Modifications of the Cubic Phase Function

By introducing a symmetric pair of time instants, a modification of the cubic phase (CP) function, named as the quartic phase (QP) function, is proposed to estimate the quadratic FM signal. The performance in terms of estimate bias and variance is presented via the first-order permutation principle. Two extensions are presented for multiple components and the cubic FM signals. Both theoretical analysis and numerical examples confirm that the QP function and its extensions provide a number of advantages, such as • lower asymptotic mean-square error (MSE) for the estimate of the third-order phase parameter at high SNR; • a better capability of discriminating multicomponent signals; • a lower SNR threshold for the estimates of the cubic FM signal.