Analysis of Decimation on Finite Frames with 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. In this study, similar results are proven for finite unitarily generated frames. We introduce a process called alternative decimation on finite frames that is compatible with first and second order Σ∆ quantization. In both cases, alternative decimation results in exponential error decay with respect to the bit usage.

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

“…The question we seek to address is whether it is possible to translate decimation from A/D conversion to finite frame quantization. This adaptation proves to be non-trivial, as the r-th order Σ∆ quantization does not yield much more than linear error decay rate for finite frames in general as opposed to polynomial degree r, [3,24].…”

“…With the introduction of alternative decimation, the author was able to adapt signal decimation to finite frames up to the second order Σ∆ quantization [24], yielding quadratic error decay rate with respect to the oversampling ratio. This paper further generalizes the concept of decimation and extends the decimation on finite frames to arbitrary polynomial degrees.…”

“…In this paper, we build on our past result in Theorem 3.2 to formulate and prove Theorem 3.5. Specifically, Theorem 3.2 is an extension of signal decimation to finite frames up to the second order Σ∆ quantization in [24], and Theorem 3.5 further generalizes Theorem 3.2 to arbitrary orders. We shall show that for any stable r-th order Σ∆ quantization, the adapted decimation to be introduced in Section 3 coupled with the quantization scheme yields reconstruction error decay rate of polynomial degree r with respect to the oversampling ratio.…”

“…In particular, we would like to obtain polynomial error decay rate with respect to the oversampling ratio ρ while compressing the data to the order of O(log(1/ρ)). In [24], the author made an Figure 1. Illustration of the first order decimation scheme for A/D conversion.…”

“…In Section 3, we first define alternative decimation and state the result of it in Theorem 3.2, which is proven in [24]. Then, we define adapted decimation and state our main results in Theorem 3.5.…”