[Math] amplitude of sine wave with multiple frequencies

harmonic-analysistrigonometry

I'm having some troubles determining the amplitude/magnitude of the following equation.

$$
A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3)
$$
Since each part is at a different frequency, i cannot sum the magnitudes of each part.

I have also thought about using variations of the double/triple angle formulae and some basic trigonometric identities, so that I can write the equation under a single frequency, but by doing so, I am introducing some higher order terms, which seem to negate the ability to sum the amplitudes together.

For example, the term with $5\omega$ would look like
$$16\cos^5(\omega t)-20\cos^3(\omega t)+5\cos(\omega t)$$

I suppose another method would be to plot the first equation and then record the amplitude, but i would like a more generalised approach to solve this problem. All input is welcome and appreciated.

Best Answer

If you want the absolute maximum amplitude and the $\beta$s are such that the peaks can line up, you can just sum the individual waves. They don't line up perfectly very often and less often as more sine waves are included. Another approach is to do a root sum square of the individual amplitudes. This gives you a better feeling for the real amplitudes you will see, but will not be the maximum. If you go to Alpha you can plot it or again. The first has all the peaks lining up at $t=0$, the other has some delays added.

Related Solutions

[Math] Sum of Sinusoids with Same Frequency = Sinusoid (proof)

To avoid any confusion, let us state that

a (pure) sine has the form $A\sin(\omega t)$,
a (pure) cosine has the form $A\cos(\omega t)$,
a sinusoid has an arbitrary phase and one of the equivalent forms $A\sin(\omega t+\phi)$ or $A\cos(\omega t+\psi)$ - where $\phi$ and $\psi$ differ by a quarter turn.

So the sine and cosine are special cases of the sinusoid.

By the well-known addition formula, $$A\sin(\omega t+\phi)=A\sin(\omega t)\cos(\phi)+A\cos(\omega t)\sin(\phi)=A'\sin(\omega t)+A''\cos(\omega t).$$

This means that

a sinusoid can be expressed as a linear combination of a sine and a cosine,
conversely, a linear combination of sine and cosine can be represented as single sinusoid$^*$,
a linear combination of two or more sinusoids can be expressed as a linear combination of a sine and a cosine, hence can be expressed as a single sinusoid.

These properties no more hold if you mix sinusoids of different periods.

$^*$This is done by solving the system $$A\cos(\phi)=A'\\A\sin(\phi)=A'',$$ i.e. $$A=\sqrt{A'^2+A''^2}\\\tan(\phi)=\frac{A''}{A'}.$$

You will soon discover that complex numbers are intensively used in harmonic analysis, based on Euler's formula $e^{ix}=\cos(x)+i\sin(x)$.

Sinusoids can be represented as the imaginary part of $Ae^{i(\omega t+\phi)}=Ae^{i\phi}e^{i\omega t}=Ze^{i\omega t}$, where $Z$ is a complex number, that carries both the amplitude and the phase ($Z=A$ is real for a sine, $Z=iA$ is imaginary for a cosine).

Using this notation, adding sinusoids becomes a trivial matter:

$$Z_0e^{i\omega t}+Z_1e^{i\omega t}+Z_2e^{i\omega t}=(Z_0+Z_1+Z_2)e^{i\omega t}.$$

[Math] Sine wave with increasing frequencies in Matlab

If you look carefully at

y=sin(2*pi*(1+(5/100)*t).*t);

you'll see that the variable $t$ appears to the second power. You've graphed a section of $$ y = \sin(C t^2) $$ rather than $$ y = \sin(C t). $$

Looking closely at your picture, you can see that the spacing of the "humps" changes from left to right, so what you've shown can't possibly be a true sinusoid.

General matlab debugging hint: it's so easy to do stuff -- especially big stuff, like building large matrices, or functions represented by 100,000 samples --- that it's easy to delude yourself into thinking you must have done it right. Always write your code so that you can look at corresponding small cases during debugging -- 20 samples of a function of low frequency rather than 20,000 samples of a high-freq function, or the 2 x 2 output matrix from a small example rather than the 200 x 200 output from a large one.

Best Answer

Related Solutions

[Math] Sum of Sinusoids with Same Frequency = Sinusoid (proof)

[Math] Sine wave with increasing frequencies in Matlab

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