Digital Alias-free Signal Processing

15

DFT of Nonuniformly Sampled Signals

In the classical case of processing periodically sampled signals on the basis of the DFT, the results of the performed DFT reflect the structure of the respective signals in the frequency domain. In other words, the DFT of periodically sampled signals leads to signal decomposition and to obtaining their spectra. Therefore it is normal to expect that when the DFT of nonuniformly sampled signals are calculated the results will also represent the spectra of these signals. However, these expectations are typically not fulfilled. As soon as the sampling procedure is randomized, the DFT of the respective signals become strongly sampling-dependent. As explained in the previous chapter, the basis functions for the DFT under the conditions of nonuniform sampling become unorthogonal. If DFT could be performed as unorthogonal transforms then the preference should be given to such an approach. However, that is not always possible. Direct calculations of DFT have to be undertaken while clearly realizing that these transforms will not complete the process of signal decomposition into their components. Estimating the Fourier coefficients then actually leads to acquiring intermediate signal processing results containing valuable information. Therefore the outcome of the DFT under these conditions should not always be automatically regarded as spectrograms of the respective signals. Some comments on this are given in Section 15.1. To obtain spectrograms showing ...