FDA-MIMO-Based Integrated Sensing and Communication System with Complex Coefficients Index Modulation for Multi-Target Sensing
Jiangwei Jian Bang Huang Wenkai Jia Mingcheng Fu Wen-Qin Wang Qimao Huang





	










FDA-MIMO-Based Integrated Sensing and Communication System with Complex Coefficients Index Modulation for Multi-Target Sensing
    Jiangwei Jian, Student Member, IEEE,
		Bang Huang,
		Wenkai Jia,
		Mingcheng Fu,		
		Wen-Qin Wang, Senior Member, IEEE
		and Qimao Huang
	This work was supported by National Natural Science Foundation of China 62171092, and in part by the Postdoctoral Program for Innovation Talents under Grant BX20240054. (Corresponding author: Wen-Qin Wang).
	Jiangwei Jian, Wenkai Jia, Mingcheng Fu, and Wen-Qin Wang are with School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, P. R. China. (Email: jianjiangwei@std.uestc.edu.cn; mrwenkaij@126.com; fumcheng@163.com; wqwang@uestc.edu.cn).
		
	Qimao Huang is with School of Physics, University of Electronic	Science and Technology of China, Chengdu, 611731, P. R. China. (Email: huangqimao@std.uestc.edu.cn). 
	
	Bang Huang is with the Computer, Electrical and Mathematical Science and Engineering (CEMSE) division in King Abdullah University of Science	and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia (Email: huangbang@std.uestc.edu.cn).
		

    September 9, 2024
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§ ABSTRACT 	
The echo signals of frequency diverse array multiple-input multiple-output (FDA-MIMO) feature angle-range coupling, enabling simultaneous discrimination and estimation of multiple targets at different locations. In light of this, based on FDA-MIMO, this paper explores an sensing-centric integrated sensing and communication (ISAC) system for multi-target sensing. On the transmitter side, the complex coefficients index modulation (CCIM) scheme is designed, which carries extra bits by selecting complex coefficients from the coefficient vector. At the sensing receiver, we propose the FDA-MIMO-based spatial spectrum multi-target estimation (SSMTE) method, which first jointly estimates the angle and distance of targets and then estimates the velocities. To reduce the sensing computational complexity, the low-complexity spatial spectrum estimation (LCSSE) algorithm is proposed. LCSSE reduces the complexity without degrading the sensing performance by converting the joint angle-range search into two one-dimensional searches. To address the range ambiguity caused by frequency offset, a frequency offset design criterion (FODC) is proposed. It designs the integer and fractional components of the frequency offset to ensure the ambiguity distance exceeds the maximum sensing range, thereby alleviating parameters pairing errors. Moreover, the closed-form expressions for the bit error rate (BER) tight upper bound and the Cramér-Rao bound (CRB) are derived. Simulation results show that the proposed system excels in multi-target sensing and communications.

	






	Multi-target sensing, frequency diverse array multiple-input multiple-output (FDA-MIMO), complex coefficients index modulation (CCIM), spatial spectrum multi-target estimation (SSMTE), low-complexity spatial spectrum estimation (LCSSE), frequency offset design criterion (FODC).








§ INTRODUCTION

The sharing of spectrum and hardware between radar and communications, termed integrated sensing and communication (ISAC), is emerging as a new trend in next-generation wireless networks <cit.>. On one hand, the ISAC technique enhances system spectrum efficiency and reduces hardware costs. On the other hand, it offers pervasive communication, sensing, and intelligence, fostering various emerging applications like autonomous driving, smart homes, and intelligent communications <cit.>. Depending on differences in application focus, ISAC technology can be categorized into  sensing-centric, communication-centric, and communication-sensing trade-off designs <cit.>.

The sensing-centric design regards the targets sensing performance as the primary function, which is the focus of this paper. As early as 1960, Mealey et al. proposed an ISAC system by embedding communication data within radar pulse intervals <cit.>. Subsequently, the application of LFM waveforms in ISAC schemes provided attractive sensing performance due to their larger pulse width-time product <cit.>. However, the phased-array radar used lacked waveform diversity gain. To address this, multiple-input multiple-output (MIMO)-based ISAC systems with waveform diversity degrees of freedom (DoFs) have garnered widespread attention <cit.>. Specifically, <cit.> designed the beampattern for the MIMO radar, where the mainlobe is only used for target sensing, and the level of sidelobe reflects communication symbols. A more promising approach is to combine index modulation (IM) technique with MIMO to increase communication rates. In this regard, <cit.> embedded communication data into spatio-spectral passbands and stopbands, and optimized the beampattern to guarantee the sensing performance. Besides,<cit.> proposed activating partial transmit antennas to carry additional index bits, achieving target perception through sparse array. As a further study, an ISAC system based on frequency agile radar was proposed in <cit.>, which utilized frequency offset selection to convey index bits and proposed the multi-target sensing method. Later, <cit.> extended this work by jointly IM in frequency and spatial dimensions, which further improved the multi-target estimation accuracy via compressed sensing.


On the flip side, the communication-centric and trade-off designs focus on the high communication performance and sensing-communication balance, respectively. In terms of communication-centered design, in <cit.>, the sensing function was attached to the spread spectrum communication systems to realize the dual function. Motivated by this, the orthogonal frequency division multiplexing (OFDM) based ISAC system was proposed in <cit.>, which estimates targets by processing the OFDM echo signals in the fast-slow time domain. Later, MIMO was combined with OFDM, proposed as MIMO-OFDM-based ISAC systems, to improve the communication and sensing performance. These encompass subcarriers allocation and optimization <cit.>, channel interference exploitation <cit.>, precoding <cit.>, waveform optimization <cit.>, and uplink design <cit.>. In the trade-off design, the focus lies in optimizing joint sensing and communication metrics through precoder design. This includes optimizing metrics such as the users sum rate and sensing beampattern optimization <cit.>, Cramér-Rao bound (CRB) and communication signal-to-interference-plus-noise ratio (SINR) optimization <cit.>.

However, aforementioned ISAC systems mainly relied on phased arrays (PAs), whose   steering is only angle-dependent without range information. An emerging frequency diverse array (FDA)-MIMO technique extends the DoFs of signal processing to the angle-range dimension by introducing a frequency offset among the adjacent elements <cit.>. Inspired by this, FDA-MIMO radar has been applied in high-resolution target estimation <cit.>, target detection <cit.>, range clutter suppression <cit.>, mainlobe interference suppression <cit.>, and exhibited superior radar performance to the PA-based MIMO in the range dimension. Moreover, FDA-MIMO can also benifit communications. <cit.> described how the angle-range coupling character of FDA was utilized to guarantee communication security for specific location users, which cannot be achieved by the PA. <cit.> employed the inherent frequency offset resources of FDA as IM entities, further enhancing the communication rates and bit error rate (BER) performance.

FDA-MIMO has demonstrated attractive performance in both radar and communications, driving its incorporation into ISAC systems <cit.>. Specifically, <cit.> embedded communication bits into the spreading sequence of each pulse, yielded satisfactory sensing performance. Another approach involved embedding constellation symbols into multiple sub-pulses witin one pulse, enabling simultaneous communication and sensing <cit.>. Further, <cit.> proposed embedding phase modulation symbols into FDA-MIMO radar waveforms and optimized the transmit beamforming to achieve the sensing-communication performance balance. Moreover, <cit.> convey extra bits by permutating the transmit frequency offsets, which improved communication rates and CRB performance. However, the challenges of enhancing communication rates and accurately estimating targets persist. To address this, <cit.> proposed the frequency offset permutation index modulation (FOPIM) scheme, which involved selecting and permutating frequency offsets to carry additional bits, along with the target estimation method. This approach achieved superior sensing performance compared to MIMO-based ISAC systems.


Nevertheless, the aforementioned FDA-MIMO-based ISAC system failed to consider the multi-target sensing, and how to suppress the range periodicity during multi-target estimation remains an open question. Moreover, the FOPIM method only activates partial frequency offsets, leading to the spectrum wastage. Motivated by this, this paper explores the FDA-MIMO-based ISAC system in multi-target scenarios and proposes a complex coefficients index modulation (CCIM) transmission scheme independent of frequency offsets. The main contributions of our work are listed as follows:


  1)
In this work, we propose the complex coefficients index modulation (CCIM) scheme to enhance the communication rate. In the CCIM method, each antenna selects a complex coefficient from a normalized complex coefficient vector to transmit additional bits and conveys an independent quadrature amplitude modulation (QAM) symbol. Additionally, the closed-form expressions for the system BER tight upper bound are derived to evaluate the communication performance.


  2)
At the sensing receiver, a spatial spectrum multi-target estimation (SSMTE) method is proposed for multi-target sensing. Specifically, within the target-containing range bins, the angles and ranges of targets are jointly estimated in the spatial spectrum of FDA-MIMO. Subsequently, the least squares (LS) is employed to estimate the velocity. The SSMTE method suffers from high complexity due to its angle-range two-dimensional (2-D) search. To tackle this issue, the low-complexity spatial spectrum estimation (LCSSE) approach is proposed, which dramatically reduces the complexity by converting the 2-D angle-range search into two one-dimensional (1-D) searches. Simulation results show that LCSSE and SSMTE methods have similar sensing performance.


  3)
The FDA-MIMO exhibits periodic variation in its steering vector with range, resulting in range ambiguity in target estimation. To tackle this issue, the frequency offset design criterion (FODC) is designed in this paper. FODC designs the integer and fractional components of each transmit frequency offset to ensure that the range periodicity of the steering vector exceeds the maximum sensing distance, thereby mitigating range ambiguity in multi-target estimation. Moreover, we derive closed-form expressions for the system CRB performance.

	
	
	

The rest of this paper is organized as follows. Section <ref> proposes the CCIM approach for the FDA-MIMO-based ISAC system. Section <ref> discusses the signal processing of system sensing and communication receivers. Section <ref> analyzes the theoretical performance of the system CRB, complexity and BER. Finally, simulation results are discussed in section <ref>, and the paper is concluded in section <ref>.


Notations: ^T, ^ and ^† stand for the transpose, conjugate and conjugate transpose operations, respectively. 𝐈_N denotes the identity matrix of order N. ⌊·⌋, ! and Γ (·) denote the floor function, factorial and Gamma function, respectively. Tr[·] represents the trace operation and j=√(-1). Re{·} and Im{·} are the real part and the imaginary part operators, respectively. ⊙ and ⊗ stand for the Hadamard product and Kronecker product operations, respectively. ⌊·⌋ _LCM denots the least common multiple operation. * represents the convolution operation. diag() denotes the vector diagonalization operation.






§ SYSTEM MODEL


This paper considers an ISAC system as shown in Fig. <ref>. The FDA-MIMO base station (BS) equips N transmit antennas and M receive antennas for sensing G targets, while serving a communication user equipped U antennas. On the transmitter side of ISAC systems, IM techniques are widely adopted to enhance the system communication rates <cit.>. Although some recent works have combined FDA with IM for ISAC systems, they carried additional information by activating some frequency offsets <cit.>, which resulted in a waste of spectrum resources. To overcome this drawback as well as to further enhance the communication rate, the CCIM method is proposed in this paper.


The proposed CCIM scheme carries extra bits by combining the constellation symbols with elements in a complex coefficient vector. Specifically, we generate a normalized complex coefficient vector c=[ c_1,⋯ ,c_j,⋯ ,c_J ] ^T∈ℂ ^J× 1 with c^†c/J=1, where the elements are shared with the communication user. We define a K pulse repetition interval (PRI) as a coherent processing interval (CPI) and each PRI is of length T. In the kth PRI, the trasmitted constellation symbol vector is denoted as 𝐱_k=[ x_1^k,⋯ ,x_n^k,⋯ ,x_N^k ] ^T∈ℂ ^N× 1, where x_n^k denotes the L-ary unit energy QAM symbol. Then, the CCIM symbol is designed as

    x̃_n^k=c^Tς_n^k=c_i_n,kx_n^k,

where 

    ς_n^k=[ 0,⋯ ,i_n,kthx_n^k,⋯ ,0 ] ^T∈ℂ ^J× 1,

denotes the complex coefficient selection vector of the nth antenna at the kth PRI. c_i_n,k stands for the complex coefficient selected from c, whereas i_n,k representing the index of c_i_n,k in c. From (<ref>) and (<ref>), we can claim that the proposed CCIM method can carry N× (⌊log _2J ⌋ +log _2L) bits in one transmission.

The transmitter, namely FDA-MIMO BS, is considered as a uniform linear array (ULA). The transmit frequency of the nth BS antenna is designed as

    f_n=f_c+(n-1) Δ f,

where f_c denotes the common carrier frequency, whereas Δ f denotes the frequecy offset increment. Following the proposed CCIM scheme, the transmit signal of the nth antenna at the kth PRI is expressed as 

    s_n^k(t) =ϱ (t-kT) x̃_n^ke^j2π [ f_c+(n-1) Δ f ] t,

where k=0,1,⋯ ,K-1. ϱ (t) is the unit energy baseband waveform with the pulse duration T_W, which satisfies the following orthogonality <cit.>:

    ∫_-∞^∞ϱ (t)ϱ ^†(t-τ )e^j2π( m-m') Δ ftdt={[ 1,m=m'; 0,m m';        ]. ,∀τ.









§ SYSTEM COMMUNICATION AND SENSING FUNCTIONS


In this section, we model the received signals of the communication and sensing receivers, as well as design signal processing methods.



 §.§ Sensing receiver

We assume that the locations of G point targets in Fig. <ref> is { (R_1,θ _1) ,⋯ ,(R_g,θ _g) ,⋯ ,(R_G,θ _G) } and the propagation path with the base station is the line-of-sight <cit.>. Then, on the BS side, the received signal of the mth antenna in the kth PRI can be written as

    y_m^k( t ) =   ∑_n=1^N∑_g=1^Gξ _ge^j2πℱ _gts_n^k(t-τ _m,n,g)
    ≈   ∑_n=1^N∑_g=1^Gξ _ge^j2πℱ _gtϱ  (t-kT-τ _g)x̃_n^ke^j2π (f_c+Δ f_n) t
       × e^-j2πΔ f_n2R_g/ce^j2πf_c(n-1) d_1sinθ _g/ce^j2πf_c( m-1) d_2sinθ _g/c,

where Δ f_n=(n-1)Δ f denots the frequency offset of the nth transmit antenna. τ _m,n,g=2R_g-(n-1) d_1sinθ _g-(m-1) d_2sinθ _g/c represents the delay between the nth transmit antenna and the mth receive antenna for the gth target. d_1 and d_2 denote the spacing of neighboring elements in the transmit and receive arrays, respectively. c represents the light speed. ξ _g is the reflection coefficient of the gth target, which absorbs the constant term e^-j2π f_c2R_g/c <cit.>. Note that the approximation ϱ (t-kT-τ _m,n,g) ≈ϱ (t-kT-τ _g) is considered in (<ref>) under the narrow-band assumption. The terms e^j2πΔ f_n(n-1) d_1sinθ _g/c, e^j2πΔ f_n(m-1)d_2sinθ _g/c are tiny enough to be ignored <cit.>. ℱ _g≈2v_g/cf_c and v_g denote the Doppler shift and velocity of the gth target, respectively. Note that, similar to <cit.>, the Doppler spreading from the frequency offset is ignored in this paper.


The sensing receiver structure is shown in Fig. <ref>, which can also be deployed to the communication user without additional design. The received signal is first down-converted with e^-j2π f_ct, and followed by the N-channel demodulator. In the nth channel, the down-converted signal is multipled by e^-j2πΔ f_nt and then match-filtered by ϱ (t). Following this, the filtered signal of the nth channel of the mth receive antenna can be expressed as <cit.>

    y_m^k=    y_m^k(t) e^-j2π f_cte^-j2πΔ f_nt *ϱ (t)
    		
    
    		=   ∑_g=1^Gξ _ge^j2πψ _g^kx̃_n^ke^-j2π (n-1) Δ f2R_g/c
       × e^j2πf_c(n-1) d_1sinθ _g/ce^j2πf_c(m-1) d_2sinθ _g/c.

where ψ _g^k=ℱ _g (k-1) T <cit.>. Then, the N outputs of the mth receive antenna can be stacked into a vector as

    𝐲_m^k   =[ y_m,1^k,⋯ ,y_m,n^k,⋯ ,y_m,N^k ] ^T
    		
       =∑_g=1^Gξ _ge^j2πψ _g^k𝐱̃_^k⊙𝐚_T (R_g) ⊙𝐚_T(θ _g) e^j2πf_c (m-1) d_2sinθ _g/c,

where 

    𝐚_T(R_g) =[ 1,⋯,e^jφ _n(R_g),⋯ ,e^jφ _N(R_g)] ^T,

and

    𝐚_T(θ _g) =[ 1,⋯,e^jϕ _n(θ _g),⋯ ,e^jϕ _N(θ _g)] ^T,

represent the transmit range and angle steering vectors, respectively. Note that φ _n(R_g) =-2πΔ f_n2R_g/c and ϕ _n( θ _g) =2πf_c(n-1) d_1sinθ _g/c in (<ref>) and (<ref>). 𝐱̃_^k=[ x̃_1^k,x̃_2^k,⋯ ,x̃_N^k] ^T denotes the emitted CCIM symbol vector.

Further, the demodulated outputs of all channels of the M antennas can be written as

    𝐲^k   =[ 𝐲_1^k^T,⋯ ,𝐲_m^k^T,⋯ ,𝐲_M^k^T ] ^T
    		
       =∑_g=1^Gξ _ge^j2πψ _g^k𝐚_R( θ _g ) ⊗[ 𝐱̃^k⊙𝐚_T( R_g ) ⊙𝐚_T( θ _g ) ]+𝐧_k,

where 

    𝐚_R(θ _g) =[ 1,⋯ ,e^jω _m( θ _g ),⋯ ,e^jω _M(θ _g)] ^T,

stands for the receive steering vector with ω _m(θ _g) =2πf_c(m-1) d_2sinθ _g/c. 𝐧_k∼𝒞𝒩  (0,σ _1^2𝐈_M× N) represents the receive noise vector with σ _1^2 denoting the noise power.

From (<ref>), one can observe that the communication symbol term, 𝐱̃^k, degrades the sensing performance. To address this dilemma, we introduce a sensing compensation vector, as

    𝐛_R=1_M⊗𝐱̃_^k,

where 1_M∈ℂ ^M× 1 denotes the all-one vector. Consequently, the compensated receive signal in the kth PRI can be expressed as 

    𝐲̃_^k   =𝐲_^k⊙𝐛_R
    		
       =∑_g=1^Gξ _ge^j2πψ _g^k𝐚_R( θ _g ) ⊗ [ 𝐚_T( R_g ) ⊙𝐚_T( θ _g ) ]+𝐧_k.


Equations (<ref>) and (<ref>) show that at the sensing receiver, the interference of communication symbols can be removed by compensating the received data using prior communication information. In other words, the proposed system does not need to consider communication and sensing balance. This is facilitated by the orthogonality between the transmitted waveforms, which allows the receiver to process the data from each demodulation channel separately. Similar methods are reported in <cit.>.


In the sequel, our focus turns to the estimation of range, angle, and velocity. Although conventional MIMO radars are capable of angle, range, and velocity estimations, the issue of how to pair the estimated parameters remains open. The FDA-MIMO can fill this gap well. From (<ref>), one can observe that the range and angle parameters are coupled <cit.>. With this observation, this paper proposes the FDA-MIMO-based SSMTE method. 

Specifically, as pointed by <cit.>, the matched filtering (also named as pulse compression) of the signal in (<ref>) will produce peaks in the range bins, where the targets are located. Thus, we can obtain the coarse range estimations of G targets as { r_1,⋯ ,r_g,⋯ ,r_G } <cit.>. r_g=𝒫 _gΔ r represents the principal range of the gth target, where 𝒫 _g and Δ r=c/2B represent the range bin number and bin size, respectively. Note that B=Δ f denotes the bandwidth of the baseband signal. In other words, the error margin of the true range R_g and the coarse estimate r_g for the gth target is within a bin size, i.e., (R_g-r_g) ∈ [0,Δ r]. 

Then, we estimate the angle of the gth target. Based on the spatial spectrum definition of FDA-MIMO <cit.>, we construct the joint range-angle spatial spectrum estimation as

    [ R̂_g,θ̂_g ] =
    		
       argmax_[            θ̂_g∈ [-90,90] ,; R̂_g∈ [r_g-Δ r/2,r_g+Δ r/2];                             ]1/| 𝐚_TR^†(R̂_g,θ̂_g) 𝐐^-1𝐚_TR(R̂_g,θ̂_g) |,

where R̂_g and θ̂_g denote the estimations of R_g and θ _g. 𝐐=1/K∑_k=1^K𝐲̃_^k(𝐲̃^k) ^† denotes the sampling covariance matrix of the received signal within one CPI <cit.>,

    𝐚_TR(R̂_g,θ̂_g) =𝐚_R(θ̂_g) ⊗ [ 𝐚_T(R̂_g) ⊙𝐚_T(θ̂_g) ] ,

denotes the transmit-receive steering vector. 𝐚_R(θ̂_g), 𝐚_T(R̂_g), 𝐚_T(θ̂_g) are calculated by (<ref>), (<ref>), (<ref>), respectively.

To estimate targets velocities, the received date of K PRI are stacked into a matrix, as

    𝐘   =[𝐲^1,⋯ ,𝐲^k,⋯ ,𝐲^K]
    		
       =𝐀𝐃+𝐍,

where 𝐀=[ 𝐚_TR^(R_1,θ _1) ,𝐚_TR^(R_2,θ _2) ,⋯ ,𝐚_TR(R_G,θ _G) ] ∈ℂ ^MN× G denotes the targets manifold matrix. 𝐃=[ ψ_1(ℱ _1) ,ψ_2(ℱ _2) ,⋯ ,ψ_G(ℱ _G) ] ^T∈ℂ ^G× K is the targets Doppler phase matrix with ψ_g(ℱ _g) =[ e^j2πψ _g^1,e^j2πψ _g^2,...,e^j2πψ _g^K ] ^T. 𝐍=[ 𝐧_1,𝐧_2,⋯ ,𝐧_K ] ∈ℂ ^MN× K represents the receive noise matrix over K snapshots.

Then, the LS method estimation is employed to estimate velocities. With the angle and range estimations in (<ref>), we can write the estimated manifold matrix as 𝐀̂=[ 𝐚_TR( R̂_1,θ̂_1 ) ,𝐚_TR( R̂_2,θ̂_2 ) ,⋯ ,𝐚_TR( R̂_G,θ̂_G ) ]. The targets Doppler phase matrix is estimated as

    𝐃̂=𝐃min𝐘-𝐀̂𝐃 ^2.

Solving (<ref>) yields 𝐃̂=( 𝐀̂^T𝐀̂ ) ^-1𝐀̂^T𝐘. Let 𝐃̂_F be the first K-1 columns of 𝐃̂ and 𝐃̂_B be the second to Kth columns of 𝐃̂. One can observe that the Doppler phase matrix is a Vandermont matrix. Therefore, there exists a rotation vector 𝐝(ℱ̂) =[ e^j2πℱ̂_1T,⋯ ,e^j2πℱ̂_gT,⋯ ,e^j2πℱ̂_GT ] ^T∈ℂ ^G× 1 satisfying

    𝐃̂_B^T=𝐃̂_F^T𝐄_ℱ̂,

where 𝐄_ℱ̂=diag[ 𝐝( ℱ̂ ) ]. Then, 𝐄_ℱ̂ is calculated as 𝐄_ℱ̂=( 𝐃̂_F𝐃̂_F^T ) ^-1𝐃̂_F𝐃̂_B^T. 

Finally, the velocity of the gth target is estimated as

    v̂_g=c·angle (κ _g,g)/4f_cπ T,

where κ _g,g denotes the gth diagonal element of 𝐄_ℱ̂. angle( ·) means the phase-taking operation.




 §.§ Low-complexity Spatial Spectrum Estimation

Inspecting (<ref>) reveals that the SSMTE method requires the joint search of 2-D spatial spectrum, which suffers from high complexity. To address this problem, we propose the LCSSE algorithm, which estimates the target angle, distance and velocity by the three-time 1-D search, respectively. Specifically, decomposing the denominator term of (<ref>) yields

    (R_g,θ _g)    =𝐚_TR^†(R_g,θ _g) 𝐐^-1𝐚_TR( R_g,θ _g ) 
    		
       =𝐚_T^†(R_g) 𝐙(θ_g ) 𝐚_T(R_g),

where 𝐙(θ_g) =[ 𝐚_R(θ _g) ⊗diag(𝐚_T(θ _g))]^†𝐐^-1[ 𝐚_R(θ _g) ⊗diag(𝐚_T(θ _g)) ]. Then, 𝐙(θ_g) is chunked as 

    𝐙(θ _g) =[ 
    			z_1(θ _g)   𝐳_2(θ _g)
    𝐳_3(θ _g)   𝐳_4(θ _g)
    ] ,

where z_1(θ _g) =z_1,1(θ _g), 𝐳_2(θ _g) =[ z_1,2(θ _g) ,z_1,3(θ _g) , ⋯ ,z_1,N(θ _g) ], 𝐳_3(θ _g) =𝐳_2(θ _g) ^†, 𝐳_4(θ _g) =[ 
	z_2,2(θ _g)   ⋯   		z_2,N(θ _g)
⋮   ⋱   ⋮

	z_N,2(θ _g)   ⋯   		z_N,N(θ _g)
]. 

Let 𝐚_T(R_g) =[1,𝐚_T(R_g) ^T] ^T with 𝐚_T(R_g)=[ e^jφ _2(R_g),e^jφ _3(R_g),⋯ ,e^jφ _N(R_g) ] ^T. The cost function ( R_g,θ _g ) is formulated as

    (R_g,θ _g) =   z_1(θ_g) +𝐚_T^†(R_g) 𝐳_2^†(θ_g) +𝐳_2(θ_g) 𝐚_T(R_g)
    		
       +𝐚_T^†(R_g) 𝐳_4(θ_g) 𝐚_T(R_g) .


The partial derivative of (R_g,θ _g) with respect to 𝐚_T(R_g) yields

    ∂ (R_g,θ _g)/∂𝐚_T(R_g)=2𝐳_2^†(θ _g) +2𝐳_4(θ _g) 𝐚_T(R_g) .


Let (<ref>) equals to 0, we have 𝐚_T( R_g ) =-𝐳_4(θ _g) ^-1𝐳_2^†(θ _g). Applying 𝐚_T( R_g ) to (<ref>), the angle θ _g is then estimated by

    [θ̂_g] =argmax_θ _g∈ [-90,90]1/|z_1(θ _g) -𝐳_2(θ _g) 𝐳_4(θ _g) ^-1𝐳_2^†(θ _g)|.

The targets' angle estimations are obtained by searching for the peaks of (<ref>), named by θ̂:(θ̂_1,⋯ ,θ̂_g,⋯ ,θ̂_G  ).

Although the targets' angle parameters have now been obtained through (<ref>), however, their corresponding distances are unknown. To handle this, the angle estimations are brought into (<ref>) one by one. Thereafter, the distance corresponding to θ̂_g is estimated as

    [R̂_g] =argmax_R_g∈A_r1/| 𝐚_TR^†(R_g,θ̂_g) 𝐐^-1𝐚_TR(R_g,θ̂_g) | ,

where A_r=[r_g-Δ r/2,r_g+Δ r/2] ∪⋯∪ [r_G-Δ r/2,r_G+Δ r/2] denotes denotes the distance search area.

In summary, in the proposed LCSSE algorithm, the paired angle and distance estimations of the targets are obtained by (<ref>) and (<ref>), which are recorded as (R̂_1,θ̂_1) ,⋯ , (R̂_G,θ̂_G). Finally, the velocity estimations for each paired target are calculated by (<ref>).




 §.§ Frequency Offset Design Criterion for Resistance to Range Estimation Ambiguity


Recalling back to (<ref>), an unexpected observation is that the transmit range steering vector is a periodic function of distance with period c/2Δ f, i.e., 𝐚_T(R_g) =𝐚_T(R_g+𝔦c/2Δ f), where 𝔦 denotes a positive integer. This characteristic can lead to errors in the pairing of targets angles and ranges. For example, if the positions of target 1 and target 2 are (R_1,θ _1) and (R_1+c/2Δ f,θ _2), respectively. Then, the range of target 2 may be estimated to be R_1 by (<ref>). Namely, the target 1 may be misestimated as (R_1,θ _2).

To address this problem, an effective way is to design the frequency offset of each transmit antenna, so that the phase does not flip periodically over the desired range. Take this into mind, we propose the frequency offset design criterion for resistance to range estimation ambiguity.

Specifically, we design the frequency offset of the nth antenna as Δ f_n=ε _nΔ f. Note that to guarantee the orthogonality between the transmitted signals, the following condition Δ f_n+1-Δ f_n⩾Δ f should be satisfied. We split ε _n as

    ε _n=𝔦 _n+𝔮 _n,

where 𝔦 _n and 𝔮 _n denote the integer and fractional parts of ε _n, respectively. Then, the nth element in 𝐚_T(R_g) is rewritten as 

    e^-j2πΔ f_n2R_g/c=e^-j2π𝔦 _nΔ f2R_g/ce^-j2π𝔮 _nΔ f2R_g/c.

For the e^-j2π𝔦 _nΔ f2R_g/c term, the distance period is r_𝔦 _n=c/2Δ f. On the other hand, the distance period of the e^-j2π𝔮 _nΔ f2R_g/c term is r_𝔮 _n=c/2Δ f𝔮 _n. Let the distance period of e^-j2πΔ f_n2R_g/c be r_n, then r_n should be a positive integer multiple of r_𝔦 _n and r_𝔮 _n. By guiding of this, we have 

    r_n=ϖ _nr_𝔦 _n=ς _nr_𝔮 _n,

where ϖ _n and ς _n denote the positive integers to be determined, respectively. In other words, (<ref>) holds at ϖ _n/ς _n=1/𝔮 _n.

Therefore, the period of the transmit range steering vector 𝐚_T(R_g) in the range dimension is equal to the least common multiple (LCM) of the distance periods of all its elements, as

    r   =⌊ r_1,⋯ ,r_n,⋯ ,r_N ⌋ _LCM
       =c/2Δ f×⌊ϖ _1,⋯ ,ϖ _n,⋯ ,ϖ _N ⌋ _LCM.

In other words, r and r_n should remain positive integer multiples, i.e., r/r_n=𝔦.

In practice, the maximum system sensing range is cT/2. Then the frequency offsets should be designed according to (<ref>)-(<ref>) such that r⩾cT/2 to ensure that no distance ambiguity occurs in the interest range.





 §.§ Communication receiver

Let ( R̅_u,θ̅_u ) be the location of the communication user. Similar to (<ref>), the received signal of the uth antenna in the kth PRI can be expressed as

    y_u^k( t ) =   ∑_n=1^Nh_u,ns_n^k( t-τ _u,n)
    ≈   ∑_n=1^Nh_u,nϱ( t-kT-τ _u ) x̃_n^ke^j2π( f_c+Δ f_n^) t
       × e^-j2πΔ f_n2R̅_u/ce^j2πf_c( n-1 ) d_1sinθ̅_u/c,

where τ _u,n=R̅_u-(n-1) d_1sinθ̅_u-( u-1 ) d_3sinθ̅_u/c and h_u,n∼𝒞𝒩 (0,σ _C^2) stand for the delay and the channel coefficient between the nth transmit antenna and the uth receive antenna. d_3 represents the adjacent spacing of the receiver. Note that the term e^-j2π f_c2R̅_u/c  is absorbed into the term h_u,n.

The receiver structure of the communication user is shown in Fig. <ref>. Similar to the signal demodulation process of the sensing receiver, the output sampled signals of the nth channel of the uth receive antenna is

    y̅_u,n^k   = y_u^k( t ) e^-j2π f_cte^-j2πΔ f_nt*ϱ(t)  +n_u,n
       =x̃_n^kh_u,n+n_u,n,

where n_u,n∼𝒞𝒩 (0,σ _2^2) is the receive noise. Note that the constant term e^-j2π( Δ f_n^2R̅_u/c-f_c(n-1) d_1sinθ̅_u/c) is absorbed into h_u,n. 

Inspecting (<ref>) reveals that the baseband signals from the N transmit antennas can be separated at the receiver. Leveraging this property, we can combine all demodulated outputs from the same transmit antenna to improve the system BER performance. Specifically, we stack the outputs of the nth channels of U receive antennas into a vector as

    𝐲̅_n^k   =[ y̅_1,n^k,⋯ ,y̅_u,n^k,⋯ ,y̅_U,n^k] 
    		
       =c_^Tς_n^k𝐡_n^+𝐧_u^,

where 𝐡_n^=[h_1,n,⋯ ,h_u,n,⋯ ,h_U,n] ^T stands for the channel vector between the nth transmit antenna and the receiver. 𝐧_u^=[ n_1,n,⋯ ,n_u,n,⋯ ,n_U,n] ^T represents the receive noise vector.

Finally, the maximum likelihood decoder is used to estimate the index and constellation bits emitted by the nth antenna as

    [ î_n,k,x̂_n^k] =argi_n,k,x_n^kmin𝐲̅_n^k-c_i_n,k^x_n^k𝐡_n^ ^2,

where î_n,k, x̂_n^k denote the estimations of i_n,k, x_n^k. Note that the transmitted index and constellation bits from all transmit antenna are sequentially estimated by (<ref>) and (<ref>). 



§ SYSTEM PERFORMACE ANALYSIS


In this paper, the widely used BER and CRB metrics are considered to evaluate the system communication and sensing performance, respectively. In this section, closed expressions for the system BER and CRB are derived. Moreover, the system sensing complexity is analyzed.





 §.§ System CRB Analysis

Within one CPI, the noiseless data matrix for a target located at (θ _g,R_g) can be rewritten as

    Π   =ξ _g{𝐚_R(θ _g) ⊗[ 𝐚_T(R_g) ⊙𝐚_T(θ _g) ] }ψ_g^T(ℱ _g)
    		
       =ξ _g𝐖,

where 𝐖=𝐚_TR(θ _g,R_g) ψ_g^T(ℱ _g) and 𝐚_TR(θ _g,R_g) is denoted in (<ref>).

For convenience, define the unknown parameter vector as

    ρ= [Re{ξ _g } ,Im{ξ _g } ,R_g,θ _g,ℱ _g] ^T
    		.


According to the CRB definition <cit.>, the estimation accuracy lower bound of ρ is given by the diagonal elements of 𝐅^-1. 𝐅∈ℂ ^5× 5 represents the Fisher information matrix of the received signal, whose (x,y)th element is given by <cit.>

    F_x,y   =2Re{Tr[ ∂Π^†/∂ρ _xΛ^-1∂Π/∂ρ _y] }
       =2Re{Tr{∂ (ξ _g𝐖) ^†/∂ρ _xΛ^-1∂ (ξ _g𝐖)/∂ρ _y}} 
    		,

where Λ=σ _1^2I_N× M represents the noise covariance matrix. ρ _x/y denotes the x/yth element of ρ. Substituting (<ref>), (<ref>) into (<ref>), we have

    ∂ (ξ _g𝐖)/∂Re{ξ _g }=𝐖
    		,


    ∂ (ξ _g𝐖)/∂Im{ξ _g }=j𝐖
    		,


    ∂(ξ _g𝐖)/∂ R_g=ξ _g( 𝐚_R(θ _g) ⊗[ 𝐚̇_T(R_g) ⊙𝐚_T(θ _g)] ) ψ_g^T(ℱ _g)
    		,


    ∂ (ξ _g𝐖)/∂θ _g
    		=   ξ _g{𝐚̇_R(θ _g) ⊗[ 𝐚_T(R_g) ⊙𝐚_T(θ _g) ] . ψ_g^T(ℱ _g)
    		
       +. 𝐚_R(θ _g) ⊗[ 𝐚_T(R_g) ⊙𝐚̇_T (θ _g) ] }ψ_g^T(ℱ _g)
    		,


    ∂ (ξ _g𝐖)/∂ℱ _g=ξ _g𝐚_R(θ _g) ⊗[ 𝐚_T(R_g) ⊙𝐚_T(θ _g) ] ψ̇_g^T(ℱ _g) 
    		,

where 𝐚̇_T(R_g) =𝐄_T,R𝐚_T(R_g), 𝐚̇_R(θ _g) =𝐄_R,θ𝐚_R(θ _g), 𝐚̇_T(θ _g) =𝐄_T,θ𝐚_T(θ _g) and ψ̇_g (ℱ _g) =𝐄_ℱψ_g(ℱ _g) with

    𝐄_T,R=-j4π/cΔ fdiag{ 0,1,⋯ ,N-1 } 
    		,


    𝐄_R,θ=j2π f_cd_2/ccosθ _gdiag{ 0,1,⋯ ,M-1 } 
    		,


    𝐄_T,θ=j2π f_cd_1/ccosθ _gdiag{ 0,1,⋯ ,N-1 } 
    		,


    𝐄_ℱ=j2π Tdiag{ 0,1,⋯ ,K-1 } 
    		.


Then, the Fisher information matrix can be rewritten as (<ref>),

where ζ _R=Λ^-1/2∂𝐖/∂ R_g, ζ _θ=Λ^-1/2∂𝐖/∂θ _g, ζ _ℱ=Λ^-1/2∂𝐖/∂ℱ _g and ζ =Λ^-1/2𝐖.

Further, 𝐅^-1 is represented by 

    𝐅^-1=1/2[ 𝐅_11   𝐅_12
    𝐅_21   𝐅_22
    ] ^-1=1/2[ ×   ×
    ×   𝐃^-1
    ] ,

where 𝐅_11∈ℂ ^2× 2, 𝐅_12∈ℂ ^2× 3, 𝐅_21∈ℂ ^3× 2 and 𝐅_22∈ℂ ^3× 3 are the chunking matrices in 𝐅. Note that the diagonal elements of 𝐃^-1 contain the  estimation information for angle, distance, and velocity. According to the chunked matrix inverse formula <cit.>, 𝐃 is calculated as

    𝐃=𝐅_22-𝐅_21𝐅_11^-1𝐅_12=[ 
    			D_11   		D_12   		D_13
    
    			D_21   		D_22   		D_23
    
    			D_31   		D_32   		D_33
    ] ,

where 𝐃_i,j denotes the i,jth element in 𝐃.

Finally, the CRBs of angle, distance, Doppler frequency estimations are given by

    CRB_R_g=1/2( 𝐃)( [ 
    			D_22   		D_23
    
    			D_32   		D_33
    ] ),


    CRB_θ _g=1/2( 𝐃)( [ 
    			D_11   		D_13
    
    			D_31   		D_33
    ] ) ,

and

    CRB_ℱ=1/2( 𝐃)( [ 
    			D_11   		D_12
    
    			D_21   		D_22
    ] ) ,

respectively, where (·) denotes the determinant operator.



 §.§ Complexity Analysis of System Sensing Methods

We analyze the complexity of the proposed SSMTE and LCSSE algorithms by counting the required multiplication operations. The complexity of computing 𝐐^-1 is 𝒪{ K(NM) ^2+(NM) ^3 }. For mathematical convenience, let s_r and s_θ denote the number of distance and angle search steps within a range bin, respectively. Then the angle-distance estimation complexity in (<ref>) is 𝒪{s_rs_θ( N^2M^2+NM+1 )}. For velocity estimation, the computational complexity of 𝐃̂ and 𝐄_ℱ̂ are 𝒪{ G^3+2G^2NM+GMNK } and 𝒪{ 2G^3+2G^2 (K-1)  }, respectively. The complexity of computing G target speeds in (<ref>) is 𝒪{ 4G }. Assuming that there are G'⩽ G targets located at different range bins, the complexity of the SSMTE method is calculated by

    𝒪{[ KN^2M^2+N^3M^3+s_rs_θG'(N^2M^2+NM+1);          +3G^3+2G^2(K+NM-1) +GMNK+4G;                                      ]}  .


For the LCSSE method, the computational complexity of 𝐙(θ _g) is 𝒪{ N^2M^2+NM }. Then, The complexity of (<ref>) is 𝒪{ s_θ[(N-1) ^3+(N-1) ^2+(N-1) +1] }, while (<ref>) costs 𝒪{ s_rG' (N^2M^2+NM+1) }. Finally, the complexity of the LCSSE method is computed as

    𝒪{[ KN^2M^2+N^3M^3+s_θ [(N-1) ^3+(N-1) ^2+N];                    +s_rGG' (N^2M^2+NM+1);              +3G^3+2G^2(K+NM-1) +GMNK+4G;                                          ]}.







 §.§ System BER Upper Bound Analysis

Referring back to Section <ref>, we know that at the BS side, the information bits carried by one antenna can be categorized into index bits μ _I=log _2J and constellation bits μ _C=log _2L. Therefore, the system average bit error rate (ABER) is formulated as

    P_CCIM=∑_n=1^N(P_I,nμ _I+P_C,nμ _C)/N(μ _I+μ _C )
    		,

where P_I,n, P_C,n denote the average error probability (ABEP) of the index and constellation bits carried by the nth transmit antenna, respectively.

We first derive P_I,n. One can observe that there are C_μ _I^e events that e∈[ 1,μ _I ] bits errors out of μ _I bits. The misestimated index has the same probability of being the remaining J-1 complex coefficients. Thus, the ABEP of the indexed bits can be modeled as

    P_I,n   =P_IM/(2^μ _I-1) μ _I∑_e=1^μ _IeC_μ _I^e
       =2^μ _IP_IM/2 (2^μ _I-1)
    		,

where P_IM denotes the probability that the selected complex coefficient is incorrectly detected.

The P_IM can be derived by the union bounding technique. Specifically, calling back to (<ref>), the conditional pairwise error probability (PEP) that i_n,k is erroneously detected as i_n,k^' on 𝐡_n can be formulated as 

    Pr(i_n,k→ i_n,k^'|𝐡_n^)
    		
       =Pr( 𝐲̅_n^k-c_i_n,k^x_n^k𝐡_n^ ^2>𝐲̅_n^k-c_i_n,k^'^x_n'^k𝐡_n^ ^2 ) 
    		
       =Q( √(κ/2σ _2^2)) 
    		,

where κ = c_i_n,k^'^x_n'^k𝐡_n -c_i_n,k x_n^k𝐡_n ^2.

Since the elements in 𝐡_n follow i.i.d. 𝒞𝒩 (0,σ _C^2), κ in (<ref>) can be rewritten as κ =∑_u=1^2Uϖ _u^2, where ϖ _u^2∼𝒞𝒩 (0,σ _κ^2) with

    {[ σ _κ^2=| c_i_n,k^' x_n'^k-c_i_n,k x_n^k|^2σ _C^2/2,if i_n,k^' i_n,k^;      σ _κ^2=| x_n'^k-x_n^k|^2| c_i_n,k|^2σ _C^2/2,if i_n,k^'=i_n,k .;                                                                      ] 
    		.


From (<ref>), one can observe that κ follows the chi-square distribution with 2U degrees of freedom (DoF), whose probability density function (PDF) is written as

    f_κ( x ) =1/2^UΓ( U ) ( σ _κ^2) ^Ux^U-1exp( -x/2σ _κ^2) 
    		.

Averaging (<ref>) on κ gives that

    Pr(i_n,k→ i_n,k^')=∫_0^∞Q( √(κ/2σ _2^2)) f_κ( x )dx
    		
       =. [ P( α) ] ^U∑_u=0^U-1( [ U-1+u;     u;       ].) [ 1-P( α) ] ^u 
    		,

where

    P(α )=1/2( 1-√(α/1+α)) 
    		,

with α =σ _κ^2/2σ _2^2.

According to the union bound technique, the tight upper bound of P_IM can  be expressed as <cit.>

    P_IM⩽1/JL∑_[ i_n,k,i_n,k^'; i_n,k i_n,k^';               ]∑_x_n^k,x_n'^kPr(i_n,k→ i_n,k^')
    		.


Next, we derive the ABEP of the constellation bits, i.e., P_C,n. We found that P_C,n consists of two parts, one is the index bits are correctly detected but the QAM symbol is incorrectly detected. The other part is the case where the index bits are incorrectly detected, resulting in incorrect detection of the QAM symbol. Take this into mind, we have

    P_C,n=J-1/JP_IM+( 1-P_IM) P_QAM
    		,

where P_QAM denotes the ABEP of the constellation bits when the complex coefficient estimate is wrong, which is derived in the sequel.

We take a G-ary QAM symbol that can be split into two pulse amplitude modulation (PAM) symbols: υ-ary PAM of the I-signal and ω-ary PAM of the Q-signal, L=υ×ω. The conditional probability that the qth bit errors in the I-signal component can be expressed as

    P_υ( q | γ _j,l) =2/υ∑_i=0^(1-2^-q) υ -1{  (-1) ^⌊i· 2^q-1/μ⌋.
       ×( 2^q-1-. ⌊i· 2^q-1/. υ.+1/2⌋. ) . Q( (2i+1)ε√(2γ _j,l)) } 
    		,

where ε =√(3/υ ^2+ω ^2-2) denotes the minimum norm distance between two constellation points. γ _j,l^= c_j^x_l^𝐡_n^ ^2/σ _2^2 denotes the instantaneous total received signal to noise ratio (SNR) on the 𝐡_n when transmit the complex coefficient c_j and the constellation symbol of x_l, the PDF of which is given by <cit.>

    f_γ _j,l(x)=1/2^UΓ( U ) ( | c_jx_l|^2σ _C^2/2σ _2^2) ^Ux^U-1exp( -σ _2^2/| c_jx_l|^2σ _C^2x )
    		.


By averaging (<ref>) on γ _j,l, we have

    P_υ^j,l (q) 
       =2/υ∑_i=0^ (1-2^-q) υ -1{ (-1) ^⌊i· 2^q-1/υ⌋( 2^q-1-. ⌊i· 2^q-1/. υ.+1/2⌋. ) .
       ×. . [ P( α _j,l' ) ] ^U∑_u=0^U-1( [ U-1+u;     u;       ].) [ 1-P (α _j,l') ] ^u } 
    		,

where 

    P( α _j,l' ) =1/2( 1-√(α _j,l'/1+α _j,l')) 
    		,

with α _j,l'=[ (2i+1)ε] ^2| c_j x_l|^2σ _C^2/σ _2^2.

Similarly, the error probability of the qth bit in the ω-ary PAM component can be expressed as

    P_ω^j,l(q) 
       =2/ω∑_i=0^(1-2^-q) ω -1{ (-1) ^⌊i· 2^q-1/ω⌋( 2^q-1-. ⌊i· 2^q-1/ω+1/2⌋. ) .
       ×. [ P (α _j,l') ] ^U∑_u=0^U-1( [ U-1+u;     u;       ])[ 1-P(α _j,l') ] ^u } 
    		.


Therefore, with emission complex coefficients of c_i, the ABEP of the constellation symbol x_n is calculated as

    P_j,l =1/log _2L[ ∑_q=1^log _2υP_υ^j,l(q)+∑_q=1^log _2ωP_ω^j,l(q)] 
    		.


Further, the ABEP of P_QAM is obtained as

    P_QAM=1/JL∑_j=1^J∑_l=1^LP_j,l
    		.


Substituting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), the system ABER can be expressed as 

    P_CCIM=2^μ _IP_IM/2 (2^μ _I-1)μ _I+[ 1/2P_IM+(1-P_IM) P_QAM] μ _C/μ _I+μ _C
    		.






§ SIMULATION RESULTS


In this section, we perform Monte Carlo simulations to evaluate the proposed ISAC system performance and verify analytical results. Unless specified, the main parameters used in the experimental study are set to f_c=10 GHz, d_1=d_2=c/f_c, T=60 μ s, T_W=20 μ s, Δ f=2MHz. σ _C^2=1. The SNR of the sensing receiver and the communication receiver are denoted as 1/σ _1^2 and 1/σ _2^2, respectively. 



 §.§ Sensing Simulation

In this subsection, the root mean square errors (RMSE) and hit rate are adopted to evaluate the system sensing performance. A hit is proclaimed if the sum of the angle, distance and velocity estimation errors for the three targets in Fig. <ref> is less than 0.2 <cit.>. The RMSE is defined as

    RMSE=1/G∑_g=1^G√(1/M∑_m=1^M ( ρ _g-ρ̂_g,m) ^2),

where M stands for the number of Monte Carlo trails. ρ _g denotes the true value of R_g, θ _g or v _g, while ρ̂_g,m representing the estimation of ρ _g in the mth trail.  

The frequency offsets design criterion proposed in Section <ref> and widely used linear frequency offsets <cit.> are denoted as 'FODC' and 'LFO' in simulations, respectively. Note that the 'FODC' transmit frequency offsets are set as {{ 0,1,2,3.17,4.2,5.2 }×Δ f }MHz, while the 'LFO' transmit frequency offsets are set as  {{0,1,2,3,4,5}×Δ f  }MHz. The proposed sensing methods are compared with FDA-MIMO-based frequency offset permutation index modulation (FOPIM) scheme <cit.>. 




To evaluate the proposed frequency offsets design criterion, in Fig. <ref>, we compare the target recovery performance of the SSMTE and LCSSE algorithms under FODC and LFO. We set up 3 targets, as { 10.55,40.9m,8.62m/s}, { 10.55,89.6m,20.42m/s}, { 32.01,115.9m,36.5m/s}, respectively.  From Fig. <ref> (a) and Fig. <ref> (c), with LFO, one can observe that both SSMTE and LCSSE methods suffer from range ambiguity, where the targets' distances are estimated to other range bins, leading to incorrect parameter estimation and pairing. Fig. <ref> (b) and Fig. <ref> (d) show that the targets can be correctly estimated when using FODC. The reason for this benefit is shown in Fig. <ref>. 



Fig. <ref> gives the signal transmit-receive space spectrum with LFO and FODC. When adopting the linear frequency offset scheme, calling back to (<ref>), the range period of the transmit range steering vector is c/2Δ f=75m. Therefore, when estimating a certain target, other targets will form false targets with high peaks in the spatial spectrum, as shown in <ref>(a). This leads to target range estimation errors and target parameters pairing errors. In contrast, the proposed frequency offsets design criterion greatly increases the range period. Hence, targets in other range bins can hardly form peak values in the estimated range bin, reducing the probability of misestimation.



Fig. <ref> compares the hit rates of the proposed methods with FOPIM and MIMO schemes. Note that the frequency offset pool size for the FOPIM scheme is set to N <cit.>. Fig. <ref> indicates that the hit rates of the proposed sensing approaches with FODC are improved by increasing the snapshots number. 
This is because more snapshots yield more accurate covariance matrix estimation results, thus improving the parameter recovery performance. 
Under the FODC method, the hit rate of the LCSSE method is approximately equal to that of the SSMTE method.
The hit rates remain 0 for SSMTE with LFO, LCSSE with LFO and FOPIM methods, indicating their inability to estimate parameters for multiple targets simultaneously. The phenomenon arises due to the distances of target 1 and target 3 differ by one period, causing range ambiguity and hindering parameter pairing in the FOPIM and LFO-ralated schemes.





In Fig. <ref>, we compare the multi-target RMSE performance among different schemes. We find that the SSMTE and LCSSE methods have the similar range and angle estimation accuracies at middle to high SNRs. At very low SNRs, the LCSSE method is slightly better than the MSSTE method. This is because the angle and range errors of the SSMTE method are coupled. That is, in the low SNR region, the angle and range estimation errors are significant and affect each other. In contrast, in the LCSSE method, the targets range estimation does not affect the angle estimation. 
On the other hand, Fig. <ref>(a) shows that the angle estimation error of FOPIM stays around 0.45° after a brief drop, which is much higher than the proposed methods. This is attributed to the fact that the FOPIM method relies only on a simple receiver beamformer to estimate the angle, which has a very low angular resolution. There are two reasons why the FOPIM's range estimation error in <ref>(b) remains high: 1) the large angle estimation error causes a large distance estimation accuracy; 2) the FOPIM method is unable to pair multi-target angles and distances, resulting in mis-paired distance estimates.



Fig. <ref> illustrates RMSEs and root CRBs for the proposed system with different numbers of transmit antennas when FODC is employed. Note that the frequency offsets are set as {{ 0,1,2,3.17,4.2,5.2, 6.2, 7.2 }×Δ f }MHz and {{ 0,1,2,3.17,4.2,5.2, 6.2, 7.2, 8.2, 9.2}×Δ f }MHz when N=8 and N=10, respectively. One can see that the estimation accuracy of target angle, range and velocity improves with the increasing number of transmit antennas. The SSMTE and LCSSE have the similar accuracies with different N. 

One can see that the angle and distance estimation performance is close to that of the CRB, but the velocity estimation differs significantly from the CRB. This is because in the proposed methods, the angle and distance estimation errors are substituted into the velocity estimation, which reduces the velocity estimation accuracy, while the velocity CRB is independent of the angle-distance estimation error. Nevertheless, at N=6, SNR=0dB, the velocity RMSE of the proposed method is 0.026m/s, which meets most civil sensing scenarios.



Fig. <ref> compares the computational complexity of the proposed SSMTE and LCSSE algorithms for different N. The complexity of both methods increases with the number of range bins (G') to be estimated. The complexity of the LCSSE is two orders of magnitude lower than that of the SSMTE method, thanks to its conversion of a 2-D angle-range joint search into two 1-D searches. Considering the sensing performance comparisons in Fig. <ref> to Fig. <ref>, we conclude that the LCSSE approach is the wiser choice to sense targets in the proposed system.









 §.§ Communication Simulation

In this subsection, we investigate the communication performance of the proposed CCIM scheme. Note that “Ana" and “Sim" represent the BER theoretical upper bound and the Monte Carlo simulation results in the following figures, respectively.


Fig. <ref> compares the BER of the proposed CCIM method with FOPIM <cit.> and traditional MIMO <cit.> methods for varying number of receive antennas. Note that the CCIM method carries 16 bits, and for fairness, the frequency offset pool size for FOPIM is set to 4, and the modulation order for both the FOPIM and MIMO methods is set to 8. From Fig. <ref>, we observe that the system BER is improved with the increasing U, which stems from the fact that the higher receive diversity gain. The simulations of the CCIM method match well with the theoretical results, which verifies the BER analysis. Moreover, MIMO shows the worst BER performance among the three schemes. This is because its transmit symbols are coupled to each other, resulting in a small judgment domain, which deteriorates the BER performance. 

Another interesting finding is that the CCIM method outperforms the FOPIM method when the number of receiving antennas is small (U=1,2). However, when U increases, the BER of the proposed CCIM method is worse than FOPIM. This can be explained as follows, referring Eq. (38) in <cit.> gives that the frequency offset permutation estimation error probability (P_perm) of the FOPIM scheme is governed by the frequency offset combination estimation error probability (P_comb). P_comb remains high with low U, resulting in a high overall index bit error probability. As U increases, P_comb decreases dramatically. Furthermore, comparing (42) in <cit.> and (<ref>) in this paper reveals that the judgment domain spacing of the FOPIM method is larger than that of CCIM, which results in a lower BER for FOPIM than for CCIM when larger U.




Fig. <ref> shows the BER comparison results with different number of transmit antennas. Note that the parameter configurations for MIMO and FOPIM methods are the same as for the CCIM scheme. As N increases, the BER performance of the CCIM scheme gradually outperforms that of the FOPIM scheme. Moreover, the BER of the FOPIM and MIMO approaches increase with increasing N, whereas the CCIM's BER  remains with increasing N. This is because as N increases, the FOPIM method suffers a higher error probability in estimating frequency offsets. On the other hand, (<ref>) gives that bits carried by every transmit antenna are decoded independently in the CCIM method, with no dependence on N. Therefore, we conclude that CCIM can achieve higher communication rates without loss of BER performance by increasing the number of transmit antennas.



In Fig. <ref>, we study the BER of the CCIM scheme with the different size of the complex coefficient set. It is seen that the BER of the proposed CCIM approach rises as the size of the complex coefficient set becomes larger. For every doubling of J, the BER performance decreases about 4 dB. The reason for this phenomenon can be found in (<ref>), where the index bits misestimation probability P_IM increases with increasing J, leading to a deterioration in the system BER performance.


Fig. <ref> compares the bits per pulse among different ISAC schemes: proposed CCIM, FOPIM <cit.>, FRaC <cit.>, JCRS <cit.>, MAJoRCom <cit.>. In the simulation, the total bandwidth of FOPIM is set equal to that of CCIM, namely, the size of FOPIM's frequency offset pool is set to N. To be fair, we set J=N for CCIM to have the same index resource. JCRS has a waveform set size equal to J. MAJoRCom uses a separate frequency for each antenna, while FRaC activates N-2 antennas. Studying Fig. <ref> finds that FOPIM scheme carries more bits than FRaC, JCRS and MAJoRCom schemes, the reason for this phenomenon has discussed in <cit.>. Moreover, Fig. <ref> depicts that the proposed CCIM method outperforms the FOPIM approach in terms of bits per pulse performance. This observation can be elaborated as follows: in Fig. <ref> the bits per pulse for CCIM and FOPIM are N× (⌊log _2N ⌋ +log _2L) and Nlog _2L+⌊log _2N! ⌋ +⌊log _2C_N^N⌋, respectively. Since N^N>N!, we see that CCIM carries more bits than that FOPIM method. 







§ CONCLUSION


This paper investigated the FDA-MIMMO-based ISAC system in a multi-target sensing scenario. Specifically, to improve the communication rate, a CCIM scheme was proposed at the transmitter, which carried the extra bits by selecting complex coefficients from a complex coefficient vector. To estimate targets, the SSMTE method was proposed. By performing 2-D searches of the spatial spectrum over target-containing range bins, the angle and range of targets can be estimated. Then, the targets velocities were estimated by the LS method. However, the 2-D search has high complexity. To address this, we designed the LCSSE method to reduce the complexity by converting the 2-D search into two 1-D searches.

On the other hand, the FDA-MIMO's range steering vector changes periodically with distance, resulting in range ambiguity. To address this issue, the FOCD scheme was proposed, which adjusted the integer and fractional parts of each transmit frequency offset to enlarge the range periodicity, thereby mitigating range ambiguity in multi-target estimation. Besides, the closed-form expressions for CRB, complexity and BER upper bound are derived. Simulation results illustrated that the LCSSE method dramatically reduced the complexity of SSMTE with no degradation in sensing accuracy. Moreover, the proposed FDA-MIMO-based ISAC system outperforms the FOPIM based ISAC system in terms of multi-target sensing performance. 












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