 Hello everyone, welcome to this session. I am Deepali Vadkar working as an assistant professor at WIT-Solapur. In this session, we are going to discuss the classification of the system part 3. At the end of this video lecture, student will be able to classify time variant and time invariant systems as well as stable and unstable systems and invertible and non-invertible systems. Okay, so these are the contents. The classification of the systems will also discuss the time variant, time invariant system, then the stability of the systems and will discuss the invertible and non-invertible systems. So let us see which systems are called as a time variant and time invariant. The time variant systems are those systems in which the input and output characteristics of these systems vary with the time. Okay, so if the input varies with the time, then the output characteristics also varies with that time. Then in that case, that system is called as a time variant systems. And if the input and output characteristics of the system does not vary with the time, there is no effect on this output characteristic of the system with respect to time change in the input signal, then that systems are called as a time invariant system. Okay, so time invariant systems are those systems in which the input-output characteristic does not changes with respect to time. Okay, so the condition for time invariant system is y of t, capital T which is equal to y of t minus t. Okay, and the condition for time variant system is y of t, capital T is not equal to y of t minus t. Okay, so let us see what is this y of t, comma t, it is nothing but transform of x of t minus t, means change in a input. Okay, and y of t minus t, it is the output change. Okay, so if the input change is equal to the output change, then that system is a invariant system. Okay, and if the change in a input is does not equal to the change in the output, then that system is called as a time variant systems. Okay, similarly in case of discrete time signals, the condition for time invariant system is y of n, comma k which is equal to y of n minus k. And time variant system y of n, comma k which is not equal to y of n minus k. Okay, so we know that y of n, comma k is equal to transform of x of n minus k. Okay, so which is the change in a input and y of n minus k it is the output change. Okay, so this is the condition for time invariant and time variant in case of discrete time systems. Let us see the example which is related to time variant systems, y of t is equal to t into x of t. Okay, so in this case y of t is nothing but what? The transform of x of t means input. Okay, so transform of x of t input signal. Okay, so this is we can say it as a t into x of t. Now the output due to delayed input is, let us see the change in output due to delayed input. Okay, so here the y of t, comma capital T which is equal to transform of x of t minus t. So this if the input is delayed by this capital T then the output is equal to t into x of t minus capital T. Okay, now if the output is delayed by the capital T then we get y of t minus t which is equal to t minus capital T into x of t minus capital T. So here if you compare these two. Okay, so y of t minus t and if you compare y of t, comma t. So in both cases what happens? Here the y of t, comma t is a t into x of t minus t and y of t minus t is equal to t minus t into x of t minus capital T. Okay, so if you observe this equation these are not equal. Okay, so we can say that this system is a time variant system. Now the time invariant system, the system itself does not changes with the time. Okay, so that systems are called as a time invariant. Okay, so suppose the x of t input to the systems and the output from the system is a y of t. Then if we apply the delay of capital T seconds then the input to the system is x of t minus t. Okay, so this will be the input signal and the output from the system for the same system it is y of t minus capital T. This is the output from the system. Okay, so these type of systems are called as a time invariant systems. Now let us see the example y of t is equal to e raise to x of t. Okay, so in this case we can say that y of t is equal to transform of x of t which is equal to e raise to x of t. Okay, now the output due to delayed input. Okay, so we will see the output due to delayed input. It is y of t comma t is equal to transform of x of t minus t which is equal to e raise to x of t minus t. Okay, so put t is equal to here t minus t. Okay, so if we substitute here we get this equation. Okay, now if the output is delayed by t. Okay, so delay output by t then what will happen y of t minus t is equal to e raise to x of t minus t. Okay, now if we observe these two equations this y of t minus t and y of t comma t. If we observe these two equations then from this we can say that y of t comma t is equal to y of t minus capital T. Okay, so this satisfies the condition for time invariant system. So we can say that this system is a time invariant. Now the linear time variant systems means what? If this system is both linear and time variant. Okay, so if the satisfy the condition for linearity as well as time variant then it is called as time variant system LTV systems. Okay, if the system is both linear as well as time invariant then that system is called as a linear time invariant system. That is LTI systems. Okay, now the next classification is a stable and unstable system. In case of stable system system is said to be stable only when the output is bonded for bonded input. Okay, and unstable for bonded input if the output is unbounded in that system then it is said to be unstable. Okay, so let us see the example here for a bounded input this system produces the output which is bounded. Okay, so if for a bounded signal the amplitude is of finite. Okay, so here in this case the amplitude is of finite. So we can say this is the bounded output. Okay, so such type of systems are called as a stable systems and unstable system means what? Here for a bonded input if the output is unbounded. Okay, so here the amplitude is infinite for infinite time. Okay, so this amplitude is increasing. Okay, and it is the infinite for infinite time it is not bonded. So in this case the for a bonded input this system produces the unbounded output such type. Okay, so such systems are called as unstable systems. Then the invertible and non invertible systems the system is called as a invertible if it produces a distinct output signals for a distinct input signal. Okay, in case of invertible system if the input X of t is a recovered at the output of the system. Okay, then such type of system are called as a invertible system. Okay, so here X of t is input given to the first system its output is a y of t. If we apply y of t to the second systems then output is X of t. So such type of systems are called as a invertible systems. Okay, input is recovered at the output of the systems. Okay, so these systems are invertible systems. So let us see the example. Here two systems are cascaded. Okay, input to the first system is X of t. Okay, and output from the first system is a R into X of t. Now this y of t is a input applied to the second system. Okay, the second system its output is which is equal to 1 upon R into y of t but y of t value is a R into X of t. Okay, so if we substitute this value here then we get X of t, R get cancelled and X of t. Okay, so output from the second system it is a X of t which is nothing but input. Okay, so here the input is recovered at the output. Okay, so such type of systems are nothing but invertible systems. Non-invertible systems, let us see the non-invertible system one of the example. Here the y of t is equal to cos into X of t. Okay, so for the input signal separated by 2 pi. Okay, the system gives the same output. So in this case we cannot say that the input is recovered at the output. Okay, so such type of systems are called as a non-invertible system. Okay, so from this we have seen the invertible and non-invertible system. Okay, so invertible system means the input is recovered at the output of the system. Such systems are invertible systems and non-invertible system means here if the input is not recovered at the output then such systems are non-invertible systems. These are the references. Thank you.