  Many phenomena are described by bivariate signals or bidimensional vectors in
applications ranging from radar to EEG, optics and oceanography. The
time-frequency analysis of bivariate signals is usually carried out by
analyzing two separate quantities, e.g. rotary components. We show that an
adequate quaternion Fourier transform permits to build relevant time-frequency
representations of bivariate signals that naturally identify geometrical or
polarization properties. First, the quaternion embedding of bivariate signals
is introduced, similar to the usual analytic signal of real signals. Then two
fundamental theorems ensure that a quaternion short term Fourier transform and
a quaternion continuous wavelet transform are well defined and obey desirable
properties such as conservation laws and reconstruction formulas. The resulting
spectrograms and scalograms provide meaningful representations of both the
time-frequency and geometrical/polarization content of the signal. Moreover the
numerical implementation remains simply based on the use of FFT. A toolbox is
available for reproducibility. Synthetic and real-world examples illustrate the
relevance and efficiency of the proposed approach.
