6. Sampling and interpolation for bandlimited vectors A vector x CN is called ba

Question

6. Sampling and interpolation for bandlimited vectors A vector x CN is called bandlinited when there exists ko E {0, 1, . . . . N-1 } such that its discrete Fourier transform (DFT) coefficient sequence xx satisfies N-1 k=0 Xd[A1-0 for all k with 0.5(ko + 1) k N-0.5(ko + 1). (6) The smallest such ko is called the bandwidth of . A vector in x e CN that is not bandlimited is called full band. The set of vectors in x e C with bandwidth at most ko Is a subspace. For x in such a bandlimited subspace, find a linear mapping . Cko-CN so that the system described by 4" achieves perfect recovery x-x Hints: We can view 'r as a cascade of the sampling system y = interpolation system x = y. 'r and an Total for Question 6: 5

Explanation / Answer

Methods of developing real and complex interpolatory representations of a bandlimited signal in which the quadrature components of the signal are explicitly exhibited are presented. Applied to the case of a bandpass signal, the real formulation is a new interpolatory result that has useful computational properties. Exact and approximate envelope and phase calculations are used to demonstrate some of these properties. In a similar application, the complex formulation gives the wellknown result of Goldman and Woodward.

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