I was reviewing an electronics textbook the other day, and it made an offhand comment that "sinusoidal signals of the same frequency always add up to a sinusoid, even if their magnitudes and phases are different". This gave me pause; is that really so? Even with different phases?

Using EE notation, a sinusoidal signal with magnitude , frequency and phase is [1]. The book's statement amounts to:

The sum is also a sinusoid with the same frequency, but potentially different magnitude and phase. I couldn't find this equality in any of my reference books, so why is it true?

## Empirical probing

Let's start by asking whether this is true at all? It's not at all obvious that this should work. Armed with Python, Numpy and matplotlib, I plotted two sinusoidal signals with the same frequency but different magnitudes and phases:

Now, plotting their sum in green on the same chart:

Well, look at that. It seems to be working. I guess it's time to prove it.

## Proof using trig identities

The first proof I want to demonstrate doesn't use any fancy math beyond some basic trigonometric identities. One of best known ones is:

Taking our sum of sinusoids:

Applying (id.1) to each of the terms, and then regrouping, we get:

Now, a change of variables trick: we'll assume we can solve the following set of equations for some and [2]:

To find , we can square each of (1) and (2) and then add the squares together:

Using the fact that , we get:

To solve for , we can divide equation (2) by (1), getting:

Meaning that:

Now that we have the values of and , let's put them aside for a bit and get back to the final line of our sum of sinusoids equation:

On the right-hand side, we can apply equations (1) and (2) to get:

Applying (id.1) again, we get:

We've just shown that the sum of sinusoids with the same frequency is another sinusoid with frequency , and we've calculated and from the other parameters (, , and )

## Proof using complex numbers

The second proof uses a bit more advanced math, but overall feels more elegant to me. The plan is to use Euler's equation and prove a more general statement on the complex plane.

Instead of looking at the sum of real sinusoids, we'll first look at the sum of two complex exponential functions:

Reminder: Euler's equation for a complex exponential is

Regrouping our sum of exponentials a bit and then applying this equation:

The value inside the square brackets can be viewed as a complex number in its rectangular form: . We can convert it to its polar form: , by calculating:

In our case:

And:

Therefore, the sum of complex exponentials is another complex exponential with the same frequency, but a different magnitude and phase:

From here, we can use Euler's equation again to see the equivalence in terms of sinusoidal functions:

If we only compare the imaginary parts of this equation, we get:

With known and we've calculated earlier from the other constants

Note that by comparing the real parts of the equation, we can trivially prove a similar statement about the sum of cosines (which should surprise no one, since a cosine is just a phase-shifted sine).

[1] | Electrical engineers prefer their signal frequencies in units of radian per second. We also like calling the imaginary unit |

[2] | If you're wondering "hold on, why would this work?", recall that
any point (x,y) on the Cartesian plane can be represented using
polar coordinates with magnitude and angle . |