 So, welcome to the 29th session on signals and systems. In this session, we answer a question related to the stability of a linear shift invariant system by looking at its impulse response. So, let us frame the question first. Given a system to be linear and shift invariant, investigate its stability by looking at impulse response and do this for continuous and discrete variable systems. Now, let us once again put down the input-output relationship. For the continuous case, we have two relationships. So, we can use either of the relationships, either using computativity, whichever is convenient can be used. Remember that. So, of course, let me write down the discrete case also, I write it on the same page to make it easier. So, I will write in a different color. Discrete case. Now, you see, what does stability mean? Let us look at the definition of stability first. So, stability means a bounded input gives a bounded output. So, let us look at the discrete case first. As usual, that is easier to handle. This expression is easier and I am ultimately interested in mod yn to study the bound of the output and therefore, I would need to look at mod yn which is summation k going from minus plus infinity hk xn minus k all with an absolute value taken. But here, we invoke the principle mod of a plus b in general is less than or equal to mod a plus mod b and carry this to the summation which means mod yn is less than equal to summation k going from minus to plus infinity mod hk xn minus k and this is the same as summation k going from minus to plus infinity mod hk mod xn minus k and given the input is bounded mod xn minus k is less than equal to mx for some mx strictly less than plus infinity and greater than equal to 0. Thus, in fact, let us go back to what we wrote previous that will be easier. You see, if you look at it, this quantity is less than or equal to mx and mx is a non-negative finite quantity, finite is important and therefore, we clearly have mod yn is less than equal to mx summation k going from minus to plus infinity mod hk. So, it is very clear that if summation k going from minus to plus infinity mod hk is finite, then mod yn is finite. It is very clear. So, here we are. We now have a sufficient condition once again. If I ensure that the absolute sum as we call it, let us look at that quantity. It is called the absolute sum. You know the word absolute sum of a sequence is now being introduced here. Absolute sum means taking the absolute value of each sample and then adding up all these absolute values. So, the absolute sum of the impulse response is finite. It is very clear that yn needs to be bounded and we can also find the bound. So, in fact, let us get back here. If hn is absolutely summable that is summation k going from minus to plus infinity mod hk is strictly less than infinity and maybe you can call it equal to mh, then mod yn must be less than or equal to mx times mh for all n and hence bounded. So, this is a sufficient condition. But we now need to ask if this question, if this condition is necessary and that we shall do in the next discussion. Thank you.