Parameter Estimation and Waveform Fitting for Narrowband Signals

Abstract: Frequency estimation has been studied for a large number of years. One reason for this is that the problem is easy to understand, but difficult to solve. Another reason, for sure, is the large number of applications that involve frequency estimation, e.g radar using frequency modulated continuous wave (FMCW) techniques where the distance to the target is embedded in the frequency, resonance sensor systems where the output signal is given as the frequency displacement from a nominal frequency, radio frequency identification systems (RFID) where frequency modulation is used in the communication link, etc. The requirement on the frequency estimator varies with the application and typical issues include: accuracy, precision or (bias) processing speed or complexity, and ability to handle multiple signals. A lot of solutions to different problems in this area has been proposed, but still several open questions remain. The first part of this thesis addresses the problem of frequency estimation using low complexity algorithms. One way of achieving such an algorithm is to employ a coarse quantization on the input signal. In this thesis, a 1-bit quantizer is considered which enables the use of low complexity algorithms. Frequency estimation using look-up tables is studied and the properties of such an estimator are presented. By analyzing the look-up tables using the Hadamard transform a novel type of lowcomplexity frequency estimators is proposed. They use operations such as binary multiplication and addition of precalculated constants. This fact makes them suitable in applications where low complexity and high speed are major issues. A hardware demonstrator using the table look-up technique is designed and a prototype is analysed by real measurements. Today, the interest of using digital signal processing instead of analog processing is almost absolute. For example, in testing analog-to-digital converters an important part is to fit a sinewave to the recorded data, as well as to calculate the parameters that in least-squares sense result in the best fit. In this thesis, the sinewave fitting method included in the IEEE Standard 1057 is studied in some detail. Asymptotic Cramér-Rao bounds for three- and four model parameters are derived under the Gaussian assumption. Further, the sinewave fitting properties of the algorithm are analyzed by the parsimony principle. A novel model order selection criterion is proposed for waveform fitting methods in the case of a linear signal model. A generalization of this criterion is made to include the non-linear sinewave fitting application. For multiple sinewave fitting applications two iterative algorithms are proposed. The first method is a combination of the standardized sinewave fit algorithm and the expectation maximization algorithm. The second algorithm is an extension of a single sinewave model to a multiple sinewave model employing the standardized sinewave fitting algorithm. Both algorithms are analysed by numerical means and are shown to accurately resolve multiple sinewaves and produce efficient estimates. Initialization issues of such algorithms are included to some extent.