Derive expression for total modulated power and net modulation index.

Introduction

Modulation is a process in communication systems where some characteristic of a carrier signal is varied in accordance with an information-bearing signal, known as the modulating signal. The two key parameters in modulation are the total modulated power and the net modulation index.

The total modulated power is the sum of the power in the carrier signal and the power in the sidebands. It is a measure of the total power required to transmit the modulated signal.

The net modulation index is a measure of the extent of variation in a carrier signal according to the information being sent. It is a key parameter in determining the bandwidth and power requirements of a modulated signal.

Total Modulated Power

The formula for total modulated power in amplitude modulation is given by:

Pt = Pc(1+ (m^2/2))

where Pt is the total modulated power, Pc is the carrier power, and m is the modulation index.

The derivation of this formula begins with the expression for the modulated signal in the time domain, which is given by:

s(t) = Ac(1+mu(t))cos(2πfct)

where Ac is the amplitude of the carrier signal, u(t) is the modulating signal, and fc is the carrier frequency.

Squaring this signal and taking the expectation gives the power:

P = E[s^2(t)] = E[Ac^2(1+2mu(t)+m^2u^2(t))cos^2(2πfct)]

Using the trigonometric identity cos^2(x) = (1+cos(2x))/2 and taking the expectation gives:

P = Ac^2/2 + Ac^2E[mu(t)] + Ac^2E[m^2u^2(t)]/2

Assuming the modulating signal is a sinusoid with peak amplitude Am, we have:

E[mu(t)] = mA_m/2 and E[m^2u^2(t)] = m^2A_m^2/2

Substituting these values into the power expression gives:

P = Ac^2/2 + mA_mAc^2/4 + m^2A_m^2Ac^2/4

Simplifying this gives the formula for the total modulated power:

Pt = Pc(1+ (m^2/2))

Net Modulation Index

The formula for the net modulation index in frequency modulation is given by:

β = √(β1^2 + β2^2)

where β is the net modulation index, and β1 and β2 are the modulation indices of the individual signals.

The derivation of this formula begins with the expression for the modulated signal in the frequency domain, which is given by:

S(f) = AcJn(β1)cos(2πfct + nφ1) + AcJn(β2)cos(2πfct + nφ2)

where Jn is the Bessel function of the first kind of order n, and φ1 and φ2 are the phases of the individual signals.

The peak frequency deviation is given by:

Δf = max|f - fc|

The modulation index is then given by:

β = Δf/fm

where fm is the modulating frequency.

Substituting the expressions for the modulated signal and the peak frequency deviation into this formula gives:

β = √(β1^2 + β2^2)

Conclusion

Understanding the concepts of total modulated power and net modulation index is crucial in analog and digital communication. These parameters determine the bandwidth and power requirements of a modulated signal, which are key considerations in the design and operation of communication systems.

In practical applications, these concepts are used in the design of modulators and demodulators, the selection of appropriate carrier frequencies, and the analysis of signal quality and system performance.

Summary

Modulation is a process in communication systems where some characteristic of a carrier signal is varied in accordance with an information-bearing signal. The total modulated power is the sum of the power in the carrier signal and the power in the sidebands. The net modulation index is a measure of the extent of variation in a carrier signal according to the information being sent.

Analogy

Modulation is like adjusting the volume of a radio. The total modulated power is the total power required to transmit the modulated signal, similar to the total power output of a radio. The net modulation index is a measure of how much the volume of the radio signal is adjusted, similar to the extent of variation in a carrier signal.