Fwhm

Overview

The Full Width at Half Maximum (FWHM) is a parameter used to describe the width of a peak in a distribution, function, or signal. It is defined as the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value. FWHM is commonly used in various fields such as spectroscopy, optics, signal processing, and medical imaging.

Definition

In mathematical terms, if a function \( f(x) \) has a maximum value \( f(x_0) \), the FWHM is the distance between the points \( x_1 \) and \( x_2 \) such that:

\[

f(x_1) = f(x_2) = \frac{1}{2} f(x_0)

\]

The FWHM is given by:

\[

\text{FWHM} = x_2 - x_1

\]

Applications

Spectroscopy

In spectroscopy, FWHM is used to describe the width of spectral lines. It provides information about the resolution of the spectroscopic instrument and the physical properties of the sample, such as temperature and pressure. Narrower FWHM values indicate higher resolution and less broadening of the spectral line.

Optics

In optics, FWHM is used to characterize the bandwidth of light sources, such as lasers and LEDs. It is an important parameter in determining the coherence length and the quality of the light source. A smaller FWHM indicates a more monochromatic light source.

Signal Processing

In signal processing, FWHM is used to describe the temporal or frequency width of a signal. It is a critical parameter in the design and analysis of filters, where it helps to define the bandwidth of the filter.

Medical Imaging

In medical imaging, FWHM is used to describe the resolution of imaging systems, such as MRI and CT scanners. It helps in quantifying the spatial resolution and the ability of the imaging system to distinguish between two closely spaced objects.

Calculation

The calculation of FWHM depends on the shape of the peak. For a Gaussian function, the FWHM can be calculated using the standard deviation \( \sigma \) as follows:

\[

\text{FWHM} = 2 \sqrt{2 \ln 2} \cdot \sigma \approx 2.355 \cdot \sigma

\]

For other peak shapes, numerical methods or fitting techniques may be required to determine the FWHM.

Importance

FWHM is a crucial parameter in many scientific and engineering applications. It provides a quantitative measure of the width of a peak, which is essential for understanding the characteristics of the system or phenomenon being studied. In medical applications, FWHM is vital for ensuring high-quality imaging and accurate diagnosis.

See Also

• Spectral line width
• Resolution (optics)
• Bandwidth (signal processing)
• Gaussian function

External Links

• [FWHM Calculator]
• [Spectroscopy Techniques]