DC component ($⟨f⟩$)

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 averageThe 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 $f=a_0/2+{\sum }_{n\ge 1}(a_n\cos (2\pi nt/T)+b_n\sin (2\pi nt/T))$, 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.^[ref]

Write $f=\mu +g$, so $⟨g⟩=0$. For a real signal with finite mean square,because the cross term $2\mu ⟨g⟩$ 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 $⟨f^2⟩=19$. Its varying part has root mean square $\sqrt{19-0.5^2}=\sqrt{18.75}\approx 4.330$. 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.