GENERALIZED SAMPLING - STABILITY AND PERFORMANCE ANALYSIS

Generalized sampling provides a general mechanism for recovering an un known input function f(x) is an element of H from the samples of the r esponses of m linear shift-invariant systems sampled at 1/mth the reco nstruction rate, The system can be designed to perform a projection of f(x) onto the reconstruction subspace V(phi) = span {phi(x-k)}(k is a n element of Z); for example, the family of bandlimited signals with p hi(x) = sine (x), This implies that the reconstruction will be perfect when the input signal is included in V(phi): the traditional framewor k of Papoulis' generalized sampling theory. Otherwise, one recovers a signal approximation f(x) is an element of V(phi) that is consistent w ith f(x) in the sense that it produces the same measurements. To chara cterize the stability of the algorithm, we prove that the dual synthes is functions that appear in the generalized sampling reconstruction fo rmula constitute a Riesz basis of V(phi), and we use the corresponding Riesz bounds to define the condition number of the system. We then us e these results to analyze the stability of various instances of inter laced and derivative sampling. Next, we consider the issue of performa nce, which becomes pertinent once we have extended the applicability o f the method to arbitrary input functions, that is, when H is consider ably larger than V(phi), and the reconstruction is no longer exact. By deriving general error bounds for projectors, we are able to show tha t the generalized sampling solution is essentially equivalent to the o ptimal minimum error approximation (orthogonal projection), which is g enerally not accessible. We then perform a detailed analysis for the c ase in which the analysis filters are in L-2 and determine all relevan t bound constants explicitly. Finally, we use an interlaced sampling e xample to illustrate these various calculations.