A warm welcome to the 19th lecture on the subject of wavelets and multirate digital signal processing. In this lecture, we shall go further on the uncertainty product or the time bandwidth product as we called it in the previous lecture to proceed to evaluate and to bound it to put certain fundamental limits on it and then to see how close we can get to those limits. Therefore, I have titled the lecture today as evaluating and bounding sigma t squared sigma omega squared. To put the discussion in perspective, let us recall very briefly what we did in the previous lecture. You will recall that we had talked about these quantities sigma t squared sigma omega squared. We had also noticed that this product sigma t squared sigma omega squared is a very strong invariant. It is invariant to translation, it is invariant to modulation both in time, translation in time, modulation in time. It is invariant to multiplication of the dependent variable by a constant or scaling the dependent variable or in other words multiplying the function by a constant. Most interesting of all is that it is invariant to a scaling of the independent variable that is striking. So, you see ultimately what is left is just the shape. The time bandwidth product is a direct function of the shape. The shape can then get compressed or expanded, it can get scaled in the vertical direction, it can be shifted, it can be modulated by an e raised to power j omega t kind of term rotating complex number or a phasor. All these do not affect the time bandwidth product sigma t squared sigma omega squared as we called it. Now, our objective today is to ask a very important question which lies at the heart as I said of wavelets and multirate digital signal processing namely how small can we make this product? Let us put down the question first. Essentially how small can the time bandwidth product be? Once again we need to emphasize a couple of points. Why are we talking about the time bandwidth product instead of just the time variance or the frequency variance? We must understand that making the time variance small or the frequency variance small is not a difficult job at all. In fact, one can do it by scaling the independent variable. If you compress a function in time, the time variance is compressed by the square of that factor. If you compress a function in frequency, the frequency variance is compressed by the square of that factor. So, compressing in time or frequency separately is not a difficult job at all. In fact, the problem is that when you compress in one domain, you are expanding in the other domain by the same factor. There is a cancellation effect as far as the time bandwidth product goes. So, compressing or expanding a function does not change its time bandwidth product, but it definitely changes the time variance and the frequency variance individually. This is what we saw in the previous lecture. In that sense, physically what it means is that nothing stops you from narrowing down as much as you desire in one of the domains. If you desire to focus on a small region of time, you can do it on as small region of time as you desire. If you focus on a certain region of frequency, you can make that region of frequency on which you focus as small as you desire. There is no problem. So, focusing in one domain is not a problem. You can do it as much as you desire. The problem is focusing in both the domains together. Any tool that we use, by tool I mean a function used for analysis. Any tool that we use has a time bandwidth product essentially based on its shape. So, when you use the same shape, you are bound by that number. No matter how much you squeeze or expand, you are not going to be able to focus in both the domains simultaneously. In fact, by focusing in one domain, you are going to compromise on your ability to focus in the other. This is the general tussle as we said between time and frequency. Now, the next natural question to ask is, how small can we make this tussle? If sigma t squared, sigma omega squared could have been 0 for example, could have been. It would be wonderful. Nothing needed to be done. You could make any of them as small as you desire and the other one as small as you desire too. So, you could focus in two domains simultaneously. But as I said, nature does not allow this and it is a fundamental property of nature that it does not allow this. What we are now going to answer is, what is the lower bound on this product? Meaning, to what extent no matter what tools you have, can you really focus in both the domains together? So, what is the ideal towards which we must strive and why is it that we have a fundamental limitation to work with which always poses a challenge before us? Anyway, to answer that question, let us recall some of the expressions that we had derived in the previous lecture. We had looked at an expression for the time bandwidth product based on entirely time domain quantities and let us put that expression down clearly once again. We had shown that the time bandwidth product without loss of generality can be written ultimately as follows. It is the L2 norm of Txt the whole squared times the L2 norm of the derivative again the whole squared divided by the L2 norm of x squared multiplied by the same. So, in other words, you could write the L2 norm of x raised to the power of 4 if you like by combining these two terms. Anyway, I am just writing them separately to emphasize that this is associated with this and this with this notionally. Now, of course, I must also again mention the slight abuse of notation here. When we write T times x t, we are referring to the whole function as an object and not to a specific value of t the same holds here. Anyway, this was just to recall what we did yesterday and remember we had done this after doing a little bit of preparation on the function so to speak. Having noted that translation in time and in frequency has no effect on the time bandwidth product, we had said we could shift the function in time so that it is centered in time. Subsequently, we can shift it in frequency without affecting the time function so that it is centered in frequency. In all this, the time bandwidth product does not change. So, here we have a function which has been assumed to be centered in time and frequency and now we are working with that function and this is without any loss of generality. Anyway, coming back then to the calculation of these quantities, let us once again give a vectorial interpretation to this. Now, you see look at the numerator. The numerator is a product of two squared norms. Let us focus our attention on the numerator. So, numerator here is the norm of T x t the whole squared multiplied by the norm of d x t d t the whole squared in L 2 R. Now, let us take recourse to a very fundamental principle that we know in vector analysis. Treat T x t and d x t d t as vectors as we often do generalized vectors. So, let us call them v 1 vector and v 2 vector. Now, if we recall a basic principle of inner products, the inner product of the product of the product of v 1 with v 2 is essentially of the form the magnitude of v 1 times the magnitude of v 2 times the cosine of the angle between v 1 and v 2. Theta is the angle between v 1 and v 2 and therefore, it is obvious in very low dimensional spaces which can be used and of course, be taken to higher dimensional spaces also that if you consider the magnitude squared of the dot products. Let me use the dot product notation. It is of course, the product of the magnitude squared of v 1 and v 2 multiplied by cos squared theta and cos squared theta is always less than 1 in fact, between 0 and 1. Here of course, we are talking about theta real and that is the interpretation which we always have even if we are talking about complex functions. The angle here is assumed to be real. You see what it means therefore, is that the modulus of the dot product squared is always less than or equal to the modulus of v 1 squared times the modulus of v 2 squared. This is a very simple, but a very important principle. In fact, this principle can be generalized to these generalized vectors that we are talking about functions viewed as vectors. In fact, this is a very important property or a very important theorem in functional analysis. It is often called the Cauchy-Schwarz inequality. It says and let me write that down in formal language. It says the inner product, the magnitude squared of the inner product of two functions.  The magnitude squared of the dot product of f 1 and f 2 is less than equal to the product of the squared norms in l 2 r of f 1 and f 2, the Cauchy-Schwarz inequality as it is often known. Now, of course, this vectorial interpretation makes this property obvious, but one can also prove this formally in functional analysis without taking recourse to this visualization in the language of vectors. However, since it is not our objective to review this basic concept from functional analysis, I mean to give a detailed proof, we shall give it this vectorial interpretation and be satisfied. So, we will take the Cauchy-Schwarz inequality as true and proceed from there. Now, please remember that we are assuming in this process that these functions belong to l 2 r. So, let us go back to our uncertainty time bandwidth product. Consider them x t, so that t x t belongs to l 2 r and d x t d t belongs to l 2 r. What if they do not? For example, we took the very simple case of a rectangular pulse a while ago. We saw that if we consider this function, not true for this function, because d x t d t does not belong to l 2 r here. Why so? d x t d t has two impulses. d x t d t would have an appearance like this and impulse there and a downward impulse at the end of the pulse. This is what d x t d t would look like here and impulses are not square integrable. Impulses do not have finite energy. I shall just spend a minute in justifying this, because we have so far been informally saying it. We have also proved this indirectly by looking at the frequency variance in this case, but we also must understand the direct proof here. After all, what is an impulse? An impulse is the limiting case of this, an object which lies on a width of delta with a height of 1 by delta. This is capital delta t if you please and an impulse or a unit impulse to be more precise is essentially something like a limit as delta tends to 0 of delta delta t. Now when you take the square of this, so delta delta t squared it looks like this and now if you take the integral of delta delta t squared d t over all t, then it diverges when you take the limit diverges. In fact, this integral is essentially 1 by delta. 1 by delta squared into delta. So, 1 by delta and this is divergent. This limit does not exist. So, therefore, the impulse is not square integrable. We must make a note of this. This is an important conclusion. Infinite energy, it contains infinite energy and that is manifested also in calculating the frequency variance for this rectangular pulse. Anyway, that a part. We have now agreed that if that is the case, anyway that lower bound does not arise, where these quantities diverge in spite of x belonging to L 2 R. Remember, since x belongs to L 2 R, in the beginning we had this denominator. So, you know you had this denominator right in the beginning. Norm x 2 x in L 2 R to the power 4 effectively and this is guaranteed to be finite because we have confined ourselves to x 2 into L 2 R. So, if this is in L 2 R, the denominator is finite and positive obviously otherwise it does not make sense. I mean you are not going to take a trivial function. So, for a non-trivial function, this is strictly positive and finite and then if any one of these is infinite, there is no question of finding a lower bound.  So, it is no harm then that we have in considering finite quantities in the numerator. So, with that little remark about our restriction, let us take them to be finite and proceed and use the Cauchy Schwarz inequality. So, using Cauchy Schwarz inequality, what do we have? Now, we will use it the other way. What we have in the numerator is the norm squared in L 2 R of t x t times the norm squared of d x t d t in L 2 R and from Cauchy Schwarz inequality, we go further. This must be greater than or equal to the magnitude of the dot product squared. Let us write this down. This dot product is essentially the following integral. Remember, we need not confine ourselves to real functions. We should not because we are allowing a modulation in time. So, we must allow complex functions here. We have of course, centered the functions. That is a different issue. We need to center them. That we have done. Now, let us essentially look at this integral a little more carefully. Now, we are talking about this as an entire complex number. This whole thing is a complex number from here to here. If you take the magnitude squared of a complex number, so let that complex number be z. It is obvious that the modulus squared of z is greater than the modulus or greater than or equal to in general, the modulus squared of the real part of z. We will use that property here. Of course, the same thing holds good for the imaginary part too, but we are interested in the real part here. In particular, we have the numerator of the time bandwidth product is greater than or equal to modulus real part of this integral. The complex conjugate is above this entire thing. This is what we have here. Now, a remark about this part, this complex conjugate. We are taking the derivative of a possibly complex function x of t with respect to the real variable t. So, the complex conjugate of dx t dt is also the derivative of the complex conjugate of x t. What I am saying in effect is that because t is a real variable, dx t dt bar is also dx bar t dt and I shall employ that in this expression first. So, this quantity now is equal to the modulus of the real part of the following. The real part operates on the whole integral. Now, we again look at the real part. You see, the real part is operating on an integral with respect to t. t is a real variable. So, the element of integration is real. This is a real function. This is possibly a complex function. So, the real part can be taken right inside the integral and brought to operate on this only. The rest of it does not require you to say real part explicitly. So, this expression is equal to integral from minus to plus infinity t times the real part of x t dx bar t dt dt the whole squared. Now, we take note of this. How do we calculate the real part of a complex function complex number in general? By adding the complex number and its conjugate and dividing by 2. So, essentially what we are saying is real part of x t dx bar t dt is half x t dx bar t dt is half x t dx bar t dt plus the complex conjugate of this. And what is that complex conjugate? It is x bar t dx t dt. Simple. And now, we can see a product rule has been employed here. Essentially what we have here is the derivative of x t into x bar t. That is an important observation. So, here this is equal to half d dt of x t x bar t. And x t x bar t is indeed the modulus of x t the whole squared. A very beautiful observation. And now, we will put back that observation into the time bandwidth product. So, therefore, the numerator of the time bandwidth product is thus greater than or equal to modulus of half integral from minus to plus infinity t times d dt mod x t squared d t. And this is the whole squared. Let us take a minute to reflect on this. How do we evaluate this integral? Well, that is easy. We can evaluate this integral by parts. And to evaluate by parts, we must first make the integral indefinite. So, let us simply consider the indefinite integral corresponding to this. Clearly, this is this term minus this term. And now, we can substitute the limits. So, we can put back the limits of minus to plus infinity here. And put this from minus to plus infinity first. And then this from minus to plus infinity as well. Now, let us focus our attention on each of these terms individually. Let us take the first term t times mod x t the whole squared. Evaluate this integral by parts. Now, let us evaluate it at plus infinity. And then from its subtract, evaluate it at minus infinity. So, let us focus our attention on t mod x t the whole squared. Evaluate it at the two limits and subtract it. Now, we have agreed that t squared mod x t squared d t from minus to plus infinity is finite. We have agreed on that. We said if that is not true, anyway we have an infinite bound. There is no question of lower bounding there. It is the worst case that we can deal with. So, if this is finite, obviously you see if the integral must be finite, the function must decay towards 0 at the ends. If the function does not decay to 0 asymptotically as you go towards plus infinity and as you go towards minus infinity, you can see that there is going to be an infinite range over which the function has a finite positive value which would make the integral diverge. So, in order that this integral converge t squared mod x t squared d t, the function t squared mod x t squared must decay as t tends to plus infinity and minus infinity. Let us make that observation. In order that this integral converge, t squared mod x t squared tends to 0 as t tends to plus infinity and t tends to minus infinity. Now, it is also true that for t greater than 1 and of course as t tends to infinity, t is definitely going to be greater than 1. So, for t greater than 1 and certainly so and I am talking about t squared actually or mod t. So, certainly so as mod t tends to infinity, t tends to plus infinity or minus infinity. You must have t squared greater than t. What this means is that t squared mod x t squared tending to 0 as t tends to plus or minus infinity guarantees t mod x t squared t tends to minus infinity t squared tending to 0 as t tends to plus or minus infinity and therefore, that first term has vanished. So, we are left only with the second term and the second term is very familiar to us. In fact, this is minus the norm of x in l to r the whole squared. So, simple and therefore, we have a very beautiful conclusion here. We are saying the numerator is always greater than or equal to modulus half into minus the norm of x in l to r the whole squared which is one fourth times this times this and therefore, overall the time bandwidth product is greater than equal to one fourth norm x in l to r to the power 4 divided by the norm x in l to r to the power 4 which is one fourth a very very fundamental conclusion. The time bandwidth product can never be less than one fourth 0.25. So, 0.25 is the very lowest value of the time bandwidth product that you can get and what we concluded here has nothing to do with the tools available at a particular time with the technology available at a particular time or with the machines and the political situation or whatever. It is fundamental to signal processing. In fact, so fundamentally is this result of P F derived in different manifestations it is seen in different subjects. What we call the uncertainty limit in physics is just another version of this. People talk about the inability to locate position and momentum simultaneously. Actually it is just another version of this. I shall just give a small hint as to how you might connect this idea of time bandwidth product to the concept of position momentum uncertainty and to do that actually we will go back to the expression for the time bandwidth product that we had derived. You see, you notice that we have shown that essentially the time bandwidth product relates to a time variance relating to the norm of t x t the whole square and you are assuming the object is centered. So, now think of x t as descriptive of an object. Remember that we have talked about a one dimensional mass. Now, here we are talking about x t then as a kind of mass distribution of the object or rather if not mod it is if not x t it is really mod x t square which is like a mass distribution of that object. So, division the denominator part is essentially a normalization make the object unit mass if you like, but here in the time variance the numerator is indicative of the uncertainty in position. What is the variance of the object as far as it is spread around the center is concerned that is what the time variance tells you. Now, if you look at the other term there the frequency variance. So, time variance involves this frequency variance involves this you know involving this the derivative of x t with respect to t. Now, x t or mod x t square as I said is indicative of in some sense the presence of that mass or that body on t. So, d x t d t is indicative of the movement of that body the change with respect to time. So, here in some sense the change it is indicative of a change. So, in a broad sense you can see the connection between position and momentum uncertainty in where it is an uncertainty in how it changes. Let me not dwell too much further on this to interpret precisely how this is the uncertainty principle in physics. I think we should leave it to a physicist a person who specializes in that, but this is what I have just given you is an intuitive indication that is all. However, it is just to bring out the various meanings that this time bandwidth product or this uncertainty limit has it has meanings in different subjects. Anyway coming back to a subject with which we are dealing this is the whole basis of the subject of wavelets of that matter time frequency methods. What it tells us is that no matter what we do we are not going to get any function which has finite energy and which can be confined beyond a certain point in the two domains time and frequency simultaneously that is bad news. What was worse is that if you looked at the hard case it is not confined at all that time bandwidth product was infinity. So, now we naturally ask the next question is there a function which gives us this 0.25 or is it something that we should never seek that is not very difficult to answer. In fact that can be answered again by using a vectorial interpretation. When does the numerator become equal in the Cauchy Schwarz inequality to the expression that we derived the optimal function in the sense of time bandwidth product means the Cauchy Schwarz inequality becomes an equality. Now, when would that become an equality it would become an equality if the cos square theta term is 1 that is the cos square theta term is 1 and what you mean by the cos square theta term being 1 what really is theta theta is an angle between these two so called vectors t x t and d x t d t. If you want the angle to be such that cos square is 1 these two vectors must be collinear they must be along the same line so to speak. So, therefore, in the language of functions we need the vectors so called vectors t x t and d x t d t to be collinear 1 dimensional what you mean by them being collinear they must be linearly dependent and what you mean by them being linearly dependent any of them must be a multiple of the other. So, in other words d x t d t is some constant let us call that constant gamma 0 times t x t the solution of this equation would give us the so called optimal function. So, we solve this how do we solve it by a simple change of the or this redistribution of the derivative. So, essentially what we are saying is d x by x is equal to gamma not t d t and if we integrate we get gamma 0 t square by 2 or log natural x well you must have a constant here a constant of integration and this is of course log natural x. So, in other words we have log natural x is of the form gamma 0 t square by 2 plus c 0 let us raise both sides using the natural base e. So, e raise the power l n x is e raise the power this and therefore, we have e raise the power l n x is just x. So, x of t is some constant e raise the power c 0 this is a constant let us call this constant c 0 till day times e raise the power gamma 0 t square by 2. So, let us write that down again and let us make a remark on gamma 0 x t is of the form c 0 till day times e raise the power gamma 0 t square by 2. You want x t to belong to l 2 r and therefore, you want mod x t square which is essentially mod c 0 till day square times mod e raise the power gamma 0 t square by 2 to be integrable. Now, this is possible if you have a constant only if gamma 0 has a negative real part. If gamma 0 has a positive real part this is going to be a Gauss so called Gaussian that grows in time it is not going to be square integrable and therefore, we must choose gamma 0 with a negative real part and we can choose any one example of that. So, in other words one function which reaches the lower bound which is optimal in this sense is with gamma 0 equal to minus 1 and there we have we can also choose of course, c 0 till day to be 1 because it does not matter if you scale a function the time band with product is unaffected and therefore, one optimal function in the sense of time band with product that is with time band with product equal to 0.25 is the Gaussian a very interesting conclusion. The Gaussian is optimal the Gaussian seems to arise in many situations it has a reason in this having noted this we shall conclude today's lecture and proceed in the next lecture to delve further into this issue of how close we can get to the optimal with other functions. Thank you.


you know i think when you have an african mom you by force will learn you will learn the language you will learn how to cook you will learn the culture but from a dad passing that true at least my dad always said that no let her learn dutch your dad was ghanian yes and your mom dutch so you're a mixed race yes yes you're also a mixed race yes and they're also mixed race it's not easy being mixed race it comes with difficulty sometimes because at one part you don't belong to the white people and at one part you don't belong to the black people why is it like that well in the Netherlands when somebody asks me like oh where are you from i'll say oh i'm then i'm i'm dutch i'm from Amsterdam and they say no you're not and say yeah i'm dutch i was born in the Netherlands no but i mean where are you really from and then they'll start touching their skin like you know where are you from the signs and i said well i'm from the Netherlands my dad's ghanian so all you're not really dutch well i'm born here so i'm dutch so you always have to explain yourself right well when we came to Ghana and people ask us where are you from we say oh i'm half ghanian half dutch or oh i'm ghanian they'll say wow and there are no questions there's just like welcome that's the only thing yeah wow so you feel a sense of belonging yes yes yes hello where's that champong family my name is my name is Anna a champong and i'm Kevin Isaac a champong and we're the champong family we make videos about our move to Ghana welcome back to the youtube channel it's your favorite village boy mr ghanababy and today i'm joined by one and only youtube family in Ghana yes hello the a champong family yes we're here everywhere people are telling me that you cannot come to a crab without meeting the champong family because there is one crab have i committed i need to meet it before i leave so nice to have you on the internet you don't know that we're getting a lot of love yeah yeah it's coming now yeah why why people love you guys so much i mean it's it's mainly because we are just who we are and we show everything yeah we're a family thing and and we show everything how it is wow yeah you're based in Ghana right now yes but where were you before Netherlands oh wow Holland yeah i you speak dutch yes oh yeah i speak dutch oh look at this vast a lot of italians that is dutch that's very german gentleman yes and then you have dutch holland so it's different the neighbors from each other i'm sorry next time i'm gonna do better don't worry are you guys from Ghana because the name of the champion is a ghanian name yeah so my father's from Ghana and her father's from Ghana yes which makes us ghanian yeah and your mom dutch and we are both born and raised in the Netherlands like Sophie Jane as well this is gonna be an interesting story just sit here and i have so many questions to ask you okay okay you were born in Netherlands yes Amsterdam to be precise yeah and what are you both of you doing in Ghana looking for a better life it's not only looking for a better life but i mean living the better life educating our children about Ghana uh giving them an opportunity which we feel like this is something they gain an experience which is it will have a big impact on my life i hope they'll thank us eventually for it yeah i don't understand i don't understand what you're saying she's saying living a better life looking for a peaceful life you are saying that giving the opportunity for the kids to know who they are or something yeah yeah true know their heritage and all that are you trying to say you didn't get a chance to know where you're coming from or something well uh the way i've been brought up i didn't learn a lot about my ghanian part uh i forgot my father but he didn't explain everything about traditions and those things and i've been to Ghana before a period twice for a short period and i have a lot of friends who have gone multiple times but for me it's not the same so i haven't got all the education about gana and about the ways and i felt like and we felt like it's time to do our parts and to bring it to them or to bring them to here yeah we wanted to pass that heritage because you know african heritage is very good and very their traditions their culture is so rich and we want to pass that true to our children especially when she started asking like why am i brown who is grandpa where am i coming from why am i not just looking the same like other kids then we let her know that she's african and she needs to be proud but we weren't really brought up that way so how can we how can we give that to her if we don't have that that's why we decided to come and learn as a whole family this is interesting and how has the experience been like so far wonderful ups and downs and yeah it's it's it's been a rough ride with a lot of eye-openers yeah but it's been very fun and educational yeah and also very nice to just come together as a family you know migrating to a different continent is quite difficult because you leave your family you leave your friends you leave the people you know and then you only come with us for so that made us like i think we connected better yeah and i think maybe i can even say that gana saved our marriage wow no let's let's go the gana saved your marriage yeah how well you know waiting together for a long time it gets tough sometimes you know he's always working i'm always home with the kids and he needs to work a lot so he will come home late and then the kids were already in bed i did everything so sometimes i'm frustrated and then sometimes i take it out on him or he takes it out on me so yeah so which means you are busy working every day every day which is normal which is well nine to five and weekends you're free but yeah but then still at that time we're building a house and do a lot of construction things and you're in the range a lot of things so basically you're an autopilot doing things and we won't have any time to so basically you're both your whole house in netherland and you decided to leave all that behind and come to gana yeah you see the way your mom and that sacrifice for you they did that because of you you know that yes you're living in gana as a family yeah the kids were not born in here yeah what are the pros and cons living in gana as a family like this okay for us i think because they didn't speak the language when they came here it was a bit difficult they were only known to dutch language so when they came here they struggled a bit at the beginning but i think through school and friends and playing they learned it in about three months would you say three months that you could speak english by that time like two months two months yeah they were really fast yeah like before they couldn't speak any english no oh no hi how are you fine oh okay and uh it's the same oh wow okay that's a plus amazing yeah yeah and um i mean coming here to gana of course it's not always the same like kids used to play outside and kids are playing outside here you know covid now has changed that thing yeah but still things are possible it's you just have to manage yeah but there are a lot of pros like the weather is one for instance she she always doesn't like to wear socks and are wearing like these shirts underneath her shirt she was always complaining that it doesn't nice it doesn't it's not that but when you're here just wear whatever you want sometimes they shower in the rain they'll be just outside when it's raining and they'll feel free um what else they can swim whenever they want i think the sun does something to you i mean when you're of color and you you have sun 247 it does something to you it gives you a good five yeah good five you just wake up open a curtains and be like wow i'm gonna enjoy the day you just smile because of the sun and and the heat it's it's amazing it makes you happy and and it's also makes you happy because in old land it's raining every day so you can't go and swim somewhere but in gana again yeah okay and uh how is your experience in gana you didn't tell me do you enjoy living in gana yes yes you have so many friends in here how many friends do you have in gana i don't know you don't know i have too much friends too much friends what are your friends i can't even count them wow you can count your ghanian friends yes okay they teach you the tree they don't teach you a little bit a little bit mom wants to teach me oh mommy's teaching her a bit you speak the tree language no no i don't but i have some basic words that i'm not trying to teach them i think you know when we came we should have told everybody that don't speak english to us speak tree but then the english is there so whenever we now tell people that no no no speak tree they don't they just speak english to us so that's something we really need to work on i'm trying now i mean whenever we go out and going for shopping or other errands yeah yeah you try you try yeah you know like you guys are living in gana as a family and i know there are so many people out there who are looking forward to bring their family back in gana if you should advise people like that what will it be it is really nice for kids to live here it's really nice in general for anybody to live here like i would say there in the netherlands you're you're a bit confined to your home and nobody's really talking to you but here everybody is out and you talk you go to the grocery store you meet new people it's it's very nice well and the people love you they do they do yeah you know she has uncountable friends yeah she's really enjoying the thing i love i really love about gana and i think it's all all over africa but i've noticed it in gana really people accept you as you are yeah people are glad that you're here yeah it's it's the love you get is so so refreshing so warm um i can i can recommend everybody to just come and and see and experience it because it's so nice and especially bringing your family here having a small kids feel feel the warmth of other people embracing them and and feeling welcome that's so beautiful oh i just want to know right now you in gana yes you tell me you're living your best life yes hey how can you live your best life in gana what do you do like what do you do in gana in general we saved for it to live here but we do we now do youtube yeah youtube is what we do yeah we create we we start with a hobby educating people and educating our family and friends and i have seen your youtube channel and i believe that you guys are doing an incredible job thank you got like 10 000 subscribers yes we gotta take it to the next level yeah of course come tell them tell them you know what tell them that it's by force to come and subscribe to their champion family tell them right now it's by force uh-huh you know what we're gonna do they are sharing their experiences in gana and i believe that we need to support people like this i mean their story can inspire so many out there what are you waiting for as i always do it's by force i'm not going to tell you oh please go and those guys no we gotta support our own this is when black lives matters comes in when you start supporting each other we're going to be stronger together so what i'm going to do right now i'm going to put the youtube channel link in the description it's by force 10 000 subscribers we're going to take them to 20 000 and after 20 000 they want to go to 30 000 it's none of our business let's talk about cost of living living in gana's family yeah is it expensive or cheaper um it depends yeah that's a real difficult question but i think um the value of what you get here is way more but um if you can leave your western life behind and leave your western mentalities behind the cost of living is way cheaper here but if you can't leave that behind and you still want to have that western food um the western things around you like all imported that's that comes with the cost yeah that's that's good and it also depends on where you stay yeah i mean akra is more expensive than takra yeah thank you so much but you know like i have so many questions but i don't want this video to belong but i just want both of you to share a message you know if you should advise the african diaspora who are looking forward to visit live in africa you'll find a message to them well so basically come to gana come explore come and see what it can do for you but mainly come and bring something bring yourself here bring value edit and try to build up this country in a way everybody benefits from it not only you but everybody in your surroundings that's my main message when you're black when your skin color has any type of blackness in it you need to come to africa don't matter which country but you just need to visit at least once it will heal your soul oh that's a beautiful message and i just want to ask is it worth it to live and invest in africa very much very very much yeah i've seen a lot of things around me people just starting a very small business and that blew up i mean africa is the future take the risk africa is the future take the risk but i would say that africa is present how what are you gonna say say something before we go


Cześć, bardzo dużo. Witajcie. Gorąco zazwyczajmy na sesejce. Dzisiaj pobieramy na Alicja Miedlić. Chciałbym o projekcie. Jest to małacka plaktyczna. Plaktyczna jest plaktyczna na otwórzanym plaktycznym. I to jest nie tylko otwórzanym plaktycznym, O, dobrze. Cześć. O, perfect. Zjednę. Ja zaczynamy kilka slajdy, tylko kilka, do co dać Wam nadąsztanie tez systemem i potem Ala will present So melodic, multi-cloud optimization platform, it's fully open source, built by several partners as open source project, and the aim of the melodic is to optimize the deployment. Why to use melodic? First thing is because it is probably the simplest way to deploy application to the multi-cloud. It is the unified way to deploy virtual machines, containers, serverless and also big data frameworks. The deployment is fully automatic, we will show that live. And the deployment is optimized, so that's the unique thing of the melodic. Melodic is fetching the offers from the different cloud providers and automatically optimize the deployment. First thing is the model of the application. We are developed our own cloud modeling language, which is cloud agnostic called camel cloud application modeling and execution language. The unique thing of that language is that we can model user requirements and user constraints. Thanks to that we are able to optimize the application deployment. And another unique thing about melodic is the ability to decide what is the best deployment. Because usually we want to deploy applications and we have certain goals. And the goals are usually contradictionary. We can optimize some kind of the trade-off between the cost and performance, cost and security. If we want to optimize only cost, then the simplest thing is just simply not to deploy application at all, because the cost will be zero, but of course the performance and other features also will be zero. So we need to find the trade-off between the performance and cost, security and cost, and other features of the application. And melodic is optimizing the deployment of data based on the collected metric, and not only technical metrics like CPU usage, memory usage, but also on the business metric, which are defined in the application, like average time of the processing user request or average time of the job calculation. And how the melodic work. First step is to model application in the camera language, and it unfortunately needs to be done manually, but it is done once. You need to model your application. And then application is deployed. Initial deployment of application is done using the standard parameters, which are in the camel model. And then initial deployment is done to the selected cloud providers. Then melodic started to collect metric from the running application, and based on the values of the metric, is able to optimize deployments. So to add resources, to remove resources, to move some resources from one region to another, or from one cloud provider to another. Also melodic is continuously fetching the cloud provider's offers from the top living free cloud providers, and also from the open stack based cloud providers. And based in the current offer is able to move some components of the application. We will show that live. And we are continuously developing melodic. So we are adding new features. One of these features is polymorphic adaptation. So we want not only to add or remove resources, but also to use the most appropriate resources for the given task. For example, if we have the application components, which is using GPU or FPGA, then we can decide to deploy the component on the machine with the GPU. If it's heavy use, if it's not heavy use, then we can deploy that machine with the CPU, which is cheaper, but of course the performance is lower. So that's the one concept, which we are working on. And the second one is proactive adaptation. Because currently melodic is reactive, so fetching the metrics from the application and based on the value of the metrics, add or remove the resources. So for example, if the number of the users is too high or the request processing time is too high, then melodic add or remove the resources. But in the proactive adaptation, we want to predict. We want to predict that the number of the users is rising, so we want to add these resources proactively to be able to handle peaks without the degradation of performance and the ergonomy of using the application. So that is the second concept, which we are working on. And how the melodic works in practice. AI Investments is one of the early adopters of melodic and is using melodic for optimizing its cloud deployment. AI Investments is a Polish company, which created the software for using various machine learning methods to optimize the investment portfolio. And AI Investments is focused on time series forecasting, financial time series forecasting and optimization. And of course, both of these tasks requires a lot of resources. And the optimization of cloud resources is crucial for that type of the company. And typical business goals for the AI Investments is to train a certain number of the machine learning models, like, for example, 50 predictions models in one hour. And how it works. The business investments analysis in AI Investments starts to train models, so send the request to train the models. And initially the on-pervices resources are used, because they are available, they are already involved, so in place. And the training starts on that resources, but usually it's not enough. So melodic is calculating how long this training will take on the on-pervices resources. And if it's too long, so longer than the business expectation, then additional resources are deployed automatically to the cloud. And these resources are automatically used to processing the data. Again, melodic is tracking the metrics. And if the expected time for the processing is one hour or below, then new resources are not added. And these resources are used for the processing. If the processing is finished, then all of the resources are automatically removed from the cloud to save the cost. So as you can see, the use case is very simple. We are just to add the resources or remove resources in the proper way, but the benefits are really significant. Now, for the AI investments, the difference between the not optimized deployment, like cloud-only deployment, the cloud-only is in the first row at the top of the table, or on-pervices-only, is about $175,000. So this is the difference between the optimal and the two extremes or the most expensive variant, which is not optimized. So as you can see, there are significant cost savings using that approach and thanks to that, investment can use more computational power and achieve better results. This is a typical example of the AI investments results based on the AI optimized deployment. So as you can see, the benefits of that approach are huge. Melodic is open source. As I said, you can download it here. It is released under the Mozilla Public License 2.0 and can be used. I think now, Ala will present how the melodic works live. That was, as I said, my theoretical introduction and now Alicia will show how it works. Ala is a team leader of the development team melodics. Techniczna kwestia, Ala jest prawdopodobnie najlepszą osobą. Tak. Dzień dobry wszystkim. W moim częściach prezentacji would like to present you how our project works in practice. Now I need to switch off the presentation and show you the melodic web page. So here we can see the initial view of melodic deployment process. And now we can see that our deployment process is being performed now. But at first I would like to show you how to configure our platform and how to start such process. And also we will observe not only the first initial deployment process, but also a regularization process and metric predictions because we not only react on current real situation, but we would like to predict our application performance, predict metrics and react in advance to ensure the most optimal deployment for us. So melodic now is performing application deployment. So in the meantime I would like to describe you how melodic works in background. The first step in melodic usage is to define provider settings. In this step we need to define which cloud providers we would like to use in our deployments. And also we need to provide credentials to our cloud accounts. So for example, in case of AWS we need to provide our developers credentials, secret and user. Also it is possible to configure our private properties which we would like to use during our deployment. For example, set of private images which we would like to use and also for example preferred security group. On my virtual machine when my melodic platform is installed I have already configured as you can see credentials and settings for two cloud providers for AWS and also for OpenStack. But melodic supports also Azure as well as GCP and OctaWave quite small Polish cloud provider. So in this place we can configure all of our cloud accounts. After that we can start the whole deployment process. In this place we need to upload to our XMI format. So the first step is to model our application in camel model in camel language which was mentioned by Paweł at the beginning also. Camel is human understandable form. We model our application in camel and after that we need to transform it to XMI format. XMI is the form understandable for melodic and we need to attach such XMI file here. Just at the beginning of our presentation I have already attached such model for our application which we are deploying now and here also we have information that our model our model also can have secure secure variables because it is possible to put some secure content to our model and not define it directly but put it in a specific way and after that just after uploading the XMI model we are asked for providing these values these secure values and they are stored in secure store. Also here it is possible to configure our model in proper way we have dedicated tool for that UF creator so it means utility function creator because it is really important to properly configure to properly model our application in order to ensure the most optimal deployment and here we have various possibilities to configure our model for example by function, by template and by plotting here we have for example our application model and we can configure how to use specific metrics during our application optimization but now we can back to our process so in this step we have model of our application we attach this model it is of course validated and after that stored in database and here we have the last step which is needed to perform manually by us just before automatic deployment so in this step we need to choose which application we would like to deploy and which providers you would like to use for example we can use all possible providers, we can choose only one today I would like to focus on deployment of AWS so just before our presentation I choose only AWS with my properties i just after that we can go to the last step where we only need to push the big green button start the whole process and after that we can relax and observe how melodic performs the deployment of our application but as I mentioned we started the whole process at the beginning because the whole deployment is quite complex and also because of that time consuming so I needed to do that a few minutes before now we can come back to our deployment process and we can see that it is already done because all boxes are checked as done and I would like to I would like to describe this process so the first step is fetching offers in this step melodic fetches offers from previously selected cloud providers in our case from AWS and after that from these offers we can also observe the details of these offers so we have our clouds hardware locations and images and from these offers melodic chooses the best configuration for us but how melodic know what is our configuration so we have our conflict problem which was defined in camel model in case of our application we have spark based application which is modeled as two components so worker and master we have one spark master which manages the application performance and we can have many spark workers and the task of melodic is to ensure the proper, the most optimal number of these workers our conflict problem we defined that we would like to have from one to maximum 10 workers here we have our requirements from worker course and we defined that we would like to have exactly one spark master and also we have requirements for providers for both spark master and spark worker and after that melodic performs next step so the rezoning part here it finds the most optimal solution and we can observe it here so in this case we have at the beginning one worker with eight course and also we have one master after that it is time for deploying in this step melodic performs deployment of our the best solution so real virtual machines are being created in our providers in our case of course we would like to see the result of this deployment and here we will see of our already created virtual machines so here we can see detailed information about these VMs so IP addresses provider also detailed information about localization of these VMs and technical information about hardware image and so on of course we would like to monitor performance of our application in order to do that we can click monitor application and we are directed to Grafana Grafana is the tool for monitoring applications and displaying statistics also we use this tool to monitor applications deployed by melodic of course each application has on metrics on parameters to measure so we need to create dedicated dashboards for each application also our application so general application has on dashboard and we can see here how our application works and what it is in the background of our application on this dashboard we can see at first number of instances so now we have two instances, one worker and one master the limitation of traffic lights means if our application with current configuration will finish on time or not the red light so it means that current configuration is not sufficient and we need to provide additional resources also we can observe here estimated time left estimated time at time left because in our requirements we defined our expected time and we can see that our estimated predicted time is higher than time left also here we can see a number of course if we now we can see also also another matrix so our predictions and current information if we finish on time or not here we have average average length of one task also estimated time matrix, which means if we finish on time or maybe we will finish too soon ok, so now we we are expect that we need to we need to have additional resources and in the background melodic performs for us reconfiguration process and after this reconfiguration process we are expect to have more efficient worker and after that the performance of our application should be better and we will see it soon but now it is it is the end of our deployment deployment by melodic presentation so we observe the reconfiguration process and and the whole process is done fully automatically by melodic ok, so thank you very much for your attention and if you have any questions please feel free to ask but maybe individually because I think that we are overtime a bit


Now, I will indeed, as Mary trailed out, I will talk a little bit about what the international financial institutions are doing and maybe can do even better to help the situation. But I want to start with a few remarks about how we see the economic situation and being the last speaker, I have the advantage of being able to pick up and draw together a few points that have already been made. Indeed, it is a gloomy picture, as indeed painted by Gorn Svilanovich and Vladimir Iglygorov. I have to say, we do share that gloom at the moment. We try to be optimistic about the region and I remain optimistic about the long-term future. But certainly, if you look at the short term, we had in 2012 negative growth in most countries in the western Balkans. For 2013, we published our new forecast a couple of weeks ago. We don't expect much growth. It's, for most countries, somewhere between 1 and 2%, with the exception of Bosnia-Herzegovina, where we put in 0.1%, so positive, but just barely. And Croatia, which has really been in stagnation or recession for the last four years, we expect this year another decline in GDP minus 0.3%. And in fact, when I talk to my friends in Zagreb, people in the economics and investor community, they think that actually we're being a bit optimistic with that figure. So notwithstanding the great achievement of Croatia on the verge of accession, the economy really is in a weak state. For 2014, we expect a little bit of pickup. But 2014, in economic forecasting terms, is a long way ahead and we could get this completely wrong, as we often do. But we do expect some pickup typically between 2% to 3% growth for most countries, but with a lot of what we call downside risk. And I think even if we do get to, or even 3%, it's well below the potential that this region has. And Goran Selanovich mentioned the relative GDP per capita compared to the EU average. It's for these countries, when you adjust for purchasing power, then Eurostat publishes these estimates regularly. It's between 30% and 40% of the EU average. So these countries should be growing faster. We know from economic theory and from experience that poorer countries should and can grow faster than rich ones provided the right conditions are in place. Now, why has there been no serious recovery from the major crisis that occurred in 2008 and 2009? Well, I think it's a number of factors and some have already been mentioned. One can point to the collapse of foreign direct investment. I think that's been important. One can point to the external weaknesses and the eurozone, which is the main market for most of these countries. That is indeed important as well. One can look at the financial sector. And although the financial sector has managed to hold up fairly well, and I will come back to that in a couple of minutes, we have a trinity of financial sector problems. A trinity being very weak or even negative credit growth. So it's really harder now for businesses to get credit than it used to be. This ongoing process of deleveraging, so basically the foreign subsidiary banks in the region paying back their debts to their parent banks abroad. That's an ongoing process, which again of course feeds into the weak credit growth. And the third part of this trinity is the non-performing loans, which are at fairly frightening levels in some countries. In Serbia, for example, around 20% of all banking loans are non-performing and it's even higher in Albania. So I think these are factors, but we would argue that actually what we have learned over the last few years is that the boom years that Vladimir spoke about in the, say, five to six years before the crisis that they masked a failure to address some serious structural and institutional reforms. So things seem to be going well, but indeed the foundations were not really being put in place at that time for sustainable long-term growth. Now, so one can see this from a variety of indicators and there are all sorts of indicators on the business environment and progress and transition and corruption and competitiveness and so on, some of which we at the EBRD produce, some of which are produced by other institutions, but although they can be confusing taken in total, they do paint a consistent picture of a region that is lagging behind on structural reform. So what we argue is that really it's not, I think, that the region needs a totally new economic model, but the only way forward, if it's to get back to economic growth, is an enhanced focus on structural reforms within a regional context. And I take very much the point Vladimir made at the end about some of the disappointing aspects of regional cooperation, but I think if those problems can be addressed, there could be long-term benefits. Now, coming to the role of international financial institutions, so IFIs, my view, of course I'm not an impartial observer here, I work for one of them, the EBRD, but my view is that IFIs have played an important constructive role in alleviating some of the worst effects in the crisis in the last few years. And it's not just about money and putting money into the region, but although that has been very important, and most of the main institutions, including the EBRD, have stepped up their investments in the region. So the EBRD is very much a counter-cyclical institution. When regions are doing badly, we do more, and when they're doing well, we step back and let others invest. So we have been investing more, and others have been investing more. But it's also, I think, about putting in place some structures that enhance cooperation both within the region and between the region and outside countries and institutions. And let me mention those three initiatives that I think have really helped in the last few years. One is what's called the Vienna Initiative. So the Vienna Initiative was really about making sure that we didn't have a banking crisis in the western Balkans and more widely southeastern Europe and some other countries as well. And it was about ensuring that the major foreign banks that dominate the banking sectors in these countries, that they remained engaged, that they didn't all rush for the exit and pull their money out of the countries, which could have caused a panic. And that was a very important cooperative effort between IFIs like ourselves and the IMF and World Bank and others. Host countries, so in the western Balkans, home countries in Austria, France, Italy, Greece and so on that are home to these big banking groups and regulators in both home and host countries. It worked well and it helped prevent this, it helped prevent a crisis. The second one was the joint action plans that institutions like ourselves, the European Investment Bank and the World Bank came up with, which again were really about showing strong commitment to the region and about promising, pledging major investments into the region. I think that the initial joint action plan in 2009 was a very strong signal and there's a new action plan for all of central and southeastern Europe announced by those three institutions at the end of 2012. And thirdly has already been mentioned the western Balkan investment framework and others around this panel and Mary certainly know intimately how this works and how it has really I think been a very successful cooperative venture in terms of making sure that funding, donor funding, whether it's lending from institutions like the BOD or grant financing from Bellatoros that it goes to the right projects. Now having said all of that I want to finish with a few remarks about how IFIs can maybe work better. I'm speaking of course my personal opinion on these and I think there are three ways in which the way we are involved in the region can be made even more effective. One would be I think even more cohesion than we have already on strategy development and this is an idea that came to me in preparing these notes because it's fresh in my mind from the EBRD annual meeting a couple of weeks ago in Istanbul where we indeed sat down with some of our colleagues from the World Bank and the European Investment Bank and the IFC also which is part of the World Bank to think about what are we doing in each countries and can we really mobilise our timeframes in a consistent way and really come up with a very coherent growth strategy for countries in this part of the world. A second way I think we can do more is to have more rigorous analysis of what really is holding back growth in this region. A more rigorous diagnosis of the obstacles to growth and the constraints because I mentioned earlier some of these cross-country indicators on reforms and business environment and so on but they don't really tell you on a country by country basis what are the key blockages to reform and I think again there's a lot of analytical power in these institutions and it can be deployed maybe more effectively than it has been before and the third way would be to pay more attention to social issues so I take very much it wasn't news to me but I really appreciate what Goran said and Vladimir about the problems of unemployment in the region. It really is indeed a frightening set of figures when you look at the unemployment rates. Possibly some of them are a little bit exaggerated because there's a lot of informal employment and so on which is maybe not captured but notwithstanding that there is indeed a serious jobs problem and as Pavel Dobble who said at the start it's really about jobs growth and reforms and jobs I think rightly being the first of that trinity being mentioned. So I think IFI's and here the EBRD we are doing some serious thinking about how we can build analysis of social issues to do with things like youth unemployment to do with things like gender differences as one into our analysis of projects and indeed this is becoming an increasingly important part of our overall strategy. So in conclusion I don't want to end on that very gloomy note because I do really feel that this region has a bright long term future. I'm not sure if in ten years time we'll have three countries or whatever as members of the European Union but I really do feel that in ten years time this will be a much more prosperous region than it is now but it's going to be still I think a difficult two to three years that we face in front of us.


of a four-to-fly position conducted by a soldier from Fibro and our Marines from the Fibro-Italian first Marines ground combat on it and also supported by 103 battery and direct by-sports assets from the US Marine Corps. That's co-located with and they've been providing supporting fires and supportive maneuver. Also co-located with Fibro ARs 81s is our 81s platoon against supporting fires and supportive maneuver and you can hear over our right shoulder we also have heavy machine guns that have been isolated and injected the entire attack. Sea Company has just taken a trench position here and there's Major Jack Meadow who will be flanking around with another platoon that have engaged in depth further. Relationships are absolutely paramount. Relationships equals trust. Trust equals combat capability. Propability here is the absolute key for this to be the culminating activity to really display the activities we've been doing, the capability we've got to but equally as important to display that trust.