 We now come to one more such independent property that we need to ask in the context of systems and that is relevant when we are talking about the independent variable of time, not so much when we are talking about the independent variable of space or any other kind of independent variable. But when the independent variable is time, then we do have a notion of fixed directionality. You cannot move backward in time, you can only move forward. If you are dealing real time, then you need to move only forward. If you are dealing offline, then you can of course use samples from the future too. So we ask whether a system is causal or not causal, that is another independent property. Now we say a system is causal if the following happens. You perform two experiments on the system with two inputs. So you have a system S, not LSI, I am not saying it is LSI. So you give it an input x1n and you give it an input x2n and record the two corresponding outputs. The only catch is that x1 and x2 are identical up to some n equal to n0. X1n is equal to x2n for all, now this please remember I will not keep writing this again in future for all n less than equal to n0 for some integer n0. Causality means causality means and is meant by y1n is also equal to y2n for all n less than equal to n0 and for all such x1, x2 and n0. That means take any such pair of inputs which are identical up to some n equal to n0 and apply them to the system in two different experiments. Study the output, the output is identical up to that point n0 if the system is causal and vice versa. The system is causal only if these outputs are identical for all n less than equal to n0 and this happens for any such choice of inputs x1, x2 and any point n0. We need to spend a minute in reflecting on what this means. What this means is if I have two inputs which are identical in all respects up to a point in time, a causal system does not show any difference in its output up to that point in time. Needless to say there could be differences afterwards if there are differences in the input. And other way of understanding this is the system never looks into the future. The system has no idea whether the inputs could be different in future and therefore in the benefit of doubt it remains identical in its output up to the point where the inputs are identical. Now of course the word causal suggests that causal refers to cause and effect. So an identical cause produces an identical effect. There is a relationship of you know you can talk about cause and effect only if there is an ordering. If there is a one after the other relationship in time then you can talk about cause and effect otherwise cause and effect is not very well understood. If an effect comes before the cause then it is not an effect at all. So that is why we say a system is causal if it follows the principle of cause and effect. Now again causality is independent of linearity, shift in variance or stability and that is therefore a fifth possible property that a system could have or not have is that right. Now we can see that if you want dependence only on the past that is what we are trying to say in effect. You know if you look at the convolution expression y n, if you look at the convolution expression and if we wish that y of n at any point n have nothing to do with future samples that means nothing to do with negative k is here. You see when would y of n involve future samples when minus k is positive or k is negative and that means you know for all negative k h k needs to be 0. It is very easy to see that if h k is equal to 0 for all negative k then y of n depends only on x n and x n minus k for positive k that means all past samples so to speak. I leave it to you as an exercise to prove this more formally I have of course given you an informal argument, but I leave it to you to prove formally that means show that it is necessary and sufficient prove formally and LSI system is causing if and only if its impulse response obeys h n equal to 0 for all n less than 0. So by proving it formally I mean that you must show it is necessary and sufficient that means you must first assume this condition holds and show that it is sufficient for two identical inputs to produce an identical output up to that point in time and then also take the counter part of it namely if I have two inputs if I have any set of inputs which are identical up to point in time and if I want the outputs to be identical that cannot happen unless all the impulse response samples at negative indices are 0. So I leave it to you as an exercise to prove this formally anyway we have now identified five independent properties of systems let us list them additivity homogeneity or scaling shift invariance stability causality needless to say for a system that is linear and shift invariant looking at the impulse response to tell us everything and therefore it tells us whether the system is stable and whether it is causing and we have also identified how we can do so. Look at the impulse response look at all the negative located samples if they are all 0 the system is causing look at the impulse response take look at its absolute sum of the absolute sum converges then the system is stable and finally we take one more property of systems namely the property of memory not quite independent not quite independent that is why I did not say 6 property the property of memory. Now we say a system has no memory if and only if yn has to do only with xn and no other xn-k for k0 equal to 0 so it is a point by point relationship that is called a system without memory an example is yn is mod xn or yn is equal to xn or for that matter even xn plus 8 if you like or yn is xn squared these are all systems without a memory and of course it is very easy to give examples of systems with memory just involve some other term like xn-1 or xn plus 1 and there you have memory with you as I said memory is not entirely independent of the other properties in fact you know if a system is memory less it is automatically causal they are not entirely independent the systems without memory or memory less systems are exam are one class of causal systems that is of course a bit of a relationship there.