Use of the symmetrical number system in resolving undersampling aliases

Abstract

Two algorithms are presented which allow for the unambiguous resolution of multiple undersampled frequency components in a signal. Digital signal processing is usually governed by the Nyquist criterion which limits the amount of information that can be unambiguously stored and recovered digitally to a spectral width no larger than half the sampling frequency. Both algorithms resolve a spectrum beyond Nyquist by using additional information. The first method samples a signal more than once using a different sampling frequency each time. The second method utilizes a single sampling frequency which is used to sample both the signal and a band-limited version of the signal. When using multiple sampling frequencies, each sampling frequency yields a digital sequence which, in turn, has a unique spectrum when the Discrete Fourier Transform (DFT) is applied. The bin and amplitude information from each of the resulting undersampled spectra is then recombined to resolve the original spectrum. In like manner, when using a single sampling frequency the spectra of both the signal and its band-limited version are recombined to obtain the solution. Given a sampling frequency, both algorithms allow for the unambiguous resolution of a signal with a spectral width at least twice as large as that predicted by Nyquist.