A maximum likelihood CDMA receiver using the EM algorithm and the discrete wavelet transform

A Maximum Likelihood ML method for joint estimation of amplitude phase time delay and data demodulation in a single user direct sequence spread spectrum communica tion system is developed The likelihood function is an alytically intractable so a recursive estimation algorithm is considered The Expectation Maximization EM algo rithm has been used in similar problems however in this case it is not computationally e cient Recently a variant of the EM algorithm called Space Alternating Generalized EM SAGE has been derived In this work we apply the SAGE algorithm to the sequence estimation problem in a way which results in simple sequential updates of all the es timated parameters An important feature of the proposed algorithm is the use of a discrete wavelet decomposition of the received signal as a su cient statistic The consequence is that all the information is still available to the receiver while the complicated estimation problem is considerably simpli ed Computer simulations of a single user system were performed It is shown that the algorithm has fast convergence and essentially achieves optimal performance INTRODUCTION The emergence of new technologies of multi user wireless communication systems requires advanced signal process ing methods for improved e ciency and reliability In or der to optimally decode the desired information the receiver should bene t from knowledge of the nuisance parameters of the received signal which typically consist of the am plitude phase and time delay Usually these parameters are estimated by a combination of several techniques each specialized to a particular parameter For example carrier phase and time delay estimation are mostly done with a Phase Locked Loop PLL and a Delay Locked Loop DLL respectively A considerable research activity has been directed at improving the performance of these basic syn chronization techniques e g by using decision feedback in a Data Aided Loop DAL con guration Clearly an optimal receiver is one which jointly estimates the nuisance parameters as well as the data symbols In this work we consider the problem of Maximum Like lihood ML estimation of all the parameters given the re ceived signal This problem is analytically intractable even for the simple AWGN channel hence the need for a recur sive estimation algorithm The Space Alternating General ized EM SAGE algorithm which has recently been devel oped in is a variant of the EM algorithm both are recursive algorithms which generate a sequence of parame ter estimates whose likelihood increases monotonically The SAGE algorithm is more exible than the EM algorithm because it is possible to update subsets of the parameter re sulting in simpler updates and faster convergence We have chosen to focus on the single user problem although the same approach can be generalized to the multi user case One of the basic concepts in estimation theory is that a signal can be represented by a su cient statistic The sig nal expansion on an orthonormal wavelet basis is one such possible representation This choice has been motivated by the excellent time frequency localization properties of the wavelet bases The SAGE algorithm has therefore been formulated in terms of the decomposition of the received signal in the discrete wavelet transform domain The re sult is a causal fully digital receiver which makes a single pass on the information without requiring any bu ering or delayed processing The principle of using a su cient statistic in a similar problem has been discussed in This paper is organized as follows In section we de ne the system model and brie y review the SAGE algorithm and the concept of a hidden data space In section we develop the single user algorithm and outline a strategy of choosing the hidden data spaces In section we describe a Fourier based method for numerically solving the maximiza tion step of the algorithm The recursive implementation of the algorithm is given in section We conclude with sim ulation results and a performance comparison with other techniques SYSTEM MODEL AND SAGE REVIEW We consider the following single user complex baseband CDMA model