Abstract:

Digital applications inherently rely on sampling a continuous-time 
signal to obtain a discrete-time representation. Many of the limitations 
encountered in current digital-to-analog converters stem from the traditional 
assumption that data must be acquired at the Shannon-Nyquist rate, corresponding 
to twice the bandwidth. A major drawback of the Shannon paradigm is that
natural signals can often be better represented by models beyond bandlimited. 
Furthermore, ideal point-wise sampling and sinc interpolation are difficult to 
implement.

In this talk we present several extensions of the Shannon theorem which
accommodate a broader class of input signals as well as nonideal sampling
and nonlinear distortions.  This framework is based on viewing sampling 
in a broader sense of projection onto appropriate subspaces, and then choosing
the subspaces to yield interesting new possibilities such as below Nyquist
sampling of sparse signals, pointwise sampling of non bandlimited signals,
and perfect compensation of nonlinear effects.

We begin by presenting a broad class of sampling theorems for signals
confined to an arbitrary subspace in the presence of non-ideal sampling, 
and nonlinear distortions. Next, we develop minimax recovery techniques that
best approximate an arbitrary smooth input signal, using simple interpolation 
kernels. Finally, we discuss sparse analog signals that can be represented 
by a disjoint set of bands in some transform domain. Combining traditional 
sampling ideas with results from the field of compressed sensing we show how 
to reconstruct an analog multi-band signal from minimum-rate samples when the 
band locations are unknown.