‫Welcome back.
‫Before we go on, I want to mention something that I should have mentioned before.
‫So you've learned what a plus domain is.
‫So this is your LA plus domain in the LA plus the main you have a real number axis and the imaginary
‫no axis.
‫However, when you're real polls are zero, then you only have complex polls and then you are only on
‫this line here.
‫And so when you're only on this imaginary number line on this imaginary number axis, then this also
‫has a name.
‫This is a frequency domain here and this is what the four year transformation is about.
‫So the four year frequency domain is this red line here.
‫So the domain has two dimensions, the real number axis and then the imaginary number axis.
‫However, the four years frequency domain.
‫That's this red line when you forget about the real number axis.
‫And perhaps if you have studied signals, then you know that when you are in the frequency domain.
‫Then this is your frequency domain and then here you have the amplitude of a signal, but then when
‫you are in the frequency domain, then you also have to consider the phase of a signal.
‫But both of them have a frequency domain here.
‫And that frequency domain is this domain here when you assume that your ex or your real number is zero.
‫So if we assume these polls here and then we're going to write a general solution, then it would be
‫like this eight times the Euler's number to the power of X times T, which is this one times the oil
‫and no to the power of I times Y times T, which is this one here.
‫And then plus B times the oil, the number to the power of X times T and then times the oil the No to
‫the power of minus eight times Y.
‫So that's for this and this one, including the minus sign.
‫We saw this form when we computed a solution for a differential equation that had complex polls or complex
‫lambdas.
‫And so if you are on this line here, then that means that you're X equals zero and that means that
‫this thing becomes one and also this thing becomes one.
‫And so you only have imaginary polls then that will give you cosine and sine terms.
‫And so in that case, your solution will be like this, you will have a times, then callsign and then
‫Y times T and plus I times sine plus Y times T, and then you will have plus B and then you will have
‫cosine Y times T and then minus I times sine and then Y times T.
‫And of course we can then rearrange these equations like we did before.
‫But in this video I want to show you something else.
‫Note that you have Ys here and here and also here and here.
‫So what does that tell you.
‫Well the greater the Y the greater the frequency.
‫Right.
‫So if you have a small Y then your wave could be something like this.
‫But if you have a big Y, then your wave could be something like this.
‫So this constant here, it increases the frequency of your wave.
‫So if your polls are somewhere here like this, then you might have a wave like this, but if your polls
‫are somewhere here further apart, then your wave could be something like this because your imaginary
‫part is greater here that will increase the frequency of your wave.
‫So what you need to understand is that the larger imaginary parts you choose for your poles, the greater
‫the frequency of oscillation of your system will be.
‫And so if you want to be purely in the frequency domain, that is not altered by these damping effects
‫here.
‫In other words, if you want to avoid this and this, that would either damp your signal or make it
‫greater like this.
‫So here are your real number would be less than zero.
‫And then here it would be greater than zero.
‫If you want to avoid this damping effect and then this explosive effect, then your real number has
‫to be zero and then you will be on this line, which actually is your frequency domain.
‫So when I think about lipless versus Fauria domain, then Tamela plus domain is a more general domain.
‫It includes one more dimension.
‫Your for your domain is only this imaginary number access and then your love plus domain is your real
‫number access and your imaginary number access.
‫So it's a two dimensional domain.
‫So that's what I wanted to point out.
‫Thank you very much.