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Quaternion‐valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector‐valued signal and image processing. However, the sampling theory of quaternion‐valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a  σ ‐bandlimited quaternion‐valued signal in QFT sense can be reconstructed from the samples of output signals of  M  linear systems based on QFT. Quaternion linear canonical transform is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for  σ ‐bandlimited quaternion‐valued signal in quaternion linear canonical transform sense. Examples are given to illustrate our results. Copyright © 2017 John Wiley & Sons, Ltd.

Generalized sampling expansions associated with quaternion Fourier transform