Editing Bandwidth (signal processing)

{{Short description|Range of usable frequencies}}
{{about|the concept in signal theory and processing measured in hertz|use in computing and networking expressed in bits per second|Bandwidth (computing)|other uses|Bandwidth (disambiguation)}}

[[Image:Baseband.svg|right|300px|thumb|Amplitude (a) vs. frequency (f) graph illustrating [[baseband]] bandwidth. Here the bandwidth equals the upper frequency.]]

'''Bandwidth''' is the difference between the upper and lower [[Frequency|frequencies]] in a continuous [[Frequency band|band of frequencies]].  It is typically measured in [[unit of measurement|unit]] of [[hertz]] (symbol Hz).

It may refer more specifically to two subcategories: ''[[Passband]] bandwidth'' is the difference between the upper and lower [[cutoff frequencies]] of, for example, a [[band-pass filter]], a [[communication channel]], or a [[signal spectrum]]. ''[[Baseband]] bandwidth'' is equal to the upper cutoff frequency of a [[low-pass filter]] or baseband signal, which includes a zero frequency.

Bandwidth in hertz is a central concept in many fields, including [[electronics]], [[information theory]], [[digital communication]]s, [[radio communication]]s, [[signal processing]], and [[spectroscopy]] and is one of the determinants of the capacity of a given [[communication channel]].

A key characteristic of bandwidth is that any band of a given width can carry the same amount of [[information]], regardless of where that band is located in the [[frequency spectrum]].{{efn|The information capacity of a channel depends on [[Noise (electronics)|noise]] level as well as bandwidth – see [[Shannon–Hartley theorem]]. Equal bandwidths can carry equal information only when subject to equal [[signal-to-noise ratio]]s.}} For example, a 3&nbsp;kHz band can carry a telephone conversation whether that band is at baseband (as in a [[plain old telephone service|POTS]] telephone line) or [[modulated]] to some higher frequency.  However, wide bandwidths are easier to obtain and [[signal processing|process]] at higher frequencies because the {{section link|#Fractional bandwidth}} is smaller.

== Overview ==

Bandwidth is a key concept in many [[telecommunications]] applications. In [[radio]] communications, for example, bandwidth is the frequency range occupied by a modulated [[carrier signal]]. An [[FM radio]] receiver's [[tuner (radio)|tuner]] spans a limited range of frequencies.  A government agency (such as the [[Federal Communications Commission]] in the United States) may apportion the regionally available bandwidth to [[broadcast license]] holders so that their [[signal (electronics)|signals]] do not mutually interfere. In this context, bandwidth is also known as [[channel spacing]].

For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance.  A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In the case of [[frequency response]], degradation could, for example, mean more than 3&nbsp;[[decibel|dB]] below the maximum value or it could mean below a certain absolute value. As with any definition of the ''width'' of a function, many definitions are suitable for different purposes.

In the context of, for example, the [[sampling theorem]] and [[Nyquist rate|Nyquist sampling rate]], bandwidth typically refers to [[baseband]] bandwidth. In the context of [[Nyquist rate|Nyquist symbol rate]] or [[Shannon-Hartley]] [[channel capacity]] for communication systems it refers to [[passband]] bandwidth.

The '''{{vanchor|Rayleigh bandwidth}}''' of a simple radar pulse is defined as the inverse of its duration.  For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz.<ref>Jeffrey A. Nanzer, ''Microwave and Millimeter-wave Remote Sensing for Security Applications'', pp. 268-269, Artech House, 2012 {{ISBN|1608071723}}.</ref>

The '''{{vanchor|essential bandwidth}}''' is defined as the portion of a [[signal spectrum]] in the frequency domain which contains most of the energy of the signal.<ref>{{cite book|last=Sundararajan|first=D.|title=A Practical Approach to Signals and Systems|url=https://books.google.com/books?id=1Oo55lFE6UoC&pg=PA109|date=4 March 2009|publisher=John Wiley & Sons|isbn=978-0-470-82354-5|page=109}}</ref>

== ''x'' dB bandwidth ==

[[Image:Bandwidth 2.svg|right|300px|thumb|The magnitude response of a [[band-pass filter]] illustrating the concept of −3 dB bandwidth at a gain of approximately 0.707]]

In some contexts, the signal bandwidth in [[hertz]] refers to the frequency range in which the signal's [[spectral density]] (in W/Hz or V<sup>2</sup>/Hz) is nonzero or above a small threshold value.  The threshold value is often defined relative to the maximum value, and is most commonly the {{no wrap|[[3 dB point]]}}, that is the point where the spectral density is half its maximum value (or the spectral amplitude, in <math>\mathrm{V}</math> or <math>\mathrm{V/\sqrt{Hz}}</math>, is 70.7% of its maximum).<ref>
{{cite book
|title=Network Analysis
|edition=3rd
|last=Van Valkenburg
|first=M. E.
|year=1974
|pages=[https://archive.org/details/networkanalysis00vanv/page/383 383–384]
|publisher=Prentice-Hall
|isbn=0-13-611095-9
|url=https://archive.org/details/networkanalysis00vanv/page/383
|access-date=2008-06-22
}}</ref> This figure, with a lower threshold value, can be used in calculations of the lowest sampling rate that will satisfy the [[sampling theorem]].  

The bandwidth is also used to denote '''system bandwidth''', for example in [[Electronic filter|filter]] or [[communication channel]] systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth.

The 3&nbsp;dB bandwidth of an [[electronic filter]] or communication channel is the part of the system's frequency response that lies within 3&nbsp;dB of the response at its peak, which, in the passband filter case, is typically at or near its [[center frequency]], and in the low-pass filter is at or near its [[cutoff frequency]]. If the maximum gain is 0&nbsp;dB, the 3&nbsp;dB bandwidth is the frequency range where attenuation is less than 3&nbsp;dB. 3&nbsp;dB attenuation is also where power is half its maximum. This same ''half-power gain'' convention is also used in [[spectral width]], and more generally for the extent of functions as [[full width at half maximum]] (FWHM).

In [[electronic filter]] design, a filter specification may require that within the filter [[passband]], the gain is nominally 0&nbsp;dB with a small variation, for example within the ±1&nbsp;dB interval. In the [[stopband]](s), the required attenuation in decibels is above a certain level, for example >100&nbsp;dB. In a [[transition band]] the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1&nbsp;dB-bandwidth. If the filter shows amplitude ripple within the passband, the ''x''&nbsp;dB point refers to the point where the gain is ''x''&nbsp;dB below the nominal passband gain rather than ''x''&nbsp;dB below the maximum gain.

In signal processing and [[control theory]] the bandwidth is the frequency at which the [[Closed-loop transfer function|closed-loop system gain]] drops 3&nbsp;dB below peak.

In communication systems, in calculations of the [[Shannon–Hartley]] [[channel capacity]], bandwidth refers to the 3&nbsp;dB-bandwidth. In calculations of the maximum [[symbol rate]], the [[Nyquist sampling rate]], and maximum bit rate according to the [[Hartley's law]], the bandwidth refers to the frequency range within which the gain is non-zero.

The fact that in equivalent [[baseband]] models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as <math>B = 2W</math>, where <math>B</math> is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and <math>W</math> is the positive bandwidth (the baseband bandwidth of the equivalent channel model).  For instance, the baseband model of the signal would require a [[low-pass filter]] with cutoff frequency of at least <math>W</math> to stay intact, and the physical passband channel would require a passband filter of at least <math>B</math> to stay intact.

== Relative bandwidth ==
{{see also|Antenna (radio)#Bandwidth|Antenna measurement#Bandwidth}}

The absolute bandwidth is not always the most appropriate or useful measure of bandwidth.  For instance, in the field of [[Antenna (radio)|antennas]] the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency.  For this reason, bandwidth is often quoted relative to the frequency of operation which gives a better indication of the structure and sophistication needed for the circuit or device under consideration.

There are two different measures of relative bandwidth in common use: ''fractional bandwidth'' (<math>B_\mathrm F</math>) and ''ratio bandwidth'' (<math>B_\mathrm R</math>).<ref>{{cite book |last1=Stutzman |first1=Warren L. |first2=Gary A. |last2=Theiele |title=Antenna Theory and Design |edition=2nd |location=New York |year=1998 |isbn=0-471-02590-9 }}</ref>  In the following, the absolute bandwidth is defined as follows,
<math display="block"> B = \Delta f = f_\mathrm H - f_\mathrm L</math> 
where <math>f_\mathrm H</math> and <math>f_\mathrm L</math> are the upper and lower frequency limits respectively of the band in question.

=== Fractional bandwidth ===
Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency (<math>f_\mathrm C</math>),
<math display="block"> B_\mathrm F = \frac {\Delta f}{f_\mathrm C} \, .</math>

The center frequency is usually defined as the [[arithmetic mean]] of the upper and lower frequencies so that,
<math display="block"> f_\mathrm C = \frac {f_\mathrm H + f_\mathrm L}{2} \ </math> and
<math display="block"> B_\mathrm F = \frac {2 (f_\mathrm H - f_\mathrm L)}{f_\mathrm H + f_\mathrm L} \, .</math>

However, the center frequency is sometimes defined as the [[geometric mean]] of the upper and lower frequencies,
<math display="block"> f_\mathrm C = \sqrt {f_\mathrm H f_\mathrm L} </math> and
<math display="block"> B_\mathrm F = \frac {f_\mathrm H - f_\mathrm L}{\sqrt {f_\mathrm H f_\mathrm L}} \, .</math>

While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous.  It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency.<ref>Hans G. Schantz, ''The Art and Science of Ultrawideband Antennas'', p. 75, Artech House, 2015 {{ISBN|1608079562}}</ref>  For [[narrowband]] applications, there is only marginal difference between the two definitions.  The geometric mean version is inconsequentially larger.  For [[wideband]] applications they diverge substantially with the arithmetic mean version approaching 2 in the limit and the geometric mean version approaching infinity.

Fractional bandwidth is sometimes expressed as a percentage of the center frequency ('''percent bandwidth''', <math>\%B</math>),
<math display="block"> \%B_\mathrm F = 100 \frac {\Delta f}{f_\mathrm C} \, .</math>

=== Ratio bandwidth ===

Ratio bandwidth is defined as the ratio of the upper and lower limits of the band,
<math display="block">B_\mathrm R= \frac {f_\mathrm H}{f_\mathrm L} \, .</math>

Ratio bandwidth may be notated as <math>B_\mathrm R:1</math>.  The relationship between ratio bandwidth and fractional bandwidth is given by,
<math display="block">B_\mathrm F = 2 \frac {B_\mathrm R - 1}{B_\mathrm R + 1} </math> and
<math display="block">B_\mathrm R = \frac {2 + B_\mathrm F}{2 - B_\mathrm F} \, .</math>

Percent bandwidth is a less meaningful measure in wideband applications.  A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1.  All higher ratios up to infinity are compressed into the range 100–200%.

Ratio bandwidth is often expressed in [[octave]]s (i.e., as a [[frequency level]]) for wideband applications.  An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, <math display="block">\log_2 \left(B_\mathrm R\right) .</math>

== Noise equivalent bandwidth ==
[[File:Enbw.svg|right|300px|thumb|Setup for the measurement of the noise equivalent bandwidth <math>B_n</math> of the system with frequency response <math>H(f)</math>.]] 
{{further|Spectral leakage#Noise bandwidth}}

The '''noise equivalent bandwidth''' (or '''equivalent noise bandwidth (enbw)''') of a system of [[frequency response]] <math>H(f)</math> is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing <math>H(f)</math> when both systems are excited with a [[white noise]] source.
The value of the noise equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals <math>|H(f)|</math> at its center frequency,<ref name="Jeruchim">{{cite book |last1=Jeruchim |first1=M. C.|last2=Balaban |first2=P.|last3=Shanmugan |first3=K. S. |title=Simulation of Communication Systems. Modeling, Methodology, and Techniques. |publisher=Kluwer Academic |isbn=0-306-46267-2 |edition=2nd|date=2000}}</ref> but it can also equal the peak value of <math>|H(f)|</math>. 

The noise equivalent bandwidth <math>B_n</math> can be calculated in the frequency domain using <math>H(f)</math> or in the time domain by exploiting the [[Parseval's theorem]] with the system [[impulse response]] <math>h(t)</math>.  If <math>H(f)</math> is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then: 

<math display="block"> B_n = \frac{\int_{-\infty}^{\infty} |H(f)|^2 df}{2|H(0)|^2} =  \frac{\int_{-\infty}^{\infty} |h(t)|^2 dt}{2\left|\int_{-\infty}^{\infty} h(t)dt\right|^2} \, . </math>

The same expression can be applied to bandpass systems by substituting the [[Baseband#Equivalent baseband signal|equivalent baseband]] frequency response for <math>H(f)</math>. 

The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.

== Photonics ==

In [[photonics]], the term ''bandwidth'' carries a variety of meanings:
*the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
*the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
*the gain bandwidth of an optical amplifier
*the width of the range of some other phenomenon, e.g., a reflection, the phase matching of a nonlinear process, or some resonance
*the maximum modulation frequency (or range of modulation frequencies) of an optical modulator
*the range of frequencies in which some measurement apparatus (e.g., a power meter) can operate
*the [[Bit rate|data rate]] (e.g., in Gbit/s) achieved in an optical communication system; see [[bandwidth (computing)]].

A related concept is the [[spectral linewidth]] of the radiation emitted by excited atoms.

== See also ==
{{Wiktionary|bandwidth}}
*[[Bandwidth extension]]
*[[Broadband]]
*[[Noise bandwidth]]
*[[Rise time]]
*[[Spectral efficiency]]
*[[Spectral width]]

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

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[[Category:Signal processing]]
[[Category:Telecommunication theory]]
[[Category:Filter frequency response]]
[[Category:Spectrum (physical sciences)]]