1. An analog-to-digital converter apparatus, the apparatus sensing an analog signal having a frequency, the apparatus comprising: a time encoder having at least an input, an output, and a pulse frequency, the analog signal frequency of the analog signal to be sensed being higher than the pulse frequency of the time encoder, the time encoder being configured to generate an excitation vector including a plurality of projection values of the analog signal into a plurality of testing basis functions; the analog signal being connected to the input of the time encoder; a pulse domain demultiplexer having at least an input and a plurality of outputs; the output of the time encoder being connected to the input of the pulse domain demultiplexer; a pulse-to-asynchronous digital converter having at least a plurality of inputs and a plurality of outputs, the plurality of outputs of the pulse domain demultiplexer being connected to the plurality of inputs of the pulse-to-asynchronous digital converter in a first predetermined sequence; an asynchronous-digital-to-synchronous digital converter having at least a plurality of inputs and an output, the plurality of outputs of the pulse-to-asynchronous digital converter being connected to the plurality of inputs of the asynchronous-digital-to-synchronous digital converter in a second predetermined sequence; and a signal processor having at least an input and an output, the output of the asynchronous-digital-to-synchronous digital converter being connected to the input of the signal processor, the signal processor processing a plurality of known basis functions, wherein the signal processor being configured to output an estimate of the analog signal at the output of the signal processor.