### 3.1 The performance degradation of PFA when *Δr* ≠ 0

According to Equations (8 to 10), one notes that when the RC locates at the cell of the RP, the phase term of the dechirped signals can be expressed in the polar format. Thus we can use the standard PFA algorithm to obtain a focused ISAR image, as shown in Figure 4. However, in practical applications, since the target is non-cooperative and the error of range measurement is unavoidable in the radar system, after translational motion compensation, RC is usually not coincided with RP. Assuming that the range difference between RC and RP is *Δr* = *r*_{
a
} - *r*_{0}, Equations (8) and (9) can be expressed as:

{R}_{\mathit{ps}\_k}^{\prime}=\mathit{\Delta}r+{x}_{k}sin\theta \left({t}_{m}\right)+{y}_{k}cos\theta \left({t}_{m}\right),

(16)

\begin{array}{c}{s}_{R4}^{\prime}\left(f,\theta \right)={\displaystyle \sum _{k=1}^{K}{\sigma}_{k}\mathrm{rect}\left\{\frac{f}{B}\right\}}\\ \times exp\left\{-j\frac{4\pi}{c}\left({f}_{c}+f\right)\left(\mathit{\Delta}r+{x}_{k}sin\theta \left({t}_{m}\right)+{y}_{k}cos\theta \left({t}_{m}\right)\right)\right\}.\end{array}

(17)

Taking scattering point *p*_{
k
} for example, its echo can be formulated as:

\begin{array}{c}{s}_{R4\_k}^{\prime}\left(f,\theta \right)={\sigma}_{k}\mathrm{rect}\left\{\frac{f}{B}\right\}\times exp\left\{-j\frac{4\pi}{c}\left({f}_{c}+f\right)\left(\mathit{\Delta}r+{x}_{k}sin\theta \left({t}_{m}\right)\right.\right.\\ \phantom{\rule{18em}{0ex}}\left(\right)close="\}">\left(\right)close=")">+{y}_{k}cos\theta \left({t}_{m}\right)\end{array}\n

(18)

Then, the phase term of Equation (18) is expressed by:

\begin{array}{l}{\mathit{\Phi}}_{k}^{\mathit{\prime}}\left(\theta ,{K}_{R}\right)={K}_{R}\left(\mathit{\Delta}r+{x}_{k}sin\theta \left({t}_{m}\right)+{y}_{k}cos\theta \left({t}_{m}\right)\right)\\ \phantom{\rule{5em}{0ex}}={K}_{R}\mathit{\Delta}r+{x}_{k}\cdot {K}_{R}sin\theta \left({t}_{m}\right)\\ \phantom{\rule{6.5em}{0ex}}+{y}_{k}\cdot {K}_{R}cos\theta \left({t}_{m}\right).\end{array}

(19)

Equation (19) indicates that the disturbed phase term *K*_{
R
}*Δr* can cause the performance degradation when the PFA algorithm is applied. For clarity, we provide some simulation results in Figure 5 to indicate the degradation of PFA image from the imprecise RC. The target is the one in Figure 4. The new given dechirped signals in Figure 5 have the RC highlighted by the red dashed line, which locates at (0 m, 10 m) in the target coordinate system. The RP is highlighted by the white solid line in Figure 5a, locating at (0 m, 0 m). The ISAR image achieved by the RDA is shown in Figure 5b and the imaging result achieved by PFA is given in Figure 5d. Compared to the image in Figure 4d, the blur in Figure 5b was not fully overcome by PFA.

### 3.2 Analysis and simulation of the disturbance system in PFA

According to the simulation above, the blur in the cross-range dimension of the image by PFA occurs when the RC is deviated from the RP. In this subsection, the influence of the disturbed phase term *K*_{
R
}*Δr* on PFA will be analyzed in detail, by adopting the equivalence of a disturbance system. From Equation (10), since:

\left\{\begin{array}{l}{K}_{x}={K}_{R}sin\theta \\ {K}_{y}={K}_{R}cos\theta \end{array},,\right.

(20)

the echo phase term of *p*_{
k
} with respect to (*K*_{
x
}, *K*_{
y
}) can be formulated as:

{\mathit{\Phi}}_{k}^{\mathit{\prime}}\left({K}_{x},{K}_{y}\right)=\sqrt{{K}_{x}^{2}+{K}_{y}^{2}}\cdot \mathit{\Delta}r+{K}_{x}{x}_{k}+{K}_{y}{y}_{k}.

(21)

Supposing that the PFA interpolation has been carried on the echo in Equation (18), the echo {s}_{R4\_k}^{\prime}\left({K}_{x},{K}_{y}\right) of (*K*_{
x
}, *K*_{
y
}) in a Cartesian coordinate system can be obtained. Taking 2D FFT on the {s}_{R4\_k}^{\prime}\left({K}_{x},{K}_{y}\right), one can obtain the 2D spectrum of scattering point *p*_{
k
}:

\begin{array}{l}{S}_{R4\_k}^{\prime}\left(X,Y\right)={\displaystyle \int \int {s}_{R4\_k}^{\prime}\left({K}_{x},{K}_{y}\right)\cdot {e}^{-j2\pi X\cdot {K}_{x}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}}\\ ={\displaystyle \int \int {\sigma}_{k}\mathrm{rect}\left\{\frac{f}{B}\right\}exp\left\{j{\mathit{\Phi}}_{k}^{\mathit{\prime}}\left({K}_{x},{K}_{y}\right)\right\}\cdot {e}^{-j2\pi X\cdot {K}_{x}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}}\\ =\int \int {\sigma}_{k}\mathrm{rect}\left\{\frac{f}{B}\right\}exp\left\{j\left(\sqrt{{K}_{x}^{2}+{K}_{y}^{2}}\cdot \mathit{\Delta}r+{K}_{x}{x}_{k}+{K}_{y}{y}_{k}\right)\right\}\\ \phantom{\rule{16em}{0ex}}\cdot {e}^{-j2\pi X\cdot {K}_{x}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}\end{array}

(22)

Obviously, when *Δr* = 0, the 2D spectrum in Equation (22) has the form of the 2D Sinc function:

{S}_{R4\_k}^{\prime}\left(X,Y\right)\left|{}_{\Delta r=0}\right.=\mathrm{sinc}\left(\frac{2\mathit{\Theta}}{\lambda}\left(X-{x}_{k}\right),\frac{2B}{c}\left(Y-{y}_{k}\right)\right).

(23)

If *Δr* ≠ 0, the 2D spectrum of scattering point *p*_{
k
} is given by:

\begin{array}{l}{S}_{R4\_k}^{\prime}\left(X,Y\right)=\int \int {\sigma}_{k}\mathrm{rect}\left\{\frac{f}{B}\right\}exp\left\{j\left({K}_{x}{x}_{k}+{K}_{y}{y}_{k}\right)\right\}\\ \phantom{\rule{14.7em}{0ex}}exp\left\{j\cdot \sqrt{{K}_{y}^{2}+{K}_{x}^{2}}\cdot \mathit{\Delta}r\right\}\cdot {e}^{-j2\pi X\cdot {K}_{x}}\\ \phantom{\rule{25.5em}{0ex}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}\\ \phantom{\rule{6.3em}{0ex}}=\int \int {s}_{R4\_k}\left({K}_{x},{K}_{y}\right)\cdot exp\left\{j\cdot \sqrt{{K}_{x}^{2}+{K}_{y}^{2}}\cdot \Delta r\right\}\cdot {e}^{-j2\pi X\cdot {K}_{x}}\\ \phantom{\rule{27.8em}{0ex}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}\end{array}.

(24)

Let \psi \left({K}_{x},{K}_{y}\right)=exp\left\{j\cdot \sqrt{{K}_{y}^{2}+{K}_{x}^{2}}\cdot \mathit{\Delta}r\right\} and its 2D FFT spectrum is *Ψ*(*X*, *Y*), then {S}_{R4\_k}^{\prime}\left(X,Y\right) satisfies:

\begin{array}{l}{S}_{R4\_k}^{\prime}\left(X,Y\right)={\displaystyle \int \int {s}_{R4\_k}\left({K}_{x},{K}_{y}\right)\cdot \psi \left({K}_{x},{K}_{y}\right)\cdot {e}^{-j2\pi X\cdot {K}_{x}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}}\\ \phantom{\rule{8em}{0ex}}={S}_{R4\_k}^{\prime}\left(X,Y\right)\left|{}_{\mathit{\Delta}r=0}\right.\otimes \mathit{\Psi}\left(X,Y\right)\end{array},

(25)

where operation symbol ⊗ denotes the operator of the 2D convolution. The characteristics of *Ψ*(*X*, *Y*) are the key point of the analysis of the influence on {S}_{R4\_k}^{\prime}\left(X,Y\right). We define *Ψ*(*X*, *Y*) as:

\begin{array}{l}\mathit{\Psi}\left(X,Y\right)={\displaystyle \int \int \psi \left({K}_{x},{K}_{y}\right)\cdot {e}^{-j2\pi X\cdot {K}_{x}}\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{x}d{K}_{y}}\\ \phantom{\rule{3.9em}{0ex}}={\displaystyle {\int}_{\left[{K}_{y1},{K}_{y2}\right]}\left({\displaystyle {\int}_{\left[{K}_{x1},{K}_{x2}\right]}\psi \left({K}_{x},{K}_{y}\right)\cdot {e}^{-j2\pi X\cdot {K}_{x}}d{K}_{x}}\right)\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{y}}\\ \phantom{\rule{3.9em}{0ex}}={\displaystyle {\int}_{\left[{K}_{y1},{K}_{y2}\right]}\left({\displaystyle {\int}_{\left[{K}_{x1},{K}_{x2}\right]}exp\left\{j\cdot \sqrt{{K}_{x}^{2}+{K}_{y}^{2}}\cdot \mathit{\Delta}r\right\}\cdot {e}^{-j2\pi X\cdot {K}_{x}}d{K}_{x}}\right)\cdot {e}^{-j2\pi Y\cdot {K}_{y}}d{K}_{y}}\end{array}.

(26)

From Equation (26), the spectrum of *Ψ*(*X*, *Y*) is independent on the location (*x*_{
k
}, *y*_{
k
}) of the scattering point. As a result, when *Δr* ≠ 0, the degraded PFA is equivalent to the output of the standard PFA model passing through a disturbance system denoted by *Ψ*(*X*, *Y*). The schematic diagram of the disturbance system *Ψ*(*X*, *Y*) is shown in Figure 6.

According to Figure 3, the value zone of *K*_{
x
} and *K*_{
y
} after the PFA interpolation is:

\left\{\begin{array}{l}{K}_{x}\in \left[{K}_{x1},{K}_{x2}\right]=\left[-{K}_{R1}\cdot tan\left(\mathit{\Theta}/2\right),{K}_{R1}\cdot tan\left(\mathit{\Theta}/2\right)\right]\\ {K}_{y}\in \left[{K}_{y1},{K}_{y2}\right]=\left[{K}_{R1},{K}_{R2}\cdot cos\left(\mathit{\Theta}/2\right)\right]\end{array}\right..

(27)

Since *ψ*(*K*_{
x
}, *K*_{
y
}) have the symmetrical format with respect to *K*_{
x
} and *K*_{
y
}, we only need to analyze the FFT distribution of either one. The formula *g*(*x*) with respect to *x* is given by:

g\left(x\right)=exp\left\{j\left(\mathit{\Delta}r\cdot \sqrt{{a}^{2}+{x}^{2}}\right)\right\},

(28)

where *a* and *Δr* are constant. When *Δr* > 0, the FFT of *g*(*x*) is given by:

G\left(X\right)={\displaystyle {\int}_{{x}_{1}}^{{x}_{2}}exp\left\{j\left(\mathit{\Delta}r\cdot \sqrt{{a}^{2}+{x}^{2}}\right)\right\}{e}^{-j2\pi X\cdot x}dx}.

(29)

Figure 7a,b shows the curves of the *g*(*x*) phase term under the conditions *x*_{2} > *x*_{1} ≫ 0 and *x*_{2} = - *x*_{1}, respectively. By comparing between Figure 7a and b, one can observe that the phase term is approximately linear to *x* when *x*_{2} > *x*_{1} ≫ 0, while it has quadratic phase if *x*_{2} = -*x*_{1}. Furthermore, the smaller *x* is, the more distinct the quadratic feature of *g*(*x*) phase term is. For a limited time signal with linear phase and rectangular envelope, its spectrum performs as a Sinc function [23], since the signal can be approximated by a point frequency signal. The signal with quadratic phase is somewhat like a LFM signal, and its spectrum will be widened around the center frequency. In a nutshell, the spectrum of *G*(*X*) is related with the integral region [*x*_{1}, *x*_{2}]. Simulations in Figure 8 are given to analyze of the disturbance system *Ψ*(*X*, *Y*). Take the signal model in Section 3.1 for example, the interval of *K*_{
x
} and *K*_{
y
} can be calculated as:

\left\{\begin{array}{l}{K}_{x}\in \left[-{K}_{R1}\cdot tan\left(\mathit{\Theta}/2\right),{K}_{R1}\cdot tan\left(\mathit{\Theta}/2\right)\right]=\left[-58.84,58.84\right]\\ {K}_{y}\in \left[{K}_{R1},{K}_{R2}\cdot cos\left(\mathit{\Theta}/2\right)\right]=\left[586.43,666.85\right]\end{array}\right..

(30)

For generality, we take the scattering point *p*_{
k
} at (*x*_{
k
}, *y*_{
k
}) = (13, 9) for example, as shown in Figure 8a. By applying RD and PFA algorithms to the simulated standard PFA signal, the obtained ISAR images of *p*_{
k
} are shown in Figure 8b,c, respectively. The blurring in range cells and Doppler cells caused by the MTRC in the RD model is compensated. The 2D spectrum of disturbance system *Ψ*(*X*, *Y*) is shown in Figure 8d. Figure 8e,f,g,h illustrates the one-dimensional (1D) distributions of *Ψ*(*X*, *Y*) along cross-range axes *X* and range axes *Y*. Since *K*_{y 2} > *K*_{y 1} ≫ 0 and *K*_{x 2} = -*K*_{x 1}, one can see from Figure 8d, e, f, g, h that the spectra in the cross range are much wider than the ones in the slant range. According to the definition and characteristics of 2D convolution [24, 25], the output {S}_{R4\_k}^{\prime}\left(X,Y\right)\left|{}_{\mathit{\Delta}r=0}\right. of the ideal PFA image will be widened in the cross range caused by the disturbance system *Ψ*(*X*, *Y*). As a result, the final output is blurred in the cross range and focused in slant range, (Figure 8i).

Furthermore, the influence of *Δr* and *Θ* values on the disturbance system *Ψ*(*X*, *Y*) is analyzed. When *Θ* = 0.2 rad is fixed, the distributions of *Ψ*(*X*) and *Ψ*(*Y*) under different values of *Δr* are shown in Figure 9, where *Δr* ∈ {2, 6, 10, 14, 18} m. With the distance of RC and RP increasing, *Ψ*(*X*, *Y*) is widened in the cross range and shifted in the slant range. The distributions of *Ψ*(*X*) and *Ψ*(*Y*) under different values of rotation angle *Θ* is shown in Figure 10, where *Θ* ∈ {0.05, 0.1, 0.15, 0.2, 0.25, 0.3} rad and *Δr* = 10 m. Similarly to the results in Figure 9, *Ψ*(*X*, *Y*) is widened in the cross range and shifted in the slant range with *Θ* increasing. Particularly, when the rotation angle *Θ* is small enough, the spectrum of *Ψ*(*X*, *Y*) in the cross range trends to the impulse function. At this level, one can obtain the formulas

\left\{\begin{array}{l}{K}_{x}={K}_{R}sin\theta \approx {K}_{R}\cdot \theta \\ {K}_{y}={K}_{R}cos\theta \approx {K}_{R}\cdot \left(1-{\theta}^{2}/2\right)\\ \sqrt{{K}_{x}^{2}+{K}_{y}^{2}}\approx {K}_{y}\end{array}\right..

(31)

Then, the phase term of *P*_{
k
} echo is approximately equal to:

\begin{array}{l}{\mathit{\Phi}}_{k}\left(\theta \right)={K}_{R}\left(\mathit{\Delta}r+{x}_{k}sin\theta +{y}_{k}cos\theta \right)\\ \phantom{\rule{1.75em}{0ex}}={K}_{R}\mathit{\Delta}r+{K}_{x}{x}_{k}+{K}_{y}{y}_{k}\\ \phantom{\rule{1.75em}{0ex}}\approx {K}_{y}\cdot \mathit{\Delta}r+{K}_{x}{x}_{k}+{K}_{y}{y}_{k}\end{array}.

(32)

Apparently, *Δr* only produces translation in the cross range without the blurring in the slant range. Actually, the signal model becomes the RD model when *Θ* is small.

From the analysis and simulations above, we can note that, because of the range measurement errors and the target's non-cooperative motion, the disturbed phase term remained after data-driven translational compensation and the blur in the cross-range dimension is induced. Furthermore, with the increasing of *Δr* and *Θ*, the blur in the cross-range dimension becomes serious. As a conclusion, the accurate rotation center is essentially significant for PFA imaging for ISAR targets and only accurate estimation of RC can overcome the blur after PFA.