# Principle of the Proposed Algorithm

In this section, the principle and processing steps of the proposed algorithm are introduced. The aim of the implementation of CZT is to avoid the use of Doppler centroid estimation algorithms. Thus, the processing of DBS algorithm is simplified, and Doppler centroid estimation error is circumvented.

Transform Eq. 4.2.9 into the range-frequency domain, it yields

where *f _{r}* denotes range frequency. In Eq. 4.3.1, the last exponential term is the RCM in range-frequency domain. It can be noted that the RCM is the correlation of range frequency and azimuth time.

Perform transform of *t _{a}* as

According to Sect. 2.5, Eq. 4.3.2 is the definition formula of Keystone transform. Keystone transform is capable of adjust the RCM without the estimation of Doppler centroid, which circumstances the Doppler centroid estimation processing steps and the impact of the Doppler centroid estimation error.

However, Keystone transform also faces the problems of Doppler centroid aliasing and interpolation. The Doppler centroid aliasing does not happen if

where K_{v} represents the PRF-to-velocity ratio. According to the parameters in Table 4.1, K_{v} can be calculated as 31.69. In a typical airborne SAR, the value of K_{v }is larger than 32, therefore the Doppler centroid aliasing can be circumstanced.

The interpolation is also an important problem of Keystone transform. The interpolation is not suited for real-time processing, thus it is not suited for the DBS modification. Therefore, a novel DBS algorithm based on CZT is proposed in this chapter [12]. The proposed algorithm can be operated by using complex multiplication and FFT, which is highly suitable for real-time processing.

CZT is an extensive z-transform which non-uniformly samples along a spiral [13]. The CZT of signal x(n) can be defined as

*?2n*

where z *= e*~^{}}~, and it denotes the N uniformly sampling on the unit circle. The definition of CZT is

where A_{0} is the vector radius length of starting sampling point, *в _{0}* is the phase of starting sampling point, W

_{0}is the extensional ratio of spiral, represents the angle difference between adjacent sampling points, and M is the complex spectrum samples.

Substitute Eq. 4.3.5 into Eq. 4.3.4, it yields

According to the Bluestein equation, Eq. 4.3.6 can be derived as

2 2

where *g(n) =* x(n)A~”W2, *h(n) =* W^{-}^. Perform range compression in the range frequency domain, and set the signal as *s _{rc}(t_{a},f*

_{r}). Perform CZT to the azimuth signal in the range-frequency azimuth-time domain, the RCMC can be achieved. Set the signal after CZT as

*s*

_{rc}(s_{a}, f_{r}), the processing steps of the CZT is shown in Fig. 4.3.

RCMC can be realized by CZT rapidly by setting up parameters of CZT as

Therefore, the proposed algorithm requires simple processing steps, and can be operated by FFT and complex multiplication. Processing steps of CZT is split and combined together with the DBS algorithm, which makes it very suitable for real-time project.

Simulations are performed to prove that RCM can be completely corrected by the proposed algorithm. Simulation parameters are listed in Table 4.1. Nine point targets are simulated, and conventional DBS algorithm [5] is performed as a comparison.

**Fig. 4.3 ****Flowchart of RCMC based on CZT**

**Fig. 4.4 **Comparison results of simulations. **a **Conventional algorithm simulation. **b **Proposed algorithm simulation. **c **Range slice of both algorithms. **d **Azimuth slice of both algorithms

Figure 4.4a, b show the simulation results of nine point targets imaging of conventional and proposed algorithms. The squint angle in the simulation is 18°. Targets in Fig. 4.4a are smeared in azimuth since RCM is not corrected. According to Fig. 4.4b, RCM are completely corrected by the proposed algorithm, and the imaging resolution is significantly improved. Figure 4.4c, d further illustrate the compression performances of both algorithms. Thus, the imaging performance of the proposed algorithm is proved.