Center frequency

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Image:Bandwidth.png

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.

<math>

f_0 = \sqrt{f_1 \cdot f_2} </math>

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:

<math>

f_0 \approx (f_1 + f_2)/2 </math>

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:

<math>

f_0 \approx 1499.993 \, \mathrm{kHz} </math>

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

<math>

f_0 = 1500 \, \mathrm{kHz} </math>

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

<math>

\Delta f \approx \frac{B^2}{8 f_0} </math>.

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.

External links

• Calculations and comparisons between the geometric mean and the arithmetic mean

de:Mittenfrequenz