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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.

A dynamical observable is a function $\phi :X\to \mathbb{R}$ or $\mathbb{C}$ that assigns a measurable quantity to each state of a dynamical system. If the dynamics is $T:X\to X$, the observed sequence from state $x$ is $\phi (x),\phi (Tx),\phi (T^{2}x),\dots$. The state space may be high dimensional even when this record is scalar.

For a harmonic oscillator with state $(q,p)$, position $\phi (q,p)=q$ is one dynamical observable; energy is another. Position generally oscillates, while energy is constant along an ideal conservative trajectory. Composition $\phi \mapsto \phi \circ T$ describes evolution of the measured quantity instead of evolution of states. Its time average, $n^{-1}{\sum }_{j=0}^{n-1}\phi (T^{j}x)$, is the object compared with a space integral in ergodic theory.

An observable $h$ is a flow coboundary if it is the derivative of a function $u$ along a flow map. An integrated formulation isWhen $u$ is differentiable along the flow, this is $Fu=h$, where $F$ is the generating vector field. Spatial regularity must be specified separately: $u$ may be smooth or merely Hölder continuous while still being differentiable along every flow line.

The function $c(t,x)=u({\Phi }_{t}x)-u(x)$ is a cocycle over a dynamical system. Its identity follows by adding the two differences from $x$ to ${\Phi }_{t}x$ and from ${\Phi }_{t}x$ to ${\Phi }_{s+t}x$. Being a coboundary says that the accumulated observable can be recovered from endpoint values.

For a periodic orbit of period $T$, the endpoint values coincide, so its integral of $h$ is zero. This is the periodic orbit obstruction. If $u$ is bounded, then the time average of $h$ also vanishes:For a transitive Anosov flow, the Livšic theorem identifies when the periodic-orbit obstruction is sufficient. This is stronger than checking just one average or finitely many periodic trajectories.

A time average measures an dynamical observable along a single trajectory of a dynamical system. For a flow ${\Phi }_t$, define $A_{T}h(x)=T^{-1}{\int }_0^{T}h({\Phi }_{t}x)dt$. If the limit as $T\to \infty$ exists, it is the long-time average at initial state $x$. For a discrete map $T$, the corresponding expression is $n^{-1}{\sum }_{j=0}^{n-1}h(T^{j}x)$.

If $h({\Phi }_{t}x)=\cos t$, the average tends to zero although the observable never settles to a constant. The Birkhoff ergodic theorem supplies almost-everywhere existence for integrable observables under a probability-preserving map. Equality with a spatial average requires the appropriate ergodicity assumption; existence alone does not imply it.