Sampling theory has been a topic of extensive research over the past decade, which has led to a reexamination of some of the foundations of Shannon's theory, and development of more general formulations with immediate relevance to signal processing and communications. Recently, it has been shown that it is possible to develop sampling schemes for a large class of non-bandlimited signals, namely certain signals of finite innovation rate [1,2]. A common feature of these signals is that they allow for a parametric representation with a finite number of degrees of freedom, and can be perfectly reconstructed from a finite set of samples. We consider one possible application of our sampling results to be the problem of timing synchronization and channel estimation in wideband communication systems, such as ultra-wideband and CDMA systems.
Synchronization in wideband communication systems is a crucial task that imposes serious restrictions on system performance. Vast literature has appeared recently, addressing both algorithmic and implementation issues of various synchronization techniques, with a clear trend toward eliminating the necessity for analog components as much as possible and performing all processing digitally. Even though many high-performance schemes have already been proposed, their application in real time systems is often not feasible due to their high computational complexity. Furthermore, almost all of them use the Nyquist sampling rate, which requires very fast and expensive A/D converters and therefore high power consumption. This problem becomes critical in ultra-wideband systems, where in digital-oriented solutions A/D converters must operate in the gigahertz range.
We propose a low-complexity timing synchronization method that uses low-rate uniform sampling and well-developed algorithmic solutions. Specifically, we extend some of our sampling results  to the problem of multipath delay estimation in wideband channels and develop an algorithm for estimating unknown propagation delays from a low-dimensional subspace of a received signal . Our approach leads to reduced computational requirements and lower power consumption compared to existing techniques, thus allowing for a practical hardware implementation and all the benefits of a digital design.