Hello, Michael. Hello, Michael. He's just checking in on you guys making sure you guys are all right. All right, folks Thank you for coming by to learn a little bit about Momentum. All right, let's get a working definition. Shall we in physics momentum is the quantity of motion of a moving body measured as a product of its mass and Velocity another way to say that the impetus gained by a moving object as a runner I can appreciate momentum. You are pushing yourself your body weight forward with your legs and your leg strength and Whether it's uphill or downhill and I love momentum, especially downhill, right? So in life Yesterday yesterday on the vlog go check it out I finally finally was able to get a little organized a little bit in the backyard with the assembly of the shed And I'm gonna continue today tonight after work. I'm not stopping guys I am a little tired of being unorganized and Allowing a little bit of clutter to overtake my inner happiness. Yes, I get stressed out a little bit when stuff is everywhere I don't like it. It's probably a little bit of OCD whatever that's called. I don't even know what it's called But anyway Momentum I want to use momentum from yesterday to today to tomorrow and I want to bring you along the journey First of all, let's look at the clutter Taking the camera off the tripod. No steady shots. All right. This is our patio. This is our patio. Look at this This is too much. It's too much and the shed and the shed, which is looking better But you know, there's my pile of running shoes Looking good, right? The shed is looking good. This is organized now. This is exciting folks. This is exciting And then there's the stuff under the bed Enough is enough And then momentum is paused when little man need you, right? Little man need you. Pause, hit that, pause button, pause. Ready? I can still remember How that music used to make me smile Dance that I could make those people dance and maybe they'd be happy for a while But February made me shiver with every paper I've delivered Bad news on the doorstep. I couldn't take one more step I can't remember if I cried when I read about his widowed bride But something touched me deep inside The day, the music Momentum thwarted, not by the banana peppers, by a sick child Ah, I feel so bad. He's got a stomach, pretty bad stomach ache. So anyway Momentum paused for a few minutes on the organization Little Don McLean, don't mind if I do, don't mind if I do Alright, the momentum is back We're literally getting a dump truck here Do you want the little one or the big one? Momentum, got it wrapped up, got it wrapped up, got it wrapped up Folks, momentum is so exciting. It's amazing When you have momentum, it's like nothing can stop you You have another gear that other people only dream about Well, right now I feel like I have momentum to get things organized Like at a whole other level that I haven't been organized at in a while So, Seth and I, it's way past his bedtime Joseph's feeling a little under the weather Seth and I are off to Home Depot to buy two screws Just two screws, actually no, four screws to hang something up in the shed So we can get more organized, so come along It's just like, we got momentum, we got momentum You know what I'm doing right now, you better believe it It's coming, just be patient, it's coming Ready, Seth? Momentum, it's happening in here folks Organization is happening The patio is finally cleaning up After yesterday's vlog all about organizing the momentum of putting the shed together And getting stuff in the shed and organized in the shed I just told myself, tonight I cannot go to bed until I get the patio clean And then, guess what? Tomorrow I will start I won't finish, but I will start, you know what? If you guys have been watching, a long time you know that I have one task to finish before the winter It's, it's, it's, I know, I know But like, I've got the momentum, now that I'm not running 100 mile races It's like, I've got a little bit of time to get some of these tasks done And I'm using that momentum mentally and physically to a certain extent But more mentally to tell myself, I did the shed, wow It was actually kind of hard to put that shed together But somehow I persevered and, you know, got a little angry at times But I persevered and I did it somehow And now tonight, woo, the patio is clear The table is clear, we can eat dinner out there again Why not? I'll just show you And so I'm definitely one of those people who feel overwhelmed when there's like a little too much to do Or there's a little too much clutter and I can't like think clearly And so, yes, I feel like now that the shed is done, now that the patio is clear Come on, come on, come on, let's do this, alright? So, folks, if you have any questions, any thoughts about momentum and the concept of momentum Comment below, question of the day, maybe this is the second one I can't remember But another one if it isn't, have you ever implored momentum in your life? In a sport, in an activity, in a hobby, in a relationship, in a... I don't know, anything, anything Like, does momentum even cross your brainwaves? And on that topic, the vlog, folks If you're a YouTuber, you've got to put one foot in front of the other You've got to start getting that snowball going And you've got to get the momentum started You've got to put one foot forward That's how you start, that's how you start And yes, there will, over time, be a snowball effect The momentum will pick up and pick up exponentially at some point If you're patient, alright? I'm telling you, I'm telling you, it'll happen So, I love you guys Thanks for learning a little bit about momentum If you learned anything, let me know, I appreciate it You guys rock Let's seek a little beauty, shall we? Shall we? See ya, too


Okay, during the break I was asked a few questions. I guess I missed some, to mention some details or I should elaborate on some points. The first one here is that when I was talking about microfuel distribution problem I forgot to mention that this distribution is for ideal plasma and when you do take into account the interaction between plasma particles of course the distribution will be different. For the stronger plasma coupling is, the further this curve will deviate from the ideal plasma case but it's not principle difference, I mean it will look more or less like the same. There will be peak at same value and there will be tail sometimes with exponential tail when it's actually easy to understand if your radiator is charged that other perturbers cannot approach it very close, there is a say classical minimal approach distance, right? And after that the far tail will be determined only by Maxwellian tail so it will be exponential decay here. The other point I probably should mention that the natural broadening exercise line shape it's nice to show how you get Lorenzen but usually in practice natural broadening can be safely ignored except for x-ray emission when the Einstein coefficient become large and so the natural broadening could become important. And in general natural broadening includes also not only just spontaneous radiative decay but also phenomena like Auger decay in which case it could be even stronger for mid-z elements, okay? Say the shape of ka-alpha lines, I don't want to elaborate much but it's mainly determined by this natural phenomena Auger and spontaneous radiative decay. And the last point when I was talking about this ergodicity arguments of course that assumes the particles in my ensemble are independent radiators which coming back to the analogy with a throwing dice if you look at your neighbor and try to after all cheat with the dice that the result will be different. So yes we are talking about independent radiators that means no coherent plasma radiation here that's the assumption, okay? So it's nice that we talked right now about natural broadening because basically what it says that the width of a line is determined by the time of life of its level or when both levels are unstable it's of course sum of two contributions. There is probably a bit of what I will explain you that's an expression for the line broadening of isolated lines in a minute I will explain what I mean by what people mean by isolated lines. So what it says that width of a line can be calculated according to the following rules. There are two main contributions one comes through in elastic processes and if you look at one of these terms basically to see its integral f sorry for confusion here f is not the field f is distribution of velocities it's Maxwellian distribution usually. So it's just sigma and v that you know that the rate of collisional process of course when you integrate over the velocity distribution you get temperature rate for a given temperature. So it's a rate of all in elastic processes of all processes that cause electron to go from level u up a level to to some another level and the same of course for the low level of the transition we count for all transition for all collisions that cause electron to go to jump from L to to some another L prime level. So actually this part is should be for people working with collisions straight forward evident. There is something more which is less evident and here is the difference of of elastic scattering amplitudes of the upper and lower level so it can be the total broadening the total width can be written as a sum of this in elastic part and elastic part. Okay so it's here they are written explicitly like I said it's just very similar to the it's very similar to what we saw for spontaneous for natural line broadening and yeah the meaning of isolated lines is that the broadening of a given level is small compared to distance in energy to other levels with strictly speaking to only we you need to consider only those levels that cause that stark broadening that broadening okay. So once we have lines that basically do not overlap one with another in the spectrum one can assume the isolated lines case so it's just some of inverse values of time of life of upper and lower levels the inelastic part okay. In principle one can generalize it to other impact other impact processes like ionization recombination etc. So the width is just the inelastic part is sum of the population rates of upper and the population rates of the lower levels again that was derived as human isolated lines when the lines are not isolated you can no longer use this simple expression for calculating a stark line broadening a few words about what's called a standard theory of line broadening that was developed by a few famous scientists around sixties including Hans Grimm and in Russia there was Sobelman and a few more of course I don't want to limit it to just these two of course contributed to development of this theory and as I said it's sixties no computers no no computers comparable to the powerful ones we have today so people had to invent ways of doing with the problem of n-body basically n-body problem analytically so the standard theory basically says we will talk separately about consider separately ions and electrons so ions are static quasi static so for quasi static it's very simple as I showed you an example with limon alpha just take for each given value of electric field of micro field calculate the spectrum then do convolution with the field distribution and you get the line shape fine electrons are different and electrons are so fast you saw in the simulation electrons are usually much faster than ions and they are so far they can be treated like a collision the impact approximation which yeah I forgot to mention in the previous slide when I said it's very much similar to to the derivation we did for for spontaneous emission in which case we got Lorenzen so the results of this impact mechanism like impact excitation ionization etc. give rise also the Lorenzen shape and that basically what you see here if you take a real part of this complex denominator actually it's operator you will get Lorenzen modified a general Lorenzen shape in which let's not go into the details okay but the basic idea of standard theory of line broadening is strict separation of slow particles which are usually irons and fast particles which are usually electrons in principle it could be the case when irons are also so fast that they can be safely considered in the same impact approximation fine but the real problem is with intermediate cases when irons do move and the motion is important but they are not really in the impact regime and there is no easy there was no easy analytical way found by that time to deal with that the problem of inability to to to to work with ion motion was that the main driving reason behind invention of using of computer simulations for line shape broadening problem the pioneers is Roland Stamm who was by that time I guess postdoc of Waslumber, Dietrich Waslumber and the idea is actually very simple you just simulate plasma right just put and body solve and body problem puts sufficiently large number of particles in a sufficiently large box let them move and interact you can calculate the line you can calculate electric fields as a function of time just give it as a input for the Schrodinger equation time-dependent Schrodinger equation you solve it you get anything from Schrodinger equation including the evolution of the dipole operator calculate autocorrelation function take four years and that's all so very simple except that it quite time-consuming and it is still time-consuming today and you can imagine it was a real a real enterprise to do it in the end of 70s but these guys did it and the results were very impressive really so I'm just going to show you exactly the same just in the graphical way so computer simulations are still basically the same today of course computers are different and you take more particles that everything is more accurate etc etc but the main the the main structure of the computer simulation of line shape rodent is basically the same so you have something which I call particle field generator which just and body and body simulations of various complexity and in principle it may include also external magnetic or electric fields these fields also can be time-dependent what you get as a result as the output of this part of the simulation code is something like that field history has just I plotted just absolute value let's say but of course it's vector you have all projection so you have three-dimensional field and then you solve yeah you solve Schrodinger equation time-dependent equation and the result you get evolution time evolution operator and you get evolution of dipole operator taking Fourier transform that's actually the the simplest apart here you get the spectrum how do you solve it usually you one nobody solves Schrodinger equation of course in the coordinate space it's always matrix so that's the Schrodinger equation we can rewrite it in form of the time-development operator that's in the in the interaction representation the dipole operator is calculated like that when you have the evolution operator handy and then one either do the indeed autocorrelation function like I've shown you or say I in my simulation is doing just directly taking Fourier of dipole and taking square of absolute value and there is also some fine details about slight polarization and dependence on the on the well-known dependence on the energy of the transition but basically that's the same so the idea as I say is very simple there could not be a simpler model for line shape calculation except that it is very time consuming so obviously people are trying to to invent to find ways of dealing with this problem using a simpler means I don't have time to to describe everything I will be a bit biased here and show you some my results and my colleagues if you look at the yeah obviously when you solve Schrodinger equation the the larger number of states you have in it the more problematic is the more difficult is to solve difficulty in terms of computer time of course so that the higher number of components in a given transition is that the harder is to solve it and hydrogen or hydrogen like transition between high end lines read the transition I especially problematic here because the number of components goes like you know as n squared and you can easily get to a completely and manage and manageable problem even with today's computers just to put you into the some context I once calculated using computer simulations Balmer 15 line from n equal 15 to n equal two and for a single line shape for a single density single temperature it took about two months of of CPU time okay okay on the other hand high end lines are important for diagnostics I don't want to go into the details so but interesting point is if you look at a spectrum now it's static stark effect of some high end in this case limon nine between n equal nine to n equal one transition you look separately at pi and sigma components you can pay attention that each one that the components the intensities of each pi and sigma components separately form a lie on a parabola and if you do some of pi and sigma and usually that the case in plasma that the micro fields has no specific direction so you can safely average so obviously average of these two mirrors parabola is just a rectangular shape so that was the idea let's use rectangular line shape instead of very complex stark pattern well so that's it you defined a line shape static static line shape as a rectangular it's zero beyond that values and the constant value inside of course it should be normalized to have we always wanted to normalize to one of course area normalized so that's numerical coefficient which depends on the stark constant of this of given transition and in the case of hydrogen it can there is a well-known expression in all textbooks about atomic about quantum atomic physics there is simple generalization for not just limon but in principle any transition from n to n prime they're just straightforward one so again we have rectangular line shape for fixed value of electric field so next step if you want to do it in the quasi static approximation let's do a convolution with a field distribution and in the case of in the case of yes so the field distribution is given by that and I do by w of f and I do that simple convolution and you remember this normalization that is used in the that was obtained in deriving the Holtzmark field distribution so I have zero again and that's the Holtzmark constant and we can go to dimensional as units both for the field it's the same better we saw when I plotted the Holtzmark distribution and instead of using these energies we or frequencies we normalize it to let's say typical stark splitting for the same given field for Holtzmark field and then everything can be expressed in a very nice way and which becomes so there is one single numerical integration here and in fact if we talk about ideal plasma which is always good to begin with this can be done analytically and that's the result is a very simple expression and it's a universal expression that should explain in principle any line shape in the quasi static approximation for sufficiently high n what is sufficiently high is a separate question of course but in practice quite surprisingly for me even though I began all this work for a really high n transition often even you know and actually what's important is not n but delta n difference between upper and lower and so even for delta n as low as 2 the results are very nice okay so let's see how this function looks like so I plot that with a green dashed line and I custom in line shape literature if the line shape is symmetric let's just plot only one half of it the rest will be symmetric so that's this universal function that and I plot the same true calculations of two lines two high lines lineman nine and lineman ten and you see that except of the central region the rest is just extremely nice extremely nice feet it's not feet it's in analytical derivation but in fact if you do include some other broadening mechanisms which always exist like Doppler or electron impact broadening or whatever they will smooth the central part as well so the result will often be practically indistinguishable that's why it also as I said quite surprisingly work even for relatively low principle quantum numbers okay so that's quasi static and but you do want to take in touch into account dynamics that's basically the point when I said when people started to use computer simulations to to find a way to somehow deal with ion motion but there was a few other approaches to deal with ion motion and one of them that I want to briefly describe is what's called frequency fluctuation model initially it was developed by in something like 20 or even more years ago but it was recently reformulated basically it's the same but but the mathematical formulation is is a cleaner one so what it says the field the microfield fluctuates fluctuates with different frequency but we can find a some typical one well there is quite obvious choice for the typical frequency you just take typical velocity divide our typical distance inter particle distance and there must be a frequency okay so what is the typical velocity is of course maxwellian one and typical distance and just density to the power of minus three it's in denominator so here's plus one third and well forgetting about numerical factors and once you know once you assume the field fluctuates with some frequency you say okay so it fluctuates and one you have once you have electrons sitting at one component stark component of a fixed electric field there are many components like you saw in lima nine for example that's a whole forest of components so it sits with field static but then feels fluctuates and electron jumps to another component and it does it randomly that the assumption of this ffm method the jumping is random from any state from any stark component you jump randomly to another one and in mathematical terms it can be expressed by this actually it's very simple expressions so the line shape is calculated like that that's expression okay where the only computationally intensive but it's nothing compared to computational intense intensive simulation of course it just again it's just simple integral numerical integration you start from quasi static line shape which is in the case of high end I showed you can be done really simple then do this integration and you can imagine that describes something which it's kind of diffusion and that's exactly the the notion here for frequency fluctuation you diffuse the static line shape with a typical frequency which is defined by here and then there is some another operation I don't want to explain the details but if you look in this paper it has a very nice explanation how you derive it just I don't have time now but it's quite straightforward so let's try to apply this method to this I forgot to say that this substitution of complex line shape with just simple rectangular I call it quasi contiguous because you can forget about distances between there are so many components and forget about each discrete component describe it as a one a contiguous entity the rectangular one okay so if you plug in that the expression becomes very simple for that J this J integration can be done analytically and you get this universal universal expression it's called it's universal when once you do the scaling of all parameters to the to the typical frequency and the typical stark static stark effect of for a given transition but when you do the scaling all transition any transition is described by exactly the same expression I want to show you some application of this method you you've had already a few times in the previous days about continuum lowering and actually continuum lowering is a bit of confusing topics there are different ways of of thinking about continuum lowering one is what is continuum for example one can say it's once you stop to see bound spectrum another approach say no continuous when the electrons are free which is not the same it's the same when you are considering a single atom but in plasma it's different the electron could be not free you must apply some work to get it out of plasma yet the spectrum that it produces is no longer discrete okay so that's a confusion and some approaches to continuum lowering that is good when you really understand you want to define with it what is the limit of free electrons it's not necessarily the same when you see the transition from from discrete spectrum to continuum one I want to describe you some approach that was given in a book of Hans Grimm by the way he he cites in this context a paper which he co-authored but in that paper I didn't find this explanation so so I I quote is that the text book because I didn't find it explicitly explained in any other paper in any paper okay so to do let's calculate bound bound spectrum for a series we want to calculate it for a series and continue up to the some principle quantum number at which the broadening of that line will be comparable to the distance between the neighbor levels and that's the reason behind the English teller effect English teller criterion that you've heard already a few times before okay and we may continue actually if for a few more principle quantum numbers just to make sure we have a smooth transition and we will assume free bound edge at the the deposition of the last member of the series we at which we stop we can we do convolution of the free bound continuum which is in is a you know it's drawn as a sharp edge we do convolution with the same line broadening we got for the last members of the of the bound bound series and some up I want to point to stress that there is no ionization potential depression is assumed here I just want to calculate the the the line broadening of a series and that's the result for lineman series hydrogen lineman series in modestly dense plasma 10 to the 17 particles per cubic centimeter one EV temperature and I do not post here Boltzmann trivial Boltzmann factor comparison and that is the series one calculation is done with this QC FFM very fast method that took about one second the other calculation is computer simulation is actually believed one of the most massive computer simulation of line shape ever done that's almost 400 fully interacting states and it took about one month on today computer so the speed factor if you use the simple analytical expression is about 3 million and you see how nicely they compare actually there is a small difference here but there is no continuum wave function in computer simulation that was that would increase the size of my Hamiltonian to infinity which is not double so but as long as we are talking about bound bound series that the agreement is extremely nice so now we want to compare with real measurements and that's actually the only time I show you something wavelength because that's how experimental list presents a result okay so that high end series in again hydrogen it's Balmer series okay very nice experiments I do not think there are comparable experimental studies of the same accuracy so it's extremely nice data really and it's not easy in general to reproduce and now I'm trying to reproduce it using the inferred value that were given from my belief Thompson scattering so we have densities and I'm doing this it's not computer simulation it's just this simple analytical calculation which takes one second actually it can be made to milliseconds if I wish with some computer tricks program tricks and just see how similar they are really even up to a very very fine details like here for the lowest density 1.8 10 to the 16 years you have epsilon and then you have what it goes I don't remember right and then something a bit here and you do here do see something a bit here so actually the agreement is extremely nice and again that the approach is very simple and it's so simple that well I wrote a program I intended to make it public by the time of of this school I didn't have time but I I'll try to do it in the coming months so it will be distributed freely of course that as a simple program and it runs extremely fast and you can calculate and hydrogen at least hydrogen like of course it's not for for isolated lines for the lights and there are different methods but for hydrogen like especially for high end which is very difficult to calculate just analytically I promise you will have this tool soon I just need to find time to do it to publish a paper and so they will be attached extra material okay I want to to mention data source when you can find pre computed or you can compute line shapes so one is star to be database which is for isolated lines there are quite a few species and transitions it's no it's nowhere complete but but it's a large database so probably you should start if you need to find the stock brown for isolated line you should go to this point and by the way like I said there's always a question how strong field is when you go from quadratic limit to the linear and the same is true for isolated lines for low density it would be isolated the line for for any line but there is for sufficiently high density the line will no longer be isolated so this database also there is also also they they they quote the maximum value of density for which you can safely use the data okay you can safely assume that the line is isolated another source is the classic book of Hans Grimm there are appendices with least broadening and shifts and widths of of of neutral of very many neutral and single ionized ions there is also appendices in the same book about hydrogen but I do not recommend you to use those okay but for isolated lines it's it's good I from time to time I do the calculations of different lines and compare with Hans calculations I always surprised how accurate they are I mean I use today computer simulations and it's not computer simulation it's just this standard theory of line broadening and there are many problematic in principle problematic issues with that but most of the time there is also very accurate so it's not electronic form but we need to page through the book the other one that's plasma formulae that self advertisement you can calculate a lot of different things with it including the stark broadening of hydrogen like or Riedberg transition so for isolated lines you go to this this too for hydrogen like you go to this one and it's always good to compare you plot both of them as a function of density and they at which some point they will intersect and just do interpolation between the two maybe I'll show examples okay the last but not the least there is an east atomic spectral line broadening bibliographic database which has a lot of links a lot of entries on stark broadening literature okay a few notes on line shape accuracy well line shape calculation is very complex task and as always with the complex calculations to really give a correct value of theoretical uncertainty is sometimes even more difficult than do the calculations themselves so one should really be critical about using just average let's say source of published data online broadening I have a few rules of some that I use myself and you can consider them as useful something like that if there is a study online broadening and there is no mention of accuracy at all factor 2 factor 3 don't expect it to be more accurate than factor 2 okay if people do claim some accuracy about 20 40 percent give or take it must be discussed in explain in the paper if it's not go back to this assumption okay 10 to 20 percent there is really must be a separate section devoted to accuracy again if it doesn't exist go to here number one factor 2 3 to 10 that's must be a subject of the study the study must be that we want to compare that with that and accuracy is our point if not go to number one well sometimes you do see studies which claim less than better than 3 percent accuracy in particular comparison to experimental data but even without comparison to experimental data I do not list do not give you a suggestion advice for politically correct reasons but basically throw that trash in that paper okay one case pay attention some data some data base yes some published data when people say with it might mean full with at half max sometimes it means half with at half max so if you do not pay attention you get that factor 2 in addition to factor 2 that okay and that well sometimes it helps but who knows so please in particular in the book of Hans green weeds is half with at half max okay but for example in start be database is full with at half max please pay attention well there I know of at least one study published study by I don't say famous person ill famous person not to give names who claim that calculations of Hans green were wrong by a large factor about 2 which turned out to be he forgot or I don't know what that in those tables in Hans green book half weeks half max are given okay so don't make a similar mistake well I have few more minutes and well first of all I want to to give you a list of a short list of recommended books and reviews number one of course it the classical textbook of Hans green on spectral line broadening and then very nice and introduction to plasma spectroscopy really given in very simple terms and there is a nice chapter online broadening by a him who was the first lecture in this school and there is a nice nice review of of models for start broadening by Marco jigosos it's a recent review and there is a list also self promoting a bit that's a short mini review on computer simulation models in line shapes so maybe I'll show you in the remaining five or so minutes that's the interface for stark be database do you see do I need to increase font so it's okay yes or not increase okay well if I increase too much there would be a problem with you know working with that let's first keep here and when we go to the data will increase okay so you go to access to the data by the way this is part of the of the virtual observatory project European project so for example big hill okay in this case there is no there is no ionized helium because ionized helium would be hydrogenic and hydrogenic is not lines are not isolated that's why you do not see it here okay so you go to select in this case just one pick your density you can also enter just specific value but for the case of isolated lines if you remember just sigma nv so the broadening is proportional to the density so there is no need actually to pick anything then one of the standard one so transition as you see there are quite a few what do we want I don't know pick your choice whatever you can also select wavelength interval and the temperature or you can select all okay by the way the calculations are done and there by Sylvie it's based on the code by Sylvie Sahabriho and Milan Dimitriyevich and collaborators I believe that for the interface itself there are references that you can follow here the result the density the transition the wavelengths of the transition and the broadening coefficient not brown coefficient there is a width sorry width and the shift of the transition as you see it's function of function of temperature where do you see temperature yeah here's the temperature and for several values of temperature you have width full with half max in this case and shift of the transition and if you need something in between you can interpolate usually the grid is chosen sufficiently dense and sufficiently wide wide to cover any important temperature for specific transition and by the way they provide also fitting coefficients to fit the data if you do not want to interpolate just by simple interpolation also there is a simple expression for interpolation okay the other one I mentioned is the plasma formulary that the link actually will have a session afternoon devoted to resources for line for for atomic data something like that and maybe I will have more time to explain here now I just to show one simple example of calculations so it's a web based utility you don't need to install anything is select plasma parameters let's say 10 to 16 temperature ion parameters charge mass and temperature which is by default the same you select atomic system you select the same for radiator its charge mass and temperature it may be a percentage if it's used as a dopant finally you pick your select the atomic system actually in this case charges one means the core is charges zero means its hydrogen and let's look for for example H beta third transition from n equal 4 to 2 we go to explorer and here we are interested in the energy total stark width which is given in electron volts you may to calculate it in their centimeters there is no anstroms if you insist you need to do this stupid one over lambda etc. calculations yourself okay and you can well I will have more minutes to calculate to show you more capabilities of these two afternoon okay well I think that's all the time is over coffee break


Welcome all back to the channel Fish and Freaks and to the great and wonderful outdoors It's like we got a bluebirder right now on the Dead Sea already brought a banana because Yeah, it doesn't even matter on the Dead Sea. I'm gonna try to catch some crappie though All right, we may look for a bass or two They catch one in the process, but you know, I have been trying to catch these things up in the dirt for a while now So hopefully today is the day We're going to slide up in here to Marina gonna slide on in get that rod Now I've been catching the bigger female crappie here in the last month away from the bank Bigger ones have been suspended the last lake I went to I Did end up catching some shallow crappie, but they were just staging males This is what's really interesting You know the rare the really dark crappie They almost look like black crappie white ones I came to that realization because when I clean the crappie all of them were males there were no females and So it's it's really cool to see those crappie all colored up. So what I'm gonna do is Go up in here around some shallow Rocks and brush and stuff and I'm just going to make some Some fancast with some some jigs from around swimming Can I talk I'm so excited about crop vision and then I've got a cork rig that is is very effective this time of year when they move up shallow so We're gonna look at those two rigs. We're gonna leave the the big pole the big 10-foot scoper pole In the box today, hopefully hopefully won't have to do any of that. So This is the fun stuff. Let's see if they're up shallow Probably gonna have to do some re-rigging because because that Yeah, yeah, yeah, we're gonna have to start the day on that note Unfortunately, I'm gonna pick up the cork rig because it's this in full function. There's cast with the cork and we are Hooked up on a colored up crappie. Yes, sir. Ladies and gentlemen natural colored dangled dart So that is a white crappie, but he has got his tuxedo on he's colored up So there's some in front of me here around this dock, but obviously there's some up shallow as well We're gonna keep some of these today Not to but let's check out what we got here Let's check out what we got. Oh, yeah got an 11-inch or Gotta keep the first one That's the rule. You got to keep the first one Okay, that is Extremely good start to the day. I'm gonna make a few more casts I'm gonna show you guys this rig that I got here. Oh got him.  Oh Right there under the dock got him another little guy Which we will let go can't resist that dart. Just looks like a little minnow not doing anything. Oh Yeah under the dock Little better one. Oh goodness. He come off. Oh My gosh, I'm so pumped right now. I've been waiting on this bite like all season. It's finally here Water is 65 and the crappie are moving in baby They're moving in I see some stagers up underneath the docks They are up on the bank. It is going down So the rod I'm gonna use for this is our micro series In the it's a gold series micro series. It's a light. It's not an ultra light This is my favorite all-around crappie fishing pole For vertical jigging and just fan casting It's also my trout rod can really do a Lot of panfish species with this this pole right here the only thing I don't use it for is scoping those those crappie Because I want to keep the keep the bait out in front a little bit If I'm trying to go for some big ones on this on the scope, but this is seven foot So I like that You know typical bass rod length you get a little bit more leverage Or casting distance with it, but it's still it's a buggy whip and you can you can handle throwing those little baits so I've got The thousand size spinning reel on here 15 pound braid. I do believe I love fishing the braid because it's just so easy to cast Very very easy to cast and if if you're gonna do the whole braid on your spinning reel I Suggest using an eight strand over Like a standard four strand Four strand is more abrasive abrasive resistant But the eight strand Casts a lot better and when you're a crappie fishing You're not not really worried about abrasion resistance All right tying up my connection not here with an eight pound fluorocarbon leader I have a 16th ounce Underspin head on there. So that's good for stained water, you know swimming it around to get that extra flash and This bait right here The snacky swimmer there's two different sizes in it This is electric chicken right here and this this little combo just swimming it around is just awesome I catch everything in the lake on this Bass crappie white bass catfish, whatever Anything that is eating minnows up in the shallows and even bed fish this time here We'll probably catch a bass on this today So it's a really fun bait to just throw around if you're just trying to get bites for whatever It's really fun, but searching for cropping and getting bigger ones Like to throw the larger size snacky swimmer and you just get some big bites on it All right, pretty much the exact same setup here thousand size reel And in our goog and gold series spinning reels I've got the 15 pound 8 strand braid on there same leader 8 pound leader But I have a line through cork on this. This is a fill cork They make really nice corks And I have a bobber stop All right, like you'd use flipping You know we call it a bobber stop We use it on our flipping setups all the time for Texas rigs But today I'm actually using it for what it is named for a bobber Stop so what makes this really neat when this hits the water? It slides up and it stops so that bait is going to sit two feet under the surface But when I cast it the cork is connected to the rig So instead of having you know a clamp on bobber up here that becomes more difficult to cast You got two things they kind of spin around they go tangled whatever this all stays together So it it enables me to cast it easier. I can pitch it I can also Get it up under docks because the whole thing is staying together and then when it hits the water it goes stops right there and You're good to go. Just watch it go under. Oh Yeah Guys to me there is no more fun crappie fishing bite than fishing shallow and getting that stomp Hit it they just hit it so hard they hit it like a bass does There's one. Oh goodness gracious Bopped me Yes, sir. That's a good white crappie. I might be a female One with me. That's two bites in a row right there. Just dropped off a little deeper. So Came over a shallow spot didn't get any bites and then Dropped off cast up there by a stick There's another one. Oh, yeah That's a largemouth bass Well, it's a spotted bass. See you buddy. It's not ideal for the 16 ounce jig. I guess they are spawning So you're right up there on the mirror that one would fill a nice Really would this bag right here. This is this crappie designated plastics Bag and it is perfect for this. So this is the two and a half inch snackie I'm just gonna go with Standard monkey milk and crappie on the monkey male. I've an inch or perfect Same size Good recipe. I'll show you guys I mean, he'll measure but barely 400 yards Very good reason y'all to always carry some form of knife in your boat because that can ruin your fishing day right there I usually keep a little ranch where I can take the lock nut off the prop and make sure there's none left in there I'll have to do that off the water today, but it can break the seal and ruin your trolling motor. You don't want that Not a croppy lmb bass. I'm just gonna filet him I have caught more bass and I have croppy possible fry garter. That is a fun if you don't have any Snackie swimmers Definitely suggest the two and a half inch All right y'all. This is just turned into fish madness catching everything Getting so many bites probably had over 30 fish, you know between the crappie and the bass I'm gonna cruise into us a little calmer area. I was fishing wind We've got an overcast that's rolled in here. I'm gonna go back to that Cork rig and see if I can catch a couple more crappies I really didn't want to catch that many as far as to keep I want to catch a thousand of them But as far as to keep I don't the freezers doing pretty good right now So I'm gonna see if I can get any of these calmer water crappies. They're not fully loaded on the bank yet It's just kind of here in there, but my gosh, there's so many bass. I mean It's like they're fighting for territory Okay, something just took off with my cork. It's gonna be a bass About four or five pounder They're quote on those It has gotten so windy out here this little calm spot that I'm in is about the only place I can fish The most effective way to catch them right now by far is Throwing the snacky swimmer around this bait Literally catches everything it swims and so I've got this on the 16th ounce head And I love that combination with the two and a half. You can just swim it really slow And there's a two inch as well, which I throw on crappie quite a bit. You know when they're being finicky so You know half inch smaller, but I love throwing this because it just keeps the rod bent I mean sometimes you go out bass fishing or even crappie fishing you want to work hard to try to get some big bites Get those nice rewarding bites sometimes you just want to keep that pole bent and keep the action going and That's exactly what this little thing does Google swad calm You can use code LFG save 10% and check out but get some monkey milk. That's this color right here It's kind of like a pearl glitter color and then also Get some stain waller water colors if you got stained water electric chicken is is money I mean I catch bass and and crappie on that electric chicken as well And we got some real clear water colors like pro blue red and all those fun colors as well The cork rig I thought was gonna be the program started off strong And when the crappie are up there shallow and they're loaded and you can just cast to the same spot You just need to keep that bait in place. This is amazing and also keeping it around cover Or you don't have a long scented pole and you know just keep it on top of them This allows you to do that So it just got way too windy though The cork was just blowing around and I wasn't able to keep it in place And the most effective way to fish for him was just fan casting around because they're they're not loaded loaded up right now And I just caught I don't know how many bass in the process of crappie fishing so got about six or seven in the well Typically this time of year I keep you know five to ten every time I go and they just adds up Keep fresh crappie going into the pan and I've got a recipe that I'm working on That is really good even our kids love to eat it Stephanie is dialing it in and Next time I go out crappie fishing. I'll probably show you guys that catch-and-cook Action and show you that recipe. It's amazing. So I'm gonna sign it off for today guys. So subscribe Smash get me outdoors. I'll see you on the next one


Oh, okay. Really haven't changed one bit, have you? Guess not. It's a bias. Finally, after so long. Thanks, man. Oh, wow, it's beautification. So, did I buy this house? It's been years. Decades at least. Since I've been inside this house. I lived here when I was just a kid. With my family. My mom. My siblings. My dad. I received a call from Tobias telling me that they were finally ready with the repairs this morning. Took them long enough. Freaking repair people. It's funny. Even after what happened, and all that time away from home, it looks just like I remembered it to be. All right. Is my memory that precise? I wish mine was. It was like the furniture, and photos aren't here anymore. But hey, can't blame anybody for that. I mean, who would want to keep the previous residents' photos up on the wall? I guess I'd better use that memory for something useful. I don't have the full story. Nor do I know why it happened. But I can tell you how it started. Everything else is up to you to figure out. This is the story of the night I lost my home. The story you never knew. And almost my family. Win five nights at Freddy's. Just ended. You know, it's hard to act interested when I've seen all of this. The Jerry of Creation story mode. Hello, people of the internet, and welcome to that. Again, I've seen everything in this game before. I just thought I'd play it. Because I've never played it myself. All right. Boys, welcome to the Jerry of Creation story mode. Again, everyone in the FNAF community has seen someone play this game or played it themselves. At the very least, I know for a fact you've seen Mark play it. There's really no introduction. It's made by Nixon, the man himself. So, let's start off with the bedroom. Again, I don't think I need to introduce this game. Everyone should know it. If you don't, and I know it so. My reaction is not going to be honest and true. I thought it was mine. I just bought it, right? This is actually in the past. No. It's me. Johnny blocks. Hello, people of the house. My name is Johnny. PUNCH SOMEONE!  My response timing, you know, is off the normal. I was looking for some food. I went out of cheese balls. It's been a while since I've had cheese balls. A lot of stuff. I won't go into. Poor's bed. Yeah, I can stand. I got some legs.  Yeah, what is wrong with me? After that, Dad carried this stranger to our guest room. He thanked me for convincing my parents to let him rest in our house for a night. Or at least, why collected his thoughts? After that, Mom put me in bed. I can't remember much about Michael. But I knew he wasn't a bad person. Or at least he didn't look at me. Just because you bad guy doesn't mean you're a bad guy. I was young, so I guess I just didn't know any better. Certainly didn't know what a burglar was. That's for sure. Okay, Michael, okay. I was implying he was. I never talked to him again after that. Well, except for when the storm happened. That's when I woke up on my bed and I saw them. That's when I woke up, you know, on the couch. Alright. Now we actually get into the gameplay. The story will continue, but right now gameplay time. Press space bar to stand up and sit down. Press space bar to react with objects. And look up while sitting and hold left click to sleep. So, I know some of the controls. I got to memorize them. So I'm just going to hide them. I'm not going to show them again. Hey, alright. I can't. Look, there we go. Okay. Pop goes. Sorry. Don't look them too long. You'll feel weird and see creepy things. Why do I have these posters in my bedroom? Are you staring at the scary monsters? The worst of the all. Okay, first things first. Freddie, the frizzy brown bear. Close it through the window. Yeah.  Then there's Bonnie, the purple bunny. He'll be there. He's not shy at all. He'll knock on the door to let you know. Oh no, I got to turn the light off.  You can't see you well in the dark. Oh, but make sure you're sitting down when you do that. You right? Then there's the chicken. Check the crust often. If she's peeking through, make sure to disable the lights. It doesn't want to be seen, but ignoring her doesn't help either. If she does catch you, she'll scare you and stun you. I know this is a lot to take care of, but you okay to just listen carefully? Yeah, how old am I here? You're explaining this to me. You'll be right beneath your bed. Awesome. You'll hear scraping to the bottom of the crib. Be watching as you hook on the back of your bed. Because sooner or later, you'll be right there next to the bed. Go over here. Right there. That too. Whoa. That's nice. I mean, that makes sense. Like how I use my mind close co-ins and turn off the lights. No. Ow. Oh, I'm sure of it. I know you can do it. Hangs up the phone. I know what to do. I know how all these guys work. So... Yeah, now we just hang out here. We play the waiting game. I think usually Freddy is the first to show up. Maybe. Cool. Oh, there he is. Hey, what's up, man? Got to wait. There we go, he's gone now. Cool. That was easy. Freddy is probably one of the easiest ones. Bonnie... Okay, Bonnie can trick you sometimes. Because he'll knock on the door just like that.  So it's kind of a guessing game. You got to play with Bonnie. Chica, super easy. The only thing you really got to worry about with Chica is your light. Because if you have this off, your insanity will go up. And you can tell how insane you are by looking at this light right here. I forget what it's called. It's the temperature thing. The temperature dial. Thermometer. And if that blinks orange, yellow, red, or any of those colors, you're going to go insane. The importance of my house screaming. Great. Also, who has a clock that only counts the hours? Like, that's really got to suck. Anyways. Already at one. Doing pretty good for ourselves. It's really just Freddy, but now I feel like they should start to get more aggressive. Especially Bonnie. There we go. I didn't come in that time. See? It's always a guessing game with him. I'm just waiting on Chica now because I know she's going to activate soon. Oh, right there. I called it. I'm going to turn that back on because she left. So did Freddy. Alright, cool. I came in that time. Alright, neat. Go away. Thank you. Awesome. So, now you should have a basic understanding of almost everyone's mechanics. Still haven't seen Foxy, but he usually appears much later in the night. Hello. Cool. Awesome. Nice seeing you. Whoa, hey, that's Foxy. Maybe he doesn't appear much later in the night. And he's gone. Awesome. Actually, no, I still want to keep that on. The annoying part about Foxy is that everyone can still attack you and come at you, even while you're quote-unquote sleeping. So, he can be a bit of a pain sometimes. But he's usually not that bad because it doesn't appear a lot. Maybe only a few times a night. Yeah, Freddy is. And he's gone. Awesome. So, it's difficult, but it's usually manageable. At least in the first section of the game. Alright, Chica.  Go on. Good girl. I feel like I'm talking to my dog.  Oh, for good luck. Nice. So, you can see it's kind of starting to change colors. It's still green, but it's starting to turn that yellowish-green color. But it's...man, I thought it would be at least three. But, oh, hey, there's Freddy again. We're starting to get into the more difficult part of the night, because now almost halfway there is Freddy. So, now they're going to start to get more aggressive, come at you more often. Three, there we go. Yeah, see that yellowish-greenish color? Kind of like those eyes up there that I can't click. Oh, hey, Chica. He's there. Awesome, boys. Good job, everyone. Round of applause. See? Turning a bit yellow. I don't want to say anything now. We're getting pretty lucky with Foxy. Alright, he's gone. Freddy is coming. Someone's screaming again, but who needs that? That's good that he didn't come in, because I really needed to get rid of Freddy. Because he can attack you pretty fast, actually. Pretty, pretty, pretty fast. Said pretty of it. See, now it's yellow. So, really got to keep this on as much as possible. Because, oh, hello. I didn't even hear you. Because if we don't, we're going to go insane. And we'll die, and we don't want that. Who needs death? We don't want death. You can also say we're starting to shake a bit. Think if we keep staring at this light, we'll calm down. Maybe not. Maybe that's not true. Never mind. Okay, he's here. I thought staring at the light helps. Everything's under control. 4am. Nobody's really doing anything. So that's really good. I hate if she can. Yeah, see now it's turning that orange color. Awesome. Get that light back on. Now we're really starting to shake. He's here. Gotta get back up, turn that thing on. Freddy is there. Got him. Alright, now it's starting to turn that reddish orange color. He's gone. Everything's all good. Stare at the light a bit. God dang chica. See now my field of view is starting to get very blurry. Oh god. What was that? What was that? Did I blink? Was that just a really long blink? Or was that me?  Good, he's not coming in. So good with Freddy almost. Awesome. I really gotta get Freddy. Awesome, awesome, awesome.  Turn that on. Five boys. We at five. Starting to go a little bit. Alright, that's good, that's good, that's good. That's starting to go a little bit insane here, but that's fine. Everything's under control. No one's coming in. Someone's coming in. Alright, that's good, get that back on. Everything's good, everything's good. Everything's starting to go a little bit. Black and white, that's all good. We don't need that because we're all good. Now we're nice and calm. I don't know how long he was there, but that definitely was not good. I need to stop talking so I can actually hear and comprehend what is happening in my general area. I don't think staring at the light helps. I'm just gonna throw that out there, in case it does. Maybe it slows it down. Maybe that's what it do. That's good. Things good. Still shaking a bit. I'm a little bit nervous. I've got final exams in the morning tomorrow. I really shouldn't be staying up this late. That's chica, so that's fine. Yes! Two years of Five Nights at Freddy's August 8th, 2016. This video commemorates the occasion by releasing a brand new behind the scenes screenshots. Learn how to make your own games, news, and entertainment. That was what? Second try? Third? I think that was our second try. I think we died once. But that's fine. Did we die? I feel like we did once. But anyways, that was the bedroom. It was hard, but still a little bit of the challenge. Next is the living room, but that is for another episode. Sorry if this was pretty long, but I just kept rambling on and on as I was playing, so I'm probably not gonna be able to have as many cuts to make this video shorter. But it's a really fun game, and I'm sure you guys wanna see a lot of gameplay. But thank you everyone so much for watching. If you like the video, I'll see you all on the flipside. Goodbye!


So, last class we have discussed about the signal size. Suppose we have a signal t in this direction and x of t is in this direction. Suppose signal is like this way, this is x of t. This signal tends to 0 as t tends to infinity. So, in that situation size of the signal is measured by the energy content in that signal. So, we have seen how to find out the energy content in the signal. Suppose the signal x does not tends to 0 just like this way something like this x of t which does not x of t which does not tends to 0 as t tends to infinity. Then we cannot give the size of the signal by a energy signal. So, we have to define by a what is called power signal. And power signal we have defined how to find out the signal size in power signal form that is limit t tends to infinity 1 by 2 t minus t 2 plus 3 x of t mod square d t. That mod is taken for if the signal is a complex then mod otherwise x t whole square is a positive quantity always in this way. This is nothing but a average power this expression that we have seen it. Then directly we went to find out the system signal norm in case of system what is the what do you mean by system in case of system what do you mean by signal norm. So, suppose we have a system which is described in the trans function domain where u is the input in time domain u of t and s domain u of s y is the output in time domain and this that system that signal may be a scalar may be vector. And we have defined and this signal is continuous and piecewise continuous that is assumption is made and this is the general definition of the p norm l norm is nothing but a absolute value of this signal from minus infinity to plus infinity that quantity when p is equal to 1. When p is equal to 2 we call is a 2 norm of a signal in case of scalar signal it is like this way note we represent the double line both sides x 2 norm is minus infinity to plus infinity in this expression you put p is equal to 2. So, this is square is nothing but a square of the signals magnitude and this 1 by half the whole that is missed here 1 by half. So, this is nothing but a square root of square root of the sum of this squares of the signals and l 2 l infinity norm is simply called infinity norm of a signal is nothing but a supremum value of this signal over the time t tends to infinity supremum that means list upper bound. Then what do you mean by list upper bound we have seen it that if you have a signal like this way the list upper bound of this thing is 1. Again that this value is will be that function value of signal function will be always less than 1 and which upper bound is 1. So, it is the list upper bound is of the signal is 1 that is what and is another way and this is a supremum and infimum value of a signal is nothing but a greatest lower bound. That means lower bound what is the lower bound greatest lower bound that is called the infimum of a signal. So, similarly this can be extended if the signal is a vector where the x of t has a n components are there they are also we can represent into similar manner that is what we have then we have discussed norm of a system just now we have considering in case of system what is the norm of a signal we have discussed. Now, we will discuss norm of a system and if you see once again if you have a system g of x when it is accepted with a input which is a sinusoidal a sin omega t the output also we expect a sinusoidal with different magnitude and different phase angle, but the frequency will be same because it is a linear see linear time invariant systems. Suppose now see input signal is a sin omega t and output signal will get it a mod of g j omega sin omega t plus 5 and what is the gain input output by input if you see gain is g j omega. So, the system norm in a psi was a system in a trans function is usually measured by either h infinity norm or h 2 norm that let us see what does it mean Percival's theorem. That means suppose we have a signal is there if you want to find out the what is the energy contained in that signal again this you can find out by using Percival's theorem. Now, let us call we have a two signal x and y it that voltage and current if you consider it is nothing, but a power integration of power minus infinity what does it mean. Then ultimately we have shown it this one if x and y are y is equal to x then we have seen the integral of minus infinity to plus infinity is the energy contained in the signal is equal to nothing, but a 1 by 2 pi minus infinity to plus infinity mod of x omega whole square d of this again. So, this indicates the this x of this is the Fourier transform of the signal. So, what this indicates this indicates that this indicates that we can write that x of t x of t small x of t norm this square is nothing, but a 1 by 2 minus infinity to plus infinity Fourier transform of that signal square d omega that is what we got it plus plus if you see last class we have minus infinity is nothing, but a 1 by t minus infinity to omega d omega. What is this indicates shows this indicates Percival's theorem tells that energy contained in that signal in time domain it just implies the energy contained in the signal in time domain it can be related with a energy contained in that signal in frequency domain descriptions. This is the frequency domain you find out the Fourier transform of the signal mod of this whole square mod of this whole square means you is nothing, but a this same thing whole square over the interval of frequency minus infinity what is the energy contained this signal is same as in frequency domain and the time domain. So, this Percival's theorem the Percival's theorem relates the time domain of the energy time domain of the energy in a signal to the frequency domain description to the frequency domain descriptions. The Fourier transform of the signal mod of this whole square and integrate minus infinity to plus infinity with agree. Now, we will go to the system norm next is what topics is the system norm. So, let us call we have a system g of s and output if the system is excited with the input u of t and output is this is the output and the s to norm of the system g of s we want to find out what is the s to norm of g of s. We assume this trans function is strictly proper trans function proper trans function if you do not assume this one then we will not be able to get the h and p norm of the system g of s is not a finite it will be infinite that is what this assumption is made it and most of the particular system is strictly proper trans function and low pass filter type. So, and also assume the system is asymptotically stable asymptotically stable this system can be described also into a state space model state space description of the system what is the x dot of is equal to a x of t plus b u of t and our output y of t is equal to c x of t. This is the state space description of the system and we know the correspondingly that g of s if you want to find out the trans function model g of s from the knowledge of the matrices a b c we can find out c s i minus a inverse b and since there is a part d part is there d we assume 0 because of the function the trans function is strictly proper trans function. So, the d term will not be there since it is a strictly proper trans function. So, now what do you mean by that system norm of this one that I told you that g of s physical interpretation of g of s is nothing but a the gain of the system at different frequency if omega is equal to some omega 1 then magnitude of g j omega is nothing will give you the gain of the system at that particular frequency. So, the h 2 norm of the system this is defined as and we will see the physical meaning of that one now 1 by twice pi integration minus infinity to plus infinity t r trace this means trace of a matrix g j omega h or star g j omega d omega this indicates note g j omega h indicates sometimes it is denoted by g j omega h indicates sometimes it is denoted by g j omega star is nothing but a g of minus j omega whole transpose in complex because g of j omega is a complex system and complex matrix or scalar quantity we assume from now onwards let us call in general g of j omega is a matrix means we if you have a multi input multi output case the dimension m cross n m cross p agree this is the p cross m you just write it p cross m indicates number of outputs is p number of inputs is m agree this and in case of the multi input multi output case h infinity norm of this one is 1 by 2 pi minus infinity to minus infinity trace of this matrix and if you recollect that our definition of what is called Frobenius this is called Frobenius norm norm is nothing but a we mention it here nothing but a this is summation of and this dimension of matrix if you consider m cross n this is nothing but a i is equal to 1 to m j is equal to 1 to n this is 1 to m 1 to n then a i j whole square then whole of this is nothing but a is same as if somebody wants to write it this whole expression is same as trace of a matrix a transpose a is same as transpose trace of a into a transpose this is a Frobenius norm here also same thing if you have a g is a transformation of multi input multi output then you find out the Frobenius norm of that one agree and then do the integration minus infinity to plus infinity this and 1 by 2 pi this indicates is nothing but a energy contained in the signals because c is the that is g of s is the matrix. So, this can be written as 1 by 2 pi 1 by 2 pi integration minus infinity to plus infinity trace in terms of you can write it trace g of minus j whole transpose in place of this one I am writing this then g of j omega and that is differentiated and that is if somebody wants to write it in terms of s you can also write into this one 1 by 2 pi minus j infinity to plus j infinity this is infinity g of s minus g of s transpose g of s d s and this quantity if you see this quantity is nothing but a Frobenius norm trace of this one is nothing but a g of j omega whole Frobenius norm of the matrix agree.  So, i is equal to 1 to p this is 1 to p this is j is equal to 1 to m number of inputs is we have considered m number of outputs is p in this see here in this expression. So, one can write into this form also that this expression this is the trace that is missed here trace t or trace of that one. So, this thing the trace of this I can easily trace of this one I can easily nothing but a Frobenius norm of g or summation of i is equal to 1 to p or j is equal to 1 to p that each element of the transfer function at a frequency omega is equal to omega 1 that is also and one can write into this is also in time domain g of s norm is equal to h to h to norm g of t is nothing but a this is equal to we can write it integration of integration of 0 to infinity trace g of tau transpose g of tau whole thing then d tau this is the and this physically this mean a measure of energy a measure of the square root square root this is the square root of the square root of the integral of the integral square value of the output signal when the system is excited with a impulse input when the input is an impulse. So, I know if the g of s is a transfer function if we excited this system with a impulse input the output response of this one is nothing but a Laplace inverse of g of s is that one. So, this physically this indicate that square root of square root of the integral square this is minus infinity to plus infinity you can write it minus infinity minus infinity to plus infinity and if you consider this is a causal system then it will be a coming from 0 to infinity the square root of the integral square of the output signal when the input is impulse. So, now question is how to compute that things or how to compute this in terms of g s how to compute this one next question is how to compute this integration of that one. So, that is the next question or you can write it g of s norm 2 norm is equal to root over 1 by twice pi i and integration the close integral of this one trace g of minus s transpose g of s whole d s now question is how to compute this expression to how to compute this expression to find that h 2 norm of a of the transfer functions of the systems. So, this is and this can note this can be done by using the Cauchy integral principle. So, the contour integral that run up the imaginary axis and then around the infinite semicircle in the left half plane left half s plane and the contribution of the integral from the for the left half semicircle is 0. The contribution integral contribution the contribution to the integral from this semicircle equals to 0 if g is strictly proper transfer functions. So, now we have to integrate because whole semicircle part that left half of the spline is 0. So, only the things we have to integrate from minus j on the along the imaginary axis from j from minus j infinity to plus j infinity. So, the our problem now boils down to this competition of this one equals by using the residue term equals the sum of the residues of g minus s transpose g s at its stable or stable poles or left half poles of s. So, this let us call we taken one example we taken one example then see how to find out the we taken one example and how to find out g of that norm of norm find out the norm of h 2 norm of g of s is 1 by suppose the system is 1 plus tau s this is. So, h 2 norm this indicates that you have to find out 1 by twice pi j minus j to infinity to plus infinity g of minus s transpose g of s d s. So, I told you the whole complete left half of the spline semicircle again that part is 0 only along the imaginary axis and along the imaginary axis nothing but a sum of the residues compute for this matrix at stable poles at stable poles.   So, you have to find out the residue at stable poles a the value of a will be equal to 1 by 2 tau by using the residue. So, what is the value of that one this composition of that one is that whole thing compositions is find out the residue of that one along this one.  So, this implies that our g of s the norm of this 2 is equal to square root of 1 by 2 tau is nothing but a 1 by square root of 2 tau, but if it is a multi input multi output case then it becomes very tedious and complex. So, that can be handled by using in state pace model for multi input multi output case for multi input multi output case composition of g j g of s this is not straight forward is not straight forward. So, how to compute this one. So, let us call g of s which is a multi input multi output which a m is p is the number of outputs p is the p is the number of whatever m is the number of inputs. And that state phase representation state phase representation converting to state phase representation that x dot is equal to a x plus b u and y is equal to c x of t. And since I have considered number of outputs is p and number of inputs is m then let us call this is the state is n. So, first you find out the either controllability Gramian matrix. So, how to find out that for multi input multi output case h infinity sorry h 2 norm of the systems h 2 norm of the system it does not it means the energy contained in the system for different signals of this one. I told you g of s is a when the input is the what is called the impulse input the output that what you are getting corresponding to input is the output signal. And what is the output signal contents in the system that can be find out with the help of h infinity h 2 norm of a g s for a impulse input. So, now two methods two way we can two ways to compute g of s when g s is the multi input multi output case. So, let us call we first find out the controllability Gramian the controllability can be u e is equal to 0 to infinity. And we are assuming the system is a causal that e of t this b b transpose a of t whole transpose d of t. So, this is the controllability Gramian and this matrix must be a this whole expression must be a value will must be a your positive nonzero values. If it is nonzero value for all t then system is controllable if you recollect that one when we are considering the linear time invariant systems then with the knowledge of system matrices b a we found out a b then a square b then then a n minus 1 b the rank of this matrix must be equal to n where n is the dimension of the matrix a n cross n. So, this in other words you can check the Gramian matrix whose value is greater than 0 for all t then system is controllable in the sense that n is there exist some controller over the interval t 0 to t f by which we can transfer that state from initial condition to the final state with the help of controller. And there exist some control if this condition is satisfied. So, what will do it. So, you solve this Lyapunov equation. So, this is the Lyapunov equation you solve it or you can consider that this is a what is called algebraic eddy type equation type also. So, solve this one once you solve this one then find out that norm of that is nothing but a h 2 norm of the system is nothing but a trace of c u c transpose trace of a matrix trace of a matrix is nothing but a sum of all diagonal elements. So, that will give you the what is called h 2 norm of the systems the physical interpretation h 2 norm of system means the system is there you are excited the system with a impulse input then output response is nothing but a Laplace transform or Laplace inverse of inverse Laplace transform of g of s and that is the in time domain what is the energy content in the output signal that it can be found out with this one. This is method one for whether it is a single input or single input single output multi input multi output does not matter you have to solve this equation. Method two this is called controllability Grameon matrix. Now, we will consider the observability Grameon matrix that s is equal to 0 to infinity that you can get it from this one a is replaced by a transpose b is replaced by c transpose. If you a replaced by a transpose it will be a t whole transpose then b is replaced by c transpose then c transpose c into a a t whole a is transpose by a is replaced by a transpose a transpose whole transpose is a a and d t. And this Grameon observability Grameon matrix this is called observability Grameon matrix if the observability Grameon matrix value is greater than 0 for all t then system is observable that means we can estimate the state information at time t is equal to 0 by processing the input output data of the system. Then we can estimate the initial value of the states if the system is observable observability Grameon matrix is greater value is greater than 0 because whole quantity is a scalar quantity for greater than 0 for all t. So, with this one I can also compute the what is called solve the again the Riccati equation that Riccati equation is a transpose s that s a in this equation in this equation replace a by a transpose u in state of Grameon matrix u observability Grameon matrix. So, this plus c transpose c is equal to 0. So, this is the Lyapunov type equation one can solve by standard technique by solving this one or you can use for solving this one you can use algebraic Riccati equation solution if you know we have discussed when you are considering the LQR problem or LQG problem in our optimization technique that is dynamic optimization technique you can use that method to solve this one. So, once you get it this one that h 2 norm of the system this is nothing, but a trace of a t r you write it b transpose s b. So, this way you can compute it. So, next is your this is the h 2 norm and next is our h infinity norm of a system g of s. Let us call we assume because h infinity norm concept is coming from the singular value for multi input multi input that concept is different. So, let us consider we have a single input single output case u of t y of t is the output. So, it is a single input single output systems it is defined as g of s that whole infinity is equal to supremum value of this function omega not omega u of t not equal to 0 y of t. This is the time domain I am expressing u of t when input is impulse input the output response of the impulse output response y of t due to the impulse input this or any input you can write it the what is the energy content into the system output signal energy content also divided by energy content in the input signal that ratio that what value of u when u is not equal to 0 that this quantity will be maximum that that supremum value of this ratio will give you the h infinity norm. But, if you talk about the system in terms of frequency domain this is the input and this is the output and if you consider for stable system the h infinity norm of the system g s is nothing but a g of s supremum w g of j omega. What does it mean that g of j omega you find out the frequency response of the system when the omega is varying from 0 to infinity and find out at what frequency it gives the maximum value of this amplitude or gain and that is nothing but a h infinity norm of a system when this is single input single output case. So, you restrict our discussion with a single input single output case because multi input multi output case the concept of singular values decomposition singular value decomposition technique will be adopted here singular values concept of singular values will be adopted here this quantity the whole quantity is the maximum maximum gain of the system or in fact it is a resonance at what frequency resonance is occur the maximum peak is occurred this is and. So, this is about the system mark because system norms signal norm will be used when we are discussing the robust design of controllers h infinity controllers or h 2 controllers when we will discuss this and that concept will be used. Now, so far we have seen that if you have a system is there if you excite the system with input and you will get the output the system stability is studied from the knowledge of that what is called based on input output stability criteria we are checking the stability of the systems that is in turn it will call input output stability. So, in true sense the system stability is studied we have to study all the internal signal of the system must be stable then it will be a call the system is stable in true sense, but input output stability does not give the clear picture of the system stability. So, let us see with an example that we have a systems let us call we have a systems g of s is equal to s minus 1 s sorry s plus 1 s plus 1. So, this system there is a pole zero cancellation is there, but stable pole zero cancellation you see the stable pole zero cancellation is there this implies mathematically, mathematically in distinguishable it cannot distinguish from the constant transaction g of s is equal to 1 same thing I can do, but here is stable pole zero cancellation is there. So, long stable pole zero cancellation is there there is no problem except the system properties will change either it may not be controllable completely or it may not be observable completely, but in similar wise like similarly if g s is equal to s minus 1 s plus sorry s minus 1 s minus 1 still this is mathematically indistinguishable from the constant transaction this mathematically it is same as the constant transaction because if you cancel this one this will be a 1 only, but let us call for the sake of our interest we assume that this pole at 1 and pole at pole at 1 and 0 at 1 they are not exactly overlapping that slightly change in state of pole at 1 I am taking pole at 0.9999 and now immediately you will say the system is unstable because they are not canceling each other the system is unstable because you can logically you can think of it the system parameters may deteriorate with day day years. So, this parameters after that it may not be 1 exactly it may be 0.99. So, let us call this type of pole 0 cancellation unstable 0 and unstable pole cancellation is not allowed. So, unstable pole 0 cancellation in practical plant stroke controller is not acceptable. Let us see why it is not this example when it is a s is equal to instead of 1 it will be 0.99991 you can say system is unstable for any kind of signals even the signal magnitude is very small. Let us call we have a system this systems this is the output y s is the output and this signal is feedback here negative feedback this is R of s reference signal and this signal is E of s and this signal we are considering E of s. Now, see this one that E of s if you can find out the trans function between the this signal and this signal this is the output signal this is the input signal trans function this will you will get s plus 1 plus s plus 2 R of s indicates stable signal E of s this signal is stable, but if you find out the trans function between R s between y s and R s that means U s signal you want to check it then this is nothing but a this you know expression this expression you multiplied by 1 by s. So, this is considered C of s this is G of s if I consider. So, it is a C of s nothing but a 1 plus G of s into C of s into R of s and that value will come if you see 1 minus s this is I have written C of s 1 by s and that value is always already there this is s plus 2 into s plus 1. So, this you can say E of s is nothing but a E of s is nothing but a 1 by G of s into C of s into R of s and put the value of C of s yes it will get it and see this signal this R of s this signal this is this indicates for any kind of input or noise or signal R s reference input indicates the signal U of s is unstable. So, this signal the internal signal internal signal is unstable for this type system. So, the concept of internal stability is and other things is coming to the picture when you will study the stability of the systems. So, internal stability of systems. So, as I mentioned the input output stability input output stability does not granted does not guarantee that the other signal other internal signal internal signal are bounded. In this example you see this internal stability you have to check all input signal all summer output signal you have to check it all internal signal this is the signal this is also signal all this thing if you see the output this this cancel and you will find it is stable, but before that this will be a unstable signal. So, in general now if you have a block diagram like this way C s is our controller then we have a block of that one this is U of s this is E of s and this is our plant G of s this is our output I will explain that one this is the reference input and this is the controller and this is the plant and this is your D e o input disturbance this is called input disturbance input disturbance this is called output disturbance output disturbance output disturbance output disturbance which is denoted by d y and this is your d n measurement noise this d n is the measurement noise. So, now we say we have a system reference input signal then controller then plant the controller output is corrupt with the input disturbance and output of the system actual output of the system is corrupted with a disturbance output disturbance and the measured signal is corrupted with a noise generally the reference input disturbances are a low frequency signals and measurement and noise are generally high frequency signals. So, you have to study the stability of the system it does not mean the stability of this system this this this signals we do not have any control, but inherently inside the system may act. So, if you just study the stability from input and output that y of t that does not mean the system is stable or not that input output stability does not grant t that other stability signal inside the system is also stable there if you study the stability between the this in signal and this signal even it is stable it does not mean that all other internal signal also will be bounded or stable. So, generally we take the output from each summer output is the output of the signals is summer output. So, this is u p of s this is this summer output is this and summer output is y this is also getting y this is the output. So, you have a how many outputs are there there are four summers are there 1 2 then 3 then 4 let us call this signal is y n. So, you have to study the stability of this one. So, let us call I am considering the system is single input single output system. Same thing you can extend for multi output multi output system provided if you take care of the matrix operations vector operation during the process. So, let us call we consider input signal or r this is the input signal where in time to r then d u d suffix u d y and your d n this is the input signal input signal input signal. Then output signal is your e u p y d y and y n these signals are not in your hand again the output signal is if you see here the output signal is summer output this one this one y this one and this one that output signal. So, we have to study the stability of the system internal stability of the system that means we have to find out the input output relationship.  Now, this is our output Laplace transform Laplace domain this output this input is this is the input and this is the output this input are mapped with a some matrix in a output signal the output signal the input signal. So, this input are mapped through the system in a another set of signal which will call output signal we will find out what is the matrices are there naturally you can see there are 4 inputs 4 outputs the dimension of the transform matrix will be 16 4 by 4 4 by 4 that means 16 elements are there you have to find out that 16 elements here. So, we will discuss tomorrow the rest of the part.


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