Computer hardware

Detector

Algorithm

Voltage

Engineering

Quantization (signal processing)

Discrete mathematics

Bandwidth (computing)

Embedded system

Companding

Telecommunications

Economics

Reset (finance)

Computer science

Digital signal processing

Bounded function

Bandlimiting

Signal processing

Oversampling

Mathematics

Noise shaping

Image (mathematics)

Sampling (signal processing)

Electronic engineering

Bottleneck

Mathematical analysis

Nyquist–Shannon sampling theorem

Orthogonal frequency-division multiplexing

Upsampling

Artificial intelligence

Financial economics

Channel (broadcasting)

Modulo

Converters

Fourier transform

Computer vision

Electrical engineering

IEEE Transactions on Signal Processing

Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analog-to-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC voltage thus leading to a significant information loss. In this paper, we develop an alternative paradigm for sensing and recovery, called the Unlimited Sampling Framework. It is based on the observation that when a signal is mapped to an appropriate bounded interval via a modulo operation before entering the ADC, the saturation problem no longer exists, but one rather encounters a different type of information loss due to the modulo operation. Such an alternative setup can be implemented, for example, via so-called folding or self-reset ADCs, as they have been proposed in various contexts in the circuit design literature. The key task that we need to accomplish in order to cope with this new type of information loss is to recover a bandlimited signal from its modulo samples. In this paper we derive conditions when perfect recovery is possible and complement them with a stable recovery algorithm. The sampling density required to guarantee recovery is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only. Our recovery guarantees extend to measurements affected by bounded noise, which includes the case of round-off quantization. Numerical experiments validate our approach. Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.

On Unlimited Sampling and Reconstruction