Perturbation stability of coherent Riesz systems under convolution operators
Consider transmission signals \[ f(x)= \sum^\infty_{k=-\infty} \sum^{N-1}_{l=0} c_{k,l}g_l(x- ak) \] represented with respect to some Riesz basis \(\{g_{k,l}\}\). After transmission over a communication channel, the received signal \(r(x)\) can be split into a linearly transformed version and statistically independent additive noise \[ r(x)= Kf(x)+ n(x), \] where the authors assume that the channel distortion corresponds to a translation invariant system \(Kf(x)= (h* f)(x)\), where \(h\) is a function with some practical relevant properties. The orthogonal perturbation of \(g\) by \(K= K_h\) is defined by \(d_{g,h}= \|K_hg-\langle K_hg, g\rangle/\langle g,g\rangle g\|_{L_2}\). The authors prove upper and lower bounds on the orthogonal perturbation formulated in terms of spectral spread and temporal support of \(g\), respectively. Further, they compare the three types of coherent families \[ \bullet\text{ Gabor systems, }\bullet\text{ Wilson bases, }\bullet\text{ wavelet bases} \] with respect to their performance under orthogonal perturbation. Finally, they illustrate the performance of these Riesz bases in a realistic setup when perturbed by a realization of a twisted copper cable impulse response. It follows that Gabor bases are best matched to a set of convolution operators which satisfy typical practical constraints.
scientific article; zbMATH DE number 1836815