Integrating time signals in frequency domain – Comparison with time domain integration

Abstract

Integrating sampled time signals is a common task in signal processing. In this paper we investigate the performance of two straightforward integration methods: (i) integration in the frequency domain by a discrete Fourier transform (DFT), division by jω followed by inverse DFT (IDFT) back to the time domain, and (ii) a method using a weighted overlap-add (WOLA) technique which is developed in the paper. These two methods are compared with two time domain methods: (a) the trapezoidal rule, and (b) an optimized IIR filter. It is shown that the intuitive method of a straightforward DFT/IDFT is a very good method which is recommended for data lengths exceeding 16K samples, provided data are short enough to allow a single DFT. The IIR filter integration is shown to have very similar accuracy and can also be recommended. The WOLA integration method is shown to perform well in most cases for steady-state conditions. For cases with short transients it should, however, be avoided. A signal integrated by the WOLA method is further shown to be incoherent with the signal before integration. This suggests that the WOLA method should be avoided in cases where coherence between the signals before and after integration is important. It is also demonstrated by a simulation example that integration by the trapezoidal rule should be avoided, as it gives biased results, particularly for higher frequencies.