Performance Analysis of Data Transmission for Joint Radar and Communication Systems

*is paper investigates the problem of data transmission for the joint radar and communication systems (JRCSs). *e 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.


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][5][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][8][9][10][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) e first one aims to develop a dual-function system that simultaneously realizes radar and communication functions [7,11,[13][14][15][16][17][18][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. e 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. is 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) e second direction focuses on the spectral coexistence between radar and communication systems. e authors in [21] present the fundamental spectrumsharing 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. e 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. e authors in [31] analyze the problem of power minimization-based radar waveform design considering the existence of communication signals on the same band. e 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. is is indeed the main topic of this paper. e 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 ΔR and scanning speed f θ , 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 is paper is organized as follows. e received signals at radar and communication receivers are modeled in Section 2. e 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: C denotes the set complex. C N (C N×N ) denotes the set of N × 1 vector (N × N matrix) with entries from C. (·) * , (·) T , and (·) H denote the optimality, transpose, and complex conjugate transpose, respectively. e symbol det(·) indicates the determinant of a square matrix. diag · { } is the diagonal matrix with all elements on the main diagonal. e complex Gaussian distribution is denoted by CN (·, ·). Finally, (x, 0) + represents the positive part of x.

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. e integrated transmitter consists of M r antennas for transmitting radar signal, M c antennas for transmitting communication signal, and N r antennas for receiving the reflected radar signal. e received signal at the communication receiver with N c 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 K subcarriers. Moreover, the frequency spacing Δf among the K 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 M r � M c � M, N r � N c � N in this paper.

Radar Receiver
as the transmitted radar waveform on the k-th subcarrier, and assume K > M; K > N. Let y n � [y n (1), y n (2), . . . , y n (K)] be the received signal of the n-th receive antenna, then we have where h n � [h 1,n , h 2,n , . . . , h M,n ] T are the target response from the m-th integrated transmitter to the n-th receiver, S∈ C K×M is the transmitted waveform matrix with S � [s (1), s (2), . . . , s (K)] T , w n ∈ C 1×K is the noise vector, and w n � [w n (1), w n (2), . . . , w n (K)].
Let Y � [y T 1 , y T 2 , . . . , y T N ], then Y ∈ C K×N can be given by where H ∈ C M×N is the target scattering matrix, For simplicity, we assume the columns of H and W are independent and identically distributed (i.i.d), with distribution □□(0, Σ H ) and □□(0, Σ w ), respectively.
Radar mutual information has been used as a parameter in many scenarios to characterize the radar system [32,[35][36][37]. e more radar mutual information is, the better performance of radar is, which can be derived from the mutual information between the input S and output Y, namely, where h(·) denotes the differential entropy of a random variable.

Communication
, then Y ∈ C K×N can be given by where H ∈ C M×N is the channel response matrix, . For simplicity, we assume that the columns of H, H r , and W are i.i.d with distributions CN(0, Σ), CN(0, Σ r ), and CN(0, Σ w ), respectively. e mutual information between the input S and output Y is given by us, the communication data rate can be given by where K c denotes the number of communication subcarrier.

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. e ISFB scheme is depicted in Figure 2(a). In this scheme, the total K subcarriers are partitioned into two disjoint categories: the K r radar-only subcarriers and the K c communication-only subcarriers; then, we have where B 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. e MUFB scheme is depicted in Figure 2(b). In this scheme, the occupied bandwidth of radar and communication bands satisfies B ≤ (K r + K c )Δf ≤ 2B in this scheme.
e K c subcarriers are split into two categories: the K 1 communication-only subcarriers where only communication signal exists and K 2 mix-used subcarriers where both radar and communication signals exist (K 1 + K 2 � K c ). Combing equation (13), the values of K 1 and K 2 can be, respectively, given by  at is, radar and communication signals can be transmitted on the same subcarriers. e 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. is 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).

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.

Radar Echo Data.
In this section, we analyze the amount of RED under a certain range resolution ΔR and scanning speed f θ . Since the radar receiver is turned off during pulse transmission, the minimum detection range R min can be given by [38] where c denotes the speed of light and T p is the pulse duration. Under a certain detection probability p d , the maximum detection range R max can be given by [38] where D 0 (1) denotes the detection factor, σ is target cross section, T pri denotes the pulse repetition interval (PRI), and k 0 , T 0 , and A e denote Boltzmann constant, absolute temperature, and antenna effective area, respectively. e received radar signals are sampled beginning at t 1 � 2R min /c and ending at t 2 � 2R max (f θ )/c + T p . According to the Nyquist sampling theorem, the sampling interval T sam should not be larger than 1/(K r Δf), and for convenience, we adopt T sam � 1/(K r Δf) in this paper, which means the sampling number N s equals to the number of range cell, that is, and for a chirp signal, we have e number of pulse transmitted in 1 second is e amount of radar echo data in 1 second can be expressed as where N bit denotes the quantify bits without apparent saturation.
From (15), we can find that, with fixed PRI T pri , pulse duration T p , and quantify bit N bit , the amount of RED D (ΔR, f θ ) is determined by range resolution ΔR and maximum detection range R max . e smaller ΔR (or larger R max ) is, the larger amount of D (ΔR, f θ ) is. Moreover, the smaller f θ is, the larger R max is. erefore, there will be a larger D (ΔR, f θ ) with a smaller f θ .

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 P c is allocated to the K c subcarriers according to the corresponding channel conditions without interference from the radar system. Let S c ∈ C K c ×M 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 en, with the constraint of total communication power P c , the maximization problem of CDR can be formulated as where the constant multiplier K c Δf is omitted. Since log 2 det (Σ w ) does not depend on the transmitted waveform S c , problem (17) is equivalent to the following problem: To obtain the optimal solution of (18), a lemma is firstly introduced.
Let the eigendecomposition of Σ be where R Σ � diag(ζ Σ,1 , . . . , ζ Σ,K c ). erefore, the optimal solution of problem (18) can be obtained by solving where ζ s,k is the k-th diagonal element of S c S H c . 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]. e corresponding Lagrangian form of the above optimization problem is where μ k and ] are Lagrangian multipliers. en, the optimal solution ζ * s,k can be obtained, i.e., where ] can be obtained by solving 3.3. Data roughput 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. erefore, define J I as the data throughput for the ISFB scheme, and we have where D (ΔR, f θ ) is given in equation (15). On one hand, J I � D (ΔR, f θ ) means that the CDR C I can meet the requirement of data transmission and the RED can be transmitted timely and there is no transmission delay. On the other hand, J I � C I means the amount of radar echo data D(Δf, f θ ) is larger than the CDR C I , the integrated system needs several PRIs to transmit all the received radar echo data, and the corresponding transmission delay T I is defined as

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 ΔR and scanning speed f θ 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. e CDR is maximized along with the interference from the radar, which is determined by the range resolution and scanning speed.

Communication Data Rate for MUFB Scheme. Let
S r ∈ C K 2 ×M be the transmitted radar waveform matrix on the K 2 mix-used subcarriers, i.e., S r � [s(1), s(2), . . . , s(K 2 )] T . en, from the communication system's point of view, the interference matrix Γ ∈ C K c ×K c caused by radar signal is erefore, the CDR of the MUFB scheme can be given by en, with the constraint of total transmitted communication power P c , the maximization problem of CDR can be formulated as where we have omitted the constant multiplier K c Δf. Since log 2 det( Γ + Σ w )) does not depend on the transmitted waveform S c , the optimal solution of problem (32) can be obtained by solving Define η � diag η 1 , . . . , η K c as the diagonal elements of interference matrix Γ, and the interference from the radar system to the communication system can be given in the following proposition.

proposition 1. Let the eigendecompositions of Σ H and Σ r be
and where R H � diag(ξ H,1 , . . . , ξ H,K r ) and R r � diag(ζ r,1 , . . . , ζ r,K c ). en, the interference from the radar system on the k-th subcarrier can be given by where K � K r + K 1 + 1 and ξ s,K−k denotes the optimal allocated radar power on the k-th subcarrier given by Proof. See Appendix. As a result, let ς s,k be the k-th diagonal element of S c S H c for the MUFB scheme, the optimal power allocation solution ς * s,k 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 μ k and λ are Lagrangian multipliers. en, the optimal solution ς * s,k can be obtained, i.e., where ] can be obtained by solving where D (ΔR, f θ ) is given in equation (15). On one hand, when J M � D (ΔR, f θ ), the RED D can be transmitted timely without transmission delay. On the other hand, when J M � C M , there will be transmission delay, which can be defined as

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 ΔR and scanning speed f θ . 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). e performance of the proposed adaptive optimization algorithm and that of the APA are labeled as "AOA" and "APA," respectively. We assume ξ H,k , ζ r,k , and ζ Σ,k obey where G r and G c denote antenna gains of radar and communication, respectively; λ r,k and λ c,k denote wavelengths of radar and communication on the k-th subcarrier, respectively; R rt ranges between the integrated transmitter and interest of target; R tc ranges between interest of target and communication receiver (without loss of generality, we assume R rt � R tc � R max ); R rc ranges between the integrated transmitter and communication receiver; β H,k , β r,k , and β Σ,k are generated from Gaussian distributions CN ∼ (0, σ 2 H,k ), CN ∼ (0, σ 2 r,k ), and CN ∼ (0, σ 2 Σ,k ), respectively, where σ 2 H,k , σ 2 r,k , and σ 2 Σ,k 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 R max versus f θ under p d � 0.85, 0.9, and 0.95. As expected, a monotonic decreasing behavior of the maximum detection range R max can be observed when the scanning speed f θ is larger. Also, we can see that the R max gets smaller as the p d gets larger. is is because a larger p d needs a larger detection factor, leading to a smaller R max . Figure 4 depicts the amount of RED D for different values of ΔR and f θ . D is determined by sampling interval T sam and the number of sampling N s , as discussed in Section 3.1. We can see that, on one hand, D is getting larger with the decrease of ΔR. A smaller ΔR needs a larger size of the bandwidth of radar band leads to a smaller sampling interval T sam according to the Nyquist sampling theorem. erefore, the smaller ΔR is, the larger D is. On the other hand, a smaller f θ leads to a larger detection area, as shown in Figure 3, which will increase the number of sample N s . us, there will be a larger D along with a smaller f θ .
Under f θ � 300°/s, 200°/s, 100°/s, and 50°/s for both AOA and APA, the CDR for the ISFB and MUFB schemes versus range resolution ΔR 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 ΔR gets larger. is is because the larger ΔR results in the smaller number of radar subcarrier K r . en, more subcarriers are used by the communication system, leading to a larger CDR C I . Also, the CDR in this scheme has no relationship with the scanning speed f θ , 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 ΔR gets larger. e larger ΔR is, the smaller number of mix-used subcarrier K 2 is. en, the smaller interference from the radar system to the communication system leads to a larger CDR C M . Also, we can notice that a smaller f θ leads to a larger CDR C M by reducing the received radar power. As a consequence, the smaller interference to communication leads to a higher CDR C M . (iii) By comparing Figures 5(a) and 5(b), the achieved CDR of the MUFB scheme is larger than the ISFB scheme. is 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. is 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.
For both AOA and APA, Figure 6 depicts the data throughput for different values of ΔR under f θ � 300°/s: (i) For the MUFB scheme, the achieved maximum data throughput is 2.36 × 10 7 bit/s corresponding to ΔR � 4 m. In the case of ΔR ≤ 4 m, the amount of RED D is larger than the CDR C M ; then, the data throughput T M is determined by CDR C M , which cannot meet the requirement of data transmission and leads to data retention. However, in the case of ΔR ≥ 4 m, the CDR C M is larger than the amount of RED D and the data throughput J M is determined by D, which means the RED can be transmitted timely. (ii) For the ISFB scheme, the maximum data throughput is 1.59 × 10 7 bit/s corresponding to ΔR � 6 m. e data through J I is determined by the amount of RED D when Δ ≤ 6 m and the CDR C I when Δ ≥ 6 m. (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. is 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 f θ is then discussed.  ≥ 10m), 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 ΔR (such as ΔR ≤ 6m), 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.    Mathematical Problems in Engineering 7(b), we can see that a significant gain in the data throughput can be achieved by employing the AOA is is because, when the AOA is employed, the values of CDR are much larger, as shown in Figure 5. Figure 8 depicts the range resolution ΔR corresponding to the maximum data throughput for different values of f θ for both schemes. A nonincreasing behavior can be observed when the scanning speed f θ is larger. is is because the data throughput is determined by f θ and ΔR, the larger f θ (or

Mathematical Problems in Engineering
ΔR) the smaller the data throughput. us, with a certain requirement of maximum data throughput, a smaller ΔR can be obtained with the increase of f θ . We can see that the ΔR of the MUFB scheme is smaller than that of the ISFB scheme, which means that the performance of the MUFB scheme is better. is 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 ΔR). 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.     Mathematical Problems in Engineering much smaller than that employing AOA, which implies the former has a better performance.

Conclusion
In this context, the performance of data transmission for the JRCS has been evaluated. Two spectral coexistence schemes are, respectively, proposed. e first one named the ISFB scheme processes the radar and communication signals independently. e 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.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.