Frequency Spectrum of Amplitude Modulation (Waveform and Equation Derivation) [HD]

This video explains- Frequency Spectrum of Amplitude Modulation, am Waveform and Equation Derivation. Time domain and frequency domain waveforms with equations, along with basics of amplitude modulation.
What is Amplitude Modulation (AM)
Definition of Amplitude Modulation (AM)- Amplitude Modulation, is a system, where the maximum amplitude of the carrier wave varies, according to the instantaneous value (amplitude) of the modulating (message or baseband) signal.
Note:-
In case of Frequency Modulation (FM) or Phase Modulation (PM), the frequency or phase respectively, of the carrier wave varies, according to the instantaneous value of the modulating (message) signal.
In the previous post, I discussed; what is amplitude modulation and its various properties. We talked about only the time domain analysis of the amplitude modulation. We saw here, how the waveform changes with time but here, we will discuss the frequency domain analysis of the amplitude modulation. Here you will see, how the spectrum of modulating signal, carrier signal and modulated wave looks like. I will also derive here the equation of AM wave in frequency domain. So let's start...

Amplitude modulation (Time Domain Equations)

As per the discussion in previous post on Amplitude Modulation, we know that-

x(t) Modulating signal (also called as Message signal or Baseband signal)
c(t) = Acos(wct) Carrier wave
s(t) = [A+x(t)]cos(wct)
=x(t) cos(wct)+ Acos(wct) Amplitude Modulated wave…(i)

Here ‘A’ is the amplitude of the carrier wave and
‘wc’ is the angular frequency of the carrier wave

Fourier Transform of waves (Spectrum i.e. frequency domain representation of the waves)

Here all the equations are in time domain, so to get the spectrum of all these waves, i.e. frequency domain representation of the waves, we need to find out the Fourier transform (FT) of the waves.
The Fourier Transform (FT) will convert the Time Domain into Frequency Domain.

We are now going to find the Fourier Transform (FT) of the Amplitude Modulated Wave, s(t).
F[s(t)]= F[x(t) cos(wct) + Acos(wct)]
= F[x(t) cos(wct)] + F[Acos(wct)]…(ii)
The equation (ii), has two parts, Lets say..
x(t) cos(wct) Part 1 &
Acos(wct) Part 2
So let's find out the Fourier Transform of each of these parts separately.
First,we will start with finding the Fourier transform of the part 2 of equation (ii).

On doing all the calculations, we derived the Fourier Transform (FT) of Part 1 of AM Wave, that is given below and also in the image as equation (iii)
FT of [Acos(wct)]= Pi A[Delta(w+wc)+Delta(w-wc)]…(iii)
We will need this later, after finding the FT of another part of the Amplitude Modulated Wave.
So one important thing that you should notice here is that, this equation (iii) means, Fourier transform of “Acos(wct)” has two impulses of strength (Amplitude) "Pi A” at “+- Wc”.
Now let’s calculate the Fourier transform of another part.
By frequency shifting theorem of Fourier transform we know that, if the Fourier transform of x(t) is X(w) then the Fourier transform of [e^(jwct)x(t)], i.e. if we multiply x(t) signal with [e^(jwct) then its Fourier transform would be X(w-wc), let’s call this as equation (iv).
In the same way we can write
FT of [e^(-jwct)x(t)] is X(w+wc)…(v)
So from equations (iv) and (v),
The FT of x(t)[cos(wct)] is,
F[x(t){cos(wct)}] =1/2[X(w-wc)+X(w+wc)]…(vi)

This equation (vi) means, that on multiplying x(t) by
cos(wct), the spectrum of X(w) shifts by (+-wc).
Now we will combine, the Fourier Transforms of both parts of the Amplitude modulated wave, Part 1 and Part 2 of equation (ii),
that we calculated in the equations (iii) and (vi).
So the final equation of Amplitude modulated wave in Frequency Domain (Fourier Transform of the AM wave)

If F[s(t)] is S(w) then
S(w) = 1/2[X(w-wc)+X(w+wc)] + Pi A[Delta(w+wc)+Delta(w-wc)]

Waveform (Time Domain) and Spectrum (Frequency Domain) of the Amplitude Modulation

So this was the Mathematical part including equations and derivations, now we will see the waveform of these waves in time domain and in frequency domain (Spectrum).

Here I will draw the waveform (time domain) and spectrum (Frequency domain) of the modulating signal x(t), carrier signal c(t) and amplitude modulated wave s(t).

From the waveforms (time domain),we can see that
Modulating signal x(t) can have (as in this case)
Multiple Frequencies and
Variable Amplitude

Carrier wave c(t) (a high frequency wave used as a carrier of the message signal (modulating signal) has (in this case)
Constant amplitude
Constant frequency (High frequency)

Amplitude Modulated wave s(t) has-
Variations in Carrier wave Amplitude, that varies as per instantaneous value of the modulating signal (message signal).

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Видео Frequency Spectrum of Amplitude Modulation (Waveform and Equation Derivation) [HD] канала Engineering Made Easy