Recursive algorithms for digital communications using the discrete wavelet transform
Ilan Sharfer, Alfred O. Hero · Deep Blue (University of Michigan) · 1996
The goal in digital communications is to efficiently and reliably transmit digital information through a channel. In order to optimally decode the transmitted symbols, it is often necessary to have good estimates of the channel parameters. In this thesis a class of recursive algorithms which perform joint Maximum Likelihood (ML) estimation of the channel parameters and the symbol sequence is developed. The channel parameters are assumed to consist of an unknown fixed complex gain (amplitude and phase), and time delay. The digital information is transmitted using binary or M-ary phase modulation and received in the presence of an additive white Gaussian noise. The algorithms presented in this work are based on a decomposition with respect to an orthonormal wavelet basis. This is motivated by the fact that the wavelet decomposition retains all the information in the observation, while facilitating processing by a recursive algorithm. In addition, the localization properties of the wavelet basis enable local updates of the symbol parameters. This results in an efficient digital algorithm with low complexity and small delay. In particular, the algorithm has a complexity per iteration that is quadratic in the number of users when applied in a multiuser system, while the optimal receiver which uses the matched filter outputs as a sufficient statistic has an exponential complexity in the number of users. The class of algorithms studied in this thesis applies to both single user and multiuser systems. First, a coordinate ascent algorithm for a single user system is developed. While the direct maximization of the likelihood function is analytically intractable, the recursive algorithm has simple updates which involve polynomial rooting for time delay estimation, discrete search for symbol estimation, and an analytical solution for gain estimation. The algorithm uses a look-up table for retrieving Fourier series coefficients of the transmitted pulse shape decomposition with respect to the wavelet basis. Simulations show fast convergence and attainment of estimation bounds. Next, the algorithm is extended to the multiuser case. Two versions have been developed: one using grouped coordinate ascent, and the other using the EM algorithm. Simulation results for a two-user system are shown. In addition, the problem of initializing the algorithm is addressed. Finally, an analysis of the single user algorithm is performed under some reasonable assumptions. It is shown that the algorithm has a fixed point at the true gain and time delay parameters in the limit of a large observation time.