Construction of signal-dependent Cohen's class time-frequency representations using iterative blind deconvolution

In this thesis we restate the time-frequency kernel design problem as a blind deconvolution. It is assumed that there exists an ideal time-frequency representation which satisfies positivity and reduced cross term (CT) interference. That ideal TFR has been convolved with an unknown distortion function, yielding the Wigner TFR, which in the general case exhibits CT interference and negativity, making it difficult to visually interpret it.; The proposed solution to the restated kernel design problem is the application of the iterative blind deconvolution method (IBDM) to TFA. The IBDM is commonly used in image restoration problems; however, the class of images used by the IBDM differ from TFRs in two important aspects: (1) IBDM assumes that images are always positive, while TFRs exhibit negative values in the general case and (2) images are always compact supported since the physical dimensions of the image acquisition sensor are finite. However, due to Fourier transform properties, TFRs cannot be compactly supported in the time-frequency domain (TFD). Despite these differences, we show that the IBDM can be successfully applied to TFA if we pretend that TFRs are images. However, the TFA case is more flexible than the image processing case in which constraints are always applied in the spatial domain. In TFA, constraints can be applied in the time-frequency domain (TFD) and/or the ambiguity domain (AD). This flexibility produces two algorithm variants: (1) constraint application on the TFD for the TFR and constraint application on the AD for the kernel (case 1) and, (2) constraint application on the TFD for both the TFR and the kernel (case 2). As shown in this work, these two variants lead to different results in the general case. As a result of application of IBDM to TFA, we get quasi-positive TFRs with reduced CT interference at the cost of sacrifying the marginals. This sacrifice is necessary due to the low pass and fast decaying kernels obtained on the ambiguity domain (AD).