Characterize Fourier coefficients

[schniefen]

Homework Statement

Consider the function $p(t)=\sin{(t/\tau)}$ for $0\leq t <2\pi \tau$ and $p(t)=0$ for $2\pi \tau \leq t < T$, which is periodically repeated outside the interval $[0,T)$ with period $T$ (we assume $2\pi \tau < T$). Which restrictions do you expect for the Fourier coefficients $a_j$ and which Fourier coefficient do you expect to be largest?

Relevant Equations

For even functions, $a_j=a_{-j}$. For odd functions, $a_j=-a_{-j}$. Also, I use the complex Fourier series, i.e. $\sum_{j=-\infty}^{\infty} a_j e^{i2\pi jt/T}$. Note that for even and odd functions the coefficients are real and imaginary respectively.

I would try to determine whether $p(t)$ is even or odd. This would be so much easier if the values of $\tau$ and $T$ would be specified, but maybe it's possible to do without it, which I'd prefer. If for example $\tau=1/2$ and $T=2\pi$, then $p(t)=\sin{(2t)}$ for $0\leq t <\pi$ and $p(t)=0$ for $\pi \leq t < 2\pi$. Then $p(\pi)=0$ and $p(-\pi)=p(-\pi+2\pi)=p(\pi)=0$. The function is even (so $a_j=a_{-j}$ and $a_j$ is real).

I am unsure which Fourier coefficient will be largest. Possibly it has something to do with the frequency $1/\tau$ and the $2\pi j/T$ in the exponent of $e$ in the Fourier series. I am unsure.

 

[osilmag]

You are on the right track. With the $e{}$ in your Fourier series you will have a periodic $\delta$ function.

 

[pasmith]

$p$ is neither even nor odd: it consists of a single complete period of $\sin(t/\tau)$ over $0 \leq t \leq 2\pi \tau$ followed by a constant zero over $2\pi \tau < t < T$. Thus $|f(-t)| = 0 \neq |f(t)|$ for $0 \leq t \leq 2\pi \tau$. The function is real, so $a_{{-}j}$ and $a_j$ are complex conjugates. The average is zero, so $a_0$ is zero.

What happens if $T$ is an integer multiple of $2\pi \tau$? What happens if this only approxiamtely true?

In this case it is easy to compute the coefficients $a_j$ expressly in order to confirm your hypotheses.