Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

It is known that the Karhunen-Lo`{e}ve transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes results in sinusoidal basis functions. The same sinusoidal bases come out of the independent-component analysis (ICA) and actually correspond to processes with completely independent samples. In this paper, we relax the Gaussian hypothesis and study how orthogonal transforms decouple symmetric-alpha-stable (S$alpha$S) AR(1) processes. The Gaussian case is not sparse and corresponds to $alpha=2$, while $0<alpha<2$ yields processes with sparse linear-prediction error. In the presence of sparsity, we show that operator-like wavelet bases do outperform the sinusoidal ones. Also, we observe that, for processes with very sparse increments ($0<alphaleq 1$), the operator-like wavelet basis is indistinguishable from the ICA solution obtained through numerical optimization. We consider two criteria for independence. The first is the Kullback-Leibler divergence between the joint probability density function (pdf) of the original signal and the product of the marginals in the transformed domain. The second is a divergence between the joint pdf of the original signal and the product of the marginals in the transformed domain, which is based on Steins formula for the mean-square estimation error in additive Gaussian noise. Our framework then offers a unified view that encompasses the discrete cosine transform (known to be asymptotically optimal for $alpha=2$) and Haar-like wavelets (for which we achieve optimality for $0<alphaleq1$).

A New Method for Constructing Large Size WBE Codes with Low Complexity ML Decoder

In this paper we wish to introduce a method to reconstruct large size Welch Bound Equality (WBE) codes from small size WBE codes. The advantage of these codes is that the implementation of ML decoder for the large size codes is reduced to implementation of ML decoder for the core codes. This leads to a drastic reduction of the computational cost of ML decoder. Our method can also be used for constructing large Binary WBE (BWBE) codes from smaller ones. Additionally, we explain that although WBE codes are maximizing the sum channel capacity when the inputs are real valued, they are not necessarily appropriate when the input alphabet is binary. The discussion shows that when the input alphabet is binary, the Total Squared Correlation (TSC) of codes is not a proper figure of merit.

Errorless Codes for Over-loaded CDMA with Active User Detection

In this paper we introduce a new class of codes for over-loaded synchronous wireless CDMA systems which increases the number of users for a fixed number of chips without introducing any errors. In addition these codes support active user detection. We derive an upper bound on the number of users with a fixed spreading factor. Also we propose an ML decoder for a subclass of these codes that is computationally implementable. Although for our simulations we consider a scenario that is worse than what occurs in practice, simulation results indicate that this coding/decoding scheme is robust against additive noise. As an example, for 64 chips and 88 users we propose a coding/decoding scheme that can obtain an arbitrary small probability of error which is computationally feasible and can detect active users. Furthermore, we prove that for this to be possible the number of users cannot be beyond 230.