 need to answer whether there is something that we should impose upon the system to make this summation convergence. One thing is clear, if this summation converges, so let us see, if this summation converges then we have a very interesting, in fact we have done what we wanted to. When we given a complex exponential of frequency omega, angular frequency of normalized angular frequency omega to the system, what we get out is the same complex exponential but multiplied by a complex constant. There is no other change, that means each complex exponential is dealt with in a decoupled way. So now if I have a sum of complex exponentials, of course it is a minor, it is a very simple thing to see that if I, I can multiply both sides by any complex number here. So I can multiply, you know let me in fact rewrite this. So if I take a times e raised to the power j omega n and give it to the same LSI system, I would of course get a h omega e raised to the power j omega n, a is any complex number, provided h omega converges. That is the million dollar question, when would it converge? But at least you know it is soothing to see that provided we have convergence, we have got what we wanted. If I have a linear combination of complex exponentials with different angular frequencies, what is going to emerge is the output to each of these complex exponentials treated independently and the output to each of these complex exponentials is going to be the same complex exponential multiplied by an appropriate constant which depends only on the angular frequency. Of course it depends on the impulse response, but since the system is the same, the response is the same for that angular frequency. So we have got where we wanted, with the only catch that we do not know when this would converge, can we guarantee that this summation converges all the time. Now to check convergence, let us explore the expression itself. In fact, convergence ultimately has to do with the magnitude, the phase is irrelevant. So let us look at the magnitude of h omega. The magnitude of h omega is obviously the magnitude, h k e raised to the power minus j omega k. And this is obviously less than or equal to the sum of the magnitude. That is you see the modulus of a plus b is always less than or equal to the modulus of a plus the modulus of b. And you can keep extending this to an infinite summation. This is less than equal to the modulus of h k times the modulus of e raised to the power minus j omega k. But of course the modulus of e raised to the power minus j omega k is obviously 1 and therefore the modulus of h omega is less than equal to summation k going from minus to plus infinity mod h k. So we have what we clearly see is a sufficient condition. The sufficient condition is that this is finite. So it means if the impulse response sequence is absolutely summable. Absolutely summable means the sum of the absolute value of the samples is finite. If the impulse response is absolutely summable then we shall make a remark. Then the output to a complex exponential of phasor converges any angular frequency. You see there are please note that this condition is sufficient. The way we proved it we have only proved sufficiency and in fact it is indeed just sufficient. The fact that it is only sufficient is a very subtle point. We will understand that better later. But we need to appreciate the meaning of this condition a little better. Mathematically of course we see it. It is an absolute sum. It is a sum of absolute values. Yes, there is a question. Yes. So the question is in this page here how did I conclude? No, no what I am saying is if I want I would have shown is that mod H omega is less than or equal to this absolute sum here. So I am saying a sufficient condition for this to converge is that this is finite. So this is a sufficient condition. I am not saying that this is necessarily finite. But if this is finite then this converges. That is what I am saying. So I was saying we need to now answer what is the physical interpretation. What do we mean by this being absolutely summative? And in fact we will get a hint if we only look at the general convolution expression once again. We shall do that and we shall carry on the discussion in this and the next lecture. So let us take that LSI system with a general input. We have Xn we have the same impulse response Hn we are asking what was the fact that and we of course we have been told that this is true given. So of course we know what Yn is. Of course Yn is summation k from minus to plus infinity Xk Hn minus k. But now you will agree with me that I can also write this term as Hk Xn minus k because convolution is commutative. So let us take the modulus of Yn. Modulus of Yn is the modulus of summation k going from minus to plus infinity Hk Xn minus k. And of course this is less than or equal to summation k from minus to plus infinity mod Hk mod Xn minus k as usual because mod A plus B is less than equal to mod A plus mod. Now let us look back at this. You see here suppose we manage to put a bound on this. Bound means we put a supremum on it. Supremum means we identify a finite non-negative number such that none of these magnitudes can be more than that number. In other words let us assume that mod Xn is less than or equal to some Mx where Mx is the quantity greater than or equal to 0 for all n. In other words the input is bounded by Mx. We say this is the terminology that we use. We say the input is bounded by Mx. Obviously the output is also bounded now. Mod Yn is clearly then less than or equal to Mx times summation k going from minus to plus infinity mod Hk. If this is finite the output is also bounded and in fact here too we have what is called a constructive proof not just an existential proof. So what we have shown is that if that condition absolute summability of the impulse response is satisfied then a bounded input results in a bounded output. That is a serious conclusion and not only have we concluded that we have also shown what the output bound is. That is what I mean by constructive. You can calculate the output bound or at least you can calculate one output bound from the input bound could be better. Now this leads us to one more property that may or may not be possible in the system in LSI systems and that is called the property of stability. In fact we define in the next lecture the idea of stability in terms of inputs and outputs being bounded. There are different notions of stability and we shall talk more about these notions and the connection to the impulse response in the next lecture.