 So, much so then for the properties of systems and now we are going to focus our attention largely on linear shift invariant system and we know why we have also convinced ourselves why now. We know that if I have a linear if a system is linear and shift invariant give it at a complex exponential a rotating phasor it produces a rotating phasor of the same angular velocity the catch is that you need you see we need that h omega to converge and what we have seen is that for stable systems it definitely converges. So, one thing that we can say with certainty now and we say it formally is a steeple LSI system always has a frequency response and we introduce this new term now frequency response. Frequency refers to the frequency of the rotating complex number which refers to the frequency of the rotating complex number. So, we call this the frequency response of that LSI system actually please note we have to be very careful in our wording we have said a stable LSI system always has a frequency response we are not saying that only stable systems have a frequency response and that is a very subtle point which we do not want to get into it this point in time, but we shall dwell on it a little later as we proceed in the course. For the time being let us look at you know systems which have a frequency response and definitely stable LSI systems are such now you know the term frequency response is meaningful only if you have a complex exponential giving you a complex exponential of the same frequency otherwise that term has no meaning. So, you know it is very clear that we have a very interesting situation here and we will now give it an entirely different interpretation we have a situation where this quantity summation n going from minus to plus infinity h n e raised the power of minus j omega n converges and we can reinterpret this we can interpret this as summation n going from minus to plus infinity h n times e raised the power j omega n complex conjugate. Now, we are reinterpreting it in this manner because we want to draw a parallel we want to give an entirely different interpretation what is this number h omega this number h omega is the number by which that phasor is multiplied when it goes into that LSI stable LSI system. Now, you see when a force acts on a system the system response by the component of the response in the direction of the force for example, if you have an elastic body it may behave differently to deformations in different directions. So, when you have a force which tries to deform an elastic body and if you know how the elastic body responds to deformations in different perpendicular directions how would you deal with the situation you would resolve the force into the different directions along which you know the body's response and then calculate the response individually for each of these directions and what do you do you multiply the component of that force in the particular direction by the component of the response in that direction right. So, force component that direction multiplied by the response component in that direction that is how you get net in that direction you consider a net in all such directions and you add this is an informal way of explaining how you deal with resolution of forces resolution of agents the same thing is true of any now you must think in terms of agents here. So, here you may think of the rotating phasor as an agent and how do you calculate the response how do you calculate how do you project how do you find the component of a response in a particular direction you take a unit vector in that direction and you take a dot product of the response with a unit vector in that direction how do you take a dot product you take a dot product by writing each of those vectors in a standard orthogonal axis set and multiply component by component. So, let us take for example, two vectors in two dimensional space suppose you had this vector v in the two dimensional space span by this paper and you wish to find a component of this vector v in the direction of the unit vector u you know what it is the component along u is simply the dot product of v and u multiplied by u and how would you find the dot product if you happen to represent this vector u let us assume that your perpendicular axis are as follows this is the axis x1 cap and here you have the axis x2 cap perpendicular to it perpendicular axis I am showing you vectors along the perpendicular axis then you could express v as v1 x1 cap plus v2 x2 cap and u also as u1 x1 cap plus u2 x2 cap and then you have the dot product of v and u is v1 u1 plus v2 u2. So, how do you calculate a dot product you multiply the corresponding components of the vectors in each perpendicular direction and then add up all such products. In the next lecture we wish to do exactly the same, but generalizing this idea through context of systems. So, you wish to interpret the quantity h omega that we have just arrived at in terms of a dot product and we wish to then carry on that interpretation further to lead us to an interpretation of how LSI systems behave and how one can decompose the behavior of LSI systems into responses along different angular frequencies omega and then come to a conclusion about the very operation of convolution itself in a different domain. Thank you.