Accurate parameter estimation for amplitude modulated sinusoidal signals incorporating smoothness constraints

Amplitude modulated sinusoidal signals arise in a wide variety of applications, e.g. in nuclear magnetic resonance spectroscopy [1], Doppler radar [2], and wide-band frequency-modulated continuous-wave radar signal processing. In some cases the type of amplitude modulation, e.g. exponential or polynomial, is known a priori, whereas in most practical cases such detailed information is not available. In this paper we derive estimators for the frequency, phase, and the modulation when only prior knowledge about the worst-case variation/smoothness of the modulation is available. No assumption on the type of modulation in terms of a parametric model is made. The estimators are compared to the best possible performance given by the Cramér-Rao lower bound (CRB) derived in [3]. Furthermore, they are shown to outperform the well-known “squaring loop” estimator [4, pp. 186], [5] that does not incorporate the smoothness of the modulation in terms of variance, threshold level (i.e. minimum necessary SNR for reliable operation), and restriction of the allowable range of signal frequencies in comparison to the sampling frequency. The presented method to incorporate constraints e.g. on the smoothness of the linear parameters in a parameter estimation problems with linear and nonlinear parameters is suitable to be applied in general constrained parameter estimation problems.