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Nonorthogonal waveform assisted DOA estimation for joint MIMO sensing and communication
EURASIP Journal on Advances in Signal Processing volumeÂ 2023, ArticleÂ number:Â 22 (2023)
Abstract
Conventional directionofarrival (DOA) estimation methods for multipleinput multipleoutput (MIMO) joint sensing and communication system normally pursue high estimation accuracy and resolution by imposing orthogonal waveforms. However, such operation results in a deterioration of communication performance. In this paper, we propose a nonorthogonal waveform assisted DOA estimation algorithm, where an augmented virtual array is derived by exploiting the nonorthogonal MIMO communication waveforms, while a high communication rate can still be maintained. To estimate the roundtrip sensing channels of each subcarrier, we utilize the transmitted symbols as pilot symbols, and obtain all the channel coefficients with a minimum mean square error solver. A virtual channel matrix can be formulated with these channel coefficients, which can be regarded as the samples of an augmented virtual array. Based on that, the subspace processing can be conducted for DOA estimation with fully nonorthogonal waveforms. Furthermore, the rank deficiency property of the equivalent signal matrix of the virtual array is analyzed when the distance of targets are identical. To address the problem, a Toeplitz reconstruction method is proposed to restore the rank of the rankdeficient equivalent signal matrix for DOA estimation. Simulations show that the proposed nonorthogonal waveform assisted DOA estimation algorithm outperforms the conventional methods in terms of resolution and accuracy, while maintaining a satisfactory computational efficiency.
1 Introduction
With the rapid development of fifthgeneration (5G) and beyond networks, the concept of joint sensing and communication has gained growing popularity due to its ability to improve spectrum efficiency while reducing hardware and energy costs [1,2,3,4,5,6]. In a joint sensing and communication system, the communication infrastructure can be multiplexed for locating noncooperative targets with a wide sensing coverage [7,8,9]. More importantly, since higher frequencies have been adopted in communication base stations [10], the finite concentrated propagation paths will be characterized by a channel in the angular domain [11]. As such, noncooperative target localization can be achieved by estimating directionofarrival (DOA) of echoes using the communication waveform [12,13,14,15,16,17,18]. Therefore, it is important to develop suitable DOA estimation methods for the joint multipleinput multipleoutput (MIMO) sensing and communication system to enable an effective target localization function.
The mainstream DOA estimation methods for joint MIMO sensing and communication can be categorized into two types. For the first type, the communication performance is the primary consideration. Thus, the transmitted waveforms between different transmit antennas are nonorthogonal to obtain a high communication rate, and the interelement spacings of the transmit and receive arrays are both set toÂ halfÂ signalÂ wavelength following the MIMO communication framework [19]. In [20], DOA estimation of noncooperative targets is achieved by incorporating an additional sensing beam, which consists a duplication of the current communication symbols to avoid interfering the communication process. Moreover, the DOA estimation performance within the multibeam framework with different power allocation fraction is analyzed in [21, 22]. Different from the above methods which employ the echo of downlink signals from the base station, the uplink signals from a specific mobile user with known location are utilized for DOA estimation without the requirement of fullduplex capability for the transceiver [23]. More recently, a DOA estimation method for a 5G New Radio (NR) system for bistatic sensing is proposed in [24], where the transmit and the receive arrays are separately located at different base stations or user equipments. Although the first type of methods minimize the impact of additional sensing function on the communication performance, the DOA estimation performance for a MIMO system traditionally depends on the derivation of an augmented virtual array. However, the nonorthogonality of communication waveforms used in these methods prevents the virtual array derivation since the signals transmitted by different antennas cannot be separated at the receiver side, thereby limiting the DOA estimation performance.
On the other hand, the second type of DOA estimation methods focus on enhancing the accuracy and resolution of DOA estimation. In particular, in order to derive virtual arrays, orthogonal waveforms are adopted between different transmit antennas, and the interelement spacing is configured the same as MIMO radar [25]. Meanwhile, the communication symbols are embedded in these orthogonal waveforms. An interleaving frame structure is designed to divide the subcarriers into several orthogonal groups [26, 27]. As such, a virtual array can be derived from the received symbols for DOA estimation. Furthermore, the beam scanning strategy is combined with the transmit power compensation for DOA estimation [28]. Moreover, the sparse antenna array configuration is exploited to achieve finer angular resolution for target localization [29]. To ensure the qualityofservice for multiple mobile users, the virtual array is derived from the orthogonal waveforms transmitted to different users instead of different antennas in a hybrid beamforming architecture [30]. However, all these methods require orthogonal transmitting to derive the virtual array at the cost of reducing the communication rate to certain extent. Therefore, it is still challenging to develop an effective DOA estimation method with a minimum communication performance loss in the joint MIMO sensing and communication system.
To solve this problem, a sparse signal recovery (SSR)based DOA estimation method is proposed with a partially orthogonal signal processing model [31]. To be specific, most subcarriers are shared by all the transmit antennas to exploit the full spectrum effectively, while each antenna possesses an exclusive subscarrier to derive the virtual array. While DOA estimation methods exploiting fully nonorthogonal waveforms are investigated in [32, 33], they deal with the imperfect orthogonality through prewhitening operation, rather than utilizing the nonorthogonality of MIMO communication waveforms to ensure the communication rate. Hence, DOA estimation with fully nonorthogonal communication waveforms is an open and urgent task.
In this paper, we propose a nonorthogonal waveform assisted DOA estimation algorithm for joint MIMO sensing and communication system, where a virtual array is derived from the fully nonorthogonal waveforms for DOA estimation with a high communication rate. The transmitted symbols are employed as pilot symbols to sense the noncooperative targets, based on which the roundtrip sensing channels of all the subcarriers are estimated separately. In order to derive an augmented virtual array, the estimated channels are reconstructed according to the geometry relationship between each pair of transmit and receive antenna. Then, the roundtrip sensing channel coefficients of all the subcarriers are modeled into a virtual channel matrix, which can be regarded as the equivalent signals received by the virtual array. Furthermore, to solve the problem of rank deficiency of the equivalent signal matrix brought by the targets located at identical distance, we propose a Toeplitz reconstruction approach to compensate the rank of the equivalent signal matrix. Finally, the DOAs can be effectively estimated by the subspace processing for the virtual channel matrix. The proposed algorithm is compared to the conventional DOA estimation methods for joint MIMO sensing and communication system, which verify its superior estimation performance.
The rest of this paper is organized as follows. In Sect. 2, the architecture of the joint MIMO sensing and communication is introduced. Then, the DOA estimation algorithm with nonorthogonal waveform is proposed, and the computational complexity is analyzed in Sect. 3. We demonstrate simulation results in Sect. 4, and draw our conclusion in Sect. 5.
Notations: We use a, \(\varvec{a}\) and \(\varvec{A}\) to denote scalar, vector and matrix, respectively. The (i,Â j)th element in the matrix \(\varvec{A}\) is denoted by \({\varvec{A}}(i,j)\), the ith row in \(\varvec{A}\) is denoted by \(\varvec{A}(i,:)\), and the jth column in \(\varvec{A}\) is denoted by \(\varvec{A}(:,j)\). The rank of matrix \(\varvec{A}\) is denoted by \(\text{rank}(\varvec{A})\). The operators \([\cdot ]^{\text{T}}\), \([\cdot ]^{\text{H}}\) and \([\cdot ]^{\mathrm {*}}\) represent transpose, Hermitian transpose and conjugation, respectively. We denote a diagonal matrix as \(\text{diag}(\cdot )\). The Frobenius norm is denoted by \(\Vert \cdot \Vert _\text{F}\), the minimum is denoted by \(\text{min}(\cdot )\), the statistical expectation is denoted by \(\text{E}\{\cdot \}\), and the rounding down operator is denoted by \(\lfloor \cdot \rfloor\). \(\otimes\) and \(\odot\) respectively denote the Kronecker and Hadamard products. Moreover, \(\textbf{0}\), \(\textbf{1}\) and \(\varvec{I}\) are the zero matrix, allones matrix and the identity matrix, respectively.
2 Joint MIMO sensing and communication system
As shown in Fig.Â 1, we consider a joint MIMO sensing and communication system equipped with \(N_t\) transmit antennas and \(N_r\) receive antennas, operating within the 5G Frequency Range 2 (FR2). Following the deployment of MIMO antenna array, the interelement spacing of the receive array is \(d_r = \lambda /2\) with \(\lambda\) being the signal wavelength, while the interelement spacing of the transmit array is \(d_t=N_r d_r\). The transmitted waveforms are utilized to communicate with mobile users and illuminate the noncooperative targets, while the echoes reflected by these farfield targets are processed to estimate their DOAs.
An orthogonal frequencydivision multiplexing (OFDM) frame with M OFDM symbols on K subcarriers is considered, where the total duration of each symbol regarding the cyclic prefix (CP) is denoted by \(T_{\text{sym}}=T+T_{\text{CP}}\). Here, T is the elementary symbol duration, and \(T_{\text{CP}}\) is the duration of CP. The mth OFDM symbol transmitted by the array can be represented as
where \(\Delta f=1/T\) is the subcarrier spacing, and \(\varvec{s}[k,m] = [s_{0}[k,m], s_{1}[k,m],\cdots,s_{N_t1}[k,m]]^{\text{T}}\) represents the complex data symbol vector modulated by phaseshift keying (PSK) or quadrature amplitude modulation (QAM), with \(m=0,1,\cdots ,M1\) and \(k=1,2,\cdots ,K\) representing the indices of the OFDM symbol and subcarrier, respectively.
Assume that there are Q noncooperative farfield targets with directions \(\varvec{\theta }= [{{\theta }}_1,{{\theta }}_2,\cdots,{{\theta }}_{Q}]^{\text{T}}\). The echoes reflected by these targets are sampled at the \((m T_{\text{sym}}+T_{\text{CP}}+k \frac{T}{K})\)th time slot for the mth OFDM symbol. Then, the CP symbols are removed, and a Kpoint discrete Fourier transform (DFT) is applied to the obtained sample set of the mth OFDM symbol. As such, the discrete representation of the received symbols in the frequency domain can be defined as
where \(\varvec{n}[k,m] \sim \mathcal {C N}\left( \textbf{0}, \sigma _{n}^{2} \varvec{I}\right)\) is the additive white Gaussian noise vector, and
is the roundtrip sensing channel of the kth subcarrier. Here, \(\varvec{B} = \text{diag}(\beta _1, \beta _2,\cdots ,\beta _Q)\) denotes a transmission attenuation matrix,
is the roundtrip delay of the kth subcarrier, \(R_q\) is the distance between the qth target and the deployed system, and \(\varvec{A}_{t}(\varvec{\theta })= \left[ \varvec{a}_{t}\left( \theta _{1}\right) , \varvec{a}_{t}\left( \theta _{2}\right), \cdots, \varvec{a}_{t}\left(\theta _{Q}\right) \right]\), \(\varvec{A}_{r}(\varvec{\theta }) = \left[ \varvec{a}_{r}\left( \theta _{1}\right) , \varvec{a}_{r}\left( \theta _{2}\right) , \cdots , \varvec{a}_{r}\left( \theta _{Q}\right) \right]\) are the transmit and receive steering matrices, where
are the steering vectors corresponding to the qth target, \(q=1,2,\cdots,Q\). In the millimeter wave scenario considered in this paper, the subcarrier bandwidth is much smaller than the carrier frequency, i.e., \(\Delta f \ll f_c\), which means that the difference between different subcarrier wavelength can be neglected, i.e., \(\lambda = \frac{c}{f_{c}}\approx \frac{c}{f_{c}+k \Delta f}\). Here, \(f_{c}\) and c denotes the carrier frequency and the speed of light, respectively. Meanwhile, the uplink signals are not considered in (2), since the echoes of the downlink signals are expected to arrive before the end of the guard period (GP) under time division duplex (TDD) mode in FR2 band.
To exploit the available bandwidth, the deployed MIMO system transmits fully nonorthogonal waveforms as shown in Fig.Â 2c. Since all the K subcarriers are utilized by each transmit antenna, a maximum bit rate can be achieved as
In comparison, for the orthogonal and partially orthogonal transmitting principle demonstrated in Fig.Â 2a and Fig.Â 2b, the bit rates are \((\log _2 4) K/T_{\text{sym}}\) and \((\log _2 4) (K1)N_t/T_{\text{sym}}\), which are both smaller than R in (6). Therefore, the fully nonorthogonal transmitting principle provides the highest communication rate. However, such a transmitting principle poses challenges on the derivation of augmented virtual array, which is essential to the MIMO system for enhancing source estimation performance [34,35,36]. To be specific, for the conventional methods with an orthogonal transmitting principle as shown in Fig.Â 2a, the mth transmitted OFDM symbol can be represented as
where \(\tilde{\varvec{P}}=[\varvec{I}_{N_t},\varvec{I}_{N_t},\cdots ,\varvec{I}_{N_t}]\in \mathbb {C}^{N_t \times K}\) represents a subcarrier selection matrix, obeying a blockwise interleaving frame structure. In particular, \(\tilde{\varvec{P}}(:,k)\) has only one nonzero element, indicating the index of the transmit antenna of the kth subcarrier. Hence, symbols transmitted on the kth subcarrier all come from a certain transmit antenna. Moreover, \(\tilde{\varvec{P}}(n_t,:)\) has \(\frac{K}{N_t}\) nonzero elements, indicating the subcarrier index \(k \in \left\{ n_t+z N_t+1 \mid z=0,1,\cdots ,\frac{K}{N_t}1\right\}\) possessed by the \(n_t\)th antenna of the transmit array. Here, z denotes the block index of the subcarriers as shown in Fig.Â 2a. Since different subcarriers can be separated by a set of matched filters at the receive array, \({d}_{n_r}[n_t+z N_t+1,m]\) can be obtained for \({s}_{n_t}[n_t+z N_t+1,m]\) at the \(n_r\)th receive array antenna. After that, the sensing channel between the \(n_t\)th transmit antenna and the \(n_r\)th receive antenna is computed by an elementwise division as \(i_{n_t,n_r}[z,m]=\frac{{d}_{n_r}[n_t+z N_t+1,m]}{{s}_{n_t}[n_t+z N_t+1,m]}\). Consequently, the sensing channel
can be derived, which corresponds to a virtual array of size \(N_t N_r\). Note that the virtual array derived from (8) is deviated from a uniform linear array (ULA), which is caused by various propagation delays on different subcarriers. Meanwhile, for the partially orthogonal transmitting principle as shown in Fig.Â 2b, the mth transmitted OFDM symbol can be represented by \(\tilde{\varvec{x}}[m]\) in (7) by substituting \(\tilde{\varvec{P}}\) in (7) with \(\acute{\varvec{P}}= [\varvec{I}_{N_t},\textbf{1}_{N_t \times (KN_t)}]\). Here, each column in \(\acute{\varvec{P}}(:,1:N_t)\) has only one nonzero element, which means that each transmit antenna possesses an exclusive subcarrier. As such, the virtual array derivation is also available in the partially orthogonal scenario.
However, in the considered fully nonorthogonal signal model as defined in (1) and (2), there are no exclusive subcarriers for each transmit antenna. As a result, the symbols transmitted by different antennas are mixed together in \(\varvec{d}[k,m]\), such that the construction of virtual array in (8) is no longer valid.
3 Proposed DOA estimation algorithm
In this section, we derive an augmented virtual array in the fully nonorthogonal scenario as shown in Fig.Â 3, based on which a nonorthogonal waveform assisted DOA estimation algorithm is proposed for the deployed joint MIMO sensing and communication system. Furthermore, we analyze the rankdeficiency problem of the equivalent signal matrix when targets are located at the same distance, and propose a suitable Toeplitz reconstruction solution.
3.1 Virtual channel matrix derivation for DOA estimation
The sensing channel, which depicts the attenuation and phase shift of the transmitted symbols during the roundtrip propagation, contains all the information for virtual array derivation. Thus, to derive the virtual array, we estimate the roundtrip sensing channel \(\varvec{G}_k\) in the nonorthogonal scenario through a minimum mean square error (MMSE) solver. First, we represent (2) in a compact form as
where \(\varvec{D}_k \!=\! [{\varvec{d}}[k,0],{\varvec{d}}[k,1],\cdots \!,{\varvec{d}}[k,M\!\!1]]\) is the OFDM demodulated symbol matrix, \(\varvec{S}_k \!=\! [{\varvec{s}}[k,0],{\varvec{s}}[k,1],\cdots \!, {\varvec{s}}[k,M\!\!1]]\) is the transmitted complex symbol matrix, \(\Delta \varvec{G}_k\) is the channel estimation error of the kth subcarrier, and \(\varvec{N}_k\) is the noise matrix which satisfies \(\text{E}\left\{ \varvec{N}_k \varvec{N}^{\text{H}}_k\right\} =\sigma ^{2}_n M \varvec{I}\). Then, the MMSE estimation for the sensing channel of the kth subcarrier can be expressed as
where \(\varvec{W}_0\) minimizes the mean square error (MSE) of the estimation of sensing channel as
For the estimator \(\varvec{W}\), the sensing channel estimation error \(\Delta \varvec{G}_k\) can be represented as
where \(\varvec{R}_{\varvec{G}_k}= \text{E}\left\{ \varvec{G}_k \varvec{G}^{\text{H}}_k \right\}\) is the matrix of sensing channel correlations. Then, the optimal \(\varvec{W}\), i.e., \(\varvec{W}_0\), can be obtained from \(\partial (\Delta \varvec{G}_k) / \partial \varvec{W}=0\) as
Hence, the estimated sensing channel on the kth subcarrier can be obtained by subtituting (13) into (10) as
Note that, we choose to estimate the roundtrip sensing channel with MMSE estimator due to its conceptual simplicity and the effectiveness in utilizing the prior information. Nevertheless, there are some other channel estimation techniques that can be adopted, such as the linear minimum mean square error (LMMSE) method [37] with \(\hat{\varvec{G}}^{\text{LMMSE}}_k={\varvec{D}_k}{\varvec{S}^{\text{H}}_k}\left( \text{E}\left\{ \varvec{S}_k \varvec{S}^{\text{H}}_k \right\} +{\sigma _{n}^{2}} M \varvec{R}^{1}_{\varvec{G}_k}\right) ^{1},\) and least square (LS) method [38] with \(\hat{\varvec{G}}^{\text{LS}}_k={\varvec{D}_k}{\varvec{S}^{\text{H}}_k}\left( \varvec{S}_k {\varvec{S}^{\text{H}}_k}\right) ^{1}\).
It can be observed from (3) that, the roundtrip sensing channel \(\varvec{G}_{k}\) is related to the transmit steering matrix \(\varvec{A}_{t}\) and the receive steering matrix \(\varvec{A}_{r}\). Due to the interelement spacing settings inherited from the MIMO radar deployment, the Kronecker product of \(\varvec{A}_{t}\) and \(\varvec{A}_{r}\) would yield an equivalent virtual array. In this light, the estimated roundtrip sensing channel \(\hat{\varvec{G}}_k\) can be vectorized as
Here, \(\varvec{h}_{k}\) corresponds to an augmented virtual array \(\mathbb {S}=\{0,d_r,\cdots ,(N_t N_r1)d_r\}\), where
is the steering matrix of the virtual array \(\mathbb {S}\), and
is the steering vector corresponding to the qth noncooperative target.
Unlike the virtual array formed in (8), the augmented virtual array \(\mathbb {S}\) corresponds to a single subcarrier, leading to the same propagation delay of all the virtual array antennas. Thus, the augmented virtual array \(\mathbb {S}\) is a ULA. Moreover, since the vectorized virtual channels \(\left\{ \varvec{h}_1,\varvec{h}_2,\cdots ,\varvec{h}_K\right\}\) estimated on different subcarriers are independent, we can formulate a virtual channel matrix
where
is the matrix of propagation delay, and \(\Delta \tilde{\varvec{G}}=[\text{vec}(\Delta \varvec{G}_1),\text{vec}(\Delta \varvec{G}_2),\cdots ,\text{vec}(\Delta \varvec{G}_K)]\) is the sensing channel estimation error matrix. The formulation of \({\varvec{H}}\) can be regarded as sampling the equivalent Q signals utilizing K uncorrelated observations simultaneously with the virtual array \(\mathbb {S}\). The covariance of virtual channel matrix is then calculated as
Here, we define \(\varvec{\Xi } :=\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}}\) as an equivalent signal matrix, and \(\varvec{\Xi }\) is fullrank when the distances of targets are different, i.e., \(R_i\ne R_j\) if \(i\ne j\), for i, \(j \in \{1,2,\cdots , Q\}\). This enables the subsequent computation of DOAs. However, when the targets have the same distance, \(\varvec{\Xi }\) becomes rankdeficient, and the corresponding solution will be elaborated in the next section.
In order to retrieve angle information from \(\varvec{\Gamma }\) in (20), the subspace processing is incorporated, where a spectrum can be constructed utilizing the orthogonality between the signal subspace and noise subspace, and angles can be then estimated by peak searching of the spectrum. Thus, we perform eigendecomposition on \(\varvec{\Gamma }\) as \(\varvec{\Gamma } = \varvec{U}\varvec{\Sigma }{\varvec{U}}^{\text{H}}\), where the diagonal matrix \(\varvec{\Sigma }\) contains \(N_t N_r\) eigenvalues, and \(\varvec{U}\) is the matrix with corresponding orthogonal eigenvectors as its columns. Since there are Q nonzero eigenvalues in \(\varvec{\Sigma }\), the matrix \(\varvec{U}\) can be expressed as \(\varvec{U}=[\varvec{U}_1~\varvec{U}_2]\), where \(\varvec{U}_1 \in \mathbb {C}^{N_t N_r \times Q}\) contains Q eigenvectors corresponding to Q largest eigenvalues, and \(\varvec{U}_2 \in \mathbb {C}^{N_t N_r \times (N_t N_r  Q)}\) contains the eigenvectors corresponding to the rest eigenvalues. Since the basis vectors in \(\varvec{U}_2\) fall into the null space of the signal matrix, the estimation of DOAs \(\hat{\theta}_q\) can be obtained by searching the peaks of the spectrum function \(\frac{1}{\left\ \varvec{a}^{\text{H}}_{\mathbb{S}} (\theta) \varvec{U}_2\right\_\text{F}^{2}}\) as
where \(\theta \in [\pi ,\pi ]\) is the spectrum searching angle. Although the deployed system in this paper is consisted of linear arrays, the proposed algorithm can be extended for the estimation of azimuth and elevation angles of the noncooperative targets when Lshaped array or planar array is utilized.
3.2 DOA estimation for targets at identical distance
When two targets has the numerically identical distance, the estimation of their DOAs using the proposed algorithm becomes a tricky problem. This is due to the fact that, the equivalent signal matrix \(\varvec{\Xi }\) in (20) is no longer fullrank under this circumstance. In particular, \(\varvec{\Xi }\) obeys the following property.
Theorem 1
For any i, \(j \in \{1,2,\cdots , Q\}\) (\(Q < K\)) and \(i\ne j\), if \(R_i = R_j\), then \(\varvec{\Xi }\) is rankdeficient.
Proof
As the transmission attenuation matrix \(\varvec{B}\) contains nonzero elements on the diagonal, it is fullrank, i.e., \(\text{rank}(\varvec{B})=Q\). Thus, we have \(\text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}}) \le \text{min} \left\{ \text{rank}(\varvec{B}^{\text{H}}), \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}) \right\} = \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}})\) and we also have \(\text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}) \le \text{min} \left\{ \text{rank}(\varvec{B}), \text{rank}(\varvec{E}\varvec{E}^{\text{H}}) \right\} = \text{rank}(\varvec{E}\varvec{E}^{\text{H}})\), which lead to \(\text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}})\le \text{rank}(\varvec{E}\varvec{E}^{\text{H}})\). Likewise, it holds that \(\text{rank}(\varvec{E}\varvec{E}^{\text{H}})\le \text{min}\! \left\{ \text{rank}(\varvec{B}^{\mathrm {1}}), \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}) \right\} \!= \! \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}) \! \le \! \text{min}\!\!\left\{ \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}}),\text{rank}(\varvec{B}^{\mathrm {H}}) \right\}\), leading to \(\text{rank}(\varvec{E}\varvec{E}^{\text{H}})\le \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}})\). Therefore, we have \(\text{rank}(\varvec{E}\varvec{E}^{\text{H}})= \text{rank}(\varvec{B}\varvec{E}\varvec{E}^{\text{H}}\varvec{B}^{\text{H}})\). We then focus on the rank of \(\varvec{E}\varvec{E}^{\text{H}}\). According to (4) and (19), \(\varvec{E}\varvec{E}^{\text{H}}\) is represented as
We can see that for \(i\ne j\), if \(R_i=R_j\), the ith and the jth columns of \(\varvec{E}\varvec{E}^{\text{H}}\) will be identical, such that the rank of \(\varvec{E}\varvec{E}^{\text{H}}\) will collapse into \(Q\!\!1\). Thus, \(\varvec{\Xi }\) is rankdeficient.
The rank deficiency of \(\varvec{\Xi }\) invalids the eigendecompositionbased DOA estimation in (21). Hence, we propose the Toeplitz reconstruction to compensate the rank of the equivalent signal matrix \(\varvec{\Xi }\). To be specific, let \(\varvec{\Lambda }\) be the following blockwise Toeplitz matrix,
where L equals to \(\lfloor \frac{N_t N_r +1}{2} \rfloor\). Then, the autocorrelation of \(\varvec{\Lambda }\) can be formulated as
where
is a selection matrix, \(l = 1, 2, \cdots , L\). Here, l denotes the index of the selected part of the augmented virtual array, and it is numerically identical with the index of the first element of the selected array. By substituting (18) into (24), \(\tilde{\varvec{\Gamma }}\) can be reformulated as
Here, \(\varvec{\Psi }=\frac{1}{L} \sum \limits _{l=1}^{L} \varvec{J}_l \Delta \tilde{\varvec{G}} \Delta \tilde{\varvec{G}}^{\text{H}} \varvec{J}^{\text{H}}_l\) represents the transformed autocorrelation of \(\Delta \tilde{\varvec{G}}\), and \(\varvec{A}_l = [ \varvec{a}_l\left( \theta _{1}\right) , \varvec{a}_l\left( \theta _{2}\right) , \cdots , \varvec{a}_l\left( \theta _{Q}\right) ]\) represents the steering matrix of the lth selected array with L elements, where the steering vector is
as the selected part of \(\varvec{a}_{\mathbb {S}}\) in (17). Since the distance between a certain element of the lth selected array and the corresponding element in the first selected array is decided by \((l1)d_r\), the steering vector for a fixed direction \(\theta _q\) for these two arrays is proportional by the scale factor \(e^{j \frac{2 \pi }{\lambda } (l1)d_r \sin \theta _{q}}\). Therefore, the steering matrix of the lth selected array can be represented as
where \(\varvec{F}_l\) denotes a phase shift diagonal matrix with Q scale factors. Then, by substituting (28) into (26), \(\tilde{\varvec{\Gamma }}\) is further expressed as
A comparison between (20) and (29) reveals that, when \(\varvec{\Xi }\) is not fullrank, the rank of \(\tilde{\varvec{\Gamma }}\) is restored to Q by sacrificing the size of aperture. By substituting \({\varvec{\Gamma }}\) with \(\tilde{\varvec{\Gamma }}\) for eigendecomposition, the true DOAs can be estimated even at presence of targets with identical distance.
3.3 Computational complexity analysis
In this section, we compare the computational complexities of the proposed algorithm to those of the MUSICbased method [22], interleaving frame structurebased method [27] and the SSRbased method [31]. We assume that the distances of the targets are not identical, and thus the Toeplitz reconstruction is not considered for the proposed algorithm in this subsection.
The proposed algorithm involves sensing channel estimation and subspacebased DOA estimation, whose computational complexities are \(\mathcal {O}\left(K N^{2}_t M+K N^{2}_t\! N_r + \!N^{3}_t+N_t N_r M \right)\) and \(\mathcal {O}\left( K N^{2}_t N^{2}_r+N^{3}_t N^{3}_r + N^{2}_t N^{2}_r P \right)\), where P denotes the number of spatial sampling grids. Since the interval of spatial sampling grids is set to \(0.01^{\circ }\) in this paper, the number of spatial sampling grids P equals to 18,Â 000, which is significantly larger than the other terms. Thus, the computational complexity of the proposed algorithm can be measured by \(\mathcal {O}\left( N^{2}_t N^{2}_r P \right)\).
In comparison, the computational complexities of the interleaving frame structurebased method, the SSRbased method, and the MUSICbased method are \(\mathcal {O}(K N_r M+ N^{2}_t N^{2}_r P + N^{2}_t N^{2}_r M+ N^{3}_t N^{3}_r)\), \(\mathcal {O}\left( N_t N_r M P^{2}\right)\) and \(\mathcal {O}\left( K N^{2}_r M + N^{3}_r+ N^{2}_r P \right)\), respectively. Note that, method of exhaustion is employed to minimize the \(\ell _1\)norm, and the estimation results from all the symbols are averaged for the SSRbased method in this manuscript. Since the computational complexities of the interleaving frame structurebased method and the MUSICbased method are dominated by the spectral peak searching procedures, they can be calculated as \(\mathcal {O}\left( N^{2}_t N^{2}_r P\right)\) and \(\mathcal {O}\left( N^{2}_r P\right)\), respectively. It is clear that the computational complexity of the SSRbased method is \(M P/(N_t N_r)\) times higher than that of the proposed algorithm. Meanwhile, although the computational complexity of the MUSICbased method can be smaller than that of the proposed algorithm, the DOA estimation performance of the MUSICbased method is limited, since the virtual array derivation with larger aperture is not considered.
4 Simulation results and discussion
In this section, we evaluate the performance of the proposed nonorthogonal waveform assisted DOA estimation algorithm. The quadrature phase shift keying (QPSK) modulation is employed for OFDM symbols. The targets are characterized by (distance, direction, attenuation). The system parameters are listed in Table 1, in which the subcarrier spacing and the carrier frequency obey the standard 38.1011 of 3rd Generation Partnership Project (3GPP) for 5G wireless communication [39]. For simplicity, the CP symbol duration is neglected.
Assume that two farfield targets are respectively located at \((100m,\theta _1,0.1)\) and \((50m,\theta _2,0.1)\), where \(\theta _1\) and \(\theta _2\) are uniformly distributed in \([{20}^\circ ,{60}^\circ ]\). The proposed algorithm is compared with MUSICbased method with fully nonorthogonal transmitting principle [22], interleaving frame structurebased method with orthogonal transmitting principle [27] and the SSRbased method with partially orthogonal transmitting principle [31]. The interval of spatial sampling grids for all the methods is set to \(0.01^{\circ }\). The rootmeansquare error (RMSE) for DOA estimation
is adopted as the performance metric, where L represents the number of MonteCarlo trials, \({\theta }_{q, l}\) denotes the DOA of the qth target in the lth trial, and \(\hat{\theta }_{q, l}\) denotes its estimation. \(L=400\) MonteCarlo trials are performed for each simulated scenario.
4.1 DOA estimation accuracy
First, since the number of deployed OFDM symbols and subcarriers are two important parameters that affect the DOA estimation accuracy, we present the DOA estimation accuracy with different number of employed OFDM symbols and subcarriers in Figs.Â 4 and Â 5, respectively. The signaltonoise ratio (SNR), i.e., the ratio of average power of the received symbols to the noise power as \({\text{E}\left\{ \varvec{d}^{\text{H}}\varvec{d}\right\} }/{\text{E}\left\{ \varvec{n}^{\text{H}}\varvec{n}\right\} }\), is set to 0 dB. As shown in Fig.Â 4, the accuracy of proposed DOA estimation increases as the number of employed OFDM symbols increases. The reason is that the error of roundtrip sensing channel estimation decreases with the increasing number of employed OFDM symbols, which benefits the accurate virtual channel matrix formation in (15) and (18). In contrast, the MUSICbased method does not calculate the sensing channel nor synthesize a virtual array. Meanwhile, for the SSRbased method and the interleaving frame structurebased method, each OFDM symbol is separately considered, which means that the increasing number of OFDM symbols will not benefit the DOA estimation performance. The result in Fig.Â 5 suggests that the proposed algorithm enjoys an improved estimation accuracy when more subcarriers are incorporated in the system. Note that, the SSRbased method has better DOA estimation accuracy than the proposed algorithm for \(K < 16\) since the true distance values of the targets are utilized as a prior for the SSRbased method. However, the accuracy of the SSRbased performance is achieved at the cost of high computation complexity. More importantly, for both the interleaving frame structurebased method and the SSRbased method, the increasing number of subcarriers do not contribute to the improvement of estimation accuracy, because the subcarriers are independently considered for the virtual array formulation in (8).
Then, in Fig.Â 6, we compare the DOA estimation accuracy of the evaluated methodsÂ with differnet SNR. It is clear that the proposed algorithm outperforms the other methods. Benefiting from the construction of virtual channel matrix, the augmented virtual array can be derived under the fully nonorthogonal scenario. Thus, more pilot symbols can be effectively utilized for the proposed algorithm. Although the virtual arrays are also derived in the SSRbased method and the interleaving frame structurebased method, they require orthogonal or partially orthogonal transmitting principles, thus the number of utilized pilot symbols is limitedÂ with respect toÂ the fixed number of OFDM symbols and subcarriers.
4.2 Angular resolution
We also compare the angular resolution of all evaluated methods in Fig.Â 7. In particular, three targets are assumed, whose locations are \((50m,{44}^\circ ,0.1)\), \((100m,{46}^\circ ,0.2)\), and \((80m,{48}^\circ ,0.3)\), respectively. The SNR is set to 0 dB. Except the proposed algorithm, the other compared methods fail to distinguish all the targets. For the methods that consider the virtual array derivation in the orthogonal and partially orthogonal scenarios, the synthesized virtual arrays are not ULAs, as illustrated in Sect. 2. The estimation performance of the interleaving frame structurebased method is thereby affected. Although the SSRbased method does not rely on the ULA, its resolution is limited by predefined searching grid. In contrast, since each row of the virtual channel matrix in (18) corresponds to a single subcarrier, the virtual array derived in the proposed algorithm is a ULA, such that high DOA estimation resolution can be achieved.
In scenarios with targets at identical distance, it is necessary for the proposed algorithm to implement Toeplitz reconstruction, as shown in Fig.Â 8. Four farfield radar targets are assumed, which locate at \((50m,{44}^\circ ,0.1)\), \((100m,{46}^\circ ,0.2)\), \((80m,{48}^\circ ,0.3)\) and \((50m,{50}^\circ ,0.2)\), respectively. With Toeplitz reconstruction, the rank of the equivalent signal matrix \(\varvec{\Xi }\) in (20) is restored, such that the proposed algorithm can effectively estimate the DOAs of targets which locate at the identical distance.
4.3 Computational complexity
To demonstrate the computational complexities of the proposed algorithm and the compared methods intuitively, a comparison of computation time is depicted in Fig.Â 9. It is clear that the proposed algorithm has better computational efficiency than the SSRbased method and the interleaving frame structurebased method. The MUSICbased method is computationally faster than the proposed algorithm since the dimension of the sample covariance matrix for the MUSICbased method is lower than that for the proposed algorithm. However, the MUSICbased method suffers from a deteriorated DOA estimation performance, while the computation time of the proposed algorithm is still comparable.
5 Conclusion
In this paper, we have addressed the DOA estimation problem in a fully nonorthogonal joint MIMO sensing and communication scenario, while maintaining a high communication rate. To enhance the DOA estimation performance, an augmented virtual array with enlarged aperture is formulated by effectively using the coefficients of the roundtrip sensing channels. Then, by treating the virtual channel matrix as the equivalent signal of the virtual array, the subspace processing is conducted for DOA estimation. To tackle the rankdeficiency problem of the equivalent signal matrix at the presence of targets with identical distance, the Toeplitz reconstruction is proposed to restore the rank. Simulation results demonstrate the superiority of the proposed algorithm over conventional methods in both estimation resolution and accuracy.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Abbreviations
 A/D:

Analogtodigital
 CP:

Cyclic prefix
 D/A:

Digitaltoanalog
 DFT:

Discrete Fourier transform
 DOA:

Directionofarrival
 FR2:

Frequency Range 2
 GP:

Guard period
 LMMSE:

Linear minimum mean square error
 LS:

Least square
 MIMO:

Multipleinput and multipleoutput
 MSE:

Mean square error
 MMSE:

Minimum mean square error
 OFDM:

Orthogonal frequencydivision multiplexing
 P/S:

Serialtoparallel
 PSK:

Phaseshift keying
 QAM:

Quadrature amplitude modulation
 QPSK:

Quadrature phase shift keying
 RMSE:

Rootmeansquare error
 SNR:

Signaltonoise ratio
 S/P:

Paralleltoserial
 SSR:

Sparse signal recovery
 ULA:

Uniform linear array
 5G:

Fifthgeneration
 5G NR:

5G New Radio.
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Acknowledgements
The authors would like to thank the editor and the anonymous reviewers for their helpful comments and suggestions.
Funding
This work was supported in part byÂ theÂ National Natural Science Foundation of China (No. U21A20456, 62271444, 61901413), Zhejiang Provincial Natural Science Foundation of China (No. LZ23F010007), Zhejiang University Education Foundation Qizhen Scholar Foundation, and the 5G Open Laboratory of Hangzhou Future SciTech City.
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Lin, L., Zheng, H., Zhou, C. et al. Nonorthogonal waveform assisted DOA estimation for joint MIMO sensing and communication. EURASIP J. Adv. Signal Process. 2023, 22 (2023). https://doi.org/10.1186/s13634023009766
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DOI: https://doi.org/10.1186/s13634023009766
Keywords
 DOA estimation
 Joint sensing and communication
 Nonorthogonal waveform
 Virtual channel matrix