Second-order Sigma–Delta ( ΣΔ) quantization of finite frame expansions

Abstract

The second-order Sigma–Delta ( ΣΔ) scheme with linear quantization rule is analyzed for quantizing finite unit-norm tight frame expansions for R d . Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1 / N 2 , where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second-order scheme only gives approximation error of order 1 / N . For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second-order ΣΔ scheme.