 Hello everyone, I am K. R. Biradhar, Assistant Professor, Department of Electronics and Telecommunication Engineering, Valchandr Institute of Technology, Salaf. Today I am going to discuss the topic on Operations on Signals. Let us start with the learning outcomes first. At the end of this session, students will be able to perform various operations on signals, identify how signal and amplitude and time changes after every operation. Operations on signals. There are six different operations, those are amplitude scaling, signal addition, signal multiplication, time shifting, time reversal and time scaling. First three are related to amplitude and later three are related to time. Later three is also called the transformation of independent variables t, amplitude scaling. The amplitude scaling of continuous time signal x of t is y of t is equal to a into x of t, where a is a constant. Now, you can take this example x of t. So, x of t is having an amplitude is equal to 2, which varies from t minus 1 to plus 1. Here, if I want to calculate 2x of t, here a is equal to 2, then amplitude will be doubled, that is amplitude is 2, which is doubled, but time is not going to change, time is same, that is minus 1 to plus 1, amplitude will be doubled is equal to 4. Similarly, if I want to get that is 0.5x of t, here a is equal to 0.5. That means, again from t is equal to minus 1 to plus 1, so amplitude will give half of it. Initially amplitude is x of t is 2, then it becomes 1. Signal addition. So, if you want to do signal addition, at least two signals are required, two or more is required signals for addition. The addition of two signals is obtained by adding their amplitudes at every instant of time, given x of t. So, x of t amplitude is equal to 1, which varies from minus 1 to 2. Another signal is required y of t, so its amplitude is equal to 1, which varies from minus 2 to 1. So, corresponding x of t plus y of t at every instant of time, we shall see. From minus 2 to minus 1, this amplitude equal to 0, this amplitude equal to 1, 0 plus 1 is equal to 1. From t is equal to minus 1 to 0, this is 1, this is also 1, 1 plus 1 is equal to 2. From 0 to 1, this is 1 and this is also 1, 1 plus 1 is equal to 2. From 1 to 2, this is 1, this is 0, so 1 plus 0 is equal to 1. You can see that is nothing but x of t plus y of t. Then signal multiplication is similar to signal addition, here also at least you require two signals. The multiplication of two signals is nothing but multiplication of their corresponding amplitudes. For example, x of t and y of t are given, then their product is x of t and multiplied by y of t. So, x of t you can see, which varies from minus 1 to 2, from minus 1 to 0, its amplitude is 0.5 and 0 to 2 it is 1. Similarly, y of t minus 1 to 1, its amplitude is 2, corresponding product you can see. From minus 1 to 0, this is 0.5, this is 2 actually, 0.5 into 2 is equal to amplitude is 1. From 0 to 1, this is 1, this is 2, 1 into 2 is 2. 1 to 2, 1 to 2, this is 1, this is 0, 1 into 0 is 0. After that, there is no signal. This is a product of x of t and y of t. Then time shifting, the time shifted version of continuous time signal x of t is y of t is equal to x of t minus capital T, where capital T denotes the time shift. If t is greater than 0, then the euform is shifted to right, that is there is a delay. If t is less than 0, then the euform is shifted to left advanced. You can see this example, this is a x of t, its amplitude is equal to a, which varies from 0 to 6. If I want to calculate x of t minus capital T, here t is positive, that means there is, which is greater than 0, there is a shifted right, there is a delay. So, how much it is delay, which will plus t times. So, initially the first point is at 0, it shifted to t and second point is at 6, it shifted to t plus 6. Similarly, if capital T is negative, then x of t minus of minus t, it becomes x of t plus t. There is a advanced or shifting towards left. So, how much it is going to shift in this case is x of t plus t, it initially it was 0 point will shift to minus t. So, initially the second point is at 6, it will shift to minus t plus 6. The time reversal next one, the signal x of minus t is time reversal of x of t. The time reversal is also called time folding. It is denoted by folding the signal about t equal to 0. You can see this x of t, it varies from 0 to 2. You can find x of minus t, it is exactly mirror image of x of t. If you put a mirror in front of x of t, you are going to get the x of minus t. So, wherever the plus is there, that will become minus that means plus 2 will become minus 2 ok. Next one is time scaling. In time shifting, you are not going to change that is shape of the signal, either it will not, neither it will expand nor it will compress the signal, but when you look for a time scaling, the time scaling may be a time expansion or compression takes place. Here the expansion or shape of the signal is going to change along the x axis. The xat is the time scaling of x of t. If magnitude of a is less than 1, then expansion of the signal. If magnitude of a is greater than 1, then compression of the signal. If it is greater than x of t, which varies from minus 2 to plus 2, then x of 2 t, here a is equal to which is greater than 1, then it must be a compression of the signals. How much it is going to compress? Divide it along the x axis of this x of t. So, if I divide by 2, it becomes minus 1. First point you are going to get. If I divide 0 by 2, 0 only we are going to get. If I divide 2 by 2, the second point you are going to get is at 1. Next similarly, x of t by 2, here a is equal to 0.5, that is 1 by 2 means 0.5, which is less than 1. There is a expansion of the signal takes place. How much it is going to expand? That you take the reciprocal of 1 by 2, it is 2, multiply by original x of t with 2. So, you are going to get here minus 4, here plus 4, 0. So, the signal varies from minus 4 to plus 4. So, then there is a combinational operations, that is a combination of time shifting, time reversal, time scaling etcetera, even amplitude also. If there is a combinational operations of independent variable t, then their hierarchy of operation is follows. You need to maintain or follow that is hierarchy of operations. So, consider an example x of minus 3 t plus 1. This involves time scaling, time shifting and time reversal, which one you need to perform first? Whether you need to time reverse or time scaling or time shifting? We need to follow this hierarchy of operation. First one is always you need to do time shifting, later is time reversal and third one is time scaling. So, second one, then third one, time reversal and time scaling can be interchanged. Identify the type of operation required and hierarchy of operation for the following signals. So, you need to identify what is the different operations available with this expression 2 in u of minus 2 t minus 1 and what is its hierarchy? Think pass the video and think and write your answer. This expression involves there are several time shifting operations in amplitude also. Time reversal is there because there is a minus sign, time shifting is there because minus 1, time scaling is there because of 2 will be multiplied with this, there is an amplitude scaling also. So, which one you need to perform first? Always you need to go for a time shifting first, minus 1 you need to shift and then time scaling and time reversal lastly amplitude. As I said you previously time scaling and time reversal can be interchanged. Always start with the time shifting first. Consider CT continuous time signal x of t, find x of minus 2 t plus 1, x of t which varies from minus 1 to 2, so its amplitude is 1 and 2. If I want to find x of minus 2 t plus 1, so I will follow that is intermediate steps here. First I will shift that is nothing but r plus 1, r plus 1 means t, t plus 1 means you are going to advance it by one sample. So, this varies from minus 1 to 2 will now change to minus 2 to 1 if I shift 1 minute left. Next is I reverse the time of this one x of minus t plus 1 means you are going to reverse all the reverse of this one mirror you take the mirror image of this signal you are going to get minus 1 to t 2. Next one is you need to get that is nothing but a scaled by 2 this signal will be scaled by 2 you are going to get here it is from minus 1 to 2 now it becomes minus 0.5 to 1. As I discussed earlier you need to divide it by 2 for entire values of t in this equation. If I divide it by minus 1 by 2 it becomes minus 0.50 this is 0.5 and this is 1. These are the references I referred for to prepare the above video thank you.