ISAR imaging of target with complex motion based on novel approach for the parameters estimation of multi-component cubic phase signal

Abstract

Inverse synthetic aperture radar (ISAR) imaging of target with complex motion is very important in the radar signal processing domain. In this case, the received signal can be characterized as multi-component cubic phase signal (CPS), and the high quality instantaneous ISAR images can be obtained by the parameters estimation approach. The match Fourier transform (MFT) has been proposed for the parameters estimation of linear frequency modulated (LFM) signal, and it has been used successfully in the field of ISAR imaging. In this paper, the third-order match Fourier transform (TMFT) is proposed as an extension of the traditional MFT for the parameters estimation of cubic phase signal, and the asymptotic statistical performance is analyzed theoretically with the derivation of asymptotic statistical results for the estimated parameters. Finally, the TMFT algorithm is used as a tool for the improvement of inverse synthetic aperture radar (ISAR) images quality of target with complex motion, and the results of simulated and real data validate the effectiveness of the TMFT algorithm.

Appendix

The Appendix within this paper derives the asymptotic MSEs for the estimation errors in (10), and the following computations are implemented based on the first-order perturbation principle:

(1) The computation of constant A

(11)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =-60\omega _0^2 -80\alpha _0^2 -36\beta _0^2 -12\beta _0 \omega _0 +72\beta _0 \alpha _0 +120\alpha _0 \omega _0 -720 \\ c_1 =120\alpha _0^2 +180\beta _0 \omega _0 -252\beta _0 \alpha _0 -120\alpha _0 \omega _0 +108\beta _0^2 \\ c_2 =40\alpha _0^2 +60\omega _0^2 -120\alpha _0 \omega _0 -117\beta _0^2 -120\beta _0 \omega _0 +180\beta _0 \alpha _0 \\ c_3 =120\alpha _0 \omega _0 -180\beta _0 \omega _0 -120\alpha _0^2 +180\beta _0 \alpha _0 \\ c_4 =132\beta _0 \omega _0 +126\beta _0^2 -252\beta _0 \alpha _0 +40\alpha _0^2 \\ c_5 =72\beta _0 \alpha _0 -108\beta _0^2 \\ c_6 =27\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(12)

For large N, we can approximate A as follows:

$$\begin{aligned} A\approx -\frac{b^{2}N^{2}}{360}c_6 N^{6}=-\frac{3}{40}N^{8}b^{2}\beta _0^2 \end{aligned}$$

(13)

(2) The computation of constant B

(14)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =6300-210\alpha _0 \omega _0 +450\alpha _0 \beta _0 +385\omega _0^2 -162\beta _0^2 -555\beta _0 \omega _0 -1050j\omega _0 \\ c_1 =-240\beta _0 \omega _0 -1050\alpha _0^2 +513\beta _0^2 +2100\alpha _0 \omega _0 -665\omega _0^2 -105\alpha _0 \beta _0 -6300 \\ c_2 =1575\alpha _0^2 +378\beta _0^2 -490\omega _0^2 -3570\alpha _0 \beta _0 +1050j\omega _0 -1050\alpha _0 \omega _0 \\ c_3 =525\alpha _0^2 -2457\beta _0^2 -2100\alpha _0 \omega _0 +560\omega _0^2 +3675\alpha _0 \beta _0 -975\beta _0 \omega _0 \\ c_4 =-1575\alpha _0^2 +1512\beta _0^2 +2100\alpha _0 \beta _0 -2970\beta _0 \omega _0 +1260\alpha _0 \omega _0 \\ c_5 =525\alpha _0^2 +1512\beta _0^2 -3570\alpha _0 \beta _0 +1440\beta _0 \omega _0 \\ c_6 =-1728\beta _0^2 +1020\alpha _0 \beta _0 \\ c_7 =432\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(15)

For large N, we can approximate B as follows:

$$\begin{aligned} B\approx -\frac{b^{2}N^{3}}{3150}c_7 N^{7}=-\frac{24}{175}N^{10}b^{2}\beta _0^2 \end{aligned}$$

(16)

(3) The computation of constant C

(17)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =-\frac{1}{5}\alpha _0 \beta _0 +\frac{1}{21}\alpha _0 \omega _0 +\frac{37}{210}\beta _0 \omega _0 -\frac{1}{21}\omega _0^2 +\frac{1}{2}+\frac{2}{225}\alpha _0^2 +\frac{1}{10}\beta _0^2 \\ c_1 =\frac{1}{10}\alpha _0 \beta _0 +\frac{1}{14}\beta _0 \omega _0 -\frac{47}{105}\alpha _0 \omega _0 -3-\frac{3}{10}\beta _0^2 +\frac{1}{3}\alpha _0^2 \\ c_2 =-\frac{1}{5}\alpha _0 \omega _0 -\frac{41}{28}\beta _0 \omega _0 +\frac{11}{24}\omega _0^2 +\frac{13}{2}-\frac{23}{45}\alpha _0^2 -\frac{41}{120}\beta _0^2 +\frac{11}{6}\alpha _0 \beta _0 \\ c_3 =\frac{21}{10}\alpha _0 \omega _0 -\frac{1}{2}\omega _0^2 -6+2\beta _0^2 -\frac{9}{10}\alpha _0^2 -\frac{25}{12}\alpha _0 \beta _0 \\ c_4 =-\frac{4}{3}\alpha _0 \omega _0 +\frac{109}{30}\beta _0 \omega _0 -\frac{1}{4}\omega _0^2 +2+\frac{877}{450}\alpha _0^2 -\frac{71}{60}\beta _0^2 -\frac{63}{20}\alpha _0 \beta _0 \\ c_5 =-2\beta _0 \omega _0 -\frac{19}{15}\alpha _0 \omega _0 +\frac{1}{2}\omega _0^2 +\frac{28}{5}\alpha _0 \beta _0 -\frac{1}{3}\alpha _0^2 -\frac{27}{10}\beta _0^2 \\ c_6 =-\frac{53}{28}\beta _0 \omega _0 +\frac{52}{35}\alpha _0 \omega _0 -\frac{9}{56}\omega _0^2 +\frac{137}{40}\beta _0^2 -\frac{19}{15}\alpha _0^2 -\frac{1}{2}\alpha _0 \beta _0 \\ c_7 =-\frac{27}{70}\alpha _0 \omega _0 +\frac{27}{14}\beta _0 \omega _0 -\frac{13}{4}\alpha _0 \beta _0 +\frac{9}{10}\alpha _0^2 \\ c_8 =-\frac{19}{42}\beta _0 \omega _0 +\frac{121}{60}\alpha _0 \beta _0 -\frac{11}{6}\beta _0^2 -\frac{9}{50}\alpha _0^2 \\ c_9 =-\frac{11}{30}\alpha _0 \beta _0 +\beta _0^2 \\ c_{10} =-\frac{1}{6}\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(18)

For large N, we can approximate C as follows:

$$\begin{aligned} C\approx b^{2}N^{2}c_{10} N^{10}=-\frac{1}{6}N^{12}b^{2}\beta _0^2 \end{aligned}$$

(19)

(4) The computation of constant D

(20)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =-360-24\alpha _0^2 -30\omega _0^2 +44\alpha _0 \omega _0 +18\beta _0 \omega _0 \\ c_1 =-90\alpha _0 \beta _0 +90\beta _0 \omega _0 +60\alpha _0^2 +18\beta _0^2 -76\alpha _0 \omega _0 \\ c_2 =135\alpha _0 \beta _0 -56\alpha _0 \omega _0 -90\beta _0 \omega _0 +30\omega _0^2 -63\beta _0^2 \\ c_3 =45\beta _0^2 -90\beta _0 \omega _0 +45\alpha _0 \beta _0 +64\alpha _0 \omega _0 -60\alpha _0^2 \\ c_4 =24\alpha _0^2 -135\alpha _0 \beta _0 +72\beta _0 \omega _0 +45\beta _0^2 \\ c_5 =45\alpha _0 \beta _0 -63\beta _0^2 \\ c_6 =18\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(21)

For large N, we can approximate D as follows:

$$\begin{aligned} D\approx -\frac{b^{2}N^{3}}{180}c_6 N^{6}=-\frac{1}{10}N^{9}b^{2}\beta _0^2 \end{aligned}$$

(22)

(5) The computation of constant E

(23)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =2520+168\omega _0^2 -112\alpha _0^2 -180\beta _0^2 -276\beta _0 \omega _0 +360\beta _0 \alpha _0 \\ c_1 =-5040-252\beta _0 \alpha _0 +1260\alpha _0 \omega _0 +360\beta _0^2 -616\alpha _0^2 +102\beta _0 \omega _0 -462\omega _0^2 \\ c_2 =-2268\beta _0 \alpha _0 +2265\beta _0 \omega _0 -252\omega _0^2 +1120\alpha _0^2 +315\beta _0^2 -840\alpha _0 \omega _0 \\ c_3 =-1260\alpha _0 \omega _0 +2520\beta _0 \alpha _0 -1575\beta _0^2 +280\alpha _0^2 -885\beta _0 \omega _0 +378\omega _0^2 \\ c_4 =1260\beta _0 \alpha _0 -1809\beta _0 \omega _0 +945\beta _0^2 +840\alpha _0 \omega _0 -1008\alpha _0^2 \\ c_5 =-2268\beta _0 \alpha _0 +963\beta _0 \omega _0 +336\alpha _0^2 +945\beta _0^2 \\ c_6 =648\beta _0 \alpha _0 -1080\beta _0^2 \\ c_7 =270\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(24)

For large N, we can approximate E as follows:

$$\begin{aligned} E\approx -\frac{b^{2}N^{3}}{2520}c_7 N^{7}=-\frac{3}{28}N^{10}b^{2}\beta _0^2 \end{aligned}$$

(25)

(6) The computation of constant F

(26)

where

$$\begin{aligned} \left\{ {\begin{array}{l} c_0 =1200\alpha _0^2 -432\alpha _0 \beta _0 -1480\alpha _0 \omega _0 -12600 \\ c_1 =-840\alpha _0^2 -2740\alpha _0 \omega _0 +37800+7668\alpha _0 \beta _0 -6930\beta _0 \omega _0 -2520\beta _0^2 +3150\omega _0^2 \\ c_2 =-25200+13150\alpha _0 \omega _0 -2142\alpha _0 \beta _0 -7560\alpha _0^2 -5670\beta _0 \omega _0 +6930\beta _0^2 -2100\omega _0^2 \\ c_3 =-25452\alpha _0 \beta _0 +22050\beta _0 \omega _0 -1550\alpha _0 \omega _0 +8400\alpha _0^2 -3150\omega _0^2 +2205\beta _0^2 \\ c_4 =-11070\alpha _0 \omega _0 +19782\alpha _0 \beta _0 -17640\beta _0^2 +4200\alpha _0^2 +2100\omega _0^2 \\ c_5 =-15120\beta _0 \omega _0 +13482\alpha _0 \beta _0 +4890\alpha _0 \omega _0 -7560\alpha _0^2 +8820\beta _0^2 \\ c_6 =-17208\alpha _0 \beta _0 +2160\alpha _0^2 +5670\beta _0 \omega _0 +8820\beta _0^2 \\ c_7 =4302\alpha _0 \beta _0 -8505\beta _0^2 \\ c_8 =1890\beta _0^2 \\ \end{array}} \right. \end{aligned}$$

(27)

For large N, we can approximate F as follows:

$$\begin{aligned} F\approx -\frac{b^{2}N^{3}}{12600}c_8 N^{8}=-\frac{3}{20}N^{11}b^{2}\beta _0^2 \end{aligned}$$

(28)

(7) The computation of variance of d

(29)

(8) The computation of variance of e

(30)

(9) The computation of variance of f

(31)

(10) The computation of covariance for d and e

$$\begin{aligned} E\left[ {de^{{*}}} \right]= & {} 2b^{2}\sigma ^{2}\sum _{n=0}^{N-1} {\sum _{m=0}^{N-1} {\sum _{k=0}^{N-1} {\left[ {\left( {n-k} \right) \left( {m^{2}-k^{2}} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) } \right. } } } \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) ^{2} \nonumber \\&\times \,\left( {\omega _0 +2\alpha _0 m+3\beta _0 m^{2}} \right) -j\left( {n-k} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) \nonumber \\&\times \,\left( {2\omega _0 m+2\omega _0 k+8\alpha _0 mk+6\beta _0 mk^{2}+6\beta _0 m^{2}k} \right) +j\left( {m^{2}-k^{2}} \right) \left( {\omega _0 +2\alpha _0 m+3\beta _0 m^{2}} \right) \nonumber \\&\times \,\left( {\omega _0 +2\alpha _0 k\!+\!3\beta _0 k^{2}} \right) \left( {2\omega _0 \!+\!2\alpha _0 k\!+\!2\alpha _0 n\!+\!3\beta _0 k^{2}+3\beta _0 n^{2}} \right) \nonumber \\&\left. {+\,\left( {2\omega _0 m\!+\!2\omega _0 k+8\alpha _0 mk+6\beta _0 mk^{2}+6\beta _0 m^{2}k} \right) \left( {2\omega _0 +2\alpha _0 k+2\alpha _0 n+3\beta _0 k^{2}+3\beta _0 n^{2}} \right) } \right] \nonumber \\\approx & {} \frac{99}{700}N^{14}b^{2}\sigma ^{2}\beta _0^4 \end{aligned}$$

(32)

(11) The computation of covariance for d and f

$$\begin{aligned} E\left[ {df^{{*}}} \right]= & {} 2b^{2}\sigma ^{2}\sum _{n=0}^{N-1} {\sum _{m=0}^{N-1} {\sum _{k=0}^{N-1} {\left[ {\left( {n-k} \right) \left( {m^{3}-k^{3}} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) } \right. } } } \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) ^{2} \nonumber \\&\times \,\left( {\omega _0 +2\alpha _0 m+3\beta _0 m^{2}} \right) -j\left( {n-k} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) \nonumber \\&\times \,\left( {3\omega _0 m^{2}\!+\!3\omega _0 k^{2}\!+\!6\alpha _0 m^{2}k\!+\!6\alpha _0 mk^{2}\!+\!18\beta _0 m^{2}k^{2}} \right) \!+\!j\left( {m^{3}-k^{3}} \right) \left( {\omega _0 \!+\!2\alpha _0 m\!+\!3\beta _0 m^{2}} \right) \nonumber \\&\times \,\left( {\omega _0 \!+\!2\alpha _0 k\!+\!3\beta _0 k^{2}} \right) \left( {2\omega _0 \!+\!2\alpha _0 k\!+\!2\alpha _0 n\!+\!3\beta _0 k^{2}\!+\!3\beta _0 n^{2}} \right) \nonumber \\&\left. +\,\left( {3\omega _0 m^{2}\!+\!3\omega _0 k^{2}\!+\!6\alpha _0 m^{2}k+6\alpha _0 mk^{2}+18\beta _0 m^{2}k^{2}} \right) \right. \nonumber \\&\left. \times \left( {2\omega _0 +2\alpha _0 k+2\alpha _0 n+3\beta _0 k^{2}+3\beta _0 n^{2}} \right) \right] \nonumber \\\approx & {} \frac{13}{80}N^{15}b^{2}\sigma ^{2}\beta _0^4 \end{aligned}$$

(33)

(12) The computation of covariance for f and e

$$\begin{aligned} E\left[ {ef^{{*}}} \right] \!= & {} \!2b^{2}\sigma ^{2}\sum _{n=0}^{N-1} {\sum _{m=0}^{N-1} {\sum _{k=0}^{N-1} {\left[ {\left( {n^{2}-k^{2}} \right) \left( {m^{3}-k^{3}} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) } \right. } } } \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) ^{2} \nonumber \\&\times \left( {\omega _0 +2\alpha _0 m+3\beta _0 m^{2}} \right) -j\left( {n^{2}-k^{2}} \right) \left( {\omega _0 +2\alpha _0 n+3\beta _0 n^{2}} \right) \left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) \nonumber \\&\times \,\left( {3\omega _0 m^{2}\!+\!3\omega _0 k^{2}\!+\!6\alpha _0 m^{2}k\!+\!6\alpha _0 mk^{2}\!+\!18\beta _0 m^{2}k^{2}} \right) \!+\!j\left( {m^{3}-k^{3}} \right) \left( {\omega _0 \!+\!2\alpha _0 m\!+\!3\beta _0 m^{2}} \right) \nonumber \\&\times \,\left( {\omega _0 +2\alpha _0 k+3\beta _0 k^{2}} \right) \left( {2\omega _0 n+2\omega _0 k+8\alpha _0 nk+6\beta _0 nk^{2}+6\beta _0 n^{2}k} \right) \nonumber \\&+\,\left( {3\omega _0 m^{2}+3\omega _0 k^{2}+6\alpha _0 m^{2}k+6\alpha _0 mk^{2}+18\beta _0 m^{2}k^{2}} \right) \nonumber \\&\left. {\times \left( {2\omega _0 n+2\omega _0 k+8\alpha _0 nk+6\beta _0 nk^{2}+6\beta _0 n^{2}k} \right) } \right] \nonumber \\\approx & {} \frac{171}{700}N^{16}b^{2}\sigma ^{2}\beta _0^4 \end{aligned}$$

(34)

By the first-order perturbation principle and combined with above definitions, the following relationship can be set up:

$$\begin{aligned} \left\{ {\begin{array}{l} d+A\delta \omega +D\delta \alpha +E\delta \beta =0 \\ e+D\delta \omega +B\delta \alpha +F\delta \beta =0 \\ f+E\delta \omega +F\delta \alpha +C\delta \beta =0 \\ \end{array}} \right. \end{aligned}$$

(35)

Then, we can obtain (10) based on the settlement of (35) and the computations from (11) to (34).