Orthogonality of complex exponentials

For integers $m$ and $n$,This orthogonality extracts matching coefficients when two Fourier series are integrated over one period.

For the trigonometric polynomial $F(t)={\sum }_{k=1}^N{b}_{k}e(kt)$, orthogonality of complex exponentials givesIndeed, expand each square and use ${\int }_0^{1}e(jt)dt=0$ for every nonzero integer $j$, and one for $j=0$.