How do musical frequencies relate to the numbers in this signal decomposition?

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    Frequencies
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Discussion Overview

The discussion revolves around the mathematical steps involved in signal decomposition related to musical frequencies, specifically examining the transition between different steps in the decomposition process and the derivation of frequency values from cosine functions. The scope includes mathematical reasoning and conceptual clarification.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant questions the transition from step 1 to step 2 in the signal decomposition, indicating confusion over the mathematical manipulation involved.
  • Another participant points out that 654 can be expressed as 2 times 327, noting that the cosine function is even, which may relate to the transformation.
  • Several participants express uncertainty about the origin of the numbers 392 and 262, questioning how these values are derived and what happens to the constant π in the context of frequency calculations.
  • A participant mentions the relationship between frequency in Hz and radian frequency, suggesting that frequency is obtained by dividing the radian frequency by 2π.
  • There is a reference to the cosine addition formula, indicating that participants are trying to understand how to apply it to find the decomposed components of the signal.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the mathematical steps or the derivation of specific values, indicating that multiple competing views and uncertainties remain in the discussion.

Contextual Notes

Participants highlight potential missing assumptions regarding the mathematical transformations and the definitions of frequency, as well as unresolved steps in the derivation process.

bobsmith76
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My book reads

step 1 = 2 cos(654πt)cos(-130πt)
step 2 = 2 cos(2 * 327 * π * t)cos(130πt)
step 3 = f2 - f1 = 392 - 262 = 130 Hz

t = seconds
π = pi

I don't see how they get from step 1 to step 2, nor do I understand how they get from step 2 to step 3.
 
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I don't see how they get from step 1 to step 2
654 = 2*327, and cos() is an even function.
 
ok, where did the numbers 392 and 262 come from? What happened to pi?
 
bobsmith76 said:
ok, where did the numbers 392 and 262 come from? What happened to pi?

Frequency (in Hz) is equal to the radian frequency divided by 2 pi.

cos(a-b) + cos(a+b) = 2 cos(a) cos(b)

You are given the "a" and the "b" and you need to find the corresponding "a+b" and "a-b" of the decomposed (into individual cosine wave) signal.
 

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