Test: Frequency Modulation

Concept:
Standard expression of FM signal 

Instantaneous frequency (f_i in Hz)

Where

From the standard expression

Peak frequency deviation: 

Calculation:
Given;

at, t = 1.5 sec ; m(t) = -10

Peak frequency deviation:
|m(t)|_max = 10

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:

Concept:
As per Carson’s rule, the bandwidth of a frequency modulated signal is defined as:
B.W. = 2 (Δf + f_m)
Δ f = Frequency deviation of the modulated wave
fm = frequency of the modulating signal

Calculation:
Given B.W. = 60 kHz and Δ f = 20 kHz
60k = 2 (20k + f_m)
30k = 20k + f_m
f_m = 10 kHz

Frequency Modulation:

• Frequency Modulation is a modulation in which the frequency of the carrier wave is altered according to the instantaneous amplitude of the modulating signal, keeping phase and amplitude constant.
• So, the variation in carrier amplitude and carrier phase does not affect the signal in the receiving end.
• Line of sight (LoS) is a type of propagation that can transmit and receive data only where transmit and receive stations are in view of each other without any sort of an obstacle between them. Eg: FM radio, microwave, and satellite transmission.
• Frequency Modulation works on the Line of sight propagation.

Types of FM detection:

• Slope detection
• Phase-locked loop detection
• Foster Seeley detection
• Ratio detector
• Quadrature detectors.

Foster Seeley discriminator:

• Discriminator circuits can generate electrical output directly proportional to the frequency deviation from the unmodulated RF carrier frequency.
• The simplest circuit could be a balanced slope detector.
• It makes use of two resonant circuits; one off-tuned to one side of unmodulated RF carrier frequency and the other off-tuned to the other side of it.
  
• Another class of FM detectors known as Quadrature detectors uses a combination of two Quadrature signals.
• The Most commonly used FM detectors namely the foster-Seeley FM discriminator and the ratio detector operates on the principle of quadrature.

Foster Seeley discriminator is used in the demodulation of FM signal and it uses a double-tuned circuit with primary and secondary tuned to the same frequency.

Concept:
In FM (Frequency Modulation), the modulation index is defined as the ratio of frequency deviation to the modulating frequency.
Mathematically, this is defined as:

m_f = Modulation index
Δf = Frequency deviation
f_m = Modulating frequency

Calculation:
Given Δf = 1000 Hz = 1 kHz
f_m = 1 kHz

• A phase-locked loop is a control system that generates an output signal whose phase is related to the phase of an input signal.
• The circuit can track an input frequency or it can generate a frequency that is a multiple of the input frequency.
• The basic elements of a PLL circuit are phase comparator/detector, a loop filter, voltage controlled oscillator (VCO).

The figure shows a schematic of a phase-locked loop:

Concept:
According to Carson rule signal, BW is given as:
B.W = (β + 1) 2fm
∵ β = Δf/f_m
B.W. = 2[Δf + f_m]          
If multiple frequencies are available in modulating signal then,
B.W. = 2(β + 1) f_max 

Calculation:
Given: 
Δf = 10 kHz
f_m = 2 kHz
B.W. = 2[Δf + fm]   
B.W. = 2(10 + 2)
BW = 24 kHz
Hence option (3) is the correct answer.

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 message signal.
• Frequency Modulation: The frequency of the carrier is varied according to the message signal.
• Phase Modulation: The Phase of the carrier is varied according to the message signal.
  

The Bandwidth of an FM signal is approximated by Carlson’s rule and is given by
B.W. = 2(Δf + f_m).
Where Δf = maximum frequency deviation from the carrier frequency and f_m = message signal frequency.

Calculation:
Given, S_FM(t) = 10 cos (2π × 10⁸ t + 0.5 sin (10⁴πt))
The instantaneous frequency is given by:

⇒ Maximum instantaneous frequency

Maximum frequency deviation

So, Approximate bandwidth according to Carson’s rule
⇒ 2(Δf + f_m) = 2(2.5 + 5) = 2 (7.5 K) = 15 K