Phase-magnitude relations and phaseless reconstruction for time-frequency
and time-scale representations
Nicki Holighaus(1) , Zdeněk Průša(2) , Günther Koliander(1)(1) Acoustics Research Institute, Austrian Academy of Sciences, Austria, {nicki.holighaus,guenther.koliander}@oeaw.ac.at(2) Lewitt GmbH, Austria, zdenek.prusa@oeaw.ac.at

Abstract
It is widely accepted that the phase and magnitude components of redundant time-frequency and time-scale
representations are not independent and carry equally important information. For the short-time Fourier andwavelet transforms with an appropriate mother wavelet, this dependence can be made explicit in the so-calledphase-magnitude (PM) relations. More explicitly, the PM relations establish a one-to-one mapping from thegradient of the logarithmic magnitude of the transform to the gradient of its phase. The PM relations for the
short-time Fourier transform were previously studied by Portnoff and later rediscovered by Auger et al. Recently,
we showed that similar relations hold for wavelet transforms as well. In both cases, the PM relations can beemployed to enable signal reconstruction from magnitude-only transform coefficients by combining them withan adaptive integration scheme. The resulting algorithms often perform on par or better than more expensive,iterative algorithms for phaseless reconstruction. We demonstrate their application in the audio domain. Basedon joint work with Luis Daniel Abreu, Peter Balazs, Pavel Rajmic and Peter L. Søndergaard
Keywords: audio processing, phaseless reconstruction, time-frequency

ACKNOWLEDGEMENTS
This work was supported by the Austrian Science Fund (FWF): Y 551-N13, I 3067-N30, and P 31225-N32 andthe Vienna Science and Technology Fund (WWTF): MA16-053.

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