 So, last class we have discussed the vector and matrix norm of a constant matrix or constant vector norm. The most general P norm is defined like this way and when P is equal to 1, we call it is a L 1 norm or 1 norm. This is nothing but the sum of all elements in the vectors absolute value, sum of all absolute values of these vectors elements. L 2 norm is nothing but the L 2 norm when P is equal to 2, the L 2 norm is nothing but a distance of the vector from the origin. This is also called a clean norm. Next we come into when P is equal to infinity that we will call L infinity norm or simply infinity norm of a vector that is nothing but a maximum absolute value out of the all elements of the vectors that is the L infinity norm simply infinity norm of a vector. Next we came into the picture that a matrix which elements are constant then how to find out its different norms and different applications. So, first is Frobenius norm. This is nothing but a that matrix whose Frobenius norm how to compute is nothing but a all column vectors this, this, this you put it in sequence. So, that whole thing will come a vector of dimension m into n cross 1. So, find out the what is called the norm 1 of this vector, norm 2 of this vector. So, this is called the Frobenius norm Frobenius norm of the matrix that means this column then put below this column and below this column and so on. Whatever the you get it vector that vector will be a dimension m into n into 1 and that vector what is called norm 2 is equal to the is called the Frobenius norm of a matrix. L 1 norm of a matrix is nothing but a maximum column sum. So, if you have a m cross n matrix dimension you have a n columns are there out of n columns whose absolute value sum of all elements on that particular column you added. Then this will out of all this column which one is maximum that is called l 1 norm. Similarly, l infinity norm is nothing but a the maximum row sum that means you absolute value of the all row elements you added out of this which one is maximum it is called l infinity norm. Then l 2 norm of a matrix is nothing but a what is called maximum Eigen value of this A transpose A and this A transpose A is a symmetric matrix is all Eigen values are real and that is a positive semi definite matrix. So, this all Eigen values will be a positive non negative numbers. So, you out of this which one is maximum that will give you the l 2 norm of a matrix. So, this same idea will now extend for a signals. So, now our question is the signal and system norms. So, the in any signal the energy is present when any signal is presented over a interval of time the energy contained in that signal how we will compute that we will call the size of a signal or strength of the signal. So, the energy contained or present in a continuous time or discrete time signal or over an interval of time is given by. So, energy over the interval t 1 to t 2 t 2 is greater than t 1 is divided by t 1 integration of t 1 t 2 and x of mod of t square. But mod of y I have taken mean suppose that signal is complex then you have to take absolute value of that signal for the time being I am considering it is a scalar signal. So, this d t in the interval t is equal to t 1 t 2 how it is energy suppose we have a once resistant whose value is 1 ohm and if I send a current to this one is I. So, what is the power I square into r and r is 1. So, simply it is a I square and I square I is a signal. So, in that case if you consider the x of t is the I then I square into r r is we consider 1. So, this is the power then over the interval of this t 1 to t 0 is nothing but a your energy contained in this signal over the interval t 1 to t 2. This is in case of continuous time session the equivalent version of discrete time case the energy present in that interval over the interval and that in the discrete time n 1 comma n 2 and n 2 is greater than n 1 is equal to that integration term will be now replaced by a summation term n 1 2 n 2 mod of x of k square. This is I told you again that absolute value if it is complex then absolute value is a that meaning if it is not a real signal is there then absolute value of this square itself a positive value is this one. So, this now total energy of a continuous time signal we can write the total energy of a continuous time signal is its energy calculated over the infinite energy calculated over the interval t is equal to minus infinity to plus infinity. So, minus infinity to plus infinity this continuous time case now continuous time signal we denote the total energy x over the interval minus infinity to plus infinity the expression of state only integration limits is changed. So, this infinite the integration minus infinity to this will be finite provided that x tends to 0 when t tends to infinity. So, this will converge this if x of t tends to 0 as t tends to infinity. So, this energy value will be finite if this condition is satisfied if this condition is not satisfied then we have to give some other meaningful definition of energy size or energy strength. So, we will use the what is called energy signal strength as a energy when the signal x of t tends to 0 as tends to infinity. So, the signal which is meaningful when x tends to x x does not tends to 0 as t tends to infinity then we have to use another definition of energy content in that signal that is called power signal. So, similarly in discrete time case similarly in discrete time case discrete time signal or sequence e x is equal to minus of k is equal to minus infinity to plus infinity mod of x of k whole square. So, the signal e is said to be e of x suppose you denote this thing is e of x this thing you denoted by e of x and this thing also you denoted by e of x. So, the signal e is said to be finite energy whenever e of x less than infinity. So, this will possible if the signal tends to 0 as t tends to infinity if it is not true that means x does not tends to 0 as t infinity then we cannot express the signal energy. So, you have to give the meaningful definition of power energy. So, definition of power energy is the it is defined as energy per unit time. This definition or this meaningful definition will give when the signal does not tends to 0 as t tends to infinity that means in that situation the definition of energy power energy will be infinite and infinite has no meaning of this energy. So, we will consider in that case is power energy time. So, the by definition this the average power of a continuous time signal x t over the period over the interval t is equal to minus infinity to plus infinity is given by that e p x this is for continuous time p x is equal to limit t tends to infinity 1 by 2 t minus t 2 t mod of x t whole square d t. The discrete time discrete time case p of x is limit k tends to infinity 1 by twice k plus 1 summation of k is equal to minus infinity to plus infinity or you can say then it is a mod of x k whole square or you can say this is in state of write it k is equal to capital minus k capital k and capital k tends to infinity in this way can write capital k tends to infinity. So, this is capital k in this way instead of writing infinity to minus infinity you can write because I have written capital k tends to infinity. So, 1 by 2 k plus 1 minus k to plus k that is 2 k that is 0 is there in the time index. So, it is a 2 k plus 1. So, this is in the case of discrete time case. So, when we will use the meaningful definition of power energy when the signal does not tends to 0 as tends to infinity t tends to infinity then we will use the meaningful definition of power energy. An energy we will use the definition of energy signal this is the power signal power signal not power signal. The energy signal definition when we will use it when the signal tends to 0 as tends to infinity then we will use the meaningful definition of energy signal. So, that is next now I will leave in exercise to show that whether this is a power signal or energy signal. So, I have just discussed first is energy signal this is the definition of energy signal and this is the definition of power signal. When we will use the definition meaningful definition of power signal or energy signal just we have discussed it. Now, we have a what is called continuous time signal this is the continuous time signal that x of t is equal to 5 when t is minus 2 and 0 elsewhere elsewhere. That means signal if you see this one if it is a 1 2 3 1 2 3 0 minus 1 minus 2 minus 3 minus 4 the signal is like this way this is a 5 1 2 3 4 5 let us call this is 5. So, signal is continuing up to 2 and this is up to 3 after that it is 0. So, here so t tends to infinity either side this is 0. So, this is we will use the definition of what is called the energy signal what is the energy content in the signal and next time when it is not that x of t does not tends to 0 as tends to infinity. Then we will use the power energy definition what is the power content in that signal average power. So, if you work out this one then your e x is equal to minus infinity to plus infinity mod of x of t in this case since it is a real signal we need not to take the mod x t of square is enough. So, this is nothing but a is a 25 minus infinity to plus infinity then d of t then this is nothing but a 225 if you that limits is will be minus 32 plus 2. So, it is a 2 minus minus 3 and this will come 5 into 25 1 25. So, this is finite whereas, power is as definition of power t tends to infinity this will be a 0 in case of average power this average power is tends to 0 this is 0. So, we have used the signal is energy signal. Now, if you use the what is called another example you take it suppose we have the signal x of t is equal to 10 cos pi by 2 t plus pi by 4 look this signal is a periodic signal and its frequency is and its frequency is what is this frequency is pi by let us say this is your omega omega is what pi by 2 is equal to twice pi f. So, pi pi cancel f is equal to 1 by 4 therefore, implies t is equal to 4 time period is 4 since is a periodic you need not to go do it is enough to go for integration average power integration for a 1 complete cycle. So, our in this case our e of x is equal to if you call e of x minus infinity to plus infinity x of t square d t and if you just this is the energy signal if you use it I told you this does not tends to this signal does not tends to 0 as t tends to infinity. So, if you apply the energy signal this value will come this is a 10 it is 100 integration minus infinity to infinity then cos square pi of t plus pi by 4 and this results is infinity that means infinite energy signal which does not have a meaningful definition when I will use the energy signal. But if in case of power signal what will do it 1 by t 2 t or because it is a I just consider the from 0 to t is defined if you consider this is defined from t greater than 0 less than infinity this one this is defined into this. So, it is a periodic nature of that one. So, this value if you just calculate this one 0 to t then cos square pi by 2 t plus pi by 4 d t actually by definition it should be what if you see by definition I write it by definition t tends to infinity 2 t minus t 2 plus t. So, I need not to go for since it is a periodic signal again. So, you take the complete one cycle. So, complete one cycle if you take it is nothing but a 0 to t 1 by t cos square pi by 2 t and pi by d t and this value you will get it that value you will get it if it is a after that you will get it this value is 50 unit. This is cos square you just converted cos square is 1 minus 1 plus that is this formula use it that cos 2 alpha is equal to cos square alpha minus sin square alpha. So, twice of this one that means 1 plus cos twice alpha by 2 that you convert into a 1 plus cos 2 alpha by 2 and then integrate you will get 15 unit check. So, this committee even if you do not consider that it is just defined from minus infinity to plus infinity then this is the power signal that what is the power contents in that signal we have to express, but you cannot express is what the energy content in that signal because this become infinite it is no meaning full of that one. So, next is state wave will come to the system norm we have if you have a system signal norm of a system signal norm of a system. Now, we are coming to the system suppose we have seen earlier our lecture that if you have a dynamic system is there we can form we can write the dynamic equation either in different model either transaction model or state based model. Let us call we have converted we have a system which is a trans function model g s input and our g s is the trans function and this is our input u of t and if you consider in Laplace transform this is u of s this is the input of this is the output is y of t this is output and in Laplace domain it is y of s the Laplace transform of y of t. Now, this signal and output is y of t this two signal norm of a system what does it mean. So, signal norm of a system systems are usually just now I mentioned that we can describe this thing in the two model either trans function model or state based model. If you do is a any model first you have to write the differential equation based on the physical laws of the systems once you write the differential equation then our job is to convert either in trans function model or state based model again. So, if it is a state based model that another order differential equation you have to convert into n faster differential equation. If you convert into a trans function model then you take the Laplace transform both side by assuming all initial conditions are 0. So, the system are usually described by differential equation that contains signals x let us call u of t over the period of time over a certain interval of time interval of time with varying amplitude. So, our job is to find out the signal is measured by a number our job is to measure the strength of the signal by a number again or size of the signal by a number u of t you know u of t or y of t or y of t strength of the signal or size of the signal or strength of the signal we have to find out and let us call the signal if it is a scalar signal let us call u of t let us call u of t or y u of t that we denoted by in u of t or y u of t we are denoted by x of t u of t or y u of t let it is denoted by it is denoted by x of t. So, the signal now onwards I am writing x of t now the signal if it is a scalar this is a scalar and then that means one signal is there and in real or it can be x of t can be a vector n cross 1 and vector this is a vector of dimension n cross vector of dimension n cross 1 and we assume the signal x or scalar or x t vector are continuous piecewise these are these and these are continuous piecewise or you can say piecewise continuous this is assumption is made. Now, we will find the strength of the signal as we did it for constant vector or constant vector of dimension n. So, a vector case let us call instead of a vector case or you first with scalar case scalar signal p norm just like is a norm of x of t p this general definition is that minus infinity to plus infinity norm of x of t that 1 by p sorry p this is a signal for this and p is greater than equal to 1 greater than equal to 1 or less than infinity this is a integer. So, l 1 norm in signal case or we can so simply one norm is x of t this infinity minus infinity to plus infinity x of t d t. So, this signal must be integrable over the period of minus infinity to minus infinity the absolute value of the signal must be integrable. So, this the absolute value this is the absolute value of absolute value of x of t. Now, l 2 norm simply we call 2 norm is so what is the physical significance of this one it is the energy this is the energy content because if you consider a resistance of unit resistance and passing a current x of t then i square r into i square into r is the power over the interval this energy. So, this is the representation of energy content in the signal that provided x tends to 0 as t tends to infinity then we will consider the energy content in the signal this way and next is l infinity to norm or is simply called infinity norm. So, that denoted by x of t infinity and that is defined as supremum t r what is it mean infinity norm of a signal you see this absolute value I told you if it is a complex signal then only you need absolute value. Here because signal may be plus minus and that obviously you have to take the absolute value of this one whether it is a real or complex this one. So, what is it mean this one so if you plot x versus the time at what time you will get the maximum value of this one that means it will always x value will be less than that maximum value always it cannot cross that value. So, supremum name is the list upper bound the list upper bound or it is called the tightest or it is called this is also called the tightest possible upper bound. What does it mean let us take in one example and see let us call we have a signal example we have a signal let us call example x of t is equal to 1 minus e to the power of tau t. This signal you can say you have a simple resistance you have a simple resistance and inductance you are excited with a dc voltage and this is the current which I am considering x of t is the signal and current flowing through the signal is like this way and so if you plot this one suppose you can consider voltage signal is one this is signal is one and l also one then you see this is the nature of this curve will get it like this way and this is the x of t. Now, you see the maximum value of this one is I cannot say maximum value why I here I can say the least upper bound of this signal is one because here you see at this point whatever the value is there 0.8 after some time the value will be greater than 0.8. So, that is also maximum again after sometime its value is greater. So, this is also we cannot say anything about the maximum value of this one. So, it has a we can say that this signal supremum value of this signal is one may mean least upper bound of this signal is least upper bound it cannot go beyond that one whenever it is t even greater than this, but all is the tightest possible upper bound. I am telling in before I am tight bound of upper bound of this signal is one agree, but I cannot say the maximum value of this signal is one this one because at this point if it is the maximum value is one after sometime time the value will change it more than this. So, you cannot say in this case will say this is the what is called least upper bound of this one. So, supremum there is another term is the infimum. Infimum means greatest upper bound again this is the greatest lower bound infimum greatest lower bound greatest lower bound. So, let us call we are considering because this definition is coming if you see the L infinity norm of a signal symbol signal is nothing, but a you draw the signal let us call this is the signal and you are telling this is the most upper bound of the signal. That means least upper bound that the least upper bound of this signal is that one that is called supremum of this signals. So, in case of vector signal. So, once again I am telling you this one that infinity norm of a signal is the supremum value absolute value of the supremum value of this one agree. So, this indicates meaning of little meaning a least upper bound I am telling in beforehand the least upper bound of the signal is that much agree. For this example least upper bound of the signal least upper bound of the signal least upper bound of the signal is one or supremum value of the signal is one. So, vector signal again vector signal instead of a scalar now we have a vector it has a n components vector contents and components and we will just give the definition as it as this way p norm all these things. So, p norm in case of vector is x summation of minus infinity to plus infinity integration minus infinity sorry this is not infinity this is I tell you this is minus infinity to plus infinity then signal. Then how many signals we have n signal because x is a vector suppose x of t is a vector whose dimension is n cross 1 this is a vector whose dimension is n cross 1. So, we will write it that I is equal to 1 to n mod x i of t mod p whole that whole thing 1 by p. So, please norm of this one whole thing by 1 by p here missing is 1 by p p is equal to 1 this is that p is equal to 2 this is then that this is a norm of that one agree. So, now what does it mean this one for p is equal to greater than equal to less than equal to 1 what does it mean that one this indicates the absolute value of x all you take it p is equal to i is equal to 1 to n p let us call p is equal to 1. So, you add it all this thing i is equal to 1 and all then integrate minus infinity to plus infinity and this is called the one if you do one norm how you define x of t p is equal to 1 is equal to integration minus infinity to plus infinity summation of i is equal to 1 to n mod of i this. So, this because p is equal to 1 i is equal to 1. So, this is and it is a d t integration of this this is also d t. So, this is called one norm, but one norm is absolute value of you just see i is equal to 1 i is equal to 1 absolute value i x 1 absolute value over that interval of time then x of t x 2 i is equal to x 2 of t add all this thing then integrate over the minus infinity to plus infinity that is called vector in case of vector one norm. Now, the most important is L 2 norm or L 1 norm previously that is L 2 norm we simply call 2 norm of a vector or signal 2 norm of a signal and signal is a vector is a vector signal. So, what is this 2 norm x of 2 this minus infinity to plus infinity x transpose this mind is x is a vector x transpose t x of t this d t and whole half. So, what is this mean sum because x is a signal it has a n components that means n signals are there is signals energy containing signal and sum each component of the vector energy content and we are making the sum of this. In other words it is nothing but a if you see this one it is nothing but a the r m s values of a signal if you have a more than one signal let us call it is x is equal to 1 d i square r over the interval into r is 1 if you can is nothing but a that your power. So, this is nothing but a this quantity is nothing but this norm is a root means root you see the root this is going to root mean square. So, this norm sum of this norm indicates the sum of energy contains in the vector signal. So, if you expand this one let us see this one if you expand this one minus infinity to plus infinity x transpose x is a row vector this is column vector it is nothing but a x 1 square t plus x 2 square t and dot x n square t whole d t then you are taking the half of that one. So, this each indicates the energy energy content in that signal vector signal. So, next is l 2 norm l infinity norm or is simply it is called infinity norm of a vector infinity norm of a vector signal. So, infinity norm of a vector signal is what supremum t is greater than equal to 0 max i then mod i of t this and i varies from 1 to dot dot n what does it mean say the infinity norm of a vector means that x of t absolute value you vary 1 to n x 1 x 2 over the period t 0 to t greater than 0 out of this which one is the maximum you note down then x is equal to 2 over the period 0 to t greater than 0 what is the maximum you note down in this way x i is equal to n over the period t greater than 0 you note down the maximum out of this which one is least upper bound that is will give you the infinity norm of s that signal vector agree. So, this has a some important concept which will be utilized in case of our system signal system signal norm. So, next is norm for norm measure of systems. So, you see this one scalar case when x of t is a scalar and our norm p norm of a signal is minus infinity to plus infinity absolute value p of that one, but it is a more meaningful when p is equal to 1 this is a absolute value of signal over the period t minus infinity to this value meaningful of this one is for scalar case is this one. So, you see this one scalar case when x of t is a scalar that means x square x square is a signal multiplied by a resistance r is equal to 1. So, it is a power and I am integrated over that interval minus infinity to plus infinity. So, the energy content over the period of time this. So, this is energy content in the signal. So, power content in the signal when you will use the expression if that x of t signal does not tends to 0 as t tends to infinity then you will use the expression for power energy or power signal concept. So, next is that norm of a system what does it mean the system norms are actually the input output gains of the system. So, suppose you have a system g s y of t is the output u of t is the input. Now, as you know the frequency response of a system what we do it we have a physical systems the system is accepted with a sinusoidal input again keeping its amplitude fixed we are changing the frequency and see the output which is also in time domain the output of signal also what is called in sinusoidal form. But it magnitude will be different from the input signal and also phase angle will be different the frequency of input signal output signal is remain same. So, when input is accepted with a system then output it is mapped to a another signal when input is accepted with the system sinusoidal input it is mapped to the another signal. In case of multi input multi output case that when input is accepted with a multi input signal that signal is mapped into a output multi output signals. So, if you see in the frequency response of the if u is equal to a sin omega t then output will be that y of t will be a g j omega mod sin omega t plus phi again mod of this one is mod of this one sin omega t. So, our gain of this one the input amplitude is a output amplitude is a into g. So, g w j omega mod is the gain so system norm actually are actually nothing but input output gains. So, the now question is what is the size of a system norm the size of the size of a system in a transfer function t f transfer function form is usually measured by h 2 and h infinity norm. The size of a system is nothing but h 2 and h infinity norm. So, before that I discuss the h 2 h infinity norm I will discuss that our important theorem what is called perse say h 2 h infinity norm. What is that theorem let us call we have a two signals I want to integrate that two signals from minus infinity to plus infinity we have a two signals x of t another signal is y of t and way we want to integrate product of this two signal over the interval. So, let us call x of t is voltage and i of t is the current in a particular branch in a circuit in a particular branch a voltage across that branch is x of t and current flow in through this one this. So, this product of this one is the instantaneous power. So, I am integrating over the minus infinity to plus infinity it gives the energy. So, I want to compute that one let us call this equation is one what is the Perseverance theorem tells that we will see here. So, what we made assumption of this one the this Laplace transform of voltage and current exist Laplace transform of both the signals Laplace transform both the signals both the signals exist that is our assumption. Now, you see this one y of t y of t I can write if the Laplace transform of y of t y of s then I can write inverse of that Laplace transform is 0 j is omega 2 j omega there is a omega is infinity omega tends to infinity. So, I am stated j this is equal to y of s is equal to s of t d t. Note if Laplace transform of y of s y of t is equal to y of s then y of t I can write it is 0. It is nothing but a y of t Laplace inverse of that y of t I can write it nothing but a y 2 pi j and minus j infinity to plus j infinity y of s e to the power s t d t. If Laplace transform of y t y s the inverse of Laplace transform y of t is equal to this. So, that if you express in this is let us call equation number 2 if you express this into a 1 from 1 I is equal to I am replacing that y of t by this expression y of t by this expression y of t by this equation 1. So, 2 pi j integration minus infinity to plus infinity then x of t minus j infinity to plus j infinity y of s e to the power of s t d t into d s d s into s of d. So, by rearranging this j minus j infinity to plus infinity y of s and minus infinity to plus infinity x of t e to the power of s of t d t. And by definition of Laplace transform we can write it this equal to this and we have a left with d s. So, we can write it 1 plus 2 pi j integration of minus j infinity this is infinity plus j infinity this we can write y of s into x of minus s into d s. If x and y are the same signal that means y if y s is y of t is replaced by x of t if x of t if y of t is equal to x of t then x of t is equal to x of t. From this equation from 3 we can write that integration of minus infinity to plus infinity x square of t d t is equal to 1 by twice pi j minus infinity minus j infinity to plus infinity x of s into x of minus s d s. And this is ultimately we can write it this is equal to 2 pi j j is equal to minus infinity to plus infinity x of j omega x of minus j omega d omega which is nothing but a twice pi minus infinity to plus infinity this I can write it mod of this whole square d omega. So, this indicates the energy content in time domain relation between the energy content in time domain to energy content in frequency domain or frequency description the same. So, this is nothing but a Fourier transform of that signal x j x omega is nothing but a Fourier transform of that signal. So, we will stop it here. So, we have made a relation that Perceval theorem this is called Perceval theorem is tells that this theorem relates the time domain energy content in the signal in same as the frequency domain energy content the same signal energy content in the frequency domain also in the same in terms of Fourier transform. So, we will stop it here today.