= DC component
{c}
{title2=$\langle f\rangle$}
The DC component of a periodic signal is its constant, zero-frequency part. For an integrable signal $f$ of period $T>0$, it is the <time average>
$$
\mu=\langle f\rangle=\frac1T\int_{t_0}^{t_0+T}f(t)\,dt.
$$
The result is independent of $t_0$ when a whole period is used. In a complex <Fourier series> with coefficients $c_n=T^{-1}\int_0^T f(t)e^{-2\pi int/T}\,dt$, the DC component is $c_0$. In the convention $f=a_0/2+\sum_{n\geq1}(a_n\cos(2\pi nt/T)+b_n\sin(2\pi nt/T))$, it is $a_0/2$, not $a_0$. The constant character and <Fourier coefficient> at zero are part of the compact-group <Fourier series> description.<245C, Notes 2: The Fourier transform by Terence Tao>[Section 2, constant character and Corollary 22, Fourier expansion]{ref}
Write $f=\mu+g$, so $\langle g\rangle=0$. For a real signal with finite mean square,
$$
\langle f^2\rangle=\mu^2+\langle g^2\rangle,
$$
because the cross term $2\mu\langle g\rangle$ vanishes. Thus total <root mean square> combines the DC component and the varying part quadratically. A meter that removes the DC component and a meter reporting total <root mean square> need not give the same number.
For an original rectangular waveform, take $f=8$ for one quarter of each period and $f=-2$ for the rest. Then $\mu=8/4-2(3/4)=0.5$, whereas $\langle f^2\rangle=19$. Its varying part has <root mean square> $\sqrt{19-0.5^2}=\sqrt{18.75}\approx4.330$. The offset is small even though the instantaneous values are not.
An observation window that covers a noninteger number of periods can give a different sample average. For example, a zero-mean <sinusoid> observed over only part of a cycle need not have a zero window average. Finite-window averaging therefore estimates the periodic DC component only under an appropriate sampling model; a nonzero finite record mean can also reflect drift or an incomplete cycle.
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