Signal reconstruction from partial or modified linear time-frequency representations

Abstract

The first section of this thesis reviews such signal dependent techniques and describes a method of inversion for the Reassigned Spectrogram through use of its link to the phase of the Short Time Fourier Transform (ST-FT).

In order to achieve this inversion, signal reconstruction is required using only knowledge of the phase of the ST-FT.  Such an inversion is shown to be possible to  within an overall amplitude constant given mild conditions upon the degree of overlap used in the computation of the ST-FT.  Complimentary to signal reconstruction from knowledge of the phase of the ST-FT, is reconstruction from its magnitude.  Two distinct approaches are described.  The first previously given in the literature, iteratively applies the Minimum Least Squares (MLS) inversion of the complete (amplitude and phase) ST-FT.  The second uses an analogous approach to that used for signal reconstruction from phase.  Although the first approach proves more robust to non-valid input TF, both of these techniques require a similar degree of overlap to the phase case, and both reconstruct the signal to within an overall phase constant.  The description of the theory concludes with a discussion of the set of Generalised Wavelet Transforms (GWTs), of which both the SF-FT and WT are members.  After defining the set of GWTs, descriptions are given for the MLS inversion for complete GWT information, and for signal reconstruction from either phase or amplitude.

The thesis concludes by using the MLS based technique to create signals from modified or synthetic spectrograms generated using heart sounds.  The first application of this is to extend the duration of heart sounds.  Temporal extension in this fashion has the ability to extend the signals in time without affecting the spectrum of the sound.  In addition, it does not require the use of a model as in matching pursuit based methods described in the literature.  The second application is to create at time-series from a synthetic spectrogram constructed by averaging the spectrograms of a patient’s heart murmur.  Such averaging cannot take place in the time-domain owing to the random nature of the flow noise in a murmur.

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