Joint Estimation of DOA and Frequency with Sub-Nyquist Sampling in a Binary Array Radar System

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1 Joint Estimation of DOA and Frequency with Sub-Nyquist Sampling in a Binary Array Radar System arxiv:80.390v eess.sp 26 O 208 Zhan Zhang, Ping Wei, Lijuan Deng, Huaguo Zhang School of Information and Communication Engineering, University of Eleronic Science and Technology of China, Chengdu 673, China zzhenry5@63.com Science and Technology on Communication Information Security Control Laboratory, Jiaxing, China Abstra Recently, several array radar struures combined with sub-nyquist techniques and corresponding algorithms have been extensively studied. Carrier frequency and direion-ofarrival DOA) estimations of multiple narrow-band signals received by array radars at the sub-nyquist rates are considered in this paper. We propose a new sub-nyquist array radar architeure a binary array radar separately conneed to a multicoset struure with branches) and an efficient joint estimation algorithm which can match frequencies up with corresponding DOAs. We further come up with a delay pattern augmenting method, by which the capability of the number of identifiable signals can increase from to Q Q is extended degrees of freedom). We further conclude that the minimum total sampling rate 2B is sufficient to identify K Q narrowband signals of maximum bandwidth B inside. The effeiveness and performance of the estimation algorithm together with the augmenting method have been verified by simulations. Index Terms DOA estimation, frequency estimation, sub- Nyquist sampling, binary array radar I. INTRODUCTION In array radar signal processing, joint frequency and direion-of-arrival DOA) estimation problems have received extensive attention. odern signals have relatively high carrier frequencies and are distributed over a comparatively wide sperum range. According to Nyquist Sampling, samplers will face high conversion speed problem and considerable data pressure, which facilitates the rapid development of sub- Nyquist sampling techniques 2. Therefore, how array radars can jointly estimate frequencies and orientations of signals effeively at sub-nyquist Sampling has been extensively studied. In the literature, many methods such as 3, 4 have efficiently achieved joint estimations of DOAs and corresponding carrier frequencies at Nyquist sampling. However, these methods cannot work well at sub-nyquist sampling. The approaches such as 5, 6 overcome this defe and can make joint estimation at sub-nyquist sampling. Nevertheless, Zoltowski et al present 5 to decompose the entire sperum into overlapping sub-band via such a struure each antenna Identify applicable funding agency here. If none, delete this. followed by a filter bank and a mixer) and each band has to be examined separately by two paths a dire path and a delayed path), which causes more hardware resources and quite slow processing speed. On the other hand, Ariananda et al 6 put forward to conne a multi-coset struure 7 to each antenna of the minimum redundancy linear array RA) 8 to implement joint estimation with sub-nyquist sampling. This array arrangement also causes high consumption. Recently, Liu et al come up with 9, 0 to jointly estimate frequencies and DOAs with sub-nyquist sampling for more sources than sensors. However, these methods also bring high hardware pressure; the total sampling rate, i.e., the sum of sampling rates for all channels, is fairly high even though the sampling rate of each channel is lower than the Nyquist sampling rate. To overcome the problems of high hardware resource consumption and high total sampling rate, Kumar et al have suggested a simple sub-nyquist sampling architeure, 2 together with the corresponding algorithm. This architeure is that each receiver has only two paths, a dire path and a delayed path, followed by an ADC separately. The corresponding algorithm is based on USIC algorithm 3 and ESPRIT algorithm 4, which can realize joint estimations of DOAs and frequencies at a fairly low total sampling rate. Due to the limitation of Kruskal rank 5, in order to avoid the blurring problem of frequency and DOA estimation, this algorithm cannot be applied the in uniform linear array ULA), but in circular shape array and reangular nested array 2, which illustrates some certain limitations for the array layout. oreover, based on the generalized eigendecomposition of the matrix beam, its frequency estimation is too sensitive to noise. Thus, a two-dimensional multiresolution algorithm has to be applied to solve performance problems. The paper aims to to achieve joint estimations of frequencies and corresponding DOAs effeively at a very low total sampling rate in an array radar system, where the number of array elements is fewer than that of signals. We propose a binary array radar struure and corresponding joint estimation algorithm with high efficiency, which can automatically pair frequencies and DOAs to overcome the ambiguity problem

2 . oreover, an augmenting method similar to literature 0 is presented, which can be applied to estimate more signal parameters with fewer channels. The paper is organized as follows: in the next seion, we introduce the scenario together with signal model and present a binary array radar architeure. In seion III, we describe our algorithm as well as an expansion of time-delay manifold method, and give the boundary of the minimal total sampling rate. Seion IV shows simulation results and offers some further discussion. Finally, seion V comes to the conclusion. II. SIGNAL ODEL AND PROPOSED ARRAY STRUCTURE In this seion, we first describe the scenario including the signal model and some specific assumptions. Then a binary array radar struure is introduced, and corresponding array received signal model is given. A. Signal odel Consider such a scenario where K uncorrelated, narrowband, far-field signals spreading over a very wide sperum range from different orientations are received by a radar array. These K signals, considered as time domain wide multiband signals BS), i.e., many disjoint narrow-band signals within a wide-band sperum, are denoted as x t) and can be expressed as x t) = s k t) e j2πf kt ) Then, the time delay expression of x t) with time difference τ can be written as x t τ) = s k t τ) e j2πf kt τ) s k t) e j2πfkt τ) = s k t) e j2πf kτ where s k t), s k t), k = {, 2,..., K} denote the k th baseband signal and corresponding modulated signal with carrier frequency f k, respeively. The reason of the approximation in 2) is the narrow-band assumption. We further make other assumptions similar to on the above signal model: The baseband signals {s k t)} K are supposed to be mutually orthogonal and are bandlimited to B k = B k /2, B k /2, k = {, 2,..., K}. We further assume that k = {, 2,..., K}, B k B. The BS x t) is assumed to be bandlimited to F = 0, /T and T denotes the Nyquist sampling interval of x t). We further assume that /T B, that is to say, the total bandwidth of signals is far less than the total sperum range. Assume that the information bands do not overlap, i.e., {I S i f)) I S j f)) = : i, j {, 2,..., K}} where S i f) denotes the Fourier transform of modulated signal s i t) e j2πfit and I S i f)) represents the support region of S i f). This assumption satisfies that the carrier frequencies {f i } K i= are distin and corresponding signal speral bands do not overlap. 2) B. Proposed Array Architeure 2 c t 2 c t x t x t x t x t 2 2 t= nlt, T = f nyq x t x t c t x t x t x t x t 2 2 t= nlt, T = f nyq x t x t c t Fig.. Proposed Array Radar Struure Binary Array) x n) x2 n) ) x n x n) x2 n) ) x n Fig. depis the receiving array radar struure proposed in this paper. This array struure has only two array elements, and the first one is considered as the reference element. The distance between the two elements is denoted as d, and d is equal to the half-wavelength of the signal whose frequency is the Nyquist sampling rate f nyq. Each array element connes separately to an identical multi-coset struure including branches followed by samplers at LT sampling intervals, where L and T denote the sampling rate reduion faor and the Nyquist sampling interval, respeively. Denote the minimal delay unit as τ and the delay coefficients of these channels are denoted as C = c,..., c, 0 = c < < c. The delay pattern can be similarly set up to the array pattern of a sparse array, such as RAs 8, coprime arrays. We can express the output signal of the m th path of the reference element as x m t) and further define the signal model of the other array element as the form ). Therefore, the outputs of the m th path of the reference element and the other one x m t), x m t), can be expressed as x m t) = K s k t) e jω kc m + n m t) x m t) = K s k t) e jω kc m e jφ k + n m t) where ω k = 2πfk τ, φ k = 2πdfk sin θ k )/c, and n m t), n m t) are additive noise of corresponding paths. θ k is the DOA of the k th signal. ω k is the unit phase difference caused by the delay unit, including only frequency information f k ; φ k is the unit phase difference derived from the array struure, including joint frequency and angle information {f k, θ k }. oreover, the additive noise of all channels obeys zero-mean Gaussian distribution with variance σ 2 and is statistically independent of signals. 3)

3 Consider the received signals of all channels written as matrix form x t) A t n t) x all = = s t) + 4) x t) A t D φ n t) where A t ω) = a t ω ) a t ω K ) is timedelay manifold matrix including only the frequency informa- K tion, D φ = diag ) e jφ,..., e jφ K is a diagonal matrix including frequency and DOA information, and n t), n t) are the matrices of the noise, whose rows represent an additive noise of the different paths. III. PROPOSED APPROACH In this seion, we put forward an approach to identify as many signals as possible at sub-nyquist sampling in a binary array radar struture. This approach contains a joint estimation algorithm and an expansion method for increasing the number of identifiable signals. A. Frequency Domain and Time Domain Analyses According to, at sub-nyquist struures, time domain data is typically converted to frequency domain data for analysis. Therefore, the discrete time Fourier transform of sampled signal of the m th path at f s = f nyq /L sampling rate can be expressed as X ) m e j2πft = K e jω kc m S p k f) + N m p f) X ) m e j2πft = K e jω kc m e jφ k S p k f) + N m p f) where S p k f) = S i Z k f if s ), k {, 2,...K}, f 0, f sub ) denotes the aliased sperum of k th signal, and Nm p f s ) = i Z N m f if s ), m {, 2,..., } represent the aliased noise sperum. Thus, The speral data of all channels can be reformulated as matrix form X all = X f) X f) = A t A t D φ S f) + N f) N f) where S f) = S p f), Sp 2 f),..., Sp K f)t and N f) = N p f), N p 2 f),..., N p f)t. We form the following covariance matrix R Xall = E { X all X H } R XX R all = X X = A t A t D φ W = AWA H + Lσ 2 I 2 R XX A t A t D φ R X X H + Lσ 2 I 2 where A = a f, φ ), a f 2, φ 2 ),..., a f k, φ K ) and a f k, φ k ) contains the information of frequency and corresponding DOA of the k th signal; W = E { S f) S H f) } = diag W,..., W K ) and W k denotes the k th signal power. Notice that noise power goes up to Lσ 2, because the noise power sperum is folded under sub-nyquist sampling. 5) 6) 7) B. Proposed Algorithm for Estimation of Carrier Frequency and DOAs Due to the diagnose matrix D φ, we can exchange the positions of D φ and W in the matrix block in 7), and the correlation matrix R Xall becomes the following form R Xall = A twa H t A t WD H φ A H t A t WD φ A H t A t WA H t + Lσ 2 I 2 8) Notice that the two matrix blocks R XX,R X X on the diagonal of the correlation matrix R Xall in 8) have the same form i.e., A t WA H t +Lσ 2 I, which contains only frequency information. According to the USIC algorithm 3, Use eigendecomposition on the matrix blocks to obtain corresponding matrices Ĝ composed of eigenveors corresponding to the noise subspace. The scanning funion of peseudo sperum can be written as follow ˆP ω) = 9) ω) ĜĜ H a t ω) a H t As a result, use twice USIC algorithm on the two matrix blocks to obtain two Pseudo sperums, and estimate carrier frequencies twice. Average two frequency estimatons, and the final estimates of the frequencies { ˆf k } K can be obtained. According to 7), A contains all information of frequencies and corresponding DOAs, which have been paired and cannot become ambiguous. Use eigen-decomposition on the correlation matrix R Xall to obtain corresponding matrix Û composed of eigenveors corresponding to the noise subspace. Utilize the following scanning funtion to obtain corresponding DOAs ˆP ˆfk, φ) = ) 0) a H ˆfk, φ) ÛÛ H a ˆfk, φ where k = {, 2,..., K}. Based on USIC algorithm, this scanning funion 0) is combined with frequency estimation { ˆf k } K to obtain the corresponding DOA estimation. This complete joint estimation algorithm is named after JDF4BA and its specific processing steps are as follows: Step Operation TABLE I ALGORITH JDF4BA ) Calculate covariance matrix R Xall according to 7); 2) Eigen Decompose matrix blocks R XX, R X X to get Ĝ; 3) Use 9) to estimate the frequencies twice; 4) Average the two frequency estimation to acquire a final frequency estimation { ˆf k } K ; 5) Eigen Decompose on R Xall to obtain Û; 6) According to { ˆf k } K, orederly utilize 0) to get joint estimation { ˆf k, ˆθ k } K, notice K <. C. Expansion of Time-Delay anifold Since the JDF4BA algorithm utilizes the orthogonal charaeristic between the noise subspace and the manifold veor, JDF4BA algorithm can estimate up to signals. The

4 aim of this subseion is to achieve the breakthrough of the number of identifiable signals on such a binary array radar i.e., the more number of identifiable signals. According to 8), we first make column veorization on 4 matrix blocks of R Xall, and the outputs are expressed as r XX = vec R XX ) = A t A t ) w + Lσ 2 i r X X = vec R X X) = A t A t ) D φ w r XX = vec R XX ) = A t A t ) D φ w r X X = vec R X X) = A t A t ) w + Lσ 2 i ) where w = W,..., W K T and i = veci )., vec ) represent Khatri-Rao produ and column veorization, respeively. The Khatri-Rao produ of the manifold matrix is aually a virtual extension of the manifold matrix. Hence, design matrix transformation operator as Ξ to rearrange virtual manifold matrix and obtain the continuous manifold matrix A c t, shown as follow A c t = Ξ A t A t ) = a c t ω ) a c t ω K ) 2) where a c t ω k ) = e jω kq ),..., e jω kq ) T. Q denotes the degree of freedom of expansion. Therefore, via matrix transformation operator, we can obtain z c XX = Ac tw + Lσ 2 Ξi z c X X = Ac td φ w z c XX = A c td φ w z c X X = A c tw + Lσ 2 Ξi 3) According to 0, define the extraion matrix as Γ i = 0Q i ), I Q, 0 Q Q i), i = {, 2,..., Q} and Q 2Q ) carry out the following process z c XXi = Γ iz c XX z c X Xi = Γ iz c X X z c XXi = Γ i z c XX z c X Xi = Γ i z c X X 4) Rearrange the above expressions and obtain new correlation matrix blocks as follows R v XX = z c XXQ zc XX = A v t WA v t ) H + Lσ 2 I Q R v X X = z c X XQ zc X X = A v t WD H φ A v t ) H R v XX = z c XXQ z c XX = A v t WD φ A v t ) H R v X = X z c X z XQ c X X = A v t WA v t ) H + Lσ 2 I Q 5) where A v t = a v t ω ),..., a v t ω K ) denotes the virtual manifold matrix, and a v t ω k ) = e jωk0,..., e jω kq ) represents the virtual manifold veor. Colle the above correlation matrix blocks and form a new correlation matrix R v X all = Av t WA v t ) H A v t WD H φ A v t ) H +Lσ 2 I 2Q A v t WD φ A v t ) H A v t WA v t ) H 6) It can be seen that the struure of the new correlation matrix is aually similar to that of the correlation matrix R Xall in 8), and frequencies and DOAs in the new manifold matrix have been already matched up, and the dimension of the correlation matrix increases from to Q. Thus, we can use JDF4BA algorithm on these new correlation matrix R v X all to identify Q signals, and this expansion of time-delay manifold method is named after ET method. The specific steps for JDF4BA algorithm combined with ET method to identify signals is shown in TABLE II below Step Operation TABLE II ALGORITH JDF4BA WITH ET ) Calculate covariance matrix R Xall according to 7); 2) ake column veorization on matrix blocks of R Xall according to ); 3) Utilize 2) to sele the continuous manifold matrix and obtain 3); 4) Use extraion Γ i and rearrange veors according to 5); 5) Colle the new matrix blocks to obtain new correlation matrix R v X according to 6); all 6) Use algorithm JDF4BA on new correlation matrix R v X to all get joint estimation { ˆf k, ˆθ k } K, notice K < Q. D. Analyses of sampling rate reduion faor and the number of identifiable signals With the proposed method on the binary array, the joint estimation technique is effeive if i. Q K + ii. L /BT. The algorithm JDF4BA based on USIC algorithm can estimate the carrier frequencies and corresponding DOAs of BS signal, by utilizing the noise subspace. Therefore, the dimension Q of expansion of time-delay manifold has to be larger than the number K of signals. For the second condition, due to assumptions from II-A, f s should satisfy that the periodic sperums i.e., S p k f) = S i Z k f if s ), k {, 2,...K}, f 0, f nyq ) do not alias. This is possible if f s B or L /BT, the maximum bandwidth is assumed to be B according to the assumption. IV. SIULATION In this seion, numerical simulations are condued to confirm the validity and performance of the proposed approach. According to 6, the RA has the maximum virtual aperture in the condition of the same array element among the sparse arrays; therefore, the delay pattern we set up

5 Fig. 2. Pseudo sperum via algorithm JDF4BA with ET the bandwidth of each signal B = 25Hz, SNR = 0dB, and the sub-nyquist sampling rate f sub = 25Hz, i.e., the sampling rate reduion faor L = 400): a) Pseudo sperum of frequency estimation according to 9); b) Pseudo sperum of DOA estimation 0). Performance comparison all are complex-valuded sinusoid signals, the Nyquist sampling rate f nyq = 0GHz, and the sub-nyquist sampling rate f sub = 00Hz, i.e., the sampling rate reduion faor L = 00): c) Frequency estimation performance comparison; d) DOA estimation performance comparison. C = c,..., c, 0 = c < < c is similar to the array element position of RA. In our simulation we choose the Nyquist sampling rate f nyq = /T = 0GHz, the minimal delay unit τ = T, the number of paths following each array element = 4, and the delay pattern C = 0,, 4, 6. Hence, the path number of virtual time-delay channels of delay pattern C i.e., the dimension of expansion, is Q = 7. The Root ean Square Error RSE) of parameters is defined as RSE = N mk Nm i= K u i k û i k) 2 where the superscript i refers to the i th trail, and N m denotes the number of onte Carlo tests. u i k and ûi k are the true parameter and estimated parameter in the i th trail, respeively. The Signal to Noise Ratio SNR) is defined as SNR = E x m t) 2 )/σ 2 ). In the first simulation, the aim is to verify whether the algorithm JDF4BA with ET method is valid and whether the conclusion of III-D is corre. We set up 6 QPSK signals of which the maximum information bandwidth inside is B = 25Hz with following different carrier frequencies in GHZ) {.22, 2.77, 4.32, 6.54, 7.64, 8.48} and corre-

6 sponding DOAs in degrees) {45, 20, 0, 30, 0, 20}. We choose the sub-nyquist sampling rate f sub = 25Hz, same as the maximum information bandwidth B, corresponding to sampling rate reduion faor L = 400. As shown in Fig.2 a), according to the 9), the persudo sperum of frequency has 6 peaks so that we can obtain frequency estimation of signals. Seeing Fig. 2 b), we then use 0) to get 6 persudo sperums of DOA, combined with { ˆf k } K. Finally, we can obtain { ˆf k, ˆθ k } K. According to simulation settings, the total sampling rate fsub total = 2 f sub = 200Hz is less than the total Nyquist sampling rate fnyq total = 2 f nyq = 80GHz. The second numerical simulation is to test the performance of algorithm JDF4BA with ET method with different SNRs. We set up 6 complex sinusoid signals with following carrier frequencies {.22, 2.77, 4.32, 6.54, 7.64, 8.48} and corresponding DOAs {0, 20, 30, 30, 50, 80}. We choose the sub-nyquist sampling rate f sub = 250Hz, corresponding to sampling rate reduion faor L = 40. The number of onte Carlo is 200. The conditions for comparison are the algorithm JDF4BA with ET method at the Nyquist sampling rate, and the algorithm JDF4BA without ET method at the sub-nyquist sampling rate as well as the Nyquist sampling rate. Due to algorithm JDF4BA without ET method, the number of identifiable signals is = 3. Therefore, in the numerical simulation of the algorithm without ET method, the signals we set up is the first three signals in set {f k, θ k } K. As shown Fig.2 c) d), we can see the performance of frequencies and DOAs via JDF4BA with ET is not as good as the performance of only JDF4BA under the same sampling rate. The reason is that ET method is in exchange for slight sacrificing performance to be able to identify more signals. oreover, only if the sub-nyquist sampling rate f sub is larger than the maximum bandwidth B, the performance is almost identical to that at the Nyquist sampling rate, because the information bands of signals are complete at sub-nyquist sampling. Therefore, the performance is not related with the sampling rate if the sampling rate is larger than the maximum bandwidth. V. CONCLUSION In this paper, we put forward a binary array radar struure each array element has branches) and a novel approach algorithm JDF4BA and ET method) to jointly estimate frequencies and corresponding DOAs at sub-nyquist sampling. Algorithm JDF4BA can achieve auto-pairing between frequencies and corresponding DOAs under the sub-nyquist sampling rate f s B, thus avoiding ambiguity. Furthermore, ET method increases the number of identifiable signals from to Q. As a consequence, the binary array radar struure with this approach just needs 2 paths to estimate Q signals with the total sampling rate fsub total = 2B, which means fewer number of array elements only 2), fewer number of channels, and smaller total sampling rate under the same number of signals, compared with the previous works. Simulation results validate the effeiveness and performance of the present approach. ACKNOWLEDGENT This work was supported by the Fundamental Research Funds for the Central Universities Grant No. ZYGX206Z005, and ZYGX206J28). REFERENCES. Unser, Sampling-50 years after shannon, Proceedings of the IEEE, vol. 88, no. 4, pp , April Y. C. Eldar, Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, A. N. Lemma, A.. van der Veen, and E. F. Deprettere, Analysis of joint angle-frequency estimation using esprit, IEEE Transaions on Signal Processing, vol. 5, no. 5, pp , ay X. Wang, X. Zhang, J. Li, and J. 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