Sub-nyquist sampling techniques

Abstract

A number of novel theoretical methods have been developed in an attempt to analyse data produced by sampling a signal at below the Nyquist rate and the limitations of the approaches have been investigated. A technique is developed that allows, under specified conditions, the frequency and amplitude of a band-limited sinusoidal signal (with no harmonics) to be determined when the signal is sampled simultaneously with three uniform samplers at below the Nyquist rate. The three samplers operate at slightly different rates. Each has its output ideally low-pass filtered with a cut-off frequency at half the sampling rate. The frequencies of the signals output from the ideal filters are analysed to determine the input sinusoid parameters. The frequency of the sinusoid can also be found within a calculated tolerance when approximate filter output frequencies are known. Two approaches extending this technique for a band-limited periodic signal consisting of more than just the fundamental, enable the frequencies of the harmonics to be found for the signal, but there is the possibility that other erroneous harmonics may be identified as part of the signal. The probability of this occurring can be reduced by uniformly sampling simultaneously with a greater number of samplers. This probability cannot reach zero. Furthermore, as the number of samplers increases or the number of signal harmonics increases, the computational workload imposed in determining the harmonic frequencies rises dramatically. The approaches are rendered impractical and sampling at irregular intervals is suggested as an alternative to using a very large number of uniform samplers. A modified discrete Fourier transform and its inverse are developed to allow an estimated spectral analysis of a continuous periodic signal sampled at irregular intervals. Additive pseudo-random sampling and periodic sampling with dither are rigorously defined as two proposed irregular sampling schemes. The periodicity and symmetrical properties of the modified transform are derived for the two schemes. Consistently alias-free spectral analysis of a band-limited periodic signal is demonstrated using additive pseudo-random sampling with a maximum sampling rate below the Nyquist rate. This does not apply when using periodic sampling with dither.