An approach to the sensitivity and statistical variability of biquadratic filters

Abstract

In attempting to predict the behavior of a filter during and at the end of its life, one is led to the study of sensitivity and then one must compare worst-case and expected results. This paper shows that sensitivity can be expressed as a sum of terms, where each term is the product of two sensitivity functions. One is the frequency-dependent sensitivity of the gain to the transfer function coefficients (the gain-to-coefficient sensitivity) and the other is the well-known coefficient-to-component sensitivity. The gain-to-coefficient sensitivity clearly shows that the gain of a biquadratic function is far more sensitive to changes in the resonant frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_0</tex> than to changes in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> only near the 3-dB frequencies. The gain is actually less sensitive to changes in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_0</tex> near <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_0</tex> . It is also shown that coefficient-to-component sensitivities for resistors and capacitors have no effect on the mean value of the change in the gain, but have marked effects on the standard deviation.