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Geometric Mean vs. Arithmetic Mean in Bandpass Filters

  1. Nov 14, 2013 #1


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    Gold Member

    Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean?

    I read in this book that

    1. All the lowpass elements yield LC pairs that resonate at ω = 1.
    2. Any point of the lowpass response is transformed into a pair of points of the bandpass filter. The frequencies of the pair of points are reciprocals. This means that, after frequency scaling, we can write


    where f1 and f2 are the scaled frequncies of the transforms of a single LP point, and f0 was scaled from ω = 1. This effect tells us that the bandpass filter has geometric-mean symmetry.

    I get that you need to use the geometric mean when multiplying (scaling) elements but I'm still having a hard time seeing why we can't just use the arithmetic mean in the case of defining the bandpass filter center frequency.
  2. jcsd
  3. Nov 17, 2013 #2


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    Staff: Mentor

    I'll try a qualitative justification. If you plot the separate responses (dB vs. frequency) on a logarithmic frequency scale (the x-axis), the centre of the response does indeed seem to be about midway. But I said this is when plotted on a logarithmic scale, so using the arithmetic mean would miss the mark by a long way.

    To throw in some figures. Suppose we have one corner frequency at 1Hz, and another at 100Hz. On a log paper plot, midway between these values corresponds to 10Hz and that's about where the response peaks. The response certainly does not peak near 50Hz.
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