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Full width at half maximum

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In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y-axis which are half the maximum amplitude. Half width at half maximum (HWHM) is half of the FWHM if the function is symmetric. The term full duration at half maximum (FDHM) is preferred when the independent variable is time.

FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. The convention of "width" meaning "half maximum" is also widely used in signal processing to define bandwidth as "width of frequency range where less than half the signal's power is attenuated", i.e., the power is at least half the maximum. In signal processing terms, this is at most −3 dB of attenuation, called half-power point or, more specifically, half-power bandwidth. When half-power point is applied to antenna beam width, it is called half-power beam width.

Specific distributions

[edit]

Normal distribution

[edit]

If the considered function is the density of a normal distribution of the form $f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(x-x_{0})^{2}}{2\sigma ^{2}}}\right]$ where σ is the standard deviation and x₀ is the expected value, then the relationship between FWHM and the standard deviation is^[1] $\mathrm {FWHM} =2{\sqrt {2\ln 2}}\;\sigma \approx 2.355\;\sigma .$ The FWHM does not depend on the expected value x₀; it is invariant under translations. The area within this FWHM is approximately 76% of the total area under the function.

Other distributions

[edit]

In spectroscopy half the width at half maximum (here γ), HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height ⁠1/πγ⁠ can be defined by $f(x)={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\quad {\text{ and }}\quad \mathrm {FWHM} =2\gamma .$

Another important distribution function, related to solitons in optics, is the hyperbolic secant: $f(x)=\operatorname {sech} \left({\frac {x}{X}}\right).$ Any translating element was omitted, since it does not affect the FWHM. For this impulse we have: $\mathrm {FWHM} =2\operatorname {arcsch} \left({\tfrac {1}{2}}\right)X=2\ln(2+{\sqrt {3}})\;X\approx 2.634\;X$ where $arcsch$ is the inverse hyperbolic secant.

References

[edit]

This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.

^ Gaussian Function – from Wolfram MathWorld

External links

[edit]

FWHM at Wolfram Mathworld

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Full width at half maximum

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The full width at half maximum (FWHM) is a quantitative measure used to describe the width of a peak or distribution in various scientific contexts, defined as the separation between the two points on a curve where the intensity or amplitude is exactly half of its maximum value.^[1] This parameter is particularly valuable because it provides a straightforward, shape-independent way to assess the breadth of features like spectral lines, resonance peaks, or Gaussian distributions without requiring complex fitting.^[2] In physics and spectroscopy, FWHM serves as a key indicator of resolution, where narrower widths signify higher ability to distinguish closely spaced features, such as in atomic emission lines or instrumental response functions.^[3] For instance, in the analysis of Lorentzian or Breit-Wigner line shapes—common in quantum mechanics for unstable particle decays or resonances—the FWHM directly relates to the lifetime or decay rate of the system, with narrower profiles indicating longer-lived states.^[4] Similarly, in optics and astronomy, it characterizes beam spreads or point spread functions, as in radio telescope primary beams where the FWHM defines the effective field of view.^[5] FWHM is also integral to signal processing and statistics, often applied to Gaussian profiles where it approximates 2.355 times the standard deviation (σ), aiding in the evaluation of data spread or filter bandwidths.^[2] In medical imaging, such as magnetic resonance spectroscopy, it measures resonance linewidths to assess spectral quality and tissue properties.^[6] Its ubiquity stems from computational simplicity: it can be directly extracted from experimental data by identifying the half-maximum points, making it robust for both theoretical modeling and practical measurements across disciplines like dosimetry and materials science.^[7]

Definition and Properties

Definition

The full width at half maximum (FWHM) is defined as the width of a peaked function or signal measured at the points where its value equals half of the maximum amplitude. This parameter quantifies the extent of the central portion of the peak by identifying the two points where the intensity or height is half the maximum value, and taking the distance between them along the independent variable axis, such as wavelength, frequency, time, or position.^[8]^[9] It applies broadly to any unimodal function or probability density exhibiting a single prominent peak, serving as a robust indicator of width without requiring assumptions about the underlying shape.^[10] The concept originated in the field of spectroscopy as a standard measure for characterizing the broadening of spectral lines, reflecting factors like instrumental resolution or physical processes such as Doppler effects.^[9] Over time, it was adopted in statistics and data analysis to describe the spread or variability in peaked distributions, providing an intuitive metric for central tendency that complements measures like standard deviation.^[11] For instance, in the Gaussian distribution, commonly encountered in natural phenomena, the FWHM offers a practical way to gauge dispersion.^[11] Visually, consider a generic bell-shaped curve representing a signal intensity versus some variable: the peak rises to a maximum, and the FWHM is determined by drawing a horizontal line at half the peak height, which intersects the curve at two points; the separation between these intersections captures the core width of the feature. This approach emphasizes the region's where the signal is most significant, effectively quantifying spread or duration in a manner independent of the precise peak asymmetry or tail extension.^[12]^[13] FWHM differs from other width measures, such as the full width at base (FWB), which extends from one baseline intersection to the other across the entire structure, including extended tails that may arise from noise or secondary effects. By focusing solely on the half-maximum level, FWHM avoids overestimation from outliers or low-level broadening, making it particularly valuable for peaked profiles where the central region dominates the functional behavior.^[14]^[15]

Key properties

The full width at half maximum (FWHM) exhibits invariance under vertical scaling, such that multiplying the entire peak by a constant factor to alter its height leaves the FWHM unchanged, as the half-maximum level scales proportionally with the peak amplitude. This property renders FWHM particularly advantageous for analyzing signals where intensity variations occur due to experimental conditions, such as differing sample concentrations or detector sensitivities, without affecting the width assessment. For instance, in spectral analysis, this ensures consistent quantification of broadening mechanisms regardless of signal strength.^[16] In scenarios involving convolution, such as when a true signal is broadened by an instrument response function, the FWHM of the resultant peak approximates the square root of the sum of the squares of the individual FWHMs for profiles with similar shapes, like Gaussians. This approximate additivity facilitates the deconvolution of instrumental effects from intrinsic properties, enabling more accurate determination of underlying widths in fields like X-ray photoelectron spectroscopy.^[17] FWHM demonstrates sensitivity to peak asymmetry, where non-symmetric profiles—often featuring extended tails on one side—yield a larger measured width at half maximum than their symmetric equivalents, capturing the influence of factors like phonon interactions or chemical shifts. This characteristic highlights deviations from ideal symmetric broadening, providing diagnostic value for identifying asymmetries in experimental data. Additionally, the units of FWHM match those of the independent variable, such as nanometers for wavelength in optical spectra or seconds for temporal pulses, ensuring direct interpretability in context-specific measurements.^[18]^[19]

Mathematical Formulation

General expression

The full width at half maximum (FWHM) of a continuous unimodal function

f(x)

is defined as the distance between the two points on either side of the peak where the function value equals half its maximum amplitude. Specifically, if

f(x)

attains its maximum value

f_{\max}

at position

x_0

, then the FWHM is given by

\Delta x = x_{1/2}^+ - x_{1/2}^-

, where

x_{1/2}^-

and

x_{1/2}^+

are the solutions to the equation

f(x_{1/2}^\pm) = f_{\max}/2

with

x_{1/2}^- < x_0 < x_{1/2}^+

.^[20]^[21] To compute the FWHM, first identify the maximum value

f_{\max}

and its location

x_0

. Then, solve the equation

f(x) - f_{\max}/2 = 0

numerically to find the roots

x_{1/2}^-

and

x_{1/2}^+

flanking the peak, and subtract these points to obtain the width.^[21]^[22] For functions normalized such that

f(x) \leq 1

with

f(x_0) = 1

, the half-maximum level simplifies to 0.5, so the roots satisfy

f(x_{1/2}^\pm) = 0.5

.^[20] This definition assumes an isolated peak; for functions with multiple or overlapping peaks, the FWHM applies only after isolating the target peak through techniques such as baseline subtraction or multi-component curve fitting to deconvolve contributions. For instance, the Gaussian function admits a closed-form expression for its FWHM.

Relations to other parameters

The full width at half maximum (FWHM) serves as a measure of dispersion in peaked functions and probability distributions, relating directly to other parameters that characterize spread or scale. In the Gaussian distribution, defined by the probability density function

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)

f(x)=σ2π

1exp(−2σ2(x−μ)2), the FWHM is given exactly by $2 \sqrt{2 \ln 2} \, \sigma \approx 2.355 \sigma$

σ≈2.355σ, where $\sigma$ is the standard deviation.^[23] This connection highlights FWHM as approximately 2.355 times the standard deviation, providing a practical link between the robust width metric and the variance-based measure of dispersion.^[11] For the Lorentzian (or Cauchy) distribution, with density $f(x) = \frac{1}{\pi \gamma} \frac{1}{1 + \left( \frac{x - \mu}{\gamma} \right)^2 }$ , the FWHM equals $2\gamma$ , where $\gamma$ denotes the half-width at half-maximum parameter.^[24] This exact equality underscores the FWHM's role as the defining scale in Lorentzian profiles, common in resonance phenomena. A related measure, the full width at quarter maximum (FWQM), for a Gaussian is $2 \sqrt{2 \ln 4} \, \sigma = 4 \sqrt{\ln 2} \, \sigma \approx 3.330 \sigma$

σ=4ln2

σ≈3.330σ, yielding FWQM $\approx 1.414 \times$ FWHM.^[23] In spectroscopy, the FWHM approximates the equivalent width (EW) for narrow line profiles, where EW $\approx$ (line depth) $\times$ FWHM, as the rectangular area of height equal to the maximum absorption and width equal to FWHM closely matches the integrated line area.^[25] This simplification holds when the line is unresolved and symmetric, facilitating quick estimates of line strength without full integration. For the Voigt profile, arising as the convolution of Gaussian and Lorentzian components, the FWHM combines both contributions but lacks a simple closed form. In parameterized families of distributions, such as location-scale families, the FWHM scales linearly with the scale parameter, preserving the shape while proportionally adjusting the width; for instance, if a distribution is scaled by a factor $c > 0$ , its FWHM multiplies by $c$ .^[26] This proportionality ensures FWHM's utility as a scale-invariant descriptor up to the scaling factor, applicable across rescaled variants like standardized normals or gamma distributions with fixed shape.^[26]

Applications in Distributions

Gaussian distribution

The Gaussian distribution, or normal distribution, is defined by its probability density function

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right),

f(x)=σ2π

1exp(−2σ2(x−μ)2), where $\mu$ is the mean (location parameter) and $\sigma > 0$ is the standard deviation (scale parameter).^[27] To derive the full width at half maximum (FWHM), identify the points where $f(x)$ equals half its maximum value. The maximum density occurs at $x = \mu$ , with $f(\mu) = \frac{1}{\sigma \sqrt{2\pi}}$

1. Setting $f(x) = \frac{1}{2} f(\mu)$ simplifies to $\exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right) = \frac{1}{2}$ . Taking the natural logarithm yields $\frac{(x - \mu)^2}{2\sigma^2} = \ln 2$ , so $|x - \mu| = \sigma \sqrt{2 \ln 2}$

. Thus, the FWHM is the distance between these points: $\text{FWHM} = 2 \sigma \sqrt{2 \ln 2} \approx 2.3548 \sigma.$

≈2.3548σ. ^[28]^[23] This measure quantifies the effective width of the bell-shaped curve, providing a practical indicator of spread that approximates the range containing a substantial portion of the probability mass, such as roughly 76% within $\mu \pm \frac{\text{FWHM}}{2}$ , which aids in statistical inference like estimating confidence intervals.^[11] For example, with $\sigma = 1$ and $\mu = 0$ , the FWHM is approximately 2.355, spanning from $x \approx -1.177$ to $x \approx 1.177$ where the density drops to half maximum.^[28]

Lorentzian distribution

The Lorentzian distribution, also known as the Cauchy distribution in probability theory, has the probability density function

f(x) = \frac{\gamma}{\pi \left[ (x - x_0)^2 + \gamma^2 \right]},

where

x_0

is the location parameter representing the peak center, and

\gamma > 0

is the scale parameter that corresponds to the half-width at half-maximum.^[29]^[30] To derive the full width at half maximum (FWHM), first note that the maximum value of the density occurs at

x = x_0

, where

f(x_0) = \frac{1}{\pi \gamma}

. The half-maximum value is thus

\frac{1}{2\pi \gamma}

. Setting

f(x) = \frac{1}{2\pi \gamma}

yields

\frac{\gamma}{\pi \left[ (x - x_0)^2 + \gamma^2 \right]} = \frac{1}{2\pi \gamma},

which simplifies to

(x - x_0)^2 + \gamma^2 = 2\gamma^2

, so

(x - x_0)^2 = \gamma^2

and

x - x_0 = \pm \gamma

. Therefore, the points at half-maximum are separated by

2\gamma

, giving FWHM =

2\gamma

.^[30]^[31] Unlike the Gaussian distribution, the Lorentzian exhibits heavy tails that decay as

1/x^2

, leading to undefined mean and all higher moments except in a principal value sense, with the variance specifically diverging to infinity. As a result, the FWHM serves as the primary measure of scale and spread for the Lorentzian, rather than variance or standard deviation, which are unavailable.^[29] In physics, the Lorentzian distribution commonly describes resonance phenomena, such as the lineshape of unstable particles or atomic transitions, where the parameter

\gamma

(or

\Gamma = 2\gamma

as the full width) relates to the lifetime

\tau

of the resonant state via the energy-time uncertainty principle:

\Delta E \cdot \Delta t \geq \hbar/2

, with

\Delta E \approx \hbar / \tau

corresponding to the energy width

\Gamma

.^[32]^[33]

Applications in Physics and Engineering

Spectroscopy

In spectroscopy, the full width at half maximum (FWHM) quantifies the broadening of spectral lines, providing insight into the physical processes affecting atomic or molecular transitions. It is typically measured in units of wavelength (e.g., nm) or frequency (e.g., GHz), representing the width of the intensity profile where the signal drops to half its peak value. This parameter is crucial for characterizing the resolution and lifetime of excited states in emission or absorption spectra. Spectral line broadening arises from various mechanisms, each contributing to the observed FWHM. Doppler broadening, due to the thermal motion of emitting or absorbing particles, results in a Gaussian profile with FWHM given by

\Delta \nu_D = \frac{2\nu_0}{c} \sqrt{2kT \ln 2 / m}

ΔνD=c2ν02kTln2/m

, where $\nu_0$ is the central frequency, $c$ the speed of light, $k$ Boltzmann's constant, $T$ temperature, and $m$ the particle mass. Natural broadening, stemming from the finite lifetime of quantum states via the uncertainty principle, produces a Lorentzian profile with FWHM $\Delta \nu_N = \frac{1}{2\pi \tau}$ , where $\tau$ is the excited state lifetime. Pressure broadening, caused by collisions in dense gases, also yields a Lorentzian shape, with FWHM proportional to the gas pressure and collision cross-section. In many real-world spectra, such as those from stellar atmospheres or laboratory plasmas, the observed profile is a Voigt function, the convolution of Gaussian (Doppler and instrumental) and Lorentzian (natural and pressure) components. The FWHM of the Voigt profile, $\Delta \nu_V$ , can be approximated for cases where one component dominates or when widths are comparable as $\Delta \nu_V \approx \sqrt{(\Delta \nu_G)^2 + (\Delta \nu_L)^2}$

, where $\Delta \nu_G$ and $\Delta \nu_L$ are the Gaussian and Lorentzian FWHMs, respectively; more precise calculations often require numerical methods. This composite broadening complicates direct interpretation, necessitating profile fitting to disentangle contributions. Measuring intrinsic FWHM requires accounting for instrumental resolution, which convolves with the sample's true profile and sets a lower limit on observable widths (typically 0.1–1 nm for common spectrometers). Deconvolution techniques, such as least-squares fitting or Fourier transform methods, are employed to extract the sample's broadening from the measured spectrum, enabling accurate determination of physical parameters like temperature or pressure. For instance, in atomic emission spectroscopy, high-resolution instruments achieve FWHMs below 0.01 nm, allowing separation of isotopic shifts. A key application is in analyzing atomic spectra, where narrow FWHMs indicate long-lived excited states; for example, visible transitions in alkali atoms like sodium exhibit natural linewidths corresponding to lifetimes around $10^{-8}$ s, yielding FWHMs of about 10 MHz. Broader lines, such as those in molecular spectra under high pressure, can reach GHz widths, reflecting collisional dephasing. These measurements underpin fields like laser cooling and precision metrology.

Signal processing

In signal processing, the full width at half maximum (FWHM) characterizes the temporal extent of pulses and the spectral breadth of waveforms, providing a robust measure for transient signals in engineering applications such as communications and radar systems. For Gaussian pulses, which approximate many ideal waveforms in RF and optical domains, the FWHM pulse duration

\tau_\text{FWHM}

relates to the frequency bandwidth

\Delta \nu_\text{FWHM}

through the time-bandwidth product

\tau_\text{FWHM} \Delta \nu_\text{FWHM} = 0.441

for transform-limited cases, establishing a fundamental limit on how narrow a pulse can be in both time and frequency. This relation, derived from Fourier transform properties, enables engineers to predict signal distortion or resolution based on pulse shaping.^[34] In the context of filters and system responses, FWHM quantifies bandwidth by defining the range where the power frequency response exceeds half its peak value, often aligning with the conventional 3 dB cutoff for bandpass designs.^[35] For instance, resonant filters exhibiting a Lorentzian lineshape use FWHM to denote the decay-limited bandwidth, influencing selectivity in analog and digital signal processing circuits.^[36] Chirped pulses, where frequency varies linearly across the duration due to dispersion, introduce asymmetry that alters the effective FWHM relative to unchirped profiles. In such cases, the intensity autocorrelation trace shows characteristic sidelobes or uneven wings, causing the autocorrelation FWHM to overestimate the true pulse duration by factors depending on chirp rate, necessitating advanced retrieval methods for accurate characterization. A practical example occurs in ultrafast laser systems for signal generation, where sub-picosecond FWHM values signify ultrashort pulses suitable for high-speed applications; mode-locked Ti:sapphire lasers, for instance, routinely achieve ~70 fs FWHM durations, enabling terahertz-rate sampling in photonic signal processing.^[37]

Imaging and resolution

In optical imaging systems, the full width at half maximum (FWHM) of the point spread function (PSF) serves as a key metric for assessing spatial resolution, quantifying the extent to which a point source is blurred into a finite-sized image. The PSF represents the response of the imaging system to an ideal delta function input, and its FWHM directly indicates the smallest resolvable feature size, as features smaller than this width become indistinguishable due to diffraction-limited blurring. For instance, in diffraction-limited systems with a circular aperture, the PSF takes the form of an Airy disk, where the FWHM is approximately 0.51 λ / NA, with λ denoting the wavelength of light and NA the numerical aperture of the objective.^[38] This measure is preferred over the Rayleigh criterion in many practical contexts because it provides a more direct gauge of the intensity profile's width at half maximum, enabling precise comparisons across instruments.^[39] In confocal microscopy, the FWHM of the PSF exhibits distinct differences between lateral and axial directions, reflecting the anisotropic nature of the illumination and detection geometry. Laterally, the FWHM typically achieves resolutions around 180–200 nm for visible wavelengths and high-NA objectives (e.g., NA = 1.4), benefiting from the pinhole's rejection of out-of-focus light to sharpen the in-plane response. Axially, however, the FWHM is poorer, often reaching 500 nm or more, due to the elongated PSF along the optical axis caused by the finite depth of focus and refractive index mismatches in specimens.^[40] These disparities limit volumetric imaging capabilities, prompting techniques like two-photon excitation to improve axial performance while maintaining lateral sharpness.^[38] Deconvolution algorithms leverage the measured FWHM of the PSF to distinguish instrumental blur from the intrinsic size of imaged features, enhancing resolution beyond the diffraction limit in post-processing. By modeling the observed image as the convolution of the true object with the known PSF—whose FWHM is experimentally determined or simulated—deconvolution reverses this blurring effect, effectively narrowing the apparent feature widths. For example, if the measured FWHM of a sub-resolution particle exceeds the PSF's FWHM, the excess can be attributed to the particle's true size, allowing quantitative separation through iterative fitting or inverse filtering.^[41] This approach is particularly valuable in fluorescence microscopy, where it restores contrast in densely labeled samples without additional hardware.^[42] Astronomical imaging exemplifies FWHM's role in large-scale applications, as seen with the Hubble Space Telescope (HST), where the PSF's FWHM for point sources like stars measures approximately 0.05 arcseconds in high-resolution channels such as the High Resolution Channel (HRC) of the Advanced Camera for Surveys (ACS). This fine PSF enables the resolution of fine details in distant galaxies and exoplanet transits, with the FWHM varying slightly across the field of view due to optical aberrations but remaining diffraction-limited at visible wavelengths.^[43] In medical imaging modalities like optical coherence tomography, similar FWHM-based PSF characterization supports blur correction for subsurface tissue visualization.^[44]

Estimation Methods

Analytical approaches

Analytical approaches to computing the full width at half maximum (FWHM) for well-known functional forms typically involve direct algebraic solving of the equation

f(x) = f_{\max}/2

to find the points where the function reaches half its peak value, with the difference between those points yielding the width. For parameterized functions that are quadratic near the peak, such as those exhibiting small asymmetries, a parabolic approximation can be applied by fitting

f(x) \approx h - a (x - \mu)^2

around the maximum, leading to an exact solution for the half-width

d = \sqrt{h / (2a)}

d=h/(2a)

and thus FWHM = $2d = \sqrt{2h / a}$

. This method is particularly useful for rough estimates in spectroscopy when the peak curvature is well-captured by a second-order Taylor expansion, assuming deviations from symmetry are minor. For more complex profiles like the Voigt function, which lacks a closed-form expression for FWHM due to its nature as a convolution of Gaussian and Lorentzian components, empirical approximations provide efficient computation. A widely used formula, accurate to within 0.02% relative error across all ratios of component widths, is given by Olivero and Longbothum as
$\text{FWHM}_V \approx 0.5346 \, \text{FWHM}_L + \sqrt{0.2166 \, \text{FWHM}_L^2 + \text{FWHM}_G^2},$

,
where $\text{FWHM}_L$ and $\text{FWHM}_G$ are the full widths at half maximum of the Lorentzian and Gaussian components, respectively. This approximation is exact in the limits of pure Lorentzian ( $\text{FWHM}_G = 0$ , where $\text{FWHM}_V = \text{FWHM}_L$ ) and pure Gaussian ( $\text{FWHM}_L = 0$ , where $\text{FWHM}_V = \text{FWHM}_G$ ), with maximum errors occurring at intermediate mixing ratios but remaining below 0.02% for Lorentzian dominance (when $\text{FWHM}_G \ll \text{FWHM}_L$ ). Symbolic computation software, such as SymPy, enables exact analytical solutions for FWHM in cases where the functional form permits symbolic root-finding, like quadratic or higher-order polynomials, by leveraging algorithms to solve transcendental equations when possible.

Numerical and experimental techniques

In numerical estimation of the full width at half maximum (FWHM) from discrete experimental data, interpolation methods offer practical approaches to refine measurements beyond raw point sampling. Linear interpolation estimates the FWHM by connecting the two nearest data points above and below the half-maximum intensity with a straight line and solving for the intersection widths. Spline interpolation, using cubic or higher-order polynomials fitted piecewise across the peak, provides a smoother approximation, reducing errors in irregularly sampled or noisy datasets; this local method has been shown to yield more accurate results than global fits in gamma camera profiles. These techniques are essential in fields like nuclear imaging, where pixel resolution limits direct measurement precision.^[45] Recent comparative studies have evaluated multiple methods for FWHM estimation, including direct measurement and those based on standard deviation approximations, highlighting their performance in various signal processing contexts.^[10] Fitting algorithms enable robust FWHM determination by modeling the peak shape against theoretical profiles. Least-squares optimization fits Gaussian or Lorentzian functions to the data, iteratively adjusting parameters such as peak center, amplitude, and width to minimize residuals; for Lorentzian peaks, nonlinear variants like the Levenberg-Marquardt algorithm efficiently handle the asymmetric form. The χ² minimization serves as a statistical measure of fit quality, incorporating data uncertainties to weight points and avoid overfitting, particularly useful in low-signal regimes. In spectroscopy applications, such fittings often outperform simple interpolation by accounting for underlying physical broadening mechanisms.^[46]^[36] Experimental techniques for reliable FWHM measurement address common data artifacts. Noise reduction via averaging multiple spectral acquisitions enhances the signal-to-noise ratio, enabling clearer half-maximum identification without introducing bias. Baseline subtraction corrects for instrumental drifts or backgrounds by fitting low-order polynomials to off-peak regions or employing asymmetric least-squares algorithms that preserve peak integrity while suppressing trends. For peak isolation in Fourier-domain data, apodization with window functions like Hamming or Blackman reduces Gibbs ringing artifacts, though it trades some resolution for sidelobe suppression. Voigt profile fitting may be referenced briefly in spectroscopic contexts to convolve Gaussian and Lorentzian components for hybrid line shapes.^[47]^[48] Uncertainty in FWHM estimates arises from noise, sampling, and model assumptions, necessitating resampling methods for quantification. Bootstrap techniques generate pseudo-datasets by resampling with replacement from the original measurements, computing FWHM repeatedly to derive confidence intervals via the standard deviation of the distribution. Monte Carlo approaches simulate noise addition based on measured variances, propagating errors through the estimation process to yield probabilistic error bars, such as those on the order of 5-10% of the FWHM value in typical spectroscopic setups. These methods provide empirical validation of precision without assuming parametric distributions.^[49]

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Full width at half maximum

Specific distributions

Normal distribution

Other distributions

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Full width at half maximum

Definition and Properties

Definition

Key properties

Mathematical Formulation

General expression

Relations to other parameters

Applications in Distributions

Gaussian distribution

Lorentzian distribution

Applications in Physics and Engineering

Spectroscopy

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