Optimum Delay and Sequence Estimation from Incomplete Data

Ahstruct -The performance of digital communication systems can be degraded severely when symbol synchronization is imperfect. This is due to the contamination of the decision statistics computed by the receiver from neighboring symbol5 and the inability of some conventional receivers to combat this interference. This inability often is due to the design of conventional receivers, which assumes that the timing information provided by the synchronization subsystem is perfect. An easily implementable dynamic programming algorithm is introduced which estimates sequences and timing delay jointly from the contaminated decision statistics available to the receiver. The new algorithm alleviates the neeh for a symbol synchronizer, it is easily implemented digitally, and it is quite robust to timing errors. I. INTRODUCTION The symbol synchronization subsystem, whose function is to dictate to the receiver where in time symbols are located, is an important and integral part of every practical receiver. The timing information is utilized subsequently by the receiver to compute its decision statistics and decide what the transmitted symbols are. Extensive work on describing the performance degradation due to imperfect timing information has been done in the past and can be found, for example, in the excellent books of Lindsey and Simon [l] and Stiffler [2]. Results reported in these references indicate that essentially disastrous results are obtained when the standard deviation of the timing jitter exceeds some small value. We will see later that a receiver designed to make decisions in the presence of timing errors has some practical advantages and is quite robust to such errors. Other pertinent approaches to the synchronization problem can be found in [3]-[lo]. In [3] the problem of decision-directed timing recovery from samples taken at the symbol rate is studied. These samples are used to generate an error signal that is then used to adjust the phase of the sampler to reduce the error. Although we still look at data obtained at the symbol rate, our main interest here is not in timing recovery but rather in sequence estimation in the presence of timing errors. The algorithms for sequence estimation described later make no attempt to adjust the phase of the sampler, although certainly this is possible. All results derived in the sequel are optimal in a maximum likelihood (ML) sense, in contrast to [3] where no such optimality is claimed. Gardner [4] looks at the timing recovery problem from observations of two samples per symbol interval and realizes the advantage of avoiding decision-directed detection compared to [3].