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EURASIP Journal on Advances in Signal Processing

, 2012:234

First Online: 12 November 2012Received: 28 June 2012Accepted: 23 October 2012


Different kinds of quaternion signal detection problems in continuous-time by using a widely linear processing are dealt with. The suggested solutions are based on an extension of the Karhunen-Loève expansion to the quaternion domain which provides uncorrelated scalar real-valued random coefficients. This expansion presents the notable advantage of transforming the original four-dimensional eigen problem to a one-dimensional problem. Firstly, we address the problem of detecting a quaternion deterministic signal in quaternion Gaussian noise and a version of Pitcher’s Theorem is given. Also the particular case of a general quaternion Wiener noise is studied and an extension of the Cameron-Martin formula is presented. Finally, the problem of detecting a quaternion random signal in quaternion white Gaussian noise is tackled. In such a case, it is shown that the detector depends on the quaternion widely linear estimator of the signal.

KeywordsDetection Quaternion random signals Series expansion AbbreviationsQWLQuaternion widely linear


QKLQuaternion Karhunen-Loève

QWGNQuaternion white Gaussian noise.

Electronic supplementary materialThe online version of this article doi:10.1186-1687-6180-2012-234 contains supplementary material, which is available to authorized users.

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Author: Jesús Navarro-Moreno - Juan Carlos Ruiz-Molina - Antonia Oya - José M Quesada-Rubio



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