v2007.11.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 277whose upper bound on DCT basis coefficient cardinality cardz ≤ k isassumed known; 4.29 hence a critical assumption that transmitted signal s issparsely represented (k < n) with respect to the DCT basis. Nonzero signalcoefficients in vector z are assumed to place each chosen basis vector abovethe noise floor.We 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 (653)+ 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 invariant over the window’s duration n .Signal to noise ratio within this window is defined[ ]∥ s 1:l ∥∥∥∥SNR = ∆ s n−l+1:n20 log(654)‖η‖In absence of noise, knowing the signal DCT basis and having a goodestimate of basis coefficient cardinality makes perfectly reconstructing gaploss easy: it amounts to solving a linear system of equations and requireslittle or no optimization; with caveat, number of equations exceeds sparsityof signal representation (roughly l ≥ k) with respect to DCT basis.But addition of a significant amount of noise η increases level ofdifficulty dramatically; a 1-norm based method of reducing cardinality, forexample, almost always returns DCT basis coefficients numbering in excessof minimum cardinality. We speculate that is because signal cardinality 2lbecomes the predominant cardinality. DCT basis coefficient cardinality is anexplicit constraint to the optimization problem we shall pose: In presence ofnoise, constraints equating reconstructed signal f to received signal g arenot possible. We can instead formulate the dropout recovery problem as a4.29 This simplifies exposition, although it may be an unrealistic assumption in manyapplications.