ISAR Imaging of maneuvering Targets based on a new Algorithm Modified Fast Chirp Fourier Transform for Cubic Chirps

AbstractInverse Synthetic Aperture Radar (ISAR) imaging of targets with complex motions can be modeled as cubic chirps. Modified Discrete Chirp Fourier Transform (MDCFT) is used to process the cubic signal to obtain ISAR image in two dimensions (2D). In this paper, a new algorithm named Modified Fast Chirp Fourier Transform (MFCFT) is proposed. This algorithm gives accurate results as, and the computation time is very much reduced, compared to MDCFT. The simulation is carried out for ship and Airplane targets with different scattering points. The obtained results show an enhancement both in terms of resolution and computation time in the proposed method hence named Modified Fast chirp Fourier Transform.

Where.f, γ and T, denote the carrier frequency, chirp rate and pulse width respectively, t is the full time given by t 1 +t 2 , t 1 being the fast time and t 2 , the slow time. t 2 is given by n/prf and n=0,1,…N. Here, prf is pulse repetition frequency and N is the number of accumulating pulses. The received echo signal is given by where R i (t 2 ) is the distance between i th scatter and Radar at time t 2 and σ i represents the reflection coefficient. For this received signal keystone formatting is applied to remove the migration through range cell [12]. Assume that if the number of scatters in a certain range bin is I, the azimuth echoes corresponding to this range bin is expressed as the cubic phase signal [13][14][15][16][17][18]. The azimuth echo signal is written as where A i denote the amplitude, θ i the initial phase, f i the center frequency, α i the chirp rate, and β i is derivative of chirp rate of azimuth echoes associated with the i th scatter respectively. R i (t 0 ) is initial range at time t 0 .
Therefore, the discrete expression of ) ( 2 ' t S is S(n) and is given by

III.ALGORITHMS FOR RIC
To process the received echo signal for cubic chirps after MTRC, some transforms has to be used in order to obtain the final ISAR image in 2D. Discrete chirp Fourier transform and MDCFT have been used in the recent times to obtain the image. Following sections detail the three methods including the proposed one i.e., DCFT, MDCFT and MFCFT. A comparison of the three methods is also presented.

A. DCFT
DCFT was developed for quadratic terms and extended for cubic chirps which is represented by [11] 1 , , 0 where x(n) is chirp signal having signal length N and k, l and m represent the constant frequency, chirp rate and derivative of chirp rate respectively. If the signal parameters are not integers or not very close to integers then, DCFT will suffer from the picket-fence effect. In DCFT since k, l, m values lies in between 0 to N-1, the parameter estimation will not be accurate. Depending on the basic properties of DCFT, if N is a prime number then only the peak is clearly identified and if it is not prime then it leads to picket fence output. For example consider the two cases. Case (i). If k=36, l=98, m=25 and N=127(prime number) then the output for DCFT is shown in Fig.1. Case (ii). If k=36, l=98, m=25 and N=128(not prime number) then the picket fence output is observed for DCFT in Fig.2. From above Figs.1&2, one can say that DCFT has a limitation in identifying the peak if N is non prime value. To avoid the severe picket-fence effect and increase the accuracy of parameter estimation, MDCFT is proposed.
If k=36, l=98, m=25 and N=128 then the output for MDCFT is shown in Fig.3. From Fig.3, it can be said that the peak is clearly identified. The ISAR imaging based on MDCFT for cubic chirps is implemented with the following steps as follows: i. After applying MDCFT to (7) i.e., s(n) in each range bin, the equation is given by ( )   where S(k,l,m) is the MFCFT of s(n). ii.
The projection of S(k,l,m) will produce a Doppler image in 3D with center frequency f i . iii.
Doppler image I D (k) can be obtained by the energy accumulation along axes of l and m given by , , ( In this Doppler image I D (k) all the main lobes of S(k,l,m) are clearly identified. iv.
Now the Doppler images of all range bins are added to form the ISAR image of the target for cubic chirp signal. v.
It may be noted that the proposed algorithm will be found to be more and more effective as higher and higher order terms are considered because exponent of N is defered with. In this way the 2D ISAR image is obtained for MFCFT where the computation time is very much reduced when compared to MDCFT.

IV.SIMULATION RESULTS
The simulation parameters used in the programming are given in Table. I  Here the simulation has been carried out initially for a ship target with 85 scattering points and for Airplane target with 84 scattering points. Case i. Simulated ship target is shown in Fig.6. The amplitudes of all the echoes are considered as equal. ISAR image in 2D obtained using MDCFT is shown in Fig.7 and that using MFCFT is shown in Fig.8. In both the cases, the hot spots are clearly identified, but, in case of MFCFT the simulation time is very much reduced when compared to MDCFT. Case ii. Here an Airplane with 84 scatters is considered as shown in Fig.9. The ISAR image using MDCFT is shown in Fig.10 and that using MFCFT is shown in Fig.11. Here also the hot spots are observed clearly in the two methods but the computation time is reduced in MFCFT. V. CONCLUSIONS In this paper, ISAR imaging using MFCFT is discussed for cubic chirp signals. The quality of the image obtained in MFCFT is comparable with that of MDCFT. Additionally the computation time is significantly reduced here. This method will be found to be more effective as higher and higher order terms need to be considered. The advantage in this method is that exponent terms of the signal length remain to be unity.