Determining Frequencies that Exist in a Signal

Click For Summary

Discussion Overview

The discussion revolves around determining the frequencies present in a signal defined by the modulation of two cosine functions, specifically in the context of amplitude modulation. Participants explore the mathematical derivation of frequencies resulting from the product of these cosine functions and clarify concepts related to modulation types.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a mathematical derivation showing that the signal can be expressed as a sum of two cosine functions, leading to the conclusion that the frequencies present are f1, f2, |f2-f1|, and f1+f2.
  • Another participant agrees with the mathematical derivation but emphasizes that the resulting expression represents the upper and lower sideband frequencies in modulation.
  • A third participant contextualizes the frequencies by explaining that in radio signals, f2 represents the audio signal, while f1 is the RF carrier, suggesting that the modulation results in a band of frequencies around the carrier.
  • A later reply challenges the initial interpretation of the frequencies, asserting that in the specific case of double-sideband suppressed-carrier modulation, only the sums and differences of the frequencies exist, not f1 and f2 themselves.

Areas of Agreement / Disagreement

There is disagreement regarding which frequencies exist in the signal. Some participants assert that f1 and f2 are present, while others argue that only the sums and differences should be considered in the context of double-sideband suppressed-carrier modulation.

Contextual Notes

Participants highlight the distinction between ordinary amplitude modulation and double-sideband suppressed-carrier modulation, indicating that the interpretation of the frequencies depends on the type of modulation being discussed.

Who May Find This Useful

This discussion may be useful for students studying signal processing, modulation techniques, or those interested in the mathematical aspects of frequency analysis in communications.

Kleric
Messages
1
Reaction score
0

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.
 
Last edited:
Physics news on Phys.org
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∏Fusbt) + Cos(2∏Flsbt)

when

Fusb = upper side band frequency = f1+f2
Flsb = lower side band frequency = f1-f2
 
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
 
Kleric said:

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.
 

Similar threads

  • · Replies 2 ·
Replies
2
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 4 ·
Replies
4
Views
3K
Replies
4
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 6 ·
Replies
6
Views
5K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K