 Hello everyone, welcome to this session. I am Dipali Vadkar working as an assistant professor at WIT Solapur. In this session, we are going to discuss the classification of the system part 2. At the end of this video lecture, student will be able to classify causal and non-causal system, then student will be able to classify linear and non-linear systems. These are the contents. The classification of the system, causal and non-causal system, linear and non-linear system. So, let us see the causal and non-causal system. The system is said to be causal if its output depend upon the present and past inputs and does not depend upon the future input. So, these type of systems are called as causal systems. So, for the present input, the output is depend upon the present value of the input as well as the past value of the input, but it does not depend upon the future value of the input. So, this is the one of the example of the causal system. Non-causal system, the output depends upon future inputs also. So, this is the example of the non-causal system. Here, the output of this signal or system depends upon the present value and it also depends upon the future value of the input system. So, the output of this system depends upon the future value of the input signal. So, such systems are called as non-causal systems. So, let us see the example of causal system. Here y of t is equal to 2 into x of t plus 3 into x of t minus 3. Now, substituting value of t here in this equation and we will see the output of this equation. If we substitute value t is equal to 0 here, then the system output is y of 0 is equal to 2 into x of 0 plus 3 into x of 0 minus 3 means it is minus 3. So, here the output y of 0 is depend upon the present value of the input as well as past value of the input in this case. Now, for t is equal to 1, the system output y of 1 is equal to 2 into x of 1 plus 3 into x of minus 2. So, here the 1 minus 3 it is x of minus 2. So, in this case also the output for t is equal to 1 is depends on the present value of the input as well as here the past value of the input. Now, we will see the third case. If we put t is equal to minus 1, the system output is y of minus 1 is equal to 2 into x of minus 1 plus 3 into x of minus 1 minus 3 which is equal to minus 4. So, here again the output is depend upon the present value and past value of the input. So, minus 1 is the present value and minus 4 is the past value of the input signal. So, in the all cases for t is equal to 0, t is equal to 1 or t is equal to minus 1, the output of this system is depend upon the present value as well as the past value only. So, we can say that this system is a causal system. Now, let us see the second example. This is the discrete time system. So, here y of n is equal to x of n plus x of n minus 2. Now, substitute all the value of n here, output n is equal to 0, then the system output is y of 0 is equal to x of 0 plus x of minus 2. So, here the output is depend upon the present value of the input and past value of the input. Now, put n is equal to 1, then the system output is y of 1 is equal to x of 1 plus x of 1 minus 2 minus 1. Then here again output is depend upon the present value of the input and past value of the input. Put n is equal to minus 1, the system output is y of minus 1 is equal to x of minus 1 plus x of minus 1 minus 2, it is minus 3. So, here we can observe that the output is depend upon the present value of the input as well as past value of the input. So, in all cases for all values of n, the output is depend upon the present and past values of the input only. So, we can say that this system is also causal system. Now, next is a non-causal system. Let us see the one of the example. Here y of t is equal to 2 into x of t plus 3 into x of t minus 3 plus 6 into x of t plus 3. So, substitute all the values of t here, first put t is equal to 0, the system output is y of 0 is equal to 2 into x of 0 plus 3 into x of 0 minus 3 minus 3 plus 6 into x of 0 plus 3 x of 3. So, here we can observe that for the present value of the output, output is depend upon the present value of the input, past value of the input as well as future value of the input. Now, put the value of t is equal to 1, then the observe the output y of 1 is equal to 2 into x of 1 plus 3 into x of 1 minus 3 minus 2 plus 6 into x of 1 plus 3 x of 4. So, here also the output is depend upon the present value of the input, past value of the input and future value of the input. Put t is equal to minus 1 and observe the output y of minus 1 is equal to 2 x of minus 1 plus 3 into x of minus 1 minus 3 minus 4 plus 6 into x of 2. So, here also the output is depend upon the present value, past value and future value of the input. So, we can say that this system is a non-causal system because the output is depend upon the future value also. So, we can say that this system is a non-causal system. So, let us see the significance of causal and non-causal system. In case of causal system, it does not include the future input samples. So, causal system is its output is depends on present as well as past value of the input signal. So, such types of system are practically realizable. Now, in case of non-causal system, this system is depends on the future value. So, practically it is not realizable, but if the signals are stored in the memory and at later time, they are used by the system, then such signals are treated as advanced or future signals. Now, the question is find whether the following signal is a causal or non-causal? y of t is equal to x of t plus x of t plus 2. So, pause the video for a while and think whether the system is a causal or non-causal. Yes, it is a non-causal because here the output is depend upon present value as well as the future value of the input. So, we can say that this system is a non-causal system. That is a linear system. The system is a linear if the superposition holds means if it obeys the principle of superposition, then that system is called as a linear system. So, here the input, arbitrary input for this system is x 1 of t and output for that y 1 of t. The another system, the arbitrary input is x 2 of t and its output is y 2 of t. Then if x of t is equal to a 1 into x 1 of t plus a 2 into x 2 of t and y of t is equal to a 1 into x y 1 of t plus a 2 into y 2 of t, then we can say that that system is a linear system. So, if it obeys the superposition principle, then in that case this system is called as a linear system. Here the a 1, a 2 are nothing but constant. Let us see the example of linear system. Here y of t is equal to x of t by 2. So, let x 1 of t is input, then y 1 of t is equal to x 1 of t by 2, say it as equation 1. Similarly, x 2 of t is input, then output y 2 of t is equal to x 2 of t by 2, say this equation 2. For the system to be linear, the response this a into x 1 of t plus b into x 2 of t should be a into y 1 of t plus b into y 2 of t. So, that is it should obeys the property of superposition. Let us see, a into x 1 of t plus b into x 2 of t, say it is equal to m of t. It is the new input which causes the output n of t. Then, so the n of t we can say that it is equal to m of t by 2, which is equal to a into x 1 of t by 2 plus b into x 2 of t by 2, say this equation 3. Substituting equation 1 and 2 in equation 3, we get that n of t is equal to a into y 1 of t plus b into y 2 of t, say this as a capital B. Now, if you observe the equation a and equation b, we can conclude that this system possesses the property of superposition. So, we can say that this system is a linear system. Now, let us see the non-linear system. In the non-linear system, if it does not obeys the superposition principle, then that system is called as a non-linear systems. So, let us see the one of the example of non-linear system. y of t is equal to x square of t. Now, here we can verify whether this system is non-linear or not. So, y 1 of t is equal to transform of x 1 of t, which is equal to x 1 square of t. y 2 of t is transform of x 2 of t, which is equal to x 2 square of t. Then, transform of a 1 x 1 of t plus a 2 x 2 of t, which is equal to a 1 x 1 of t plus a 2 x 2 of t, its square. Now, which is not equal to a 1 y 1 of t plus a 2 y 2 of t. So, a 1 y 1 of t means a 1 x 1 square t plus a 2 x 2 square t. But this equation is a 1 square into x 1 square t plus 2 a 1 a 2 x 1 x 2 t plus a 2 square into x 2 square of t, which is not equal to this. It does not obey the superposition principle. So, we can say that this system is a non-linear system. So, these are the references. Thank you.