1 00:00:00,520 --> 00:00:01,360 ‫Welcome back. 2 00:00:02,110 --> 00:00:08,200 ‫Before we go on, I want to mention something that I should have mentioned before. 3 00:00:08,980 --> 00:00:12,580 ‫So you've learned what a plus domain is. 4 00:00:12,730 --> 00:00:20,320 ‫So this is your LA plus domain in the LA plus the main you have a real number axis and the imaginary 5 00:00:20,320 --> 00:00:21,320 ‫no axis. 6 00:00:22,060 --> 00:00:31,870 ‫However, when you're real polls are zero, then you only have complex polls and then you are only on 7 00:00:31,870 --> 00:00:33,690 ‫this line here. 8 00:00:34,510 --> 00:00:45,550 ‫And so when you're only on this imaginary number line on this imaginary number axis, then this also 9 00:00:45,550 --> 00:00:46,330 ‫has a name. 10 00:00:47,440 --> 00:00:53,510 ‫This is a frequency domain here and this is what the four year transformation is about. 11 00:00:54,460 --> 00:00:59,170 ‫So the four year frequency domain is this red line here. 12 00:01:00,070 --> 00:01:07,210 ‫So the domain has two dimensions, the real number axis and then the imaginary number axis. 13 00:01:07,870 --> 00:01:10,870 ‫However, the four years frequency domain. 14 00:01:12,070 --> 00:01:18,160 ‫That's this red line when you forget about the real number axis. 15 00:01:19,220 --> 00:01:25,880 ‫And perhaps if you have studied signals, then you know that when you are in the frequency domain. 16 00:01:26,910 --> 00:01:36,600 ‫Then this is your frequency domain and then here you have the amplitude of a signal, but then when 17 00:01:36,600 --> 00:01:43,550 ‫you are in the frequency domain, then you also have to consider the phase of a signal. 18 00:01:44,430 --> 00:01:48,540 ‫But both of them have a frequency domain here. 19 00:01:49,080 --> 00:02:01,010 ‫And that frequency domain is this domain here when you assume that your ex or your real number is zero. 20 00:02:02,300 --> 00:02:09,770 ‫So if we assume these polls here and then we're going to write a general solution, then it would be 21 00:02:09,770 --> 00:02:21,380 ‫like this eight times the Euler's number to the power of X times T, which is this one times the oil 22 00:02:21,380 --> 00:02:28,760 ‫and no to the power of I times Y times T, which is this one here. 23 00:02:28,970 --> 00:02:38,660 ‫And then plus B times the oil, the number to the power of X times T and then times the oil the No to 24 00:02:38,660 --> 00:02:42,070 ‫the power of minus eight times Y. 25 00:02:42,770 --> 00:02:46,730 ‫So that's for this and this one, including the minus sign. 26 00:02:47,630 --> 00:02:59,390 ‫We saw this form when we computed a solution for a differential equation that had complex polls or complex 27 00:02:59,390 --> 00:03:00,140 ‫lambdas. 28 00:03:01,010 --> 00:03:11,480 ‫And so if you are on this line here, then that means that you're X equals zero and that means that 29 00:03:11,810 --> 00:03:18,830 ‫this thing becomes one and also this thing becomes one. 30 00:03:20,090 --> 00:03:28,610 ‫And so you only have imaginary polls then that will give you cosine and sine terms. 31 00:03:29,670 --> 00:03:41,610 ‫And so in that case, your solution will be like this, you will have a times, then callsign and then 32 00:03:41,730 --> 00:03:56,220 ‫Y times T and plus I times sine plus Y times T, and then you will have plus B and then you will have 33 00:03:56,820 --> 00:04:05,940 ‫cosine Y times T and then minus I times sine and then Y times T. 34 00:04:06,600 --> 00:04:12,360 ‫And of course we can then rearrange these equations like we did before. 35 00:04:12,870 --> 00:04:16,170 ‫But in this video I want to show you something else. 36 00:04:16,950 --> 00:04:22,400 ‫Note that you have Ys here and here and also here and here. 37 00:04:22,440 --> 00:04:23,850 ‫So what does that tell you. 38 00:04:24,690 --> 00:04:27,260 ‫Well the greater the Y the greater the frequency. 39 00:04:27,270 --> 00:04:27,570 ‫Right. 40 00:04:27,870 --> 00:04:32,150 ‫So if you have a small Y then your wave could be something like this. 41 00:04:32,700 --> 00:04:38,040 ‫But if you have a big Y, then your wave could be something like this. 42 00:04:38,650 --> 00:04:42,900 ‫So this constant here, it increases the frequency of your wave. 43 00:04:43,940 --> 00:04:52,640 ‫So if your polls are somewhere here like this, then you might have a wave like this, but if your polls 44 00:04:52,640 --> 00:05:02,030 ‫are somewhere here further apart, then your wave could be something like this because your imaginary 45 00:05:02,030 --> 00:05:05,900 ‫part is greater here that will increase the frequency of your wave. 46 00:05:06,920 --> 00:05:16,910 ‫So what you need to understand is that the larger imaginary parts you choose for your poles, the greater 47 00:05:16,910 --> 00:05:20,600 ‫the frequency of oscillation of your system will be. 48 00:05:21,650 --> 00:05:29,930 ‫And so if you want to be purely in the frequency domain, that is not altered by these damping effects 49 00:05:29,930 --> 00:05:30,370 ‫here. 50 00:05:31,130 --> 00:05:41,750 ‫In other words, if you want to avoid this and this, that would either damp your signal or make it 51 00:05:41,750 --> 00:05:43,760 ‫greater like this. 52 00:05:44,770 --> 00:05:50,210 ‫So here are your real number would be less than zero. 53 00:05:50,500 --> 00:05:53,160 ‫And then here it would be greater than zero. 54 00:05:54,160 --> 00:06:01,570 ‫If you want to avoid this damping effect and then this explosive effect, then your real number has 55 00:06:01,570 --> 00:06:08,530 ‫to be zero and then you will be on this line, which actually is your frequency domain. 56 00:06:09,760 --> 00:06:19,480 ‫So when I think about lipless versus Fauria domain, then Tamela plus domain is a more general domain. 57 00:06:19,690 --> 00:06:22,120 ‫It includes one more dimension. 58 00:06:22,990 --> 00:06:32,620 ‫Your for your domain is only this imaginary number access and then your love plus domain is your real 59 00:06:32,620 --> 00:06:36,170 ‫number access and your imaginary number access. 60 00:06:36,220 --> 00:06:38,110 ‫So it's a two dimensional domain. 61 00:06:38,590 --> 00:06:40,480 ‫So that's what I wanted to point out. 62 00:06:40,780 --> 00:06:41,590 ‫Thank you very much.