 So in our syllabus, amplitude modulation is something which is taken up with greater detail. So let us talk about amplitude modulation. All of you please write down. I hope all of you are sitting with paper and pen and not with something to eat. Write down amplitude modulation. These things are very important with respect to your school exams. So this is just theoretical analysis of amplitude modulation. This is just theory. So this is the carrier wave equation. Let us say Ct is Ac sin of omega Ct. This is the carrier wave. And let us assume that the message wave is this, Am sin of omega m into t. Now I know that the reason why I have to do this modulation is because omega m is very, very less. So automatically this is given that omega C is very large compared to omega m. Now the way theoretically we should do amplitude modulation is like this. You will understand why it is like this in some time. Right now let us first take it as it is. So this is how it should get modulated. Ac plus Am sin of omega mt multiplied by sin of omega Ct. Now if I take Ac common then I will get 1 plus amplitude of message divided by amplitude of carrier wave into sin of omega mt multiplied by sin of omega Ct. Okay fine. There is this term called modulation index which is defined as Am by Ac. There is a mu which is called modulation index which is defined as Am by Ac. This is modulation index. Since we are defining everything with respect to the variation of change in amplitude, right? So amplitude has a special importance. If the carrier wave amplitude is very large, the fluctuation in amplitude because of the message signal becomes very less. So that is why the ratio of amplitude is some importance. So that is why there is a physical quantity named as modulation index Am by Ac. So let us write this down as Ac1 plus mu times sin of omega mt into sin of omega Ct. Now I want you to open up the brackets, all of you open up the brackets and find out how many waves are there in this equation. Okay. How many waves are there? Sin omega Ct. You have to tell me how many waves are there? One wave, two waves, three waves or four waves. Kindly message how many waves are there in this equation. How many waves do you see here? This equation represents how many waves. Correct. It has three waves. How it has three waves? You can see that this you can further expand as Ac sin of omega Ct plus mu Ac by 2 cos of omega C minus omega m into t minus mu Ac by 2 cos of omega C plus omega m into t. Getting it? So you can see that you have three wave equations here. This, this and that. All right. So that is why I am saying that this particular expression has three waves of what frequencies of omega C, omega C minus omega m and omega C plus omega m. Okay. And if you draw amplitude versus frequency graph over here, suppose this is amplitude, you will see that this particular frequency which is the least one, omega C minus omega m. Okay. This has amplitude of Ac, okay. And this one has the amplitude of mu Ac by 2, okay. What we do? We intentionally keep value of mu to be greater than 1, okay. It is very large so that the amplitude get influenced by message the most, okay. So if amplitude should get influenced by message most, then amplitude of message should be larger than amplitude of carrier wave. So that is why mu Ac by 2 has larger amplitude. So this is amplitude and on x axis you have frequency, okay. And then you have another wave having amplitude Ac and frequency omega C plus omega m, okay. The middle one, this one has a frequency of omega C, okay. Now why we have drawn this particular graph is that you can see that if you modulate it like this, if your modulated signal is like this, you are occupying a band of this frequency. You are occupying a band of omega C minus omega m and omega C plus omega m, fine. So depending on the message frequency, automatically a small bandwidth is assigned to a particular message if you modulate it like this, okay. So your message is between these two frequencies and it is encapsulated between these two frequencies, okay. So that is the reason why your message is secured and it cannot get interfered by other waves because other waves will have a different frequency because message will have a different frequency. Different message will have a different message frequency. Carrier wave could be having same frequency but since message wave has different frequency the range omega C minus omega m and omega C plus omega m, this range will be different for different messages, okay. So this is how theoretically amplitude modulation is done, okay. Let's take a small example on whatever we have learned till now, okay. So here is the question, a message signal, there is a message signal of frequency, message frequency is given as 10 kilohertz, okay. And the peak voltage of this message signal, as in the amplitude of the message signal is given guys, the peak voltage is 10 volt for this message, okay. And you are using a carrier wave of frequency 1 megahertz, error wave frequencies 1 megahertz and the peak voltage of carrier wave is 20 volts, okay. You need to find these things, these two things, modulation index, find out modulation index and then B part, you have to find out side bands. Side bands are the range of the frequencies once you do the modulation. All of you quickly solve the A part first and message in the chat box. The formula for modulation index is what? Amplitude of message divided by amplitude of carrier wave, this is what, by 10 volt is the amplitude of message and 20 volt is the amplitude of carrier wave, right. So 10 divided by 20 which is 0.5, okay. Now can you get the side bands, what is the range of frequency? This is a question related to amplitude modulation, side frequencies, we know the side frequencies will be carrier wave frequency minus message wave frequency to carrier wave frequency plus message wave frequency, okay. So you need not always find angular frequency, although the formula is for the angular frequency but you can find out using linear frequencies also. So carrier wave is 1 megahertz which is like 1000 kilohertz minus 10. So this is 990 kilohertz is the first side band and 1000 plus 10 that is 1010 kilohertz this is the second side band. So the bandwidth, this particular message can occupy is 1010 minus 990. So total bandwidth will be 20 kilohertz difference in the frequencies, okay. Side bands are 990 and 1010 kilohertz, alright. So like this you have to do this particular question.