Generalized Sampling Theorem for Bandpass Signals
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Generalized Sampling Theorem for Bandpass Signals Ales Prokes Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic Received 29 September 2005; Revised 19 January 2006; Accepted 26 February 2006 Recommended for Publication by Yuan-Pei Lin The reconstruction of an unknown continuously defined function f (t) from the samples of the responses of m linear timeinvariant (LTI) systems sampled by the 1/mth Nyquist rate is the aim of the generalized sampling. Papoulis (1977) provided an elegant solution for the case where f (t) is a band-limited function with finite energy and the sampling rate is equal to 2/m times cutoff frequency. In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals. In the first part, a generalized sampling theorem (GST) for bandpass signals is presented. The second part deals with utilizing this theorem for signal recovery from nonuniform samples, and an efficient way of computing images of reconstructing functions for signal recovery is discussed. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
A multichannel sampling involves passing the signal through distinct transformations before sampling. Typical cases of these transformations treated in many works are delays [2– 6] or differentiations of various orders [7]. A generalization of both these cases on the assumption that the signal is represented by a band-limited time-continuous real function f (t) with finite energy was introduced in [1] and developed in [8]. Under certain restrictions mentioned below, a similar generalization can be formed for bandpass signals represented by a function f (t) whose spectrum F(ω) is assumed to be zero outside the bands (−ωU , −ωL ) and (ωL , ωU ) as depicted in Figure 1(a) while its other properties are identical as in the case of bandpass signals. If a signal is undersampled (i.e., a higher sampling order is used), then its original spectrum components and their replicas overlap and the frequency intervals (ωL , ωU ) and (−ωU , ωL ) are divided into several subbands, whose number depends for given frequencies ωU and ωL on the sampling frequency ωS , [4, 9]. For the introduction of GST, the number of overlapped spectrum replicas has to agree with the sampling order m, with the number of subbands inside the frequency ranges (ωL , ωU ) and (−ωU , −ωL ), and with the number of linear systems. As presented in [9], to meet the above demands, m must be an even number and the sampling frequency ωS = 2π/TS , where TS is the sampling period, and bandwidth
ωB = ωU − ωL has to meet the following conditions: ωL k0 ωL = = , ωU − ωL ωB m
(1)
where k0 is any positive integer number, and 2 ωS = . ωB m
(2)
An example of a fourth-order sampled signal spectrum in the vicinity of positive and negative original spectral components, if conditions (1) and (2) are fulfilled, is shown in Figure 1(b) and Figure 1(c). Figure 2 shows a graphical interpretation of the sampling orders m = 2 to 6 in the plane ωS /ω
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