Test: Phase Modulation - Electronics and Communication Engineering (ECE) MCQ

A digital synthesizer is a synthesizer that uses digital signal processing (DSP) techniques to make musical sounds. 

• Phase modulation is widely used for transmitting radio waves and is an integral element of many digital transmission coding schemes that support an ample range of wireless technologies such as GSM and Wi-Fi. 
• It is also used in satellite television. 
• Phase modulation is used in digital synthesizers for generating waveform and signal and not for transferring signals. (option 3 is incorrect)
• Phase modulation is very useful in radio waves transmission.

The frequency deviation in a phase-modulated carrier is proportional to the amplitude and frequency of the modulating signal.

Derivation:
A general expression for a phase-modulated wave is:
x_PM (t) = A cos [2πf_ct + k_pm(t)]
The instantaneous angle is given as:
ϕi(t) = 2πf_ct + k_p m(t)
The instantaneous frequency (in Hz) will be obtained as:

Frequency deviation is given by:

For a single tone modulation:
m(t) = A_m sin ω_mt
The frequency deviation becomes:

Analysis:
Let the carries signal S_c(t) = A_c sin ω_ct
The message signal m(t) = A_m sin ω_m t
So, the expression of FM is:

If m: modulation index then:

S_fm(t) = A_c sin [ω_ct – m cos ω_m t]
So hence given expression is not FM signal.
Now,
The expression of PM is:
S_pm(t) = A_c sin (ω_ct + k_p m(t)]
= A_c sin (ω_c t + k_p A_m sin ω_m t)
If m is the modulation index, then:
S_pm(t) = A_c sin (ω_c t + m sin ω_m t)      
∴ m = k_p A_m
Hence given expression is of PM signal.   

• Phase modulation (PM) is a modulation pattern that encodes information as variations in the instantaneous phase of a carrier wave
• Phase modulation is widely used for transmitting radio waves and is an integral part of many digital transmission coding schemes that underlie a wide range of technologies like Wi-Fi, GSM and satellite television
• Phase Modulation is generally not used for transmission because it is complex for decoding

A wave has 3 parameters Amplitude, Phase, and Frequency. Thus there are 3 types of modulation techniques.

• Amplitude Modulation: The amplitude of the carrier is varied according to the amplitude of the message signal or the intensity of light.
• Frequency Modulation: The frequency of the carrier is varied according to the amplitude of the message signal or the intensity of light.
• Phase Modulation: The Phase of the carrier is varied according to the amplitude of the message signal or the intensity of light.
  

Given:
The angle modulated signal is given by
ϕ_m(t) = cos(ω_ct + 5 sin (1000πt) + 10 sin (2000πt))
instantaneous frequency

Where ϕt is the phase of angle modulated signal.

In the above expression
f_c is the carrier frequency and additional term is the deviation (2500 cos 1000πt + 10, 000 cos cos 2000 πt)
f_d = 2500 cos 1000πt + 10, 000 cos cos 2000 πt
(f_D)_max = max (2500 cos 1000πt + 10, 000 cos cos 2000 πt)
(f_D)_max = (2500 + 10, 000)Hz = 12.5 kHz 

The output of FM modulator is of form

After passing the modulating signal through the differentiator we give  as the input to the frequency modulator, i.e.

The expression of the output signal will be:

The differentiation of the integral will simply give the original signal.
We get a phase modulated signal which is of the form:

The frequency deviation in a phase-modulated carrier is proportional to the amplitude and frequency of the modulating signal both.
Derivation:
A general expression for a phase-modulated wave is:
x_PM (t) = A cos [2πf_ct + k_pm(t)]
The instantaneous angle is given as:
ϕi(t) = 2πf_ct + k_p m(t)
The instantaneous frequency (in Hz) will be obtained as:

Frequency deviation is given by:

For a single tone modulation:
m(t) = A_m sin ω_mt
The frequency deviation becomes:

Concept:
If m(t) is the message single & c(t) = A_c cos(ω_ct) is the Carrier signal, then the General expression for a Phase modulated signal is:
S_PM(t) = A_c cos[ω_ct + k_pm(t)]
Where k_p = Phase Sensitivity (rad/Volts)

Analysis:
SPM(t) = A_ccos[ω_ct + kpm(t)] = A_c cos(θ_i)
Instantaneous Phase θ_i(t) = ω_c(t) + k_pm(t)
Instantaneous frequency ωi(t) = 

• Phase deviation Δϕ = kp m(t)
• Frequency deviation Δω = 
• Deviation ratio 

According to Carson formula, the Bandwidth of a phase-modulated signal is:
BW = 2(β + 1) ω_m = 2(k_pA_m + 1)ω_m
Hence the Bandwidth depends on both A_m & ω_m
So, Option (3) is Correct.

Concept:
The instantaneous angular frequency will be obtained by differentiating the instantaneous phase as:   if θ (t) = 2π f_ct + k m(t)

Calculation:
Given Φ(t) = cos (2π f_ct + km(t))
f_c = 50 kHz
K = 10 π

Phase of the modulated signal is given by:
θ (t) = 2π f_ct + k m(t)
The instantaneous angular frequency will be obtained by differentiating the instantaneous phase as:

is the slope of the message signal:
maximum positive slope (between 0 ≤ t ≤ 1) is 2
Maximum negative slope (between +1 < t < 3) is:
= -2/2 = -1
f_{i max} = 50 + 5(2) = 60 kHz
f_{i min} = 50 – 5(1) = 45 kHz