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An amplitude spectral density is the square root of a power spectral density. For an electric potential difference expressed in volts and frequency in hertz, its units are $\mathrm{V}/\sqrt{\mathrm{Hz}}$.

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 $\sqrt{\pi /2}$.

Low-frequency noise with an approximately inverse-frequency power spectral density over a specified range. It can dominate the flat spectrum of a weakly confined mechanical coordinate or an electronic readout at low frequencies. A $1/f$ law cannot hold with finite total variance over unlimited frequency range, so its measured bandwidth and low-frequency cutoff matter. An observed excess alone does not uniquely identify the microscopic source.

For power spectral density $P(f)=C/(f^2+f_c^2)$, the low-frequency plateau is $P_0=C/f_c^2$ and the high-frequency asymptote is $C/f^2$. Their intersection on a log-log plot gives the displayed corner frequency. Equivalently $P(f_c)=P_0/2$. Use the range unaffected by additional low-frequency noise or high-frequency filtering; a measurement chain can otherwise shift an apparent corner.

The Hasselmann stochastic climate model represents slowly responding climate variables driven by fast weather fluctuations as well as external forcing. Klaus Hasselmann's 1976 paper separates fast and slow variables, retains fluctuations lost by simple averaging, and explains why stabilizing climate feedbacks are needed to prevent indefinitely growing variability. Its general framework is broader than a single-temperature equation.^[ref]

Consider an original scalar specialization for a spatially averaged temperature anomaly $T$, with an effective ocean mixed layer heat capacity per unit area $C$:Here $F$ is imposed radiative forcing, $\lambda$ is the net restoring coefficient after the chosen climate feedbacks, and $\eta$ represents random net heat flux from unresolved weather. The anomaly energy per unit area is $CT$, so its tendency equals the net input. The weather term represents exchange with omitted degrees of freedom; it does not assert that an isolated ocean creates energy. $F$ and $\eta$ have units $\mathrm{W}{\mathrm{m}}^{-2}$, and $\lambda$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

First set $\eta =0$, start from $T(0)=0$, and switch to constant $F_0>0$. The integrating factor $e^{\lambda t/C}$ givesFor $C=2.0\times 10^8\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $\lambda =2.0\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and $F_0=4.0\mathrm{W}{\mathrm{m}}^{-2}$, the relaxation time is $10^8\mathrm{s}$, about $3.17$ years using 365.25 days per year. At $t=\tau$, $T=2(1-e^{-1})\mathrm{K}\simeq 1.264\mathrm{K}$, with eventual warming $2\mathrm{K}$. Doubling $C$ doubles the response time but leaves this equilibrium unchanged. This separates thermal inertia from climate sensitivity.

Now set $F=0$ and idealize the weather forcing as Gaussian white noise withThe Dirac delta function has inverse-time units, so $D$ has units $(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$. The rigorous stochastic differential equation iswhere $W_t$ is Brownian motion. This is an Ornstein-Uhlenbeck process. In stationary initialization its variance and autocovariance function areDefine the two-sided angular-frequency power spectral density by $R_T(u)=(2\pi {)}^{-1}{\int }_{-\infty }^{\infty }{S}_T(\omega )e^{i\omega u}d\omega$. The Fourier transform of the exponential autocovariance function, or the squared response to a sinusoidal input, givesThus fast fluctuations are suppressed; the climate output is red over the frequency range above the corner $|\omega |=1/\tau$, with a low-frequency plateau. This is a colored noise spectrum without a periodic driver. The corner is $f_c=1/(2\pi \tau )$ in cycles per second, not $1/\tau$. The continuous white input is only an approximation on times longer than weather's correlation time.

For the same $C,\lambda$ and $D=10^8(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$, $v_{\infty }=0.25{\mathrm{K}}^2$ and the stationary standard deviation is $0.50\mathrm{K}$. From fixed $T(0)=0$, the variance grows as $v_{\infty }(1-e^{-2t/\tau })$, even when the forced expected value is predictable. Integrating $S_T/(2\pi )$ over all angular frequencies returns $D/(C\lambda )$, checking the normalization independently. If $\lambda =0$ the unforced variance instead grows as $2Dt/C^2$; if $\lambda <0$, deviations amplify. Neither case has the stationary spectrum above.

A red spectrum alone cannot establish attribution of recent climate change: internal variability, changing forcing and measurement effects must be distinguished. This scalar model supplies a calculable baseline for variability and persistence, not a universal spectrum for all climate timescales or a full attribution analysis.

For detector $k$, model the calibrated data as $d_k(t)=h_k(t;\theta )+n_k(t)$, with a shared source parameter vector $\theta$, separate detector responses, propagation delays and noise $n_k$. Under an independent stationary Gaussian process approximation with one-sided power spectral density $S_k(f)$, define the noise-weighted inner product and likelihood functionStrain power spectral density has units ${\mathrm{Hz}}^{-1}$; the displayed inner product is dimensionless when the Fourier transform integrates over seconds. A matched filter rewards agreement over the full evolving wave phase, weighting frequency bands by their noise. One conspicuous peak is insufficient: a candidate should also have the right distribution of signal across frequency bands and compatible projections in the network.

The parameter paper used coherent Bayesian inference, numerical relativity-informed inspiral–merger–ringdown models and explicit calibration uncertainties. It found a strong distance–inclination correlation; source-frame masses additionally require converting the detector-frame masses by a cosmological redshift.^[ref] Conceptually, multiply the likelihood function by a prior distribution to obtain a posterior distribution. Changes in waveform families, residual structure after subtraction, and checks with a generic transient reconstruction test parts of the modeling assumptions. Detection significance and parameter uncertainty answer different questions.

Add the random thermal force to the deterministic optical trap relaxation time model. In one stationary coordinate of a harmonic optical trap, the overdamped Langevin equation isHere $\xi$ is Gaussian white noise understood through its integrals, $\delta$ is the Dirac delta function, and $\zeta$ is the linear friction coefficient. The covariance coefficient has units ${\mathrm{N}}^2\mathrm{s}$; multiplying by $\delta (t-t^{\prime })$, of units ${\mathrm{s}}^{-1}$, gives squared force. This fluctuation-dissipation relation for a Langevin particle makes the stationary variance ${\sigma }^2=k_{B}T/k$, consistent with the equipartition theorem. The solution is an Ornstein-Uhlenbeck process with autocovariance functionIndeed, at positive lag the noise has zero correlation with the initial position, so $\zeta C_x^{\prime }=-k{C}_x$; its initial value is ${\sigma }^2$. This supplies both a decay time and an amplitude, making a stronger calibration test than a position histogram alone.

Specify the Fourier transform convention before using a power spectral density. For frequency $f$ in hertz, define $S_x^{(2)}(f)={\int }_{-\infty }^{\infty }{C}_x(t)e^{-2\pi ift}dt$ and the one-sided density $S_x^{(1)}(f)=2S_x^{(2)}(f)$ for $f>0$. Integration of the exponential givesThe position power spectral density has units ${\mathrm{m}}^2/Hz$; its integral is a variance, not a standard deviation. At $f=f_c$ it is half its zero-frequency plateau. Angular frequency conventions require a corresponding $2\pi$ conversion in both axis and integration measure. If $x=\beta (V-V_0)$ with position sensitivity $\beta$ in $m/V$, then $S_V^{(1)}=S_x^{(1)}/{\beta }^2$ in ${\mathrm{V}}^2/Hz$ before independent detector noise is added.

For an original invented calibration example, choose $a=0.500\mu \mathrm{m}$, dynamic viscosity $\eta =1.00\times 10^{-3}\mathrm{Pa}\mathrm{s}$, $T=300\mathrm{K}$, $k=0.0200pN/nm=2.00\times 10^{-5}N/m$, and $\beta =100nm/V$. The unbounded no-slip Stokes drag law givesThe corresponding voltage standard deviation is $0.14391\mathrm{V}$, and the position power spectral density plateau is $0.390369\mathrm{n}{\mathrm{m}}^2/Hz$, or $3.90369\times 10^{-5}{\mathrm{V}}^2/Hz$ in the detector. Recovering both $k=k_{B}T/{\sigma }^2$ and $k=2\pi \zeta f_c$ is an internal model check; actual agreement also tests the assumed temperature, drag and detector response.

The generator uses the exact stationary update $x_{j+1}=\rho x_j+\sigma \sqrt{1-{\rho }^2}{Z}_j$, with $\rho =e^{-\Delta t/\tau }$ and independent standard normal random variables $Z_j$. For a demeaned block of $N$ samples and its unnormalized discrete Fourier transform $X_j$, the two-sided periodogram is $\Delta t|X_j{|}^2/N$; doubling positive bins except the zero and Nyquist endpoints gives a one-sided density. Summing those bins times $\Delta f=1/(N\Delta t)$ returns the block mean square by discrete Parseval identity. Independent detector noise, finite exposure and aliasing would alter a measured spectrum and must enter a fit before interpreting $f_c$.

Ideal spectrum. Write $P_i(f)=C/(f^2+f_c^2)$, with $C>0$. The overdamped harmonic-trap power spectrum is a Lorentzian line shape:Thus its log-log plot is horizontal at low frequency and approaches slope $-2$ at high frequency, changing around $f_c$. The left panel below compares two characteristic frequencies with the same illustrative $C$; a common numerator is explicitly assumed for this sketch. Their high-frequency tails coincide and their low-frequency plateaux differ as $f_c^{-2}$.

The right panel plots all nine original data points, including the very-low-frequency excess and the highest-frequency suppression. It also shows the ideal curve inferred only from the requested intermediate range.

Graphical corner estimate. Between roughly $1$ and $100\mathrm{Hz}$ the measured power spectral density is approximately the plateau $P_0=10^{-6}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. At $10^4$ and $10^5\mathrm{Hz}$,so the descending asymptote is $C/f^2$ with $C\simeq 1{\mathrm{V}}^2\mathrm{Hz}$. The graphical corner frequency of a Lorentzian spectrum is where these asymptotes intersect:Independently, the measured value $P_m(1000\mathrm{Hz})=5\times 10^{-7}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}=P_0/2$ is exactly the half-plateau condition $P_i(f_c)=P_0/2$. Both graphical readings agree. The inferred ideal curve is approximately

Low-frequency excess and recording duration. The two lowest-frequency measurements increase approximately as $1/f$, whereas the ideal trapped oscillator tends to a constant plateau. This resembles flicker noise or slow apparatus drift superposed on the motion. The transimpedance amplifier, detector or environment could contribute; the spectrum alone does not identify the source uniquely. It would be incorrect to attribute that whole excess to the ideal Brownian oscillator.

A finite record of duration $T$ has ordinary Fourier frequency resolution of order $1/T$. Resolving the lowest tabulated frequency $0.01\mathrm{Hz}$ therefore requires at leastThis is a formal lower bound corresponding to about one cycle; a reliable noise-spectrum estimate normally needs a substantially longer record and averaging.

Sampling. The largest reported frequency is $f_{\max }=10^6\mathrm{Hz}$. The Nyquist–Shannon sampling theorem therefore gives the Nyquist lower boundIn practice choose a rate above $2\mathrm{MS}{\mathrm{s}}^{-1}$, with an analogue anti-aliasing filter suppressing unmeasured content above half the sampling rate. Exact equality cannot recover an arbitrary pure sinusoid precisely at the boundary for every phase; it is the limiting lower-bound answer rather than a comfortable practical sampling design.

High-frequency departure and amplifier design. Extrapolating the fitted $f^{-2}$ tail to $10^6\mathrm{Hz}$ predicts approximately $10^{-12}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$, while the measured value is $10^{-15}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. The observed drop from $10^5$ to $10^6\mathrm{Hz}$ has effective logarithmic slope $-5$, much steeper than the ideal $-2$.

A detector or amplifier bandwidth limit, or intentional filtering before digitization, can explain such suppression. For a linear filter the spectral density transformation under a linear filter is $P_{\mathrm{out}}=|H{|}^2{P}_{\mathrm{in}}$, so the last point corresponds approximately to an additional relative voltage gain $\sqrt{10^{-3}}=0.0316$, if attributed entirely to the readout transfer function. One additional point does not determine a unique filter order or identify the circuit responsible.

Rolling off an amplifier's gain beyond the useful signal band reduces integrated high-frequency noise, suppresses interference, and helps prevent aliasing in the analogue-to-digital converter. Limiting high-frequency loop gain can also improve feedback stability and suppress unwanted oscillation. The cutoff must retain the frequencies needed for the measurement; excessive filtering would distort the desired fluctuation spectrum itself.

Electrical noise reduction starts by identifying what is unwanted in the measured signal and where it enters. Increasing gain amplifies both signal and input noise; it does not by itself improve the input signal-to-noise ratio. Separate intrinsic component fluctuations, amplifier noise and external pickup before choosing a remedy.

Physical sources. A resistor at temperature $T$ has classical one-sided open-circuit thermal electrical noise density $S_V=4k_{B}TR$ per hertz. In a flat measurement bandwidth $B$, its rms voltage is $\sqrt{4k_{B}TRB}$. Lowering an appropriate resistance, reducing temperature or narrowing the useful bandwidth can reduce this noise, with loading and signal-gain tradeoffs considered. These are power spectral densities; an amplitude spectral density is their square root.

Discrete independent charge transfers give shot noise, with one-sided current density $S_I=2eI$ in the simple Poisson process model. Reducing leakage or unwanted bias current can help, but reducing a photocurrent may also reduce the desired signal. Charge correlations can change the simple shot-noise law. At low frequency, flicker noise and drift often dominate and are not described by a white spectrum.

Amplifier and impedance choice. An operational amplifier has equivalent input voltage and current noise. In a simplified uncorrelated model, a resistive source contributes the input-referred voltage densityHere $e_n,i_n$ are the voltage and current amplitude spectral densities and $Z_s$ is the source electrical impedance; feedback components add further terms in a full circuit. Low current noise matters particularly for a high-impedance source, while voltage noise can dominate for a low-impedance source. Minimize needless source impedance, leakage and long sensitive connections without sacrificing the required transduction gain or bandwidth. Bias-current offsets and thermal drift also require calibration; they are not removed simply by quoting a random-noise density.

Pickup and bandwidth. Shield sensitive circuits against electric-field coupling, minimize loop area to reduce changing magnetic-flux pickup, and use twisted signal/return pairs or suitable coaxial connections. Differential readout rejects common-mode interference when impedances and amplifier responses are adequately matched. Plan the signal returns to avoid unwanted circulating ground currents, and decouple supply rails locally so supply fluctuations do not enter the signal path. Mechanical vibration, cable motion and temperature changes may be converted into electrical noise as well.

A low-pass filter removes out-of-band noise before digitization; an anti-aliasing filter is needed because digital filtering cannot undo already aliased content. Avoid a narrower passband than the physical signal requires. For a known periodic signal, modulation and a lock-in amplifier select a narrow frequency interval, often moving the measurement above a strong flicker noise region. A notch can reject a narrow interference line if that does not remove essential signal information.

Independent noise contributions add in variance, not directly in rms amplitude. Averaging $N$ independent equal-variance readings reduces the standard error by $N^{-1/2}$, but slow correlated drift and systematic errors need separate controls. Good noise reduction preserves the information-bearing signal while reducing unwanted coupling, intrinsic fluctuations or the admitted noise bandwidth.

Johnson-Nyquist noise is equilibrium electrical fluctuation caused by thermal agitation of charge carriers. A resistor can generate a fluctuating terminal voltage even when its mean current is zero. In the classical frequency range $hf\ll k_{B}T$, the Fluctuation-dissipation theorem relates this noise to dissipative resistance. The one-sided voltage power spectral density of an unloaded resistor isEquivalently its short-circuit current spectrum is $4k_{B}T/R$. Over a flat bandwidth $B$ the rms voltage is $\sqrt{4k_{B}TRB}$. A real measurement instead integrates $S_V(f)|H(f){|}^2$ over positive frequency, with loading and the filter's effective noise bandwidth included. A flat spectrum does not imply an infinite physically measurable variance; the circuit and the quantum high-frequency regime limit the ideal white-noise model.

Cooling or narrowing the accepted bandwidth reduces thermal noise. Reducing resistance lowers its open-circuit voltage noise but raises its short-circuit current noise, so the appropriate choice depends on source impedance and amplifier loading. Thermal agitation is not a fluctuating externally applied voltage that can be removed by perfect shielding alone.

Shot noise arises from discrete transfer events. In the simplest shot noise counting model, equal charges $q$ arrive independently at constant rate $r$. A gate of duration $t$ contains a Poisson number $K$, with mean and variance $rt$. Therefore mean current is $I=qr$, and the averaged-current variance is $qI/t$. The corresponding one-sided low-frequency spectrum isfor a flat accepted bandwidth. Electron current has $q=e$. Vacuum emission or photodetection can approximate independent arrivals; correlations in a conductor can suppress or enhance the fluctuations. Discreteness by itself does not guarantee Poisson statistics.

For a fixed mean current the ideal shot-noise formula is not proportional to temperature, whereas Johnson noise is. Increasing current raises absolute shot noise but lowers its fractional size, since $\sqrt{2qIB}/I\propto I^{-1/2}$. Both can appear approximately white within a useful frequency band, and independent noise powers add after referring them to a common measurement point. Their spectra must use the same convention: two-sided densities are half these one-sided values. A real low-current experiment can also contain amplifier noise, low-frequency drift and correlated transfer noise, so a mean current or resistance alone does not specify its complete noise budget.

Temperature changes affect measurements through several routes. Thermal expansion changes lengths and alignments; temperature-dependent resistance and refractive index change electrical and optical calibrations; temperature differences between dissimilar junctions generate Seebeck effect voltages. These slow environmental changes are distinct from unavoidable equilibrium thermal fluctuations and Johnson-Nyquist noise. A useful strategy identifies the coupling to the measured signal before choosing a remedy.

Reduce environmental coupling by enclosing the apparatus against drafts, using thermal insulation and radiation shields, isolating heat sources, and minimizing dissipated power. A vacuum enclosure suppresses convection but leaves thermal conduction and thermal radiation paths to control. A large heat capacity reduces rapid temperature excursions. In a simple thermal model with heat capacity $C_T$ and thermal conductance $G_T$ to a reservoir, the relaxation time is $C_T/G_T$, so adequate settling time is necessary after changing the apparatus. A temperature sensor close to the sensitive element and stable negative feedback heating or cooling can regulate its temperature; poorly placed sensors or excessive feedback gain can leave gradients or cause oscillations.

Electrical thermal noise has the classical low-frequency one-sided voltage power spectral densityfor a resistor measured over a flat bandwidth $B$ without significant loading. More generally replace $B$ by the effective noise bandwidth. Lower temperature or narrower bandwidth reduces the noise, and resistance choices must account for source loading and the amplifier's own voltage and current noise. A lock-in amplifier uses modulation and narrowband detection to reject much low-frequency drift and out-of-band noise, while thermal noise inside the accepted band remains. Temperature stabilization does not remove equilibrium fluctuations.

Use a matched reference or differential bridge to reject common environmental changes. Keep junctions isothermal and choose matched materials to reduce thermoelectric offsets. For a resistance measurement, reversing a small test current separates a stable offset from the wanted voltage:Reversal must be fast enough that the offset stays approximately fixed, with enough settling time and sufficiently little self-heating. Recording the sample temperature permits a measured calibration slope $dS/dT$ to correct a signal to a reference temperature; the correction's calibration and thermometer uncertainties must be propagated. Independent white noise fluctuations decrease under averaging, whereas correlated temperature drift may not decrease as the inverse square root of the number of readings. Temperature records and repeated reference measurements help distinguish the two.

A power spectral density distributes the mean square of a signal over frequency. Its integral over the specified frequency domain gives that mean square. State whether the convention is one-sided or two-sided and whether frequency is measured in hertz or as angular frequency. For a zero-mean stationary signal, the mean square is its variance; spectral density of a stationary process gives the discrete-time convention.

Connect one physical resistor $R$ across an ideal lossless capacitor $C$, with no applied source and an ideal readout that draws no electric current. Represent the resistor by its noisy Thévenin equivalent. The resulting RC low-pass frequency response gives the stationary capacitor voltage power spectral densityIts mean square follows by the effective noise bandwidth calculation:This is also the equipartition theorem: the mean capacitor energy is $\frac{1}{2}C⟨v_C^2⟩=\frac{1}{2}{k}_{\mathrm{B}}T$. Raising $R$ makes the low-frequency spectrum taller and its width narrower in exactly compensating proportions. It changes the equilibration time $RC$, not the final variance.

The integral to infinity belongs to the ideal classical model. For a physical electrical circuit, assume thermal equilibrium, linear components with negligible extra dissipation, and a corner far below the frequencies where parasitic effects, frequency-dependent electrical resistance, or $hf\sim k_{\mathrm{B}}T$ invalidate the model; the omitted high-frequency tail must be negligible. A finite-bandwidth instrument measures only part of this variance. Its input capacitance contributes to the total $C$, and its finite input electrical resistance introduces both loading and its own thermal electrical noise.

For $T=300\mathrm{K}$ and $C=1\mathrm{nF}$, the predicted equilibrium variance is $4.141947\times 10^{-12}{\mathrm{V}}^2$ and the root mean square voltage is $2.0352\mu \mathrm{V}$. At $R=10\mathrm{k}\Omega$, $RC=10\mu \mathrm{s}$ and $B_{\mathrm{n}}=25\mathrm{kHz}$. At $R=100\mathrm{k}\Omega$, $RC=100\mu \mathrm{s}$ and $B_{\mathrm{n}}=2.5\mathrm{kHz}$; the equilibrium root mean square voltage is identical.

To obtain the growth curve, prepare $v_C(0)=0$ exactly and connect the resistor at $t=0$ while maintaining its temperature. The RC low-pass filter's causal impulse response is $h(t)=e^{-t/(RC)}/(RC)$ for $t\ge 0$. Classical ideal white noise with two-sided power spectral density $2k_{\mathrm{B}}TR$ givesThe factor $2k_{\mathrm{B}}TR$ here is the two-sided time-domain noise intensity; the equilibrium spectral integral above used the one-sided $4k_{\mathrm{B}}TR$. They give the same long-time limit. The variance approaches equilibrium with time constant $RC/2$, although a deterministic initial voltage decays with time constant $RC$. If the capacitor already starts in thermal equilibrium, its initial variance supplies the missing exponentially decaying term and the total variance stays constant.

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.

For an original model, set $q=1.602176634\times 10^{-19}\mathrm{C}$, $I=1\mu \mathrm{A}$, $R_F=100\mathrm{k}\Omega$, $G=10$, and the exponential low-pass filter's $\tau =100\mu \mathrm{s}$. Assume an ideal detector without charge multiplication and omit all additive noise for this calculation. ThenDoubling $I$ doubles the voltage variance and multiplies its standard deviation by $\sqrt{2}$. Halving $\tau$ also doubles the variance, provided the detector remains approximately white throughout the enlarged passband. Using $f_c$ in place of $B_{\mathrm{n}}$ underestimates the variance by the factor $2/\pi$. These are calculated predictions, not measured values or settings claimed for the laboratory below.

To read the dimensionless figure, set $a=\lambda \tau$ and $x=\tau i_{\mathrm{filtered}}/q$, with frequency $\nu =f\tau$ in cycles per unit $t/\tau$. A gate of duration $\tau$ has count mean and variance $a$. The continuously filtered $x$ instead hasThe count and the filtered electric current are different observables. The factor $1/2$ comes from the exponential weighting, not from non-Poisson arrivals. The dimensionless count illustration is a separate choice of rate; it does not assert that the $1\mu \mathrm{A}$ example has only four Electrons per $100\mu \mathrm{s}$.