Distortion Estimates for Adaptive Lifting Transforms with Noise – Support Document

Distortion Estimates for Adaptive Lifting Transforms with Noise – Support Document Fabio Verdicchio and Yiannis Andreopoulos APPENDIX A A.1. Validation of Observation 1 Assuming the experimental settings described in Section V.A, we consider the approximation of (14), i.e. the relative impact of the term ∆ ∆ M v in the synthesis error ∆x . The approximation of (14) involves only one lifting step, hence the experimental assessment is separately carried out for the predict and update step as follows. When = M P , the vector x comprises the samples of the input signal and the coefficient vector p = v x holds the predictstep output. Conversely, when = M U , the input vector is p = x x and the coefficient vector is given by u = v x . The objective is to assess the relative impact of neglecting the term ∆ ∆ M v in the expression of ∆x given by (13), hence we compute the ratio ∆ ∆ ∆ M v x for several signals x . Prior to synthesis, several perturbation patterns ∆M are generated (each with a given mismatch probability ρ ) and quantization is applied to the coefficient vector v (thereby inducing noise ∆v ). This leads to a population of synthesis errors ∆x . Sample results are given in Figure A1 for several probabilities of mismatch 0.02 , 0.14 ρ   ∈     . The graphs in the figure report both the average value of the relative approximation error ∆ ∆ ∆ M v x , using dots, and the standard deviation, using bars. As shown in the figure, the approximation of (14) incurs less than a 10% error on average.