Title :
Burst Erasures and the Mean-Square Error for Cyclic Parseval Frames
Author :
Bodmann, Bernhard G. ; Singh, Pankaj K.
Author_Institution :
Dept. of Math., Univ. of Houston, Houston, TX, USA
fDate :
7/1/2011 12:00:00 AM
Abstract :
This paper investigates the performance of frames for the linear, redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors. We assume that the distribution of bursts is invariant under cyclic shifts and that the burst-length statistics are known. In analogy with rate-distortion theory, we wish to find frames of a given size, which minimize the mean-square reconstruction error for the encoding of vectors in a complex finite-dimensional Hilbert space. We obtain an upper bound for the mean-square reconstruction error for a given Parseval frame and in the case of cyclic Parseval frames, we find a family of frames which minimizes this upper bound. Under certain conditions, these minimizers are identical to complex Bose-Chaudhuri-Hocquenghem codes discussed in the literature. The accuracy of our upper bounds for the mean-square error is substantiated by complementary lower bounds. All estimates are based on convexity arguments and a discrete rearrangement inequality.
Keywords :
BCH codes; Hilbert spaces; cyclic codes; linear codes; mean square error methods; random codes; rate distortion theory; Bose-Chaudhuri-Hocquenghem code; burst distribution; burst-length statistics; complex finite dimensional Hilbert space; cyclic Parseval frames; discrete rearrangement inequality; invariant under cyclic shift; linear encoding; mean square reconstruction error; random burst error; rate distortion theory; redundant encoding; vector encoding; Encoding; Hilbert space; Mean square error methods; Measurement uncertainty; Polynomials; Probability; Upper bound; Burst erasures; codes; error bounds; frames; mean-square error (MSE);
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2011.2146150