This paper deals with two related problems, namely distance‐preserving binary embeddings and quantization for compressed sensing. First, we propose fast methods to replace points from a subset Χ ⊂ ℝn, associated with the euclidean metric, with points in the cube {±1}m, and we associate the cube with a pseudometric that approximates euclidean distance among points in Χ. Our methods rely on quantizing fast Johnson‐Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise shaping, and include sigma‐delta schemes and distributed noise‐shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in m, depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson‐Lindenstrauss maps while preserving ℓ2 norms. Second, we again consider noise‐shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for sigma‐delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise‐shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. © 2019 Wiley Periodicals, Inc.

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This paper studies the impacts of low-bit quantization to wideband signal reconstruction with a sub-Nyquist sampling scheme Modulated Wideband Converter (MWC). Several real types of analog-to-digital converters (ADCs) are firstly simulated to the MWC and their results on wideband signal reconstruction are compared based on the low-bit criterion of the ADCs. Assessing the wideband reconstruction performances, this study proposes a trade-off between the oversampling factor, the number of bits and the type of quantizer, which can achieve the performance of reconstruction as close as possible to the ideal reconstruction.

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