how to find out how many Fourier coefficients there are (which are not zeros)

Question

given a real periodic (with period $T_0$) signal $x(t)$ with fourier transform in which $$X(jw)=0\ \ \forall |w|\ge {6\pi \over T_0}$$ I know that the fourier series will have finite coefficients (5 in this case). how do I know in general cases, how many there are?

Answer 1

A $T$-periodic function has discrete frequency components at

$$f_n=\frac{n}{T},\quad n=0,1,2,\ldots\tag{1}$$

If you know that the function is band-limited with bandwidth $B$, this means that only the components at frequencies $f_n<B$ are non-zero. From (1) you get for the indices of the non-zero frequency components

$$n<BT\tag{2}$$

In your example the bandwidth is given in radians, so $B=3/T$, and from (2) you get $n<3$. You probably also count the negative frequency components which leaves you with indices $n\in\{-2,-1,0,1,2\}$ giving a total of $5$ coefficients. You might as well only count the non-negative frequency components, which would result in $3$ coefficients. Note that for a real-valued signal, the coefficients with negative indices are just complex conjugates of the coefficients with the corresponding positive index, so they are in fact redundant.