 Now let us interpret the so called frequency response that we wrote the last time in the language of the dot product. So we have h omega that we wrote the last time is summation k going from minus to plus infinity h k e raised to power minus j omega k and it is obvious that this is the dot product of the sequence h with the sequence e raised to the power j omega m. Now this dot denotes the independent variable in the sequence you know so what we are when we write a dot like this what we mean is that we are treating the whole sequence as an object we are not taking an individual sample you know so the dot means otherwise if we write n we could write n if we are careful but when we write n the tendency is to think of the nth sample only and here we are not talking about a specific sample we are talking about the whole sequence as an object that is very important and that is why we put a dot there more than the question of notation it is the philosophy behind this that matters that is why we have to be careful with notation it is not so important that we should be very fussy about notation all the while but the thought behind the reasoning should not be lost so when we take a dot product it is not enough to look only at a part of the sequence and dot product involves the whole sequence that is what is being emphasized here anyway. Now you see there is a very interesting interpretation that we have just given this whole concept of you know the whole concept of frequency response what we have said is that if you have this LSI system S with an input e raised to power j omega n so you have given a complex exponential is the input you get h omega e raised to power j omega n coming out and h omega is essentially the inner product the dot product or the inner product of h n and e raised to power j omega n now you already know what an inner product does in conventional small dimensional physics or engineering when you take the dot product I told you of one force in a certain unit vector in a direction chosen so if you have chosen a certain direction and if you take the dot product of a force vector with a unit vector in that direction what you have done is to resolve the force in that particular direction to find the component of the force that acts in that direction. Now what we are saying here is that h omega is like the component of the impulse response that acts in the direction of e raised to power j omega n in fact it is an exact interpretation it is not just very vague or you know I mean approximate it is exactly what we are saying we are saying that if you give e raised to power j omega n to a LSI system and if h omega converges that means if the sum h k raised to power j omega minus e raised to power minus j omega k over all k converges then this quantity h omega is like the dot product of the impulse response in the direction of that complex exponential. The dot product is a number it is a complex number and the physical interpretation of that complex number is that the magnitude of that complex number multiplies the amplitude of the complex exponential and the angle of the complex number adds to the phase of that exponential.