Two complex sinusoids at integer-multiple frequencies and are orthogonal over any interval of length . This is the algebraic fact that makes the Fourier series work.
Inner product of complex functions
For two complex-valued functions over an interval , define the inner product
where the asterisk is complex conjugation. If the inner product is zero, the two functions are orthogonal over the interval. Same idea as the dot product of vectors, extended to functions, and it plays the same role: a sum of mutually orthogonal pieces is uniquely decomposable, with each piece’s contribution determined independently.
The result
Take complex sinusoids and , with and integers. Their inner product is
If , the integrand is , and the integral is . If , the integrand is a complex exponential whose argument completes full cycles over the interval of length , and any complex exponential integrated over a whole number of its periods gives zero.
Complex sinusoids at integer-multiple frequencies are orthogonal over a period .
Why it matters
Once you have the synthesis equation
finding the coefficients is straightforward. Multiply both sides by and integrate over one period. By orthogonality, only the term in the sum survives, giving the analysis equation
The integral over extracts the contribution of the -th harmonic from the sum, because every other harmonic integrates to zero. That’s why the Fourier series can decompose a signal into harmonic coefficients at all.
The same orthogonality, with sines and cosines as the orthogonal set, gives the trigonometric Fourier series. It also generalizes to the Fourier transform (Plancherel’s theorem) and to other function-space decompositions like wavelets and polynomial bases.