Aliasing effects in Fourier transforms of monotonically decaying functions and the calculation of viscoelastic moduli by combining transforms over different time periods

An analysis is presented of the discrete Fourier transform of a monotonically decaying function, represented by a sum of exponentials with negative exponents. The results are compared with those of a previous analysis based on the analytical Fourier transform, which proposed a method for extending the frequency range of the Fourier transform of experimental data by combining transforms performed over different time periods. The principle of the method is confirmed but comparison shows that results derived for the analytical transform cannot always be applied directly to the discrete transform. Modifications are therefore proposed which improve the accuracy and mitigate the aliasing effects evident in short-time transforms while keeping the computing time to a minimum. These results are particularly applicable to stress relaxation and creep in viscoelastic materials and an example from articular cartilage shows the compliance modulus over a frequency range from 10 -3 Hz to 230 Hz from one creep experiment of duration 18.7 min.