Very short theory question signals?

Question

My teacher asked us this question yesterday in the lecture but it didn't make any sense to me.

He asked: What do the coefficients of the exponential Fourier series represent? Also, what's the difference between this series and the Fourier transform for the same signal?

Thing is, I know how to find the coefficients. What does he mean by what do they represent?

Answer 1

The difference is that a Fourier series represents a periodic function. So Fourier transform and Fourier series can be compared if restricted to one period. That is, if you take a function with some compact support that fits into an interval $[a,a+\tau]$ and compute the Fourier transform, then a specific sampling of that transform will give coefficients for an exponential Fourier series with period $\tau$ that coincides with the given function inside the interval.

See also the sampling theorem and its generalizations.

Answer 2

Fourier series are studied with signals not only because they make the math required easier to do but also because the numbers you get mean something in the real world aspect of the problem. This real world piece is what he means by what they represent.