# A Cooperative SAR-Communication System using Continuous Phase Modulation Codes and Mismatched Filters

Maria-Elisavet Chatzitheodoridi, Abigael Taylor, Olivier Rabaste, Hélène Oriot

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## ► To cite this version:

Maria-Elisavet Chatzitheodoridi, Abigael Taylor, Olivier Rabaste, Hélène Oriot. A Cooperative SAR-Communication System using Continuous Phase Modulation Codes and Mismatched Filters. IEEE Transactions on Geoscience and Remote Sensing, 2022, 51, pp.5201314. 10.1109/TGRS.2022.3233198. [hal-03972438](https://hal.science/hal-03972438)

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# A Cooperative SAR-Communication System using Continuous Phase Modulation Codes and Mismatched Filters

Maria-Elisavet Chatzitheodoridi, Abigael Taylor, Olivier Rabaste, and Hélène Oriot

*Abstract*—The electromagnetic congestion due to the continuous growth of spectral demand has been skyrocketing for the past years. Joint radar-communication systems are thus attracting attention as they can alleviate the spectrum occupancy by using the same bandwidth to perform both applications. In this context, a cooperative radar-communication system, which is a specific category that uses communication codes to both transmit information and perform radar missions, can be considered. Continuous phase modulated (CPM) codes are considered in this article in order to generate high resolution radar images from an airborne radar. Since mitigating the sidelobe level energy is essential for good image quality, we resort here to optimized mismatched filters. A fast algorithm is proposed to minimize the computational time of these filters. Simulated data are generated, as well as re-synthesized SAR images, reconstructed from real chirp-based data, using CPM codes and mismatched filters. Their performance is evaluated using different comparison tools, and shows that the use of mismatched filtering and different messages embedded in the phase of each transmitted code, provides enhanced image quality.

*Index Terms*—Joint SAR-communication system, Synthetic Aperture Radar (SAR), Communications, Mismatched Filter, Continuous Phase Modulation, Integrated Sidelobe Level, Waveform Diversity.

## I. INTRODUCTION

THE radio-frequency electromagnetic spectrum has become gradually congested due to the exponential need of additional bandwidth in numerous communication applications. Concurrently, it is necessary to alleviate this limited resource in order to avoid degraded performance in different domains [1], and in particular in radar applications. Although such an issue was firstly investigated for military purposes, it appears also in various civilian domains such as autonomous vehicles, remote sensing, medical devices [2]. Combining two different applications in a joint radar-communication system can be an effective solution to deal with the spectral congestion problem.

Three main different categories of joint radar-communication systems are presented in the literature: co-existence, cooperation and co-design [3], [4]. Co-existence refers to the case when the radio-frequency applications are functioning autonomously and the secondary system is considered as an interference that needs to be mitigated. For example, LTE wireless

communication and air traffic control radar systems can co-exist through spectrum sharing [5]. Cooperation occurs when there is a mutual operation between the two systems, for example, using different signals for each application but optimizing their combined performance [6], using communication signals to perform radar applications [7], or vice versa [8]. Co-design consists in designing dual-function waveforms [9], that perform simultaneously both applications. Examples of co-design joint radar systems are for instance MIMO-OFDM systems [10], or combinations of radar-communication signals such as linear frequency modulated waveforms with phase modulation [11].

In this paper, a cooperative joint radar-communication system is introduced, the aim of which is to transmit information while performing Synthetic Aperture Radar (SAR) imaging. SAR consists in forming two-dimensional high resolution images of an area, using an airborne or spaceborne platform [12], [13]. The specificity of SAR systems consists in creating a virtual radar antenna, by exploiting the plane motion during the pulse train transmission and reception, in order to obtain a better azimuth resolution on the output image. In such a framework, linear frequency modulated signals (chirps) are mainly used due to their favorable properties for radar, along with the matched filter to perform range compression.

Since chirp signals cannot transmit information, we propose in this article a cooperative framework where this waveform is replaced with communication codes. Continuous Phase Modulated (CPM) codes [8], [14] are considered here, for their constant amplitude and their well-contained spectrum, usually desired characteristics for radar waveforms. They should yet enable both good SAR image quality and sufficient data rate. Unfortunately, CPM signals provide high integrated sidelobe level (ISL) after matched filter compression, which leads to image quality deterioration. Therefore, in order to ensure good SAR image quality, we propose to use mismatched filtering [15]–[17] for range compression, optimized to minimize the energy of the sidelobes.

In order to maximize the data rate, the binary message embedded in the pulse should change for each transmitted signal in the pulse train, and thus the mismatched filter as well. In particular, in a SAR framework, as the number of transmitted pulses is high, the number of optimization problems to solve is also large, which raises a potential
computational problem. It is thus required to solve a large number of large-scaled problems, which unfortunately cannot be computed with classical tools such as Matlab Software CVX [18], in acceptable time limits. Therefore we use our proposed solution [19], which enables to obtain the same optimal ISL solution as CVX at a highly reduced computational cost, by directly solving the dual problem.

Finally, performance results are provided both from simulated and realistic re-synthesized SAR images. Different evaluation criteria are presented in order to compare the results. This work provides a deeper analysis of our work in [20] as it contributes to a better comprehension of the communication codes and their choice as well as a strong comparison on the different elements of the image, in order to provide a global understanding of the performed work.

This article is organized in seven sections. In section II, CPM codes are presented and studied for the creation of the joint SAR-communication system. Section III provides the definition of the convex mismatched filter optimization problem, along with a fast algorithm that allows to compute the filter through the dual space in a feasible time. In Section IV, the reconstruction algorithm used to generate the re-synthesized SAR images is explained in depth, and in Section V, different evaluation tools that will be used to compare these SAR images are introduced. Section VI provides the resulting images along with their evaluation criteria and all these results are compared to each other. Finally, section VII provides a conclusion of the paper.

### Notations

<table><tr><td><i>B</i></td><td>Bandwidth</td></tr><tr><td><i>C</i></td><td>Constellation</td></tr><tr><td><i>f</i><sub>c</sub></td><td>Carrier Frequency</td></tr><tr><td><i>f</i><sub>s</sub></td><td>Sampling Frequency</td></tr><tr><td><i>h</i></td><td>Modulation Index</td></tr><tr><td><i>N</i><sub>b</sub></td><td>Number of bits per pulse</td></tr><tr><td><i>N</i><sub>s</sub></td><td>Number of different CPFSK codes</td></tr><tr><td>PRF</td><td>Pulse Repetition Frequency</td></tr><tr><td><i>t</i></td><td>(Fast) Time Vector</td></tr><tr><td><i>T</i><sub>c</sub></td><td>Chip Duration</td></tr><tr><td><i>T</i><sub>p</sub></td><td>Pulse Duration</td></tr></table>

## II. CONTINUOUS PHASE MODULATED CODE PROPERTIES FOR RADAR-COMMUNICATION APPLICATION

In communication applications, diverse coding schemes can be used to transmit the desired information. It can be embedded in the amplitude, frequency and/or phase of the code. Some of the most used communication codes in Radar-Communication systems are Orthogonal Frequency-Division Multiplexing (OFDM) codes [4], [21], [22], Phase Shift-Keying (PSK) codes [21], [23], Phase Shift Keying codes embedded in Linear Frequency-Modulated pulse (LFM-PSK) [11], [24], [25] and Continuous Phase Modulated (CPM) codes [8], [14].

The latter ones will be the selected coding scheme for our SAR-Communication application. Their formulation and some of their properties, that justify this choice, are presented in the following.

<figure><img src="image_3.png" /><figcaption>Figure 1: Phase tree for a binary ($C = 2$) constellation of a CPFSK code.</figcaption></figure>

### A. Continuous Phase Modulation Properties

Unlike PSK modulations, CPMs provide continuous phase shaping that prevents brutal phase transitions from one symbol to the following one. The CPMs embedded symbols $I = [I_1, I_2, ...]$ belong to the set $\{\pm 1, \pm 3, ..., \pm (C - 1)\}$, where $C = 2^k$ is the size of the constellation, with $k$ the number of bits per symbol.

The phase of a CPM signal is then expressed as [14]:

$$ \phi(t) = 2\pi \sum_{i \le n} I_i h_i \int_0^t g(\tau - iT_c) d\tau \\ = 2\pi \sum_{i \le n} I_i h_i u(t - iT_c), \quad nT_c < t \le (n + 1)T_c, $$

where $t$ is the time variable, $h_i$ is the modulation index in $]0, 1[$ for the $i^{th}$ sample, $I_i$ is the $i^{th}$ information symbol, $T_c$ is the chip time duration and $u(t)$ is a phase smoothing response function,

$$ u(t) = \int_{0}^{t} g(\tau) d\tau. $$

where $g$ is a frequency response function [26].

In this paper, a special category of CPM is considered, called continuous-phase frequency-shift keying (CPFSK) [14]. It is defined by a constant modulation index $h$, such that $h = h_i, \forall i$ and a rectangular smoothing function $g$. By extension the function $u$ is equal to 0 for $t < 0$, equal to $\frac{t}{2T_c}$ when $t \in [0, T_c[$ and equal to 0.5 for $t \ge T_c$.

Finally, the CPFSK signal can be formulated as $s(t) = e^{j\phi(t)}$, with the phase $\phi(t)$ expressed as [14]

$$ \phi(t) = \pi h \sum_{i<n} I_i + \frac{\pi h}{T_c} (t - nT_c) I_n, \quad nT_c < t \le (n+1)T_c. \quad (1) $$

Figure 1 illustrates the possible phase transitions of a CPFSK code for a 2-ary constellation. The final phase values are dependent on the current symbol and the modulation index $h$ as it provides the proportionality of the phase shift [26].
Note that it is usually difficult to express the bandwidth for *CPM* codes, but fortunately the Power Spectral Density (*PSD*) of *CPFSK* codes can be expressed analytically [26]. Figure 2 illustrates the analytical *PSD* of these codes for different constellations. We can remark that for high constellations like *C* = 8 and *C* = 16, the *PSD* is relatively flat and well-contained. Hence, the choice of high constellation is favorable for the spectrum from a radar point of view.

<figure><img src="image_4.png" /><figcaption>Figure 2: Analytically computed power spectral density of the *CPFSK* codes for different constellation sizes.</figcaption></figure>

## B. *CPFSK versus Other Communications Codes for a SAR-Communication System*

In this section, we will justify the choice of *CPFSK* codes for a *SAR-Communication system*. In our framework, two main properties are desired, constant amplitude and spectral containment.

A signal with a constant envelope enables the amplifier to function in a saturated mode, preventing non-linear distortions that can lead to a potentially strong deformation of the transmitted signal. Thus, by considering a constant amplitude, we can limit this effect, as well as provide an increased power efficiency as we use the full amplitude during the pulse duration [27]. Note that among the signals mentioned in the introduction of this section, all have constant amplitude, except for the *OFDM* codes, the amplitude modulation of which creates a high peak-to-average power ratio (*PAPR*) [28], which can degrade the hardware performance [4]. For this reason, even though they have attracted the *SAR* experts' attention for their potentially low range compression sidelobes [29], [30], they are not considered in the following comparison.

In order to limit the out-of-band spectral energy, we thus seek a waveform presenting a spectrum as close to a rectangular shape as possible. Such a spectrum, additionally to minimizing the out-of-band energy, prevents substantial distortions due to band-pass filtering at the transmission end that could cause amplitude modulation. Thus, we focus on the comparison between different spectra. Table I provides the parameters and characteristics

used to generate the different considered waveforms. These parameters were set so as to ensure the same bandwidth and pulse duration for all the codes. Here the binary constellation was considered for the *PSK* and *LFM-PSK* codes. The variable values among the parameters are thus the constellation, the number of embedded bits and the chip duration.

<table><caption>Table I: Parameters used for the simulations for the different waveforms.</caption><thead><tr><th>Parameter</th><th>Value for CPFSK codes</th><th>Value for other codes</th></tr></thead><tbody><tr><td><i>B</i></td><td>352 MHz</td><td></td></tr><tr><td><i>f</i><sub>s</sub></td><td>4B</td><td></td></tr><tr><td><i>T</i><sub>p</sub></td><td>1e-06 s</td><td></td></tr><tr><td><i>h</i></td><td>0.5</td><td>-</td></tr><tr><td><i>N</i><sub>b</sub></td><td>450 bits</td><td>529 bits</td></tr><tr><td><i>T</i><sub>c</sub></td><td>1e-8 s</td><td>2.8e-9 s</td></tr><tr><td><i>C</i></td><td>8</td><td>2</td></tr></tbody></table>

It can be observed in figure 3 that the *CPFSK* spectrum is better contained in the band than the other constant amplitude pulses.

As a further comparison, the out-of-band (*P*<sub>ob</sub>) power is calculated. It is expressed as [31],

$$P_{ob}(f_{ob}) = \frac{\int_{|\nu| > f_{ob}} |S(\nu)|^2 d\nu}{\int_{-\infty}^{\infty} |S(\nu)|^2 d\nu}, \quad f_{ob} \ge 0,$$

where *S* is the signal spectrum and *f*<sub>ob</sub> is the positive frequency limit after which we consider that the spectral energy is out-of-band.

Figure 3 also shows an out-of-band power of *P*<sub>ob</sub> = 0.01, which means that 99% of the power is inside *f*<sub>ob</sub>. The spectrum of each signal is plotted, and the vertical lines of the same color specify the associated bounds for the out-of-band power. Table II displays these bound values, and ideally, this power should not exceed the bandwidth value (B/2 = 176MHz as the spectrum is centered) in order to avoid losing spectral energy. The spectrum of the chirp pulse is also plotted to give a reference of the out-of-band power. We can observe that the chirp spectrum is very well contained. For the remaining constant amplitude signals, *CPFSK* codes have the least loss followed by the *BPSK* and then the *LFM-PSK* codes. Especially, it can be noticed from table II, that the *f*<sub>ob</sub> associated to the *BPSK* and *LFM-PSK* codes have exceeded the bandwidth value.

Signals exhibiting a high out-of-band power could be filtered to comply with an imposed spectral containment. But cutting the spectrum at given frequencies can lead to an energy loss after matched filtering with the input signal. To illustrate this, two band-pass filters are considered, one at 1.2B which removes much of the out-of-band energy, and a second one at 2B. Table III presents the energy loss after range compression for the 2 cuts. We can observe that for the 1.2B cut and the *LFM-PSK*, a loss of 2.5 dB on the peak is unacceptable. As expected, the best results
<figure><img src="image_5.png" /><figcaption>Figure 3: Spectra and 99% Power Bandwidth for different codes. The 99% Power Bandwidth is delimited by the vertical lines.</figcaption></figure>

<table><caption>Table II: The frequency bounds of the out-of-band power for a 99% power of the spectrum. Ideally these bounds should not exceed the bandwidth value ($\frac{B}{2}$ = 176MHz as the spectrum is centered) so that there is no spectral loss. These bounds are also illustrated in Figure 3.</caption><thead><tr><th>Waveform</th><th>Frequency Bound f<sub>ob</sub> (MHz)</th></tr></thead><tbody><tr><td>BPSK</td><td>251.9</td></tr><tr><td>LFM-PSK</td><td>332.9</td></tr><tr><td>CPFSK</td><td>168.2</td></tr><tr><td>Chirp</td><td>159.2</td></tr></tbody></table>

are provided when using the CPFSK signals, as in the worst case scenario there is a loss of 0.2 dB. It appears

<table><caption>Table III: Energy loss of the mainlobe after matched filter (MF) with the input non-filtered signal with a spectral cut at 1.2B and 2B.</caption><thead><tr><th>Signals</th><th>MF (cut at 1.2B)</th><th>MF (cut at 2B)</th></tr></thead><tbody><tr><td>BPSK</td><td>-1.35 dB</td><td>-0.68 dB</td></tr><tr><td>LFM-PSK</td><td>-2.52 dB</td><td>-0.77 dB</td></tr><tr><td>CPFSK</td><td>-0.18 dB</td><td>-0.01 dB</td></tr></tbody></table>

from this study that the CPFSK codes are more suitable for our SAR-Communication system compared to the PSK and LFM-PSK codes. In the following, we will turn our attention to the reception processing.

### III. MISMATCHED FILTERING RANGE COMPRESSION

In radar applications, range compression at the reception is performed to increase the signal-to-noise ratio (SNR) and improve the range resolution. The most common compression technique is the matched filter that maximizes the SNR. However, when applied on CPFSK signals, such a filter produces high sidelobes, as observed in Figure 4, that can degrade the performance of the radar system, and in our case, the quality of the generated

SAR image. Thereby in this section, a mismatched filter formulation for range compression that minimizes the integrated sidelobe level energy is provided. Note that this formulation is general and it can be applied to any sampled waveform.

<figure><img src="image_6.png" /><figcaption>Figure 4: Auto-correlation of a CPFSK code and a chirp pulse for the same bandwidth and pulse duration.</figcaption></figure>

#### A. Mismatched Filter Formulation

Let $s$ be a $N$-length sampled CPFSK code expressed as

$$s = [s_1 \ s_2 \ \dots \ s_N]^T, \qquad (2)$$

where $s$ is the transpose operator.

The received signal is filtered by a mismatched filter (MMF) in order to constrain the integrated sidelobe level energy. The length $K$ of the mismatched filter $q \in \mathbb{C}^K$ can be equal to or greater than the length of the signal ($K \ge N$). The output signal can be expressed as [15]

$$y = \Lambda_K^*(s) q, \qquad (3)$$

where

$$q = [q_1 \ q_2 \ \dots \ q_K]^T,$$

is the mismatched filter, and where $\Lambda_K$ is the correlation matrix of $s$ of size $(K + N - 1) \times K$, defined by

$$\Lambda_K(s) = \begin{bmatrix} s_N & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ s_1 & \cdots & s_N & \cdots \\ 0 & \cdots & \vdots & 0 \\ \vdots & \ddots & s_1 & \ddots \\ \vdots & \ddots & \vdots & s_N \\ 0 & \cdots & 0 & s_1 \end{bmatrix},$$

and $A^*$ denotes the complex conjugate of matrix $A$.

Setting $q = s$ in (3) enables to retrieve the matched filter that maximizes the SNR at the cost of higher sidelobes. Other types of filters $q$ can be defined through
an optimization problem, the cost function of which is characterized by a given desired criterion. Most papers consider either the Peak-to-Sidelobe Ratio criterion or the Integrate Sidelobe Ratio criterion [15]–[17], [32], [33].

In a SAR framework [12], a low sidelobe level after compression is essential to avoid possible image quality degradation due to the energy summation of numerous strong scatterers. Thus, we consider the Integrated Sidelobe Level (ISL) criterion. It is defined as the energy of all sidelobes [34], and it can be expressed as

$$ISL(\mathbf{y}) = \sum_{k=1}^{K+N-1} |y_k|^2 - \sum_{k \in \Omega_{ML}} |y_k|^2, \quad (4)$$

where the set $\Omega_{ML}$ corresponds to the mainlobe indices.

Hence, the mismatched filter problem corresponds to the following $L_2$-norm minimization problem [15]:

$$\begin{array}{ll}
\min_{\mathbf{q}} & \|\mathbf{F}\mathbf{y}\|_2^2, \\
\text{s.t.} & \mathbf{s}^H \mathbf{q} = \mathbf{s}^H \mathbf{s}, \\
& \mathbf{q}^H \mathbf{q} \le 10^{\frac{\beta}{10}} \mathbf{s}^H \mathbf{s},
\end{array} \quad (5)$$

where $\mathbf{F}$ is a diagonal matrix of ones except for zero values for indices in $\Omega_{ML}$, and $\beta$ is a positive constant in dB that expresses the maximum tolerated loss-in-processing gain with respect to the matched filter [15]. The cost function can be also expressed as $\|\mathbf{F}\mathbf{y}\|_2^2 = \mathbf{q}^H \mathbf{M} \mathbf{q}$, where $\mathbf{M} = \mathbf{\Lambda}_K^T(s) \mathbf{F} \mathbf{\Lambda}_K^*(s)$ is a symmetric semi-definite positive matrix.

As mentioned in [15], the first constraint allows to discard the trivial null solution. The second constraint enables to control the loss-in-processing gain. Both constraints are convex [15]. Since the objective function is also convex, the optimization problem (5) is convex with convex constraints.

## B. Fast Computation of the Mismatched Filter

The solution of the convex optimization problem (5) can be computed using a generic convex solver, such as the CVX: Matlab Software for Disciplined Convex Programming [18], as presented in [15], [16], [35], [36]. However, such a solver is unsuitable for large-scale problems due to high computational costs, as mentioned by its creators [18]. In the considered joint SAR-communications framework, we must solve many large-scale convex optimization problems, one per transmitted pulse, so the use of CVX tool becomes infeasible. This led us to find a faster algorithm to solve the optimization problem (5) [19].

The solution we propose consists in solving the dual problem. We show that the optimization of the ISL problem can be calculated analytically in the dual space so that the solution can be obtained via a simple monodimensional gradient descent algorithm. The main steps are summarized below, but more details can be found in [19]:

* First, the Lagrangian function is calculated, the gradient of which provides the expression of the optimum mismatched filter in the dual space, defined as,

$$\mathbf{q}_{\text{opt}} = -\frac{1}{2}\nu(\mathbf{M} + \lambda\mathbf{I})^{-1}\mathbf{s}, \quad (6)$$

where $\nu$ is the dual variable for the equality constraint, and $\lambda$ is the dual variable for the inequality constraint.

* In order to compute $\mathbf{q}_{\text{opt}}$, the optimal values of the two Lagrangian variables $(\lambda_{\text{opt}}, \nu_{\text{opt}})$ need to be found. For this reason, the dual function of the primal problem is maximized over the two variables, defined as,

$$g(\lambda, \nu) = -\lambda\alpha \mathbf{s}^H \mathbf{s} - \nu \mathbf{s}^H \mathbf{s} - \frac{1}{4} \nu^2 \mathbf{s}^H (\mathbf{M} + \lambda \mathbf{I})^{-1} \mathbf{s}.$$

* The $\nu_{\text{opt}}$ value is calculated analytically (see $\nu_{\text{opt}}$ in Algorithm 1) and the $\lambda_{\text{opt}}$ value is computed numerically via a mono-dimensional gradient descent algorithm. Thus, the optimal filter can now be expressed through (6).

* Note that it can be easily proven that the optimal solution in the dual space coincides with the primal solution.

### Algorithm 1: Mismatched Filter Optimization using Gradient Descent

**Input:** Loss-in-progressing gain $\alpha$, initialization $\lambda_1$, max iterations $max_i$, precision $\epsilon$,

**Output:** Optimal pair $(\lambda_{\text{opt}}, \nu_{\text{opt}})$, optimal mismatched filter $q_{\text{opt}}$;

$\mathbf{s} \leftarrow$ code generation;

$\mathbf{M} \leftarrow \mathbf{\Lambda}_K^T(s) \mathbf{F} \mathbf{\Lambda}_K(s);$

$[\mathbf{P}, \mathbf{D}] \leftarrow \text{EIG}(\mathbf{M});$

**for** $i \in [1, max_i]$ **do**

$h_i \leftarrow \mathbf{s}^H \mathbf{P}^H \text{diag} \left( \frac{1}{\text{diag}(\mathbf{D}) + \lambda_i} \right) \mathbf{P} \mathbf{s};$

$h'_i \leftarrow \mathbf{s}^H \mathbf{P}^H \text{diag} \left( -\frac{1}{(\text{diag}(\mathbf{D}) + \lambda_i)^2} \right) \mathbf{P} \mathbf{s};$

$h''_i \leftarrow \mathbf{s}^H \mathbf{P}^H \text{diag} \left( \frac{2}{(\text{diag}(\mathbf{D}) + \lambda_i)^3} \right) \mathbf{P} \mathbf{s};$

$g'_i \leftarrow -\alpha \mathbf{s}^H \mathbf{s} - (\mathbf{s}^H \mathbf{s})^2 \frac{h'_i}{h_i^2};$

$g''_i \leftarrow -(\mathbf{s}^H \mathbf{s})^2 \cdot \frac{h''_i h_i - 2h_i^2}{h_i^3};$

$\lambda_{i+1} \leftarrow \lambda_i - \frac{g'_i}{g''_i};$

**if** $|g'_i| < \epsilon$ **then**

break;

end

end

$\nu_{\text{opt}} \leftarrow -\frac{2\mathbf{s}^H \mathbf{s}}{\mathbf{s}^H(\mathbf{M} + \lambda_{\text{opt}}\mathbf{I})^{-1}\mathbf{s}};$

$\mathbf{q}_{\text{opt}} \leftarrow -\frac{1}{2}\nu_{\text{opt}}(\mathbf{M} + \lambda_{\text{opt}}\mathbf{I})^{-1}\mathbf{s}.$

This procedure is summarized in Algorithm 1. It must be applied to every different transmitted pulse.
IV. SAR IMAGE RECONSTRUCTION ALGORITHM

In the following, we will turn our attention to the formation of SAR images. In the absence of real CPFSK data acquisitions, in this section, we create re-synthesized SAR images, in order to provide as realistic results as possible in comparison to real data SAR images. More precisely, the ground reflectivity of a real-data SAR image generated with chirp signals is used to re-create synthesized images from communication codes. In the following, the fundamental SAR definition that expresses mathematically the image creation is given, as well as the details of the reconstruction algorithm that generates the re-synthesized SAR images.

A. SAR Framework

After signal reception, the noiseless raw data can be expressed as a convolution between the transmitted waveform and the ground reflectivity [12]:

$$r_u(t) = g_{r_u}(t) * s_u(t), \qquad (7)$$

where $r_u(t)$ is the received signal, $g_{r_u}(t)$ is the ground reflectivity, and $t$ denotes the fast time, related to the intra-pulse sampling time, and $u$ the slow time, related to the sampling time of the pulse train. Note that the ground reflectivity at fast time $t$ and slow time $u$ is the sum of backscatterers located at $R = \frac{ct}{2}$ from the sensor. The received data undergo the SAR processing consisting of a range compression and an azimuth compression to provide a focused image of the scene. Range compression determines the distances between the platform and the targets, and it is usually performed via the matched filter. Azimuth compression compresses the signal in the azimuth direction.

B. Details on the Image Reconstruction Algorithm

In the absence of an appropriate airborne CPFSK system, a possible way to evaluate the proposed methods would be to simulate the electromagnetic response of a scene, which is not an easy task. We propose in place to retrieve a realistic-looking ground reflectivity using existing raw data from chirp acquisitions. From these back-scattered real-data, the SAR image reconstruction algorithm generates a re-synthesized SAR image using CPFSK codes. The procedure is illustrated in Figure 5.

The raw data have been acquired with chirps, and the ground reflectivity $g_{r_u}$ can be recovered through (7). From an implementation point of view, it is extracted by deconvolving the raw data with the impulse response of the chirp signal, which is performed through the following computation in the Fourier domain,

$$\hat{G}_{r_u}(f) = \begin{cases} \frac{R_u(f)}{S_u(f)}, & f_0 - \frac{B_{\text{chirp}}}{2} < f < f_0 + \frac{B_{\text{chirp}}}{2} \\ 0, & \text{otherwise} \end{cases}$$

where $\hat{G}_{r_u}$ is the estimated ground reflectively, $R_u$ is the spectrum of the received data at slow time $u$, $S_u$ is the

chirp spectrum at slow time $u$, $f_0$ is the carrier frequency, $B_{\text{chirp}}$ is the chirp bandwidth, and $f$ is the frequency variable in range direction.

Once $\hat{G}_{r_u}$ is obtained, it is multiplied in the Fourier domain by the impulse response spectrum of the CPFSK codes along with their matched or mismatched filters in order to create the compressed CPFSK data, as illustrated in Figure 5. Finally, the azimuth compression is applied, creating the re-synthesized SAR image. This procedure is repeated on different signals and compression techniques in order to study and compare their performance.

After providing the framework and the processing, comparison techniques are presented in order to compare the resulting images.

V. COMPARISON TOOLS FOR THE RE-SYNTHESIZED SAR IMAGE EVALUATION

This section is dedicated to the definition of the four main evaluation tools used to study the performance. The Integrated Sidelobe Level Ratio (ISLR) and the Peak-to-Sidelobe Level Ratio (PSLR) are two classical radar criteria that provide two different insights on the sidelobe level. Pearson correlation coefficient is also considered, and it analyzes to which extent two SAR images are similar. Finally, the average noise level over an area of the image that has low intensity is computed, and it provides the noise sensitivity of the system for different waveforms or processing.

A. Integrated Sidelobe Level Ratio (ISLR)

The Integrated Sidelobe Ratio is defined as the ratio of the energy of all the sidelobes to the energy of the main lobe. It provides an indication of the minimum level under which a weak target may be undetected in the presence of uniformly distributed scatterers of the same level over the whole signal range [34], [37].

The ISLR is expressed as

$$ISLR(y) = 10 \log_{10} \frac{\int_{-\infty}^{a} |y(\tau)|^2 d\tau + \int_{b}^{\infty} |y(\tau)|^2 d\tau}{\int_{a}^{b} |y(\tau)|^2 d\tau}, \quad (8)$$

where $y$ is the compressed signal in range or azimuth direction and the mainlobe is contained in the interval $[a, b]$.

In its discrete form, it can be expressed as the equation (4) divided by the energy of the mainlobe.

B. Peak-to-Sidelobe Level Ratio (PSLR)

The Peak to Sidelobe Ratio is defined as the ratio between the energy of the highest sidelobe to the energy of the mainlobe. It is helpful for determining whether low amplitude targets will be detected or not when they are close to stronger ones [34], [37].

The PSLR is expressed as

$$PSLR = 10 \log_{10} \left| \frac{I_s}{I_m} \right|, \qquad (9)$$
<figure><img src="image_7.png" /><figcaption>Figure 5: SAR Image Reconstitution Algorithm</figcaption></figure>

where $I_s$ is the amplitude of the strongest sidelobe and $I_m$ is the peak intensity of the main lobe.

### C. Pearson Correlation Coefficient

The Pearson correlation coefficients, also called coherence coefficients, of two signals or images provide a degree of similarity of the inputs.

Let us consider two images A, B and d the length of the image patch of interest $D$, then the coherence coefficients between A and B are defined as following [38], [39],

$$C_{i,j} = \frac{\sum_{ij \in D} (A_{ij} - \mathbb{E}[A_D]) (B_{ij} - \mathbb{E}[B_D])^*}{\sqrt{\sum_{ij \in D} ||A_{ij} - \mathbb{E}[A_D]||^2 ||B_{ij} - \mathbb{E}[B_D]||^2}}$$

where $i, j$ the current index position and $D = \{i - d : i + d, j - d : j + d\}$, the number of indices considered for the computation of a given coefficient.

Note that we consider the amplitude of the correlation coefficients. More precisely, when the output values are converging to 1, the inputs have a strong coherence, which means that the possible variations on the pixel intensity of the two images are small so the images are similar, and when the output values are converging to 0, then the images tend to be very different [40]. In our framework, we want to compare the pixel energy of the generated images with respect to the configuration of a chirp compressed with matched filtering in order to provide the best possible combination of signal transmission and processing procedure. The idea is that an image close to the reference one will be advantageous for SAR specialists, as it will preserve significant information such as shape, contrast, etc.

### D. Average Value of Low-Energy Pixels of a SAR Image

The criterion of this subsection provides the mean value of a low energy zone of a SAR image. This value gives an insight into the average thermal noise level or equivalently to the Noise Equivalent Sigma Zero ($NE\sigma_0$) [41], as in this area the presence of signals is very low.

Let A a sub-image of the SAR image that has low intensity pixels, then we define the normalized average value over a low intensity area $N_0$ as

$$N_0 = 10 \log_{10} \mathbb{E}[A]^2.$$

When using different signals and compression techniques, the $N_0$ gives an insight into the energy difference of the system between the noise floor and the potential artifacts that are created.

## VI. EVALUATION OF GENERATED IMAGES

This section presents the performance study on simulated and re-synthesized data. The first subsection provides the characteristics of the real data SAR image used for this performance study, along with some characteristics for the CPFSK code generation. The following two subsections illustrate the impulse response of the CPFSK codes (presented in subsection II-A) while using matched or mismatched filter (provided in subsection III-A) as well as their associated re-synthesized SAR images, created using the reconstruction algorithm defined in subsection IV-B. All these images are compared to each other using the evaluation criteria presented in section V.

### A. Real Data and CPFSK Characteristics

The SAR image reconstruction algorithm takes as input the raw data of a real-data high-resolution SAR image that is acquired from SETHI [42], an airborne radar system owned by ONERA, the French Aerospace Lab. They are obtained in X-band from a pulse train of chirp signals at a PRF of 5 kHz as presented in Table IV, which also provides the main parameters and characteristics of the real system, along with the parameters of the CPFSK codes generation. Note that the bandwidth of the real SAR image is larger than the bandwidth of the generated re-synthesized SAR images in order to avoid losing information from the spectrum.

In the following, four different configurations are considered:

* One identical CPFSK pulse is transmitted for all the acquisitions, and it is compressed with the matched filter (MF),

* One identical CPFSK pulse is transmitted for all the acquisitions, and it is compressed with the mismatched filter (MMF),

* A different CPFSK code is transmitted at each pulse, and each one of them is compressed with the MF,

* A different CPFSK code is transmitted at each pulse, and each one of them is compressed with the MMF.

Based on these CPFSK configurations, it is possible to calculate the resulting data rate, defined by

$$D = \log_2(C) \frac{T_p N_s}{T_c N_{\text{pulses}}} \text{PRF}, \quad (10)$$

where $N_{\text{pulses}}$ is the total number of transmitted pulses, whereas $N_s$ expresses the number of non-identical transmitted CPFSK codes.

When the same CPFSK signal is transmitted over the whole pulse train, then $N_s = 1$ and the resulting data
<figure><img src="image_8.png" /><figcaption>(a) Chirp with MF</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) One *CPFSK* with *MF*</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) One *CPFSK* with *MMF*</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(d) Different *CPFSK* with *MF*</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(e) Different *CPFSK* with *MMF*</figcaption></figure>

Figure 6: Impulse responses for the chirp and the *CPFSK* codes for one or different transmitted codes with their associated matched or mismatched filters. The abbreviation *MF* corresponds to matched filter and the *MMF* to mismatched filter. The small window on the top right of each figure represents a zoom on the maximum point in order to be able to observe both close to the peak and farther sidelobe behaviour.

rate is 0.76kbit/s, calculated using the parameters of Table IV. On the contrary, when the *CPFSK* signals differ from pulse to pulse, then $N_s = 6522$ and the resulting data rate becomes 5Mbit/s (or 625kbytes/s).

## B. Evaluation of the Impulse Responses

Before creating the re-synthesized *SAR* images, we provide in this subsection the impulse response of the chirp and the *CPFSK* codes obtained for a single point reflector. In both cases, note that the algorithm presented

in subsection III-B is used to compute all the needed mismatched filters. The computation time gain is enormous, as the filters are computed 250 times faster than using the classical *CVX Matlab Toolbox*. More precisely, it took roughly 16 days to compute 6522 filters, instead of 11 years of computational time needed when using the *CVX* algorithm [19].

Figure 6 provides the impulse responses for a chirp and the four different *CPFSK* configurations. The first observation that can be made on the impulse responses is that the single *CPFSK* case presents range sidelobes
<table><caption>Table IV: Parameters and characteristics of the CPFSK codes and the real-data SAR images.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td colspan="2"><strong>CPFSK Characteristics</strong></td></tr><tr><td>Bandwidth <em>B</em></td><td>305 MHz</td></tr><tr><td>Pulse Duration <em>T</em><sub>p</sub></td><td>4.8e-06 s</td></tr><tr><td>Chip Duration <em>T</em><sub>c</sub></td><td>1.92e-08 s</td></tr><tr><td>Bit Number <em>N</em><sub>b</sub></td><td>1000</td></tr><tr><td>Modulation Index <em>h</em></td><td>0.4</td></tr><tr><td>Constellation <em>C</em></td><td>16</td></tr><tr><td>Number of pulses <em>N</em><sub>pulses</sub></td><td>6522</td></tr><tr><td colspan="2"><strong>Real Image Characteristics</strong></td></tr><tr><td>Band</td><td>X</td></tr><tr><td>Bandwidth <em>B</em></td><td>500 MHz</td></tr><tr><td>Azimuth Resolution</td><td>0.38 m</td></tr><tr><td>Platform velocity</td><td>124 m/s</td></tr><tr><td>PRF</td><td>5 kHz</td></tr><tr><td>Waveform</td><td>Chirp</td></tr></tbody></table>

<table><caption>Table V: PSLR and ISLR values in the range direction on the different impulse responses. ISLR<sub>small</sub> provides the ISLR values of the impulse responses when the energy is computed over a shortened window of the compression output that corresponds to the number of points considered in Table VI for the ISLR computation.</caption><thead><tr><th>Signals</th><th>PSLR (dB)</th><th>ISLR (dB)</th><th>ISLR<sub>small</sub> (dB)</th></tr></thead><tbody><tr><td>Chirp with MF</td><td>-13.2</td><td>-9.5</td><td>-9.6</td></tr><tr><td>One CPFSK with MF</td><td>-14.9</td><td>0.2</td><td>-6.8</td></tr><tr><td>Different CPFSK with MF</td><td>-14.8</td><td>-12.9</td><td>-12.9</td></tr><tr><td>One CPFSK with MMF</td><td>-15.8</td><td>-8.1</td><td>-13.4</td></tr><tr><td>Different CPFSK with MMF</td><td>-15.2</td><td>-14.0</td><td>-14.0</td></tr></tbody></table>

concentrated in the range direction. On the contrary, using different CPFSK signals leads to a spreading of the sidelobes along the azimuth axis due to the combination of the radar-communications incoherent messages embedded in the phase of the code. When using matched filtering, the sidelobe level energy is higher than the impulse responses compressed with the mismatched filters. Moreover, as seen in sub-figure 6c & 6e, mismatched filters that are longer than the signal length, which results also to longer sidelobes, allows to have a more diffused sidelobe level. The level around the mainlobe is lower, however, as it spreads on a larger area. This behavior is quantified in Table V, where the best ISLR value corresponds to the case of different CPFSK with MMF (sub-figure 6e), because of the sidelobe level optimization of the filter, as well as the energy spreading. In Table V, the highlighted values correspond to the best value of PSLR and ISLR. However, these values do not consider the spreading of the sidelobes outside the main range axis, which disadvantages the chirp. The last column of Table V provides the computation of the ISLR value over a smaller amount of points, compared to the middle column ISLR. This window corresponds to the number of points considered for the ISLR computations of Table VI that represent the evaluation criteria on the corner reflector of the re-synthesized SAR images. It is displayed for comparison reasons, as there is no clutter or noise on the impulse responses, contrary to the re-synthesized SAR images introduced in the following.

### C. Evaluation of the Re-synthesized SAR Images

By using the impulse responses previously introduced (Figure 6), together with the ground reflectivity of the real-data SAR image presented in subsection VI-A, we can finally create the re-synthesized SAR images from CPFSK communication codes. A large area of the SAR image is illustrated in figure 7 for the configurations presented previously. Sub-figure 7a illustrates the SAR image acquired with chirps, and it is used as a reference point for comparison. In the case when only one CPFSK is used (sub-figures 7b & 7c), we can observe high energy lines of the sidelobes of high-energy points along the range axis in both figures, especially the one created from matched filter. This spreading deteriorates the quality of the SAR images and can potentially mask details. On the contrary, when different CPFSK codes are transmitted (sub-figures 7d & 7e), we can remark that the sidelobe energy is spread along the 2D axis, because of an averaging of range sidelobes after azimuth compression, which prevents having these big energy lines observed before. Note that these SAR images are an attempt to create more realistic images, but they cannot entirely represent real data acquisitions.

In the following, four different sub-areas of the whole SAR image will be considered for comparison, illustrated in Figure 8. They are representative of different types of clutter (urban, natural) and include a corner reflector (sub-figure 8a), for which the results should be close to the impulse response ones. The building (sub-figure 8b) is a high energy area, so it will show how the energy is distributed with respect to the chirp one, and the lake (sub-figure 8c) is exactly the opposite configuration, as it is an area of low energy because the majority of the waves are not back-scattered to the antenna. Finally, the forest image (sub-figure 8d) is a medium energy area that will show on average the energy distribution of the generated images.

1) **PSLR and ISLR of the corner reflector:** Table VI presents PSLR and ISLR values for the corner reflector. As the pixels around the corner reflector contribute to its energy, its PSLR and ISLR values are generally higher than the ideal values of the impulse response, but they represent a more realistic framework. The ISLR of the corner reflector is computed on a given number of points inferior to the actual number over which the corner reflector's energy is spread. This choice is made because other scatterers can contribute to the sidelobe energy of the corner reflector and thus it could be no longer defined as the ISLR of the corner reflector only. Moreover, the last column of Table V displays the ISLR of the impulse response over this restricted set of points in order to be able to compare the sidelobe energy of the re-synthesized corner reflector with the impulse response. Both tables show that communication codes compressed with mismatched filter provide better values for the PSLR and the ISLR compared to the matched filter. For the ISLR, this remark also validates the criterion of the optimization problem (5), that minimizes the integrated sidelobe energy. The best
<figure><img src="image_13.png" /><figcaption>Figure 7: Selection of an area of the re-synthesized SAR images for the chirp and the CPFSK codes for one or different transmitted codes with their associated matched or mismatched filters. Note that all five sub-figures have the same intensity and orientation.</figcaption></figure>

<table><caption>Table VI: Range PSLR and ISLR for the corner reflector of every image.</caption><thead><tr><th>Signals</th><th>PSLR (dB)</th><th>ISLR<sub>small</sub> (dB)</th></tr></thead><tbody><tr><td>Chirp with MF</td><td>-11.5</td><td>-8.6</td></tr><tr><td>One CPFSK with MF</td><td>-12.9</td><td>-7.3</td></tr><tr><td>Different CPFSK with MF</td><td>-13.0</td><td>-10.6</td></tr><tr><td>One CPFSK with MMF</td><td>-13.7</td><td>-11.72</td></tr><tr><td>Different CPFSK with MMF</td><td>-13.2</td><td>-11.7</td></tr></tbody></table>

results are highlighted in Table VI. They correspond to the case when a single CPFSK code is transmitted for all pulses, with its associated mismatched filter, although the values for different CPFSK codes with their mismatched filters present very close values, especially for ISLR.

2) *Pearson Correlation Coefficient*: It is the next comparison criterion presented in subsection V-C. The image created with the chirp pulse is taken as a reference for computing this criterion. Results are presented in figures

9 to 12 for all the four areas of figure 8. Each sub-figure is created by taking the Pearson correlation coefficient on a small window of size 5x5 of the re-synthesized SAR images that are compared. Table VII represents the average value of each correlation figure over the sub-areas in order to provide a more comparative measure than the visualization. The highlighted values show the best results for the correlation on each compared area.

From these figures, it appears that the highest correlation with the reference image is obtained with the mismatched filter. The coherence is however varying according to the SNR of the area. The high-energy areas have a higher coherence since the noise has less impact on high-energy areas than in low-energy ones. A more detailed comparison is given in the following for each configuration.
<figure><img src="image_14.png" /><figcaption>Figure 8: Four different sub-areas of the SAR image. Note that the sub-images are rotated –90 degrees compared to the SAR images displayed in Figure 7.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 9: Correlation coefficients of the corner reflector for different signal configurations</figcaption></figure>

<table><caption>Table VII: Mean values of the correlation coefficients for different areas presented in Figure 8.</caption><thead><tr><th>Signals</th><th>Corner Reflector</th><th>Building</th><th>Lake</th><th>Forest</th></tr></thead><tbody><tr><td>One CPFSK with MF</td><td>0.7007</td><td>0.658</td><td>0.3682</td><td>0.655</td></tr><tr><td>Different CPFSK with MF</td><td>0.642</td><td>0.711</td><td>0.464</td><td>0.761</td></tr><tr><td>One CPFSK with MMF</td><td>0.908</td><td>0.8335</td><td>0.6035</td><td>0.882</td></tr><tr><td>Different CPFSK with MMF</td><td>0.8887</td><td>0.8745</td><td>0.682</td><td>0.9116</td></tr></tbody></table>

a) Corner Reflector: Figure 9 represents the correlation coefficients on the corner reflector area. In this configuration, the best result is obtained when the same code is transmitted at every acquisition (right up subfigure), as indicated in Table VII. This behavior can be explained, firstly because the mismatched filter minimizes the compression sidelobes and secondly due to the spreading created when multiple signals are transmitted, so that the energy is distributed in the 2D plane, creating dissimilarities with the chirp result. However, we can remark that in the case of multiple signals (right down subfigure) the sidelobes away from the mainlobe are lower, which could be explained by a sidelobe averaging after the azimuth compression. Note that the sidelobe behavior of the CPFSK codes near the mainlobe is not the same as the chirp one, because the first sidelobe of the chirp is around 2 dB higher than for the CPFSK codes.

b) Building: Figure 10 illustrates the coherence of the building image, which corresponds to an area of high

<figure><img src="image_16.png" /><figcaption>Figure 10: Correlation coefficients of the building for different signal configurations</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 11: Correlation coefficients of the lake for different signal configurations</figcaption></figure>

energy due to the materials and the geometry of the building. We can observe that the high-energy pixels that are located on the building, maintain their high energy in every configuration. The rest of the scene, using the matched filter (first column sub-figures), has low similarity compared to the chirp energy, and for the mismatched filter (second column sub-figures), the performance is close. The best similarity is reached when multiple codes are transmitted (right down sub-figure).

c) Lake: Figure 11 presents the correlation coefficient images of the lake between the chirp and the CPFSK codes with the different compression techniques. The lake is an area of low intensity as the water is a specular reflector [43]. In such an area, the main energy contribution is the neighboring pixel sidelobes and the thermal noise. This is the reason why in all of the correlation images, the similarity of the lake is low, as shown in Table VII
<figure><img src="image_18.png" /><figcaption>Figure 12: Correlation coefficients of the forest for different signal configurations</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 13: The red square area of the lake where the average value is considered. compared to other areas.</figcaption></figure>

d) *Forest*: Finally, figure 12 illustrates the correlation coefficient for the forest region. The vegetation area is a medium-intensity area. Same results as before can be observed, as the matched filter outputs provide a bad correlation coefficient when compared to the chirp image forest intensity. Between the mismatched filter outputs, still the configuration where we transmit a different pulse at each acquisition provides a better value, shown in Table VII.

3) *Average energy of the lake*: The last comparison criterion, defined in subsection V-D, is the average energy of a low-intensity area between the different codes and processing configurations. In our framework, it is performed on sampling points of the lake image, as shown in Figure 13. This criterion provides an insight into how the pixels with low energy that are close to noise level of the scene, are influenced from different codes and compression techniques. Table VIII provides the average value of the pixels of the red square in the figure 13. The chirp is considered a reference case and provides the best results among all the configurations. From the remaining cases, the configuration of different *CPFSK* codes with *MMF* presents the lowest energy, followed by the configuration

of one *CPFSK* code with *MMF*. We can observe that when using the matched filter, sidelobes from neighbouring pixels are polluting low energy points in the lake.

<table><caption>Table VIII: Average value of energy over an area of the lake.</caption><thead><tr><td>Signals</td><td>Average in dB</td></tr></thead><tbody><tr><td>Chirp with MF</td><td>-31.5</td></tr><tr><td>One CPFSK with MF</td><td>-15.3</td></tr><tr><td>Different CPFSK with MF</td><td>-17.3</td></tr><tr><td>One CPFSK with MMF</td><td>-24.7</td></tr><tr><td>Different CPFSK with MMF</td><td>-26.4</td></tr></tbody></table>

Overall, the configuration of different *CPFSK* codes with *MMF* provides good radar performance, for different level intensity areas: the *PSLR* and the *ISLR* values indicate to a relatively well-contrasted image.

## VII. CONCLUSION

Our paper proposes a joint *SAR*-communication system in order to simultaneously create radar images and transmit information. It uses continuous phase modulated communication codes, which are processed with mismatched filtering, that are optimized to provide the minimum *ISL*. The proposed codes and processing are used to generate re-synthesized *SAR* images. Different areas of the images are compared, to a reference chirp-based image, with high or low energy pixels, in order to provide a more complete evaluation framework. It is shown both on simulated impulse responses and on re-synthesized images, that when transmitting different codes per pulse and compressing them with their associated mismatched filters, the performance in terms of contrast and sidelobe energy is enhanced. Overall, this shows that transmitting information while maintaining a good *SAR* image quality is possible. A perspective on this work is to do the experimentation by transmitting the communications codes, in order to observe if the real-data results are close to the ones of the re-synthesized images.

## REFERENCES

[1] H. Griffiths, S. Blunt, L. Cohen, and L. Savy, “Challenge problems in spectrum engineering and waveform diversity,” in *2013 IEEE Radar Conference (RadarCon13)*, 2013, pp. 1–5.

[2] B. Paul, A. R. Chiriyath, and D. W. Bliss, “Survey of RF communications and sensing convergence research,” *IEEE Access*, vol. 5, pp. 252–270, 2017.

[3] Z. Feng, Z. Fang, Z. Wei, X. Chen, Z. Quan, and D. Ji, “Joint radar and communication: A survey,” *China Communications*, vol. 17, no. 1, pp. 1–27, 2020.

[4] D. Ma, N. Shlezinger, T. Huang, Y. Liu, and Y. C. Eldar, “Joint radar-communication strategies for autonomous vehicles: Combining two key automotive technologies,” *IEEE signal processing magazine*, vol. 37, no. 4, pp. 85–97, 2020.

[5] H. Wang, J. T. Johnson, and C. J. Baker, “Spectrum sharing between communications and atc radar systems,” *IET Radar, Sonar & Navigation*, vol. 11, no. 6, pp. 994–1001, 2017.

[6] B. K. Chalise, M. G. Amin, and B. Himed, “Performance tradeoff in a unified passive radar and communications system,” *IEEE SPL*, vol. 24, no. 9, pp. 1275–1279, 2017.

[7] Q. He, Z. Wang, J. Hu, and R. S. Blum, “Performance gains from cooperative MIMO radar and MIMO communication systems,” *IEEE SPL*, vol. 26, no. 1, pp. 194–198, 2018.
[8] C. Sahin, J. Jakabosky, P. M. McCormick, J. G. Metcalf, and S. D. Blunt, "A novel approach for embedding communication symbols into physical radar waveforms," in *2017 IEEE Radar Conference (RadarConf)*. IEEE, 2017, pp. 1498-1503.

[9] F. Liu, L. Zhou, C. Masouros, A. Li, W. Luo, and A. Petropulu, "Toward dual-functional radar-communication systems: Optimal waveform design," *IEEE Transactions on Signal Processing*, vol. 66, no. 16, pp. 4264-4279, 2018.

[10] Y. Liu, G. Liao, Z. Yang, and J. Xu, "Design of integrated radar and communication system based on MIMO-OFDM waveform," *Journal of Systems Engineering and Electronics*, vol. 28, no. 4, pp. 669-680, 2017.

[11] M. Bekar, C. J. Baker, E. Hoare, and M. Gashinova, "Realization of a joint MIMO radar and communication system using a PSK-LFM waveform," in *2020 IEEE Radar Conference (RadarConf20)*. IEEE, 2020, pp. 1-6.

[12] I. G. Cumming and F. H. Wong, *Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation* [*With CDROM](Artech House Remote Sensing Library). Artech House, 2005.

[13] B.-C. Wang, *Digital Signal Processing Techniques and Applications in Radar Image Processing*. Hoboken, NJ, USA: John Wiley & Sons, Inc., Aug. 2008.

[14] J. B. Anderson, T. Aulin, and C.-E. Sundberg, *Digital phase modulation*. Springer Science & Business Media, 2013.

[15] O. Rabaste and L. Savy, "Mismatched filter optimization for radar applications using quadratically constrained quadratic programs," *IEEE Transactions on Aerospace and Electronic Systems*, vol. 51, no. 4, pp. 3107-3122, 2015.

[16] A. De Maio, Y. Huang, M. Piezzo, S. Zhang, and A. Farina, "Design of radar receive filters optimized according to $L_p$-norm based criteria," *IEEE transactions on signal processing*, 2011.

[17] P. Stoica, J. Li, and M. Xue, "Transmit codes and receive filters for radar," *IEEE Signal Processing Magazine*, vol. 25, no. 6, pp. 94-109, 2008.

[18] M. Grant and S. Boyd, "CVX: Matlab Software for Disciplined Convex Programming, version 2.1," http://cvxr.com/cvx, Mar. 2014.

[19] M.-E. Chatzitheodoridi, A. Taylor, O. Rabaste, and H. Oriot, "Fast computation of the optimal integrated sidelobe level filter for large-scaled problems," in *IEEE RadarConf 2022*, 2022.

[20] M.-E. Chatzitheodoridi, A. Taylor, H. Oriot, and O. Rabaste, "A joint SAR-Communication system using continuous phase frequency shift keying codes and mismatched filtering compression," in *14th European Conference on Synthetic Aperture Radar (Eusar'22)*, 2022.

[21] A. Grami, "Chapter 7 - passband digital transmission," in *Introduction to digital communications*. Academic Press, 2016.

[22] R. v. Nee and R. Prasad, *OFDM for wireless multimedia communications*. Artech House, Inc., 2000.

[23] J. G. Proakis, *Digital communications*. McGraw-Hill, 1983.

[24] M. Nowak, M. Wicks, Z. Zhang, and Z. Wu, "Co-designed radar-communication using linear frequency modulation waveform," *IEEE Aerospace and Electronic Systems Magazine*, vol. 31, no. 10, pp. 28-35, 2016.

[25] A. Hassanien, M. G. Amin, Y. D. Zhang, and B. Himed, "A dual-function MIMO radar-communications system using PSK modulation," in *2016 24th European Signal Processing Conference (EUSIPCO)*, 2016, pp. 1613-1617.

[26] J. B. Anderson and A. Svensson, *Coded modulation systems*. Springer Science & Business Media, 2006.

[27] C. Tierney, "Adaptive waveform design for SAR in a crowded spectrum," 2019.

[28] M. C. P. Paredes and M. Garcia, "The problem of peak-to-average power ratio in OFDM systems," *arXiv preprint arXiv:1503.08271*, 2015.

[29] D. Garmatyuk and M. Brenneman, "Adaptive multicarrier OFDM SAR signal processing," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 49, no. 10, pp. 3780-3790, 2011.

[30] T. Zhang and X.-G. Xia, "OFDM synthetic aperture radar imaging with sufficient cyclic prefix," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 53, no. 1, pp. 394-404, 2014.

[31] J. K. Wolf, R. J. McEliece, J. Proakis, and W. H. Tranter, Eds., *Modulation Theory*. Boston, MA: Springer US, 2002, pp. 17-74.

[32] J. E. Cilliers and J. C. Smit, "Pulse compression sidelobe reduction by minimization of $L_p$-norms," *IEEE Transactions on Aerospace and Electronic Systems*, 2007.

[33] J. Baden and M. Cohen, "Optimal peak sidelobe filters for biphase pulse compression," in *IEEE International Conference on Radar*. IEEE, 1990, pp. 249-252.

[34] L. Xin and H. Sun, "Parameter assessment for SAR image quality evaluation system," in *2007 1st Asian and Pacific Conference on Synthetic Aperture Radar Proceedings*, APSAR 2007, 12 2007, pp. 58-60.

[35] B. Jiu, H. Liu, X. Wang, L. Zhang, Y. Wang, and B. Chen, "Knowledge-based spatial-temporal hierarchical MIMO radar waveform design method for target detection in heterogeneous clutter zone," *IEEE Transactions on Signal Processing*, 2015.

[36] F. Jia, P. Zhao, L. Zhang, and M. Zhai, "A novel method to suppress short-range clutter in airborne radar," in *RadarConf20*, 2020.

[37] A. Martinez and J. L. Marchand, "SAR image quality assessment," *Revista de Teledeteccion*, vol. 2, pp. 12-18, 1993.

[38] D. Asamoah, E. Ofori, S. Opoku, and J. Danso, "Measuring the performance of image contrast enhancement technique," *International Journal of Computer Applications*, vol. 181, no. 22, pp. 6-13, 2018.

[39] M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light*, 7th ed. Cambridge University Press, 1999.

[40] K. Hachemi, M. Visan, A. Ozer, F. Grecu, K. Anouche, M. Ju-rchescu, and M. Nouacer, "The interest of coherence from radar SAR images in the follow-up of urban extensions of Focşani city (Romania)," *Geoinformatics and Geostatistics: An Overview*, 10 2013.

[41] N. D. A. Gouveia, F. C. Alves, and L. D. O. Pereira, "Pre-processing of Sentinel-1 C-band SAR images based on incidence angle correction for dark target detection," *Remote Sensing Letters*, vol. 10, no. 10, pp. 939-948, 2019.

[42] J. Henrion et al., "SETHI: Digital radar system for signal generation and acquisition," in *International Radar Conference (RADAR)*, 2019.

[43] S. Solbø and I. Solheim, "Towards operational flood mapping with satellite SAR," *European Space Agency, (Special Publication) ESA SP*, 04 2005.