Wireless Pers Commun (2008) 46:371–383 DOI 10.1007/s11277-007-9440-7
Blind Joint Symbol Detection and DOA Estimation for OFDM System with Antenna Array Xiaofei Zhang · Bao Feng · Dazhuan Xu
Published online: 10 January 2008 © Springer Science+Business Media, LLC. 2007
Abstract An analysis of the received signal of Orthogonal frequency division multiplexing (OFDM) system with array antennas shows that the received signal has trilinear model characteristics. Trilinear decomposition-based joint symbol detection and direction of arrival (DOA) estimation for OFDM system with antenna array is proposed in this paper. The simulation results reveal that the symbol detection performance of the proposed algorithm is very close to the post-FFT receiver with perfect channel state information; DOA estimation performance is very close to least squares method, and even this algorithm supports small sample sizes. Finally this algorithm does not require the channel fading information, DOA and training sequence or pilot information, so it has blind characteristics. Keywords OFDM · Antenna array · Trilinear model · Joint processing · Symbol detection · DOA estimation
1 Introduction Orthogonal frequency division multiplexing (OFDM) is an efficient technique for high-speed digital transmission over multipath fading channels [1, 2]. OFDM has been recently emerged as a promising technique for future mobile communications. Array antennas not only combat multi-path fading, but also suppress interference signals [3]. The performance of an OFDM system in the presence of cochannel signals can be significantly improved by using an antenna array at the receiver to form an Array-OFDM system [4, 5]. An Array-OFDM system exploits the spatial domain to outperform the conventional OFDM system; such a solution has been adopted in IEEE802.16 standard. In general, the receiver structure of OFDM with antenna array can be classified into two types, namely, pre-FFT [6–8] and post-FFT types [9]. The post-FFT method is optimum in terms of maximizing signal-to-noise-and-interference power
X. Zhang (B) · B. Feng · D. Xu Electronic Engineering Department, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China e-mail:
[email protected]
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ratio (SNIR), but it requires a larger number of FFT processors and perfect channel state information. On the other hand, the pre-FFT method, which requires only one FFT processor, can reduce greatly the computational complexity by tolerating performance degradation. The methods mentioned above are non-blind methods, since they require the knowledge of direction of arrival (DOA) and channel fading information. Blind joint symbol detection and DOA estimation for OFDM system with antenna array is investigated in this paper. It was known that most of signal processing methods are based on matrix decomposition, or the bilinear model. In general, matrix decomposition is not unique since inserting a product of an arbitrary invertible matrix and its inverse in between two matrix factors preserves their product. Matrix decomposition can be unique only if one imposes additional problem-specific structural properties including orthogonality, vandermonde, toeplitz, constant modulus or finite-alphabet constraints. Compared with the case of matrices, trilinear model or trilinear decomposition has a distinctive and attractive feature: it is often unique [10]. The uniqueness of trilinear decomposition is of great practical significance, which is crucial in many applications such as psychometrics [11] and chemistry [12–14]. Trilinear decomposition is also called Parallel factor (PARAFAC) analysis. Trilinear decomposition is thus naturally related to linear algebra for multi-way arrays. In signal processing field, trilinear decomposition can be thought of as a generalization of ESPRIT and joint approximate diagonalization ideas [15, 16]. Trilinear decomposition is widely used in blind receiver detection for Direct-sequence code-division multiple access (CDMA) system [17], array signal processing [18–20], blind estimation of Multi-Input-Multi-Output (MIMO) system [21], blind speech separation [22], downlink receiver for space-time block-boded CDMA System [23] and multiuser detection for Single-Input-Multi-Output (SIMO) CDMA System [24] and polarization sensitive antennas [25]. Our work links the joint symbol detection and DOA estimation problem to the trilinear model and derives trilinear decomposition-based joint symbol detection and DOA estimation algorithm for OFDM system with array antennas. The symbol detection performance of the proposed algorithm is close to post-FFT receiver, and DOA estimation performance is very close to least squares method. Our algorithm relies on a fundamental result of Kruskal [10] on the uniqueness of low-rank three-way data decomposition; it supports small sample sizes. This paper is structured as follows. In Sect. 2, we present the data model we developed. In Sect. 3 we propose blind joint symbol detection and DOA estimation algorithm for OFDM system with array antennas and discuss identifiability. In Sect. 4, we present simulation results, and conclusions are made in Sect. 5.
2 Data Model Considering the uplink of an OFDM system, the transmitter has a single transmitting antenna, but the receiver is equipped with a uniform linear array containing I antennas. To simplify the model, it is assumed that the receiver is fully synchronized and the carrier frequency offset has been estimated and compensated. The number of subcarriers or inverse fast fourier transform (IFFT) size is N and a cyclic prefix (guard interval) of L sampling intervals is used so that one OFDM symbol has N + L samples. L is chosen to exceed the maximum delay spread. The channel is modeled as an finite impulse response (FIR) filter of length L m . The symbol sequence is first serial-to-parallel converted to produce symbol blocks of length N . We denote the kth block to be transmitted by s(k) = [s1 (k), s2 (k), . . . , s N (k)]T . The kth block is then multicarrier modulated (using IFFT) and padded with a cyclic prefix.
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The output signal of insertion cyclic prefix unit is shown as follows d(k) = Tcp F H s(k)
(1) T is a matrix which is used to add cyclic prefix. F is the fast where Tcp = (I L×N )T I NT fourier transform (FFT) matrix with N×N. The signal d(k) is transmitted through the multipath fading channel. The received baseband signal of the ith antenna is u i (t) =
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(2)
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where h i (l) is the lth path channel fading between transmit antenna and the ith receiver antenna. L m is the number of multiple paths. According to the uniform linear array characteristic, the received signal of the jth antenna is: u j (t) =
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h j (l)d(t − l),
where h j (l) = h i (l)e− j2π( j−i)d sin θl /λ
(3)
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The output signal of the removing-cyclic-prefix unit is modeled as xi (k) = Tr m u i (k)
(4)
R N ×(N +L)
where Tr m = [0 N ×L , I N ] ∈ is a matrix which is used to remove cyclic prefix. − j2π nl Li N Define Hi (n) = l=1 h i (l)e as the channel frequency response for the nth subcarrier, corresponding to the ith antenna in the array antenna. The frequency domain channel vector for the ith receiver antenna Hi = [Hi (1), Hi (2), . . . , Hi (N )]T
(5)
The frequency domain channel matrix for the array antenna receiver is H = [H1 , H2 , . . . , H I ]T ∈ C I ×N
(6)
The signal in (4) is denoted as
where
xi (k) = F H diag(Hi )s T (k)
(7)
⎤ 0 Hi (1) · · · ⎢ .. ⎥ ∈ C N ×N diag(Hi ) = ⎣ ... H (n) . ⎦ i 0 · · · Hi (N )
(8)
⎡
Assuming the space-time channel parameter is constant for K blocks. Define the source matrix S = [s(1), s(2), . . . , s(K )]T ∈ C K ×N . The output signal of the ith antenna through removing cyclic prefix is denoted as X i = F H diag(Hi )S T , i = 1, 2, . . . , I
(9)
X i can be regarded as the ith slice along spatial direction. In the presence of noise, the received signal model becomes X˜ i = X i + Wi = F H diag(Hi )S T + Wi , i = 1, 2, . . . , I
(10)
where Wi is the received noise corresponding to the ith slice.
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The noiseless signal in (9) is also denoted through rearrangements xm,k,i =
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f m,n sk,n h i,n , m = 1, . . . , N ; k = 1, . . . , K ; i = 1, . . . , I
(11)
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where h i,n stands for the (i, n) element of the matrix H, and similarly for the others. N is the number of subcarriers; I is the number of antennas in array antennas; K is the block number of source symbols. Note that (11) is a sum of triple products; it is well known as the trilinear model. The trilinear model X reflects three different kinds of diversity available: spatial, temporal and frequency diversity. Another view, likely, X i = F H diag(Hi )S T , i = 1, 2, . . . , I , can be regarded as slicing the 3-D data in a series of slices (2-D data) along the spatial direction. The symmetry of the trilinear model in (11) allows two more matrix system rearrangements. In particular Yk = H diag(Sk )[F H ]T , k = 1, 2, . . . , K
(12)
where Yk is the kth slice along the temporal direction. Sk is the kth row of the source matrix S. Similarly Z m = S diag(FmH )H T , m = 1, 2, . . . , N
(13)
where Z m is the mth slice along the frequency direction. FmH is the mth row of the source matrix F H .
3 Joint Symbol Detection and DOA Estimation 3.1 Trilinear decomposition Trilinear alternating least square (TALS) algorithm is the common data detection method for trilinear model [10]. The basic idea of TALS is as follows: (a) Each time, update a matrix using least squares conditioned on previously obtained estimates of the remaining matrix; (b) proceed to update another matrix; (c) repeat until convergence. TALS algorithm is discussed in detail as follows. The signal in (9) is also represented as ⎡
⎤ ⎤ ⎡ H F diag(H1 ) X1 ⎢ X 2 ⎥ ⎢ F H diag(H2 ) ⎥ ⎢ ⎥ T ⎥ ⎢ ⎢ .. ⎥ = ⎢ ⎥S .. ⎣ . ⎦ ⎣ ⎦ . XI F H diag(H I )
(14)
According to (14), least squares fitting is ⎡ ⎤ ⎤ ⎡ H X˜ 1 F diag(H1 ) ⎢ X˜ 2 ⎥ ⎢ F H diag(H2 ) ⎥ ⎢ ⎥ T ⎥ ⎢ min ⎢ . ⎥ − ⎢ ⎥S .. F,H,S ⎣ .. ⎦ ⎣ ⎦ . H diag(H ) X˜ F I I
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where F stands for the Frobenius norm. X˜ i , i = 1, 2, . . . , I are the noisy slices. Least squares (LS) update for S is ⎡ H ⎤+ ⎡ ˜ ⎤ F diag( Hˆ 1 ) X1 ⎢ F H diag( Hˆ 2 ) ⎥ ⎢ X˜ 2 ⎥ ⎥ ⎢ ⎥ ⎢ (16) Sˆ T = ⎢ ⎥ ⎢ . ⎥ .. ⎣ ⎦ ⎣ .. ⎦ . X˜ I F H diag( Hˆ I ) where [.]+ stands for pseudo-inverse. Hˆ i , i = 1, 2, . . . , I , denote previously obtained estimates of Hi . Similarly, from the second way of slices: Z m = S diag(FmH )H T , m = 1, 2, . . . , N , which is represented as ⎤ ⎤ ⎡ ⎡ S diag(F1H ) Z1 ⎢ Z 2 ⎥ ⎢ S diag(F H ) ⎥ 2 ⎥ T ⎥ ⎢ ⎢ (17) ⎥H ⎢ .. ⎥ = ⎢ .. ⎦ ⎣ . ⎦ ⎣ . ZN S diag(FNH ) According to (17), least squares fitting is ⎡ ⎤ ⎤ ⎡ Z˜ 1 S diag(F1H ) ⎢ Z˜ 2 ⎥ ⎢ S diag(F H ) ⎥ 2 ⎥ T ⎢ ⎥ ⎢ H min ⎢ . ⎥ − ⎢ ⎥ .. F,H,S ⎣ .. ⎦ ⎣ ⎦ . H) Z˜ S diag(F N N F where Z˜ m , m = 1, 2, . . . , N , re the noisy slices. Least squares update for H is ⎡ˆ ⎤+ ⎡ ˜ ⎤ S diag(F1H ) Z1 ⎢ Sˆ diag(F H ) ⎥ ⎢ Z˜ 2 ⎥ 2 ⎥ ⎢ ⎥ ⎢ Hˆ T = ⎢ ⎥ ⎢ . ⎥ .. . ⎣ ⎣ ⎦ . ⎦ . H Z˜ N Sˆ diag(F )
(18)
(19)
N
where Sˆ denote previously obtained estimate of S. According to (16) and (19), the matrices S and H are updated with conditioned least squares, respectively. The matrix update will stop until convergence. We can use TALS algorithm to attain the frequency domain channel matrix H and the source matrix S. TALS is optimal when noise is additive i.i.d. Gaussian [26]. TALS algorithm has several advantages: it is easy to implement, guaranteed to converge and simple to extend to higher order data. The shortcomings lies mainly in the occasional slowness of the convergence process [27]. The algorithm proposed in this paper uses the TALS with the known matrix F H , which will speed up convergence. 3.2 Identifiablity The k-rank concept is very important in the trilinear algebra. Definition 1 [10] Consider a matrix A ∈ C I ×J . If rank(A) =r , then A contains a collection of r linearly independent columns. Moreover, if every l ≤ J columns of A are linearly independent, but this does not hold for every l+1 columns, then A has k-rank k A = l. Note that k A ≤ rank(A), ∀A.
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Theorem 1 [20] Z m = S diag(FmH )H T , m = 1, 2, . . . , N, where F ∈ C N ×N, H ∈ R I ×N , S ∈ C K ×N, considering that F H is a matrix with Vandermonde characteristic, if k S + min(k H + N , 2N ) ≥ 2N + 2
(20)
then H, F and S are unique up to permutation and scaling of columns. Consider that FFT matrix F is known in practice. The permutation ambiguity can be resolved by the known matrix F. Any other H¯ , S¯ that construct Z ..m (m = 1, 2, . . . , N ) is related to H and S via H¯ = H 1 , S¯ = S2
(21)
where 1 and 2 are diagonal scaling matrices satisfying: 1 2 = I . The scale ambiguity can be resolved using automatic gain control, differential encoding/decoding, normalization and the embedded information. In our present context, for source-wise independent source signals, k S = min(K , N ); for source-wise independent channel, k H = min(I, N ), and therefore, (20) becomes min(N , K ) + min((N + min(I, N ), 2N ) ≥ 2N + 2
(22)
In practice N ≥ I , min((N + min(I, N ), 2N ) = I + N , hence the practical condition is I + min(N , K ) ≥ N + 2
(23)
If N ≤ K , the identifiable condition is I ≥ 2. If N ≥ K , the identifiable condition is K + I ≥ N +2
(24)
So this algorithm can support small samples. 3.3 DOA Estimation The channel is modeled as an finite impulse response (FIR) filter of length L m . The frequency domain channel vector H i is processed through IFFT to get the time domain channel response vector, [h i (1), h i (2), . . . , h i (L m )]. According to the frequency domain channel matrix H, The time domain channel matrix Ht is shown as follows ⎤ ⎡ h 1 (1) h 1 (2) · · · h 1 (L m ) ⎢ h 2 (1) h 2 (2) · · · h 2 (L m ) ⎥ ⎥ ⎢ (25) Ht = ⎢ . ⎥ where h j (l) = h i (l)e− j2π( j−i)d sin θl /λ .. . . .. ⎦ ⎣ .. . . . h I (L m ) h I (1) h I (2) Normalization for the matrix Ht is to get the direction matrix A. The direction vector for DOA θl is a(θl ) = [1, e− j2π d sin θl /λ , . . . , e− j2π(I −1)d sin θl /λ ]T
(26)
g = −imag(ln(a(θl ))) = [0, 2πd sin θl /λ, . . . , 2πd(I − 1) sin θl /λ]T
(27)
According to (26)
where ln(.) is natural logarithm; imag(.) is to get imaginary part of a complex number. Equation 27 should be arithmetic progression sequence. We can use least squares principle to estimate sin θl and then estimate DOAs.
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We can use TALS algorithm to attain the estimated frequency domain channel matrix Hˆ . Hˆ is processed through IFFT to get the time domain channel response matrix Hˆ t . The ˆ which also resolves the scale normalization for the matrix Hˆ t is to get the direction matrix A, ambiguity. Assuming the estimated direction vector is a(θ ˆ l ) (the lth column of the estimated ˆ and then a(θ direction matrix A), ˆ l ) is processed to get gˆ according to (27). Finally we use least squares principle to estimate sin θl . Least squares fitting is PC = gˆ where
⎡ ⎢ ⎢ P=⎢ ⎣
1 0 1 2πd/λ .. .. . . 1 (I − 1)2πd/λ
(28) ⎤ ⎥ ⎥ ⎥ ⎦
C = [c0 , c1 ]T , where c1 is the estimated value of sin θl
(29)
(30)
The least square solution for C is C = (P T P)−1 P T gˆ
(31)
ˆ In practice, the matrix We define D = (P T P)−1 P T , Eq. 31 is represented as C = D g. D is a constant matrix, so the calculation of C is very small. DOA estimation is shown as follows θˆl = sin−1 (c1 )
(32)
3.4 Joint Symbol Detection and DOA Estimation for OFDM System with Array Antenna Trilinear decomposition-based joint symbol detection and DOA estimation for OFDM system with array antenna (Trilinear-JSDE) is proposed in this paper. The detailed steps are shown as follows: Step 1: Initialize for the source matrix S and the frequency domain channel matrix H. IFFT matrix F H is known. Step 2: LS update for the source matrix S according to (16) Step 3: LS update for the frequency domain channel matrix H according to (19) Step 4: Repeat step 2 to step 3 until convergence Step 5: Make decision for the estimated source matrix S and estimate DOA according to the frequency domain channel matrix H.
4 Simulation results Let X˜ i = F H diag(Hi )S T + Wi be the received noisy data, for i = 1, 2, . . . , I , where Wi are the AWGN matrices. We define the sample SNR I T 2 H i=1 F diag(Hi )S F SNR = 10 log10 (33) I 2 W i i=1 F
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Fig. 1 Symbol detection performance under several values of K
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We performed Monte Carlo simulations to assess the performance of trilinear decomposition-based joint symbol detection and DOA estimation for OFDM system with array antenna (Trilinear-JSDE). The number of Monte Carlo trials is 1000. Binary phase shift keying (BPSK) modulated signal and additive gauss white noise are used in the simulations. The receiver is equipped with a uniform linear array containing eight antennas, and the array spacing is half wavelength. The number of subcarriers size is 32 and a cyclic prefix (guard interval) with eight sampling intervals is used in this simulation. The channel is modeled as an FIR filter of length 3, and their DOAs are 10◦ , 30◦ and 50.◦ Note that K is the block number of snapshots, and a block includes N samples (N is the number of subcarriers size). In our simulations we used N = 32, I = 8. According to Eq. 24, K ≥ N + 2 − I = 26, and then the minimum value of K is 26. Simulation 1 : The symbol detection performance of the proposed algorithm is investigated in this simulation. We compare our proposed algorithm with the post-FFT receiver. The postFFT receiver is optimum in terms of maximizing signal-to-noise-and-interference power ratio (SNIR). In contrast to Trilinear-JSDE algorithm, the non-blind post-FFT receiver assumes the perfect knowledge of DOA and channel fading information. The proposed algorithm does not require the information of DOA and channel fading. Figure 1 presents the results for for several values of K i.e. 30, 50, 100 and 200. From Fig. 1 we find that the symbol detection performance of Trilinear-JSDE algorithm is very close to non-blind post-FFT receiver. The performance of Trilinear-JSDE algorithm with K = 200 and S N R = 2 is about 0.5 dB away from non-blind post-FFT receiver. We also find that the gap between blind Trilinear-JSDE algorithm and (non-blind) postFFT receiver increases as K decreases. When K = 30, the Trilinear-JSDE algorithm is about 1 dB away from non-blind post-FFT receiver. The symbol detection performance of our proposed algorithm improves with K increasing. When K > 100, the symbol detection performance of our proposed algorithm improves slightly. Figure 1 also shows small sample results: K = 30. It is clear that Trilinear-JSDE algorithm performs well even for small sample sizes. Simulation 2 : The DOA estimation performance of the proposed algorithm is investigated in this simulation.
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Fig. 3 DOA estimation performance comparison for K = 100
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Define RMSE = |DOAe −DOA0 |2 , where DOAe is the estimated DOA and DOA0 is the perfect DOA. We compare our proposed algorithm with the least squares DOA estimation method. The least squares DOA estimation method uses the training sequence and least squares principle to estimate DOA, and it is non-blind method. Figure 2, 3 show the results for K = 50 and 100, respectively. From Figs. 2, 3 we find that the DOA estimation performance of blind Trilinear-JSDE algorithm is very close to non-blind least squares DOA estimation method. Trilinear-JSDE algorithm almost has the same DOA estimation performance to non-blind least squares method. From Figs. 2, 3, we find that angle estimation performance of Trilinear-JSDE algorithm degrades with DOA increasing. The DOA estimation performance for DOA = 10◦ is better than that for DOA = 30◦ , and The DOA estimation performance for DOA = 30◦ is better than that for DOA = 50◦ . The DOA estimation performance degrades with DOA increasing, and that is because that sin−1 (.) in (32) is the nonlinear function. Figures 4, 6 present DOA estimation performances for DOA = 10◦ , 30◦ and 50◦ under different K, respectively. From Fig. 4–6 we find that Trilinear-JSDE algorithm has better DOA
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Fig. 4 DOA estimation performance with DOA = 10◦ and several values of K
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estimation performance with K increasing. Figs. 4–6 show small sample results: K = 30. It is clear that the proposed algorithm performs well even for very small sample sizes.
5 Conclusions An analysis of the received signal of OFDM system with array antennas shows that the received signal has trilinear model characteristics. Trilinear decomposition-based joint symbol detection and DOA estimation for OFDM system with antenna array is proposed in this paper. The simulation results reveal that the symbol detection performance of the proposed algorithm is very close to the post-FFT receiver with perfect channel state information, DOA estimation performance is very close to non-blind least squares method. Joint symbol detection and DOA estimation performance of our proposed algorithm improves when the number of snapshots increases. This algorithm even supports small sample sizes. We also find that the DOA estimation performance of our proposed algorithm degrades with DOA increasing.
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Our contribution in this paper is that we link the joint symbol detection and DOA estimation problem to the trilinear model, and derives a blind joint symbol detection and DOA estimation algorithm. Compared with other methods (especially non-blind method), our proposed algorithm does not need the channel fading information, DOA and pilot or training information, so it has blind characteristics. Acknowledgements This work is supported by the startup fund of Nanjing University of Aeronautics & Astronautics (S0583-041) and Jiangsu NSF Grants BK2007192. The authors wish to thank the anonymous reviewers for their valuable suggestions on improving this paper.
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Author Biographies Xiaofei Zhang received M.S. degree in Electrical Engineering from Wuhan University, Wuhan, China, in 2001. He received Ph.D. degrees in Communication and Information Systems from Nanjing University of Aeronautics and Astronautics in 2005. From 2005 to 2007, he was a Lecturer in Electronic Engineering Department, Nanjing University of Aeronautics & Astronautics, Nanjing, China. His research is focused on array signal processing and communication signal processing.
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Blind Joint Symbol Detection and DOA Estimation for OFDM System with Antenna Array
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Bao Feng is currently postgraduate student at Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research is focused on Mobile Communication.
Dazhuan Xu was graduated from Nanjing Institute of Technology, Nanjing, China, in 1983. He received the M.S. degree and the Ph.D. in communication and information systems from Nanjing University of Aeronautics and Astronautics in 1986 and 2001, respectively. He is now a Full Professor in the College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include digital communications, software radio and coding theory.
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