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Fundamental Limits on Detection With a Dual-function Radar Communication System

This paper investigates the fundamental limits on the target detection performance with a dual-function multiple-input-multiple-output (MIMO) radar communication (RadCom) systems. By assuming the presence of a point-like target and a communication receiver, closed-form expressions for the maximum detection probability and the transmit waveforms achieving the optimal performance are derived. Results show that for the considered case, the dual-function system should transmit coherent waveforms to achieve the optimal detection performance. Moreover, the angle separation between the target and communication receiver has a great impact on the achievable detection performance.


I. INTRODUCTION
Dual-function radar communication systems (DFRC) integrate radar and communication functions with shared aperture, shared spectrum, and shared waveform [1], [2]. The use of DFRC systems reduces the number of antennas and the hardware cost as well as improving compatibility in spectrally crowded environments. Thus, the design of DFRC systems have received considerable interests in both military and civilian applications [3], [4].
The concept of DFRC system dates back to as early as 1960s. In [5], the authors proposed a position modulated pulse group to perform radar and communication function simultaneously. However, the information bit rates conveyed by the proposed method are low. To improve the data rates of the DFRC system, mixed-modulated scheme are proposed. This scheme intentionally modulates the radar signal (e.g., the linear frequency modulated signal) with a communication signal (see, e.g., [6] and the references therein). Nevertheless, the mixed-modulated signals suffer from high correlation sidelobes. Recently, there have been growing interests in designing dual-function multiple-input-multiple-output (MIMO) radar communication (RadCom) systems. In [7], the authors proposed to deliver the information bits via synthesizing the transmit beampattern of a MIMO radar system. In [8], the authors considered the design of waveforms for dual-function MIMO RadCom systems to simultaneously transmit radar signals and communication symbols. In [9], [10], efficient algorithms were developed to design constant-modulus waveforms for dual-function MIMO RadCom systems.
In this paper, we investigate the fundamental limits on the detection performance of a dual-function MIMO RadCom system. For simplicity we assume that a point-like target and a communication receiver are present. We analyze the maximum probability of detection (for a given probability of false alarm) for the dual-function MIMO RadCom system, and derive the closed-form expression for the waveforms that achieve the optimal performance. Numerical examples are provided to support the analysis.

II. PROBLEM FORMULATION
Consider a colocated dual-function MIMO RadCom system with N T transmit antennas and N R receive antennas, as illustrated in Fig. 1. Let S ∈ C NT×L denote the discrete-time baseband waveform matrix, where L is the code length. For simplicity we assume a point-like target, with the direction of arrival (DOA) denoted by θ t , and a communication receiver with DOA of θ c . Following [11], [12], we can write the received target signal as where α t is the target amplitude, b(θ t ) and a(θ t ) are the steering vectors of the receive array and the transmit array at θ t , respectively, and N is the receiver noise. Let y = vec(Y). Then y can be written as where A(θ t ) = I L ⊗ (b * (θ t )a † (θ t )), s = vec(S), and n = vec(N).
To determine the presence of target in the received signal, we establish the following binary hypothesis testing: Assume that n is white Gaussian, with zero mean and variance of σ 2 . According to the Neyman-Pearson criterion [13], we decide where T h is the detection threshold. Moreover, the detection probability of this detector is given by [14] where dt is the complementary error function, and the target SNR is given by We can observe from (5) that P D is a monotonically increasing function of SNR. Therefore, the maximization of the target SNR results in the maximization of the detection probability.
On the other hand, the signal reaching the communication receiver is given by a † (θ c )S.
Thus, to study the fundamental limits of the target detection performance of the dual-function system, we consider the following optimization problem: where θ c is the DOA of the communication receiver, d c ∈ C L×1 is the desired communication signals, and e t is the total available transmit energy. Once we obtain the optimal solution of (8), we can analyze the achievable detection performance of the dual-function system via (5).

III. WAVEFORM DESIGN AND PERFORMANCE ANALYSIS
To tackle the optimization problem in (8), we defineŜ = 1 NT a(θ c )d c , and let where B ∈ C NT×(NT−1) is a semi-unitary matrix whose column spans the null space of a † (θ c ) (i.e., a † (θ c )B = 0 and B † B = I NT−1 ), and V ∈ C (NT−1)×L is an arbitrary matrix.
By using (9), we can eliminate the equality constraint in (8) and rewrite the energy constraint as Therefore, the optimization problem in (8) can be recast as whereê t = e t − d c 2 2 /N T . It is worth noting that to make the optimization problem in (11) feasible, the transmit energy e t should be larger than d c 2 2 /N T (otherwise,ê t < 0). Note that for the trivial case that θ t = θ c , a † (θ t )B = 0 and the target SNR is given by |α t | 2 d c 2 2 /Lσ 2 . For the nontrivial case, we let q =Ŝ † a(θ t ) and u = B † a(θ t ). Then (11) can be rewritten as Note that q = B(θ t , θ c )d * c , where B(θ t , θ c ) = a † (θ c )a(θ t )/N T is the normalized transmit beampattern at θ c when the array is pointing to θ t . Expanding the objective of (12) (denoted by f (V)), we obtain Note that where V 2 denotes the spectral norm of V, the upper bound is achieved if u = u/ u 2 , and x ∈ C L×1 is an arbitrary normalized vector (i.e., x † x = 1). In addition, we can verify that and the equality holds when whereq = q/ q 2 1 . Combining the results in (15) and (17), we conclude that the maximum of f (V) is achieved if V is given by (17).
Next we derive the optimal waveforms and analyze the maximum SNR that can be achieved by the dual-function system.

A. The Optimal Waveforms
Using (9) and (17), we can write the optimal waveform matrix by Note that BB † = I − a(θ c )a † (θ c )/N . Then after some algebraic manipulations, we can rewrite S as where From (19), we can observe that when only one communication user is present, the dual-function system should transmit coherent waveforms to maximize the SNR (i.e., the system should work in a phased-array mode). In particular, each waveform is a scaled version of d c and the associated beamformer w is a linear combination of a(θ c ) and a(θ t ).

B. The Maximum SNR
Substituting (17) into (13), we can obtain that the maximum is given by Note that where G = |B(θ t , θ c )| is the normalized gain of the transmit array at the direction of the communication user. Moreover, by using the fact that BB † = I − a(θ c )a † (θ c )/N T , we can obtain Therefore, the maximum achievable SNR under a communication constraint is given by From (24), we can observe that: • If the communication user is at the sidelobe, i.e., G 2 ≈ 0, the maximum SNR is given by Note that the maximum SNR that can be achieved by a mono-function system (i.e., we only consider target detection and remove the first constraint in (8)) is As a result, the SNR loss (compared with the monofunction system) brought by the communication constraint isê t /e t . If e t d c 2 2 /N T , the SNR loss is negligible; otherwise if e t ≈ d c 2 2 /N T , the SNR loss will be significant. • If the communication user is at the same direction as the target (i.e., the mainlobe), i.e., G 2 = 1, the maximum SNR is given by and the associated SNR loss is given by d c 2 2 /(e t N T ). Interestingly, If e t d c 2 2 /N T , the SNR loss will be significant.
, the maximum SNR is given by (See the Appendix for a proof) i.e., there will be no SNR loss.

IV. NUMERICAL RESULTS
In this section, we provide numerical examples to demonstrate the performance of the dual-function system. We consider a dual-function system with N T = 16 transmit antennas and N R = 16 receive antennas. The inter-element spacings of the transmit array and the receive array are d T = λ/2 and d R = λ/2, respectively, where λ is the wavelength. The code length is L = 128. The DOAs of the target and the communication receiver are θ t = 0 • and θ c = 32 • , respectively. The desired communication signal d c is a binary phase shift keying modulated (BPSK) signal with amplitude of 0.1. The transmit energy is e t = 1.5 d c 2 2 /N T . When analyzing the detection performance of the system, the probability of false alarm is fixed to be P FA = 10 −6 .   ., d c ). We can observe that the synthesized signal overlaps the desired signal, meaning that the dual-function system can communicate with the user. Fig. 3 shows the detection probability of the dual-function system versus |α t | 2 /σ 2 (we call it input SNR hereafter). For this parameter setting, to attain a detection probability of 0.9, the input SNR should be higher than 1.8 dB.   4 shows the detection performance of the dual-function system versus θ c , where the red dashed line denotes the performance of the mono-function system. We can observe that for some directions of the communication receiver (±2.5 • for this parameter setting), the dual-function system has the same performance as the mono-function system. This is consistent with the theoretical analysis in Section III.B. If the communication receiver is at the mainlobe, there is an SNR loss of 1.76 dB, and the detection probability decreases to 0.95. For the case that the communication receiver is at the sidelobe, the SNR loss can be as large as 3.77 dB and the associated detection performance degrades significantly.

V. CONCLUSIONS
The fundamental limits on the detection performance of a dual-function MIMO RadCom system was studied in this paper. A closed-form expression for the optimal waveform matrix that maximized the SNR under a communication constraint was derived. The achievable SNR and the associated detection performance of the dual-function system was analyzed. Results showed that the angle separation between the target and the communication receiver impacted the achievable SNR significantly.
As a result, the maximum is achieved if sin ψ = cos φ = b 0 / a 2 0 + b 2 0 , and the maximum value is e t N T .