= Period average
{title2=$\langle f\rangle=\frac1T\int_{t_0}^{t_0+T}f(t)\,dt$}
The period average extracts the reference level of an <integrable function> that repeats. For a real- or complex-valued <periodic function> $f$ with positive <period of a function> $T$, assume $f$ is <Lebesgue integrable> on every bounded interval. Its average is
$$
\boxed{\langle f\rangle=\frac1T\int_a^{a+T}f(t)\,dt.}
$$
The answer is independent of the starting point $a$. Splitting a shifted interval at the old endpoint and using $f(t+T)=f(t)$ moves its extra piece back into the missing piece. The interval length must be a <period>; an arbitrary window can give a different average.
For $g(t)=2+\sin(2\pi t)+\tfrac12\cos(4\pi t)$, every length-one window has average $2$: both oscillating <integrals> vanish over that window. A half-unit window instead has average $2+(2/\pi)\cos(2\pi a)$, which depends on its start. Thus a short sample of an oscillation can bias an estimate of its reference level even when the full-cycle average is known exactly.
\Image[/period-average-full-and-partial-windows.png]
{title=A full-period average is independent of its starting point, while a shorter window is not}
In the complex <Fourier series> convention with basis $e^{2\pi int/T}$, the zero-frequency <Fourier coefficient> is $c_0=\langle f\rangle$. This assertion defines the coefficient even when an additional convergence theorem is needed to reconstruct $f$ from its series. Strang's §4.1 explicitly identifies that constant coefficient with the average.<Computational Science and Engineering by Gilbert Strang/Fourier Series for Periodic Functions>[§4.1; pp. 317]{ref}
Averaging a square measures something different from averaging its original <function>. For the illustrated $g$, <orthogonality> of the trigonometric terms gives $\langle g^2\rangle=4+1/2+1/8=37/8$, whereas $\langle g\rangle^2=4$. Its <root mean square> is therefore $\sqrt{37/8}$. Similarly, the <period average> of $\sin^2$ is $1/2$. A zero mean need not imply a zero mean square.
Back to article page
