Parametric estimation of 2D cubic phase signals using high-order Wigner distribution with genetic algorithm

Abstract

A two-dimensional (2D) high-order Wigner distribution (HO-WD) is proposed for parameter estimation of 2D polynomial phase signals (PPSs). The genetic algorithm is employed for maximization of the 2D HO-WD. In comparison to the 2D cubic phase function and classical Francos and Friedlander approach, the 2D HO-WD reduces error propagation effect which leads to lower mean squared error in estimation of signal parameters. The proposed technique is generalized for the 2D higher-order PPS.

Appendix

Consider signal (1). The 2D HO-WD of (1) can be written as

$$\begin{aligned}&\text {E}\{\text {HO-WD}\left( n,k;\omega _{n},\omega _{k};\alpha _{30} ,\alpha _{21},\alpha _{12},\alpha _{03}\right) \} \\&\quad = \underset{\tau _{n}}{\sum }\sum _{\tau _{m}}\text {E}\{x\left( n+\tau _{n},m+\tau _{m}\right) x^{*}\left( n-\tau _{n},m-\tau _{m}\right) \}\\&\qquad \times \exp \left( -j\omega _{n}\tau _{n}-j2\omega _{m}\tau _{m}-j2\alpha _{30} \tau _{n}^{3}-j2\alpha _{21}\tau _{n}^{2}\tau _{m}\right. \\&\qquad \qquad \qquad \left. -j2\alpha _{12}\tau _{n}\tau _{m}^{2}-j2\alpha _{03}\tau _{m}^{3}\right) \\&\quad =\underset{\tau _{n}}{\sum }\underset{\tau _{m}}{\sum }(\text {E}\{f\left( n+\tau _{n},m+\tau _{m}\right) f^{*}\left( n-\tau _{n},m-\tau _{m}\right) \} \\&\qquad + \text {E}\{f\left( n+\tau _{n},m+\tau _{m}\right) \nu ^{*}\left( n-\tau _{n},m-\tau _{m}\right) \}\\&\qquad +\text {E}\{\nu \left( n+\tau _{n},m+\tau _{m}\right) f^{*}\left( n-\tau _{n},m-\tau _{m}\right) \} \\&\qquad + \text {E}\{\nu \left( n+\tau _{n},m+\tau _{m}\right) \nu ^{*}\left( n-\tau _{n},m-\tau _{m}\right) \})\\&\qquad \times \exp \left( -j2\omega _{n}\tau _{n}-j2\omega _{m}\tau _{m}-j2\alpha _{30} \tau _{n}^{3}-j2\alpha _{21}\tau _{n}^{2}\tau _{m}-j2\alpha _{12} \tau _{n}\tau _{m}^{2}-j2\alpha _{03}\tau _{m}^{3}\right) . \end{aligned}$$

Since noise is zero-mean Gaussian with variance $\sigma ^{2}$, it follows E$\{f\left( n+\tau _{n},m+\tau _{m}\right)$$\nu ^{*}\left( n-\tau _{n},m-\tau _{m}\right) \}=0$, E$\{\nu \left( n+\tau _{n},m+\tau _{m}\right)$$f^{*}\left( n-\tau _{n},m-\tau _{m}\right) \}=0$ and E$\{\nu \left( n+\tau _{n},m+\tau _{m}\right)$$\nu ^{*}\left( n-\tau _{n},m-\tau _{m}\right) \}=\sigma ^{2}\delta (\tau _{n},\tau _{m})$. In an ideal case, the 2D HO-WD peaks for $\alpha _{30}=a_{30},\alpha _{21}=a_{21},\alpha _{12} =a_{12},\alpha _{03}=a_{03}$, $\omega _{n}\left( n,m\right) =a_{10} +2a_{20}n+a_{11}m+3a_{30}n^{2}+a_{12}m^{2}+a_{21}4nm$, $\omega _{m}\left( n,m\right) =a_{01}+2a_{02}m+a_{11}n+3a_{03}m^{2}+a_{21}n^{2}+a_{12}4nm$. Since the objective function is the sum of three 2D HO-WDs, calculated at (0, 0), ($n_{0},0$) and (0,$m_{0}$), the mathematical expectation of (14) for position of the HO-WD maximum equals

$$\begin{aligned} \text {ME}\approx (MNA^{2}+\sigma ^{2})+\left( \left( N-2|n_{0}|\right) MA^{2}+\sigma ^{2}\right) +\left( N\left( M-2|m_{0}|\right) A^{2} +\sigma ^{2}\right) \text {.} \end{aligned}$$

(24)