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I want to be able to be able to get the amplitude of the following function:

$$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$

I am trying to find a way to get the amplitude of this function. This is usually simple for when there is one value of $\omega$ to consider, but I'm having a hard time thinking how to go about this.

One thing I did try was to just sum $||A||, ||B||, ||C||$ together. If we ignore the phases a, b, c, this gives an idea of what the max value obtained is, but I don't think it aptly describes the amplitude of the function as it is not sinusoidal in the traditional sense and most of the time, is below this max value. I noted that that min value is not equal to the negative of the max value either.

I played around with the RMS $$\sqrt{||A||^2+||B||^2+||C||^2}$$ but i'm sure if that is an appropriate approach.

Suggestions?

  • 1
    If you are thinking to solve this as $A\cos({k\omega t+\phi})$ and you are asking for what's the value of $A$ as real number. Its not possible. Amplitude is time varying function in this case and it is hard to simplify all three terms into one term. If you had two terms, You can see how amplitude varies with time.2012-05-10
  • 0
    Where does $D$ come from?2012-05-10
  • 1
    Why the notation $\| A \|$ etc. in the first displayed formula?2012-05-10
  • 0
    @DaanMichiels Sorry, I made a mistake there. I was afraid that that would be the case. Is there not even a proper way to approximate the amplitude in this case? You mentioned that if there was two terms, we could observe this. Can we therefore use superposition? i.e. the amplitude of the first two cosines, then that sum plus the third?2012-05-10
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    @suzu You still have a $D$ in there. It was Ramana Venkata who mentioned the two-term case.2012-05-10
  • 1
    The essential problem is that there is no well-defined notion of "amplitude" for a function that is not a pure sinusoid. You can consider the RMS value of the function instead, $\big(\frac1{T}\int_0^T f(x)^2 dx\big)^{1/2}$.2012-05-10
  • 0
    Your question would be trivial in electrical engineering if it were $$A\cos(k \omega t + a)+B\cos(k \omega t +b)+C\cos(k \omega t +c)$$2012-05-10
  • 1
    Are you the same user as Suzuki who asked the very similar http://math.stackexchange.com/questions/139505/amplitude-of-sine-wave-with-multiple-frequencies/139509#1395092012-05-11
  • 0
    My previous comment that there is no well-defined notion of amplitude for non-sinusoidal functions was mistaken. A natural definition of amplitude can be obtained by taking the magnitude of the corresponding [analytic signal](https://en.wikipedia.org/wiki/Analytic_signal) (see [picture](https://en.wikipedia.org/wiki/File:Analytic.svg)). However, this will be a function of time, and is only constant when the original function is sinusoidal.2012-05-12
  • 0
    [Here's an example with a function of your form.](http://www.wolframalpha.com/input/?i=plot%20cos%202t%20%2b%20cos%203t%20%2b%20cos%205t,%20sqrt%28%28cos%202t%20%2b%20cos%203t%20%2b%20cos%205t%29%5E2%20%2b%20%28sin%202t%20%2b%20sin%203t%20%2b%20sin%205t%29%5E2%29,%20-sqrt%28%28cos%202t%20%2b%20cos%203t%20%2b%20cos%205t%29%5E2%20%2b%20%28sin%202t%20%2b%20sin%203t%20%2b%20sin%205t%29%5E2%29)2012-05-12

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