Where did the phase go from the first to...

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Where did the phase go from the first to the second formula?

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>>7812056

It was expanded into a sine and a cosine.

The purpose of the fourier series is that you can simplify it to a series of coefficients. If you have to keep track of phase as well, well then you have to keep track of more things.

The first formula and the second formula are equivalent at corresponding A_n, a_n, b_n, and phi_n. In case you didn't know, sin(x+pi/2) = cos(x)

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>>7812092

I should clarify: generally you look at just the fourier series when you don't particularly care about the phase. If you also care about the phase, you'd more likely use the fourier transform. Some math majors might get angry at me for this explanation, but as a EE that's how I like to think about it.

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>>7812449

the other guy is bs-ing you

we care about the phase.

It's just that you always need two coefficients to define a sine wave : the amplitude and the phase.

However, you can also express a sine wave as the sum of a cos and a sin with two distinct amplitudes (but it's still two coefficients)

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>>7812458

Wow you're right. Is there a formula to calculate the coefficient of the sine and cosine from the sine phase?

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>>7812490

forgot to close the parenthesis in the third line, otherwise you did well pig

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>>7812495

He probably didn't capture it since it was out of range.

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>>7812490

Now I remember that formula, thanks!

I just don't get why adding more or less cosine to a sine wave just does the same as modifying the phase (and the amplitude) of the sine wave.

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>>7812449

Yes because it's a different function. However, you can just shift the sines and cosines in your fourier transform to make it all good again.

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>>7812490

Nice

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>>7812546

the addition formula is pretty clear, what is the issue here?

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>>7812583

I get the formula it's just that I was expecting it to create a strange periodic function instead of a nice shifted sine wave.

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>>7812615

well it would happen if the frequency were different, which is almost always the case... except here.

Take another look at the formula.

A sine and a cosine are just the same function shifter by 90° right?

the coefficients in front of them are just tuning parameters saying how much closer to a sine or to a cosine the function is.

If phi=0, you get a regular sin. All the weight is put on the sine part.

If phi increases, you get something closer to a cos, and more weight is put on the cos part.

So another way to see it is to consider the shift as a weight given to each function basically.

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>>7812659

Thanks, this makes sense, I'm just too used to seeing different frequencies.

Could I just ask one last question?

As far as I understood I can use Fourier transform on a periodic function (let's say a piano note) to switch between time (direct) and frequency (inverse) domain.

But how do I use it?

Let's say I have the value of the air pressure in a point in function of the time (real value), but the Fourier transform uses complex numbers, doesn't it?

Do I just set the imaginary part to 0 and call it done or did I get it wrong?

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>>7812659

Oh well thanks anyway, this is the first discussion I had on /sci/ which didn't turn into a shitposting thread

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>>7812711

look up how to compute the fourier coefficients

it might be complex numbers, but they can represent very real functions nonetheless.

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>>7812099

I'm a math major and yes we do get "angry". Just in case anyone is interested, there's a major difference between a Fourier series and a transform. The series works with periodic functions or with discrete functions defined for all n. The transform works with integrable functions in the whole real line (or real plane, or whatever you're dealing with).

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>>7812954

I'm an engineer and we get angry as well.

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>>7812056

If you are still around and are confused, you should read Feynman Lectures Vol,. 1 Chapter 50. Crystal clear derivation of Fourier Series from scratch. Free online at Caltech.

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>>7813192

It seems to be exactly what I was looking for

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