v2010.10.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 341we then engage convex iteration, detect stalls, and randomly reinitialize thedirection vector, failure rate drops to 0% but the amount of computation isapproximately doubled.Stalling is not an inevitable behavior. For some problem types (beyondmere Ax = b), convex iteration succeeds nearly all the time. Here is acardinality problem, with noise, whose statement is just a bit more intricatebut easy to solve in a few convex iterations:4.5.1.5.2 Example. Signal dropout. [121,6.2]Signal dropout is an old problem; well studied from both an industrial andacademic perspective. Essentially dropout means momentary loss or gap ina signal, while passing through some channel, caused by some man-madeor natural phenomenon. The signal lost is assumed completely destroyedsomehow. What remains within the time-gap is system or idle channel noise.The signal could be voice over Internet protocol (VoIP), for example, audiodata from a compact disc (CD) or video data from a digital video disc (DVD),a television transmission over cable or the airwaves, or a typically ravagedcell phone communication, etcetera.Here we consider signal dropout in a discrete-time signal corrupted byadditive white noise assumed uncorrelated to the signal. The linear channelis assumed to introduce no filtering. We create a discretized windowedsignal for this example by positively combining k randomly chosen vectorsfrom a discrete cosine transform (DCT) basis denoted Ψ∈ R n×n . Frequencyincreases, in the Fourier sense, from DC toward Nyquist as column index ofbasis Ψ increases. Otherwise, details of the basis are unimportant exceptfor its orthogonality Ψ T = Ψ −1 . Transmitted signal is denoteds = Ψz ∈ R n (770)whose upper bound on DCT basis coefficient cardinality cardz ≤ k isassumed known; 4.34 hence a critical assumption: transmitted signal s issparsely supported (k < n) on the DCT basis. It is further assumed thatnonzero signal coefficients in vector z place each chosen basis vector abovethe noise floor.4.34 This simplifies exposition, although it may be an unrealistic assumption in manyapplications.