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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 zero-mean stationary Gaussian random field $V$, the unnormalized autocorrelation function of a random field isIt depends only on the separation $h$, and $C(0)$ is the pointwise variance. If $C(0)>0$, the normalized correlation is $\rho (h)=C(h)/C(0)$; the two conventions have different units.

For a mean-square continuous field, its spectral measure of a stationary random field $\nu$ obeys $C(h)=\int e^{ik\cdot h}\nu (dk)$. If $C$ is integrable, an ordinary power spectral density exists and, in the angular-wavevector convention,Thus $\nu$ has density $S(k)/(2\pi {)}^d$. A spectral measure need not have a density: a spatially constant centered Gaussian field of variance $a>0$ has $C(h)=a$ and spectral measure $a{\delta }_0$, concentrated at zero wavevector. Calling every spectrum an ordinary function would miss this case.^[ref]

Subtracting two photocurrents can suppress a shared classical intensity fluctuation when the optical responses and electronic gains are matched. Independent shot noise contributions do not cancel: their variances add in the difference. With equal charge $q$ and mean currents $I_1,I_2$, the independent-arrival one-sided current power spectral density is $2q(I_1+I_2)$. Correlated optical inputs require their covariance as well.

Independent arrivals of charges $q$ at rate $\lambda$ give a Poisson distribution for the number $N$ in time $\tau$, with mean and variance $\lambda \tau$. Since ${\overline{I}}_{\tau }=qN/\tau$ and $I=q\lambda$, its variance is $qI/\tau$. This agrees with integrating the one-sided shot noise power spectral density $2qI$ through the averaging filter: its effective noise bandwidth is $1/(2\tau )$. A two-sided convention instead assigns density $qI$ to each sign of frequency.

Dan H. Neuhart is the first named author, with Catherine B. McGinley, of the 2004 Langley study of wind-tunnel turbulence intensity. The report's title page identifies the byline and Langley affiliation. Its scientific contribution is an experimental characterization of tunnel flow with a specified signal chain, rather than an assertion that a facility has one universal turbulence number.^[ref]

The study combined hot-wire anemometry, velocity calibration and spectral analysis. The reported dynamic-data procedure retained approximately $1\mathrm{Hz}$ to $10\mathrm{kHz}$, sampled at $25.6\mathrm{kHz}$ and acquired simultaneous records. The distinction between sampling rate and retained bandwidth is essential: increasing the former does not establish which fluctuations were included in the reported intensity.^[ref]

An original ideal spectral example makes that distinction quantitative. Suppose a one-sided power spectral density is constant at $S_u=4\times 10^{-8}(m/s{)}^2/\mathrm{Hz}$ across a selected $10000\mathrm{Hz}$ band and zero elsewhere. ThenKeeping only a $1000\mathrm{Hz}$ subset of that flat band gives $u_{\mathrm{rms}}\simeq 0.006325m/s$. The changed answer reflects a changed integration band, not a measured improvement in the underlying flow. Real spectra are not generally flat; the invented example isolates the reporting issue without attributing either value to Neuhart's experiment.

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 the equilibrium generalized Langevin equation with unit mass, the random force has zero mean and the displayed autocorrelation, where $\Theta$ is the equilibrium velocity variance. If the deterministic relaxation $g$ is stable, $g(0)=1$ and $g^{\prime }=-\kappa \ast g$, then the stationary velocity autocorrelation function is $\chi =\Theta g$, equivalentlyOne proof uses the power spectral density. The even force autocorrelation has spectrum $2\Theta \mathrm{Re}\tilde{\kappa }(i\omega )$. With transfer function $H(i\omega )=[i\omega +\tilde{\kappa }(i\omega ){]}^{-1}$, the velocity spectrum is $2\Theta |H{|}^2\mathrm{Re}\tilde{\kappa }=2\Theta \mathrm{Re}H$, the spectrum of the even extension of $\Theta g$. Stability removes initial transients with finite second moment. A valid force autocorrelation must have nonnegative power spectral density; an arbitrary proposed memory kernel does not automatically satisfy this condition.

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.

Independent-record averaging and increasing detector bandwidth change a measurement in different ways. Let $Y_1,\dots ,Y_N$ be independent measurements of zero-mean noise, made with the same fixed filter, and suppose each has finite variance ${\sigma }_B^2$. Their average satisfiesThe vanishing mean is an ensemble statement; the finite-record noise fluctuations only decrease as the number of independent records grows. For correlated records the variance is instead $N^{-2}{\sum }_{i,j}\mathrm{Cov}(Y_i,Y_j)$, so merely increasing the number of samples need not give the same reduction.

Bandwidth controls which fluctuations enter a record. With two-sided angular-frequency power spectral density $S(\omega )$ and an ideal low-pass filter retaining $|\omega |\le \Omega$, the filtered variance isFor flat noise spectrum $S(\omega )=S_0>0$ in that range, this is $S_0\Omega /\pi$, and averaging $N$ independent records gives $S_0\Omega /(\pi N)$. Increasing bandwidth exposes more of the physical fluctuations; independent averaging reduces their variance. Sampling faster from one correlated trajectory is not automatically equivalent to obtaining independent records.

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.

A noise-limited dynamic range is the ratio between an upper usable signal level and a lower level defined by a noise criterion, with the observation conditions specified. The lower bound might require a chosen signal-to-noise ratio; the upper bound can be set by amplifier clipping or detector saturation. The ratio depends on bandwidth, integration time, signal convention and acceptance threshold, not only on displayed digits.

For a linear response $V=Kx$ after offset correction, let the output-noise standard deviation be ${\sigma }_V$, the minimum accepted signal be $k{\sigma }_V$, and the output ceiling be $V_{\max }$. ThenThese equations assume the stated noise and ceiling apply throughout the interval. Changing $K$ alone shifts the input interval without changing $D$ under those assumptions. A real amplifier can change noise and bandwidth with gain; the primary specification reports these dependencies separately.^[ref]

For an original voltage-amplitude example, take a usable RMS signal ceiling of $1\mathrm{V}$, RMS zero-mean noise $100\mu \mathrm{V}$, and $k=5$. The lower accepted signal is $0.5\mathrm{mV}$ and $D=2000$. As a voltage-amplitude ratio at fixed impedance, this is $20{\log }_{10}(2000)\approx 66.02\mathrm{dB}$. If the input quantity instead denotes optical power, its power ratio uses $10{\log }_{10}D$; the physical quantity must accompany a decibel statement.

Noise density and total noise are different. For a flat one-sided output power spectral density $S_V$ over a specified effective noise bandwidth $B$, the model gives ${\sigma }_V^2=S_{V}B$. Increasing $B$ fourfold doubles ${\sigma }_V$ and halves $D$ if the ceiling and threshold factor remain fixed. A spectrum with drift or nonwhite components need not obey that scaling.

A noise-limited detection threshold is consequently an operational choice. Five times RMS noise is not a universal minimum detectable signal, nor by itself a statement of false-alarm probability without a noise distribution and observation rule. To characterize a particular instrument, measure its zero-input fluctuations under the actual filtering and loading conditions, then test a stable source near the proposed lower and upper bounds.

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 $f_c$.

Write the source's noise terms consistently as ${\eta }_R,{\eta }_P$. Subtract the steady mean to get $\dot{\delta R}=-{\gamma }_R\delta R+{\eta }_R$. Use the Fourier convention $f(t)=\int d\omega e^{-i\omega t}\tilde{f}(\omega )/(2\pi )$. After initial transients decay,Fourier transforming the stationary white-noise covariance givesTherefore the RNA fluctuation power spectral density is $S_R(\omega )=q_R/({\gamma }_R^2+{\omega }^2)$. Its integral is the equal-time variance:The positivity of ${\gamma }_R$ ensures a stationary limit. The same calculation gives the autocovariance $⟨\delta R(t)\delta R(0)⟩=[q_R/(2{\gamma }_R)]e^{-{\gamma }_R|t|}$: the decay rate filters the white forcing into a finite correlation time. Here the covariance convention is $q_R\delta (t-t^{\prime })$; replacing it by $2D_R\delta (t-t^{\prime })$ would identify $q_R=2D_R$.

A linearly stable $f_0$ has no zeros of the causal dielectric function in $\mathrm{Re}p>0$. Hence the inverse Laplace transform contours can approach $\sigma =0^{+}$, retaining the causal boundary value at $p=-i\omega +0^{+}$. To obtain a stationary forced response, further assume that the excited collective perturbations actually damp. This is the decay condition in the question's long-time instruction.

A transparent evaluation of the double integral uses the stationary power spectral density. The white noise covariance $2A_k\delta (t-t^{\prime })$ has temporal spectrum $2A_k$. Linear filtering by $1/ϵ$ therefore givesThe causal solution for $\delta f_k$ in a gives, after the initial transients have decayed,The real part can be taken because the full $k$ sum pairs conjugates. Using the distribution theory identity $\mathrm{Re}{\int }_0^{\infty }{e}^{ia\tau }d\tau =\pi \delta (a)$, with a positive vanishing convergence factor, samples the temporal spectrum at $\omega =kv$:The factor $1/2$ from the one-sided time integral cancels the $2$ in the noise covariance. The result is nonnegative, as a stationary velocity-space diffusion coefficient should be.

Qualification of stability. Absence of growing modes alone does not guarantee this stationary limit. For example, the electron waterbag distribution $f_0(v)=n_e{1}_{|v|<V}/(2V)$ hasas undamped collective roots. White noise drives their field variance to grow instead of settling. Thus the formula requires decay of excited modes, not merely absence of exponentially growing ones. For weak Landau damping, the waiting time can be much longer than ${\omega }_{pe}^{-1}$, and must still be short compared with evolution of $f_0$ for the frozen-background quasilinear approximation to apply.

shot noise arises from fluctuations in discrete transfer events. For equal charges $q$ arriving independently with a constant Poisson rate $\lambda$, the charge collected during $\tau$ is $Q=qN$, withThus the relative counting uncertainty is $1/\sqrt{\lambda \tau }$. Longer integration improves the precision of the mean current even when the microscopic arrivals remain irregular.

In the frequency range where independent arrivals can be treated as instantaneous pulses, the one-sided current power spectral density isFor an ideal unit-gain rectangular bandwidth $B$, this gives $i_{\mathrm{rms}}=\sqrt{2qIB}$. boxcar averaging of Poisson shot noise over duration $\tau$ has effective noise bandwidth $B_{\mathrm{n}}=1/(2\tau )$, reproducing the counting result $i_{\mathrm{rms}}=\sqrt{qI/\tau }$. The two-sided spectral convention instead assigns $qI$ to each sign of frequency. Stating that convention prevents a factor-of-two error.

A photodiode illuminated at a stable rate provides a useful measurement: subtract the mean current, amplify its fluctuations through a calibrated linear filter, and compare the variance with the mean photocurrent. Independent detected photoelectrons predict a variance proportional to mean current. Measure the dark and amplifier backgrounds separately, check the filter's effective noise bandwidth and vary illumination to distinguish this term from classical intensity noise, whose variance can scale quadratically with mean current. The shot noise model concerns counting statistics, whereas thermal electrical noise can exist in an unbiased resistor. Walter Schottky's motivating analysis treated amplifier limits and independent electron transfers in thermionic discharge tubes.^[ref]

The Poisson result is a model, not a universal consequence of charge quantization. Correlated arrivals, detector dead time, avalanche multiplication or a non-Poisson light source can change the fluctuations. Filtering reduces integrated noise at the cost of temporal resolution. Increasing the detected event rate reduces relative shot noise but raises its absolute variance; balanced photodetection cancels shared classical fluctuations, not the independent shot noise of two uncorrelated detectors.

For independent charge arrivals described by a Poisson distribution, the one-sided Shot noise current Power spectral density is $S_I=2eI$. Treating the stated bandwidth as the Effective noise bandwidth,The expected rms noise is $5.66\mathrm{pA}$, about $0.566\%$ of the mean current. This is the full Shot noise value with Fano factor one; correlations can suppress or enhance it. The elementary charge is from the external CODATA 2014 table.^[ref]

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.