Frame-Theoretic Analysis of Robust Filter Bank Frames to Quantization and Erasures

This paper presents the theoretic analysis for the robustness of filter bank (FB) frames in infinite dimensional Hilbert space l²(BBZ) to quantization and erasures as well as studies the design of such robust frames, from the perspective of both frame and FB theory. First, a characterization of the eigenstructure for the frame operator and the induced Gram matrix of general FB frames is presented. The robustness of FB frames to erasures is investigated in detail, especially on the necessary and sufficient condition. Maximally robust frames are further analyzed by explicitly constructive methods. Moreover, the stability and even the possible FIR reconstruction of the subframe by the pseudoinverse are studied thoroughly. Next, we examine the optimal quantized FB frames. Introducing a novel notion named as frame energy, the universal optimality of tight FB frames to quantization is established, in contrast to the optimality of only equal norm tight frames shown in previous works. The added design freedom obtained by removing the equal norm constraint is explained and illustrated with examples. Finally, the effect of erasures on the quantized FB frames is studied by incorporating the probability of erasures, which shows the universal optimality of uniform tight FB frames for one erasure. The optimal FB frame is usually not of equal norm and actually its norm distribution follows the reverse waterfilling principle. An example of two-state erasure channel model is shown to further explain our result, followed by an analysis of the successive reconstruction which shows its potential application to frame-based multiple description coding.