On the Reconstruction of Bandlimited Signals from Random Samples Quantized via Noise-Shaping

Noise-shaping quantization techniques are widely used for converting bandlimited signals from the analog to the digital domain. They work by “shaping” the quantization noise so that it falls close to the reconstruction operator’s null space. We investigate the compatibility of two such schemes, specifically Σ Δ Σ Δ quantization and distributed noise-shaping quantization, with random samples of bandlimited functions. Suppose $$R>1$$ R > 1 is a real number and assume that ᵢ\ᵢ₌₁ᵐ { x i } i = 1 m is a sequence of i.i.d. random variables uniformly distributed on [-R̃,R̃] [ - R ~ , R ~ ] , where R̃>R R ~ > R is appropriately chosen. We show that by using a noise-shaping quantizer to quantize the (randomly sign flipped) values of a real-valued π π -bandlimited function f at ᵢ\ᵢ₌₁ᵐ { x i } i = 1 m , a function f f ♯ can be reconstructed from these quantized values such that f-f L²[-R, R] ‖ f - f ♯ ‖ L 2 [ - R , R ] decays with high probability as m and R̃ R ~ increase. This decay holds uniformly over all bandlimited f . We emphasize that the sample points ᵢ\ᵢ₌₁ᵐ {