Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization

Abstract

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the Σ∆ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. We build on our previous result, which extended signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling ratio, and exponential with respect to the bit-rate. ∞ −∞ f (t)e −2πıtγ dt.The Fourier transform can also be uniquely extended to L 2 (R) as a unitary transformation. An important component of A/D conversion is the following theorem:Theorem 1.2 (Classical Sampling Theorem). Given f ∈ P W [−1/2,1/2] , for any g ∈ L 2 (R) satisfying•ĝ(ω) = 1 on [−1/2, 1/2] •ĝ(ω) = 0 for |ω| ≥ 1/2 + , with > 0 and T ∈ (0, 1 − 2 ), t ∈ R, one haswhere the convergence is both uniform on compact sets of R and in L 2 .2010 Mathematics Subject Classification. 42C15, 94A08, 94A34.