 which is on amplitude modulation. So, in this lecture, we will define the amplitude modulation and its derivation for amplitude modulation. So, these are the learning outcomes. At end of this session, students will be able to define and derive the equation of amplitude modulation in electronic communication system here. So, this is before starting to the amplitude modulation, just recall what do you mean by modulation. So, the modulation is nothing, but you are going to change the characteristics of an carrier view with respect to modulating signals. So, that is known as an modulation. So, the characteristics we are going to change is either amplitude, frequency or phase here. So, if you are going to change the amplitude of an carrier signal, which is in proportion to the modulating signal, so that is known as an amplitude modulation by keeping frequency and phase constant throughout the process. Same way, if you are going to change the frequency of an carrier view with the proportional to the amplitude of an modulating signal by keeping amplitude and phase constant, it is known as an frequency modulation here. So, now, we will define the amplitude modulation. So, amplitude modulation is generally used for transmitting the information via a radio carrier wave here. In amplitude modulation, the amplitude of an carrier view is varied in a proportionate to that of a modulating signal being transmitted here. So, this was the definition of an amplitude modulation here. Now, we will derive the equation for the amplitude modulation. So, we will see the final output, what are the components in the amplitude modulation here. So, let us switch over to the class. So, now, as we know for modulation, we required two signals which is there. So, this is modulating signal. So, this is an amplitude of an modulating signal. This is T. So, one more signal that is carrier signal whose amplitude is Vc, whose amplitude is Vc here. So, this is carrier signal here. So, if you define the equation for an modulating signal, so it will be Vm sin omega m of T. This is amplitude of an the modulating signal. This will be an frequency and phase and frequencies are kept constant. So, that is why we are not going to take this into this equation here. Similarly, for carrier equation, we will be Vc by Vc sin omega c T. So, before starting to the equation, so we have to understand what do you mean by modulation index here. So, modulation index is defined that is Vm is equals to Vm by Vc that is amplitude of an modulating signal to the amplitude of an the carrier signal here. So, which lies between 0 to 1. If you have to calculate in percentage, then we have to multiply by 100. So, Vm by Vc into 100 here. So, you will get into the percentage. So, now we will have the amplitude of an am wave. So, we will derive the equation. So, for that we required one diagram. So, this is an amplitude of an am wave. So, this is Vc, this will be Vm and this total will be a that is total amplitude of an Vm plus Vc. And the equation for this modulating signal will be Vm sin omega m of T here. So, first let us derive the total amplitude of an am wave, total amplitude of an am wave. So, we have a equals to Vc plus Vm Vc plus Vm. So, Vc is equals to Vm sin omega m of T because we are going to change with respect to amplitude of an modulating signal. That is why we have to take total signal that is Vm sin omega m of T instantaneous value of an the modulating signal here. Therefore, we are going to take common Vc 1 plus Vm by Vc sin omega m of T. So, which will be Vc substituting the value of an m that is m is Vm by Vc here sin omega m of T. So, this is equation 1. So, now we will have the total instantaneous voltage that is total instantaneous voltage of an am. So, for that we have to be a sin theta instantaneous voltage here. So, now substituting the value of an a into this equation. So, you will be having Vc 1 plus m sin omega m of T into sin omega C T. So, this is also one equation of an am signal. So, now by trigonometric relations like sin x into sin y. So, we have to solve sin x and sin y. So, which is half cos x minus y minus cos x plus y. So, this is the relation. So, which gives which gives that is Vc sin omega C T plus Vm Vc by 2 cos omega C minus cos m of T minus m Vc by 2 cos omega C plus m of T. So, this is the final equation for an amplitude of an this modulation amplitude modulation. So, now from this what we can analyze that this is the same equation which is there for carrier signal. So, this is LSB that is lower side band and this is USB which is called as an upper side band here. And the amplitude of an LSB and USB are m Vc by 2 and here m Vc by 2 here. And the information contained in the amplitude modulation this side bands will be the same here with carrier signal here. So, which gives three components in amplitude modulated equation that is carrier signal LSB and USB here. So, now we will see how to plot an amplitude modulation signal in frequency domain. So, for frequency domain if I have to analyze the signal we have to make use of an spectrum analyzer here. So, in that we are going to see with respect to frequency. So, there you will look like the signal in this manner here. So, this will be f USB, this will be f LSB, this will be an FC here. And this will be amplitude of an the side bands m Vc by 2. So, if you calculate the bandwidth. So, which is f 2 minus f 1 that is upper cutoff frequency minus lower cutoff frequency upper cutoff frequency means USB that is upper side band that is omega c plus omega m. So, which we can write f c plus f m minus you can write f c minus f m here. So, from this f c plus f m minus f c plus f m. So, which get cancel you require you get an bandwidth is equals to 2 f m. So, this is the equation number 5. So, remember how much bandwidth will be required to transmit an AM signal you required to f m here. And if you want to observe the AM signal in frequency domain using a spectrum analyzer. So, you will look like the signal will be in this manner here. And in time domain your signal will be in this form here. So, this is the total AM modulated signal here. Thank you.