Multichannel Recovery of Quadrature Components of Bandpass Signals

The problem of sampling signals maintaining the theoretically minimum (average) sampling rate and allowing a separate interpolation is considered from a general point of view. The formulation will follow a recently published method for multichannel sampling; this way there is the advantage of working with functions depending only on the frequency and not on time and frequency as in other approaches. This simplifies the general expressions and the determination of separation conditions. Under the assumption of nonsingularity of certain matrices, we obtain the necessary and sufficient conditions for a separate interpolation and discuss them. Finally, an especially important case is considered as an example.     

Sampling Bandpass Signals

In a recent paper [1], Brown examined the sampling of a real finiteenergy bandpass signal having an (angular) bandwidth ? (in radians per second) at the theoretically minimum (average) rate of ?/? samples per second. Following Grace and Pitt's [2] quadrature sampling, a particular case of Kohlenberg's second-order sampling [3, 4], Brown has proved the feasibility of a separate interpolation of the in-phase and quadrature components of the signal when ?o = k?/2 (Brown's condition), where ?o is the center (angular) frequency of the signal and k is an arbitrary positive integer. Here the problem is reconsidered from a general point of view, introducing, under Brown's condition, an interpolation formula which includes that of Grace and Pitt and which extends a theorem of Populis [5-7]. We indicate the necessary and sufficient conditions to obtain a separate interpolation, offering closed formulas to obtain the interpolation functions. We also discuss the minimum oversampling rate needed whenBrown's condition is not verified.