Summing cosines of different amplitude

[radiogaga35]

Hi there

I am trying to sum many cosines of different amplitude and phase shift, but same ang. frequency (it's not a coursework question). My first thoughts are to sum them two at a time (to simplify matters?), probably using complex numbers. I tried doing it symbolically in MATLAB but it wasn't able to simplify things. Supposing the ang. frequency is 1, I know that the solution can be written:

$$\displaystyle{ C\cos (t + \delta ) + D\cos (t + \varepsilon ) = E\cos (t + \varphi)}$$

where I would have to solve for E and phi. Or equivalently:

$$\displaystyle{Ae^{i(t + \delta )} + e^{i(t + \varepsilon )} = Be^{i(t + \varphi )}}$$

where I would have to solve for B and phi.

Then I split things into two equations (one using real part/cosines, other using imag. part/sines), and eliminate B. Unfortunately this approach doesn't seem to help, as I just end up with a messy arctan of sums of sines and cosines (of different amplitudes -- i.e. back to original problem!).

Any suggestions as to a more fruitful approach?

Thank you.

 

[radiogaga35]

Ok, nevermind, got it figured out! Just used a bit of geometry

 

[Architeuthis Dux]

Hi there,

It seems like you have a good understanding of the problem and have already attempted some approaches. One possible method that may help simplify things is to use the trigonometric identity:

\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)

Using this identity, you can rewrite the equation as:

C\cos(t)\cos(\delta) - C\sin(t)\sin(\delta) + D\cos(t)\cos(\varepsilon) - D\sin(t)\sin(\varepsilon) = E\cos(t)\cos(\varphi) - E\sin(t)\sin(\varphi)

Then, you can group the cosine and sine terms separately and solve for E and phi separately. This may help simplify the equations and make it easier to solve for the unknowns. Another approach could be to use the exponential form of cosine and solve for the unknowns using complex numbers. I would suggest experimenting with different approaches and seeing which one yields the simplest solution. Good luck with your problem!