Performance Analysis of Data Transmission for Joint Radar and Communication Systems

Lu, Jianhua; Tian, Tuanwei; Tang, Yanli; Tang, Bin

doi:https://doi.org/10.1155/2021/6672394

Mathematical Problems in Engineering

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Abstract Introduction System Model Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6672394 | https://doi.org/10.1155/2021/6672394

Performance Analysis of Data Transmission for Joint Radar and Communication Systems

Jianhua Lu,^1,2Tuanwei Tian,¹Yanli Tang,¹and Bin Tang¹

Academic Editor: Fangqing Wen

Received30 Nov 2020

Revised07 Jan 2021

Accepted27 Feb 2021

Published16 Mar 2021

Abstract

This paper investigates the problem of data transmission for the joint radar and communication systems (JRCSs). The performance of the JRCS is characterized by data throughput related to the radar echo data (RED) and communication data rate (CDR). Two spectral coexistence schemes are proposed based on the degree of spectrum sharing for radar and communication, i.e., the isolated subfrequency band (ISFB) and mix-used frequency band (MUFB) schemes. Firstly, the signals of radar and communication are operated on the isolated subcarriers, enabling the received signals to be processed independently and bringing the advantage of interference avoidance. Secondly, the signals of radar and communication can be jointly operated on the same subcarriers for the MUFB scheme, which enhances the spectrum efficiency. Unlike the ISFB scheme, the CDR of the MUFB scheme is maximized along with the interference from the radar signal, and meanwhile, the allocated radar power on each subcarrier is derived by maximizing the radar mutual information. Numerical results show that the MUFB scheme significantly improves the performance of data transmission over the ISFB scheme, and a significant performance gain in the data transmission can be achieved, compared to the average power allocation case.

1. Introduction

Recent information battlefield scenarios urge modern radar equipment to deploy optimally [1,2] and exchange the estimation information [3] with each other. How to form an integrated system by combining the radar equipment with the communication equipment and solve the problems of the generalization, such as tremendous wireless data traffic, rational use of the system resources [4–6] has been the theme of increasing research over recent years. Spectrum sharing between radar and communication systems, which can overcome the shortage of spectrum resources and improve the efficiency of the spectrum significantly, is one of the essential research studies and has been a hot topic [7–11]. One factor that has to be emphasized, however, is the interference concerns caused by the cochannel spectrum sharing of radar and communication systems [12].

Two directions of coexistence between radar and communication have been developed:(i)The first one aims to develop a dual-function system that simultaneously realizes radar and communication functions [7, 11, 13–19]. In [13], a method of utilizing an orthogonal frequency division multiplexing (OFDM) signal to operate the functions of radar and communication systems is applied to the problem of radar and communication coexistence. The similar strategy can also found in [14, 15, 20]. Embed communication information into the radar waveform so that radar and communication functions can be realized by one common signal. This dual-function radar and communication (DFRC) technique includes sidelobe amplitude modulation (AM) scheme [16], waveform diversity-based scheme [17], phase shift keying (PSK) scheme [18], and multiwaveform amplitude shift keying (ASK) scheme [19].(ii)The second direction focuses on the spectral coexistence between radar and communication systems. The authors in [21] present the fundamental spectrum-sharing concepts and technologies. An overview of interference mitigation techniques is introduced in [22]. In [23], radar and communication systems are operated at the same frequency band using cognitive radio technology, to mitigate interference between the two systems. The authors in [24, 25] derive the achievable performance bounds of coexistence between radar and communication systems. A novel parameter, named estimation information rate, is proposed to measure the performance of radar systems and data transmission rate for the communication systems. In [26], to minimize the interference caused by spectrum sharing between radar and communication systems, the null-space projection (NSP) method is adopted. An IEEE 802.11ad-based radar is proposed in [27], which enables a joint waveform for both automotive radar and mm-wave communication systems. For the situation with unreliable radar estimation, the authors in [28] propose a system level interference cancellation algorithm. In [29], spectrum sharing between radar and communication based on codesign transmitted waveform is studied to minimize the effective interference from each other. Two generalized sidelobe cancellers are used in [30] to increase the communication data rate without affecting the radar effectiveness. The authors in [31] analyze the problem of power minimization-based radar waveform design considering the existence of communication signals on the same band. The problem of power allocation between multicarrier radar and communication systems is emphasized in [32, 33]. Based on the information theory, the authors in [34] design an adaptive signal for the integrated OFDM radar and communication system.

To the best of our knowledge, the performance of data transmission for the joint radar and communication systems (JRCS) has not been mentioned in the above works. This is indeed the main topic of this paper. The contributions are as follows:(1)Based on the degree of spectrum sharing for radar and communication, we present two spectral coexistence schemes, i.e., the isolated subfrequency band (ISFB) and mix-used frequency band (MUFB) schemes(2)Under certain range resolution and scanning speed , we give the amount of radar echo data (RED) and the expressions of data throughput for the ISFB and MUFB schemes, which maximize the data throughput under a certain range resolution and scanning speed along with a total transmit power constraint, are proposed(3)Choosing the maximization of data throughput along with a total transmit power constraint as optimization, we maximize the data throughput of the JRCS and drive the closed-form solution of the power allocation

This paper is organized as follows. The received signals at radar and communication receivers are modeled in Section 2. The performances of data transmission for the ISFB and MUFB schemes are analyzed in Section 3 and Section 4, respectively. Section 5 gives a numerical example, and conclusions are drawn in Section 6.

Notations: denotes the set complex. denotes the set of vector ( matrix) with entries from . , , and denote the optimality, transpose, and complex conjugate transpose, respectively. The symbol indicates the determinant of a square matrix. is the diagonal matrix with all elements on the main diagonal. The complex Gaussian distribution is denoted by . Finally, represents the positive part of .

2. System Model

As shown in Figure 1, we consider such a scenario that an integrated transmitter/receiver scans the region of interest and then transmits the scanned information to the communication receiver.The integrated transmitter consists of antennas for transmitting radar signal, antennas for transmitting communication signal, and antennas for receiving the reflected radar signal. The received signal at the communication receiver with antennas is a compound of signals scattered from the integrated transmitter and region of interest. Both radar and communication signals are assumed to be the OFDM-type multicarrier signals with subcarriers. Moreover, the frequency spacing among the subcarriers is assumed to be sufficiently large. Perfect OFDM symbol timing at the receiver is assumed, and the cyclic prefix (CP) of each received signal is removed before OFDM demodulation. Without loss of generality, we assume , in this paper.

2.1. Radar Receiver

Define as the transmitted radar waveform on the -th subcarrier, and assume ; . Let be the received signal of the -th receive antenna, then we havewhere are the target response from the -th integrated transmitter to the -th receiver, is the transmitted waveform matrix with , is the noise vector, and .

Let , then can be given bywhere is the target scattering matrix, , is the noise matrix, and . For simplicity, we assume the columns of and are independent and identically distributed (i.i.d), with distribution and , respectively.

Radar mutual information has been used as a parameter in many scenarios to characterize the radar system [32, 35–37]. The more radar mutual information is, the better performance of radar is, which can be derived from the mutual information between the input and output , namely,where denotes the differential entropy of a random variable.

2.2. Communication Receiver

Let be the transmitted communication waveform on the -th subcarrier and be the received signal of the -th communication antenna, then we havewhere are the channel responses from the -th integrated transmitter to the -th communication receiver, are the target response of link from the -th integrated transmitter to the region of interest and to the -th communication antenna, is the transmitted communication waveform matrix, , is the noise vector at the communication receiver, and .

Let , then can be given bywhere is the channel response matrix, , is the target response matrix, , is the noise matrix, and . For simplicity, we assume that the columns of , , and are i.i.d with distributions , , and , respectively.

The mutual information between the input and output is given byThus, the communication data rate can be given bywhere denotes the number of communication subcarrier.

2.3. Coexistence Scheme of Radar and Communication

In this section, we introduce the coexistence schemes of radar and communication, i.e., the ISFB and MUFB schemes, as shown in Figure 2.

(a)

(b)

The ISFB scheme is depicted in Figure 2(a). In this scheme, the total subcarriers are partitioned into two disjoint categories: the radar-only subcarriers and the communication-only subcarriers; then, we havewhere denotes the bandwidth of the total available band. Clearly, the signals of radar and communication are transmitted on different subcarriers and can be processed independently.

The MUFB scheme is depicted in Figure 2(b). In this scheme, the occupied bandwidth of radar and communication bands satisfies in this scheme. The subcarriers are split into two categories: the communication-only subcarriers where only communication signal exists and mix-used subcarriers where both radar and communication signals exist (). Combing equation (13), the values of and can be, respectively, given by

That is, radar and communication signals can be transmitted on the same subcarriers. The communication data rate (CDR) is maximized with the interference from the radar, which is determined by the range resolution and scanning speed.

By comparing Figures 2(a) and 2(b), we can see that the ISFB scheme can be regarded as a special case of the MUFB scheme. However, we still study both schemes because, in the ISFB scheme, we can treat radar and communication systems independently from a signal processing point of view. This makes the ISFB is easy to handle and more appealing for practical implementation. For the MUFB scheme, the performance of data transmission can be enhanced at the price of higher complexity (due to the additional interference to be handled).

3. Data Transmission of ISFB Scheme

In this section, we analyze the performance of data transmission of the ISFB scheme. In this scheme, part of the subcarriers is assigned to the radar and the rest is used by the communication so that the received signals of radar and communication can be processed independently.

3.1. Radar Echo Data

In this section, we analyze the amount of RED under a certain range resolution and scanning speed . Since the radar receiver is turned off during pulse transmission, the minimum detection range can be given by [38]where denotes the speed of light and is the pulse duration. Under a certain detection probability , the maximum detection range can be given by [38]where denotes the detection factor, is target cross section, denotes the pulse repetition interval (PRI), and , , and denote Boltzmann constant, absolute temperature, and antenna effective area, respectively. The received radar signals are sampled beginning at and ending at . According to the Nyquist sampling theorem, the sampling interval should not be larger than , and for convenience, we adopt in this paper, which means the sampling number equals to the number of range cell, that is,and for a chirp signal, we have

The number of pulse transmitted in 1 second is

The amount of radar echo data in 1 second can be expressed aswhere denotes the quantify bits without apparent saturation.

From (15), we can find that, with fixed PRI , pulse duration , and quantify bit , the amount of RED is determined by range resolution and maximum detection range . The smaller (or larger ) is, the larger amount of is. Moreover, the smaller is, the larger is. Therefore, there will be a larger with a smaller .

3.2. Communication Data Rate for ISFB Scheme

Since radar and communication signals are operated on the isolated subcarriers in the ISFB scheme, the total communication power is allocated to the subcarriers according to the corresponding channel conditions without interference from the radar system. Let be the transmitted communication waveform matrix for the ISFB scheme, i.e., . Since there is no interference from radar system, the CDR for the ISFB scheme can be given by

Then, with the constraint of total communication power , the maximization problem of CDR can be formulated aswhere the constant multiplier is omitted. Since does not depend on the transmitted waveform , problem (17) is equivalent to the following problem:

To obtain the optimal solution of (18), a lemma is firstly introduced.

Lemma 1 (see [39]). Define and as positive semidefinite Hermitian matrices with eigendecomposition and , respectively, the eigenvalues of and satisfy that and , and then we havewhere the lower bound is derived if and only if while the upper bound is derived if and only if with denoting the permutation matrix:

Since and are positive semidefinite Hermitian matrices, the corresponding eigendecomposition can be computed bywhere , and , . According to Lemma 1, we havewith equality achieved at .

Let the eigendecomposition of bewhere . Therefore, the optimal solution of problem (18) can be obtained by solvingwhere is the -th diagonal element of .

Since optimization problem (24) is convex, the optimal solution of which can be characterized by using the Karush–Kuhn–Tucker (KKT) optimality conditions [40, 41]. The corresponding Lagrangian form of the above optimization problem iswhere and are Lagrangian multipliers. Then, the optimal solution can be obtained, i.e.,where can be obtained by solving

3.3. Data Throughput for ISFB Scheme

Data throughput for the integrated system is a parameter characterizing the maximum data rate, which is determined by the smaller of CDR and RED. Therefore, define as the data throughput for the ISFB scheme, and we havewhere is given in equation (15). On one hand, means that the CDR can meet the requirement of data transmission and the RED can be transmitted timely and there is no transmission delay. On the other hand, means the amount of radar echo data is larger than the CDR , the integrated system needs several PRIs to transmit all the received radar echo data, and the corresponding transmission delay is defined as

4. Data Transmission of MUFB Scheme

In this section, we analyze the performance of data transmission for the MUFB scheme. Since the amount of RED related to the range resolution and scanning speed is the same for the ISFB and MUFB schemes, the description is not given again. In this scheme, radar and communication signals can be used on the same subcarriers. The CDR is maximized along with the interference from the radar, which is determined by the range resolution and scanning speed.

4.1. Communication Data Rate for MUFB Scheme

Let be the transmitted radar waveform matrix on the mix-used subcarriers, i.e., . Then, from the communication system’s point of view, the interference matrix caused by radar signal is

Therefore, the CDR of the MUFB scheme can be given by

Then, with the constraint of total transmitted communication power , the maximization problem of CDR can be formulated aswhere we have omitted the constant multiplier . Since does not depend on the transmitted waveform , the optimal solution of problem (32) can be obtained by solving

Define as the diagonal elements of interference matrix , and the interference from the radar system to the communication system can be given in the following proposition.

proposition 1. Let the eigendecompositions of and beandwhere and . Then, the interference from the radar system on the -th subcarrier can be given bywhere and denotes the optimal allocated radar power on the -th subcarrier given by

Proof. See Appendix.
As a result, let be the -th diagonal element of for the MUFB scheme, the optimal power allocation solution can be obtained by solving the following problem:Similar to the ISFB scheme, we solve the above problem by Lagrange multipliers and the corresponding Lagrange form is needed, namely,where and are Lagrangian multipliers. Then, the optimal solution can be obtained, i.e.,where can be obtained by solving

4.2. Data Throughput for MUFB Scheme

Similar to the ISFB scheme, the data throughput for the MUFB scheme is determined by the smaller of CDR and RED , that is,where is given in equation (15). On one hand, when , the RED can be transmitted timely without transmission delay. On the other hand, when , there will be transmission delay, which can be defined as

5. Numerical Examples

In this section, we provide numerical examples to demonstrate the performance of data transmission for the JRCS under different values of range resolution and scanning speed . Also, for comparison, the simulated results of average power allocation (APA) are presented (APA implies that the total power is allocated across all the available subcarriers). The performance of the proposed adaptive optimization algorithm and that of the APA are labeled as “AOA” and “APA,” respectively. We assume , , and obeywhere and denote antenna gains of radar and communication, respectively; and denote wavelengths of radar and communication on the -th subcarrier, respectively; ranges between the integrated transmitter and interest of target; ranges between interest of target and communication receiver (without loss of generality, we assume ); ranges between the integrated transmitter and communication receiver; , , and are generated from Gaussian distributions , , and , respectively, where , , and are set to be 1 in this simulation. Without no explicit specification, the setting of other system parameters is given in Table 1.

Figure 3 shows the maximum detection range versus under 0.85, 0.9, and 0.95. As expected, a monotonic decreasing behavior of the maximum detection range can be observed when the scanning speed is larger. Also, we can see that the gets smaller as the gets larger. This is because a larger needs a larger detection factor, leading to a smaller .

Figure 4 depicts the amount of RED for different values of and . is determined by sampling interval and the number of sampling , as discussed in Section 3.1. We can see that, on one hand, is getting larger with the decrease of . A smaller needs a larger size of the bandwidth of radar band leads to a smaller sampling interval according to the Nyquist sampling theorem. Therefore, the smaller is, the larger is. On the other hand, a smaller leads to a larger detection area, as shown in Figure 3, which will increase the number of sample . Thus, there will be a larger along with a smaller .

Under , , , and for both AOA and APA, the CDR for the ISFB and MUFB schemes versus range resolution is depicted in Figure 5:(i)In Figure 5(a), a monotonic increasing behavior of the CDR for both AOA and APA can be observed as the range resolution gets larger. This is because the larger results in the smaller number of radar subcarrier . Then, more subcarriers are used by the communication system, leading to a larger CDR . Also, the CDR in this scheme has no relationship with the scanning speed , since radar and communication signals are operated on isolated subcarriers.(ii)In Figure 5(b), a monotonic increasing behavior of the CDR for the MUFB scheme can be observed as the range resolution gets larger. The larger is, the smaller number of mix-used subcarrier is. Then, the smaller interference from the radar system to the communication system leads to a larger CDR . Also, we can notice that a smaller leads to a larger CDR by reducing the received radar power. As a consequence, the smaller interference to communication leads to a higher CDR .(iii)By comparing Figures 5(a) and 5(b), the achieved CDR of the MUFB scheme is larger than the ISFB scheme. This is because all available subcarriers can be used by both systems in the MUFB scheme, which leads to an increase of CDR even there being interference from cofrequency radar signal. Also, by comparing the AOA and APA, a significant gain in the CDR can be achieved by employing the AOA. This is because, when the AOA is employed, the transmit power is allocated based on the conditions of subcarriers, which enable significantly maximizing the CDR performance; however, the transmit power is averagely distributed among the available subcarriers when the AOA is employed.

(a)

(b)

For both AOA and APA, Figure 6 depicts the data throughput for different values of under :(i)For the MUFB scheme, the achieved maximum data throughput is corresponding to . In the case of , the amount of RED is larger than the CDR ; then, the data throughput is determined by CDR , which cannot meet the requirement of data transmission and leads to data retention. However, in the case of , the CDR is larger than the amount of RED and the data throughput is determined by , which means the RED can be transmitted timely.(ii)For the ISFB scheme, the maximum data throughput is corresponding to . The data through is determined by the amount of RED when and the CDR when .(iii)We can see that the maximum data throughput of the MUFB scheme is larger than that of the ISFB scheme, i.e., the performance of the MUFB scheme is better. By comparing Figures 6(a) and 6(b), a significant gain in the data throughput can be achieved by employing the AOA. This is because, when the AOA is employed, the values of CDR are much larger, as shown in Figure 5. To give further insight, the data throughput with different values of scanning speed is then discussed.

(a)

(b)

Figure 7 depicts the data throughput versus range resolution under , , , and :(i)The simulated results of the ISFB scheme employing AOA are depicted in Figure 7(a), and points , , , and correspond to the maximum data throughput of the ISFB scheme for , , , and , respectively. We can see that the maximum data throughput is getting smaller with the increase of . This is because the larger is, the smaller amount of RED is, leading a smaller maximum data throughput.(ii)The simulated results of the MUFB scheme employing AOA are depicted in Figure 7(b), and points , , , and correspond to the maximum data throughput of the MUFB scheme for , , , and , respectively. By comparing Figures 7(a) and 7(b), in the case of large (such as ), the data throughput of both schemes are the same, which is because the data throughput is determined by the amount of RED, as discussed in Figure 6. Meanwhile, in the case of small (such as ), the data throughput of the MUFB scheme is larger since a larger CDR of the MUFB scheme; that is, the performance of the MUFB scheme is much better than that of the ISFB scheme.(iii)The simulated results of the ISFB and MUFB schemes employing APA are depicted in Figures 7(c) and 7(d), respectively. By comparing the simulated results depicted in Figures 7(a) and 7(b), we can see that a significant gain in the data throughput can be achieved by employing the AOA This is because, when the AOA is employed, the values of CDR are much larger, as shown in Figure 5.

(a)

(b)

(c)

(d)

Figure 8 depicts the range resolution corresponding to the maximum data throughput for different values of for both schemes. A nonincreasing behavior can be observed when the scanning speed is larger. This is because the data throughput is determined by and , the larger (or ) the smaller the data throughput. Thus, with a certain requirement of maximum data throughput, a smaller can be obtained with the increase of . We can see that the of the MUFB scheme is smaller than that of the ISFB scheme, which means that the performance of the MUFB scheme is better. This is because, as mentioned earlier, the CDR of the MUFB scheme is much larger than that of the ISFB scheme, which can meet a higher requirement of data transmission (corresponds to smaller ). Also, take the MUFB scheme as an example, the range resolution employing AOA is much smaller than that employing APA, which implies the AOA case can achieve better performance.

Figure 9 shows the delay time for different values of under , , , and . A monotonic decreasing behavior of delay time can be observed for all schemes. It is because that the amount of RED is getting smaller and CDR is getting larger with the increase of . For , , , and , the delay time of the MUFB scheme is smaller than the ISFB scheme so that the performance of the MUFB scheme is better. For all cases, under a certain , the delay time is the largest when . It is because a smaller causes a smaller amount of data throughput corresponding to a larger delay time. Also, for all cases, the delay time employing APA is much smaller than that employing AOA, which implies the former has a better performance.

(a)

(b)

(c)

(d)

6. Conclusion

In this context, the performance of data transmission for the JRCS has been evaluated. Two spectral coexistence schemes are, respectively, proposed. The first one named the ISFB scheme processes the radar and communication signals independently. The other one, named the MUFB scheme, allows the radar and communication signals to operate on the same subcarriers. Comparing the two schemes, we can find the ISFB scheme brings the advantage of easy execution and interference avoidance while the MUFB scheme significantly improves the performance of data transmission over the ISFB scheme by maximizing the CDR with the presence of radar interference and meanwhile obtaining the optimal power allocation solution of radar power via maximizing radar mutual information. Simulation results have shown the performance of the data transmission for the JRCS, and a significant performance gain in the data transmission can be achieved, compared to the APA case.

Appendix

Proof. with the constraint of total radar power , the optimal allocation solutions of radar, , can be obtained by solving the following problem:Since does not depend on the transmitted waveform , the optimal solutions of problem (A.1) can be obtained by solvingSince and are positive semidefinite Hermitian matrices, the corresponding eigendecomposition can be given byrespectively, where , and , . Therefore, according to Lemma 1, problem (A.2) is equivalent to the following optimization problem:where is the -th diagonal element of . The optimal solutions of the above convex problem can be characterized by the KKT conditions, and the corresponding Lagrangian form is needed, namely,where and are Lagrangian multipliers. Then, the KKT conditions are given as follows:Eliminating , we haveThen, the optimal solution can be given bywhere can be obtained by solvingFor the communication-only subcarriers, there is no radar signal, , with ; meanwhile, for the mix-used subcarriers, we havefor , where is given in equation (A.9). Till now, the result of equation (36) can be obtained.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Jianhua Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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