A new sampling theorem based on the minimum energy principle

Abstract

AbstractAmong those low‐pass band‐limited waveforms which take specified sampled values at a specified finite number of sampling points, the one with minimum energy is called the minimum‐energy waveform. This paper first derives the minimum‐energy waveform and strictly proves its uniqueness. Discussion is then devoted to the minimum‐energy waveform which is determined by sampled values of a low‐pass band‐limited waveform of finite energy, and it is shown that the sequence obtained by successively adding sampling points uniformly converges to a low‐pass band‐limited waveform. In general, when a low‐pass band‐limited waveform is uniquely determined from the sampled values at the sampling points, the sampling point sequence is called complete. This paper shows that various kinds of sampling theorems can be derived, by prescribing sampled values for the low‐pass band‐limited waveform for the complete sampling point sequence given by Levinson and constructing the limiting waveform of minimum energy. Especially, a new reconstruction formula is obtained which restores the original band‐limited waveform solely from the past sampled values. It is shown that the traditional Someya‐Shannon interpolation formula can be obtained by specifying particular sampling points in the above restoration formula. The relation between the two formulas is also discussed.