Understanding Invertible and Non-Invertible Systems

Hello, welcome to GATE Academy Plus. Subject we are doing signals and systems. In module 1, the topic is our properties of systems. In this, till now we have completed causal, non-causal, time variant, invariant, static, dynamic and linear and non-linear systems. Now, the next properties in which our topic for today is, we have to consider the That is first if we see invertible and non-invertible systems, okay. So, till now we have seen that in the last class we have seen for linear non-linear systems. So, all the problems we had taken, most of them, we had taken them for continuous time. But you don't have to do anything separately. You don't have to understand that we have learned to check the properties of continuous time system. we do not know for discrete time. You just have to replace T with N. The whole procedure will remain exactly the same. Similarly, you can take the procedure, as you have checked the properties for continuous time systems, you can check for discrete time systems. So now, as we have taken the next property, if we have to check which system is invertible and which is non-invertible. So, if we define invertible systems, then the system is said to be invertible. If the system produces distinct output for distinct input, if distinct input produces distinct outputs. Means, the output for any particular input should be unique. You can say this in other words, that our output should not be same for any two inputs. output is there, so if you can analyze the input by looking at the output, if the output of the system is this and if we can calculate how the input will be, then what do you call the system? Invertible system. Means you can write more in this, if output, sorry if input can be determined by observing the output. If we can identify our input by looking at the output, then you call such systems as invertible system, in which distinct output is obtained for distinct input, we can analyze the input by looking at the output. You can also say that the system which is system follows one to one mapping. Meaning, one to one mapping, as you read in set theory, that if we assume that we have one set of values for input and the other one set of values for output, then for one value of each input set, there should be a value of a particular output set. Meaning, it should not happen that for more than one input, the particular value of our output set is being selected. So, what do you call the system that it is following? It is following one to one mapping. For example, suppose, Example, we have taken suppose we have a system for which the output input relation is given as yt is equal to 4 times x of t. You can write one more point in this that if any system is invertible then for that it is possible to design an inverse system such that if the inverse system is cascaded with the original system then the output of the overall system will be the input itself. So, for that system, if you have an invertible system, then for this system, it will be possible to design an inverse system from which if we apply x of t from here and if we cascade the inverse system with it, then the output here will be the value of the input. That means you can say that the overall transfer function of this overall transfer function is the value of the system's transfer function and the inverse of it. Transfer function multiplication, if two blocks are in series then overall transfer function is multiplied. So, the product of these two should always be unity, so that x of t into overall transfer function 1, so here again we get x of t in output. So, we call such system as invertible system, for which distinct inputs produce distinct output, you can tell the input by looking at the output and for that system you can design an inverse system. If we cascade it with the original system, then the output of the overall system is equal to the input. You can say that the value of the transfer function or the value of the gain is equal to the value of the unity. For example, if we take an example y of t is equal to 4 times x of t. So, if you want to tell the input by looking at the output, then you can tell x of t is equal to 1 by 4 times y of t. One to one mapping if you want you can check it. Suppose for values of x of t, if you check values of y of t that is equal to 4 times x of t, apply distinct inputs. Suppose x of t is 0 then yt is equal to 0, 4 times 0 that is 0, x of t is minus 1 then y of t is equal to minus 4, x of t is plus 1 then y of t is equals to plus 4. If you check any value here, then you will not get a single output for two different inputs. This means that our system is an invertible system. But if we take a system like this, suppose y of t is equal to x square of t means what is the property of the system? Whatever input we are applying, it is giving us in the output by making it square. So, if you check for this, whether there is one to one mapping between x of t and y of t, then you will see from here that if we take y of t, that is x square of t, then for the value of x of t, we will get 0 square of 0, for plus 1, we will get 1 square of 1, What value will you get for minus 1? You will get 1. For plus 2, you will get 4 value and for minus 2, what value will you get? 4. So, what are we seeing? For more than 1 inputs, our output is common, it is same. So, how will our system be? It will be a non-invertible system. Now, if we talk about inverse system, here actually you have not seen transform. But suppose if we had a system like this, y of n is equal to summation. k is equals to minus infinity to infinity x of n. This actually we call this system as accumulator. How is its output getting? From minus infinity to infinity, whatever inputs you have applied, or actually if we take minus infinity to n, this is the value of k, x of k minus infinity to n x of k. So, if we see the output for this system, then the output for this, if you see in the transformed domain, So, if Y of n for transform we will read these things further if we have given the name of this transform Y of z, okay. Or if we have X of n if we are considering the input, if our transform is X of n for input X of z then you know from accumulation property that summation k is equal to minus infinity to n X of k for this the value of transform that will be 1 upon 1 minus z inverse. into the previous transform x of z. So, if we have such a system, then for that system we can design an inverse system, for this system, which will be the inverse system of this. If we see its inverse system, then if we take such an output, where suppose we name the inverse system output as z of n, then z of n is equal to yn minus y of n minus 1. Means, where the output is, If we see from here, then the value which is the two consecutive outputs, means one is our n minus 1, this is a previous output and this is a present output. The difference of these will be if we define it as z of n. So again, if we assume that for x of z of n, if you are writing z transform z of z, then z of n is equal to z of n minus 1. So, if For example, if you have written the transform for y of n as y of z, then you will see the value of the transform of y of n minus 1 using the time shifting property. z inverse time y of z. The more time you do time shifting, the more power z is multiplied, which is the actual transform. So, if you see from here, then the overall transform of this, suppose we call it y dash of n, for inverse, then the value of y dash of z will be, 1 minus Z inverse times Y of Z. So, if you multiply these two, then this 1 upon 1 minus Z inverse and this 1 minus Z inverse will be cancelled. So, this system, we can call this system again, how is it? Invertible system. For this, we can design an inverse system, which if we cascade, then what value will we get? We will get 1. What is this system? It is an invertible system. But how is this system? It is a non-invertible system. So, you can check some values with this one to one mapping. For example, if we have a relation that y of t is equal to mod of xt. You can also check this. Take different values of x of t and according to that see the value of y of t that is magnitude of x of t. So, if you take 0 instead of x of t, then you will get 0. If you take minus 1, then you will get 1. If we take plus 1, what value will you get of yt? You will get 1. If we take plus j, what will be its magnitude? 1. And if we take minus j, what will be its magnitude? 1. So, what do we see now? There are 4 different inputs and what is the output for all of them? It is common, that is 1. So, how will this system be ours? Non-invertible system. Okay? So, it is not always necessary that you check these 1 to 1 mapping for the system. Time in, this is our invertible or non-invertible, you can tell that. So in such questions, there we use some test signals. We normally use some test signals when you are getting the product of two functions in the output and if you want to check the time invertibility for that system, whether the system is invertible or not, then we will use some test signals there. For this, normally you can take either U or U and U and U and then DELT and or you can apply U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and 2 del of n can also be applied. So, how we will use these test signals for systems? Suppose you have any output input relationship given as such that y of t is equal to suppose given we have sin t times x of t. Now if you check with one to one mapping then you will get confused. So, in this if we replace x t with sin t, then we will get a result of Suppose we have applied the input, which test signal? Apply del t. So, for x of t is equal to del t. If we see this output, what will be the value? y of t is equal to sin t into del t. Now we know the product property of impulse function. If we product any signal x t with del t, then what is this value? Because this signal will only exist at 0, so what will we do? We put the value of t in the bracket as 0. From here, we put the value of t in x. x of 0 into del of t. You saw this in the previous lecture. If you have any doubt, you can see it once again. So, if you see from here, then what will be the output value of this? y of t is equal to, what will we put in place of t? So, this function will be 0. 0. So, what will be the value? sin 0 into del of t. And what is the value of sin 0? 0, so the overall result will be 0. Again if we apply another input for x of t is equal to suppose we have applied 4 del t, okay. Or if you apply minus del t, okay, then what will be the output for that? y of t is equal to sin t into 4 times del of t. Now if these two inputs are different, first we applied del t, now what are we applying? 4 del t. But still, from the product property, the value which will be 4 times sin 0 into del of t and again sin 0's value is 0, then overall what will be our result? 0. So, what are we seeing? For two different inputs, our system is producing the same output that is 0. So, what will you call such a system? It is a non-invertible system. If we take another system, suppose we have some system y of t. or take y of n if we take that is equal to x of n into x of n minus 1. Okay. You can use any test signal or you can use it in these type of questions. Let if we see for input x of t or x of n, then our output is this, we have not used any test signal, we are simply looking for xn. Or if you want, then okay, if we look at this for u of n. Okay. So, what will be the output for un? un times u of n minus 1 and if we see for input minus of un, if we had applied un input earlier, what are we applying? Minus u of n, so what will be the output? y of n is equals to minus of u of n multiplied with minus of u of n. n minus 1 and this is minus minus plus and again what value will you get? u n times u of n minus 1. This output and this output what you are seeing is same. Whereas, inputs are different. So, for two different inputs, our output is same. So, how will our system be? It will be non-invertible. Okay. Suppose, if we check one more, if we have any function output input relation, y of n is equals to suppose, x of n plus 2 for n greater than or equals to 0 and if it is equals to x of n for n less than minus 1. This is our system for which the output of yn is x of n plus 2, when? If n is greater than 0 and if n is less than minus 1, then in that case our output is equal to x of n. Now if you want to check the invertibility for this system, then if we see for xn is equal to del n, if you see, then what will be the value of output yn? del of n plus 2 for n greater than or equals to 0. Now, where does this del of n plus 2 exist? How do you see the existence? You put the thing in this bracket as 0 and its existence, if you see, then from here you get the value of n minus 2. That means where does this function exist? On minus 2. But for whom is it defined? For n greater than 0. So, what will be the value? 0. If you see from here also, if you write y of n is equal to del of n, then you than minus 1. So, this value is for n greater than 0, if we take 2 del of n, then the output is, what is its value? y of n is equals to 2 times del of n plus 2 for n greater than equals to 0. This function is existing there at n equals to minus 2. So, it is existing at minus 2 and if we are looking for n greater than 0, then what value will we get again? 0. So, what we are looking for? For two different inputs, what is our output again? Same. So, what will you say? This system of ours, what kind of system is this? Non-invertible system. Okay. So, in this way, you can check the invertibility for the system. You can check the invertibility by saying that if our system is 1, same output is produced for more inputs, that means that system will be non-invertible. And if distinct input is producing distinct output, then that system will be invertible. So, next classification is for systems, stable and unstable systems. See, about this, you deal with different approaches in different subjects. You must have read a lot to check the stability of the system. When we study signals in depth, when we talk about transform domain, when we calculate the transform, these things will be more clear to you. As you must have read in control system, in Laplace's terms. If we define stable and unstable systems, then the common criteria we use there is bounded input, bounded output, If any system fulfils this bounded input bounded output criteria which you call BIBO stability criteria in short then we assume that system to be stable. What does bounded input bounded output mean? If we apply bounded input in any system, bounded input means such an input whose value is finite at every instant of time. So, if we see at any instant then that input is finite value, then for such input, the output of the system should be bounded. Means, for, or you can write like this, the output of the system must be bounded for bounded value. That is, if the input we are applying, if the magnitude of that applied input is between 0 to infinity, okay, it is 0 or greater than 0, but what is the magnitude? If it is smaller than infinity for x of t in between 0 and infinity, if x of t is finite, then output must be, the output which should be ours, its magnitude should be, what should be the magnitude of output? finite means this value should also be less than infinity. Now if the output of a system is bounded for bounded input, then this criteria also tells us that the impulse response of this system will change with time for 0. Bounded input bounded output implies that impulse response Must 10 0 as time t tends to infinity. As time value increases, if any system is absolutely stable and it is following bounded input bounded output, it means its impulse response will decrease with time and become 0. You can understand it like this. What is the meaning of impulse response? Response of the system for impulse input If in any system What is impulse input? It is a system that exists momentarily for you So if in our system We leave a pulse After that if we remove the input If that system is stable Then for that the output should also be there Because this is for a fraction of second And the input is gone So for that what should be the output with time When we have removed the input Then it should be zero So, if you write bounded input bounded output in time domain, then from there we also get the result that the magnitude of impulse function, impulse response should be absolutely integrable. Means, if we integrate minus infinity to plus infinity magnitude of h of tau d tau, then what should be our value? Less than infinity should be there. The impulse response of the system should be absolutely integrable. If we are taking a continuous time system, then we are using absolutely integrable word. If we take a discrete time system, then what do we write? k is equal to minus infinity to infinity. The magnitude of h of k should be less than infinity. It means it should be absolutely summable. If we talk about all this in time domain, then we give mis-type statements. Because in time domain we have impulse response. When we talk about transfer function, then we talk about frequency domain and deal with transfer function. So till now you must have read, like in control also you must have read that where all the poles of the system should lie. Left half of the S plane. So there you must have seen that the total response of the system. There you write the response normally from Ct. So this Ct is made up of two parts, a transient response. and one CT steady state response. So, if any system is stable then in that case what you see is that the transient response limit t tends to infinity, c of t transient response it must be 0. Till the time the transient of the system does not vanish, till then it will not be able to reach its steady state and till then we cannot call that system stable. So, this transient response should be 0. And when will the transient response be zero? Like if you have Ct of this form, 4 plus e to the power minus 2t plus e to the power minus 3t. So, you know this, t tends to infinity. This is an independent term from t. Which means, this is the response. This is the steady state response. And this part here is our response. This is the transient response. So, if you see here, if there is a negative sign, then this term will be zero on time t tends to infinity, transient will vanish and your system will reach steady state. But if sign is not negative here, then what happens to this value on t tends to infinity? It becomes infinite. So in that case, the system becomes unstable. So you must have seen that the terms that come in the transient response in exponential, these are directly the values of poles. That is why you say that the poles of the system, where should these poles lie in the? left half of the S plane, till this part is not negative, till then the transient response will not be zero with time and then the system will not be stable. So, when we talk about Laplace, then we will deal with these things. Now, you can understand it like this, like if we take a criteria of bounded input and bounded output, suppose we have, suppose we This is a very deep container. Assume that it is very deep. And here we have a ball. Now if you are applying a force on this ball. The force you are applying from here means that your force is the input. Now how did we apply this input? If you applied this input Finite. Finite means. Even though you pushed it very hard, but if the magnitude of force is finite, then what will happen? This ball may come here for some time, then come back here, then come back here. Then gradually with time, its distance from the mean will decrease, and finally this ball will come back and stop at its place. It means it will regain its steady state. So what kind of system is this? Absolutely stable. Because the ball cannot go out of this container, if we apply force input till its magnitude is finite. Assume that you have a system like this and you have placed the ball here. Now if this height is not too much, then you can apply a particular value of force on it till it comes back to its steady state. You did it, it went up a little, then it came down, then it oscillated and finally it will stop here. But for a finite value of input, finite magnitude of force, the ball may go out of the container and never return to its steady state. So this system is stable for a particular input, and after that input, the behavior of the system is unstable. So what do you call such a system? Marginally stable or conditionally stable system. Conditionally stable means A critical value of force is that the ball will not go beyond the force of the same magnitude but if you apply more force than that, then the ball will go beyond the system and after that, its steady state will never come back. So this is our absolutely stable system. This is absolutely stable because this system is following the bounded input and bounded output criteria. What is this? It is conditionally stable. And if we take a system like this and place the ball here, If we put the ball back, what will happen? If someone applies a little force on it, what will happen? The ball will move away from here and it is not possible that the ball will come back here. This is an absolutely unstable system. How is the system behaving for a finite magnitude of input? What is the output for this? means it cannot return to its steady state. So, what do we call such a system? It is an unstable system. So, to check these unstable-stable systems, we have these criteria. We have to apply a finite input, a bounded input, and for that we have to see the output, whether the output of the system is bounded or not. So, for this, the test signals that you use again, will remain the same. Normally, the test signals that you can use are, Xt is equal to, you can use Ut, Xt is equal to, you can use, for example, you can use the signals of sin or cos, sin t, cos t, because their values are finite. If you say Ut, then its value will remain magnitude 1, no matter what the value of time is. For sin t or cos t, we know the magnitude varies from where to where. Amplitude 1, if nothing is written behind it, this value will be between plus and minus 1, this value will also be between plus and minus 1. So, these are standard inputs which you can use as a test signal. So, check bounded input, bounded output criteria. So, suppose we have a relationship of output-input for a system. For example, y of t is equals to x square of, Now if we see here, if we apply any bounded input, bounded input means whose magnitude is finite, if we apply any finite value instead of x, no matter what the value is, 1 lakh, 2 lakh, 3 lakh, then what will be the output of its square? 2 lakh square will also be a very big value, but what is that value? It is finite. So, what will be that system? It will be stable. Or you can see that if we take for x of t is equal to u of So, the output for ut will be y of t, what will be? U of t whole square. U of t whole square means again what value will you get? U of t value is 1, so what will be the square of 1? 1 for t greater than 0. So, what can we call it again? ut. So, what we are seeing is the output for bounded input, how is it? It is bounded, so what is our system? Stable. If we have this, y of t is equal to sin t into xt. Now again if you apply any bounded input in this for, if we take the input x of t is equal to u of t, whose magnitude is 1 from 0 to infinity. And this sin t, if you see the magnitude of y of t, what will be the magnitude of y of t? So, it will be equals to sin t magnitude into magnitude of x of t. The magnitude of x of t, the magnitude of ut if you are taking ut instead of xt, then what will be the magnitude of ut? 1 for t greater than 0. And the value of sin t can never be greater than 1. Where will this value be? It will be between minus 1 and 1, means the magnitude of it will always be between minus 1 and 1. So whatever the value of Xt is, the magnitude of Yt will always be, what we are seeing from this relation is that the magnitude of Yt must be less than or equal to magnitude of X of t because the magnitude of sin t can always be equal to or less than 1. So the magnitude of its maximum is 1 into Xt, which means the magnitude of Yt will always be less than Xt and if you have applied bounded input, which means if the magnitude of Xt is less than or equal to infinity, then the magnitude of Yt will always be less than or So automatically, you will get the magnitude of yt as well as the finite. So, what can we do with this system again? This system is stable. If you assume a system, y of t is equal to or take y of n, n into x of n. Now, how will this system be? If you put for xn is equal to un, if you put un, So, what will be the output of this system? y of n is equal to n times un. If you draw n times un, it means the value will be equal to n for greater than or equal to 0. So, if you draw in respect to n, your y of n, then n is equal to 0 for 0, 1 for this value, 1 for 2, 3 for 3. So, what are we seeing? With n, This value is increasing. So, what will be the value at n tends to infinity? Infinite. What is the other name of this signal? The name of the signal is RAM function R of n. Which is increased with time. So, at time t tends to infinity or n tends to infinity, the value of this output will be infinite, that is unbounded. Whereas, the input we took was bounded. So, again this system will be called as unstable system. If we have a system y of t is equals to integration from minus infinity to t x of tau d tau, okay. If you see this, if suppose we take here for xt is equals to ut, then the value of y of t will be minus infinity to t u of tau d tau. step function, then what you get is RAM function. This is a continuous time case, this is a discrete case. So, if we see its value RAM function, then again what kind of signal is this? It is an increasing signal with time, so what will be its value at time t tends to infinity? So, this system is also an unstable system. So, in this way, you can see that the system For different test inputs, whatever systems you have, you can see the output for those systems and if the output for bounded input is bounded, then what will we call that system? Stable. Otherwise, our system will be unstable. Now if you check this, suppose y of t is equals to integration from minus infinity to infinity x t into, suppose we have taken. sin2t dt. Now if we look at this signal, until this integration is not applied, until the integration is not applied, the function xt into sin t, how was the output? It was absolutely integrable, means that output was finite for every value of time. But now if you have used the integration, then if you take ut, then you will see again what is our system? If suppose we take t from minus infinity, then we will see that we have taken the output If you take ut, then you will feel that this system is stable. But if you take xt is equal to sin2t in this system, then what will be our output? What is sin2t? It is a bounded input whose value is between minus 1 and 1. Now if you check this system, integration from minus infinity to infinity, sin2t dt. What will we write sin2t? 1 minus cos2t divided by 2. So, the integral of 1 will be t. The integration of 1 into dt is t. And when you put the limit in it, minus infinity, then you will get a term from there, infinity means the output will be unbounded. So, you check it by doing it. And finally, when you see the part of 1 or the part of 1 by 2, what do you write? 1 minus cos 4 t divided by 2 dt integration from minus infinity to t. So, from here two parts are made, one part is made 1 by 2 if we make common out, then our one part is made dt minus infinity to t and our second part is made minus integration of cos 4 t divided by, we have made this 2 out dt minus infinity to t. This function will not be infinite because its value will always be between minus 1 and 1. But this part here, you will get the integration of dt as t and when you put minus infinity in it, then the value of this part will be infinite. So, our output will be unbounded and that is why our system will be unstable. So, we have seen some examples of stable, unstable, time-invertible and vertebral systems. Now, in this, Before this, we have done linear, non-linear, time variant, invariant, causal, non-causal, static, dynamic, memoryless, static system. We have done all these. Now this is your homework. Whatever question you have, whatever book you are following, do that. Now we are not giving more time separately because our module is lagging a little. All the problems like we have taken for non-linear, linear, for all of them, you check all the other properties. You know how to check properties. If we do everything here, you will not do it yourself. Till then you will not get confidence. You do all the questions by yourself. Check all the properties in given questions. And after that see in which property you are feeling more that there is more need in this, we did not understand or there is problem in this. You comment that and we will do it again. Now we are starting our next topic that is convolution. So next topic we are starting that is convolution. Till now we have understood what are linear systems and what are time invariant systems. So, the convolution property, the topic of convolution, we use it only for linear time invariant systems. This is always applicable for the linear and time invariant systems. So, we will see the definition of convolution, how, what, and how to do it. First, we have to read two types of convolution. First for time invariant systems. continuous time systems. So, for continuous time systems, the convolution will be the convolution of continuous time signals. This is also called convolution integral. Convolution integral and this same convolution we have to study for discrete time systems also. Where you will convolve discrete time signals, there discrete time For signals, the convolution for discrete time, we call it convolution summation. Because when you use convolution for continuous time signals, then you will use integration in continuous time and you will use summation in discrete time. So, we call it convolution summation, convolution integral. Now, why do we use this convolution? So, it is used. to find output of any LTI system. This is important. Whose output we can see? We are talking about the output of only LTI systems. Convolution is used to find output of any LTI system for any arbitrary input. Kisi bhi input ke liye hum output bata sakte hai kis ka? LTI system ka using convolution provided kya hai condition? Provided impulse response of the system is known. So, what is the condition we have? What should we know? We should know about impulse response system. Impulse response means response of the system for impulse input. If you apply a delta function, if you know that in any system if we apply delta function, impulse function, then how does its output come? If you know the output for impulse, then using convolution you can define output for any arbitrary signal of LTI systems. So, LTI system, if we have, this is a continuous time LTI system, this is a LTI system which is continuous time and for this you know impulse response. From whom do you write impulse response? From ht. What does impulse response mean? If this is a system, in this if you apply what instead of x of t? Del t, then what is the output of this system? What is the name of the output? Impulse response h of t. So, if you If you know the impulse response for this LTI system, then if you are applying any input in it, then for that you can tell the output of this system. Okay, what will be the value of this output? Output yt is equal to x of t convolved with h of t. Similarly, if you have a discrete time LTI system, this is our LTI system, but how is it now? If this is a discrete time LTI system, then what should you know for this also? Impulse response means H. If you are applying in del n input, then you should know the output of the system. So for that, if you apply input X, then you can tell the value of output Y using same result, Y is equal to X convolved with H. Okay? So, If we know the impulse response, then we can tell the output for LTI systems using convolution. Now, what does the symbol we have used for convolution define in itself? Suppose, you have used any symbol 5 into 4. So, what does this multiplication mean? How do you tell the result of this 20? This symbol defines something in itself. It means that you add 5 4 times. If you do the calculation of this in the computer, So, in computer we know the meaning of addition only. If we add 5 4 times, then the result will be 20. Similarly, if you have written that our output yt is equal to x of t convolved with h of t, then the symbol of this convolution will have some mathematical meaning. We are going to see that meaning now. So, first what we will consider is, first we will consider convolution integral, convolution of continuous time signals or output of continuous LTI system. we will start by looking at this. To look at this, first we will look at by definition that how we calculate the convolution of two signals by definition, which we will call graphical convolution. After that, we will read some properties of convolution. Then we will use those properties to see how we can find our result in a very short time easily. So, first our topic will be continuous time convolution. So, for that, if we look at So, here you have written that the value of the output yt will be equal to x of t convolved with h of t. So, now we define this y of t is equal to integration from minus infinity to infinity x of tau h of t minus tau d tau. The mathematical representation of the symbol of convolution means that if you want to convolve two signals, then we have written h of t minus tau. You can also understand h of t minus tau as h of minus tau plus t. So, in convolution, you perform three operations. First, you have to find out the number of signals that are present in the symbol. The base t, which means the signal represented in respect to t, what did you do to that base? You took tau. This means that there will be no changes in the values of the signal, but the thing in respect to t, the x-axis which used to be t earlier, what did you do to that x-axis? You took tau. Secondly, you just changed the base on one function, and on the other function, see what operations have been performed. You took tau of the base, t, there is a minus sign on it, which means you have taken a reversal. and plus t is the term that means we have applied shifting also. So, if we see, we have to do three things in convolution. We have to change the base from t to tau, in the second function, we have to provide a shift and a reversal. So, when we calculate this, we will need two signals for that, which we have seen earlier. If you remember, we had drawn two signals. u of minus t plus t naught. In two cases, you had drawn this signal. When the value of t naught is positive and when the value of t naught is negative. So, you remember this signal when you had drawn. So, this value, if we had taken t naught as negative. So, this value starting from minus infinity, this u of minus t plus t naught's value is for t naught less than 0. What is this for? For t0 less than 0. When you make this signal, whether the value of t0 is positive or negative, you saw that from where this signal will start t tends to minus infinity. We have not written the magnitude. Where is its existence starting from? From minus infinity. And if this t0 is negative, then where will it end? It will end at the negative side at the t0 value. We have written t0 less than 0 on our own, so we are not writing minus t0 here. And when you draw this signal, if you draw u of minus t plus t naught, for t naught greater than zero, so you saw that our signal starts with minus infinity but now it is extended to positive side, t naught, because t naught is greater than zero. You learnt to draw these two functions. These are the values of both, u of t, so what is this value? One. We will use them again. And now if we want to take the convolution of two signals, we will see that the time shifting and reversal that we are taking, we can take this reversal on ht if we want, we can take this reversal and shifting on xt if we want, because the convolution follows our commutative property, means whether you want to convolve xt with ht or convolve ht with xt, the result will be same. So, The function which you can see is an easy function, on which you are more comfortable in taking time reversal and shifting, consider it as your second function and in which, consider it as your first function, second function and in which there is a little problem, consider it as your first function. We will see this further. So, in convolution, you have to find these two functions. How will be x of tau? How will be h of t minus tau? Now, if these two things are multiplied, then where will be the result of these two? where this function should not be zero and this function should not be zero. Meaning, if you are going to integrate this product, then you will find the common area first, where both exist with non-zero values. Wherever one is zero, that area will not be common, you can remove it. After that, we will calculate this integral by putting values, simple integration will come. So first, by definition, if we take a system, Suppose, we are taking a question to start that x of t is equals to ut and impulse response h of t is again equals to ut then find output y of t that is equals to convolution of xt with h of t, okay. We have some LTI system whose impulse response is also ut and the input applied in it is also u. So, you have to tell the output for that system using definition of convolution that is graphically you have to convolve these two signals and tell that what will be our product convolution. Now see from here we have just written that the yt signal will be our output, the value of that yt that is x of t convolved with h of t, what will be its value? Conversion from minus infinity to infinity x of tau. H of t minus tau d tau. Now, you have to find this signal x of tau. What is given to you? X of t is given. So, you will find x of tau from there. H of t is given. So, you will find h of t minus tau from there. And what values can be written? As we have given x of t is u t. Therefore, x of tau will be u of tau. Is there any doubt in this? If you are saying xt is equal to ut, then if we replace it with tau here, then we also replace it with tau here, which means that x of tau will be equal to u of tau. If you draw this, then you can draw this. In respect of tau, we have to draw x of tau, whose value is u of tau. If it is t, then you draw ut. In a similar way, if you draw x of tau, that is equal to u of tau, then you will get a signal whose value is 1 from 0 to infinity. Value is 1. This is our signal. u of tau. Now, again you have given that h of t. is again equals to u of t. So, what you have to find? Which signal? You have found x of tau. Now, what you have to find? h of t minus tau. So, if h of t is ut, then from here h of t minus tau, what will be its value? u of t minus tau, which we can write as that is equals to u of minus tau plus t. Now if you draw this signal u of minus tau plus t and remember those two signals which we have discussed just now. So if here we have tau as our base, as you draw minus t plus t naught, there were two conditions that the t naught can be positive or negative. So if you draw below this with respect to tau, if you have to draw h of t minus tau that is u of T minus tau and we are taking case for T less than 0. Means this shifting, when the value of this shifting is less than 0, then we saw how this signal will be. The start will be from minus infinity, but if we take T negative, T less than 0 is taken, if T is negative, then what will happen? This signal will end where? On the negative side, on T. What is this value? 1. Now if we draw this signal, Same signal you have drawn with respect to tau for, we are drawing h of t minus tau, that is u of t minus tau and now for what case we are drawing? For t greater than 0. So from where this signal will be? It will start from minus infinity but now how far will it exist? On the positive side, this signal will exist till t. So you have seen the value of this signal. This signal is our, now we have the value y of t is equals to integration from minus infinity to infinity x of tau is u of tau, h of t minus tau is u of t minus tau d tau. Now we have to take the integration of this product from minus infinity to plus infinity. Now if we assume this y of t in two cases, if if t is greater than 0 and if t is less than 0. In that case, if you look at t less than 0, for t less than 0, look at your diagram, h of t minus tau. How is it looking? If we are going to multiply these two signals for t less than 0, this is our x of tau and this is h of t minus tau for t less than 0. If you multiply these two, will you get any result? Is there any common area between these two signals? When this signal starts, before that this signal starts and ends. Means, where it has non-zero values, there this value is zero. Where it has non-zero values, there its value is zero. So, you can say that this product, if we see any value of t less than zero, then what will be this product? Zero. And what will be its integration? Zero. So, we can write from here, y of t is equal to zero. for t less than 0 as no common area exists. No common area exists, then what will be the value of our output? It will be 0 for t less than 0. Now, if we see for t greater than 0, okay? t greater than 0 means this signal is our x of tau, which we are writing u of tau. This is always the same signal. The second signal we have written u of t minus tau, its value is the same for t greater than zero. Now, is there any common area in both these signals? This signal is starting from zero, it exists from zero to infinity. If you look at both these, then this area is from zero to t time, it is in this signal and in this signal too. That means in this duration, the value of this signal is also one and the value of this signal is also one. So, from where we will get common area? Suppose this is some instant t, then we will get t So, we will get this signal from 0 to t and this signal will also get 1. So, if we write this result, for t greater than 0, then y of t is equals to, integration from 0 to t, from where tau value is from 0 to t, in between that our u of tau function is also 1 and u of t minus tau function value is also 1. So, 1 into 1 into d tau, means overall what is this value? D tau integration, what will be the integration of d tau? Tau and when you put the limit from 0 to t, then what value you will get? t minus 0 that is equals to So, now we have the final result that yt value will be 0 for t less than 0 and yt is equals to t for t greater than 0. You can take the sign of greater than or equals to. Because if you take equal to then you will get t value 0. So, integration will be 0 to 0, you will get that value 0. So, if you combine both these results, then you can finally write that y of t is equals to 0. for t less than 0 and it is equals to t for t greater than or equals to 0. Now, look at these values. If you want to say this statement in one step that the value of yt will be t but t greater than or equals to 0 will be t then what can you say? If something exists after 0 then for that which function you write with it? u of t, it is causal signal so you can define yt in this way. It is equals to t times u of t. If you want to define in the same statement then you can say yt is equals to t into ut and what do you call t into ut function? That is equals to RAM function r of t. If you draw t into ut, if you draw its graph in respect to t, you are drawing yt, so ut means you have to draw after 0, can you see? t is equal to 0, value is 0, 1 on 1, 2 on 2, so what will you get? You will get a straight line, whose values if you see, then what will happen? What is the slope? 1. These values 1, 2, 3, 4, you will get such values, t is equal to 1, 2, 3, 4. So, we have a standard result that if we convolve ut with ut, then what will be our result? Ram function, This is the result, we will use it further. Suppose, we take the next example, suppose you are given H of t is equal to del t and X of t is equal to u t, you have to tell the output Y of t is equal to what? Now we have written the value of the output yt, this is the LTI system, so x of t convolved with h of t. And you have written this integration from minus infinity to infinity, x of tau, h of t minus tau, d tau. Now if you see its values again, if x t is our ut, then what will be x of tau? u of tau we have seen this. If xt is ut then x of tau u of tau. If ht is our del t then h of t minus tau what will be its value? del of t minus tau. Now if we substitute these two values here then therefore y of t it is written as minus infinity to infinity. Instead of xt instead of x of tau instead of u of tau. Instead of h of t minus tau, del of t minus tau d tau. Now, you use integration property that is shifting property of impulse function, in which we used to write that if such an integral was there, minus infinity to infinity, xt into del of t minus t naught dt, so how did we find its value? We used to put 0 in this bracket. And from there, we find the value of the integration in respect to which we get the value of t. So, from here, what value of t we got? t naught and the value of t we got, the result of this entire integration of summation, put that value of x of t naught in place of x of t. This result was provided, this value should lie between the limits of this integral. In respect to which integration we have here? In respect to tau. So, you put the thing in this bracket as 0 and find the value of tau. What is the value of tau from here? t is coming in between this limit. So, what will happen to this whole result integration? The value of x of t, which is u of tau here, is at tau equals to t. If you put t instead of tau, then what is the result? u of t. So, now from here it means that we have convolved ut with some impulse function del t and what is the result? Same same function which we had convolved. So, this is our first conclusion that if we convolve two step functions then what will be the resultant? It will be a ram function. This is our second conclusion. Now if you convolve any signal with impulse function, we will get a resultant. So the resultant will be the same function. Okay? The function we have convolved, if we convolve with the impulse function, then our resultant will be the same function. But we are not talking about shifted impulse. Nor are we talking about shifting in this function. You just have to understand that if you are doing it with impulse, then the result of convolution will come in the terms of this function. If it is ut, then it will come in the terms of ut. If it was rt, then it would come in the terms of rt. So now if we see one more function, convolution, On graphical representation, graphical method say uske baad phir hum properties ko deal karenge to suppose humare paas x of t hai that is equals to e to the power t u of minus t or h of t humara given hai that is u of t. Aapko agar in dono functions ko convolve karke apna output y t calculate karna hai. To y of t ki value kya hogi? integration from minus infinity to infinity x of tau h of t minus tau d tau, okay. Again, if you have x of t given, then find x of tau. If h of t is given, then find h of t minus tau. Here, suppose if h function was here. What do you see as h function? Complicated. So, in that case, finding h of t minus tau would be a little tough. So, what you do in that case you find x of t minus tau because we have seen now we talked that you can take time shifting and reversal on any one, either you can take on h or you can take on x. Here our h of t is the easy function, so if we see here if x of t is given x of t is equal to e to the power t u of minus t, then what will happen from here x of tau what will happen? e to the power tau u of minus tau. H of t given is u t, so H of t minus tau, this will be u of t minus tau, okay. Now again we will draw these two functions, okay. Because we want to see the product of these two functions, means the output y t, what is the value of that output y t, integration from minus infinity to infinity, e to the power tau, u of minus tau, this is our x of tau function. into H of tau function that is u of t minus tau d tau. You have to take the integration of this product. So, again we have to see that where this product is zero, where it is non-zero, where is the common area. If we see the limits of this function, this which we have written u of minus tau, from where this function will exist? U of minus tau means like you take time reversal on ut means this function will exist from minus infinity to zero. This function will exist from minus infinity to tau value to zero. And this function we have seen, you will draw it in two cases. One for t greater than zero and one for t less than zero. So, if we see this function, shape wise this function will be like t greater than zero but from where it will exist from? From minus infinity to zero. If you draw the top here, you will get some function of this type. x of t minus tau, x of tau that is e to the power tau u of minus tau. u of minus tau is there, so you can see it from minus infinity to zero. How is this function shape wise? Like e to the power tau. We don't mean by shape. Because we will write the value of its magnitude directly e to the power tau. We just have to see the limits that till when this function is non-zero. Okay. And if you draw these two functions below it. You have drawn this in respect of tau. This too you have drawn in respect of tau. If you are drawing u of t minus tau for t less than zero. So you had seen this function again. This will end on t in the negative side. And if you draw this function for t. greater than 0 u of minus t that is u of t minus tau. So, now this function will end here. Now, if you want to see the common area between these two, then in the case of these two, how much common area is there? If you are defining for this function and this function, then for t less than 0, if you define the output, then the value of y of t, t less than 0, this function will remain the same in every case, but the h of t minus tau has two values, that function can be either like this or like this, depending on the values of t. So, t less than 0 is this value. Between these two, from where to where is the common area? From minus infinity to t. So, now what we have written the result of integration, this is our limit, we have written from minus infinity to t, from minus infinity to t, in between this function, what is its value? e to the power tau, okay? into, this function, its value is 1 and its value is also, 1. How many of these values are there? 1. So, into 1 into d tau. Now, if you integrate this, what value will you get? That is, integration result e to the power tau, put limits in this from minus infinity to t. So, what will be this value now? e to the power t, And minus, when we put e to the power minus infinity instead of tau, then power minus infinity will become zero, that is minus zero. So, you have this output yt value, that is equals to e to the power t, for which you have calculated? For t less than zero. One result came. Now, if you calculate for t greater than zero, then what will happen between these two? Case 2 for t greater than zero. Now if you are looking at yt, which means you have to see the common area of this signal with whom? With this signal. So where is this ending from minus infinity? At 0. And where is this signal extended from minus infinity? To t. So where is the common area? From minus infinity to 0, this is also there and from minus infinity to 0, this signal is also there. So the common area of both is 0 from minus infinity. So now how much of the limits of integration we will write from minus infinity? In the meantime, this function of ours again the value of this function 1 means the value of this overall function e to the power tau and its value is again 1. So, if we put the values of both of these, then integration e to the power tau into 1 into d tau. Now, from where the limits of integration are from minus infinity to 0. Now, if you calculate it that is equals to e to the power tau. minus infinity to 0. So, what will be this value? e to the power 0 is 1 minus if you put minus infinity then 0. Means what will be this value? Therefore, y of t is equals to 1 for t greater than 0. So, overall if you write both the conditions together then what can we write? Therefore, y of t is equals to e to the power t for t less than 0 and it is equals to 1 for t greater than or equals to 0. So, this is our result for convolution, for the two signals we took, for both these signals, this is our convolution result. So, till now we have learnt to do convolution graphically for two signals, but in the exam, suppose if you are in university exams, then it is absolutely fine, you have to do it with this method. But you can also use a second method if you are out of time or in a competitive exam. For that, what you should know is properties of convolution. Now we will deal with properties of convolution. So if we see properties of convolution. When you do properties of convolution, So, you have to remember one thing whenever you are applying convolution, you have to remember one thing that convolution is only defined for LTI systems. Means, you know how the system is linear and time invariant. So, first we are looking at some properties of convolution, after that we will check properties based on time invariancy and properties based on linearity. So, first property is commutative property. We will not take much time in this because you know that there are commutative words just like you read in digital when you read Boolean Algebra, a plus b is equal to b plus. What does commutative log mean? If we are getting yt by convolving xt with ht, then this means value, we can write it more, whether you convolve Xt with Ht or convolve Ht with Xt, the result will be same, means the convolution follows the commutative law. Means Yt, which you wrote as integration from minus infinity to infinity, X of tau H of t minus tau d tau, you can write it as minus infinity to infinity, H of tau X of tau, H of tau X of tau, H of t minus tau d tau. This is our first property that is commutative property. Second property is associative property. So associative property means we can change the order of convolution. We can change the order of convolution means if you have y of t which is getting you from three functions of convolution, x1 of t convolved with x2 of t. convolved with x3 of t. So you can write it like this, first what we see in this, the result is, first we are convolving x1 and x2 and its resultant we are convolving with x3. So if we want, we can change the order of convolution, means if we want, first we convolve x1, x2 and x3 and after that the result we convolve it with x1. So we can convolve it with x1. then also our value will be same. We have already convolved x2 and x3. If we convolve their resultant with x1, then also our value will be same. This means convolution follows associative law. You can see one more thing from this. Suppose we have such cascaded systems. Its impulse response is h1 t. its impulse response is H2T, its impulse response is H3T. If you apply input of X of T, for every system, if you look at the individual outputs, for this system, the input is known, the impulse response is known, so its output can be calculated as Y1 of T, where the value of Y1 of T will be X of T convolved with H1 of T. Now, for this system, the Y1 of T will be the input, If you know the input and impulse response, then you can also tell the output of this system. What will be the value of y2? What is the input for y2? y1. What is the value of which you have found? X convolved with h1, this is our input for this system. And with whom do we convolve this? With its impulse response, which is h2. Now, this thing y2 is the input for the third system. You know the impulse response of this also and from here we will get our final output y of t. So, what will be the value of final output y of t? Input y2 of t convolved with h3 of t and what is the value of y2 of t? x of t convolved with h1 of t, its convolution with h2 of t, this is our input for this system. And with whom you will convolve this input? With this impulse response H3, you will get Y3. So, if you want, we have just written that we can interchange the order of convolution. So, if you want, first you can convolve these three together. And if you convolve it with X, then you will get Y. Means we can write Y overall output as such that H1 convolved with Y. H2 of t convolved with H3 of t and the resultant of these, whose convolution is with? With the input X of t. This means if we want to tell an equivalent for these three cache carried systems, that from here we want to represent it from a single system, in which we apply input Xt and our output is the same as it was getting earlier, Yt. So see the value of Yt. Xt will have to be convolved with this value. So, the impulse response of this overall system, Ht, what will be the value of that Ht? The individual impulse responses will be convolved. So, when you talk about transfer function in frequency domain, and if there is cascading, then what happens there? It multiplies. But if you look at impulse responses in time domain, So if the system is cascaded, then what will be their impulse responses? If you want to represent it from a single system, then you have to take the convolution of the individual impulse responses. This is the second property, associative property. If you see the third property, that is, distributive property. So distributive property says that if somewhere we have yt, the value we are getting is x of t convolved with h1 of t plus or minus h2 of t. So you can distribute it like a into b plus c, you write ab plus ac exactly the same. Now, you can convert Xt to H1t and H2t at the same time. The middle sign here is as it is, you have to take the sign. Means we can write this as X of t convolved with H1 of t and the sign here plus or minus again X of t convolved with H2 of t. In which cases does this come? When there are parallel systems. Means suppose you have this type of, This is our H1 of t, for this system impulse response and for other system impulse response is H2 of t and you are applying an input from here X of t, the output of these two is coming, if you are applying the output of these two in a summer, which is coming with plus or minus sign, if these values are coming here, okay, both can come with any sign. So, if you see the output here, X of t itself has an input in it. So, what will be the output of x of t convolved with h1 of t? What will be the overall output of x of t convolved with h1 of t? So, what will be the overall output of x of t convolved with h1 of t? This is plus minus sign and then plus minus x of t convolved with h2 of t. So, you can write it as x of t convolved with plus minus h1 t plus minus sign will be plus minus h2 of t. So, if you write an equivalent system for this, for parallel systems, equivalent system, then you can represent it with a single system, where if you are applying x of t input and you want the same output yt, then the impulse response of the system here will be h1 t plus minus h2 of t. t, whatever the symbol with plus comes with h1 t, then take plus h1 t, minus h1 t comes with minus h1 t, similarly, whatever sign will be with h2, you have to use that sign. So, these three basic properties we have seen are the properties of convolution. Now, if we talk about properties based on linearity, that means, you said that convolution is only defined for LTI systems. So, for that, we have to use the If we are talking about only LTI system, which means the system is linear, then the linearity property will also follow convolution. So, if we look at the property based on linearity, this property says that any linear operation on any of the inputs ht or xt, which is the result in the identical operation on the output y of t. Means, what you write as y of t is equal to x of t convolved with h of t. So, the property says that if we perform any linear operation in X t or H t, then our same linear operation will be performed on y t. What are the normal linear operations? differentiation, integration. So, if we have this result, then if we differentiate Xt once, if we want to convolve X dash of t, if we convolve it with Ht, then what will happen? If Xt Ht convolution was Yt, if we differentiate one of these, then what will happen to our resultant? If the first was Yt output, then what will be its resultant? Y dash of t will be there. If we keep Xt as it is, if we perform the same linear operation on H, then also if Xt is Yt of Xt, if we perform any linear operation on Ht, then the same operation will be performed on the previous output, which means the output will be again Y dash of t. If we perform the operation on both, if we perform the differentiation on X, then also and we have differentiated the edge as well. Now what will be our overall output? The operation performed on them, how many times did it perform? Two times. So, what will be our output now? The actual output will have double differentiation. What is its use? As you saw, now see two things. The first thing you saw is that the convolution of ut with ut gives us the result of RT. Okay, to see this graphically, it took us a lot of time. And you have seen one more thing that if we convolve any function with an impulse function, then what result do we get? Same function. So, this means that if we can differentiate any function and bring it in the form of an impulse function, then our convolution will become easy. So, suppose that our xt was ut and ht was also ut, if we had to convolve any function, then if we can differentiate x, then we can find out that if we can find So, the differentiation of x, the differentiation of unit step will give us impulse function. Now, if we convolve them, x dash of t convolved with h of t, then x dash means del t convolved with h of t means ut and we know that if we convolve any function with impulse, then we will get the same function, means the resultant we will get is ut, but this result will be This is not our actual output. What is our output? The actual output is the differentiation of the actual output. So, what do we have to do for the actual output? We have to integrate it once. y'is ut. Therefore, y will be the integration of ut. And what is the integration of ut? Ram function R of t. So, this type of property is used based on linearity. If you see any function which we will get by differentiating it, impulse function, then you can differentiate it once or twice and bring it in the form of impulse and then convolve it with ht. The result that you will get will not be the actual output. The times you have used differentiation on x, the same time for actual output, you will have to convert that yt to ht. Integrate it and you will get the actual output. So this is our property based on linearity. If we see the last property, then the last property is property based on time invariance. You have to use it exactly like this. Because our system is also time invariant. So for this, you will also apply time invariance. So we are taking the last property. Property based on time invariancy. For time invariancy, we know again that if Xt convolved with Ht gives you Yt, then if the system is time invariant, it means that if we apply any time shift in this Ht, Xt or Ht, So, the output will have the same effect that the time you have shifted Xt and Xt, the same amount of shifting you will get in the output. Meaning, if we delay X of t with t0 and then if we convolve with Ht, then our resultant will also be the previous resultant, that will be shifted with the same amount with t0. Or if we had convolved Xt, if we had taken shifting in H, then also our result would be y of t minus t0. If we take in both, then x of t minus t0 would be convolved to h of t minus t0. So resultant, now our total shifting in y, what is the value of total shifting? Minus t0 minus t0, what would these two add and give us resultant? y of t minus 2t0. So this is our property based on time invariance. What is its use? For example, if you want to convolve u convolved with u. You know what is the convolution of a step with a step? Ramp, means RT. So if you take a shift in any one, then what will be the resultant? R. If you want to convolve u, convolved with u of t minus 4. So, now what would be the resultant? Step by step convolution ramp is obtained. Total shifting is added and what happens? R of t minus 7. If somewhere somewhere, u of t minus 3. convolved with del of t plus 4. What is the value? If we convolve any function with impulse, then in which form will the resultant come? The function we are convolving, means we know that our resultant is going to come in the form of ut. What is the total shifting? Minus 3 and plus 4. How much resultant came? Plus 1. So therefore, what will be the value? u of t plus 1. So we have just now The convolution we have taken is only for continuous time signals. What we have seen till now is that, graphically convolving takes time. So, we have read some properties in the last. Now, using these properties and the two standard results we have, the resultant of step and step will give the convolution ramp. If we convolve any function with impulse, then the resultant will be, same function form. By using these, we will solve some difficult problems. For them, we will see the output of the system. We will see some techniques and tricks from which we can find these things quickly. We will see the differentiation of signals because we just saw that if you differentiate the signal and come to the form of impulse, then if one of these things, Xt or Ht, is converted into the impulse, then we can calculate it very easily. So we will learn to differentiate the signals. And after that, we will start our discrete time convolution or convolution summation and will also start to see the output of discrete time systems. So, till now, the last lecture of our module 1, means the properties of systems you have done, you should practice on them also, of problems. Convolution, now we can take one or two more lectures in these. But it is a very important topic because we had talked earlier also, whenever you have to tell the output of a system in time domain, So, we can use only convolution there. So, the topic is very important. You get 1-2 questions in every exam from properties of system and convolution. Whether it is question of continuous time convolution or discrete time convolution. So, do them well and keep practicing. And if you have any doubt, you can comment. Okay, thank you very much.

Hello, welcome to GATE Academy Plus. Subject we are doing signals and systems. In module 1, the topic is our properties of systems.

In this, till now we have completed causal, non-causal, time variant, invariant, static, dynamic and linear and non-linear systems. Now, the next properties in which our topic for today is, we have to consider the That is first if we see invertible and non-invertible systems, okay. So, till now we have seen that in the last class we have seen for linear non-linear systems.

So, all the problems we had taken, most of them, we had taken them for continuous time. But you don't have to do anything separately. You don't have to understand that we have learned to check the properties of continuous time system. we do not know for discrete time.

You just have to replace T with N. The whole procedure will remain exactly the same. Similarly, you can take the procedure, as you have checked the properties for continuous time systems, you can check for discrete time systems. So now, as we have taken the next property, if we have to check which system is invertible and which is non-invertible.

So, if we define invertible systems, then the system is said to be invertible. If the system produces distinct output for distinct input, if distinct input produces distinct outputs. Means, the output for any particular input should be unique. You can say this in other words, that our output should not be same for any two inputs.

output is there, so if you can analyze the input by looking at the output, if the output of the system is this and if we can calculate how the input will be, then what do you call the system? Invertible system. Means you can write more in this, if output, sorry if input can be determined by observing the output. If we can identify our input by looking at the output, then you call such systems as invertible system, in which distinct output is obtained for distinct input, we can analyze the input by looking at the output.

You can also say that the system which is system follows one to one mapping. Meaning, one to one mapping, as you read in set theory, that if we assume that we have one set of values for input and the other one set of values for output, then for one value of each input set, there should be a value of a particular output set. Meaning, it should not happen that for more than one input, the particular value of our output set is being selected. So, what do you call the system that it is following?

It is following one to one mapping. For example, suppose, Example, we have taken suppose we have a system for which the output input relation is given as yt is equal to 4 times x of t. You can write one more point in this that if any system is invertible then for that it is possible to design an inverse system such that if the inverse system is cascaded with the original system then the output of the overall system will be the input itself.

So, for that system, if you have an invertible system, then for this system, it will be possible to design an inverse system from which if we apply x of t from here and if we cascade the inverse system with it, then the output here will be the value of the input. That means you can say that the overall transfer function of this overall transfer function is the value of the system's transfer function and the inverse of it. Transfer function multiplication, if two blocks are in series then overall transfer function is multiplied. So, the product of these two should always be unity, so that x of t into overall transfer function 1, so here again we get x of t in output.

So, we call such system as invertible system, for which distinct inputs produce distinct output, you can tell the input by looking at the output and for that system you can design an inverse system. If we cascade it with the original system, then the output of the overall system is equal to the input. You can say that the value of the transfer function or the value of the gain is equal to the value of the unity. For example, if we take an example y of t is equal to 4 times x of t. So, if you want to tell the input by looking at the output, then you can tell x of t is equal to 1 by 4 times y of t.

One to one mapping if you want you can check it. Suppose for values of x of t, if you check values of y of t that is equal to 4 times x of t, apply distinct inputs. Suppose x of t is 0 then yt is equal to 0, 4 times 0 that is 0, x of t is minus 1 then y of t is equal to minus 4, x of t is plus 1 then y of t is equals to plus 4. If you check any value here, then you will not get a single output for two different inputs.

This means that our system is an invertible system. But if we take a system like this, suppose y of t is equal to x square of t means what is the property of the system? Whatever input we are applying, it is giving us in the output by making it square. So, if you check for this, whether there is one to one mapping between x of t and y of t, then you will see from here that if we take y of t, that is x square of t, then for the value of x of t, we will get 0 square of 0, for plus 1, we will get 1 square of 1, What value will you get for minus 1?

You will get 1. For plus 2, you will get 4 value and for minus 2, what value will you get? 4. So, what are we seeing? For more than 1 inputs, our output is common, it is same.

So, how will our system be? It will be a non-invertible system. Now, if we talk about inverse system, here actually you have not seen transform.

But suppose if we had a system like this, y of n is equal to summation. k is equals to minus infinity to infinity x of n. This actually we call this system as accumulator.

How is its output getting? From minus infinity to infinity, whatever inputs you have applied, or actually if we take minus infinity to n, this is the value of k, x of k minus infinity to n x of k. So, if we see the output for this system, then the output for this, if you see in the transformed domain, So, if Y of n for transform we will read these things further if we have given the name of this transform Y of z, okay.

Or if we have X of n if we are considering the input, if our transform is X of n for input X of z then you know from accumulation property that summation k is equal to minus infinity to n X of k for this the value of transform that will be 1 upon 1 minus z inverse. into the previous transform x of z. So, if we have such a system, then for that system we can design an inverse system, for this system, which will be the inverse system of this. If we see its inverse system, then if we take such an output, where suppose we name the inverse system output as z of n, then z of n is equal to yn minus y of n minus 1. Means, where the output is, If we see from here, then the value which is the two consecutive outputs, means one is our n minus 1, this is a previous output and this is a present output.

The difference of these will be if we define it as z of n. So again, if we assume that for x of z of n, if you are writing z transform z of z, then z of n is equal to z of n minus 1. So, if For example, if you have written the transform for y of n as y of z, then you will see the value of the transform of y of n minus 1 using the time shifting property. z inverse time y of z.

The more time you do time shifting, the more power z is multiplied, which is the actual transform. So, if you see from here, then the overall transform of this, suppose we call it y dash of n, for inverse, then the value of y dash of z will be, 1 minus Z inverse times Y of Z. So, if you multiply these two, then this 1 upon 1 minus Z inverse and this 1 minus Z inverse will be cancelled. So, this system, we can call this system again, how is it? Invertible system.

For this, we can design an inverse system, which if we cascade, then what value will we get? We will get 1. What is this system? It is an invertible system.

But how is this system? It is a non-invertible system. So, you can check some values with this one to one mapping. For example, if we have a relation that y of t is equal to mod of xt.

You can also check this. Take different values of x of t and according to that see the value of y of t that is magnitude of x of t. So, if you take 0 instead of x of t, then you will get 0. If you take minus 1, then you will get 1. If we take plus 1, what value will you get of yt? You will get 1. If we take plus j, what will be its magnitude?

1. And if we take minus j, what will be its magnitude? 1. So, what do we see now? There are 4 different inputs and what is the output for all of them? It is common, that is 1. So, how will this system be ours?

Non-invertible system. Okay? So, it is not always necessary that you check these 1 to 1 mapping for the system.

Time in, this is our invertible or non-invertible, you can tell that. So in such questions, there we use some test signals. We normally use some test signals when you are getting the product of two functions in the output and if you want to check the time invertibility for that system, whether the system is invertible or not, then we will use some test signals there. For this, normally you can take either U or U and U and U and then DELT and or you can apply U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and U and 2 del of n can also be applied.

So, how we will use these test signals for systems? Suppose you have any output input relationship given as such that y of t is equal to suppose given we have sin t times x of t. Now if you check with one to one mapping then you will get confused.

So, in this if we replace x t with sin t, then we will get a result of Suppose we have applied the input, which test signal? Apply del t. So, for x of t is equal to del t. If we see this output, what will be the value? y of t is equal to sin t into del t.

Now we know the product property of impulse function. If we product any signal x t with del t, then what is this value? Because this signal will only exist at 0, so what will we do?

We put the value of t in the bracket as 0. From here, we put the value of t in x. x of 0 into del of t. You saw this in the previous lecture.

If you have any doubt, you can see it once again. So, if you see from here, then what will be the output value of this? y of t is equal to, what will we put in place of t?

So, this function will be 0. 0. So, what will be the value? sin 0 into del of t. And what is the value of sin 0?

0, so the overall result will be 0. Again if we apply another input for x of t is equal to suppose we have applied 4 del t, okay. Or if you apply minus del t, okay, then what will be the output for that? y of t is equal to sin t into 4 times del of t. Now if these two inputs are different, first we applied del t, now what are we applying? 4 del t.

But still, from the product property, the value which will be 4 times sin 0 into del of t and again sin 0's value is 0, then overall what will be our result? 0. So, what are we seeing? For two different inputs, our system is producing the same output that is 0. So, what will you call such a system? It is a non-invertible system.

If we take another system, suppose we have some system y of t. or take y of n if we take that is equal to x of n into x of n minus 1. Okay. You can use any test signal or you can use it in these type of questions. Let if we see for input x of t or x of n, then our output is this, we have not used any test signal, we are simply looking for xn. Or if you want, then okay, if we look at this for u of n.

Okay. So, what will be the output for un? un times u of n minus 1 and if we see for input minus of un, if we had applied un input earlier, what are we applying? Minus u of n, so what will be the output?

y of n is equals to minus of u of n multiplied with minus of u of n. n minus 1 and this is minus minus plus and again what value will you get? u n times u of n minus 1. This output and this output what you are seeing is same. Whereas, inputs are different.

So, for two different inputs, our output is same. So, how will our system be? It will be non-invertible. Okay.

Suppose, if we check one more, if we have any function output input relation, y of n is equals to suppose, x of n plus 2 for n greater than or equals to 0 and if it is equals to x of n for n less than minus 1. This is our system for which the output of yn is x of n plus 2, when? If n is greater than 0 and if n is less than minus 1, then in that case our output is equal to x of n. Now if you want to check the invertibility for this system, then if we see for xn is equal to del n, if you see, then what will be the value of output yn? del of n plus 2 for n greater than or equals to 0. Now, where does this del of n plus 2 exist?

How do you see the existence? You put the thing in this bracket as 0 and its existence, if you see, then from here you get the value of n minus 2. That means where does this function exist? On minus 2. But for whom is it defined? For n greater than 0. So, what will be the value? 0. If you see from here also, if you write y of n is equal to del of n, then you than minus 1. So, this value is for n greater than 0, if we take 2 del of n, then the output is, what is its value?

y of n is equals to 2 times del of n plus 2 for n greater than equals to 0. This function is existing there at n equals to minus 2. So, it is existing at minus 2 and if we are looking for n greater than 0, then what value will we get again? 0. So, what we are looking for? For two different inputs, what is our output again?

Same. So, what will you say? This system of ours, what kind of system is this?

Non-invertible system. Okay. So, in this way, you can check the invertibility for the system. You can check the invertibility by saying that if our system is 1, same output is produced for more inputs, that means that system will be non-invertible.

And if distinct input is producing distinct output, then that system will be invertible. So, next classification is for systems, stable and unstable systems. See, about this, you deal with different approaches in different subjects. You must have read a lot to check the stability of the system.

When we study signals in depth, when we talk about transform domain, when we calculate the transform, these things will be more clear to you. As you must have read in control system, in Laplace's terms. If we define stable and unstable systems, then the common criteria we use there is bounded input, bounded output, If any system fulfils this bounded input bounded output criteria which you call BIBO stability criteria in short then we assume that system to be stable. What does bounded input bounded output mean? If we apply bounded input in any system, bounded input means such an input whose value is finite at every instant of time.

So, if we see at any instant then that input is finite value, then for such input, the output of the system should be bounded. Means, for, or you can write like this, the output of the system must be bounded for bounded value. That is, if the input we are applying, if the magnitude of that applied input is between 0 to infinity, okay, it is 0 or greater than 0, but what is the magnitude?

If it is smaller than infinity for x of t in between 0 and infinity, if x of t is finite, then output must be, the output which should be ours, its magnitude should be, what should be the magnitude of output? finite means this value should also be less than infinity. Now if the output of a system is bounded for bounded input, then this criteria also tells us that the impulse response of this system will change with time for 0. Bounded input bounded output implies that impulse response Must 10 0 as time t tends to infinity.

As time value increases, if any system is absolutely stable and it is following bounded input bounded output, it means its impulse response will decrease with time and become 0. You can understand it like this. What is the meaning of impulse response? Response of the system for impulse input If in any system What is impulse input?

It is a system that exists momentarily for you So if in our system We leave a pulse After that if we remove the input If that system is stable Then for that the output should also be there Because this is for a fraction of second And the input is gone So for that what should be the output with time When we have removed the input Then it should be zero So, if you write bounded input bounded output in time domain, then from there we also get the result that the magnitude of impulse function, impulse response should be absolutely integrable. Means, if we integrate minus infinity to plus infinity magnitude of h of tau d tau, then what should be our value? Less than infinity should be there.

The impulse response of the system should be absolutely integrable. If we are taking a continuous time system, then we are using absolutely integrable word. If we take a discrete time system, then what do we write?

k is equal to minus infinity to infinity. The magnitude of h of k should be less than infinity. It means it should be absolutely summable.

If we talk about all this in time domain, then we give mis-type statements. Because in time domain we have impulse response. When we talk about transfer function, then we talk about frequency domain and deal with transfer function. So till now you must have read, like in control also you must have read that where all the poles of the system should lie. Left half of the S plane.

So there you must have seen that the total response of the system. There you write the response normally from Ct. So this Ct is made up of two parts, a transient response.

and one CT steady state response. So, if any system is stable then in that case what you see is that the transient response limit t tends to infinity, c of t transient response it must be 0. Till the time the transient of the system does not vanish, till then it will not be able to reach its steady state and till then we cannot call that system stable. So, this transient response should be 0. And when will the transient response be zero? Like if you have Ct of this form, 4 plus e to the power minus 2t plus e to the power minus 3t.

So, you know this, t tends to infinity. This is an independent term from t. Which means, this is the response. This is the steady state response. And this part here is our response.

This is the transient response. So, if you see here, if there is a negative sign, then this term will be zero on time t tends to infinity, transient will vanish and your system will reach steady state. But if sign is not negative here, then what happens to this value on t tends to infinity?

It becomes infinite. So in that case, the system becomes unstable. So you must have seen that the terms that come in the transient response in exponential, these are directly the values of poles. That is why you say that the poles of the system, where should these poles lie in the? left half of the S plane, till this part is not negative, till then the transient response will not be zero with time and then the system will not be stable.

So, when we talk about Laplace, then we will deal with these things. Now, you can understand it like this, like if we take a criteria of bounded input and bounded output, suppose we have, suppose we This is a very deep container. Assume that it is very deep. And here we have a ball. Now if you are applying a force on this ball.

The force you are applying from here means that your force is the input. Now how did we apply this input? If you applied this input Finite.

Finite means. Even though you pushed it very hard, but if the magnitude of force is finite, then what will happen? This ball may come here for some time, then come back here, then come back here.

Then gradually with time, its distance from the mean will decrease, and finally this ball will come back and stop at its place. It means it will regain its steady state. So what kind of system is this?

Absolutely stable. Because the ball cannot go out of this container, if we apply force input till its magnitude is finite. Assume that you have a system like this and you have placed the ball here. Now if this height is not too much, then you can apply a particular value of force on it till it comes back to its steady state.

You did it, it went up a little, then it came down, then it oscillated and finally it will stop here. But for a finite value of input, finite magnitude of force, the ball may go out of the container and never return to its steady state. So this system is stable for a particular input, and after that input, the behavior of the system is unstable. So what do you call such a system? Marginally stable or conditionally stable system.

Conditionally stable means A critical value of force is that the ball will not go beyond the force of the same magnitude but if you apply more force than that, then the ball will go beyond the system and after that, its steady state will never come back. So this is our absolutely stable system. This is absolutely stable because this system is following the bounded input and bounded output criteria. What is this? It is conditionally stable.

And if we take a system like this and place the ball here, If we put the ball back, what will happen? If someone applies a little force on it, what will happen? The ball will move away from here and it is not possible that the ball will come back here.

This is an absolutely unstable system. How is the system behaving for a finite magnitude of input? What is the output for this? means it cannot return to its steady state. So, what do we call such a system?

It is an unstable system. So, to check these unstable-stable systems, we have these criteria. We have to apply a finite input, a bounded input, and for that we have to see the output, whether the output of the system is bounded or not.

So, for this, the test signals that you use again, will remain the same. Normally, the test signals that you can use are, Xt is equal to, you can use Ut, Xt is equal to, you can use, for example, you can use the signals of sin or cos, sin t, cos t, because their values are finite. If you say Ut, then its value will remain magnitude 1, no matter what the value of time is. For sin t or cos t, we know the magnitude varies from where to where. Amplitude 1, if nothing is written behind it, this value will be between plus and minus 1, this value will also be between plus and minus 1. So, these are standard inputs which you can use as a test signal.

So, check bounded input, bounded output criteria. So, suppose we have a relationship of output-input for a system. For example, y of t is equals to x square of, Now if we see here, if we apply any bounded input, bounded input means whose magnitude is finite, if we apply any finite value instead of x, no matter what the value is, 1 lakh, 2 lakh, 3 lakh, then what will be the output of its square? 2 lakh square will also be a very big value, but what is that value?

It is finite. So, what will be that system? It will be stable. Or you can see that if we take for x of t is equal to u of So, the output for ut will be y of t, what will be?

U of t whole square. U of t whole square means again what value will you get? U of t value is 1, so what will be the square of 1? 1 for t greater than 0. So, what can we call it again? ut.

So, what we are seeing is the output for bounded input, how is it? It is bounded, so what is our system? Stable. If we have this, y of t is equal to sin t into xt.

Now again if you apply any bounded input in this for, if we take the input x of t is equal to u of t, whose magnitude is 1 from 0 to infinity. And this sin t, if you see the magnitude of y of t, what will be the magnitude of y of t? So, it will be equals to sin t magnitude into magnitude of x of t.

The magnitude of x of t, the magnitude of ut if you are taking ut instead of xt, then what will be the magnitude of ut? 1 for t greater than 0. And the value of sin t can never be greater than 1. Where will this value be? It will be between minus 1 and 1, means the magnitude of it will always be between minus 1 and 1. So whatever the value of Xt is, the magnitude of Yt will always be, what we are seeing from this relation is that the magnitude of Yt must be less than or equal to magnitude of X of t because the magnitude of sin t can always be equal to or less than 1. So the magnitude of its maximum is 1 into Xt, which means the magnitude of Yt will always be less than Xt and if you have applied bounded input, which means if the magnitude of Xt is less than or equal to infinity, then the magnitude of Yt will always be less than or So automatically, you will get the magnitude of yt as well as the finite. So, what can we do with this system again? This system is stable.

If you assume a system, y of t is equal to or take y of n, n into x of n. Now, how will this system be? If you put for xn is equal to un, if you put un, So, what will be the output of this system? y of n is equal to n times un. If you draw n times un, it means the value will be equal to n for greater than or equal to 0. So, if you draw in respect to n, your y of n, then n is equal to 0 for 0, 1 for this value, 1 for 2, 3 for 3. So, what are we seeing?

With n, This value is increasing. So, what will be the value at n tends to infinity? Infinite.

What is the other name of this signal? The name of the signal is RAM function R of n. Which is increased with time. So, at time t tends to infinity or n tends to infinity, the value of this output will be infinite, that is unbounded.

Whereas, the input we took was bounded. So, again this system will be called as unstable system. If we have a system y of t is equals to integration from minus infinity to t x of tau d tau, okay.

If you see this, if suppose we take here for xt is equals to ut, then the value of y of t will be minus infinity to t u of tau d tau. step function, then what you get is RAM function. This is a continuous time case, this is a discrete case.

So, if we see its value RAM function, then again what kind of signal is this? It is an increasing signal with time, so what will be its value at time t tends to infinity? So, this system is also an unstable system. So, in this way, you can see that the system For different test inputs, whatever systems you have, you can see the output for those systems and if the output for bounded input is bounded, then what will we call that system? Stable.

Otherwise, our system will be unstable. Now if you check this, suppose y of t is equals to integration from minus infinity to infinity x t into, suppose we have taken. sin2t dt. Now if we look at this signal, until this integration is not applied, until the integration is not applied, the function xt into sin t, how was the output? It was absolutely integrable, means that output was finite for every value of time.

But now if you have used the integration, then if you take ut, then you will see again what is our system? If suppose we take t from minus infinity, then we will see that we have taken the output If you take ut, then you will feel that this system is stable. But if you take xt is equal to sin2t in this system, then what will be our output? What is sin2t?

It is a bounded input whose value is between minus 1 and 1. Now if you check this system, integration from minus infinity to infinity, sin2t dt. What will we write sin2t? 1 minus cos2t divided by 2. So, the integral of 1 will be t.

The integration of 1 into dt is t. And when you put the limit in it, minus infinity, then you will get a term from there, infinity means the output will be unbounded. So, you check it by doing it. And finally, when you see the part of 1 or the part of 1 by 2, what do you write?

1 minus cos 4 t divided by 2 dt integration from minus infinity to t. So, from here two parts are made, one part is made 1 by 2 if we make common out, then our one part is made dt minus infinity to t and our second part is made minus integration of cos 4 t divided by, we have made this 2 out dt minus infinity to t. This function will not be infinite because its value will always be between minus 1 and 1. But this part here, you will get the integration of dt as t and when you put minus infinity in it, then the value of this part will be infinite. So, our output will be unbounded and that is why our system will be unstable.

So, we have seen some examples of stable, unstable, time-invertible and vertebral systems. Now, in this, Before this, we have done linear, non-linear, time variant, invariant, causal, non-causal, static, dynamic, memoryless, static system. We have done all these.

Now this is your homework. Whatever question you have, whatever book you are following, do that. Now we are not giving more time separately because our module is lagging a little.

All the problems like we have taken for non-linear, linear, for all of them, you check all the other properties. You know how to check properties. If we do everything here, you will not do it yourself. Till then you will not get confidence.

You do all the questions by yourself. Check all the properties in given questions. And after that see in which property you are feeling more that there is more need in this, we did not understand or there is problem in this. You comment that and we will do it again. Now we are starting our next topic that is convolution.

So next topic we are starting that is convolution. Till now we have understood what are linear systems and what are time invariant systems. So, the convolution property, the topic of convolution, we use it only for linear time invariant systems.

This is always applicable for the linear and time invariant systems. So, we will see the definition of convolution, how, what, and how to do it. First, we have to read two types of convolution.

First for time invariant systems. continuous time systems. So, for continuous time systems, the convolution will be the convolution of continuous time signals.

This is also called convolution integral. Convolution integral and this same convolution we have to study for discrete time systems also. Where you will convolve discrete time signals, there discrete time For signals, the convolution for discrete time, we call it convolution summation.

Because when you use convolution for continuous time signals, then you will use integration in continuous time and you will use summation in discrete time. So, we call it convolution summation, convolution integral. Now, why do we use this convolution? So, it is used.

to find output of any LTI system. This is important. Whose output we can see? We are talking about the output of only LTI systems.

Convolution is used to find output of any LTI system for any arbitrary input. Kisi bhi input ke liye hum output bata sakte hai kis ka? LTI system ka using convolution provided kya hai condition? Provided impulse response of the system is known. So, what is the condition we have?

What should we know? We should know about impulse response system. Impulse response means response of the system for impulse input.

If you apply a delta function, if you know that in any system if we apply delta function, impulse function, then how does its output come? If you know the output for impulse, then using convolution you can define output for any arbitrary signal of LTI systems. So, LTI system, if we have, this is a continuous time LTI system, this is a LTI system which is continuous time and for this you know impulse response. From whom do you write impulse response?

From ht. What does impulse response mean? If this is a system, in this if you apply what instead of x of t?

Del t, then what is the output of this system? What is the name of the output? Impulse response h of t. So, if you If you know the impulse response for this LTI system, then if you are applying any input in it, then for that you can tell the output of this system. Okay, what will be the value of this output?

Output yt is equal to x of t convolved with h of t. Similarly, if you have a discrete time LTI system, this is our LTI system, but how is it now? If this is a discrete time LTI system, then what should you know for this also? Impulse response means H.

If you are applying in del n input, then you should know the output of the system. So for that, if you apply input X, then you can tell the value of output Y using same result, Y is equal to X convolved with H. Okay? So, If we know the impulse response, then we can tell the output for LTI systems using convolution.

Now, what does the symbol we have used for convolution define in itself? Suppose, you have used any symbol 5 into 4. So, what does this multiplication mean? How do you tell the result of this 20? This symbol defines something in itself. It means that you add 5 4 times.

If you do the calculation of this in the computer, So, in computer we know the meaning of addition only. If we add 5 4 times, then the result will be 20. Similarly, if you have written that our output yt is equal to x of t convolved with h of t, then the symbol of this convolution will have some mathematical meaning. We are going to see that meaning now. So, first what we will consider is, first we will consider convolution integral, convolution of continuous time signals or output of continuous LTI system. we will start by looking at this.

To look at this, first we will look at by definition that how we calculate the convolution of two signals by definition, which we will call graphical convolution. After that, we will read some properties of convolution. Then we will use those properties to see how we can find our result in a very short time easily. So, first our topic will be continuous time convolution.

So, for that, if we look at So, here you have written that the value of the output yt will be equal to x of t convolved with h of t. So, now we define this y of t is equal to integration from minus infinity to infinity x of tau h of t minus tau d tau. The mathematical representation of the symbol of convolution means that if you want to convolve two signals, then we have written h of t minus tau. You can also understand h of t minus tau as h of minus tau plus t.

So, in convolution, you perform three operations. First, you have to find out the number of signals that are present in the symbol. The base t, which means the signal represented in respect to t, what did you do to that base? You took tau. This means that there will be no changes in the values of the signal, but the thing in respect to t, the x-axis which used to be t earlier, what did you do to that x-axis?

You took tau. Secondly, you just changed the base on one function, and on the other function, see what operations have been performed. You took tau of the base, t, there is a minus sign on it, which means you have taken a reversal. and plus t is the term that means we have applied shifting also.

So, if we see, we have to do three things in convolution. We have to change the base from t to tau, in the second function, we have to provide a shift and a reversal. So, when we calculate this, we will need two signals for that, which we have seen earlier. If you remember, we had drawn two signals.

u of minus t plus t naught. In two cases, you had drawn this signal. When the value of t naught is positive and when the value of t naught is negative.

So, you remember this signal when you had drawn. So, this value, if we had taken t naught as negative. So, this value starting from minus infinity, this u of minus t plus t naught's value is for t naught less than 0. What is this for? For t0 less than 0. When you make this signal, whether the value of t0 is positive or negative, you saw that from where this signal will start t tends to minus infinity.

We have not written the magnitude. Where is its existence starting from? From minus infinity.

And if this t0 is negative, then where will it end? It will end at the negative side at the t0 value. We have written t0 less than 0 on our own, so we are not writing minus t0 here.

And when you draw this signal, if you draw u of minus t plus t naught, for t naught greater than zero, so you saw that our signal starts with minus infinity but now it is extended to positive side, t naught, because t naught is greater than zero. You learnt to draw these two functions. These are the values of both, u of t, so what is this value? One. We will use them again.

And now if we want to take the convolution of two signals, we will see that the time shifting and reversal that we are taking, we can take this reversal on ht if we want, we can take this reversal and shifting on xt if we want, because the convolution follows our commutative property, means whether you want to convolve xt with ht or convolve ht with xt, the result will be same. So, The function which you can see is an easy function, on which you are more comfortable in taking time reversal and shifting, consider it as your second function and in which, consider it as your first function, second function and in which there is a little problem, consider it as your first function. We will see this further.

So, in convolution, you have to find these two functions. How will be x of tau? How will be h of t minus tau?

Now, if these two things are multiplied, then where will be the result of these two? where this function should not be zero and this function should not be zero. Meaning, if you are going to integrate this product, then you will find the common area first, where both exist with non-zero values. Wherever one is zero, that area will not be common, you can remove it. After that, we will calculate this integral by putting values, simple integration will come.

So first, by definition, if we take a system, Suppose, we are taking a question to start that x of t is equals to ut and impulse response h of t is again equals to ut then find output y of t that is equals to convolution of xt with h of t, okay. We have some LTI system whose impulse response is also ut and the input applied in it is also u. So, you have to tell the output for that system using definition of convolution that is graphically you have to convolve these two signals and tell that what will be our product convolution.

Now see from here we have just written that the yt signal will be our output, the value of that yt that is x of t convolved with h of t, what will be its value? Conversion from minus infinity to infinity x of tau. H of t minus tau d tau.

Now, you have to find this signal x of tau. What is given to you? X of t is given.

So, you will find x of tau from there. H of t is given. So, you will find h of t minus tau from there.

And what values can be written? As we have given x of t is u t. Therefore, x of tau will be u of tau. Is there any doubt in this? If you are saying xt is equal to ut, then if we replace it with tau here, then we also replace it with tau here, which means that x of tau will be equal to u of tau.

If you draw this, then you can draw this. In respect of tau, we have to draw x of tau, whose value is u of tau. If it is t, then you draw ut.

In a similar way, if you draw x of tau, that is equal to u of tau, then you will get a signal whose value is 1 from 0 to infinity. Value is 1. This is our signal. u of tau. Now, again you have given that h of t.

is again equals to u of t. So, what you have to find? Which signal?

You have found x of tau. Now, what you have to find? h of t minus tau. So, if h of t is ut, then from here h of t minus tau, what will be its value? u of t minus tau, which we can write as that is equals to u of minus tau plus t.

Now if you draw this signal u of minus tau plus t and remember those two signals which we have discussed just now. So if here we have tau as our base, as you draw minus t plus t naught, there were two conditions that the t naught can be positive or negative. So if you draw below this with respect to tau, if you have to draw h of t minus tau that is u of T minus tau and we are taking case for T less than 0. Means this shifting, when the value of this shifting is less than 0, then we saw how this signal will be. The start will be from minus infinity, but if we take T negative, T less than 0 is taken, if T is negative, then what will happen? This signal will end where?

On the negative side, on T. What is this value? 1. Now if we draw this signal, Same signal you have drawn with respect to tau for, we are drawing h of t minus tau, that is u of t minus tau and now for what case we are drawing? For t greater than 0. So from where this signal will be? It will start from minus infinity but now how far will it exist?

On the positive side, this signal will exist till t. So you have seen the value of this signal. This signal is our, now we have the value y of t is equals to integration from minus infinity to infinity x of tau is u of tau, h of t minus tau is u of t minus tau d tau.

Now we have to take the integration of this product from minus infinity to plus infinity. Now if we assume this y of t in two cases, if if t is greater than 0 and if t is less than 0. In that case, if you look at t less than 0, for t less than 0, look at your diagram, h of t minus tau. How is it looking? If we are going to multiply these two signals for t less than 0, this is our x of tau and this is h of t minus tau for t less than 0. If you multiply these two, will you get any result?

Is there any common area between these two signals? When this signal starts, before that this signal starts and ends. Means, where it has non-zero values, there this value is zero. Where it has non-zero values, there its value is zero. So, you can say that this product, if we see any value of t less than zero, then what will be this product?

Zero. And what will be its integration? Zero.

So, we can write from here, y of t is equal to zero. for t less than 0 as no common area exists. No common area exists, then what will be the value of our output?

It will be 0 for t less than 0. Now, if we see for t greater than 0, okay? t greater than 0 means this signal is our x of tau, which we are writing u of tau. This is always the same signal. The second signal we have written u of t minus tau, its value is the same for t greater than zero. Now, is there any common area in both these signals?

This signal is starting from zero, it exists from zero to infinity. If you look at both these, then this area is from zero to t time, it is in this signal and in this signal too. That means in this duration, the value of this signal is also one and the value of this signal is also one.

So, from where we will get common area? Suppose this is some instant t, then we will get t So, we will get this signal from 0 to t and this signal will also get 1. So, if we write this result, for t greater than 0, then y of t is equals to, integration from 0 to t, from where tau value is from 0 to t, in between that our u of tau function is also 1 and u of t minus tau function value is also 1. So, 1 into 1 into d tau, means overall what is this value? D tau integration, what will be the integration of d tau? Tau and when you put the limit from 0 to t, then what value you will get?

t minus 0 that is equals to So, now we have the final result that yt value will be 0 for t less than 0 and yt is equals to t for t greater than 0. You can take the sign of greater than or equals to. Because if you take equal to then you will get t value 0. So, integration will be 0 to 0, you will get that value 0. So, if you combine both these results, then you can finally write that y of t is equals to 0. for t less than 0 and it is equals to t for t greater than or equals to 0. Now, look at these values. If you want to say this statement in one step that the value of yt will be t but t greater than or equals to 0 will be t then what can you say? If something exists after 0 then for that which function you write with it?

u of t, it is causal signal so you can define yt in this way. It is equals to t times u of t. If you want to define in the same statement then you can say yt is equals to t into ut and what do you call t into ut function?

That is equals to RAM function r of t. If you draw t into ut, if you draw its graph in respect to t, you are drawing yt, so ut means you have to draw after 0, can you see? t is equal to 0, value is 0, 1 on 1, 2 on 2, so what will you get? You will get a straight line, whose values if you see, then what will happen? What is the slope?

1. These values 1, 2, 3, 4, you will get such values, t is equal to 1, 2, 3, 4. So, we have a standard result that if we convolve ut with ut, then what will be our result? Ram function, This is the result, we will use it further. Suppose, we take the next example, suppose you are given H of t is equal to del t and X of t is equal to u t, you have to tell the output Y of t is equal to what? Now we have written the value of the output yt, this is the LTI system, so x of t convolved with h of t.

And you have written this integration from minus infinity to infinity, x of tau, h of t minus tau, d tau. Now if you see its values again, if x t is our ut, then what will be x of tau? u of tau we have seen this.

If xt is ut then x of tau u of tau. If ht is our del t then h of t minus tau what will be its value? del of t minus tau. Now if we substitute these two values here then therefore y of t it is written as minus infinity to infinity.

Instead of xt instead of x of tau instead of u of tau. Instead of h of t minus tau, del of t minus tau d tau. Now, you use integration property that is shifting property of impulse function, in which we used to write that if such an integral was there, minus infinity to infinity, xt into del of t minus t naught dt, so how did we find its value?

We used to put 0 in this bracket. And from there, we find the value of the integration in respect to which we get the value of t. So, from here, what value of t we got? t naught and the value of t we got, the result of this entire integration of summation, put that value of x of t naught in place of x of t. This result was provided, this value should lie between the limits of this integral.

In respect to which integration we have here? In respect to tau. So, you put the thing in this bracket as 0 and find the value of tau. What is the value of tau from here?

t is coming in between this limit. So, what will happen to this whole result integration? The value of x of t, which is u of tau here, is at tau equals to t.

If you put t instead of tau, then what is the result? u of t. So, now from here it means that we have convolved ut with some impulse function del t and what is the result?

Same same function which we had convolved. So, this is our first conclusion that if we convolve two step functions then what will be the resultant? It will be a ram function. This is our second conclusion. Now if you convolve any signal with impulse function, we will get a resultant.

So the resultant will be the same function. Okay? The function we have convolved, if we convolve with the impulse function, then our resultant will be the same function.

But we are not talking about shifted impulse. Nor are we talking about shifting in this function. You just have to understand that if you are doing it with impulse, then the result of convolution will come in the terms of this function.

If it is ut, then it will come in the terms of ut. If it was rt, then it would come in the terms of rt. So now if we see one more function, convolution, On graphical representation, graphical method say uske baad phir hum properties ko deal karenge to suppose humare paas x of t hai that is equals to e to the power t u of minus t or h of t humara given hai that is u of t.

Aapko agar in dono functions ko convolve karke apna output y t calculate karna hai. To y of t ki value kya hogi? integration from minus infinity to infinity x of tau h of t minus tau d tau, okay.

Again, if you have x of t given, then find x of tau. If h of t is given, then find h of t minus tau. Here, suppose if h function was here.

What do you see as h function? Complicated. So, in that case, finding h of t minus tau would be a little tough. So, what you do in that case you find x of t minus tau because we have seen now we talked that you can take time shifting and reversal on any one, either you can take on h or you can take on x. Here our h of t is the easy function, so if we see here if x of t is given x of t is equal to e to the power t u of minus t, then what will happen from here x of tau what will happen?

e to the power tau u of minus tau. H of t given is u t, so H of t minus tau, this will be u of t minus tau, okay. Now again we will draw these two functions, okay.

Because we want to see the product of these two functions, means the output y t, what is the value of that output y t, integration from minus infinity to infinity, e to the power tau, u of minus tau, this is our x of tau function. into H of tau function that is u of t minus tau d tau. You have to take the integration of this product.

So, again we have to see that where this product is zero, where it is non-zero, where is the common area. If we see the limits of this function, this which we have written u of minus tau, from where this function will exist? U of minus tau means like you take time reversal on ut means this function will exist from minus infinity to zero. This function will exist from minus infinity to tau value to zero.

And this function we have seen, you will draw it in two cases. One for t greater than zero and one for t less than zero. So, if we see this function, shape wise this function will be like t greater than zero but from where it will exist from? From minus infinity to zero. If you draw the top here, you will get some function of this type.

x of t minus tau, x of tau that is e to the power tau u of minus tau. u of minus tau is there, so you can see it from minus infinity to zero. How is this function shape wise?

Like e to the power tau. We don't mean by shape. Because we will write the value of its magnitude directly e to the power tau.

We just have to see the limits that till when this function is non-zero. Okay. And if you draw these two functions below it.

You have drawn this in respect of tau. This too you have drawn in respect of tau. If you are drawing u of t minus tau for t less than zero.

So you had seen this function again. This will end on t in the negative side. And if you draw this function for t. greater than 0 u of minus t that is u of t minus tau.

So, now this function will end here. Now, if you want to see the common area between these two, then in the case of these two, how much common area is there? If you are defining for this function and this function, then for t less than 0, if you define the output, then the value of y of t, t less than 0, this function will remain the same in every case, but the h of t minus tau has two values, that function can be either like this or like this, depending on the values of t. So, t less than 0 is this value. Between these two, from where to where is the common area?

From minus infinity to t. So, now what we have written the result of integration, this is our limit, we have written from minus infinity to t, from minus infinity to t, in between this function, what is its value? e to the power tau, okay? into, this function, its value is 1 and its value is also, 1. How many of these values are there?

1. So, into 1 into d tau. Now, if you integrate this, what value will you get? That is, integration result e to the power tau, put limits in this from minus infinity to t. So, what will be this value now?

e to the power t, And minus, when we put e to the power minus infinity instead of tau, then power minus infinity will become zero, that is minus zero. So, you have this output yt value, that is equals to e to the power t, for which you have calculated? For t less than zero.

One result came. Now, if you calculate for t greater than zero, then what will happen between these two? Case 2 for t greater than zero.

Now if you are looking at yt, which means you have to see the common area of this signal with whom? With this signal. So where is this ending from minus infinity?

At 0. And where is this signal extended from minus infinity? To t. So where is the common area? From minus infinity to 0, this is also there and from minus infinity to 0, this signal is also there.

So the common area of both is 0 from minus infinity. So now how much of the limits of integration we will write from minus infinity? In the meantime, this function of ours again the value of this function 1 means the value of this overall function e to the power tau and its value is again 1. So, if we put the values of both of these, then integration e to the power tau into 1 into d tau.

Now, from where the limits of integration are from minus infinity to 0. Now, if you calculate it that is equals to e to the power tau. minus infinity to 0. So, what will be this value? e to the power 0 is 1 minus if you put minus infinity then 0. Means what will be this value? Therefore, y of t is equals to 1 for t greater than 0. So, overall if you write both the conditions together then what can we write? Therefore, y of t is equals to e to the power t for t less than 0 and it is equals to 1 for t greater than or equals to 0. So, this is our result for convolution, for the two signals we took, for both these signals, this is our convolution result.

So, till now we have learnt to do convolution graphically for two signals, but in the exam, suppose if you are in university exams, then it is absolutely fine, you have to do it with this method. But you can also use a second method if you are out of time or in a competitive exam. For that, what you should know is properties of convolution. Now we will deal with properties of convolution.

So if we see properties of convolution. When you do properties of convolution, So, you have to remember one thing whenever you are applying convolution, you have to remember one thing that convolution is only defined for LTI systems. Means, you know how the system is linear and time invariant.

So, first we are looking at some properties of convolution, after that we will check properties based on time invariancy and properties based on linearity. So, first property is commutative property. We will not take much time in this because you know that there are commutative words just like you read in digital when you read Boolean Algebra, a plus b is equal to b plus. What does commutative log mean?

If we are getting yt by convolving xt with ht, then this means value, we can write it more, whether you convolve Xt with Ht or convolve Ht with Xt, the result will be same, means the convolution follows the commutative law. Means Yt, which you wrote as integration from minus infinity to infinity, X of tau H of t minus tau d tau, you can write it as minus infinity to infinity, H of tau X of tau, H of tau X of tau, H of t minus tau d tau. This is our first property that is commutative property.

Second property is associative property. So associative property means we can change the order of convolution. We can change the order of convolution means if you have y of t which is getting you from three functions of convolution, x1 of t convolved with x2 of t. convolved with x3 of t. So you can write it like this, first what we see in this, the result is, first we are convolving x1 and x2 and its resultant we are convolving with x3.

So if we want, we can change the order of convolution, means if we want, first we convolve x1, x2 and x3 and after that the result we convolve it with x1. So we can convolve it with x1. then also our value will be same.

We have already convolved x2 and x3. If we convolve their resultant with x1, then also our value will be same. This means convolution follows associative law. You can see one more thing from this.

Suppose we have such cascaded systems. Its impulse response is h1 t. its impulse response is H2T, its impulse response is H3T.

If you apply input of X of T, for every system, if you look at the individual outputs, for this system, the input is known, the impulse response is known, so its output can be calculated as Y1 of T, where the value of Y1 of T will be X of T convolved with H1 of T. Now, for this system, the Y1 of T will be the input, If you know the input and impulse response, then you can also tell the output of this system. What will be the value of y2?

What is the input for y2? y1. What is the value of which you have found? X convolved with h1, this is our input for this system.

And with whom do we convolve this? With its impulse response, which is h2. Now, this thing y2 is the input for the third system.

You know the impulse response of this also and from here we will get our final output y of t. So, what will be the value of final output y of t? Input y2 of t convolved with h3 of t and what is the value of y2 of t? x of t convolved with h1 of t, its convolution with h2 of t, this is our input for this system.

And with whom you will convolve this input? With this impulse response H3, you will get Y3. So, if you want, we have just written that we can interchange the order of convolution. So, if you want, first you can convolve these three together.

And if you convolve it with X, then you will get Y. Means we can write Y overall output as such that H1 convolved with Y. H2 of t convolved with H3 of t and the resultant of these, whose convolution is with?

With the input X of t. This means if we want to tell an equivalent for these three cache carried systems, that from here we want to represent it from a single system, in which we apply input Xt and our output is the same as it was getting earlier, Yt. So see the value of Yt.

Xt will have to be convolved with this value. So, the impulse response of this overall system, Ht, what will be the value of that Ht? The individual impulse responses will be convolved.

So, when you talk about transfer function in frequency domain, and if there is cascading, then what happens there? It multiplies. But if you look at impulse responses in time domain, So if the system is cascaded, then what will be their impulse responses?

If you want to represent it from a single system, then you have to take the convolution of the individual impulse responses. This is the second property, associative property. If you see the third property, that is, distributive property. So distributive property says that if somewhere we have yt, the value we are getting is x of t convolved with h1 of t plus or minus h2 of t. So you can distribute it like a into b plus c, you write ab plus ac exactly the same.

Now, you can convert Xt to H1t and H2t at the same time. The middle sign here is as it is, you have to take the sign. Means we can write this as X of t convolved with H1 of t and the sign here plus or minus again X of t convolved with H2 of t.

In which cases does this come? When there are parallel systems. Means suppose you have this type of, This is our H1 of t, for this system impulse response and for other system impulse response is H2 of t and you are applying an input from here X of t, the output of these two is coming, if you are applying the output of these two in a summer, which is coming with plus or minus sign, if these values are coming here, okay, both can come with any sign. So, if you see the output here, X of t itself has an input in it.

So, what will be the output of x of t convolved with h1 of t? What will be the overall output of x of t convolved with h1 of t? So, what will be the overall output of x of t convolved with h1 of t? This is plus minus sign and then plus minus x of t convolved with h2 of t.

So, you can write it as x of t convolved with plus minus h1 t plus minus sign will be plus minus h2 of t. So, if you write an equivalent system for this, for parallel systems, equivalent system, then you can represent it with a single system, where if you are applying x of t input and you want the same output yt, then the impulse response of the system here will be h1 t plus minus h2 of t. t, whatever the symbol with plus comes with h1 t, then take plus h1 t, minus h1 t comes with minus h1 t, similarly, whatever sign will be with h2, you have to use that sign.

So, these three basic properties we have seen are the properties of convolution. Now, if we talk about properties based on linearity, that means, you said that convolution is only defined for LTI systems. So, for that, we have to use the If we are talking about only LTI system, which means the system is linear, then the linearity property will also follow convolution. So, if we look at the property based on linearity, this property says that any linear operation on any of the inputs ht or xt, which is the result in the identical operation on the output y of t.

Means, what you write as y of t is equal to x of t convolved with h of t. So, the property says that if we perform any linear operation in X t or H t, then our same linear operation will be performed on y t. What are the normal linear operations?

differentiation, integration. So, if we have this result, then if we differentiate Xt once, if we want to convolve X dash of t, if we convolve it with Ht, then what will happen? If Xt Ht convolution was Yt, if we differentiate one of these, then what will happen to our resultant?

If the first was Yt output, then what will be its resultant? Y dash of t will be there. If we keep Xt as it is, if we perform the same linear operation on H, then also if Xt is Yt of Xt, if we perform any linear operation on Ht, then the same operation will be performed on the previous output, which means the output will be again Y dash of t. If we perform the operation on both, if we perform the differentiation on X, then also and we have differentiated the edge as well. Now what will be our overall output?

The operation performed on them, how many times did it perform? Two times. So, what will be our output now?

The actual output will have double differentiation. What is its use? As you saw, now see two things. The first thing you saw is that the convolution of ut with ut gives us the result of RT. Okay, to see this graphically, it took us a lot of time.

And you have seen one more thing that if we convolve any function with an impulse function, then what result do we get? Same function. So, this means that if we can differentiate any function and bring it in the form of an impulse function, then our convolution will become easy.

So, suppose that our xt was ut and ht was also ut, if we had to convolve any function, then if we can differentiate x, then we can find out that if we can find So, the differentiation of x, the differentiation of unit step will give us impulse function. Now, if we convolve them, x dash of t convolved with h of t, then x dash means del t convolved with h of t means ut and we know that if we convolve any function with impulse, then we will get the same function, means the resultant we will get is ut, but this result will be This is not our actual output. What is our output?

The actual output is the differentiation of the actual output. So, what do we have to do for the actual output? We have to integrate it once.

y'is ut. Therefore, y will be the integration of ut. And what is the integration of ut? Ram function R of t.

So, this type of property is used based on linearity. If you see any function which we will get by differentiating it, impulse function, then you can differentiate it once or twice and bring it in the form of impulse and then convolve it with ht. The result that you will get will not be the actual output.

The times you have used differentiation on x, the same time for actual output, you will have to convert that yt to ht. Integrate it and you will get the actual output. So this is our property based on linearity.

If we see the last property, then the last property is property based on time invariance. You have to use it exactly like this. Because our system is also time invariant. So for this, you will also apply time invariance. So we are taking the last property.

Property based on time invariancy. For time invariancy, we know again that if Xt convolved with Ht gives you Yt, then if the system is time invariant, it means that if we apply any time shift in this Ht, Xt or Ht, So, the output will have the same effect that the time you have shifted Xt and Xt, the same amount of shifting you will get in the output. Meaning, if we delay X of t with t0 and then if we convolve with Ht, then our resultant will also be the previous resultant, that will be shifted with the same amount with t0. Or if we had convolved Xt, if we had taken shifting in H, then also our result would be y of t minus t0. If we take in both, then x of t minus t0 would be convolved to h of t minus t0.

So resultant, now our total shifting in y, what is the value of total shifting? Minus t0 minus t0, what would these two add and give us resultant? y of t minus 2t0.

So this is our property based on time invariance. What is its use? For example, if you want to convolve u convolved with u. You know what is the convolution of a step with a step? Ramp, means RT.

So if you take a shift in any one, then what will be the resultant? R. If you want to convolve u, convolved with u of t minus 4. So, now what would be the resultant? Step by step convolution ramp is obtained. Total shifting is added and what happens?

R of t minus 7. If somewhere somewhere, u of t minus 3. convolved with del of t plus 4. What is the value? If we convolve any function with impulse, then in which form will the resultant come? The function we are convolving, means we know that our resultant is going to come in the form of ut. What is the total shifting?

Minus 3 and plus 4. How much resultant came? Plus 1. So therefore, what will be the value? u of t plus 1. So we have just now The convolution we have taken is only for continuous time signals.

What we have seen till now is that, graphically convolving takes time. So, we have read some properties in the last. Now, using these properties and the two standard results we have, the resultant of step and step will give the convolution ramp.

If we convolve any function with impulse, then the resultant will be, same function form. By using these, we will solve some difficult problems. For them, we will see the output of the system.

We will see some techniques and tricks from which we can find these things quickly. We will see the differentiation of signals because we just saw that if you differentiate the signal and come to the form of impulse, then if one of these things, Xt or Ht, is converted into the impulse, then we can calculate it very easily. So we will learn to differentiate the signals.

And after that, we will start our discrete time convolution or convolution summation and will also start to see the output of discrete time systems. So, till now, the last lecture of our module 1, means the properties of systems you have done, you should practice on them also, of problems. Convolution, now we can take one or two more lectures in these. But it is a very important topic because we had talked earlier also, whenever you have to tell the output of a system in time domain, So, we can use only convolution there. So, the topic is very important.

You get 1-2 questions in every exam from properties of system and convolution. Whether it is question of continuous time convolution or discrete time convolution. So, do them well and keep practicing. And if you have any doubt, you can comment.

Okay, thank you very much.

Transcript for:Understanding Invertible and Non-Invertible Systems

Transcript for:
Understanding Invertible and Non-Invertible Systems