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

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 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 airborne radar. Since mitigating the sidelobe level energy is essential for good image quality, we resort here to optimized mismatched filters (MMFs). A fast algorithm is proposed to minimize the computational time of these filters. Simulated data are generated, as well as resynthesized synthetic aperture radar (SAR) images, and reconstructed from real chirp-based data using CPM codes and MMFs. 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.

T HE radio frequency electromagnetic spectrum has become gradually congested due to the exponential need for 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 first investigated for military purposes, it appears also in various civilian domains, such as autonomous vehicles, remote sensing, and 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 radarcommunication systems are presented in the literature: coexistence, cooperation, and codesign [3], [4]. Coexistence 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, long-term Evolution (LTE) wireless communication and air traffic control radar systems can coexist 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]. Codesign consists of designing dual-function waveforms [9] that perform simultaneously both applications. Examples of codesign joint radar systems are, for instance, Multiple Input Multiple Output (MIMO)-orthogonal frequency-division multiplexing (OFDM) systems [10], or combinations of radarcommunication signals, such as linear frequency-modulated waveforms with phase modulation [11].
In this article, 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 of forming 2-D high-resolution images of an area using an airborne or spaceborne platform [12], [13]. The specificity of SAR systems consists of 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 (MF) 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 phasemodulated (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 a high integrated sidelobe level (ISL) after MF compression, which leads to image quality deterioration. Therefore, in order to ensure good SAR image quality, we propose to use mismatched filtering [15], [16], [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 (MMF) as well. In particular, in an 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-scale 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 resynthesized 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 of the different elements of the image, in order to provide a global understanding of the performed work.
This article is organized into 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 MMF 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 resynthesized 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 this article.
The latter ones will be the selected coding scheme for our SAR-communication application. Their formulation and some of their properties, which justify this choice, are presented in the following.

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 {±1, ±3, . . . , ±(C − 1)}, where C = 2 k is the size of the constellation, with k being the number of bits per symbol.
The phase of a CPM signal is then expressed as [14] φ(t) = 2π 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 where g is a frequency response function [26].
In this article, 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 ∀i and a rectangular smoothing function g. By extension, the function u is equal to 0 for t < 0 equal to (t/2T c ) when t ∈ [0, T c ] and equal to 0.5 for t ≥ T c .
Finally, the CPFSK signal can be formulated as s(t) = e j φ(t) , with the phase φ(t) expressed as [14] φ(t) = πh (1) Fig. 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]. Fig. 2 illustrates the analytical PSD of these codes for different constellations. We can remark that, for high constellations, such as C = 8 and C = 16, the PSD is relatively flat and well-contained. Hence, the choice of the high constellation is favorable for the spectrum from a radar point of view.
In our configuration, inserting a radar system into a communication chain implies considering a bistatic system where the airborne radar transmits and receives a set of known waveforms, while the communication receiver will deal with unknown signals that must be decoded. Even if the decoding procedure is out of the scope of this work, in this paragraph, two possible solutions are suggested by the authors. First, from  Analytically computed PSD of the CPFSK codes for different constellation sizes. the CPFSK encoding procedure, it appears that N chip information symbols I = [I 1 , I 2 , . . . , I N chip ] are embedded continuously in the CPFSK phase during the pulse duration. Each symbol is mapped to k bits according to the given constellation of size C = 2 k . This enables to transmit up to k N chip bits in the pulse duration, corresponding to an overall N CPFSK tot = 2 kN chip number of different CPFSK waveforms. Decoding these information symbols from the direct path signal at reception can be performed using the Viterbi algorithm [27]. This first solution maximizes the data rate but will be efficient only if the signalto-noise ratio (SNR) at the chip level is sufficient and requires a quite demanding decoding procedure. The second possible solution consists of considering only a subset of size N sub N CPFSK tot of CPFSK waveforms. In that case, each waveform from the subset can be directly assigned to a communication symbol corresponding to a specific binary word of log 2 (N sub ) successive bits. In this way, the communication system uses the knowledge of this bank of waveforms at the reception in order to perform correlations between the received signal and all waveforms from the set, thus finding the maximum likelihood and matching it with the associated symbol. This second method provides an easier decoding processing, but the quantity of information that can be transmitted is limited (log 2 (N sub ) bits for the second solution compared to k N chip bits for the first one). Choosing one of these two solutions, thus, resorts to finding a tradeoff between data rate, and simplicity and quality of the decoding procedure. For applications where the SNR is sufficient and a high data rate is desired, the first solution is preferable. On the contrary for low SNR, the second solution should be preferred at the price of a lower data rate.

B. CPFSK Versus Other Communications Codes for an SAR-Communication System
In this section, we will justify the choice of CPFSK codes for an 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 nonlinear 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 [28]. 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) [29], 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 [30], [31], 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, in addition to minimizing the out-of-band energy, prevents substantial distortions due to bandpass 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, as they are crucial parameters for the radar system. Here the binary constellation was considered for the PSK code because, in such a way, we can assure that the bandwidth is (1/T c ). Afterward, the LFM-PSK code is defined such that its bandwidth is the same as for the PSK code. This is why the chip duration for the rest of the codes is different. In addition, the number of bits embedded in the phase changes, as the pulse duration is identical, and the chip duration is not the same as the CPFSK codes. All these parameters are linked to each other, so it is difficult and sometimes impossible to keep them identical. Finally, the variable values among the parameters are the constellation, the number of embedded bits, and the chip duration.
It can be observed in Fig. 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 ob ) power is calculated. It is expressed as [32] P where S is the signal spectrum and f ob is the positive frequency limit after which we consider that the spectral energy is out-of-band. Fig. 3 also shows an out-of-band power of P ob = 0.01, which means that 99% of the power is inside f ob . The spectrum of each signal is plotted, and the vertical lines of the same color specify the associated bounds for the out-ofband power. Table II displays these bound values, and ideally, this power should not exceed the bandwidth value [(B/2) = 176 MHz 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 to 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 f ob associated with the BPSK and LFM-PSK codes has exceeded the bandwidth value.
Signals exhibiting a high out-of-band power could be filtered to comply with an imposed spectral containment. However, cutting the spectrum at given frequencies can lead to an energy loss after matched filtering with the input signal. To illustrate this, two bandpass 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 two cuts. We can observe that, for the 1.2B cut and the LFM-PSK, a loss of 1.14 dB on the peak may be unacceptable. As expected, the best results are provided when using the CPFSK signals, as, in the worst case scenario, there is a loss of 0.1 dB. It appears 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 reception processing.

III. MISMATCHED FILTERING RANGE COMPRESSION
In radar applications, range compression at the reception is performed to increase the SNR and improve the range resolution. The most common compression technique is the MF that maximizes the SNR. However, when applied on CPFSK signals, such a filter produces high sidelobes, as observed in Fig. 4, which can degrade the performance of the radar system, and in our case, the quality of the generated SAR image. Thereby, in this section, an MMF formulation for range compression that minimizes the ISL energy is provided. Note that this formulation is general, and it can be applied to any sampled waveform.

A. Mismatched Filter Formulation
Let s be an N-length sampled CPFSK code expressed as where . T is the transpose operator. The received signal is filtered by an MMF in order to constrain the ISL energy. The length K of the MMF q ∈ C K can be equal to or greater than the length of the signal (K ≥ N). The output signal can be expressed as [15]  where and A * denotes the complex conjugate of matrix A. Setting q = s in (3) enables to retrieve the MF 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], [16], [17], [33], [34].
In an 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 ISL criterion. It is defined as the energy of all sidelobes [35], and it can be expressed as where the set ML corresponds to the mainlobe indices. Hence, the MMF problem corresponds to the following L 2 -norm minimization problem [15]: where F is a diagonal matrix of ones except for zero values for indices in ML , and β is a positive constant in dB that expresses the maximum tolerated loss-in-processing gain with respect to the MF [15]. The cost function can be also expressed as ||F y|| 2 2 As mentioned in [15], the first constraint allows discarding 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], [36], and [37]. 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 the CVX tool becomes infeasible. This led us to find a faster algorithm to solve the optimization problem (5) [19].
The solution that we propose consists of 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 in the following, but more details can be found in [19].
1) First, the Lagrangian function is calculated, the gradient of which provides the expression of the optimum MMF in the dual space, defined as where ν is the dual variable for the equality constraint and λ is the dual variable for the inequality constraint. 2) In order to compute q opt , the optimal values of the two Lagrangian variables (λ opt , ν opt ) need to be found. For this reason, the dual function of the primal problem is maximized over the two variables, defined as 3) The ν opt value is calculated analytically (see ν opt in Algorithm 1), and the λ opt value is computed numerically via a monodimensional gradient descent algorithm. Thus, the optimal filter can now be expressed through (6). 4) Note that it can be easily proven that the optimal solution in the dual space coincides with the primal solution. 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

Algorithm 1 MMF Optimization Using Gradient Descent
Input: Loss-in-progressing gain α, initialization λ 1 , max iterations max i , precision , Output: Optimal pair (λ opt , ν opt ), optimal mismatched filter q opt ; s ← code generation; acquisitions, in this section, we create resynthesized 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 recreate 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 resynthesized 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) where r u (t) is the received signal, g r u (t) is the ground reflectivity, t denotes the fast time, related to the intrapulse sampling time, and u denotes 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 = (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 MF. 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 backscattered real data, the SAR image reconstruction algorithm generates a resynthesized SAR image using CPFSK codes. The procedure is illustrated in Fig. 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: whereĜ 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 chirp is the chirp bandwidth, and f is the frequency variable in range direction. OnceĜ 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 MMFs in order to create the compressed CPFSK data, as illustrated in Fig. 5. Finally, azimuth compression is applied, creating the resynthesized 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 RESYNTHESIZED
SAR IMAGE EVALUATION This section is dedicated to the definition of the four main evaluation tools used to study performance. The ISL ratio (ISLR) and the peak-to-sidelobe level ratio (PSLR) are two classical radar criteria that provide two different insights on the sidelobe level. The 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
The integrated sidelobe ratio is defined as the ratio of the energy of all the sidelobes to the energy of the mainlobe. 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 [35], [38].
The ISLR is expressed as 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 equation (4) divided by the energy of the mainlobe.

B. Peak-to-Sidelobe Level Ratio
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 [35], [38].
The PSLR is expressed as PSLR = 10 log 10 I s I m (9) where I s is the amplitude of the strongest sidelobe and I m is the peak intensity of the mainlobe.

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, and then, the coherence coefficients between A and B are defined as following [39], [40]: where i, j is 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. The notation E[·] refers to the mean value of the elements of the given matrix on the set D.
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 [41]. 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 and contrast.

D. Average Value of Low-Energy Pixels of an SAR Image
The criterion of this section provides the mean value of a low-energy zone of an SAR image. This value gives an insight into the average thermal noise level or equivalently to the noise equivalent sigma zero (NEσ 0 ) [42], as in this area the presence of signals is very low.
Let A a subimage of the SAR image that has low-intensity pixels; then, we define the normalized average value over a low-intensity area N 0 as When using different signals and compression techniques, 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 resynthesized data. Section VI-A provides the characteristics of the real data SAR image used for this performance study, along with some characteristics for the CPFSK code generation. Sections VI-B and VI-C illustrate the impulse response of the CPFSK codes (presented in Section II-A) while using MF or MMF (provided in Section III-A), as well as their associated resynthesized SAR images, created using the reconstruction algorithm defined in Section 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 [43], an airborne radar system owned by the Office National d'Etudes et de Recherches Aérospatiales (ONERA), the French Aerospace Laboratory. They are obtained in the 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 resynthesized SAR images in order to avoid problems at the boundaries of the spectrum. The bandwidth value of the resynthesized SAR images depends on the signal characteristics. Its choice is a compromise between the need of providing a small computational cost for the MMFs by having fewer signal samples and concurrently having a sufficiently large bandwidth for the SAR. In order to be able to compare it with the CPFSK outputs in terms of range resolution, the following chirp results are computed as a resynthesized SAR image with this smaller bandwidth. In the following, four different configurations are considered.
1) One identical CPFSK pulse is transmitted for all the acquisitions, and it is compressed with the MF. 2) One identical CPFSK pulse is transmitted for all the acquisitions, and it is compressed with the MMF. 3) A different CPFSK code is transmitted at each pulse, and each one of them is compressed with the MF. 4) 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 where N 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 rate is 0.76 kbit/s, calculated using the parameters in Table IV. On the contrary, when the CPFSK signals differ from pulse to pulse, then N s = 6522, and the resulting data rate becomes 5 Mbit/s (or 625 kbytes/s).

B. Evaluation of the Impulse Responses
Before creating the resynthesized SAR images, we provide in this section 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 Section III-B is used  V   PSLR AND ISLR VALUES IN THE RANGE DIRECTION ON THE  DIFFERENT IMPULSE RESPONSES. ISLR SMALL 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 to compute all the needed MMFs. 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]. Fig. 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 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 fact that the transmitted signals with different messages embedded in the phase provide random sidelobes after compression, which are averaged after azimuth compression. More precisely, signals with different embedded messages provide different sidelobes after range compression, as illustrated in Fig. 7, where eight different codes are plotted after mismatched filtering. Their average value is also provided in order to show that the sidelobe level is decreasing even with the small number of eight signals. When using matched filtering, the sidelobe level energy is higher than the impulse responses compressed with the MMFs. Moreover, as seen in Fig. 6(c) and (e), MMFs that are longer than the signal length, which results also in longer sidelobes, allow to have a more diffused sidelobe level. The level around the mainlobe is lower, but it spreads on a larger area. These remarks on the sidelobe level difference and the compression length can also be observed in Fig. 8, which displays the range axis cut of all the subfigures in Fig. 6. Only the range axis plot is provided, as the azimuth cut is the same for all different configurations. The sidelobe behavior is also quantified in Table V, where the best ISLR value corresponds to the case of different CPFSK with MMF [see Fig. 6(e)], 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 resynthesized SAR images. It is displayed for comparison reasons, as there is no clutter or noise on the impulse responses, contrary to the resynthesized SAR images introduced in the following.

C. Evaluation of the Resynthesized SAR Images
By using the impulse responses previously introduced (see Fig. 6), together with the ground reflectivity of the real-data SAR image presented in Section VI-A, we can, finally, create the resynthesized SAR images from CPFSK communication codes. A large area of the SAR image is illustrated in Fig. 9 for the configurations presented previously. Fig. 9(a) 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 [see Fig. 9(b) and (c)], we can observe highenergy lines of the sidelobes of high-energy points along the range axis in both figures, especially the one created from MF. This spreading deteriorates the quality of the SAR images and can potentially mask details. It is mainly created because compression sidelobes are concentrated in the range axis. Thus, the sidelobes of each high-energy point affect the neighboring points, which results in image deterioration. On the contrary, for dark regions of the image, the energy of   sidelobes is low and may be lower than the noise level. Hence, the surrounding pixels are not impacted a lot, and the energy is uniformly distributed in such areas. On the contrary, when different CPFSK codes are transmitted [see Fig. 9(d) and (e)], we can remark that the sidelobe energy is spread along the 2-D 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 subareas of the whole SAR image will be considered for comparison, as illustrated in   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 resynthesized corner reflector with the impulse response. Both tables show that communication codes compressed with MMF provide better values for the PSLR and the ISLR compared to the MF. For the ISLR, this remark also validates the criterion of the optimization problem (5), which minimizes the integrated sidelobe energy. The best results are highlighted in Table VI. They correspond to the case when a single CPFSK code is transmitted for all pulses, with its associated MMF, although the values for different CPFSK codes with their MMFs present very close values, especially for ISLR.
2) Pearson Correlation Coefficient: It is the next comparison criterion presented in Section V-C. The image created with the chirp pulse is taken as a reference for computing this criterion. Results are presented in Figs. 11-14 for all the four areas of Fig. 10. Each subfigure is created by taking the Pearson correlation coefficient on a small window of size 5 × 5 of the resynthesized SAR images that are compared. Table VII represents the average value of each correlation figure over the subareas 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 MMF. 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 on lowenergy ones. A more detailed comparison is given in the following for each configuration. a) Corner reflector: Fig. 11 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 (top right subfigure), as indicated in Table VII. This behavior can be explained, first, because the MMF minimizes the compression sidelobes, and second, due to the spreading created when multiple signals are transmitted, so that the energy is distributed in the 2-D 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: Fig. 12 illustrates the coherence of the building image, which corresponds to an area of high 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 MF (first column subfigures), has low similarity compared to the chirp energy, and for the MMF (second column subfigures), the performance is close. The best similarity is reached when multiple codes are transmitted (right down subfigure). c) Lake: Fig. 13 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 [44]. 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, compared to other areas. d) Forest: Finally, Fig. 14 illustrates the correlation coefficient for the forest region. The vegetation area is a mediumintensity area. The same results as before can be observed, as the MF outputs provide a bad correlation coefficient compared to the chirp image forest intensity. Between the MMF outputs, still, the configuration where we transmit a different  Correlation coefficients of the forest for different signal configurations. pulse at each acquisition provides a better value, as shown in Table VII.
3) Average Energy of the Lake: The last comparison criterion, as defined in Section 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 Fig. 15. This criterion provides insight into how the pixels with low energy, which are close to the noise level of the scene, are influenced by different codes and compression techniques. Table VIII provides the average value of the pixels of the red square in Fig. 15. 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 MF, sidelobes from neighboring pixels are polluting lowenergy points in the lake.
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 a relatively well-contrasted image.

VII. CONCLUSION
This article proposes a joint SAR-communication system in order to simultaneously create radar images and transmit information. It uses CPM communication codes that are processed with mismatched filtering and are optimized to provide the minimum ISL. The proposed codes and processing are used to generate resynthesized 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 resynthesized images that, when transmitting different codes per pulse and compressing them with their associated MMFs, 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 resynthesized images.