00:17:00.040 for the system function. 00:17:02.430 --> 00:17:06.970 So this is now a second order system function, and we'll 00:17:06.970 --> 00:17:12.960 have two zeroes at the origin and it will have two poles. 00:17:12.960 --> 00:17:17.339 Again, there's the question of what we assume about the 00:17:17.339 --> 00:17:18.349 region of convergence-- 00:17:18.349 --> 00:17:20.240 I haven't specified that yet. 00:17:20.240 --> 00:17:25.990 But if we, let's say, assume that the system is causal, 00:17:25.990 --> 00:17:29.050 which I will tend to do, then that means that the region of 00:17:29.050 --> 00:17:32.830 convergence is outside the outermost pole. 00:17:32.830 --> 00:17:34.080 Now, where are the poles? 00:17:34.080 --> 00:17:38.920 Well, let me just kind of indicate that if-- 00:17:38.920 --> 00:17:40.010 and you can verify this 00:17:40.010 --> 00:17:43.250 algebraically at your leisure-- 00:17:43.250 --> 00:17:52.680 that if the cosine theta term is less than 1, then the roots 00:17:52.680 --> 00:17:56.110 of this polynomial will be complex. 00:17:56.110 --> 00:18:00.570 And in fact, the poles are at r e to the 00:18:00.570 --> 00:18:02.740 plus or minus j theta. 00:18:02.740 --> 00:18:08.970 So for cosine theta less than 1, then the poles are complex, 00:18:08.970 --> 00:18:13.960 and the complex poles at an angle theta and with a 00:18:13.960 --> 00:18:18.070 distance from the origin equal to the parameter r. 00:18:18.070 --> 00:18:21.550 In fact, let's just look at that. 00:18:21.550 --> 00:18:27.640 What I show here is the pole zero pattern. 00:18:27.640 --> 00:18:34.490 Assuming that r is less than 1, and that cosine theta is 00:18:34.490 --> 00:18:40.300 less than 1, and we have a complex pole pair shown here-- 00:18:40.300 --> 00:18:44.210 now if we assume that the system was causal, that means 00:18:44.210 --> 00:18:48.240 that the region of convergence is outside these poles. 00:18:48.240 --> 00:18:51.300 That would then include the unit circle, which means that 00:18:51.300 --> 00:18:54.290 the system is also stable. 00:18:54.290 --> 00:18:57.370 In fact, as long as the reach of convergence includes the 00:18:57.370 --> 00:19:02.850 unit circle, we can also talk about the frequency response 00:19:02.850 --> 00:19:03.680 of the system-- 00:19:03.680 --> 00:19:05.240 namely, we can evaluate the system 00:19:05.240 --> 00:19:08.500 function on the unit circle. 00:19:08.500 --> 00:19:13.270 We, in fact, evaluated the Fourier transform associated 00:19:13.270 --> 00:19:16.660 with this pole zero pattern last time. 00:19:16.660 --> 00:19:21.210 Recall that the frequency response, then, is one that 00:19:21.210 --> 00:19:26.620 has a resonant character with the resonant peak being 00:19:26.620 --> 00:19:33.010 roughly in the vicinity of the angle of the pole location. 00:19:33.010 --> 00:19:36.490 As the parameter r varies-- let's say, as r gets smaller-- 00:19:36.490 --> 00:19:38.710 this peak tends to broaden. 00:19:38.710 --> 00:19:41.980 As r gets closer to 1, the resonance 00:19:41.980 --> 00:19:43.230 tends to get sharper. 00:19:49.020 --> 00:19:55.100 This is now a look at the z-transform, and we see very 00:19:55.100 --> 00:19:59.000 strong parallels to the Laplace transform. 00:19:59.000 --> 00:20:02.920 In fact, throughout the course, I've tried to 00:20:02.920 --> 00:20:03.700 emphasize-- 00:20:03.700 --> 00:20:05.590 and it just naturally happens-- 00:20:05.590 --> 00:20:10.370 that there are very strong relationships and parallels 00:20:10.370 --> 00:20:14.890 between continuous time and discrete time. 00:20:14.890 --> 00:20:19.940 In fact, at one point we specifically mapped from 00:20:19.940 --> 00:20:22.960 continuous time to discrete time when we talked about 00:20:22.960 --> 00:20:27.460 discrete time processing of continuous time signals. 00:20:27.460 --> 00:20:32.760 What I'd like to do now is turn our attention to another 00:20:32.760 --> 00:20:38.000 very important reason for mapping from continuous time 00:20:38.000 --> 00:20:41.600 to discrete time, and in the process of doing this, what 00:20:41.600 --> 00:20:46.670 we'll need to do is exploit fairly heavily the insight, 00:20:46.670 --> 00:20:51.210 intuition, and procedures that we've developed for the 00:20:51.210 --> 00:20:55.150 Laplace transform and the z-transform. 00:20:55.150 --> 00:20:59.540 Specifically, what I would like to begin is a discussion 00:20:59.540 --> 00:21:06.530 relating to mapping continuous time filters to discrete time 00:21:06.530 --> 00:21:12.260 filters, or continuous time system functions to discrete 00:21:12.260 --> 00:21:14.410 time system functions. 00:21:14.410 --> 00:21:17.170 Now, why would we want to do that? 00:21:17.170 --> 00:21:21.130 Well, there are at least several reasons for wanting to 00:21:21.130 --> 00:21:25.140 map continuous time filters to discrete time filters. 00:21:25.140 --> 00:21:29.940 One, of course, is the fact that in some situations what 00:21:29.940 --> 00:21:34.140 we're interested in doing is processing continuous time 00:21:34.140 --> 00:21:38.220 signals with discrete time systems-- 00:21:38.220 --> 00:21:45.050 or, said another way, simulate continuous time systems with 00:21:45.050 --> 00:21:46.690 discrete time systems. 00:21:46.690 --> 00:21:49.530 So it would be natural in a setting like that to think of 00:21:49.530 --> 00:21:53.500 mapping the desired continuous time filter to a 00:21:53.500 --> 00:21:55.840 discrete time filter. 00:21:55.840 --> 00:21:58.480 So that's one very important context. 00:21:58.480 --> 00:22:02.880 There's another very important context in which this is done, 00:22:02.880 --> 00:22:11.650 and that is in the context or for the purpose of exploiting 00:22:11.650 --> 00:22:16.180 established design procedures for continuous time filters. 00:22:16.180 --> 00:22:18.440 The point is the following. 00:22:18.440 --> 00:22:22.240 We may or may not be processing a sample continuous 00:22:22.240 --> 00:22:25.260 time signal with our discrete time filter-- it may just be 00:22:25.260 --> 00:22:28.630 discrete time signals that we're working with. 00:22:28.630 --> 00:22:34.460 But in that situation, still, we need to design the 00:22:34.460 --> 00:22:38.280 appropriate discrete time filter. 00:22:38.280 --> 00:22:46.300 Historically, there is a very rich history associated with 00:22:46.300 --> 00:22:50.290 design of continuous time filters. 00:22:50.290 --> 00:22:54.860 In many cases, it's possible and very worthwhile and 00:22:54.860 --> 00:23:00.790 efficient to take those designs and map them to 00:23:00.790 --> 00:23:04.880 discrete time designs to use them as discrete time filters. 00:23:04.880 --> 00:23:08.650 So, another very important reason for talking about the 00:23:08.650 --> 00:23:13.120 kinds of mappings that we will be going into is to simply 00:23:13.120 --> 00:23:16.800 take advantage of what has been done historically in the 00:23:16.800 --> 00:23:18.050 continuous time case. 00:23:21.960 --> 00:23:26.160 Now, if we want to map continuous time filters to 00:23:26.160 --> 00:23:31.330 discrete time filters, then in continuous time, we're talking 00:23:31.330 --> 00:23:34.820 about a system function and an associated 00:23:34.820 --> 00:23:37.840 differential equation. 00:23:37.840 --> 00:23:43.230 In discrete time, there is the corresponding system function 00:23:43.230 --> 00:23:46.105 and the corresponding difference equation. 00:23:49.170 --> 00:23:53.960 Basically what we want to do is generate from a continuous 00:23:53.960 --> 00:23:59.640 time system in some way a discrete time system that 00:23:59.640 --> 00:24:05.650 meets an associated set of desired specifications. 00:24:05.650 --> 00:24:09.530 Now, there are certain constraints that it's 00:24:09.530 --> 00:24:13.410 reasonable and important to impose on whatever kinds of 00:24:13.410 --> 00:24:15.940 mappings we use. 00:24:15.940 --> 00:24:19.230 Obviously, we want a mapping that will take our continuous 00:24:19.230 --> 00:24:21.900 time system function and map it to a 00:24:21.900 --> 00:24:24.780 discrete time system function. 00:24:24.780 --> 00:24:28.210 Correspondingly in the time domain, there is a continuous 00:24:28.210 --> 00:24:32.520 time impulse response that maps to the associated 00:24:32.520 --> 00:24:35.610 discrete time impulse response. 00:24:35.610 --> 00:24:37.710 These are more or less natural. 00:24:37.710 --> 00:24:41.310 The two that are important and sometimes easy to lose sight 00:24:41.310 --> 00:24:45.450 of are the two that I indicate here. 00:24:45.450 --> 00:24:50.970 In particular, if we are mapping a continuous time 00:24:50.970 --> 00:24:55.130 filter with, let's say, a desired or desirable frequency 00:24:55.130 --> 00:24:59.680 response to a discrete time filter and we would like to 00:24:59.680 --> 00:25:03.360 preserve the good qualities of that frequency response as we 00:25:03.360 --> 00:25:09.460 look at the discrete time frequency response, then it's 00:25:09.460 --> 00:25:14.680 important what happens in the s-plane for the continuous 00:25:14.680 --> 00:25:19.860 time filter along the j omega axis relate in a nice way to 00:25:19.860 --> 00:25:24.310 what happens in the z-plane around the unit circle, 00:25:24.310 --> 00:25:29.170 because it's this over here that represents the frequency 00:25:29.170 --> 00:25:33.440 response in continuous time and this contour over here 00:25:33.440 --> 00:25:36.570 that represents the frequency response in discrete time. 00:25:36.570 --> 00:25:38.090 So that's an important property. 00:25:38.090 --> 00:25:41.100 We want to kind of the j omega axis to 00:25:41.100 --> 00:25:44.370 map to the unit circle. 00:25:44.370 --> 00:25:49.160 Another more or less natural condition to impose is a 00:25:49.160 --> 00:25:53.960 condition that if we are assured in some way that our 00:25:53.960 --> 00:25:59.400 continuous time filter is stable, then we would like to 00:25:59.400 --> 00:26:02.510 concentrate on design procedures that more or less 00:26:02.510 --> 00:26:07.550 preserve that and will give us stable digital filters. 00:26:07.550 --> 00:26:11.150 So these are kind of reasonable conditions to 00:26:11.150 --> 00:26:13.150 impose on the procedure. 00:26:13.150 --> 00:26:16.960 What I'd like to do in the remainder of this lecture is 00:26:16.960 --> 00:26:22.040 look at two common procedures for mapping continuous time 00:26:22.040 --> 00:26:25.370 filters to discrete time filters. 00:26:25.370 --> 00:26:28.020 The first one that I want to talk about is one that, in 00:26:28.020 --> 00:26:35.210 fact, is very frequently used, and also one that, as we'll 00:26:35.210 --> 00:26:40.990 see for a variety of reasons, is highly undesirable. 00:26:40.990 --> 00:26:44.930 The second is one that is also frequently used, and as we'll 00:26:44.930 --> 00:26:50.280 see is, in certain situations, very desirable. 00:26:50.280 --> 00:26:53.830 The first one that I want to talk about is the more or less 00:26:53.830 --> 00:27:00.260 intuitive simple procedure of mapping a differential 00:27:00.260 --> 00:27:04.790 equation to a difference equation by simply replacing 00:27:04.790 --> 00:27:07.265 derivatives by differences. 00:27:09.890 --> 00:27:13.870 The idea is that a derivative is more or less a difference, 00:27:13.870 --> 00:27:16.390 and there's some dummy parameter capital T that I've 00:27:16.390 --> 00:27:19.720 thrown in here, which I won't focus too much on. 00:27:19.720 --> 00:27:24.870 But in any case, this seems to have some plausibility. 00:27:24.870 --> 00:27:27.730 If we take the differential equation and do this with all 00:27:27.730 --> 00:27:32.410 the derivatives, both in terms of y of t and x of t, what 00:27:32.410 --> 00:27:36.560 we'll end up with is a difference equation. 00:27:36.560 --> 00:27:42.540 Now what we can use are the properties of the Laplace 00:27:42.540 --> 00:27:45.920 transform and the z-transform to see what this means in 00:27:45.920 --> 00:27:47.440 terms of a mapping-- 00:27:47.440 --> 00:27:50.940 in particular, using the differentiation property for 00:27:50.940 --> 00:27:52.390 Laplace transforms. 00:27:52.390 --> 00:27:56.970 In the Laplace transform domain, we would have this. 00:27:56.970 --> 00:28:02.400 Using the properties for the z-transform, the z-transform 00:28:02.400 --> 00:28:04.740 of this expression would be this. 00:28:04.740 --> 00:28:09.950 So, in effect, what it says is that every place in the system 00:28:09.950 --> 00:28:12.370 function or in the differential equation that we 00:28:12.370 --> 00:28:16.460 would be multiplying by s when Laplace transformed. 00:28:16.460 --> 00:28:18.000 In the difference equation, we would be 00:28:18.000 --> 00:28:21.070 multiplying by this factor. 00:28:21.070 --> 00:28:26.840 In fact, what this means is that the mapping from 00:28:26.840 --> 00:28:32.000 continuous time to discrete time corresponds to taking the 00:28:32.000 --> 00:28:38.200 system function and replacing s wherever we see it by 1 00:28:38.200 --> 00:28:43.110 minus z to the minus 1 over capital T. So if we have a 00:28:43.110 --> 00:28:47.400 system function in continuous time and we map it to a 00:28:47.400 --> 00:28:51.220 discrete time system function this way by replacing 00:28:51.220 --> 00:28:55.700 derivatives by differences, then that corresponds to 00:28:55.700 --> 00:29:01.390 replacing s by 1 minus z to the minus 1 over capital T. 00:29:01.390 --> 00:29:06.190 Now we'll see shortly what this mapping actually means 00:29:06.190 --> 00:29:09.980 more specifically in relating the s-plane to the z-plane. 00:29:09.980 --> 00:29:13.370 Let me just quickly, because I want to refer to this, also 00:29:13.370 --> 00:29:16.670 point to another procedure very much like backward 00:29:16.670 --> 00:29:20.910 differences which corresponds to replacing derivatives not 00:29:20.910 --> 00:29:23.770 by the backward differences that I just showed, but by 00:29:23.770 --> 00:29:25.640 forward differences. 00:29:25.640 --> 00:29:29.600 In that case, then, the mapping corresponds to 00:29:29.600 --> 00:29:36.030 replacing s by z to the minus 1 over capital T. It looks 00:29:36.030 --> 00:29:39.270 very similar to the previous case. 00:29:39.270 --> 00:29:43.330 So there the relationship between these system functions 00:29:43.330 --> 00:29:44.860 is what I indicate here. 00:29:47.800 --> 00:29:53.590 Let's just take a look at what those mappings correspond to 00:29:53.590 --> 00:29:56.730 when we look at this specifically in the s-plane 00:29:56.730 --> 00:29:59.250 and in the z-plane. 00:29:59.250 --> 00:30:04.920 What I show here is the s-plane, and of course it's 00:30:04.920 --> 00:30:07.350 things on the left half of the s-plane, poles on the left 00:30:07.350 --> 00:30:10.350 half of the s-plane that would guarantee stability. 00:30:10.350 --> 00:30:14.340 It's the j omega axis the tells us about the frequency 00:30:14.340 --> 00:30:21.630 response, and in the z-plane it's the unit circle that 00:30:21.630 --> 00:30:24.490 tells us about the frequency response. 00:30:24.490 --> 00:30:27.560 Things inside the unit circle, or poles inside the unit 00:30:27.560 --> 00:30:30.170 circle, that guarantee stability. 00:30:30.170 --> 00:30:35.310 Now the mapping from s-plane to the z-plane corresponding 00:30:35.310 --> 00:30:40.660 to replacing derivatives by backward differences in fact 00:30:40.660 --> 00:30:48.140 can be shown to correspond to mapping the j omega axis not 00:30:48.140 --> 00:30:51.780 to the unit circle, but to the little circle that I show 00:30:51.780 --> 00:30:55.230 here, which is inside the unit circle. 00:30:55.230 --> 00:30:58.550 The left half of the s-plane maps to the 00:30:58.550 --> 00:31:00.430 inside of that circle. 00:31:00.430 --> 00:31:01.210 What does that mean? 00:31:01.210 --> 00:31:06.500 That means that if we have a really good frequency response 00:31:06.500 --> 00:31:11.150 characteristic along this contour in the s-plane, we'll 00:31:11.150 --> 00:31:13.270 see that same frequency response 00:31:13.270 --> 00:31:14.910 along this little circle. 00:31:14.910 --> 00:31:17.030 That's not the one that we want, though-- 00:31:17.030 --> 00:31:20.420 we would like to see that same frequency response around the 00:31:20.420 --> 00:31:22.080 unit circle. 00:31:22.080 --> 00:31:26.960 To emphasize this point even more-- suppose, for example, 00:31:26.960 --> 00:31:33.230 that we had a pair of poles in our continuous time system 00:31:33.230 --> 00:31:36.000 function that looked like this. 00:31:36.000 --> 00:31:42.610 Then, where they're likely to end up in the z-plane is 00:31:42.610 --> 00:31:45.050 inside the unit circle, of course. 00:31:45.050 --> 00:31:50.080 But if the poles here are close to the j omega axis, 00:31:50.080 --> 00:31:53.930 that means that these poles will be close to this circle, 00:31:53.930 --> 00:31:58.320 but in fact might be very far away from the unit circle. 00:31:58.320 --> 00:32:01.530 What would happen, then, is that if we saw in the 00:32:01.530 --> 00:32:06.220 continuous time filter a very sharp resonance, the discrete 00:32:06.220 --> 00:32:08.900 time filter in fact might very well have that resonance 00:32:08.900 --> 00:32:12.660 broadened considerably because the poles are so far away from 00:32:12.660 --> 00:32:14.500 the unit circle. 00:32:14.500 --> 00:32:19.530 Now, one plus with this method, and it's about the 00:32:19.530 --> 00:32:25.570 only one, is the fact that the left half of the s-plane maps 00:32:25.570 --> 00:32:30.110 inside the unit circle-- in fact, inside a circle inside 00:32:30.110 --> 00:32:32.250 the unit circle, and so stability is always 00:32:32.250 --> 00:32:33.870 guaranteed. 00:32:33.870 --> 00:32:36.170 Let me just quickly mention, and you'll have a chance to 00:32:36.170 --> 00:32:38.910 look at this a little more carefully in the video course 00:32:38.910 --> 00:32:42.960 manual, that for forward differences instead of 00:32:42.960 --> 00:32:51.120 backward differences, this contour in the s-plane maps to 00:32:51.120 --> 00:32:56.380 a line in the z-plane, which is a line tangent to the unit 00:32:56.380 --> 00:33:01.250 circle, and in fact is the line that I showed there. 00:33:01.250 --> 00:33:05.890 So not only are forward differences equally bad in 00:33:05.890 --> 00:33:09.270 terms of the issue of whether they map from the j omega axis 00:33:09.270 --> 00:33:13.440 to the unit circle, but they have a further difficulty 00:33:13.440 --> 00:33:19.360 associated with them, which is the difficulty they may not 00:33:19.360 --> 00:33:21.505 and generally don't guarantee stability. 00:33:25.070 --> 00:33:29.840 Now, that's one method, and one, as I indicated, that's 00:33:29.840 --> 00:33:35.600 often used partly because it seems 00:33:35.600 --> 00:33:37.510 so intuitively plausible. 00:33:37.510 --> 00:33:40.960 What you can see is that by understanding carefully the 00:33:40.960 --> 00:33:44.500 issues and the techniques of Laplace and z-transforms, you 00:33:44.500 --> 00:33:48.280 can begin to see what some of the difficulties with those 00:33:48.280 --> 00:33:50.450 methods are. 00:33:50.450 --> 00:33:54.080 The next method that I'd like to talk about is a method 00:33:54.080 --> 00:33:57.000 that, in fact, is very commonly used. 00:33:57.000 --> 00:34:01.360 It's a very important, useful method, which kind of can be 00:34:01.360 --> 00:34:06.330 motivated by thinking along the lines of mapping the 00:34:06.330 --> 00:34:11.630 continuous time filter to a discrete time filter in such a 00:34:11.630 --> 00:34:15.580 way that the shape of the impulse response is 00:34:15.580 --> 00:34:16.620 preserved-- 00:34:16.620 --> 00:34:20.989 and, in fact, more specifically so that the 00:34:20.989 --> 00:34:26.590 discrete time impulse response corresponds to samples of the 00:34:26.590 --> 00:34:28.960 continuous time impulse response. 00:34:28.960 --> 00:34:32.800 And this is a method that's referred to as impulse 00:34:32.800 --> 00:34:35.260 invariance. 00:34:35.260 --> 00:34:40.790 So what impulse invariance corresponds to is designing 00:34:40.790 --> 00:34:46.100 the filter in such a way that the discrete time filter 00:34:46.100 --> 00:34:51.550 impulse response is simply a sample version of the 00:34:51.550 --> 00:34:57.200 continuous time filter impulse response with a sampling 00:34:57.200 --> 00:35:01.450 period which I denote here as capital T. That will turn into 00:35:01.450 --> 00:35:06.350 a slightly confusing parameter shortly, and perhaps carried 00:35:06.350 --> 00:35:08.270 over into the next lecture. 00:35:08.270 --> 00:35:11.900 Hopefully, we'll get that straighted out, though, within 00:35:11.900 --> 00:35:14.030 those two lectures. 00:35:14.030 --> 00:35:18.680 Remembering the issues of sampling, the discrete time 00:35:18.680 --> 00:35:22.970 frequency response, then since the frequency responses the 00:35:22.970 --> 00:35:27.180 Fourier transform of the impulse response is related to 00:35:27.180 --> 00:35:31.950 the continuous time impulse response as I indicate here, 00:35:31.950 --> 00:35:41.130 what this says is that it is the superposition of 00:35:41.130 --> 00:35:46.600 replications of the continuous time frequency response, 00:35:46.600 --> 00:35:51.440 linearly scaled in frequency and shifted and 00:35:51.440 --> 00:35:53.120 added to each other. 00:35:53.120 --> 00:35:56.830 It's the same old sort of shifting, adding, or aliasing 00:35:56.830 --> 00:35:58.430 issue-- the same sampling issues that 00:35:58.430 --> 00:36:01.740 we've addressed before. 00:36:01.740 --> 00:36:04.690 This equation will help us understand what the frequency 00:36:04.690 --> 00:36:06.290 response looks like. 00:36:06.290 --> 00:36:11.560 But in terms of an analytical procedure for mapping the 00:36:11.560 --> 00:36:14.540 continuous time system function to a discrete time 00:36:14.540 --> 00:36:20.020 system function, we can see that and develop it in the 00:36:20.020 --> 00:36:22.200 following way. 00:36:22.200 --> 00:36:27.000 Let's consider the continuous time system function expanded 00:36:27.000 --> 00:36:30.250 in a partial fraction expansion. 00:36:30.250 --> 00:36:33.320 And just for convenience, I'm picking first order poles-- 00:36:33.320 --> 00:36:44.540 this can be generalized multiple order poles, and we 00:36:44.540 --> 00:36:45.730 won't do that here. 00:36:45.730 --> 00:36:48.200 The same basic strategy applies. 00:36:48.200 --> 00:36:54.120 If we expand this in a partial fraction expansion, and we 00:36:54.120 --> 00:36:57.210 look at the impulse response associated with this-- we know 00:36:57.210 --> 00:36:59.940 how to take the inverse of Laplace transform of this, 00:36:59.940 --> 00:37:03.280 where I'm just naturally assuming causality throughout 00:37:03.280 --> 00:37:05.490 the discussion-- 00:37:05.490 --> 00:37:09.070 the continuous time impulse response, then, is the sum of 00:37:09.070 --> 00:37:14.610 exponentials with these amplitudes and at these 00:37:14.610 --> 00:37:17.910 complex exponential frequencies. 00:37:17.910 --> 00:37:21.570 Now, impulse invariance corresponds to sampling this, 00:37:21.570 --> 00:37:26.090 and so the discrete time impulse response is simply a 00:37:26.090 --> 00:37:29.700 sampled version of this. 00:37:29.700 --> 00:37:33.720 The A sub k, of course, carries down. 00:37:33.720 --> 00:37:35.440 We have the exponential-- 00:37:35.440 --> 00:37:39.340 we're sampling at t equals n capital T, and so we've 00:37:39.340 --> 00:37:43.550 replaced that here, and then the unit step to truncate 00:37:43.550 --> 00:37:46.550 things for negative time. 00:37:46.550 --> 00:37:50.360 Let's manipulate this further, and eventually what we want to 00:37:50.360 --> 00:37:52.530 get is a relationship-- 00:37:52.530 --> 00:37:53.230 a mapping-- 00:37:53.230 --> 00:37:58.360 from the continuous time to the discrete time filter. 00:37:58.360 --> 00:38:04.380 We have this step, and we can rewrite that now as I show 00:38:04.380 --> 00:38:08.860 here, just simply taking this n outside, and we have e to 00:38:08.860 --> 00:38:13.770 the s sub k capital T. Now this is of the form the sum of 00:38:13.770 --> 00:38:19.100 terms like A sub k times is beta to the n. 00:38:19.100 --> 00:38:23.750 We can compute the z-transform of this, and the z-transform 00:38:23.750 --> 00:38:26.840 that we get I show here-- 00:38:26.840 --> 00:38:30.680 it's simply A sub k over 1 minus e to the s sub k capital 00:38:30.680 --> 00:38:36.700 T z to the minus 1, simply carrying this term or this 00:38:36.700 --> 00:38:39.720 parameter down. 00:38:39.720 --> 00:38:42.750 So we started with a continuous time system 00:38:42.750 --> 00:38:46.800 function, which was a sum of terms like A sub k over s 00:38:46.800 --> 00:38:49.730 minus s sub k-- the poles were at s sub k. 00:38:49.730 --> 00:38:54.200 We now have the discrete time filter in this form. 00:38:54.200 --> 00:38:57.110 Consequently, then, this procedure of impulse 00:38:57.110 --> 00:39:03.400 invariance corresponds to mapping the continuous time 00:39:03.400 --> 00:39:07.840 filter to a discrete time filter by mapping the poles in 00:39:07.840 --> 00:39:10.080 the continuous time filter. 00:39:10.080 --> 00:39:15.870 According to this mapping, the continuous time filter pole at 00:39:15.870 --> 00:39:20.530 s sub k gets mapped to a pole e to the s sub k capital T, 00:39:20.530 --> 00:39:27.280 and the coefficients A sub k are preserved. 00:39:27.280 --> 00:39:31.210 That, then, algebraically, is what the procedure of impulse 00:39:31.210 --> 00:39:34.960 invariance corresponds to. 00:39:34.960 --> 00:39:39.630 Let's look at how we interpret some of this in 00:39:39.630 --> 00:39:42.180 the frequency domain. 00:39:42.180 --> 00:39:49.020 In particular, we have the expression that tells us how 00:39:49.020 --> 00:39:54.700 the discrete time frequency response is related to the 00:39:54.700 --> 00:39:57.470 continuous time frequency response. 00:39:57.470 --> 00:40:02.570 This is the expression that we had previously when we had 00:40:02.570 --> 00:40:05.490 talked about issues of sampling. 00:40:05.490 --> 00:40:07.910 So that means that we would form the discrete time 00:40:07.910 --> 00:40:12.460 frequency response by taking the continuous time 1, scaling 00:40:12.460 --> 00:40:17.100 it in frequency according to this parameter capital T, and 00:40:17.100 --> 00:40:21.190 then adding replications of that together. 00:40:21.190 --> 00:40:26.890 So if this is the continuous time frequency response, just 00:40:26.890 --> 00:40:30.460 simply an ideal low-pass filter with a cutoff frequency 00:40:30.460 --> 00:40:36.400 of omega sub c, then the frequency scaling operation 00:40:36.400 --> 00:40:42.130 would keep the same basic shape but linearly scale the 00:40:42.130 --> 00:40:47.600 frequency axis so that we now have omega sub c times T. Then 00:40:47.600 --> 00:40:52.060 the discrete time frequency response would be a 00:40:52.060 --> 00:40:57.400 superposition of these added together at multiples of 2 pi 00:40:57.400 --> 00:41:00.190 in discrete time frequency. 00:41:00.190 --> 00:41:03.350 So that's what we have here-- 00:41:03.350 --> 00:41:06.245 so this is the discrete time frequency response. 00:41:08.920 --> 00:41:11.810 This looks very nice-- it looks like impulse invariance 00:41:11.810 --> 00:41:15.610 will take the continuous time frequency response, just 00:41:15.610 --> 00:41:18.470 simply linearly scale the frequency axis, and 00:41:18.470 --> 00:41:21.210 incidentally periodically repeat it. 00:41:21.210 --> 00:41:24.360 We know that for an ideal low-pass filter, 00:41:24.360 --> 00:41:25.670 that looks just fine. 00:41:25.670 --> 00:41:28.940 In fact, for a band-limited frequency response, 00:41:28.940 --> 00:41:31.300 that looks just fine. 00:41:31.300 --> 00:41:35.355 But we know also that any time that we're sampling-- 00:41:35.355 --> 00:41:38.210 and here we're sampling the impulse response-- 00:41:38.210 --> 00:41:42.210 we have an effect in the frequency domain or the 00:41:42.210 --> 00:41:44.440 potential for an affect an effect that 00:41:44.440 --> 00:41:46.710 we refer to as aliasing. 00:41:46.710 --> 00:41:52.810 So in fact, if instead of the ideal low-pass filter we had 00:41:52.810 --> 00:41:56.030 taken a filter that was an approximation to a low-pass 00:41:56.030 --> 00:42:01.510 filter, then the corresponding frequency scale version would 00:42:01.510 --> 00:42:03.590 look as I've shown here. 00:42:03.590 --> 00:42:08.390 And now as we add these together, then what we will 00:42:08.390 --> 00:42:13.070 have is some potential for distortion corresponding to 00:42:13.070 --> 00:42:17.530 the fact that these replications overlap, and what 00:42:17.530 --> 00:42:19.660 that will lead to is aliasing. 00:42:22.330 --> 00:42:26.720 So some things that we can say about impulse invariance is 00:42:26.720 --> 00:42:29.270 that we have an algebraic procedure-- 00:42:29.270 --> 00:42:32.370 and I'll illustrate with another example shortly-- 00:42:32.370 --> 00:42:35.200 for taking a continuous time system function and mapping it 00:42:35.200 --> 00:42:37.680 to a discrete time system function. 00:42:37.680 --> 00:42:44.290 It has a very nice property in terms of mapping, the mapping 00:42:44.290 --> 00:42:48.910 from the frequency axis in continuous time due to the 00:42:48.910 --> 00:42:50.190 unit circle-- 00:42:50.190 --> 00:42:53.750 namely, to a first approximation. 00:42:53.750 --> 00:42:57.900 As long as there's no aliasing, the mapping is just 00:42:57.900 --> 00:43:01.200 simply a linear scaling of the frequency axis, although there 00:43:01.200 --> 00:43:02.910 may be some aliasing. 00:43:02.910 --> 00:43:06.640 That means, of course, that this method can only be used 00:43:06.640 --> 00:43:09.810 if the frequency response that's being mapped, or if the 00:43:09.810 --> 00:43:13.300 system that's being mapped, has a frequency response 00:43:13.300 --> 00:43:15.300 that's approximately low-pass-- 00:43:15.300 --> 00:43:16.940 it has to be approximately band-limited. 00:43:20.520 --> 00:43:23.970 Then what we have is some of potential distortion, which 00:43:23.970 --> 00:43:26.540 comes about because of aliasing. 00:43:26.540 --> 00:43:30.660 Also because of the mapping, the fact that poles at s sub k 00:43:30.660 --> 00:43:36.290 get mapped to poles at e to the s sub k capital T, if the 00:43:36.290 --> 00:43:39.960 analog or continuous time filter is stable-- 00:43:39.960 --> 00:43:43.050 meaning that the real part of s sub k is negative-- 00:43:43.050 --> 00:43:47.240 then the discrete time filter is guaranteed to be stable. 00:43:47.240 --> 00:43:51.220 In other words, the magnitude of z sub k will be guaranteed 00:43:51.220 --> 00:43:53.060 to be less than 1. 00:43:53.060 --> 00:43:56.300 I'm assuming, of course, in that discussion that we are 00:43:56.300 --> 00:43:58.730 always imposing causality on the systems. 00:44:01.700 --> 00:44:06.030 To just look at the algebraic mapping a little more 00:44:06.030 --> 00:44:09.650 carefully, let's take a simple example. 00:44:09.650 --> 00:44:15.130 Here is an example of a system, a continuous time 00:44:15.130 --> 00:44:21.690 system, where I simply have a resident pole pair with an 00:44:21.690 --> 00:44:26.000 imaginary part along the imaginary axis of omega sub r 00:44:26.000 --> 00:44:28.660 and a real part of minus alpha. 00:44:28.660 --> 00:44:34.650 And so the associated system function then is just the 00:44:34.650 --> 00:44:39.110 expression which incorporates the two poles, and I've put in 00:44:39.110 --> 00:44:42.850 a scale factor of 2 omega sub r. 00:44:42.850 --> 00:44:47.160 And now to design the discrete time filter using impulse 00:44:47.160 --> 00:44:49.990 invariance, you would carry out a partial fraction 00:44:49.990 --> 00:44:53.690 expansion of this, and that partial fraction expansion is 00:44:53.690 --> 00:44:55.990 shown below. 00:44:55.990 --> 00:44:59.550 We have a pole at minus alpha minus j omega r and at minus 00:44:59.550 --> 00:45:02.130 alpha plus j omega r. 00:45:02.130 --> 00:45:07.430 And to determine the discrete time filter based on impulse 00:45:07.430 --> 00:45:10.640 invariance, we would map the poles and preserve the 00:45:10.640 --> 00:45:14.480 coefficients a sub k, referred to as the residues. 00:45:14.480 --> 00:45:18.740 And so the discrete time filter that we would end up 00:45:18.740 --> 00:45:25.560 with as a system function, which I indicate here-- 00:45:25.560 --> 00:45:30.870 and we have, as I said, preserved the residue, and the 00:45:30.870 --> 00:45:34.820 pole is now at a to the minus alpha T, e to the minus j 00:45:34.820 --> 00:45:40.020 omega sub r T. That's one term, and the other term in 00:45:40.020 --> 00:45:44.950 the sum has a pole at the complex conjugate location. 00:45:44.950 --> 00:45:50.090 If we were to add these two factors together, then what we 00:45:50.090 --> 00:45:57.420 would get is both poles and a 0 at the origin. 00:45:57.420 --> 00:46:04.270 In fact, then, the pole is defined by its angle, and this 00:46:04.270 --> 00:46:15.050 angle is e to the j omega sub r capital T, and by its 00:46:15.050 --> 00:46:24.030 radius, and this radius is e to the minus alpha capital T. 00:46:24.030 --> 00:46:28.310 Now we can look at the frequency response associated 00:46:28.310 --> 00:46:31.600 with that, and let's just do that. 00:46:31.600 --> 00:46:40.730 For the original continuous time frequency response, what 00:46:40.730 --> 00:46:45.900 we have is simply a resonant character, as I've shown here. 00:46:45.900 --> 00:46:50.730 And if we map this using impulse invariance, which we 00:46:50.730 --> 00:46:56.920 just did, the resulting frequency response that we get 00:46:56.920 --> 00:46:59.670 is the frequency response which I indicate. 00:46:59.670 --> 00:47:03.920 We see that that's basically identical to the continuous 00:47:03.920 --> 00:47:09.430 time frequency response, except for a linear scaling in 00:47:09.430 --> 00:47:13.510 the frequency axis, if you just compare the dimensions 00:47:13.510 --> 00:47:21.210 along which the frequency axis is shown except for one minor 00:47:21.210 --> 00:47:25.500 issue, which is particularly highlighted when we look at 00:47:25.500 --> 00:47:30.180 the frequency response at higher frequencies. 00:47:30.180 --> 00:47:35.140 What's the reason why those two curves don't quite follow 00:47:35.140 --> 00:47:38.980 each other at higher frequencies? 00:47:38.980 --> 00:47:41.790 Well, the reason is aliasing. 00:47:41.790 --> 00:47:44.980 In other words, what's happened is that in the 00:47:44.980 --> 00:47:50.320 process so applying impulse invariance, the frequency 00:47:50.320 --> 00:47:53.770 response of the original continuous time filter is 00:47:53.770 --> 00:47:57.540 approximately preserved, except for some distortion, 00:47:57.540 --> 00:48:01.420 that distortion corresponding to aliasing. 00:48:01.420 --> 00:48:06.000 Well, just for comparison, let's look at what would 00:48:06.000 --> 00:48:08.490 happen if we took the same example-- 00:48:08.490 --> 00:48:10.710 and we're not going to work it through here carefully. 00:48:10.710 --> 00:48:15.100 We're not work it through it all, not even not carefully. 00:48:15.100 --> 00:48:19.560 If we took the same example and mapped it to a discrete 00:48:19.560 --> 00:48:22.720 time filter by replacing derivatives by backward 00:48:22.720 --> 00:48:28.200 differences, what happens in that case is that we get a 00:48:28.200 --> 00:48:32.030 frequency response that I indicate here. 00:48:32.030 --> 00:48:36.720 Notice that the resonance in the original continuous time 00:48:36.720 --> 00:48:41.020 filter is totally lost. 00:48:41.020 --> 00:48:44.950 In fact, basically the character of the continuous 00:48:44.950 --> 00:48:47.510 time frequency response is lost. 00:48:47.510 --> 00:48:48.410 What's the reason? 00:48:48.410 --> 00:48:49.920 Well, the reason goes back to the 00:48:49.920 --> 00:48:51.810 picture that I drew before. 00:48:51.810 --> 00:48:58.920 The continuous time filter had a pair of resident poles close 00:48:58.920 --> 00:49:00.830 to the j omega axis. 00:49:00.830 --> 00:49:03.850 When those get mapped using backward differences, they end 00:49:03.850 --> 00:49:07.850 up close to this little circle that's inside the unit circle, 00:49:07.850 --> 00:49:10.700 but in fact for this example, are far away 00:49:10.700 --> 00:49:11.950 from the unit circle. 00:49:16.350 --> 00:49:20.320 So far we have one useful technique for mapping 00:49:20.320 --> 00:49:23.600 continuous time filters to discrete time filters. 00:49:23.600 --> 00:49:27.160 In part to highlight some of the issues, I focused 00:49:27.160 --> 00:49:31.780 attention also on some not so useful methods-- 00:49:31.780 --> 00:49:35.410 namely, mapping derivatives to forward or backward 00:49:35.410 --> 00:49:37.630 differences. 00:49:37.630 --> 00:49:43.360 Next time what I would like to do is look at impulse 00:49:43.360 --> 00:49:46.470 invariance for another example-- 00:49:46.470 --> 00:49:51.360 namely, the design of a Butterworth filter, and I'll 00:49:51.360 --> 00:49:54.970 talk more specifically about what Butterworth filters are 00:49:54.970 --> 00:49:57.550 at the beginning of that lecture. 00:49:57.550 --> 00:50:01.170 Then, in addition, what we'll introduce is another very 00:50:01.170 --> 00:50:05.600 useful technique, which has some difficulties which 00:50:05.600 --> 00:50:11.360 impulse invariance doesn't have, but avoids the principal 00:50:11.360 --> 00:50:13.480 difficulty that impulse invariance does have-- 00:50:13.480 --> 00:50:14.950 namely, aliasing. 00:50:14.950 --> 00:50:19.880 That method is referred to the bilinear transformation, which 00:50:19.880 --> 00:50:24.090 we will define and utilize next time. 00:50:24.090 --> 00:50:25.340 Thank you.