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Apply the product-to-sum formula to the voltage model:The carrier wave component has peak voltage amplitude $A$; each AM sideband has peak voltage amplitude $Am/2$. These are temporal frequency spectrum components, unlike the spatial sidebands used in pattern stability. The separation $\Omega >0$ makes the three positive angular frequencies distinct, and $\Omega <{\omega }_c$ keeps the lower one positive. The span between sidebands is $2\Omega$, or $2f_m$ in hertz when $f_m=\Omega /(2\pi )$; this is the single-tone span, not a universal signal bandwidth for arbitrary messages.

A one-sided cosine-amplitude display assigns $A$ to the carrier line. A two-sided complex-exponential expansion instead assigns $A/2$ at each of $\pm {\omega }_c$, and $Am/4$ at each of the four sideband locations. The signal is identical; confusing the conventions creates a factor-of-two error. An ideal infinite-duration sinusoid gives a spectral line. A finite recording and finite analyser resolution broaden the displayed peak.

For a voltage applied across an ideal resistor of electrical resistance $R>0$, a component of peak voltage $B$ has root mean square voltage $B/\sqrt{2}$ and mean power $B^2/(2R)$ by Ohm's law. ThusThe proof of additivity is to average $s(t{)}^2/R$ over a long time: cross-products of distinct frequencies have zero mean. Consequently the whole signal's root mean square voltage is $A\sqrt{(1+m^2/2)/2}$, not simply its maximum wave envelope divided by $\sqrt{2}$. The carrier-component $A/\sqrt{2}$ remains unchanged as $m$ varies.

With $A=2\mathrm{V}$, $R=50\Omega$ and $m=0.5$, the carrier supplies $40\mathrm{mW}$ and each $0.5\mathrm{V}$ peak sideband supplies $2.5\mathrm{mW}$, giving $45\mathrm{mW}$ total. At $m=1.4$ each sideband is $1.4\mathrm{V}$ peak and supplies $19.6\mathrm{mW}$, giving $79.2\mathrm{mW}$ total. Neither case changes the carrier-component power.

The fraction of mean power in the message-bearing sidebands is $m^2/(2+m^2)$. It is $1/9$ at $m=0.5$ and $1/3$ at $m=1$. Conventional AM spends the remaining power on a carrier that enables an inexpensive envelope detector. Increasing $m$ beyond $1$ increases this fraction in the linear model but breaks ordinary envelope recovery. Removing the carrier gives suppressed-carrier amplitude modulation and changes the receiver requirement.

The bandwidth is the width of a specified frequency interval, $\Delta \nu ={\nu }_h-{\nu }_l$. Its meaning depends on how the interval is selected: a signal bandwidth and an effective noise bandwidth can differ even for the same receiving system.

The capacitive bridging complex-error bandwidth is the largest frequency over which a receiving port preserves the source phasor within a specified complex relative gain error. The criterion includes phase shift as well as amplitude. A large DC input impedance alone cannot establish this bandwidth when input capacitance matters. The manufacturer's input models explicitly include resistive and capacitive paths.^[ref]

For the scalar impedance bridging with capacitive input model, assume a positive real source electrical resistance $R_s$, receiver resistance $R_{\mathrm{in}}>0$, and input capacitance $C_{\mathrm{in}}>0$ in parallel. Let $V_s$ be the unloaded source phasor. A voltage divider givesFor tolerance $0<\eta <1$, the complex relative gain error isIts square is strictly increasing in $x^2$, since its derivative is $(1+2g)/[(1+g{)}^2+x^2{]}^2>0$. Thus satisfying $E(f)\le \eta$ defines a continuous frequency interval starting at DC, whenever it is feasible. Rearranging givesThe DC error floor is ${\eta }_0=g/(1+g)$. If $\eta <{\eta }_0$, no frequency meets the tolerance; if $\eta ={\eta }_0$, only DC does. The boxed positive bandwidth applies when $\eta >{\eta }_0$.

For an original near-limit example, choose $R_s=10\mathrm{k\Omega }$, $R_{\mathrm{in}}=1\mathrm{M\Omega }$ and $C_{\mathrm{in}}=100\mathrm{pF}$. Then $g=0.01$ and the DC error is about 0.9901 percent. A 1 percent full-complex tolerance leaves onlyAn amplitude-only 1 percent criterion, $|H|\ge 0.99$, instead allows about $2273.47\mathrm{Hz}$. The half-power (approximately $-3\mathrm{dB}$) signal bandwidth, relative to DC, is $(1+g)/(2\pi R_s{C}_{\mathrm{in}})\approx 160.75\mathrm{kHz}$. These are three distinct specifications of this same ideal model.

The square-root collapse of capacitive bridging bandwidth near the DC error floor follows by factoring the numerator:For fixed $g>0$ and $\eta \downarrow {\eta }_0$,This asymptotic equivalence explains the sharp loss of usable bandwidth when the resistive divider has already consumed nearly all the error allowance. Increasing $R_{\mathrm{in}}$ or reducing $R_s$ changes that floor; reducing only $C_{\mathrm{in}}$ rescales a feasible bandwidth but cannot remove infeasibility at DC.

The criterion also controls a complete waveform. Suppose a source waveform $v_s$ is both a square-integrable function and a bandlimited function, with spectrum supported where $|H-1|\le \eta$. For the corresponding zero-state response $v_L$, the Fourier transform of $v_L-v_s$ is $(H-1){\hat{v}}_s$. The Plancherel theorem, or the norm bound for a Fourier multiplier on L2, givesHere the Fourier transform uses the frequency-in-hertz convention, so the displayed Plancherel theorem has no extra normalization factor. This bound includes waveform changes caused by phase. It applies to the zero-state response of the stated frequency response. Any additional homogeneous solution from initially stored energy contributes its own error.

A bench check therefore needs both amplitude ratio and phase shift, with probe and cable capacitance included in the port model. Magnitude-only observations determine $|H|$ without determining $|1-H|$. The formula describes this chosen passive input network; additional amplifier dynamics require their own transfer function.

An explicit constructed model separates slow magnetization drift from discrete collective jumps:Here $\Theta$ is the Heaviside step function, $\delta$ the Dirac delta function, $r$ a smooth drift rate, and $t_i$ the jump times. The ideal discontinuities stand for events shorter than the resolved timescale; actual magnetic domain walls move over finite times. This equation stipulates jump sizes rather than deriving them from atomic spin. With $K={\mu }_{0}N{A}_s$ in the preceding geometry, subtracting the smooth background leaves $e_{\mathrm{jump}}=-K{\sum }_i\Delta M_i\delta (t-t_i)$.

Let a stable linear filter have causal impulse response $h(t)$, normalized by $\int h(t)dt=1$, and let $G$ be a constant dimensionless voltage gain. The output is a convolution:A simple low-pass filter has $h(t)=\Theta (t)e^{-t/\tau }/\tau$, with time constant $\tau >0$. Its output pulse has peak proportional to $\Delta M_i/\tau$ and signed area $-GK\Delta M_i$. Integrating the convolution proves area preservation because $\int h=1$; a wider filter response lowers peaks and lengthens tails. This ideal filter models resolution, rather than claiming the cited instrument had exactly this response.

The nearby positive jumps in the figure have additive negative pulse areas even when their tails overlap. For $\tau =0.50\mathrm{ms}$ the first of the pair still has $e^{-0.2/0.5}=e^{-0.4}\approx 0.67032$ of its initial height when the second arrives. Thresholding can merge those responses into one detected event; limited signal bandwidth also obscures their separation. The total signed area still records the sum if the complete response is integrated, but cannot identify how that sum was partitioned into events. Equal simultaneous jumps are exactly indistinguishable from a single jump of their summed size in this linear measurement. Oppositely signed simultaneous changes can cancel.

Area preservation requires the stated electronics. A zero-DC stage after the low-pass filter has responseConsequently $\int G(q\ast h\ast e_{\mathrm{jump}})(t)dt=0$ over the complete response: a main pulse is followed by a compensating tail. Integrating only a threshold-selected lobe, clipping the record, or saturating the amplifier breaks the simple pulse-area inference. Recovering a magnetization change from such a readout needs the calibrated frequency response and an appropriate reconstruction. Neither the audible crackle nor the displayed voltage is a direct staircase of magnetization.

For a linear filter with chosen nonzero reference voltage gain $H_0$, define its effective noise bandwidth byThis is the width of an ideal rectangular linear filter with gain magnitude $|H_0|$ passing the same mean square of input white noise. ConsequentlyUse the full voltage transfer from the open-circuit voltage source to the readout, including any electrical load, when applying this formula. A nominal signal bandwidth stated between two cutoff frequencies does not specify the integral: gradual tails and a nonuniform passband matter. For an ideal rectangular unit-gain passband $[f_1,f_2]$, $B_{\mathrm{n}}=f_2-f_1$. For an unloaded RC low-pass filter with $H_0=1$ and cutoff frequency $f_c=1/(2\pi RC)$,The last equality follows by substituting $x=f/f_c$ and integrating $1/(1+x^2)$ as $\arctan x$. Using the nominal $f_c$ as the noise bandwidth underestimates the variance by a factor $\pi /2$, or the root mean square voltage by $\sqrt{\pi /2}$.

The peak frequency deviation is the largest absolute value of the instantaneous frequency offset from the chosen carrier wave frequency $f_c$. For $f_i=f_c+K_{f}M\cos (2\pi f_{m}t)$, it is $\Delta f=|K_f|M$. It is a frequency, measured in hertz, rather than a wave phase or an exact signal bandwidth.

Harry Nyquist was an electrical engineer whose work connected communication engineering with physical and mathematical limits. His 1928 theory explained thermal electrical fluctuations using thermodynamics and statistical mechanics, agreeing with John Bertrand Johnson's measurements.^[ref]

In the classical regime $hf\ll k_{B}T$, an approximately constant electrical resistance $R$ at temperature $T$ produces an open-circuit voltage expected value square $⟨V^2⟩=4k_{B}TR\Delta f$ over a one-sided frequency band of width $\Delta f$. Here $k_B$ is the Boltzmann constant. The formula connects Johnson-Nyquist noise with both the physical conductor and the signal bandwidth through which it is observed.^[ref]

Heinrich Barkhausen's 1919 report used iron, a 300-turn pickup coil, moving magnet, amplified telephone readout and galvanometer. Removing iron corrected drive pickup. ^[ref]

A modern primary experiment by Gianfranco Durin and Stefano Zapperi used annealed iron–silicon strips and amorphous alloys. A 60 cm solenoid swept a triangular magnetic field intensity; a 50-turn pickup coil, wound within 1 mm, sensed the specimen. A preamplifier with a 3–20 kHz low-pass filter preceded sampling. Measurements selected the central hysteresis region dominated by magnetic domain wall motion. Above-background threshold crossings defined event durations; integrating between crossings defined event sizes. These are reported procedures, rather than data generated for the diagrams.^[ref]

For this stationary implementation, secure a thin ferromagnet and its pickup coil inside a longer drive solenoid, keeping their relative positions fixed. Drive the solenoid with a slow triangular electric current, monitor that electric current, and record the amplified, filtered induced electromotive force with a digitizer. A speaker may make suitable frequencies audible, but the voltage record is needed for quantitative analysis. Slow variation of the drive does not require smooth magnetization. A triangular drive supplies an approximately constant applied-field sweep rate along each ramp; it does not guarantee constant instantaneous $dM/dt$.

A proposed control sequence makes the inference testable. First record the electronics background with no drive. Repeat the same sweep with the specimen removed, retaining coil geometry: mutual induction from the drive can itself induce a voltage. A matching empty pickup coil wired in opposition can suppress common drive pickup, after checking its matching. Compensation of air flux by an opposed coil is an established method.^[ref] Then restore the specimen, repeat both sweep directions, reduce sweep rate, and compare the same hysteresis region with fixed amplifier gain, filter and threshold. Monitor drive irregularities and mechanical vibration; either can generate spurious pulses. These controls are recommendations for the schematic, rather than undocumented controls attributed to the modern paper.

A genuine material contribution should differ from the blank-coil record and change reproducibly with the magnetic preparation and sweep region. Quieting the drive tests drive-correlated pickup, but a held field need not eliminate all delayed material motion. Reversing the sweep is a consistency check on signed changes, not a demand that individual events retrace. A changing pulse count under a different filter or threshold cannot by itself establish changed microscopic dynamics. The useful inference is intermittent collective magnetization change within the instrument's signal bandwidth, with spatial detail left unresolved.

Input capacitance describes the capacitive part of an electrical port's small-signal input impedance, under stated bias, frequency, and termination conditions. In a simple parallel model with the $e^{i\omega t}$ convention, its electrical admittance isThe electric current drawn by the capacitance grows with angular frequency even if the DC input resistance is enormous. Analog Devices distinguishes differential input paths from common-mode paths to ground: a quoted capacitance must identify which terminals are driven and how the remaining circuit is connected.^[ref]

For an original loading example, let $R_{\mathrm{in}}=10\mathrm{M\Omega }$ and $C_{\mathrm{in}}=20\mathrm{pF}$. At $10\mathrm{MHz}$, the capacitive reactance magnitude is only $1/(2\pi f{C}_{\mathrm{in}})\approx 796\Omega$. A source with electrical resistance $R_s=1\mathrm{k\Omega }$ therefore has loaded-to-unloaded phasor ratioThe DC ratio is about $0.99990$. Thus a resistance-only reading would miss a substantial high-frequency amplitude reduction. The numbers are chosen examples, not a specification for a named device.

For this same model, the source-to-input time constant is $(R_s\Vert R_{\mathrm{in}})C_{\mathrm{in}}$ and the DC-normalized half-power signal bandwidth is its reciprocal divided by $2\pi$. A complex-error requirement is instead governed by capacitive bridging complex-error bandwidth; these specifications answer different questions.

For input capacitance inferred from measured admittance, apply a small sinusoidal electric potential difference, measure the input electric current and its phase shift, and infer $\mathrm{Im}(I/V)/\omega$ within the parallel-capacitance approximation. Fixture, cable, and probe capacitances contribute to the measured port unless removed or included explicitly. Dependence on bias or frequency can make a single fitted $C_{\mathrm{in}}$ an inadequate model.

A low-pass filter preferentially transmits low frequencies and attenuates higher frequencies. It can smooth rapid noise in a slowly varying signal, but it also suppresses wanted variations outside the useful signal bandwidth. An RC low-pass filter gives a concrete continuous-time example; a moving average implements related smoothing on a discrete-time signal.

The useful question is how much change the measurement can tolerate. Preserving the low frequencies of a temperature record while reducing a fast unwanted oscillation can improve readability. Preserving the sharp onset of a pulse requires higher frequencies, so excessive smoothing changes its timing and shape. The RC low-pass frequency response and RC low-pass step response make this signal bandwidth versus settling time tradeoff quantitative.

For weak drive in this homogeneous model, half maximum occurs at $|\Delta |=1/T_2$. The absorption full width at half maximum is thereforeThe example gives $2.653\mathrm{Hz}$ or $0.06230\mu \mathrm{T}$. At finite drive the model width is multiplied by $\sqrt{1+s}$. Field inhomogeneity and unresolved components can also broaden an actual line, so its width alone does not determine pure dephasing.

The same width appears by an original Fourier transform calculation. For $a=1/T_2$ and a free homogeneous signal demodulated exactly at resonance,This connects exponential memory to a Lorentzian absorptive line. It is a mathematical connection, not evidence that the 1946 teams performed pulsed Fourier transform spectroscopy. Instrumental signal bandwidth is a separate choice: narrowing a receiver band can reject thermal electrical noise, but too narrow a band distorts a changing signal. A sample linewidth and a receiver bandwidth are not interchangeable.

Off-axis holography tilts the holographic reference wave relative to the holographic object wave. Its interference fringes acquire a carrier spatial frequency; the two cross terms occupy different spatial frequency regions when the carrier is sufficiently large compared with the object's signal bandwidth. This permits filtering a single image term.

Gabor's lecture credits Emmett Leith and Juris Upatnieks with the later skew-reference method using helium–neon laser illumination, separating reconstructed images angularly.^[ref]

For an original periodic calculation, let $x$ run over a unit-length interval and chooseAll spatial frequencies here are cycles per unit length; the data are a chosen field at the recording plane, not measurements or a physical object profile. Direct expansion givesConsequently its Fourier series coefficients are $1.05$ at zero, $0.02$ at $\pm 8$, $0.2$ at $\pm 17$ and $0.1$ at $\pm 25$. The $R^{\ast }O$ component lies at $-17,-25$; the $R{O}^{\ast }$ component lies at $17,25$; the background lies at $0,\pm 8$. Filtering the negative-frequency cross term and multiplying by $R$ recovers $O$, including its relative wave phase.

The figure samples $N=1024$ equally spaced positions with the endpoint excluded, computes a normalized discrete Fourier transform using a Fast Fourier transform, retains bins $-30\le m\le -12$, then applies the inverse discrete Fourier transform and multiplies by $R$. The recovered real and imaginary parts coincide with the chosen $O$ to numerical precision. This exact periodic example avoids leakage by placing every frequency on an integer bin; finite nonperiodic records need allowance for broadening and an aperture.

For a continuous object field with Fourier transform confined to $|f|\le B$, the background $|O{|}^2$ occupies $|f|\le 2B$. A carrier $f_c>3B$ is a sufficient separation condition for the sideband centred at $-f_c$, and $f_c+B$ must remain below the Nyquist spatial frequency. Here $B=5$, $f_c=20$ and the Nyquist spatial frequency is $512$. This sufficient bound describes a whole bandwidth, rather than exploiting the gaps in this particular two-component field.

The same example also makes optical playback explicit. Choose $t_0=0.3$ and $\beta =0.01$. Since $|O|\le 0.3$, the bounds $0.49\le I\le 1.69$ imply $0.3049\le t\le 0.3169$, and $|\beta I|/t_0\le 0.0564$. The affine response is stipulated; its weak-modulation interpretation is approximate. In $Rt$, the desired field $0.01O$ has frequencies $3,-5$; the conjugate term $0.01R^2{O}^{\ast }$ has frequencies $37,45$; the background has frequencies $20,12,28$. Thus the same carrier that separates the recorded cross terms separates the playback components. Reillumination restores a field through diffraction; computational filtering extracts the corresponding encoded field through Fourier analysis.

A detector measures a signal, not a force. One proposed arrangement collects transmitted and scattered light onto a quadrant photodiode, using a normalized difference of opposite segments as a position-sensitive signal. A camera can supply an independent image and bead-height check. Near a chosen operating point, a voltage channel may obey $V-V_0=x/\beta$, but its linear range and sensitivity $\beta$ must be measured at that bead size, focus and collection geometry. An arbitrary scattered-light intensity is not automatically proportional to displacement.

A useful proposed control sequence is to record detector offsets with illumination blocked, calibrate image scale against known stage travel, and test the detector response with independently known bead-relative motion. Moving a free trapped bead's chamber is not by itself a known bead displacement: fluid drag and trap stiffness determine the shift. Use a separate immobilized bead or an independently imaged trajectory for the position response, then check whether the response transfers to the actual trapped bead. Repeat the calibration after changing trapping power or optics.

Measure the acquisition frequency response and choose a sample rate that resolves $f_c$, with sufficient signal bandwidth and control of aliasing. Compare thermal calibration of an optical trap with optical trap drag calibration, sweep small positive and negative flow speeds, inspect for multiple beads, and repeat at several wall distances. Check temperature and drift before interpreting a power-dependent stiffness: a changed fluid viscosity can change $f_c$ even at unchanged $k$. These are proposed diagnostics of the model, not procedures attributed to the 1970 or 1986 papers.

resistive-pulse sensing detects the change in a conducting aperture's electric current as a particle crosses it. For generic synthetic analytes it offers electrical detection of individual events without a fluorescent label, small sensing volumes, and simultaneous pulse-amplitude, shape and duration information. Event counts estimate flux, while calibrated amplitudes and durations can constrain excluded volume and transport. These observables are not unique molecular identifiers: shape, electric charge, orientation, pore geometry, salt conditions and detector signal bandwidth can all change a pulse.

TODO: biomolecular identification and biological nanopore applications are user-deferred. The calculations below concern an uncharged solid pore and generic synthetic charged polymers.

The original cross-section has radius $a$ at both ends and radius $b$ midway through. In the intended one-dimensional series-resistance approximation, put $s=(b-a)/l$ andFor constant ionic electrical resistivity $\rho$, a slice contributes $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. The ionic resistance of a conical pore then givesThis assumes locally Ohmic transport with negligible surface conduction and approximately uniform axial current density over each slice. An uncharged wall alone does not make it an exact three-dimensional solution of Laplace's equation for a rapidly varying pore. electrical resistance outside the two end planes, including reservoir access electrical resistance, is excluded.

Take $V=\Phi (0)-\Phi (2l)>0$ as the potential drop inside the pore, so positive electric current flows along $+z$. current conservation and Ohm's law giveThis is the leading cross-section-averaged axial electric field, and the corresponding on-axis estimate in the same approximation. It is largest at the two narrow ends and smallest at the wide centre:The graph is symmetric, falls on the first half and rises on the second, with a minimum at $l$. The electric potential difference check is ${\int }_0^{2l}E(z)dz=V$. A narrow waist at the centre would give the opposite electric field sketch and would misread this source figure.

Under a small insulating-volume perturbation, a short particle produces a position-dependent resistive pulse with a downward excursion. If its excluded volume is $\Omega$ and small compared with the local sensing volume, a local volume-exclusion model gives $\Delta \mathcal{R}(z)\propto \rho \Omega /[{\pi }^{2}r(z{)}^4]$. Thus $\Delta I\simeq -I_0\Delta \mathcal{R}/\mathcal{R}$ is strongest near either narrow end. A sufficiently resolved event can show two deeper end features separated by a shallower middle depression, then return to baseline. Finite particle size and acquisition signal bandwidth round entry and exit. The original illustration below shows this conditional small-particle mechanism, not a fitted amplitude. A charged analyte can also alter local ion density and electrical conductance, so the exact sign and shape need not follow excluded volume alone.

For positive drift in this coordinate, require $\mu V>0$. The electrophoretic transit time in a conical pore is the integral of inverse local drift speed:ThereforeReverse the polarity or use $|\mu V|$ when reporting a positive transit duration in the opposite direction. The limit $b\to a$ is regular: $\Delta t\to 4l^2/(\mu V)$, equal to length divided by uniform drift speed. Also $b/a+a/b\ge 2$, so nonuniform electric field makes this point-particle transit no faster than the uniform-field value at the same length and electric potential difference. The wide, weak-field interior dominates residence time.

The final length-dependent observations do not uniquely identify a microscopic mechanism. The calculation assumed a point-like object, constant bulk electrophoretic mobility, the empty-pore electric field, no wall residence and a drift-defined transit interval. An extended chain can sample entrance and reservoir fields simultaneously, alter its conformation and tension, and change the effective driving-to-friction ratio. These effects can make a bulk-mobility, local-field prediction underestimate the driving. For a small object, wall interactions, reversible trapping, or repeated entry excursions can add residence time that the deterministic formula omits. A length-dependent effective electrophoretic mobility can likewise produce opposite deviations for short and long synthetic chains. Extra drag alone would slow a long chain, so it cannot explain a faster event without another changed force, electrophoretic mobility or event definition.

Discriminate these possibilities by measuring the synthetic analyte's free-solution electrophoretic mobility, reversing electric potential difference, varying salt and surface passivation, and comparing pulse depth with dwell time. Resolved multiple occupancy or folds can alter amplitude as well as timing. Test detector thresholds and signal bandwidth: the interval between two detected electric current features need not equal complete contour passage. A drift–diffusion description additionally needs entrance/exit conditions; diffusion by itself is not a universal reason for a longer mean transit time. No unique anomalous-duration prediction follows from the provided parameters alone.

Thorlabs' silicon FDS100 experiment sends a collimated, fibre-coupled laser beam through a beam splitter to the photodiode and a reference optical power meter. Varying reverse bias and electrical load tests linearity. The measured readout approaches saturation when its electric potential difference consumes the bias.^[ref] A separate low-signal setup uses a $637\mathrm{nm}$ laser, polarizers, an optical chopper, optical attenuators and a lock-in amplifier inside a light-tight enclosure. These are actual apparatus choices, not the parameters of the constructed example above.^[ref]

For an independent dark current check, Thorlabs enclosed a photodiode in nested metal boxes to exclude light and suppress interference, used a Keithley 6487 ammeter supplying reverse bias, and monitored temperature. Only one specimen per type was tested; the nearby temperature sensor did not directly contact the semiconductor. Such a dark control measures the baseline but cannot certify spectral responsivity.^[ref]

To infer spectral responsivity at a selected wavelength, measure the radiant flux actually reaching the test photodiode, rather than just the reference arm. A reference-arm measurement calibration must include the beam splitter ratio and path losses. With signed readouts in one fixed orientation, subtract a beam-blocked baseline and usefor direct electric current readout or a calibrated linear transimpedance amplifier, respectively. Repeating at several incident powers tests whether this ratio is constant. Blocking only the source estimates ambient light plus electronic baseline; covering the detector excludes ambient light too. Their difference distinguishes optical background from dark current and readout offset. For a broad optical spectrum, the ratio is a spectrum-weighted response, not generally $R$ at one wavelength.

For independent Poisson distribution collection, the root mean square shot noise in a one-sided rectangular frequency band of signal bandwidth $B>0$ isHere $I_d$ is reverse dark current; thermal electrical noise and amplifier noise also contribute.^[ref]

Subtracting a mean dark current removes its offset, not its fluctuations: subtracting a constant leaves the variance unchanged. Consequently $I_{\mathrm{ph}}\gg I_d$ makes dark current shot noise small relative to signal shot noise; it does not by itself establish high signal-to-noise ratio. With an independently measured baseline, the variance of that estimate must also be propagated. Increasing the measurement signal bandwidth raises the admitted noise. Calibration of wavelength, reference radiant flux, readout gain and collection geometry, together with drift and linearity, determines what measurement uncertainty can be claimed.

The University of Colorado Boulder's Lab 3: Filters, §§7.1–7.4, specifies measuring the resistor and capacitor, then driving the assembled RC low-pass filter with a function generator. A two-channel oscilloscope reads actual input and output using high electrical impedance. Sweep the frequency, record the voltage amplitude ratio, and locate $1/\sqrt{2}$. A control compares a cable with a $10\times$ oscilloscope probe on a resistive voltage divider, isolating measurement capacitance before using the probe for the filter. A slow square wave then lets discharge timing estimate the time constant independently. Agreement supports the RC low-pass frequency response; source output impedance, probe loading, and instrument signal bandwidth limit it. These are prescribed measurements, with no measured dataset supplied here.^[ref]

The two-channel setup can also test the predicted phase difference directly: for positive frequency $f$, measure the output's time lag $\Delta t$ between corresponding features of the two sinusoids. For this single-stage model the phase lies between $-\pi /2$ and $0$, so choose the lag within a quarter cycle and calculate $\varphi =-2\pi f\Delta t$. At the cutoff frequency the predicted lag is one eighth of a cycle. This tests the timing part of the frequency response as well as the voltage ratio.

The step response describes a sudden persistent input change. Suppose the input becomes a constant $V_s$ at $t=0$, and the initial capacitor voltage is $v_0$. Finite electric current leaves that voltage continuous. Subtract $V_s$ from the RC low-pass filter equation: the remaining error satisfies $\tau de/dt+e=0$. Separating variables givesFor an initially uncharged capacitor and a $0$ to $3\mathrm{V}$ step in the constructed example,At one time constant, $t=103.4\mu \mathrm{s}$, the voltage is $1.8964\mathrm{V}$, about $63.2\%$ of its final value. The voltage error then continues its exponential decay. This is different from a pure delay: the output starts changing immediately, yet takes time to approach the target.

Define settling time relative to the total voltage change: require $|v_{\mathrm{out}}-V_s|\le ϵ|v_0-V_s|$ thereafter, with $0<ϵ<1$. The exponential decay is monotone, soFor $1\%$ settling, $t_{0.01}=4.6052\tau \approx 476.17\mu \mathrm{s}$; the $10\%$ to $90\%$ rise takes $\tau \ln 9\approx 227.19\mu \mathrm{s}$. Reducing the cutoff frequency by doubling $C$ doubles both times. This reciprocal relation is the useful connection between signal bandwidth and response speed: stronger suppression of rapid noise comes with slower tracking of wanted changes.

MIT's guide uses an illuminated vacuum photodiode, with collecting anode positive relative to the emitting cathode. A transimpedance amplifier gives $V_1=R_{F}I$; a multimeter reads its mean. The fluctuation path rejects below $100\mathrm{Hz}$, adds roughly tenfold voltage gain, then nominal hundredfold preamplification and a $1$–$50\mathrm{kHz}$ band-pass filter; the readout measures root mean square voltage. With diode bias off, a sinusoid at the test port calibrates $g(f)$. Restore bias, retain gain settings, vary lamp brightness, and repeat both readings. The guide warns about overload; it supplies a procedure, not measured results.^[ref]

The inference can be expressed without relying on nominal passband edges. Let $g(f)$ denote the calibrated voltage transfer after the $R_F$ conversion, so $K(f)=R_{F}g(f)$ and define $\mathcal{K}={\int }_0^{\infty }|K(f){|}^{2}df$. Under independent arrivals and an unchanged additive background,The slope of voltage variance against mean electric current can therefore estimate $q$. A free intercept accounts for a constant background; forcing the line through zero generally biases the slope. The root mean square readings must have their mean removed and then be squared. Subtracting two root mean square amplitudes does not subtract independent variances. Estimate the variance from successive records; averaging the waveforms first would suppress the fluctuation being measured.

Additional controls sharpen this interpretation. Block the light with detector bias, temperature, and gain unchanged; this preserves dark current and electronics while removing photocurrent. Repeat this dark comparison between illumination settings to check drift. If independent dark arrivals have mean magnitude $I_d$ and illuminated arrivals add $I_{\mathrm{ph}}$, their spectra add: $2q(I_d+I_{\mathrm{ph}})$. Subtracting the unchanged dark variance isolates $2q{I}_{\mathrm{ph}}\mathcal{K}$. This subtraction model assumes unchanged transfers and independence; switching off bias for gain calibration is a different control and need not preserve the dark background.

Vary measured $I_{\mathrm{ph}}$ rather than assuming lamp knob position is proportional to photocurrent. Check collection versus bias to identify a range where incomplete collection is negligible; a space charge limited tube can correlate transfers. Verify that the transimpedance amplifier remains linear and that its frequency response is unchanged over the illumination range. Changing signal bandwidth should change the excess variance through $\mathcal{K}$; compare inferred input spectra in several bands. Correlated fluctuations of the light source can add a contribution scaling approximately as $I_{\mathrm{ph}}^2$, while an electronics floor can hide the linear term at small electric currents. A fitted line over a short range alone does not establish independent transfers.

The gain calibration must include the same detector-to-readout path used for noise, or explicitly supply any omitted transfer. Detector capacitance, finite collection time, amplifier voltage and current noise, thermal electrical noise, pickup, clipping, and finite recording precision limit the inference. Current-dependent electrical impedance or heating can make a background current-dependent too. Since $\mathcal{K}$ contains squared gain, a small fractional error in gain produces approximately twice that fractional error in the predicted variance scale. Knowing a nominal gain and two cutoff settings is insufficient.

Write an ideal point-transfer model using the Dirac delta function:For this arrival stream with independent increments, distinct arrivals contribute the product of the means, which cancels when centering. The same-arrival term leaves the autocovarianceIts Fourier transform gives the two-sided power spectral density, while folding negative frequencies onto positive ones gives the one-sided convention:These spectra describe $\delta i$, excluding the spectral line from the mean direct current. The amplitude spectral density is $\sqrt{2qI}$ in $\mathrm{A}/\sqrt{\mathrm{Hz}}$; it is not a root mean square electric current until a measurement signal bandwidth is specified. Approximate white noise requires frequencies low enough that the single-transfer pulse shape and any correlations can be neglected. Neither the ideal impulses nor a flat spectrum extending to infinity describe a real detector at every timescale.

All spectra above are per hertz, with $f$ in cycles per second. One possible angular frequency convention defines a density ${\mathcal{S}}_I^{(2)}(\omega )$ integrated using $d\omega$ alone. Since $\omega =2\pi f$, it hasAnother common convention keeps the numerical density $qI$ but integrates it with $d\omega /(2\pi )$. State the integration measure as well as whether a spectrum is one-sided: a change of variable and a folding of signs are separate operations.

A signal-to-noise ratio compares a chosen measure of a signal with the corresponding noise measure. For a constant measured electric potential difference $V$ and a root mean square noise voltage $v_n$, an amplitude ratio is $|V|/v_n$; the corresponding power ratio under equal electrical resistance is its square. State the convention and measurement signal bandwidth, since changing either changes the numerical ratio.