Sigma-Delta Quantization for Fusion Frames and Distributed Sensor Networks

High order low-bit Sigma-Delta quantization for fusion frames

“…Our motivation comes from the stylized sensor network in [24]. Suppose that one seeks to measure a signal x ∈ R d over a large environment using a collection of remotely dispersed sensors s n that are constrained by limited power, limited computational resources, and limited ability to communicate.…”

“…• The base station is relatively unconstrained in power and computational resources. Mathematically, the above sensor network problem can be formulated as a quantization problem for fusion frames, e.g., [24]. Suppose that {W n } N n=1 are subspaces of R d and suppose that each A n ⊂ W n is a finite quantization alphabet.…”

“…The work in [24] constructed and studied stable analogues of Sigma-Delta quantization in the setting of fusion frames. The first order Σ∆ algorithms in [24] were stably implementable using very low-bit quantization alphabets.…”

“…The work in [24] constructed and studied stable analogues of Sigma-Delta quantization in the setting of fusion frames. The first order Σ∆ algorithms in [24] were stably implementable using very low-bit quantization alphabets. Unfortunately, the higher order Σ∆ algorithms in [24] required large quantization alphabets for stability to hold.…”

“…The first order Σ∆ algorithms in [24] were stably implementable using very low-bit quantization alphabets. Unfortunately, the higher order Σ∆ algorithms in [24] required large quantization alphabets for stability to hold. Stable high order Σ∆ algorithms are desirable since quantization error generally decreases as the order of a Sigma-Delta algorithm increases, e.g., [14].…”