Determining Frequencies that Exist in a Signal

[Kleric]

Homework Statement

Using trig identities from a calculus book or other, write out the results to the following modulations. State which frequencies exist in the signal s(t).

Homework Equations

a) s(t) = Acos(2∏f₁t) * Bcos(2∏f₂t)

The Attempt at a Solution

cos(s) * cos(t) = cos (s + t)/2 + cos (s – t)/2

s(t) = ((AB)/2)*(cos(2∏t(f₁ + f₂)) + cos(2∏t(f₁ - f₂)))

What I don't quite understand is the part that asks to state which frequencies exist in the signal. I understand that in modulation, half of the signal is shifted to the right by f₂, then the other half of the signal is shifted to the left by f₂. Which would make the answers f₂ and -f₂.

According to the answer it should be f₁, f₂, |f₂-f₁|, f₁+f₂.

 

[CWatters]

Your maths looks right.

s(t) = ((AB)/2)*(cos(2∏t(f1 + f2)) + cos(2∏t(f1 - f2)))

is similar to the sum of two waves..

= Cos(2∏F_usbt) + Cos(2∏F_lsbt)

when

F_usb = upper side band frequency = f1+f2
F_lsb = lower side band frequency = f1-f2

 

[CWatters]

In radio signal f2 would be the speech/music and f1 the RF carrier. Since speech and music are not pure tones the result is a band of frequencies either side of the carrier as shown in the diagram

http://en.wikipedia.org/wiki/Sideband

 

[rude man]

Homework Statement

Using trig identities from a calculus book or other, write out the results to the following modulations. State which frequencies exist in the signal s(t).

Homework Equations

a) s(t) = Acos(2∏f₁t) * Bcos(2∏f₂t)

The Attempt at a Solution

cos(s) * cos(t) = cos (s + t)/2 + cos (s – t)/2

s(t) = ((AB)/2)*(cos(2∏t(f₁ + f₂)) + cos(2∏t(f₁ - f₂)))

What I don't quite understand is the part that asks to state which frequencies exist in the signal. I understand that in modulation, half of the signal is shifted to the right by f₂, then the other half of the signal is shifted to the left by f₂. Which would make the answers f₂ and -f₂.

According to the answer it should be f₁, f₂, |f₂-f₁|, f₁+f₂.

The answer is wrong. f1 and f2 do not exist in the product, just the sums and differences, just as you have derived.

In ordinary amplitude modulation (think radio broadcast AM), , they do, but you have what is called "double-sideband, suppressed-carrier" modulation. Think of f1 as the carrier and f2 as the modulation, then you can see why it's called what it is.