How can we calculate the average power of a conventional AM signal?

[Dina Khaled]

Homework Statement

Trying to calculate Conventional AM average transmitted power of the modulated signal
s(t) = A (1+ka m(t) ) cos(2∏f t)
where
A: carrier amplitude
ka: amplitude sensitivity
m(t) : message signal
f: Carrier frequency

my professor's lecture notes calculate the average power as
0.5 A²( 1+< ka² m²(t)>)
where <.> is the time average.

Homework Equations

Average Power (for periodic Signals) = 1/T (∫₀^T x²(t) dt)

The Attempt at a Solution

average power for cos(2∏ft) =0.5
but applying the Average power rule I can't get this integral to be equal to the formula my professor's got!
Thanks in advance.

 

[rude man]

Rewrite s(t) as A km(t)cos(wt) + Acos(wt), w = 2 pi f.
Then do s^2(t). Hint: as you will recall, (a+b)^2 = a^2 + 2ab + b^2.

So making the appropriate substitutions for a and b,
avg{s(t)^2} = avg(a^2) + avg(2ab) + avg(b^2).
Etc.

 

[Dina Khaled]

so avg{s(t)^2} = avg{A^2 k^2 m(t)^2 cos^2(wt)} + avg{2A^2 k m(t)cos^2(wt)}+avg{A^2 cos^2(wt)}
still who says avg{A^2 k^2 m(t)^2 cos^2(wt)} = avg{cos^2(wt)} * avg{A^2 k^2 m(t)}

 

[rude man]

so avg{s(t)^2} = avg{A^2 k^2 m(t)^2 cos^2(wt)} + avg{2A^2 k m(t)cos^2(wt)}+avg{A^2 cos^2(wt)}
still who says avg{A^2 k^2 m(t)^2 cos^2(wt)} = avg{cos^2(wt)} * avg{A^2 k^2 m(t)}

Excellent question! The answer is that f(t) is assumed to be slowly-varying compared to cos(wt), and that the avergage of f(t) = 0. This is certainly the case in "ordinary" AM where f(t) is limited to a few sinusoidal (zero-avg.) KHz whereas w = 2 pi*1 MHz typical.

The prof should have stated this for you IMO.

 

[Dina Khaled]

Well, he did state that assumption, although what I understand is that the assumption that the average of the message is 0 means we will neglect the term avg{2A^2 k m(t)cos^2(wt)}.
However,my question is
if ∫ ab is not equal to ∫a ∫ b
then how come average(ab) = average(a) average(b)
Thanks !

 

[rude man]

Well, he did state that assumption, although what I understand is that the assumption that the average of the message is 0 means we will neglect the term avg{2A^2 k m(t)cos^2(wt)}.

Right.

However,my question is
if ∫ ab is not equal to ∫a ∫ b
then how come average(ab) = average(a) average(b)
Thanks !

Are you referring to the third term 2ab or the term A^2 k^2 m^2(t) cos^2(wt)? I'll assume the latter since you seem to agree that the 2ab term disappears. The answer is that m(t) varies slowly compared to cos(wt) so m(t) is approx. constant over 1 cycle of cos(wt) and so that term average boils down to A^2 k^2 <m^2>/2.