Generalized sampling: stability and performance analysis

Generalized sampling provides a general mechanism for recovering an unknown input function f(x)/spl epsiv//spl Hscr/ from the samples of the responses of m linear shift-invariant systems sampled at 1/mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace V(/spl phi/)=span {/spl phi/(x-k)}/sub k/spl epsiv/Z/; for example, the family of bandlimited signals with /spl phi/(x)=sinc(x). This implies that the reconstruction will be perfect when the input signal is included in V(/spl phi/): the traditional framework of Papoulis' (1977) generalized sampling theory. Otherwise, one recovers a signal approximation f(x)/spl epsiv/V(/spl phi/) that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(/spl phi/), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when /spl Hscr/ is considerably larger than V(/spl phi/), and the reconstruction is no longer exact. We show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation. We then perform a detailed analysis for the case in which the analysis filters are in L/sub 2/ and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations.