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Robust passive beamformer using bridge function sequences as weights Sheng Hong1b) , Dongkai Yang1a) , Kefei Liu1 , Xiaoxiang He2 , and Manos M. Tentzeris3 1
Beihang University,
37 Xueyuan Road, Haidian District, Beijing, 100191, P.R.China 2
Nanjing University of Aeronautics and Astronautics,
29 Yu Dao Street, Nanjing, 210016, P.R.China 3
GEDC/ECE, Georgia Institute of Technology, Atlanta, GA 30332-250, U.S.A
a)
[email protected] b)
[email protected]
Abstract: This letter introduces a novel passive beamformer which uses Bridge function sequence as spreading sequence weights for every antenna element. The set of Bridge function sequences have zero correlation zones (ZCZs) in their cross-correlation functions (CCFs). Due to the ZCZ properties of Bridge function sequences, the introduced passive beamformer features a much better performance on avoiding synchronization errors than the passive beamformers utilizing other spreading sequences without ZCZ. The theoretical justification of the proposed approach is presented in detail and the achievable performance is verified through computer simulations. Keywords: passive beamformer, robustness, bridge function, SNR, wrong decision probability Classification: Wireless circuits and devices References [1] F. B. Gross and C. M. Elam, “A new digital beamforming approach for SDMA using spreading sequence array weights,” Signal Process., vol. 88, no. 10, pp. 2425–2430, Oct. 2008. [2] H. L. Van Trees, Detection, estimation, and modulation theory, Part IV: Optimum array processing, John Wiley & Sons, pp. 47–49, 2002. [3] H. L. Van Trees, Detection, estimation, and modulation theory Part I, John Wiley& Sons, Inc, USA, pp. 42–52, 2001. [4] Z. Li and Q. Zhang, “Introduction to bridge functions,” IEEE Trans. Electromagn. Compat., vol. EMC-25, no. 4, pp. 459–464, Nov. 1983. [5] Z. Li and Z. Wang, “V-transform and its application to the analysis of images,” Conf. Proc. Chinese Electronic Society Conf. Walsh functions, Beijing, China, pp. 250–253, 1980. (In Chinese) [6] Z. Li and Q. Zhang, “Ordering of Walsh functions,” IEEE Trans. Electromagn. Compat., vol. EMC-25, no. 2, pp. 115–119, 1983. c
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DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
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1
Introduction
Traditional beamforming is usually implemented either in an analog manner tuning analog attenuators and phase shifters to steer directional beams or in an digital manner requiring complicated and time-comsuming adaptive digital beamforming algorithms. Aiming in alleviating the problems of previous generations of beamforming, Frank B.Gross and Carl M. Elam [1] proposed a new passive beamformer adapting spreading sequences as weights. This approach can process multiple angles of arrival simultaneously without the need of adaptive algorithms, and it is not limited by acquisition or tracking speed, while not requiring analog attenuators and phase shifters. Furthermore, that approach is quite generic as any arbitrary and/or random antenna array geometry can be incorporated into it. Still, the authors in [1] don’t state any specific guidelines for choosing spreading sequences as weights for this new beamformer and no data are shown about the beamformers’ reliability and robustness of different spreading sequences. In this letter, a new design employing Bridge function sequences as weights is applied to the structure considered in [1]. The proposed passive beamformer has the advantages that its anti-interference performance can be easily maintained to a high-level, i.e. higher processing gain can be achieved in comparison to other conventional spreading sequences when spreading sequences suffer with synchronization error interferences.
2
System structure
The generic passive beamformer is described in Fig. 1, wherein the array weights βn (t)(n = 1, 2, · · · , N ) are orthogonal spreading sequences [1]. Without loss of generality and for the proof of concept, the array used in this discussion will be a N -element uniform linear array (ULA) with element spacing d = λ/2 where λ is the wavelength of carrier wave. The incoming signals arrive at angles θl where l = 1, 2, · · · L. The total spreading array
Fig. 1. Passive beamformer using spreading sequences as array weights. c
IEICE 2009
DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
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output is called the received signal vector given by ¯ T A ¯ r [¯sr (t) + n ¯ Tx ¯ r (t) = β(t) ¯] y r (t) = β(t)
(1)
¯ T = [β1 (t), · · · , βN (t)], x ¯ r = [¯ ¯ r (t)T = [x1 (t), · · · , xN (t)], A ar1 , · · ·, where β(t) ¯rl = [1, ejkd sin(θl ) , · · · , ej(N −1)kd sin(θl ) ] ¯rL ] is the matrix of steering vectors, a a is the steering vector for angle of arrival θl , ¯sr (t) = [ejm1 (ti ) , · · · , ejmL (ti ) ]T is the vector of arriving signals’ equivalent phase modulation (PM) baseband phasors, ml (t) is the l-th emitter’s narrow-band digital baseband signal in ¯ bipolar NRZ code form, ml (ti ) is the i-th sample of one chip of ml (t), and n is the complex narrow-band noise vector. Corresponding to the actual N -element antenna array there is a second virtual array modeled in memory, which is based on the calibrated array and has N virtual outputs for each expected direction θk (k = 1, · · · , K). In most of the case, K L. The array signal memory steering vectors are created based upon K expected angles-of-arrival θk as ¯ e = [¯ ¯ eK ] A ae1 , · · · , a
(2)
¯ e is the matrix of steering vectors for expected direction θk and a ¯ ek where A is the steering vector for expected direction θk . The memory has K complex output waveforms, one for each expected direction θk . Each memory output is given by ¯ Ta ¯ek yke (t) = β(t)
(3)
According to [1], the best correlations occur when the actual angle of arrival matches the expected angle of arrival. The correlation can be used as a discriminating “merit factor” for detection. The general complex correlation output, for the k-th expected direction of arrival, is given as t+T
Rk =
t
∗
y r (t) · yke (t)dt = |Rk | ejφk
(4)
where Rk is the complex correlation at expected angle θk , and ϕk is the average correlation phase. Assuming the spreading sequence length is Nchips and each chip is of time duration τc , the entire sequence time duration is T = Nchips · τc . In general, the sequence duration should be T ≤ 1/(4Bm ), where Bm is the signal bandwidth [1]. That is to say, any chip of ml (t) should be sampled no less than 4 times in its duration, and then each sample of the chip is multiplied by the spreading sequence. Without special claim, we choose T = 1/(4Bm ) in this letter. We should derive the further result of the (4) under the condition that L incoming signals arriving at different directions with the angular spacing within the resolution of the antenna array geometry, i.e. the angular spacing of the incoming signals is larger than 2/N radians, which is the array angular resolution for the ULA when d = λ/2 [2]. By using the following orthogonality of spreading sequence βn (t), c
IEICE 2009
DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
t+T t
βu (t)βv (t)dt =
M, u = v 0, u = v
where u, v ∈ {1, · · · , N }
(5)
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we evaluate the correlation function of the realistic array output and the steering vectors for expected direction θk for i-th sample of one chip of ml (t) as follows, t+T
Rk,i =
t
∗
y r (t) · yke (t)dt
t+T t
=M
¯ T [¯ ¯ Ta ¯ek ¯ rL ] (¯ ¯ ) · β(t) ar1 , · · · , a sr (t) + n β(t)
= L
(ejml (ti ) + nl )
l=1
∗
dt
(6)
1 − ej·2π/λ·d·(sin θl −sin θk )·N 1 − ej·2π/λ·d·(sin θl −sin θk )
wherein M is the absolute value of auto-correlation peak of a spreading sequence, N is the number of array elements, ml (ti ) is the i-th sample of one chip of ml (t) where i ∈ {1, · · · , 4}, and nl = Ncl + jNsl is the complex Gaussian noise with in-phase component Ncl and quadrature phase component Nsl . Without loss of generality, we use 1st sample of one chip of ml (t) in this letter. When the expected direction of arrival θk matches some realistic direction of arrival θs where s ∈ {1, . . . , L}. In that benchmarking case, we can easily observe that the modulus of the s-th term of the equation is far larger than other terms. Hence, the formula can be described as follows, Rk,1 = M N (ejms (ti ) + ns )+M
L
(ejml (ti ) + nl )
l=1, l=s
≈ M N (ejms (ti ) + ns )
1 − ej·2π/λ·d·(sin θl −sin θk )·N 1 − ej·2π/λ·d·(sin θl −sin θk ) (7)
Easily known, we will get L local maxima in the expected direction range when all the θl are matched by θk without noise influence. Hence, we can indicate realistic incoming directions by looking for the corresponding expected directions of local maxima of |Rk,1 |. Actually, we always use Bayes detection method [3] to judge the existence of incoming directions under the noise background after obtaining all the local maxima of |Rk,1 |.
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IEICE 2009
DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
Robust beamforming scheme
In the absence of synchronization errors of the spreading sequences in the passive beamformer, i.e. the whole chips of each spreading sequence are correctly aligned with the other ones; so, the correlation output of any desired direction contains no significant interference from other directions. If there are timing errors and the spreading sequences suffer some synchronization mismatch when the system is located on a complicated electromagnetic environment, a leakage of signals from other directions into the correlation output of the desired direction occurs. This results in performance degradation as the desired signal is superimposed with components from other directions. Such performance degradation due to synchronization errors can be overcome by employing some spread sequences with anti-interference characteristics. In this paper, we present a kind of very efficient spreading sequences, Bridge function sequences, well suited to passive beamformers. 1195
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The Bridge function sequences are three-valued functions taking the values −1, 0, and +1. These functions are introduced by Zhihua Li and Qishan Zhang [4] and derived from a combination of the block pulses and the WalshHadamard functions. Based on the specific copy and shift method introduced in [4, 5, 6], the Hadamard-ordering Bridge function sequences can be obtained through the following recursive procedure. B HB P (j) = IP , H2i P (j)
⎡
⎤
1 HBi−1 (j) HB 2i−1 P (j) ⎦ ; 0 ≤ j ≤ q, i = 1, 2, · · · , q−j (8) =√ ⎣ 2 P B 2 HB (j) −H (j) i−1 i−1 P P 2 2 where P = 2j , IP is P -order unit matrix, j is shift parameter, Nchips = 2q is the length of Bridge function sequences generated. H2Bi P (j) is called the Bridge function sequences matrix. Its each column vector corresponds to a Bridge function sequence containing nonzero values (±1), which are uniformly spaced with equal “padding” zeros between adjacent elements. All of the 2q sequences in this matrix form a sequence set. The choice of a different shift parameter generates a different Bridge function sequences set, so a total of q + 1 sequence sets can be totally generated for each q. Specifically, Walsh function sequence set is generated at j = 0, while block pulse sequence is generated at j = q. It has to be stressed that different Bridge function sequences are orthogonal to each other. Due to fact that the Bridge function sequences set at different j(1 ≤ j ≤ q − 1) featuring similar numeric characteristics, without loss of generality, in this letter we use Bridge function sequences set at j = 3 when q = 8 for illustration in Fig. 2. As shown in Fig. 2, there ubiquitously exists a zero correlation zone (ZCZ) at the area surrounding origin point for arbitrary two Bridge function sequences’ cross-correlation function in a same Bridge function sequences set. Hence, the orthogonality between arbitrary two Bridge function sequences is still maintained under the condition that the initial phase difference τ of the two spreading sequences locates only in the range of ZCZ of their crosscorrelation function when some slight chip synchronization mismatch occurs between the two spreading sequences.
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IEICE 2009
DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
Fig. 2. (a) Cross-correlation function of β148 (t) and β150 (t) when using Walsh function sequences. (b) Cross-correlation function of β148 (t) and β150 (t) when using Bridge function sequences at j = 3. 1196
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Fig. 3. (a) Comparison of Correlation output Magnitude between Walsh function sequences and Bridge function sequences with SNR= 10 dB. (b) Comparison of wrong decision probability of Correlation Detection using Walsh function sequences and Bridge function sequences with different SNRs.
4
Simulation results
In the enhanced-performance passive beamforming structure presented in the previous sections, the correlation output performance using Bridge function sequences is evaluated and benchmarked in comparison to other spreading sequences without ZCZ. The test structure is a linear array which consists of six elements with an interelement spacing of half wavelength. There exist two incoming signals (L = 2), which are incident from θ1 = 0◦ and θ2 = 45◦ . Without loss of generality, we set the scenario of the occurence of a timing mismatch in the realistic array so that the second spreading sequence β2 (t) in Eq. (1) leads one chip ahead of the other weighting sequences while β2 (t) in Eq. (3) and Eq. (4) are still kept synchronized. Matlab simulations, shown in Fig. 3 (a), illustrate a sample snapshot that Bridge function sequences work so well that the proposed beamformer forms two correlation peaks at θ1 = 0◦ and θ2 = 45◦ nearly without any interference for signal-to-noise ratio of 10 dB. In comparison, the beamformer using Walsh function sequences as weights suffers a heavy interference with very large sidelobes. The statistical effect of the synchronization error described above is illustrated in Fig. 3 (b), where we set the number of the observation signal Rk,1 to 20000 for each SNR in AWGN channel when θk = θ1 and use Minimal Error Probability (MEP) [3] rule as the detection rule. The proposed beamformer using Bridge function spreading sequences with 1 ≤ j ≤ q − 1 and q = 8, are very insensitive to the synchronization error by virtue of ZCZ, while the beamformer using Walsh spreading sequences with no ZCZ suffers from performance degradation. The proposed beamformer obtains about 22 − 16.5 = 5.5 dB processing gain than the one using Walsh function sequences when wrong decision probability is Pe = 10−3 . c
IEICE 2009
DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
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Conclusion
A novel passive beamforming structure is introduced for robustness enhancement. Using Bridge function sequences as its weights, the desired signal direction is accurately detected when synchronization error occurs, essentially with lower performance loss compared with other spreading sequences without ZCZ. Hence, the introduced beamformer employing Bridge function sequences is virtually insensitive to synchronization errors as verified by simulations for realistic antenna arrays scenario.
Acknowledgments This paper is supported by China National Science Fund under grants No.60601024, and China National High-tech Research and Development Program under grant No.2007AA12Z340.
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DOI: 10.1587/elex.6.1192 Received July 15, 2009 Accepted July 23, 2009 Published August 25, 2009
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