*3.1. PTRS*

Based on the Fourier transformation (FT), the signal denoted by (2) is transformed into the 2D frequency domain. The expression is given by:

$$SS\_i(f\_{\tau}, f\_t) = P(f\_{\tau}) \int\_{-T\_s/2}^{T\_s/2} \exp\{-j2\pi(f\_c + f\_\tau)\tau\_i - j2\pi f\_t t\}dt\tag{3}$$

where *Ts* denotes the integration time; *P*(*fτ*) is the spectrum of the transmitted signal; *f<sup>τ</sup>* and *ft* are the instantaneous and Doppler frequencies.

Due to the complex expression of (1), it is difficult to calculate (3). To solve this problem, approximations are usually exploited by traditional methods. However, the residual errors would influence the imaging performance. Here, we present the numerical result, which avoids approximations.

The phase of the exponent in (3) is defined as:

$$\Psi\_i(f\_\mp, f\_l) = -2\pi (f\_\mp + f\_\mp)\tau\_l - 2\pi f\_l t \tag{4}$$

Applying the method of stationary phase [9] to (4) yields:

$$\frac{\partial \pi\_i(\widetilde{t\_i})}{\partial t} + \frac{f\_t}{f\_c + f\_\tau} = 0 \tag{5}$$

where "*ti* <sup>∈</sup> [−*Ts*/2, *Ts*/2] represents the PSP; *∂τ<sup>i</sup> <sup>∂</sup><sup>t</sup>* denotes the first derivative with respect to the slow time.

Using (1), the first derivative with respect to the slow time is given by:

$$\begin{cases} \frac{\partial \overline{v}\_{i}}{\partial t} = \frac{v^{2} + ctv^{2} \left[ \left( vt \right)^{2} + r^{2} \right]^{-0.5}}{\varepsilon^{2} - v^{2}} + \frac{\left\{ \left\{ v\left[ \left( vt \right) + d\_{i} \right] + c\sqrt{\left( vt \right)^{2} + r^{2}} \right\}^{2} + \left( c^{2} - v^{2} \right) \left[ 2\left( vt \right)d\_{i} + d\_{i}^{2} \right] \right\}^{-0.5}}{\varepsilon^{2} - v^{2}} \times\\ \left\{ \left[ v^{2}t + vd\_{i} + c\sqrt{\left( vt \right)^{2} + r^{2}} \right] \left\{ v^{2} + ctv^{2} \left[ \left( vt \right)^{2} + r^{2} \right]^{-0.5} \right\} + vd\_{i} \left( c^{2} - v^{2} \right) \right\} \end{cases} \tag{6}$$

(5) cannot be solved analytically, as (6) is very complicated. Due to this, the numerical evaluation method is used to calculate the effective solution of (5). The effective solution called the PSP is denoted by "*ti*. Substituting the numerical PSP into (4) yields:

$$\Psi\_i(f\_\tau, f\_l; \tilde{t}\_i) = -2\pi (f\_\ell + f\_\tau) \tau\_i(\tilde{t}\_i) - 2\pi f\_l \tilde{t}\_i \tag{7}$$

Examining (7), we see that the numerical PTRS avoids approximations. Besides, the PTRS is a function of the instantaneous frequency *fτ*, Doppler frequency *ft* and range *r*. The space variance makes the development of fast imaging algorithms a challenge.

Based on the series expansion, the PTRS is decomposed into the azimuth modulation and coupling term. Using the coupling term, the decoupling operation between the range and azimuth dimensions is first carried out, and hence the imaging process is decomposed into two separate filtering processes in the range and azimuth dimensions. However, it is hard to obtain both terms using series expansion, because (7) is not an analytic expression.

#### *3.2. Azimuth Modulation*

After the decoupling operation, the azimuth compression is usually performed in the range Doppler domain. It is easily concluded that the filtering function related to the azimuth compression is only a function of the range *r* and Doppler frequency *ft* [8,9]. In other words, the azimuth modulation is independent of the instantaneous frequency *fτ* [8,9]. If the azimuth modulation were obtained, the phase difference between the PTRS and azimuth modulation denotes the coupling term. In other words, the azimuth modulation should be first derived. For conventional methods, the azimuth modulation and coupling term are simultaneously obtained based on the series approximation of the PTRS with respect to the instantaneous frequency. Inspecting (7), it is impossible to perform the series approximation, because the PTRS in this paper does not possess the explicit expression. To solve this problem, the numerical evaluation method is still used to calculate the azimuth modulation. Since the azimuth modulation is independent of the instantaneous frequency [8,9], we obtain the azimuth modulation by setting *f<sup>τ</sup>* = 0 in (7). This is given by:

$$\varphi\_{\rm ac\\_j}(f\_l; r) = \Psi\_{\rm i}(f\_\tau, 0; \hat{t}\_{\hat{l}}) = -2\pi f\_c \pi\_{\hat{l}}(\hat{t}\_{\hat{l}}) - 2\pi f\_l \hat{t}\_{\hat{l}} \tag{8}$$

In (8), <sup>ˆ</sup>*ti* ∈ [−*Ts*/2, *Ts*/2] represents the PSP used by the azimuth modulation. At this point, we call it the azimuth PSP. It is independent of the instantaneous frequency [8,9]. Although we present the azimuth modulation in (8), the azimuth PSP is not expressed explicitly. Considering the fact that the azimuth modulation is independent of the instantaneous frequency, we turn our attention to the deduction of the azimuth PSP. Setting *f<sup>τ</sup>* = 0 in (5) yields:

$$\frac{\partial \pi\_i(\hat{t}\_i)}{\partial t} + \frac{f\_l}{f\_c} = 0 \tag{9}$$

Based on the numerical method, the azimuth PSP ˆ*ti* is calculated. Substituting the azimuth PSP into (8), we obtain the numerical expression of the azimuth modulation.

## *3.3. Coupling Term*

Because of the relative motion between the sonar and target, the distance between them changes with time; hence, the time delay changes correspondingly. The effect is that the received echo from the same target at different azimuth sample times will distribute at different bins along the range direction. This phenomenon is called the range cell migration (RCM), which completely describes the coupling between the range and azimuth dimensions. Due to the coupling, the imaging process cannot be simply decomposed into two separate filtering processes in the range and azimuth dimensions. The direct processing scheme is to cancel the coupling before the azimuth matched filtering.

With traditional methods, the coupling term is obtained by using the series expansion of the PTRS with respect to the instantaneous frequency. Since the PTRS shown as (7) does not own an explicit expression, the series expansion method cannot be exploited. In practice, the PTRS consists of the coupling term and azimuth modulation. Therefore, the difference between the PTRS and azimuth modulation denotes the coupling term between the range and azimuth dimensions. It is expressed as:

$$
\varrho\_i (f\_{\overline{\tau}, f} f\_l; r) = \Psi\_i(f\_{\overline{\tau}, f} f\_l; r) - \varrho\_{ac\\_i}(f\_l; r) \tag{10}
$$

Until now, the PTRS, coupling term and azimuth modulation are all deduced. The azimuth modulation is a wideband signal. After decoupling, the matched filtering is expected to perform the focusing in the azimuth dimension.
