| dc.description.abstract | Many practical signals, such as speech, music, etc., are inherently time?varying, but most of the analysis models used in applications such as coding, recognition, synthesis, etc., are based on the quasi?stationary assumption. While substantial success has been achieved in understanding signal properties using quasi?stationary models, much more needs to be done to make these models more robust and flexible. The concept of the time?varying spectrum, more generally joint time?frequency representations (TFRs), holds much promise in analyzing these information?rich time?varying signals.
Although TFRs were introduced decades ago by Gabor [51] and Ville [52], renewed interest in this area emerged about fifteen years ago with the work of Claasen and Mecklenbrauker [11], [12], [13]. While many new TFRs have been proposed, we are yet to develop a good TFR that provides effective localization of signal components in the time?frequency domain and is sufficiently robust to interfering noise. Attempting this, the thesis addresses the properties of discrete?time TFRs, such as the Cone?Kernel Representation (CKR) and Instantaneous Power Spectrum (IPS), and develops methods for cross?term attenuation and instantaneous?frequency estimation based on CKR, LWD, CTF, etc., in the presence of noise.
Research Contribution
We consider two well?known TFRs-CKR and IPS-and show a close relationship between the two. We then compare their properties, such as localization of cross?terms in the region of auto?terms, time?support property, inversion, noise immunity, etc. An inversion algorithm for IPS in the discrete domain has been derived. The performance in complex Gaussian white noise is quantified for both CKR and IPS. CKR shows better performance than IPS.
Although these distributions are known to be alias?free in the TF domain for signals sampled at the Nyquist rate, we show that aliasing may occur in other transform domains. The cross?terms of CKR and IPS, concentrated in the region of auto?terms, can be attenuated by smoothing along time in the TF domain. This smoothed TFR can be viewed as a member of Cohen抯 class with a compound kernel. Smoothing functions can be any 1?D window functions or reduced?interference kernels [44].
Studying various compound kernels (e.g., CW and IPS, CKR and CW), it is seen that IPS smoothed with a Hamming window gives good cross?term attenuation. These smoothed TFRs are therefore classified as Cohen抯?class distributions with compound kernels-i.e., the product of two kernels in the ambiguity domain. We also form compound kernels classified as convolution compound kernels, where kernels are convolved along the Doppler axis in the ambiguity domain. CKR and Born朖ordan convolution compound kernels give a localized representation for nonstationary signals.
Instantaneous?frequency estimation of nonlinear FM signals based on L?Wigner distribution and complex time?frequency distribution is addressed. In the case of L?Wigner distribution, it is shown in literature that higher?order phase derivatives are attenuated. In the complex time?frequency representation, many higher?order phase derivatives are absent. The convergence of these instantaneous?frequency estimation algorithms is also discussed. We further introduce the L?complex time?frequency distribution, where the higher?order phase derivatives present in the complex distribution are further attenuated.
Finally, for multicomponent signals in noise, a Hough?transform?based IF estimation method is suggested. An initial estimate is made from peaks in the Hough?transform domain. Further refinement is done by picking peaks from a local region in the TF domain bounded by the Hough?region representation. IF estimation after adaptive Wiener filtering is also discussed. | |
