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.

 

[rezeew]

Can you explain why these frequencies exist in the signal?The frequencies that exist in the signal can be determined by looking at the trigonometric identities used to solve the problem. In this case, we can see that the signal s(t) is a combination of two cosine functions, each with their own frequency (f1 and f2).

When we use the trig identity cos(s) * cos(t) = cos(s + t)/2 + cos(s - t)/2, we can see that the signal s(t) can be represented as a sum of two cosine functions with frequencies (f1+f2)/2 and (f1-f2)/2.

Therefore, the frequencies that exist in the signal s(t) are f1, f2, (f1+f2)/2, and (f1-f2)/2. This is because the original signal is a combination of two cosine functions at frequencies f1 and f2, and the trig identity shows that the resulting signal also contains components at frequencies that are the sum and difference of the original frequencies divided by 2.

Additionally, the absolute value of the difference between f1 and f2 (|f1-f2|) also exists in the signal. This is because the trig identity shows that the resulting signal also contains a component at the difference of the original frequencies divided by 2, regardless of whether this difference is positive or negative.

Overall, the frequencies that exist in the signal s(t) are f1, f2, (f1+f2)/2, (f1-f2)/2, and |f1-f2|.