Editing Bandwidth (signal processing) (section)

== 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>