v2010.10.26 - Convex Optimization

342 CHAPTER 4. SEMIDEFINITE PROGRAMMINGWe also assume that the gap’s beginning and ending in time are preciselylocalized to within a sample; id est, index l locates the last sample prior tothe gap’s onset, while index n−l+1 locates the first sample subsequent tothe gap: for rectangularly windowed received signal g possessing a time-gaploss and additive noise η ∈ R n⎡⎤s 1:l + η 1:lg = ⎣ η l+1:n−l⎦∈ R n (771)+ η n−l+1:ns n−l+1:nThe window is thereby centered on the gap and short enough so that the DCTspectrum of signal s can be assumed static over the window’s duration n .Signal to noise ratio within this window is defined[ ]∥ s 1:l ∥∥∥∥ s n−l+1:nSNR 20 log(772)‖η‖In absence of noise, knowing the signal DCT basis and having agood estimate of basis coefficient cardinality makes perfectly reconstructinggap-loss easy: it amounts to solving a linear system of equations andrequires little or no optimization; with caveat, number of equations exceedscardinality of signal representation (roughly l ≥ k) with respect to DCTbasis.But addition of a significant amount of noise η increases level of difficultydramatically; a 1-norm based method of reducing cardinality, for example,almost always returns DCT basis coefficients numbering in excess of minimalcardinality. We speculate that is because signal cardinality 2l becomes thepredominant cardinality. DCT basis coefficient cardinality is an explicitconstraint to the optimization problem we shall pose: In presence of noise,constraints equating reconstructed signal f to received signal g are notpossible. We can instead formulate the dropout recovery problem as a bestapproximation:∥[∥∥∥minimizex∈R nsubject to f = Ψxx ≽ 0card x ≤ k]∥f 1:l − g 1:l ∥∥∥f n−l+1:n − g n−l+1:n(773)