Phase approximation by logarithmic sampling of gain

The plotting of the frequency transfer function has been simplified for decades by using logarithmic plots. One advantage of logarithmic plots is that the mathematical operations of multiplication and division are transformed to addition and subtraction. With this method the work of obtaining the phase of a transfer function is largely graphical instead of analytical. On the other hand, we need the factors of the transfer function. In the case where we have only gain samples this might be a sensitive issue. The difficulty increases if there is available only a part of the samples. Moreover, the distance between samples might not be very small and this is the most encountered case in practical situations. The goal of this paper is to address this special situation. Therefore we establish methods for approximating the phase values from the gain samples, in nepers, equally spaced in the logarithmic frequency domain. First a general approximation formula is proved, then two quadrature formulae are derived using Newton-Cotes and Simpson rules. Numerical examples are also presented.