Editing Bandwidth (signal processing) (section)

=== 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>