__What is Amplitude Modulation (AM)__

__What is Amplitude Modulation (AM)__

__Definition__:-

__Definition__:-

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.*

__FREQUENCY SPECTRUM OF AMPLITUDE MODULATION ____VIDEO__** [HD]**

__FREQUENCY SPECTRUM OF AMPLITUDE MODULATION__

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)__

__Amplitude Modulation (Time Domain Equations)__

**Now look at the Image given below, that is self explanatory (Click the Image to enlarge)**

Amplitude Modulation Equations in time domainImage (1) |

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(w**

**c**

**t)**

*Carrier wave*

**s(t) = [A+x(t)]cos(w**

**c**

**t)**

**=x(t) cos(w**

**c**

**t)+ Acos(w**

**c**

**t)**

*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)**

**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.

**Now see the Image given below carefully (Click the image to enlarge)**

Calculation of Fourier Transform of Amplitude Modulated Wave Image (2) |

Here as per the image-

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(w**

**c**

**t)] + F[Acos(w**

**c**

**t)]…(ii)**

The equation (ii), has two parts, Lets say..

**x(t) cos(w**

**c**

**t)**Part 1 &

**Acos(w**

**c**

**t)**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)

Calculation of Fourier Transform of Amplitude Modulated Wave Image (3) |

**FT of [Acos(w**

**c**

**t)]= Pi A[Delta(w+w**

**c**

**)+Delta(w-w**

**c**

**)]…(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”.

We will see this, when we will draw the
spectrum of the waves, Now let’s calculate the Fourier transform of another
part.

**Now you need to see the Image given below-**

Calculation of Fourier Transform of Amplitude Modulated Wave Image (4) |

**According to the image-**

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, (See the derivation in Image 4 and Image 5)

Calculation of Fourier Transform of Amplitude Modulated Wave Image (5) |

**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-w**

**c**

**)+X(w+w**

**c**

**)] + Pi A[Delta(w+w**

**c**

**)+Delta(w-w**

**c**

**)]**

**You can see this equation in the image below-**

Final Equation of Amplitude modulated wave in Frequency Domain (Fourier Transform)Image (6) |

__Waveform (Time Domain) and Spectrum (Frequency Domain) of the Amplitude Modulation__

__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).

**Please see the Image given below (Click on the image to enlarge) for all the waveform in time domain and also the spectrum.**

From the waveforms (time domain) in the
image, we can see that

*can have (as in this case)*__Modulating signal x(t)__
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)

*has-*__Amplitude Modulated wave s(t)__
Variations in Carrier wave Amplitude, that
varies as per instantaneous value of the modulating signal (message signal). This
varying amplitude present in the Amplitude modulated wave contains information about
the message and is called as the

*Envelope of the wave*. So the information is contained in the envelop, that can be extracted at the receiving end by the process of demodulation.
[Since it is amplitude modulation, therefore the
amplitude of the carrier wave varies, according to the instantaneous value of the
modulating signal. 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 signal.]

**Now let’s discuss the Spectrum of these waves:-**

__Spectrum of Modulating wave X(w)__

__Spectrum of Modulating wave X(w)__
It has a triangular shape having frequencies at (+-wm)

Here wm is the highest frequency present in the modulating
signal.

__S____pectrum of the
carrier wave C(w)__

__S__

__pectrum of the carrier wave C(w)__
It has two impulses at +wm and –wm

This thing we have seen already in the
equation (iii) given below-

**FT of [Acos(w**

**c**

**t)]= Pi A[Delta(w+w**

**c**

**)+Delta(w-w**

**c**

**)]…(iii)**

Fourier Transform gives the spectrum
(frequency domain) of the waveform in time domain.

__Spectrum
of amplitude modulated wave S(w)__

__Spectrum of amplitude modulated wave S(w)__
It contains two identical and
symmetrical triangular shapes.

These two parts are the replicas of each
other.

**If**

**F[s(t)] is S(w)**

**then**

**S(w)**

**= 1/2[X(w-w**

**c**

**)+X(w+w**

**c**

**)] + Pi A[Delta(w+w**

**c**

**)+Delta(w-w**

**c**

**)]**

From the image you can easily observe
that each of the triangular part contains two sidebands known as Upper Side Band
(USB) and Lower Side Band (LSB).

Observe the image 7 that-

**w**

**c**

**>w**

**m**

Because of this, the two sidebands i.e. LSB
and USB don’t overlap.

We can easily calculate the Bandwidth of
the Amplitude modulated wave from its spectrum-

As you can see in the spectrum shown in
the image-

Bandwidth

**=**(**w****c****+w****m) - (****w****c****-w****m)****= 2 w**m (equal to the length of the base of the triangle in the spectrum)

signal and

**w****c**is the Frequency of the carrier wave.
Here I want to tell you one more thing
that the negative frequencies in the spectrum of the modulated wave are due to
the Fourier transform of exponential signal and are just for mathematical
representation. In reality these negative frequencies have no significance. But
as we know that positive and negative parts in the spectrum are replica of each
other, so all that we need can be calculated only by the positive side of the frequencies
in the spectrum.

So this was all about the frequency
domain analysis i.e. spectrum analysis of the amplitude modulated wave. For more
detailed and interesting explanation of all these concepts, I advise you to watch
my video present in this post.

__Read More-__**Go To HOME Page****FREQUENCY SPECTRUM OF AMPLITUDE MODULATION (WAVEFORMS AND EQUATIONS DERIVATION)****AMPLITUDE MODULATION (TIME DOMAIN EQUATIONS AND WAVEFORMS)**

**ADVANTAGES AND DISADVANTAGES OF DIGITAL COMMUNICATION SYSTEM**

**ADVANTAGES OF OPTICAL FIBER COMMUNICATION**

**STEP INDEX OPTICAL FIBER (MULTIMODE AND SINGLE MODE STEP INDEX FIBERS)**

**PULSE MODULATION TECHNIQUES (PAM, PWM, PPM, PCM)**

**OPTICAL FIBER: STRUCTURE AND WORKING PRINCIPLE**

**PULSE AMPLITUDE MODULATION (PAM)****COMPARISON OF PAM, PWM, PPM MODULATION TECHNIQUES**

**PULSE WIDTH MODULATION (PWM)**

**CONTINUOUS TIME AND DISCRETE TIME SIGNALS (C.T. AND D.T. SIGNALS)**

**NEED AND BENEFITS OF MODULATION**

**PULSE POSITION MODULATION (PPM)**

**OPTICAL FIBERS IN COMMUNICATION: COVERS ALL IMPORTANT POINTS**

**OPTICAL FIBER SOURCES (DESIRABLE PROPERTIES)**

**AMPLITUDE MODULATION Vs FREQUENCY MODULATION (ADVANTAGES AND DISADVANTAGES)**

**PULSE CODE MODULATION (PCM) [ADVANTAGES AND DISADVANTAGES]**

**SAMPLING THEOREM AND RECONSTRUCTION (SAMPLING AND QUANTIZATION)**

**SUPERPOSITION THEOREM (BASICS, SOLVED PROBLEMS, APPLICATIONS AND LIMITATIONS)**

**Digital Modulation Techniques (ASK, FSK, PSK, BPSK)/ Amplitude, Frequency and Phase Shift Keying**

**Conventional AM Vs DSB-SC Vs SSB-SC Vs VSB (Comparison of AM Systems)**

**Quadrature Amplitude Modulation (QAM)/ QAM Transmitter and QAM Receiver Block Diagram**

**Single-Mode Optical Fiber Advantages**

**What are Microwaves and their Applications (Uses) in various fields**

**Microwaves Properties and Advantages (Benefits)**

**Basic Structure of Bipolar Junction Transistor (BJT) - BJT Transistor - Working and Properties**

**Polar Plots of Transfer Functions in Control Systems (How to Draw Nyquist Plot Examples)**

**Generation of Binary Phase Shift Keying (BPSK Generation) - Block Diagram of Binary Phase Shift Keying (BPSK)**

**Low Level and High Level Modulation Block Diagram (AM Transmitter Block Diagram)**

**Block Diagram of CRO (Cathode Ray Oscilloscope), Components of CRO and CRT with Structure and Working**

**Slope Overload Distortion and Granular (Idle Noise), Quantization Noise in Delta Modulation**

**Frequency Translation/Frequency Mixing/Frequency Conversion/Heterodyning (Basic Concepts and Need)**

**Quadrature Phase Shift Keying Modulation (QPSK) Basics, Waveform and Benefits**

**Pulse Code Modulation (PCM) Vs Differential Pulse Code Modulation (DPCM)**
Excellent post. I learned a lot from this blog and I suggest my friends to visit your blog to learn new concept about technology.

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