# Center frequency

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Bandwidth.png
The frequency axis of this symbolic diagram would be logarithmically scaled.

The center frequency f0 (resonant frequency) is the geometric mean between the lower cutoff frequency f1 and the upper cutoff frequency f2 of a frequency band. See also: Band-pass filter. f2 - f1 is called the bandwidth B.

[itex]

f_0 = \sqrt{f_1 \cdot f_2} [itex]

Only if the bandwidth f2 - f1 is very small in comparison to the center frequency it is sometimes possible to use this arithmetic mean for calculations, but this is often calculated in mistake:

[itex]

f_0 \approx (f_1 + f_2)/2 [itex]

At radio stations (medium wave) the bandwidth is often only 9 kHz. A transmitter, which has 1500 kHz, is transmitting from 1495.5 kHz to 1504.5 kHz.
The exact formula gives:

[itex]

f_0 = 1500 \, \mathrm{kHz} [itex]

and the short formula gives in this case the very close result of:

[itex]

f_0 \approx 1499.993 \, \mathrm{kHz} [itex]

The short calculated value is always too large. If the bandwidth is given by B = f2 - f1, the difference is:

[itex]

\Delta f \approx \frac{B^2}{8 f_0} [itex].

But if for instance we are looking for the center frequency of the telephone audio band from 300 Hz to 3300 Hz, we get (3300 + 300) / 2 = "1800 Hz" for the short arithmetic mean calculation, but the root of 300 x 3300 = "995" Hz with the correct geometric mean formula. What a big difference!

That the geometric mean is not the arithmetic mean can be seen in a calculation program at the bottom in the external link. There one can compare the difference of both values.

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