# Angle Modulation | What is Angle Modulation?

In this lecture, we are going to learn about Angle Modulation, the definition of angle modulation, types of angle modulation, and applications of angle modulation. So let’s discuss the definition and then we will discuss the angle modulation in detail.

## What is Angle Modulation

• In angle modulation, the frequency or phase of the carrier signal is varied according to the message signal. In this method of modulation, the amplitude of the carrier is kept constant.

## Concept of Angle Modulation: Basic Definition

### Definition of Angle Modulation

• Angle Modulation may be defined as the process in which the total phase angle of a carrier wave is varied in accordance with the instantaneous value of the modulating or message signal while keeping the amplitude of the carrier constant.

### Mathematical Representation of Angle Modulation

• Let us consider that an unmodulated carrier signal is expressed as

c(t)=A\cos (w_ct+\theta_0) …….. (1)

Where A = Amplitude of carrier, W_c=carrier frequency, \theta_0=Some phase angle,

• Substituting w_ct+\theta_0=\phi, we get

c(t)=A\cos \phi ……. (2)

• From equation (2), the constant angular velocity w_c of the phasor C is related to its total phase angle \phi as,

\phi=w_ct+\theta_0

• Differentiating both sides of this equation with respect to t, we have

w_c=\frac{d\phi}{dt} …… (3)

• It may be noted at this point that for the unmodulated carrier, the derivative \frac{d\phi}{dt} is constant. However, this derivative \frac{d\phi}{dt} may not be a constant with time, in general i.e. this may vary with time. Therefore, the angular velocity of the phasor C would also vary with time. hence, this time-dependent angular velocity or angular frequency is known as instantaneous angular velocity or instantaneous angular frequency. This instantaneous angular frequency is denoted by w_i.
• Hence, for this case, the equation (3) becomes

\frac{d\phi}{dt}=w_i …… (4)

• From equation (4) we get,

\phi=\int w_i\;dt …… (5)

• Now, if this angle \phi is varied according to the instantaneous value of the message or modulating signal, the carrier signal is then said to be angle modulated.

## Types of Angle Modulation

We can vary this phase angle \phi in two ways and thus there are two types of angle modulation:

• Angle Modulation has several advantages over amplitude modulation such as noise reduction, improved system fidelity, and more efficient use of power.
• However, there are some disadvantages too such as increased bandwidth and the use of more complex circuits.

## Applications of Angle Modulation

Angle Modulation is being used for the following applications:

• Microwave communication
• TV sound transmission
• Satellite communication

## Phase Modulation (PM)

### Definition of PM

• Phase Modulation (PM) is that type of angle modulation in which the phase angle \phi is varied linearly with a baseband or modulating signal x(t) about an unmodulated phase angle w_ct+\theta_0.
• This means that in Phase modulation, the instantaneous value of the phase angle is equal to the phase angle of the unmodulated carrier \phi=w_ct+\theta_0 plus a time-varying component which is proportional to modulating signal.

### Mathematical Representation of Phase Modulation

• We know that an unmodulated carrier signal is expressed as,

c(t)=A\cos (w_ct+\theta_0) or

c(t)=A\cos \phi

where, \phi=w_ct+\theta_0

• Neglecting \theta_0, we get the total phase angle of the unmodulated carrier is

\phi=w_ct …… (6)

• Now, according to Phase modulation, this phase angle \phi is varied linearly with a baseband or modulating signal x(t).
• Let the instantaneous value of phase angle be denoted by \phi_i.
• Therefore,

\phi_i=w_ct+k_p.x(t)

where k_p is the proportionality constant and is known as phase sensitivity of the modulator. This is expressed in radians/volts.

• Since, the expression for unmodulated carrier wave is

c(t)=A\cos \phi

• Therefore, the expression for phase modulated wave will be

s(t)=A\cos \phi_i

• putting the value of \phi_i in above equation.

\boxed{s(t)=A\cos [w_ct+k_p.x(t)]} …… (7)

which is the required mathematical expression for a phase modulated wave.

## Frequency Modulation (FM)

### Definition of FM

• Frequency modulation is that type of angle modulation in which the instantaneous frequency w_i is varied linearly with a message or baseband signal x(t) about an unmodulated carrier frequency w_c.
• This means that the instantaneous value of the angular frequency w_i will be equal to the carrier frequency w_c plus a time-varying component proportional to the baseband signal x(t).

### Mathematical Representation of Frequency Modulation

• We know that the instantaneous frequency is given by

w_i=w_c+k_f.x(t) …… (8)

where k_f is proportionality constant and is known as the frequency sensitivity of the modulator. This is expressed in Hz/volt.

• Now let the expression for unmodulated carrier signal be

c(t)=A\cos (w_ct+\theta_0) or

c(t)=A\cos \phi

where, \phi=w_ct+\theta_0 …… (9)

• Let \phi_i be the instantaneous phase angle of the modulated signal.
• Hence the expression for frequency-modulated wave will be

s(t)=A\cos \phi_i …… (10)

where \phi_i= instantaneous phase angle.

• From equation (9), we have

\phi=w_ct+\theta_0

• on differentiation, we get

\frac{d\phi}{dt}=w_c

or \phi=\int w_c\;dt …… (11)

• Based on the equation (11), we may write the expression for instantaneous phase angle \phi_i as

\phi_i=\int w_i\;dt ……(12)

where w_i=instantaenous frequency of frequency modulated wave.

• putting the value of w_i in equation (12) from equation (8), we get

\phi_i=\int [w_c+k_fx(t)]dt=w_ct+k_f\int x(t)\;dt …… (12)

• putting the value of \phi_i in equation (10), we get the expression for frequency modulated wave will be

\boxed{s(t)=A\cos [w_ct+k_f\int x(t) dt]}

which is the required general expression for Freqeuncy Modulated wave.

## Performance comparison of FM and PM system

Hello friends, my name is Trupal Bhavsar, I am the Writer and Founder of this blog. I am Electronics Engineer(2014 pass out), Currently working as Junior Telecom Officer(B.S.N.L.) also I do Project Development, PCB designing and Teaching of Electronics Subjects.

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