Sampling expansions for multiband signals

Abstract

The class B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> of finite energy signals bandlimited to a set I is considered where I denotes a finite union of disjoint open intervals (bands) on the real frequency axis. If such a signal is sampled uniformly (constant intersample spacing T), it is shown that the following five propositions are equivalent. 1) Any signal in B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> is uniquely determined from its uniform samples taken at the rate 1/T samples/s. 2) The set of complex exponentials <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{e^{in\omegaT}}\min{-infin}\max{\infin}</tex> is complete on the set I. 3) the translated sets <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\omega + k\sigma: \omega\epsilonI}</tex> with σ = 2π/T do not intersect I for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k \neq 0</tex> . 4) Each signal in B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> possesses a mean-square convergent sampling expansion in terms of its samples at rate 1/T samples/s. 5) A generalized Parseval relation holds for each pair of signals in B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> .