1. Introduction
With the development of information theory and electronic technology, there is an urgent need for multifunctional electronic information systems, as the electromagnetism environment in space is increasingly complex and informatization is continuously increasing. Meanwhile, the development of wireless communication has brought about an explosion of wireless devices and enormous demands for high-quality services. The conflict of spectrum resources is becoming increasingly tense [
1,
2]. For example, the S-band is divided to perform reconnaissance and warning tasks in radar systems and to transmit wireless communication messages in 5th generation (5G) communication systems. To relieve spectrum pressure, two spectrum sharing strategies are proposed. The first strategy is the co-existence of radar and communication systems, both of which transmit separate signals at the same time and/or frequency [
3,
4,
5]. The main challenge is suppressing mutual interference between different systems to guarantee the performance [
6,
7,
8,
9]. The other strategy is the joint radar-communication (JRC) system. The JRC system can effectively solve the spectrum tension and system redundancy problems by sharing the radio frequency front-end and transmit signal [
10,
11,
12,
13,
14]. In recent years, the JRC system has attached significant attention due to its extensive application potential in both civilian and military fields [
15,
16,
17,
18]. In earthquake relief, for example, the JRC system which is installed on the airplane platform detects the terrain while timely transmitting detection images to ground rescue vehicles.
The critical issue of the JRC system is to create an integrated waveform that can balance the requirement of radar and communication in order to achieve better system performance [
19,
20]. There is a large volume of published studies describing the method of the integrated waveform. According to the number of transmit channels, the existing research can be divided into two main categories: single-channel-based waveform design methods [
21,
22,
23,
24,
25,
26,
27,
28,
29] and array-based waveform design methods [
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47]. In the first category, waveform design methods mainly focus on temporal and spectral resources. Linear frequency modulation (LFM) signal as one type of traditional radar signal was combined with continuous phase modulation [
21], spread-spectrum encoding [
22], and reduced phase shift keying [
23,
24] modulations to embed communications messages. In [
25,
26,
27,
28,
29], orthogonal frequency division multiplexing (OFDM) was extended to radar applications which could be used to realize target detection and information transmission. The studies of the single-channel-based method are already sufficient. However, all of these methods have inherent performance loss for either radar or communications systems, e.g., high autocorrelation sidelobe level or low data rate. Moreover, this category of methods tends to concentrate on basic functionality realization instead of specific systems and scenes. When radar targets and communication users are in different directions, these methods would be disabled to cover them effectively.
Array-based methods exploit waveform diversity and spatial diversity to overcome the above problems. Hassanien utilized an orthogonal waveform set to deliver messages [
30]. The communication symbols are modulated on waveform permutation. The data rate depends on radar pulse repetition, only a limited number of symbols can be transmitted. Hassanien also combined frequency-hop** (FH) orthogonal waveforms with phase shift keying [
31,
32]. The data transmission rate can be significantly improved by embedding in each FH. However, there are no perfect orthogonal waveforms in the radar concept. Increasing the number of orthogonal waveforms can compromise the performance of radar processing. To avoid designing orthogonal waveforms, the circulating code array was used for target detection and downlink communication [
33,
34,
35]. Multiple communication symbols are embedded in the designed basic waveform. In order to make full use of the degrees of spatial freedom, researchers turn the spotlight on the transmit beamforming. In [
36,
37,
38], the beampattern sidelobe directing to the communication user was employed to convey the message. The communication bits were transmitted by modulating the sidelobe level or phase, while the main beam was used for radar detection. In the whole coherent processing interval, the main beam remains constant. This sidelobe control scheme is limited by user numbers. It is difficult to meet the requirements of multiple sidelobes at the same time. References [
39,
40,
41] consider the downlink multi-user interference (MUI), as well as the desired radar beampattern and power distribution, in order to overcome this drawback. In [
42,
43], the radar transmit beampattern was optimized by precoding the radar signal and communication signal to ensure the radar performance while guaranteeing the signal-to-interference-plus-noise ratio (SINR) of the communication receiver. All of these methods accomplish radar functionality in the main beam and communication functionality in the sidelobe. They result in the performance trade-off of the low SINR or high autocorrelation sidelobe. Despite being regarded as a component of the beampattern, communication does not have the equal status as the radar beam.
The multibeam design approach utilizes spatial and temporal dimensions to synthesize the desired waveforms in different directions. Radar and communication simultaneously have excellent performance. This approach has the ability to control the waveform distribution in space. In addition, it is not limit the type of the integrated waveform. That is to say, the waveform performance in the temporal dimension is not bound by the traditional signal model. Reference [
44,
45] presented a far-field radiated emission design (FFRED) model that simultaneously synthesized the desired radar and communication waveforms in different directions. However, it is unsatisfactory that the conflict between peak to average ratio (PAPR) and beampattern performance gives rise to a high SLL, which would reduce energy efficiency, enlarge exposure probability, and submerge small targets. An alternative projection method was proposed in [
46] to optimize the robustness of radar match filtering in the radar beam. Such an approach has failed to address the high sidelobe level (SLL) problem. Reference [
47] extended the work of [
44] by imposing the SLL constraint and amplitude weighting. An iterative optimization with the amplitude weighting (IO-AW) algorithm was adopted to implement this method. The IO-AW method is between the normalized waveform error and the SLL of the formed beampattern with the side effect of a high mirror lobe. Moreover, the SLL is strongly dependent on the channel number and objective direction.
The integrated waveform design based on spatial dimension is a potential topic, which is wide and unexploited. With the development of the digital array, every transmitter has an independent channel to emit arbitrary waveforms. In this paper, we focus on the spatial–temporal joint waveform that can obtain classic waveforms through spatial coherent synthesis. We propose a two-stage waveform optimization algorithm based on the transmit beamforming algorithm and waveform constraints. This method can fully utilize the spatial degree of freedom of the array to obtain the transmit beampattern with low SLL and avoid extra processing in the receiver. First, optimizing the covariance matrix, a beampattern with smooth and low-level distribution in the sidelobe region is obtained. Second, under waveform synthesis constraint and constant modulus constraint, we adopt the alternating projection algorithm and cyclic algorithm (CA) to determine the integrated waveform whose covariance matrix is equal to or close to the matrix obtained in the previous stage. The performance of the proposed method is verified by simulation results, showing that this method can significantly reduce the SLL while suppressing the PAPR of the waveform. With implementation fully taken into account, the proposed method is further verified by semi-physical experiments.
The remaining part of the paper proceeds as follows.
Section 2 begins by laying out the theoretical derivations of the multibeam strategy, and establishes the spatial–temporal joint waveform model.
Section 3 is concerned with the two-stage waveform optimization algorithm employed for this study. Performance metrics are illustrated in
Section 4.
Section 5 presents the findings of the numerical simulations and semi-physical simulations. Several discussions are given in
Section 6. Finally, summarize the conclusion in
Section 7.
2. Signal Model
In this section, the multibeam waveform strategy is outlined. Moreover, we derive the mathematical expression in any direction employing the minimum-norm solution.
2.1. Multibeam Waveform Strategy
Consider a JRC system equipped with an
M-channel uniform linear array (ULA), as shown in
Figure 1. The transmitter can employ this array to synthesize the desired radar waveform in one direction and an information-carrying communication waveform in another direction during the same duration and bandwidth. The two desired waveforms work together to simultaneously complete the communication and radar tasks.
Assume that the sampled waveform emitted by the m-th element is , for and , where N is the number of samples. Let
. Under the narrowband assumption, the far-field synthesized waveform in direction
can be defined as
where
is the steering vector with
being the waveform carrier wavelength and
d being the array element spacing,
and
are the transpose and conjugate transpose, respectively.
When direction
, the synthetic waveform should be identical to the desired radar waveform
. When direction
, the synthetic waveform should be equal to the desired communication waveform
. This can be formulated as
where
is the transmit waveform matrix composed of all samples of
M channels.
Further sorting, (
2) and (
3) can be succinctly expressed as
where
is the collection of spatial steering vectors and
is the collection of desired waveforms.
Under the minimum-norm constraint, the multibeam waveform matrix can be established as
where
is the squared-Frobenius norm. The optimization model in (
5) is convex and has the close-form solution
where
is the matrix inversion.
The waveform can synthesize the desired waveforms with the minimum power in direction and .
2.2. Spatial Waveform Distribution
When all of the channels transmit the same waveform, the transmit beampattern pointing to is generated by employing the beamforming vector in the corresponding direction. In the multibeam strategy, only synthetic waveforms in two directions are assigned. The waveform in each channel is different. It is unknown how the waveform behaves in other directions. We make the mathematical derivation to examine the waveform in any direction. Additionally, spatial emission power is discussed to judge whether the main beams are formed.
When the integrated waveform
is transmitted by the ULA, the synthetic waveform in direction
is expressed as
The waveform expression can be expanded by substituting (
6)
where
and
. To simplify the analysis, the formula analysis is carried out item by item.
where
The multiplication result of item
in (
9) and item
in (
10) represents the weighting factor of the desired radar waveform and desired communication waveform. It determines the spatial synthetic waveform performance and power distribution. The expression for multiplying two items is
Now we discuss the range of two weighting factors in (
12)
When , the weighting factor is maximum, the other weighting vector , and the synthetic waveform is the desired one corresponding to direction . The same situation happens when . The weighting factor and . The synthetic waveform is equal to . When , the synthetic waveform is the sum of the damaged desired waveforms which have amplitude attenuation and phase delay. Consequently, there is no expected waveform in any direction except for the area near the desired angle.
Assume that the radar waveform and communication waveform have the same amplitude. The weighting vector value indicates the spatial power performance according to the absolute value inequality theorem,
where
is the absolute value operation.
According to (
13) and (
14), the absolute values of the weighting vectors have similar distribution as the sinc function. The maximum value is obtained only in the desired direction. The sum of the two vectors forms two main beams in the desired directions, while the sum is very low in the remaining direction. A numerical simulation of the weighting vector is analyzed to verify the above discussion. The simulation result is illustrated in
Figure 2. Moreover, the parameters are configured in
Table 1.
Table 1.
Simulation parameters.
Table 1.
Simulation parameters.
Parameter | Value |
---|
Array number | 10 |
Element spacing | /2 |
Carrier frequency | 3 GHz |
Sampling number | 1024 |
Radar direction | 36.87 |
Radar 3dB beam width | 12.69 |
Desired radar waveform | LFM |
Baseband bandwidth | 100 MHz |
Pulse width | 5.12 s |
Communication direction | −45 |
Communication 3dB beam width | 14.36 |
Modulation | QPSK |
Symbol number | 64 |
PSLR upper bounder | −6 dB |
PSLR lower bounder | −11 dB |
Main beam power difference | 3 dB |
Main beams region | |
Sidelobe region | |
Iteration number | 300 |
These mathematical derivations confirm that the multibeam strategy synthesizes the desired waveform in the specific direction with the additional effect of energy accumulation. Moreover, the synthetic waveform in the remainder of the directions is random and has low-energy compared to the desired waveform, which prevents the leakage of the waveform and transmit power.
4. Performance Metrics
In this section, some metrics are defined to evaluate the performance of the proposed method.
4.1. Beampattern Performance Metrics
In the multibeam strategy, two main beams are established to perform radar and communication functions, respectively. The sidelobe region is expected to have low SLL to avoid producing interference. Consequently, the performance of the whole beampattern directly affects the realization of the tasks. We adopt the peak of the main beam, 3 dB main beam width, PSLR, and integrated sidelobe ratio (ISLR) to evaluate it.
The peaks are the value in the radar direction and the value in the communication direction . High peak values indicate high-power efficiency when the transmit power is the same. Aside from that, the high peak value indicates a high signal-to-noise ratio (SNR) at the same distance in space.
The 3dB main beam widths also include the 3 dB radar beam width
and the 3 dB communication beam width
. Since no method involves the beam width design, they are expected to be equal to the ideal beam widths
where
is the main beam direction.
PSLR and ISLR are both used to estimate the sidelobe level. The PSLR is already defined in (
20). It measures the relative level between the highest SLL and the main beam. A high PSLR shows that the worthless waveform consumes an amount of transmission power. The high PSLR may even make the transmit array position expose. ISLR describes the efficiency of the transmit power. The ISLR is defined as
where
is the power of the whole sidelobe region
, and
is the sum power of two main beams
.
4.2. Radar Performance Metrics
From (
13) and (
14), it can be known that the synthetic waveform varies in different directions. To analyze the radar waveform distribution in the radar beam, the synthetic waveform is filtered to match the desired radar waveform. The impulse response width (IRW), matched filtering PSLR (MF-PSLR), and matched filtering ISLR (MF-ISLR) are utilized to estimate the similarity with the desired radar waveform. IRW is the 3dB width of the main beam. The maximum sidelobe to main beam ratio is defined as the MF-PSLR. MF-ISLR is the ratio of the power of the remaining region except for the first null beam width to the power of the first null beam width.
Besides the matched filtering, noise equivalent sigma zero (NESZ) is another significant metric to evaluate the radar performance. Based on the radar equation, the NESZ is defined as
where
R is the waveform emission range,
is the system noise figure,
J/K is the Boltzmann constant,
= 290 K is the standard noise temperature,
is the noise bandwidth which is equal to the waveform bandwidth,
is the transmit power, and
G is the transmit or receive antenna gain. The transmit power is associated with the PAPR of the integrated waveform [
50]. When the PAPR is close to 1, the synthetic waveform has the maximum transmit power.
4.3. Communication Performance Metrics
Bit error ratio (BER) is used to determine communication similarity while analyzing the transmission waveform distribution. Since security is a crucial requirement for communication application, the synthetic waveform is decoded in the entire direction.
SNR is another crucial metric that has an impact on communication performance. The high SNR reveals the low BER. Under the same parameter configuration, the SNR is associated with the transmission distance. Therefore, we examine how the BER degrades with the range in the communication beam.
4.4. Convergence and Computational Complexity Performance
In the BF-SLL method, the first stage is a convex function, which can be computed instantly. The convergence is not affected by this stage. The second stage is non-convex and adopts the alternating projection algorithm to optimize the integrated waveform. The function is divided into three suboptimal problems according to the constraints. Each projection minimizes the error of the iteration variable. Based on the error reduction algorithm [
51], the error is non-increasing,
where
.
Consequently, the proposed method is converged until the convergence metric is met. To compare the convergence speeds of different methods, the normalized iteration error is defined as
Additionally, the computational complexity of the proposed and conventional algorithms is analyzed. In the proposed algorithm, the computational complexity is mostly determined by the SVD operation and matrix multiplication. The specific value is expressed as
,
is the total number of iterations. In unit time, the computational complexity is proportional to the communication rate. The computational complexities of the FFRED method in [
44] and the AP method in [
46] are also expressed as
. Despite having the same expression, these three approaches’ varying convergence speeds lead to various
in real-world applications. In the IO-AW method [
47], the computational complexity is expressed as
, where
K is the number of radar targets and communication users. Typically,
K in this paper equals 2. Since this method adopts matrix vectorization, it is much more complicated than other approaches. Naturally, extreme complexity is traded for SLL minimization.
6. Discussion
In the previous sections, the BF-SLL method is proposed to simultaneously synthesize the desired waveforms with the low SLL beampattern. Simulation and semi-physical experiment results demonstrate the effectiveness of the proposed method. In the optimization Model (
19), the waveform synthesis constraint is the chief constraint, which is the foundation to realize simultaneously radar and communication functions. The multifunction realization can only be guaranteed when the desired functional waveform is synthesized. Therefore, the waveform synthesis constraint is maintained throughout all references. In addition, the data stream and modulations are connected to the waveform synthesis constraint. While the function requirement changes, the integrated waveform needs to update to satisfy it. The objective function minimizes the SLL of the whole sidelobe. It differs from maximizing the power on the two main beams. The deterioration of the sidelobe is caused by the constant modulus constraint which needs more power to provide design freedom. Although the main beams have the maximum power, there is possible to form a false peak in the sidelobe region. We expect to disperse the false peak power to the sidelobe region to decrease the SLL (e.g., Equation (
21)).
The optimization Model (
21) aims to obtain the waveform covariance matrix corresponding to the low SLL beampattern. The waveform covariance matrix is related to the power difference between the desired radar waveform and the desired communication waveform. There is little relation between the covariance matrix and the specific style of the desired waveform. Equation (
21) is convex and can be computed by the convex toolbox. Regarding the generic toolbox’s slow optimization speed, we can design a customized primal-dual interior point method. The Karush–Kuhn–Tucker (KKT) conditions of (
21) is expressed as
where
are the dual variables. By employing this customized optimization algorithm, the optimization problem can be solved quickly to save computational time.
For communication performance, multipath is a big issue in the real-world communication environment. According to the existing findings, the proposed method can be applied in the multipath environment for the following reasons. The desired communication waveform
in Equation (
3) can adopt any modulation mode. For example,
can be modulated with OFDM to combat multipath interference. Moreover, the proposed method can only complete the information transmission ability in the line-of-sight direction, while the synthetic waveform in the remaining direction is random and the transmit power is low (seen in
Figure 7 and
Figure 15). Consequently, the waveform that takes multiple paths to the receiving antenna cannot convey information or cause serious interference. We will verify this performance in future work.