 Welcome to the 11th session in the second module of the course signals and systems. We continue now with the recrysin dot product or the inner product as we called it and move on to explain how we could generalize that to the context of continuous variable signals. Now what we do for discrete variable signals? We treated each value of the independent variable for each integer as a different dimension. So, let us make that clear, let us make that formal. So, discrete signals as vectors xn as a vector means every n a different dimension and x of n is the component at the nth dimension. And in fact, now we can take one more step. You see let us just verify what happens if you describe two perpendicular components here in this language. Now what are the perpendicular components? That is very interesting. The perpendicular components, perpendicular orthogonal as we call them are essentially the unit discrete impulses. So, for example, if I have a 1 at the location capital N and 0 everywhere else, this is essentially the sequence delta n minus n where delta denotes the unit discrete impulse signals or unit discrete impulse sequence as we would like to call it for every different capital N, a different unit vector. And in fact, when we write down xn in terms of its decomposition in terms of these unit vectors, in other words, we could write down x of n as summation capital N going from minus infinity to plus infinity x capital N for each specific capital N, delta n minus capital N. We are saying the same thing, but in the language of vectors. What we are saying is xn capital N I mean is the component along dimension capital N and delta n minus n is a sequence which also forms a unit vector along dimension n. So, you know it is just like saying if you look at it for two dimensional vectors, it is just like saying this v when decomposed into its components which I am now showing in green being v1 and v2 and v1 again being a multiple of the corresponding unit vector which I am showing in blue. So, this is the unit vector here which I call u1 cap and there is a unit vector u2 cap and v1 is essentially the magnitude of v1 times u1 cap and v2 is the magnitude of v2 times u2 cap and I have v is essentially v1 plus v2 and that is essentially magnitude of v1 times u1 cap plus the magnitude of v2 times u2 cap and this is essentially very similar. What we have written here is essentially a two dimensional or a smaller dimensional version of this. So, what we wrote in two dimensions we are now writing in an infinity of dimensions a countably infinity of dimensions. Now, you know infinities have different grades, the smallest infinity is the infinity of the integers it is called the countable infinity. So, any play any kind of infinity where you can put all the elements of that infinity or all the objects of that infinity in one to one correspondence with the set of integers is called the countable infinity. The next infinity is what you call the infinity of reals of course, there are you know there are grades. So, there is the 0th grade of infinity which is the infinity of the natural numbers of the infinity of the integers then there is the infinity of the reals and there are higher infinities as well. Let us not get into all that, but at least let us appreciate that for a countable infinity we have now understood what it means to have a countably infinite dimensional vector and we have now generalized the notion of components of that vector unit vectors for that for those infinitely countably infinite dimensions. The unit vectors are essentially are familiar unit discrete impulses and you could express any sequence as a linear combination of that countably infinite set of unit vectors using the points on that sequence, the points on that sequence x capital N for every integer N are essentially the components or you know the contributions of each of these unit vectors. Now, when we the reason for emphasizing this perspective of a sequence as a vector is that it is very easy now to generalize to a continuous variable context. In fact, the generalization is almost straight forward, all that we need to do is to recognize that when you go to a continuous variable context instead of a discrete variable N you have a continuous variable T that continuous variable T can take on an uncountably infinite set of values all the reals and for each real number T you need to associate a different dimension. In fact, if we go back and look at what we call the shifting property of the impulse or essentially what we call the combination of impulses coming together to form an arbitrary signal x t let us write that expression down once again. You know that a large class of signal x of t can be written in the following way integral from minus to plus infinity x lambda delta t minus lambda d lambda. So what is happened here? The integral has replaced the summation x lambda is the component along the direction lambda or dimension lambda. And in fact, now you can all think of this continuous impulse at lambda as a unit vector so to speak along the dimension lambda. So what are we saying in this expression? We are saying that one can put together all these different continuum of dimensions indexed by lambda with the corresponding unit vectors which are the unit impulses but now in continuous time. Now you remember that unit impulse in continuous time require a little bit of work to explain. So in fact that is why we went from discrete to continuous here. It is easy to understand the connection between discrete variable signals and vectors first and then move to the continuous context. But once we understand the connection between discrete signals and vectors moving to a continuous context as you can see from here in this expression is not difficult at all. All that you are doing is to sum instead of summing over the different dimensions you are now integrating over the different dimensions. It is a replacement of the summation. So in fact now you can also define a dot product between two signals. We can talk of the dot product or inner product. Let us be formal now of let us say x1 t and x2 t. Now you know all this while we are assuming real signals. The dot product or the inner product which we will denote like this is essentially the integral of x1 t times x2 t over all t t running from minus to plus infinity. In fact we can take an example. We can find the dot product of these two signals. Let us define x1 t to be 1 between 0 and 1 and let us define x2 t to be a ramp between 0 and 1. In fact we could even take that ramp to infinity. So x2 t is equal to t times ut so to speak and we can calculate the dot product x1 x2 which is very easy. It is essentially the integral from 0 to 1 t dt and that is an easy integral to calculate t squared by 2 from 0 to 1 and that is just half. This is the dot product of the two signals x1 t and x2 t. You have now understood how to calculate dot products of continuous time signals. Now we are going to come to business. How am I going to use these dot products or inner products to get what I want? Namely to decompose signals into their sinusoidal components and then why am I talking about what I call perpendicular components? How can I think of sinusoids as perpendicular components? We will come to that in the next section. Thank you.