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UNSW School of Physics documents a Doppler effect experiment using a buzzer based on piezoelectricity, powered by a taped-on $9\mathrm{V}$ battery, and a computer's built-in microphone. Audacity records the acoustic wave. First record the stationary source to establish its baseline frequency; then attach it to a bicycle passing the stationary microphone. The published spectrogram shows a rapid decrease in frequency during passage.^[ref]

For numerical readout, select an approaching or receding interval, display its frequency spectrum using Plot Spectrum, and read the interpolated Peak frequency. The protocol recommends repeating with slightly different selections to test stability, and working outdoors to reduce reflections and standing waves. Longer windows refine frequency spacing while averaging over more motion. The baseline controls for the source's actual tone; repeated selections expose analysis-window sensitivity.^[ref]

Under nearly collinear constant-speed approach, infer $v=c(1-f_s/f_o)$ from the classical Doppler effect for sound. For recession the positive speed is $v=c(f_s/f_o-1)$. These inversions need a specified speed of sound. An offset path changes the projection, as in Doppler frequency along a trajectory; neither inversion measures the full speed near closest passage. The protocol's controls cannot remove arbitrary acceleration, background measurement noise or unresolved reflected paths. No new measurements are claimed here.

The experiment in A time of flight method to measure the speed of sound using a stereo sound card by Carlos C. Carvalho and others makes wave speed visible as a delay. Two powered microphones feed separate channels of one stereo line input. A clap beyond one end of their common axis reaches the nearer microphone first. The displayed signals show two separated onsets when enlarged; count their sample separation $N$ to obtain $\Delta t=N/f_s$ and infer speed of sound $c=d/\Delta t$ from the measured microphone spacing $d$. The reported sampling rate is $44100{\mathrm{s}}^{-1}$. Repeating at different spacings gives a distance-versus-delay plot whose slope estimates speed of sound; dependence on distance helps identify acoustic travel rather than a fixed electronic delay. These are the paper's apparatus and observed readout, not measurements made for this article.^[ref]

A controlled implementation also needs timing calibration for acoustic speed measurements. Put the microphones side by side to estimate the channel-delay offset ${\tau }_0$, swap channels to check its sign, and check the acquisition clock against an independent timing reference. Measure spacings between consistent points on the microphones, keep the source outside their span and on their axis, and fit $\Delta t=d/c+{\tau }_0$ by linear regression rather than forcing a zero intercept. Repeat pulses and compare the same onset criterion before reflections arrive. These are proposed calibration and control checks, not additional procedures claimed for the paper.

One sample corresponds to $1/f_s\simeq 22.7\mu \mathrm{s}$ for the reported sampling rate, so a delay of only a few samples has large relative measurement error. Background measurement noise, different microphone responses and threshold choice can move the apparent onset; reflections can supply a later competing pulse. Record temperature, keep air motion small, and compare several spacings. The result estimates speed of sound under those conditions; it cannot prove an exactly zero precursor below the instrument's sensitivity or establish finite propagation speed for every real medium. Both new diagrams in these sections show explicitly synthetic ideal wave equation solutions.^[ref]

An acoustic wavelength measurement makes an invisible spatial pattern readable with a ruler. The authored guide Interesting and inexpensive experiments for high school physics by Joe Wolfe places a loudspeaker near a pipe's open end, drives it with an oscillator, and slides a microphone attached to a metre rule through the pipe. An oscilloscope or computer displays the microphone signal. Its oscillation amplitude rises and falls with position; adjacent minima locate pressure nodes of a standing wave a distance $d=\lambda /2$ apart. A cap gives a pressure antinode at the closed end; an open end has a pressure node just outside it. These are pressure patterns, distinct from displacement patterns. The guide advises wavelengths longer than the pipe diameter, sufficient pipe length to locate both a node and an antinode, and frequency spectrum analysis to distinguish the intended tone from other sounds.^[ref]

The inference follows from superposition of two oppositely directed acoustic plane waves. In an ideal uniform, lossless pipe, taking equal pressure amplitudes $P$ and a rigid cap at $x=0$ givesThe time-varying pressure perturbation $p^{\prime }$ has local oscillation amplitude $2|P\cos (kx)|$. Its zeros are separated by $\pi /|k|=\lambda /2$. Unequal incident and reflected amplitudes leave nonzero minima, but their spacing remains $\lambda /2$ in this ideal model. A microphone detects that pressure variation through a changing electrical signal, rather than watching a wave crest move. Record the tone's frequency too, then infer phase speed $v_p=2fd$.

For proposed controls, keep the drive frequency and amplitude fixed during a scan, verify the frequency on the recorded signal, and repeat the scan in both directions. Fit several minimum positions $x_j=a+jd$ using linear regression to reduce dependence on one pair of positions. Check the ruler scale and use one consistent reference point on the microphone. Compare with the source switched off to locate background measurement noise; use frequency spectrum analysis to check for additional tones, and reduce drive if the trace ceases to resemble a sinusoid. Record air temperature and avoid strong air motion. These are suggested controls, not additional procedures or observations attributed to the guide.

The pipe must support approximately plane waves over the measured region: transverse structure at shorter wavelengths, the microphone's finite size, and disturbance by the ruler can compromise the interpretation. Weak reflection gives shallow minima, while damping and other reflections can shift the extrema of the simple two-wave model. Measure interior spacings rather than assuming the first minimum is exactly at the pipe lip. A short scan may not include enough minima to estimate $d$ reliably, and drift during sequential scanning can imitate spatial variation.

For an explicitly hypothetical calculation, $f=1000\mathrm{Hz}$ and $d=0.1715\mathrm{m}$ give $\lambda =0.3430\mathrm{m}$, $K\simeq 18.3183\mathrm{rad}{\mathrm{m}}^{-1}$, and $v_p=343.0\mathrm{m}{\mathrm{s}}^{-1}$. No such measurements are claimed here. If independent small measurement errors in $f$ and fitted $d$ have standard deviations ${\sigma }_f$ and ${\sigma }_d$, first-order error propagation gives ${\sigma }_{v_p}/v_p\simeq \sqrt{({\sigma }_f/f{)}^2+({\sigma }_d/d{)}^2}$. A systematic scale error or changing air conditions requires separate assessment.

An experimentally useful question is whether the disappearing oscillation follows viscous damping, and whether increasing the brake changes $b$ rather than $m$ or $k$. A published teaching protocol uses a cart on an air track, attached between two springs, with ceramic magnets providing eddy-current damping and an ultrasonic motion sensor recording position against time. Replace the magnets with equal masses in the baseline trial so the moving mass stays fixed. Obtain each spring constant from the slope of applied force versus extension, then use $k=k_1+k_2$. Time many baseline cycles to check ${\omega }_0\simeq \sqrt{k/m}$. Compare one-magnet and two-magnet free decays; the protocol calls for a fit of the full trace including its offset. ^[ref]

The following additional controls make the inference testable. Level the air track and hold its air supply fixed. Check the ultrasonic motion sensor at several stationary flag positions against a ruler, including the full planned travel range; record the offset and scale correction. Check elapsed acquisition time against an independent clock, and choose a sampling rate that resolves many samples within the shortest expected oscillation period. Record the resting equilibrium before and after each run. Release the cart from a small measured displacement with minimal initial velocity, with the driving apparatus inactive. Repeat at several release sizes and magnet settings while keeping the springs and total mass unchanged. These are proposed calibration and repeatability checks, not reported measurements from the source.

For underdamping, fit the recorded position to $y(t)=y_{\mathrm{eq}}+A{e}^{-\alpha t}\cos ({\omega }_{\mathrm{d}}t+\varphi )$ by nonlinear regression. The readout to inspect is a sequence of alternating extrema about $y_{\mathrm{eq}}$ with shrinking amplitudes, and the deviations of the fitted curve from the recorded points. The model predictsCompare the right-hand side, determined independently from mass and spring constants, with the fitted left-hand side. A good fit supports a constant-coefficient single normal mode over the observed range; it does not establish a universal microscopic force law. The published protocol denotes the decay rate by $b$ and the velocity-proportional coefficient by $c$; here they are $\alpha$ and $b$, respectively.

At critical damping or overdamping, fit the corresponding nonoscillatory response including its offset and both initial conditions, rather than forcing a damped cosine onto the data. Adjusting the brake can explore the transition, but the cited protocol does not establish that its one-magnet or two-magnet settings attain critical damping. A near-critical trace over a short interval can be hard to distinguish from neighbouring damping ratios.

Important limitations are finite sensor range, missed or reflected acoustic wave pulses, measurement noise near equilibrium, nonzero release velocity, and position-dependent magnetic braking. Changing the release size tests for quadratic drag or nonlinear spring response; inspecting deviations near turning points can expose kinetic friction. Slow drift of the offset can imitate a long overdamped tail. Report uncertainties in fitted rates and independently measured $m,k$; absence of a visible crossing only bounds the behaviour within the instrument's sensitivity and recording interval. No numerical experimental results are asserted here.

A feedback delay means a controller reacts to an earlier plant output. An accurate measurement can still be stale: when the plant output has already crossed the reference signal, a controller using the earlier value may continue pushing it away from the target. In a pure constant-delay model, $y_m(t)=y(t-\tau )$ with $\tau \ge 0$. This isolates timing from measurement noise; it does not describe every sampling or communication mechanism. ^[ref]

For the ideal scalar controlled plant $\dot{x}=u$, a zero reference signal, no control disturbance, and proportional control with constant feedback gain $k>0$, the equations areWith $\tau =0$, this is an ordinary differential equation with exponential decay $x(t)=x(0)e^{-kt}$. With $\tau >0$, it is a delay differential equation, requiring an initial history of delayed feedback. The delay stability of proportional integrator control depends on the product $k\tau$: even this simple controlled plant loses asymptotic stability at $k\tau =\pi /2$.

In a linear Gaussian state-space model with process noise covariance $Q$ and exact scalar observation $X_t=F{S}_t$, the predictive state covariance includes $Q$ even though there is no separate measurement noise. Here $R_t=G{P}_{t-1}{G}^{\top }+Q$ denotes predicted state covariance; it differs from the measurement-noise notation in the general Kalman filter equations. The observation innovation variance is $V_t=FG{P}_{t-1}{G}^{\top }{F}^{\top }+FQ{F}^{\top }$. Omitting the second term incorrectly removes the current process noise. If $V_t>0$, Gaussian conditional expectation gives the posterior covariance matrix $P_t=R_t-R_t{F}^{\top }F{R}_t/V_t$. This can be a singular matrix because the observed linear combination is then known exactly.

For a concrete example with two state components that are dimensionless quantities, take predicted mean $(10,0{)}^{\top }$, predicted covariance matrixand exact observation of the first component, $F=(1,0)$, $X=12$. The innovation is $2$, its variance is $4$, and the Kalman gain is $(1,1/2{)}^{\top }$. The posterior mean and posterior covariance matrix are thereforeThe observed component now has zero variance. The second component remains uncertain, but its posterior mean changes and its variance falls from $9$ to $8$ because its prediction error had nonzero correlation with that of the first. This is the mechanism by which measurements of one component can inform another.

A laboratory determination of the gravitational constant must connect a sensor's reading to a gravitational signal from independently characterized masses. For an ideal two-point mass force measurement, invert the model:These are signed small changes obtained by taking the logarithmic derivative, not an instruction to add standard uncertainties with signs. For independent small input uncertainties, linear propagation of uncertainty instead givesCorrelated inputs require their covariances too. In particular, common geometric calibration errors must not be treated as independent simply because several distances were recorded.

For a concrete model check, hold the actual force and masses fixed but overestimate a true $r=0.100\mathrm{m}$ by $1\mu \mathrm{m}$. The exact inferred ratio isThe resulting bias is $20.0001$ parts per million, already comparable with the relative standard uncertainty of the recommended gravitational constant. A sensor can resolve very small changes while its inferred answer remains limited by source geometry. The example concerns one idealized separation; it is not an uncertainty estimate for any actual experiment.

Making the source mass larger or bringing it closer strengthens the signal, but also makes detailed mass density and geometry important. Nearby apparatus supplies its own Newtonian gravitational field; electromagnetic shielding does not remove that attraction. Sensor offsets, changing temperature, vibration and source motion can imitate or alter a signal. The Cavendish experiment illustrates calibration of a torsion constant; a different sensor requires its own calibrated response and a full source model.

Repeating readings can reduce independent measurement noise, but a fixed scale error survives averaging. A useful experiment deliberately changes the source configuration, checks whether the signal follows the predicted change, and characterizes what moves with the source. A systematic error that changes with that configuration also survives background subtraction. Comparing methods with different sensors and different dominant errors is therefore valuable.

An inverse problem is an ill-posed problem if it fails any requirement of a well-posed problem: existence for every admissible datum, uniqueness, or continuity of the reconstructed object as the data change. These requirements depend on the chosen spaces and norms, not just the equation. Clason's regularization notes formulate this distinction explicitly.^[ref]

Complete averaging illustrates lost information:The difference mode $q_{-}=(1,-1{)}^T/\sqrt{2}$ belongs to the null space because $B{q}_{-}=0$. Even restricting both pixels to $[0,1]$ leaves infinitely many indistinguishable inputs. This is failure of uniqueness. Data such as $(0.51,0.49{)}^T$ also have no exact preimage under $B$, illustrating failure of existence when arbitrary two-component observations are admissible.

The two-pixel matrix $A$ in the inverse problem example has a weakly observed difference mode, not a null space mode. Its inverse is continuous: this finite-dimensional problem is a well-posed problem with noise-sensitive conditioning. Small positive singular values alone do not imply mathematical ill-posedness. In infinite-dimensional problems, positive singular values approaching zero can make inversion unbounded: unit solution modes then produce data tending to zero. This destroys continuity even without a null space. Regularization of an inverse problem must distinguish missing information from information recoverable only at the cost of amplifying measurement noise.

Image deblurring reconstructs a sharper image from observations described by a blur model. In the linear model, a matrix or linear operator mixes values of the unknown image; measurement noise perturbs the observed result. The inverse problem must therefore recover attenuated distinctions while controlling amplification of measurement noise, as the two-pixel example illustrates.

For the Kalman filter model, the innovation is $e_t=y_t-H_t{m}_t^{-}$. Its conditional covariance matrix before the measurement isThis includes uncertainty in both the predicted state and the measurement noise. It is a covariance in observation coordinates; $P_t^{-}$ is a covariance in state coordinates, and need not even have the same dimensions.

An inverse problem reconstructs an unknown object $x$ from observed data $y$ and a specified model $A(x)=y$. The associated forward problem predicts the observations from $x$. In image deblurring, $x$ contains sharp pixel values and a matrix $A$ mixes neighbouring pixels; in tomography, the observations are integrals through the object. The appeal is that indirect measurements can reveal otherwise inaccessible structure. The difficulty is that the forward problem may erase precisely the distinctions we want to recover.

Consider this completely computable two-pixel image deblurring model:Each measured pixel retains $55\%$ of its own value and receives $45\%$ of its neighbour's. Adding measurement noise $e=(0.01,-0.01{)}^T$ gives $y^{\delta }=(0.56,0.44{)}^T$, whose direct reconstruction is $A^{-1}{y}^{\delta }=(1.1,-0.1{)}^T$. The Euclidean norm of the reconstruction error is $0.1\sqrt{2}$, ten times the data error $0.01\sqrt{2}$. A negative second pixel appears even though both observed pixels are positive.

Despite this wrong reconstruction, its residual of an inverse problem is exactly zero: the predicted data equal the noisy observations. Thus matching the data perfectly does not certify reconstruction accuracy. The relevant question is how large a change in the unknown could hide within the measurement uncertainty. A condition number quantifies that sensitivity for an invertible matrix; the null space identifies changes that no improvement in precision can reveal.

The exact singular value decomposition explains this amplification. With the orthonormal basisthe mean mode survives unchanged while the difference mode shrinks tenfold. Direct inversion multiplies difference-mode measurement noise by ten. The condition number is ${\kappa }_2(A)={\sigma }_{\max }/{\sigma }_{\min }=10$. The beauty of the singular value decomposition is this separation into independent channels: it identifies which information survives and which needs protection by regularization of an inverse problem.

The Kalman filter alternates a Kalman prediction step with a Kalman measurement update. It carries a state estimate forward using a state-space model, then adjusts it by the part of the new observation that was not predicted: the innovation. The Kalman gain determines the adjustment from the uncertainty of the prediction and the measurement noise. A noisy measurement should have less influence than an equally surprising precise one.

For a precise discrete-time statement, let the state and observation be vectors in real vector spaces obeying the linear state-space modelThe matrices and control inputs are known. Assume a Gaussian random vector $x_0$ with mean $m_0$ and covariance matrix $P_0$, and mutually independent centered Gaussian noises $w_t,v_t$, independent of $x_0$, with covariance matrices $Q_t,R_t$. Suppose $R_t$ is positive-definite, so the observation innovation covariance is invertible. Then the recursions below give the exact posterior mean $m_t=\mathbb{E}[x_t\mid y_1,\dots ,y_t]$ and posterior covariance matrix $P_t$, hence the full conditional multivariate normal distribution. They also minimize mean squared error. Without the Gaussian assumptions, suitable second-order assumptions still give best affine state estimates, but do not generally give an exact posterior distribution. ^[ref].

For an exact scalar example, suppose the predicted state has normal distribution $N(10,4)$ and a measurement reports $y=14$ in the same units. With measurement noise variance $R=1$, the Kalman gain is $4/(4+1)=4/5$ and the posterior distribution isWith the same prediction and observed value but measurement noise variance $R=16$, the Kalman gain becomes $1/5$ and the posterior distribution is instead $N(10.8,3.2)$. Both updates move toward the observation; the more precise measurement moves the estimate further and reduces uncertainty more.

The useful compression is exact under the stated model: the two objects $m_t,P_t$ contain everything from the past observations needed for the next update. Their size depends on the state dimension, not on the length of the observation history. This makes causal state filtering practical when observations arrive continually. It also gives Linear-quadratic-Gaussian control an estimate of the state on which to act.

Rudolf E. Kalman's 1960 paper organized linear filtering around state transitions, error covariance and recursive computation, connecting estimation with optimal regulation. Its introduction acknowledges earlier work by Wiener, Bode, Shannon and others; the useful historical claim is this particular formulation, rather than that estimation began in 1960. ^[ref].

For underdamping, let $x_n,x_{n+1}$ be successive same-sign extrema of displacement relative to equilibrium, one full oscillation period apart. The logarithmic decrement isComparing extrema $N$ full cycles apart instead gives $\delta =N^{-1}\ln (|x_n|/|x_{n+N}|)$. Adjacent extrema of opposite sign are only half a cycle apart and give half this decrement. This distinction matters when inferring a damping ratio. Near critical damping, a full second same-sign extremum may be too small to resolve above measurement noise; a full-trace fit with uncertainty is then more informative than a peak ratio.

The measured output is the information about the plant output available to the controller. It may differ from the actual plant output because of measurement noise or feedback delay. In the ideal examples $y_m=y=x$, so the controller observes the actual current state exactly.

Measurement noise is a fluctuating contribution to measurement error. An additive model for a measured output is $y_m(t)=y(t)+n(t)$, where $y$ is the actual plant output. A controller using feedback control can react to $n$ as though it were a change in the plant output. This effect is absent from the ideal noiseless examples under control theory; an errors-in-variables model represents a related distinction between latent quantities and their noisy observations.

An ideal PID controller combines proportional control, integral action, and a term proportional to the derivative of the tracking error. With $k_d=0$, it gives proportional-integral control. The integral stores past error, while the derivative responds to its current rate of change. Practical implementations must account for measurement noise, which an ideal differentiator can amplify. ^[ref].

Process noise is the random contribution to a state transition in a state-space model. It represents uncertainty in how the state evolves. Measurement noise instead affects the observed value of the state. In the local-level state-space model $x_t=x_{t-1}+w_t$, $y_t=x_t+v_t$, the first disturbance changes the state itself; the second changes its measurement.

Proportional control chooses a control input proportional to the current tracking error. For the ideal controlled plant and constant reference signal $r$, its ordinary differential equation is $\dot{x}=k(r-x)+w$. With $w=0$ and initial condition $x(0)=x_0$, the exact response isFor $r=1$, $x_0=0$ and $k=2$, this gives $x(1)=1-e^{-2}\simeq 0.864665$ and $x(2)=1-e^{-4}\simeq 0.981684$. To verify the formula, substitute it into the ordinary differential equation and check the initial condition; its decaying exponential function explains the approach to the goal.

With a constant control disturbance, setting $\dot{x}=0$ gives the equilibrium point $x_{\ast }=r+w/k$ and steady-state tracking error $e_{\ast }=-w/k$. For $w=0.2$, doubling the feedback gain from $k=2$ to $k=4$ reduces the offset from $0.1$ to $0.05$. This improvement is a statement about the ideal model: it neglects feedback delay and measurement noise, so it does not establish that arbitrarily large feedback gains are safe. ^[ref]

In the first diagram's switched control disturbance experiment, $k=2$ and $w=0.2$ from $t=2$ give, for $t\ge 2$,Thus $x(4)\simeq 1.097833$ and $x(8)\simeq 1.099999$. At the equilibrium point, the controller supplies $u=-0.2$ to cancel the control disturbance. Proportional control needs a nonzero tracking error to produce that persistent correction.

Regularization replaces sensitive inversion by a controlled approximation. For a bounded linear operator between Hilbert spaces, a linear regularization is a family of bounded linear operators $R_{\alpha }$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ for each datum in the domain of the Moore–Penrose inverse of an operator. Noisy-data convergence additionally requires a suitable regularization parameter choice.^[ref] Nonlinear variational regularization can instead use nonquadratic penalties.

The noise-bias decomposition for linear regularization separates two costs: amplifying measurement noise and changing the exact-data reconstruction. Truncated singular value decomposition discards weak modes; Tikhonov regularization attenuates them smoothly. A quadratic norm penalty expresses a preference for smaller reconstructions. Neither method manufactures a missing null space component: with complete averaging, the minimum-norm least-squares solution for $(0.5,0.5{)}^T$ is $(0.5,0.5{)}^T$, even if the original image was $(1,0{)}^T$. The preference selects a representative; extra observations or justified constraints are needed to identify the original.

For Tikhonov regularization, the Tikhonov stability bound limits transmitted measurement noise of norm at most $\delta$ to $\delta /(2\sqrt{\alpha })$. Taking $\alpha \to 0$ while $\delta /\sqrt{\alpha }\to 0$ gives a sufficient rule for convergent regularization of an inverse problem. Holding $\alpha$ fixed leaves approximation bias; letting it vanish too quickly can admit overwhelming measurement noise.

For the state-space model $S_t=G{S}_{t-1}+w_t$, $X_t=F{S}_t+v_t$, assume independent Gaussian process noise and measurement noise of variances $W,V$, independent of the prior state and previous observations. A normal distribution for the filtered state, with mean $m$ and variance $P$, predicts mean $Gm$ and variance $R=G^{2}P+W$. Here $R$ denotes predicted state variance, rather than the observation-noise covariance notation used in the vector equations above. Gaussian conditional expectation then gives posterior mean $Gm+K(X-FGm)$ and posterior variance $RV/(F^{2}R+V)$, where the Kalman gain is $K=FR/(F^{2}R+V)$.

Mere weak white noise assumptions do not imply normal posterior distributions: independent process noise with a Rademacher distribution makes the next state a mixture of two shifted normal distributions. The Kalman filter can still describe a best affine state estimate under appropriate second-order assumptions, but that is distinct from an exact Gaussian conditional law.