Faster than Nyquist but Slower than Tropp
Robert Carderbank


The sampling rate of analog-to-digital converters is severely limited by underlying technological constraints. Recently, Tropp et al. proposed a new architecture, called a random demodulator, that attempts to overcome this limitation by sampling sparse, band limited signals at a rate much lower than the Nyquist rate. An integral part of this architecture is a random bi-polar modulating waveform that changes polarity at the Nyquist rate of the input signal. Technological constraints also limit how fast such a waveform can change polarity.

We propose an extension of the random demodulator that uses a run-length limited (RLL) modulating waveform, and which we call a constrained random demodulator (CRD). The RLL modulating waveform changes polarity at a slower rate. We demonstrate that a CRD enjoys theoretical guarantees similar to the RD and that these guarantees are directly related to the power spectrum of the modulating waveform. Further, we show that the relationship between the placement of energy in the spectrum of the input signal and the placement of energy in the power spectrum of the modulating waveform has a major effect on the reconstruction performance of signals sampled by a CRD.