Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals
We consider the problem of the spectral content of a complex improper signal and the time-varying behaviour of this spectral content. The signals considered are one-dimensional (1D), complex-valued, with possible correlation between the real and imaginary parts, i.e., improper complex signals. As a consequence, it is well known that the ‘classical’ (complex-valued) Fourier transform does not exhibit Hermitian symmetry and also that it is necessary to consider simultaneously the spectrum and the pseudo-spectrum to completely characterize such signals. Hence, an ‘augmented’ representation is necessary. However, this does not provide a geometric analysis of the complex improper signal. We propose another approach for the analysis of improper complex signals based on the use of a 1D Quaternion Fourier Transform (QFT). In the case where complex signals are non-stationary, we investigate the extension of the well-known ‘analytic signal’ and introduce the quaternion-valued ‘hyperanalytic signal’. As with the hypercomplex two-dimensional (2D) extension of the analytic signal proposed by Bülow in 2001, our extension of analytic signals for complex-valued signals can be obtained by the inverse QFT of the quaternion-valued spectrum after suppressing negative frequencies. Analysis of the hyperanalytic signal reveals the time-varying frequency content of the corresponding complex signal. Using two different representations of quaternions, we show how modulus and quaternion angles of the hyperanalytic signal are linked to geometric features of the complex signal. This allows the definition of the angular velocity and the complex envelope of a complex signal. These concepts are illustrated on synthetic signal examples. The hyperanalytic signal can be seen as the exact counterpart of the classical analytic signal, and should be thought of as the very first and simplest quaternionic time-frequency representation for improper non-stationary complex-valued signals.
KeywordsQuaternions complex signals Fourier transform analytic signal
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