Derive the Power Relation for Single Tone AM Wave and Multiple-tone AM Wave

Definition

It may be observed from the expression of AM wave that the carrier component of the amplitude modulated wave has the same amplitude as unmodulated carrier.

In addition to carrier component , the modulated wave consists of two sideband components. It means that the modulated wave contains more power than the unmodulated carrier.

However, since the amplitudes of two sidebands depend upon the modulation index, it may be anticipated that the total power of the amplitude modulated wave would depend upon the modulation index also. In this section, we shall find the power contents of the carrier and the sidebands.

As we know  the general expression of AM wave is given as :

s(t) = A cos ω_ct + x(t) cos ω_ct       ………………..(1)

The total power P of the AM wave is the sum of the carrier power P_C and sideband power P_S.

Carrier Power

The carrier power P_C is equal to the mean-square (ms) value of the carrier term A cos ω_ct i.e.

P_C =  mean square value of A cos ω_ct

Sideband Power

The sideband power P_S is equal to the mean square value of the sideband term x(t) cos ω_ct i.e.

P_S =  mean square value of x(t) cos ω_ct

In AM generation , a Band pass filter (BPF) or a tuned circuit tuned to carrier frequency ω_c is used to filter out the second integral term.

Therefore,

However, the total sideband power P_S is due to the equal contributions of the upper and lower sidebands. Hence, the power carried by the upper and lower sidebands will be

Power of a Single-Tone Amplitude-Modulated (AM) Signal

Definition

We know that the power content of the AM signal when modulating-signal is a random signal and may consist of several frequency components. Likewise we can find power content of single-tone Amplitude Modulated (AM) signal.

Mathematical Expression

Let us consider that a carrier signal A cos ω_ct is amplitude-modulated by a single-tone modulating signal x(t) =V_m A cos ω_mt.

Then the unmodulated or carrier power

P_C =  mean square (ms value)

Power Content In Multiple-tone Amplitude Modulation (AM)

Definition

A multiple-tone amplitude modulation is that type of modulation in which the modulating signal consists of more than one frequency components.

Mathematical Expression

Let us consider that a carrier signal A cos ω_ct is modulated by a baseband or modulating signal x(t) which is expressed as :

x(t) = V₁ cos ω₁t + V₂ cos ω₂t + V₃ cos ω₃t          ………………….(9)

We know that the general expression for AM wave is

s(t) = A cos ω_ct + x(t) cos ω_ct       ………………..(10)

Putting the value of x(t), we get

s(t) = A cos ω_ct + [V₁ cos ω₁t + V₂ cos ω₂t + V₃ cos ω₃t ] cos ω_ct

or,

The expression for AM wave in equation (12) can further be expanded as under:

s(t) = A cos ω_ct + m₁ A cos ω_ct cos ω₁t + m₂ A cos ω_ct cos ω₂t + m₃ A cos ω_ct cos ω₃t

Now we know that the total power in AM is given as

P_t = carrier power + sideband power

P_t = P_C + P_S …………(13)

The carrier power P_C is given as

Total or Net Modulation Index for Multiple-Tone Modulation

Definition

Let us consider that m_t is the total or net modulation indexes for a multiple-tone modulation.

Mathematical Expression

We know that for a multiple-tone modulation, the total power is expressed as :

Where, m₁ , m₂ , …… m_n are the modulation indexes for different modulating signals.

The power for AM wave is also expressed as

This is the desired expression for the total or net modulation index.