Center frequency

The center frequency f ₀ is the geometric mean of the lower f ₁ and the upper f ₂ limit frequency (crossover frequency ) of a frequency band with a certain filter bandwidth , also known under the term bandpass .

Basics

The center frequency of the filter bandwidth B = f ₂ - f ₁ is calculated from:

{\ displaystyle f_ {0} = {\ sqrt {f_ {1} \ cdot f_ {2}}}}

The arithmetic mean is often used incorrectly , although the frequencies in the frequency bands are logarithmically related. For example, the center frequency of the telephone audio frequencies from 300 Hz to 3400 Hz is not (3400 + 300) / 2 Hz = 1850 Hz, but the square root of 300 Hz x 3400 Hz = 1010 Hz.

The center frequency of linearly arranged spectra, as they can be found in antenna technology, for example, must nevertheless be calculated as an arithmetic mean:

${\ displaystyle f_ {0} = {\ frac {f_ {1} + f_ {2}} {2}}}$

properties

By defining the center frequency, the ratios of the limit frequencies to the center frequency are the same:

{\ display style {\ frac {f_ {1}} {f_ {0}}} = {\ sqrt {\ frac {f_ {1}} {f_ {2}}}} = {\ frac {f_ {0}} {f_ {2}}}}

If f ₁ , f ₀ and f _{2 are marked} on a logarithmic frequency scale, then f _{0 is} in the middle:

{\ displaystyle \ log {f_ {0}} - \ log {f_ {1}} = \ log {f_ {2}} - \ log {f_ {0}}}

Use as an approximation

The bandwidth f ₂ - f ₁ is often small compared to the center frequency. Then you can use the arithmetic mean for the calculation as a good approximation:

{\ displaystyle f_ {0} \ approx {\ frac {f_ {1} + f_ {2}} {2}}}

With many medium wave transmitters z. B. the bandwidth is only 9 kHz. A transmitter that is specified as 1500 kHz transmits in the band from 1495.5 kHz to 1504.5 kHz. The approximation formula gives

{\ displaystyle f_ {0} \ approx 1500 \, \ mathrm {kHz}}

while the exact formula is used to determine:

{\ displaystyle f_ {0} = 1499 {,} 993 \, \ mathrm {kHz}}

The value calculated with the approximation formula is always too large. If you designate the bandwidth with B , the deviation of the approximation formula is approximately

{\ displaystyle \ Delta f \ approx {\ frac {B ^ {2}} {8 \ cdot f_ {0}}}}

in the example given, this is less than 7 Hz.

literature

Michael Dickreiter, Volker Dittel, Wolfgang Hoeg, Martin Wöhr (eds.), "Handbuch der Tonstudiotechnik", 8th, revised and expanded edition, 2 volumes, publisher: Walter de Gruyter, Berlin / Boston, 2014, ISBN 978-3- 11-028978-7 or e- ISBN 978-3-11-031650-6
Gregor Häberle, Heinz Häberle, Thomas Kleiber: Expertise in radio, television and radio electronics. 3rd edition, Verlag Europa-Lehrmittel, Haan-Gruiten, 1996, ISBN 3-8085-3263-7
Karl Hermann Huber: Filtering and sifting of audio frequencies. 1st edition, Frech Verlag, Stuttgart, 1974, ISBN 3-7724-0162-7
Warren L. Stutzman: Antenna Theory and Design . 3rd edition, Wiley Verlag, Weinheim, 2012, ISBN 978-0470576649

Web links