The Non-Linear Instantaneous Least Squares Approach to Signal Parameter Estimation

Doctoral thesis, 1998

Estimation of parameters of non-stationary signals observed in noise is a challenging and difficult task, motivated by applications such as radar and mobile tele-communication. Here, a novel method for signal parameter estimation named the Non-linear Instantaneous Least Squares (NILS) approach is presented, which can be applied to linear and non-linear signals, and to both uniformly and non-uniformly sampled data. It is shown that the NILS approach is related to some well known signal parameter estimation techniques, such as Non-Linear Least Squares (NLLS), signal-subspace fitting and linear prediction based estimation approaches. The NILS estimator can be used to construct Time Frequency Representations (TFR) that are signal-model based and adaptive. One class of NILS TFRs can be interpreted as members of the well known Cohen's class. It is shown that different TFRs in Cohen's class correspond to different NILS criterion-functions. Hereby, a link is also established between TFRs of Cohen's class and signal parameter estimation, and the concept of kernels in the time-frequency domain is given an interpretation as different criterion-functions in the time-domain. Considerable attention has been paid to the estimation of Polynomial-Phase Signal (PPS) parameters due to numerous applications of the model. The NILS approach applied to estimate PS parameters is studied in some detail. A theoretical analysis of the PPS parameters is presented for the case of a mono-component PPS. It is shown that the PPS parameter estimates tend to their true values as the SNR or the number of data points tend to infinity. A theoretical expression for the covariance matrix of the PPS parameter estimates is derived and the performance of the NILS PPS parameter estimates is compared to that of other estimation techniques. The numerical results show that the Signal to Noise Ratio (SNR) threshold of the NILS estimator is considerably lower than that of the other approaches, that the estimates can be made efficient and confirms that the PPS estimates are consistent. Also, results on aliasing and identifiability of sampled mono-component PPS parameters are presented. Finally, an auto-focusing algorithm for Syntethic Aperture Radar (SAR) images based on the NILS approach and a PPS model is proposed.