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Reachability in control theory asks about a particular initial state $x_0$ and target $x_1$: does some admissible control input make $x(T)=x_1$ at the declared time $T$, or at some allowed duration? Controllability requires this for every pair in the declared state space. This is a property of state evolution, distinct from graph reachability.

For a linear time-invariant system, solving the ordinary differential equation by a matrix exponential givesThe first term is the unforced endpoint. Reachability in control theory asks whether the control input can supply exactly the displacement $x_1-e^{AT}{x}_0$. The timing matters because the same command is carried by different amounts of subsequent dynamics. ^[ref]

Here is a contrasting mathematical example with one shared control input:The controllability matrix has rank of a matrix one. Since $\frac{d}{dt}(z_1-z_2)=0$, the difference is invariant under every allowed control input. From $(0,0)$ the reachable set is the diagonal $z_1=z_2$ when input amplitudes are unrestricted: $(1,1)$ is reachable at $T=2$ using $u=1/2$, while $(1,0)$ is impossible at every duration. Having a control input appear in both state equations did not give independent control of both states. This reachable set is pictured alongside the bounded double integrator below.

For a linear time-invariant system, a sinusoidal wanted signal at ${\omega }_s$ and an unwanted sinusoidal component at ${\omega }_n$ acquire their separate frequency response factors. The ratio of their output amplitudes, divided by their input amplitude ratio, is $\eta =|H({\omega }_n)|/|H({\omega }_s)|$. An amplitude attenuation factor of $F=1/\eta$ corresponds to a factor $F^2$ in squared-amplitude power ratios under a common impedance convention. The unwanted component's own absolute attenuation is $|H({\omega }_n)|$, which need not equal its attenuation relative to the wanted signal.

Signal processing transforms or analyzes signals to extract information, change a representation, or improve a measurement. Examples include smoothing a sensor record, selecting a band of frequencies, and decomposing a recording through Fourier analysis. A signal is a mathematical representation of a physical record; deciding which variation is meaningful and which is noise is part of the problem.

A linear filter offers a simple entry point. Its output is assembled from weighted copies of its input. A moving average can reduce rapid variation, but it also changes timing and can erase useful detail. The worked example below makes that tradeoff explicit rather than treating a smoother plot as automatic improvement.

Two complementary views make the subject appealing. An impulse response describes the effect of a localized input; a frequency response describes the effect of a sustained oscillation. For a linear time-invariant system, both describe the same operation, connected by convolution and Fourier analysis. ^[ref].

Model an ideal resistor of constant electrical resistance $R>0$ at uniform absolute temperature $T$ by its Thévenin equivalent: a zero-mean random voltage source in series with a noiseless $R$. In the classical range its one-sided, open-circuit voltage power spectral density, per hertz, iswhere $k_{\mathrm{B}}$ is the Boltzmann constant. Thus a positive-frequency interval of width $df$ contributes $4k_{\mathrm{B}}TRdf$ to the mean square voltage. This is approximately white noise only where the electrical resistance is effectively constant and $hf\ll k_{\mathrm{B}}T$, with $h$ the Planck constant. The ideal constant spectrum cannot describe arbitrarily high frequencies.^[ref]

For a real voltage, an equivalent two-sided power spectral density is $S_v^{(2)}(f)=2k_{\mathrm{B}}TR$ over signed frequencies. With a real linear time-invariant system of frequency response $H(f)$, the same output mean square isAngle brackets denote an expected value; for this zero-mean noise the mean square is its variance. Do not integrate $4k_{\mathrm{B}}TR$ over both signs of frequency: that counts the same fluctuations twice. All integrals here use frequency $f$ in hertz, not angular frequency $\omega =2\pi f$. The amplitude spectral density is $\sqrt{4k_{\mathrm{B}}TR}$ in $\mathrm{V}/\sqrt{\mathrm{Hz}}$, whereas the root mean square voltage requires a specified measurement signal bandwidth.

A transfer function packages the response of a linear time-invariant system into a rule for turning a plant input into a plant output. Instead of solving a differential equation separately for every signal, take a Laplace transform, multiply by $G(s)$, then transform back. This also makes it possible to compare slow and fast oscillations through the continuous-time frequency response.

Here consider a continuous-time, causal, finite-dimensional linear time-invariant system with one plant input $u(t)$ and one plant output $y(t)$. With every initial state set to zero, its transfer function is defined byThe ratio $G(s)=Y(s)/U(s)$ is used where $U(s)\ne 0$ in a common Laplace convergence region; $G$ describes the system, independently of that particular plant input. Zero initial state matters: stored motion or another nonzero initial condition can contribute to the plant output even when the plant input vanishes. ^[ref]

The first-order low-pass filter below shows the same transfer function predicting both a step and a sustained sinusoid. Its pole gives a decay rate, its value at zero gives the eventual gain, and its values along the imaginary axis give the continuous-time frequency response. These descriptions make a useful bridge between a physical relaxation model and control theory calculations.