View source for Half-power point

{{Short description|Electronics reference point}}
The '''half-power point''' is the condition at which a system's output [[Electric power|power]] has dropped to half of its peak value; that is, at a level of approximately {{val|-3|u=[[Decibel|dB]]}}.{{efn|name=power}} 

In [[Filter (signal processing)|filters]], [[optical filter]]s, and [[electronic amplifier]]s,<ref>{{cite book|last1=Schlessinger|first1=Monroe|title=Infrared technology fundamentals|date=1995|publisher=M. Dekker|location=New York|isbn=0824792599|edition=2nd ed., rev. and expanded.|url=https://books.google.com/books?id=QPBQ5w4X8RkC&dq=half-power+point&pg=PA113}}</ref> the half-power point is also known as '''half-power bandwidth''' and is a commonly used definition for the [[cutoff frequency]]. 

In the characterization of [[Antenna (radio)|antennas]] the half-power point is also known as '''half-power beamwidth''' and relates to measurement position as an angle and describes [[Directional antenna|directionality]].

==Amplifiers and filters==
This occurs when the output [[voltage]] has dropped to <math>\tfrac{1}{\sqrt{2}} \approx \text{0.707}</math> of the filter's nominal passband voltage{{efn|name=level|Exact: <math>20\log_{10}\left(\tfrac{1}{\sqrt{2}}\right) \approx -3.0103\, \mathrm{dB}</math>}} and the power has dropped by half.{{efn|name=power|Exact: <math>10\log_{10}\left(\tfrac{1}{2}\right) \approx -3.0103\, \mathrm{dB}</math>}} A [[bandpass]] amplifier will have two half-power points, while a [[Low-pass filter|low-pass]] amplifier or a [[High-pass filter|high-pass]] amplifier will have only one. 

The [[Bandwidth (signal processing)|bandwidth]] of a filter or amplifier is usually defined as the difference between the lower and upper half-power points. This is, therefore, also known as the {{val|3|u=dB}} bandwidth.  There is no lower half-power point for a low-pass amplifier, so the bandwidth is measured relative to [[direct current|DC]], i.e., {{val|0|u=Hz}}. There is no upper half-power point for an ideal high-pass amplifier, its bandwidth is theoretically infinite.<ref>In practice there is no high-pass with infinite bandwidth. All high-passes are bandpasses, but, if properly designed, with the upper half-point so high that it does not affect the application.</ref> In practice the [[stopband]] and [[transition band]] are used to characterize a high-pass.

==Antenna beams{{anchor|Antennas|Beams}}==
{{broader|Beam width}}
[[File:Richt-kar1.png|thumb|A 'polar' diagram showing beamwidth]]

In antennas, the expression ''half-power point'' does not relate to frequency: instead, it describes the extent in space of an antenna beam.  The half-power point is the angle off [[antenna boresight|boresight]] at which the antenna gain first falls to half power (approximately {{val|-3|u=dB}}){{efn|name=power}} from the peak. The angle between the {{val|-3|u=dB}} points is known as the ''half-power beam width'' (or simply ''beam width'').<ref>{{citation |url=http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |title=Antenna Introduction / Basics |access-date=2017-08-08 |archive-date=2017-08-28 |archive-url=https://web.archive.org/web/20170828230007/http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |url-status=dead }}</ref>

Beamwidth is usually but not always expressed in degrees and for the horizontal plane.
It refers to the ''[[main lobe]]'', when referenced to the peak [[effective radiated power]] of the main lobe.
Note that other definitions of [[beam width]] exist, such as the distance between nulls and distance between first [[side lobe]]s.

===Calculation===
The beamwidth can be computed for arbitrary antenna arrays. Defining the array manifold as the complex response of the <math>\mathrm{m}</math> element antenna array as {{nowrap|<math>\mathrm{A}(\theta)</math>,}} where <math>\mathrm{A}(\theta)</math> is a matrix with <math>\mathrm{m}</math> rows, the beam pattern is first computed as:<ref>{{cite book|first1=H. L.|last1=Van Trees|title=Optimum Array Processing|publisher=Wiley|location=New York|year=2002}}</ref><ref>E. Tuncer and B. Friedlander (Editors), "Classical and Modern Direction-of-Arrival Estimation", Academic Press, 2009.</ref>

<math display="block">\mathrm{B}(\theta) = \frac{1}{\mathrm{m}}\mathrm{A}(\theta_0)^{*}\mathrm{A}(\theta)</math>

where <math>\mathrm{A}(\theta_0)^{*}</math> is the conjugate transpose of <math>\mathrm{A}</math> at the reference angle {{nowrap|<math>\theta_0</math>.}}

From the beam pattern {{nowrap|<math>\mathrm{B}(\theta)</math>,}} the antenna power is computed as:

<math display="block">\mathrm{P} = |\mathrm{B}|^{2}\,.</math>

The ''half-power beamwidth'' (HPBW) is then found as the range of <math>\theta</math> where {{nowrap|<math>\mathrm{P} = 0.5\mathrm{P_{max}}</math>.}}

== See also ==
* [[Antenna aperture]]
* [[Angular resolution]]
*[[Full width at half maximum]]

==Notes==
{{notelist}}

==References==
{{reflist}}
{{FS1037C MS188}}

[[Category:Electronic amplifiers]]
[[Category:Antennas (radio)]]