SIGNAL SEPARATION USING CIRCULAR ARRAYS. I. Jouny

Transcription

1 SIGNAL SEPARATION USING CIRCULAR ARRAYS I Jouny Department of Electrical Engineering Lafayette College Easton, PA ABSTRACT Signal separation algorithms can be utilized to improve channel capacity of mobile communication systems with frequency re-use Blind source separation of signals received using circular antenna arrays is compared with that received using linear antenna arrays Circular antenna arrays can be better concealed than their linear counterpart, and do have a 360 degrees field of view Techniques for improving the performance of source separation algorithms for circular arrays through interpolation are implemented The interpolated arrays are proposed as alternative base station receivers for mobile communication systems 1 INTRODUCTION Antenna arrays at the base station of a cellular network have been proposed as alternatives to single antenna receivers for the purpose of increasing capacity through frequency re-use Spatial-temporal signal separation using adaptive array processing techniques increases channel capacity by allowing multiple use of the same frequency within the same cell Linear antenna arrays have been primarily examined for source separation purposes Unfortunately, linear arrays have a limited field of view and are also esthetically unappeiding An attractive alternative is to use circular antenna arrays that have a 360 degrees field of view, and can perhaps be disguised more appropriately However, source separation using circular arrays is less efficient than using linear arrays because of inherent signal nonstationarity as a function of range In this paper a comparison is made between the performance of signal separation algorithms received by linear arrays and circular arrays Algorithms for improving the performance of :source separation via circular arrays are then implemented These algorithms rely on mapping a circular array into a linear array via interpolation or refocusing techniques Various scenarios of source separation are considered in this paper showing the utility of circular arrays at the base station of a mobile cellular network 2 LINEAR ANTENNA ARRAYS FOR MOBILE COMMUNICATIONS Linear antenna arrays that are capable of resolving mobile communication signals based on their angular distribution have been proposed for increasing channel capacity and improving communication quality Antenna arrays that direct nulls against co-channel users have been proposed in [3] A more general approach appears in [5] where beams are directed either towards the mobile user or a group of mobiles A method for separating mobile users at the same frequency using linear antenna arrays was developed in [4] In all of these algorithms, the angular position of the mobile is either known a priori or estimated using a direction finding algorithm The concept is based on extracting signal components of each of the mobiles from a vector (array) of observations%= [~1, %2,, z~], z(t) = A(0)s(t) + n(t) (1) where A(6) = [al(~), an(0)] represents the array manifold with a steering vector a(0) = [1, e~2=asin@f/c, ej2r(n-l)a Sinof/c]~ (2) Where A is the inter-element spacing and the parameters f and c represent the operating frequency and the speed of light respectively It is well known the direction of arrival 6 can be estimated using 8 = max@tr [PAE$ WE,] (3) where Tro indicates the trace of a matrix, PA is the orthogonal projection onto the space of A(6), W is the weight vector, and E8 is the signal subspace constructed using the largest eighvalues of the second order cumulant or the autocorrelation matrix Rrz = E{zz* }, The mobile signal is the estimated using i(t) = We(t) (4) The above linear antenna array source separation and estimation algorithm is one of many known array processing algorithms that rely on complete knowledge of

2 the array manifold and assume additive Gaussian noise Linear antenna arrays have been further exploited for suppression of interference and multipath propagation 3 BLIND SOURCE SEPARATION An alternative to using hypothesized values of the angular position of each mobile is to estimate such directional vectors via blind identification This particularly appealing in scenarios of array mismatch, improper calibration, wave-front distortion, etc Blind beamforming and source separation relies on statistical assumptions such as independence of sources [2] The blind source separation algorithm utilized in this study is based on the fourth order cumulant of the observation vector The algorithm developed in [2] is known as the Jade algorithm and can be summarized as follows [2] 1) Whiten the observed signal x by generating the signal vector z (where z = WZ ) whose aut ocorrelat ion Rz is a diagonal matrix 2) The fourth order cumulant of the whitened signal it then computed Q = ~{ ZZ*ZZ*} (5) Let AI, M1, A2, M2,, &, Mn be the m most significant eigenvalues and eigenvectors of Q (m being the nu;mber of sources) 3) The matrices uhaln!flu (6) UHA1M2U UHAm Mm u are then jointly diagonalized [2] using a unitary matrix U where where H denotes the complex conjugate thhispose 4, The sources are then separated using (7) $ = [(FVW)-WJ] lz (8) The above algorithm assume complete statistical independence of sources as well as their multipath components An adaptive blind source separation algorithm based on joint diagonalization is also developed in [2] The overall estimated spatial transfer function for source separation purposes is P = UWA (9) If source separation was accomplished completely, then the matrix P would include only one none-zero term at each row The performance of source separation is then measured by the mean rejection level [2] ~Pq = ~{lppq12} (lo) where ipq represents the level at which the qt h source is included in the pt h estimated signal 4 CIRCULAR ARRAYS FOR MOBILE COMMUNICATIONS The interest in uniform circular antenna arrays has been significant because of their 360 degrees field of view and their numerous applications in military early warning and support systems, cellular networks, and surveillance The core of source separation using antenna arrays relies on the inversion of a covariance matrix that combines time and space information within a coherent processing interval (CPI) Low rank well conditioned covariance matrix is essential to robust, stable, and effective separation In uniform linear arrays, low rank is controlled by the correlation between received samples from pulse to pulse Well conditioning is normally available but can be attained in a variety of ways least of which is diagonal loading Unfortunately in uniform circular arrays (UC!A S) pulse-to-pulse correlation is lost simply because of the nonuniform separation between elements when projected in space (see Zatman [6] for more details) This phenomenon leads to a data covariance matrix with higher rank, poorer conditioning, and consequent y a less effective source separation By inducing curvature to an antenna array, the uniform separation between elements when projected on the range/time axis is lost thus resulting in a different signal scenario with different statistical features in each range bin or equivalently a nonstationary signal as a function of range This problem can be solved using a technique that [10] relies on the interpolation of a virtual uniform linear array from a uniform circular array and applies source separation to the interpolated array This techniques assumes that the manifold of a virtual linear array is related to the manifold of a circular array via a linear over-determined mapping This mapping can thus be estimated using simple least square projection of the manifold of the virtual array onto that of a circular array It is recommended that such projection is associated with individual sectors of the field of view of the array The size of the virtual array is normally less than that of the circular array Singular value decomposition is used to estimate the dimension of the

3 virtual array and in selecting the number of angular mapping sectors The above blind source separation algorithm (JADE[2]) was implemented using signals received using a uniform linear array and signals received using circular arrays wit h or without interpolation Three hypothetical signals were used and received using an array of 4 elements The signals are (11) _L(ek + o,5e~7) z(t) = ~5) 3(t) = a;5;b where 0, -y, a are uniformly distributed over [0, 27r] and a, b is a sequence of random binary data (using bipolar representation) Figures 1 and 2 show (from top to bottom) one of the original signals, the recovered signal using a linear array, the recovered signal using a circular array without interpolation, and the recovered signal using a circular array with interpolation Fig,ure 3 shows a comparison between the normalized rejection levels for linear, circular, and circular (with interpolation) arrays This graph is generated using 30 experiments per noise level and using 400 samples per experiment Clearly the performance of circular arrays is inferior to its linear counterpart, but can be improved significant y through interpolation Array curvature also increases the signals nonstationarity level which results in an inferior source separation performance As curvature increases the size of the interpolated virtual array is reduced leading to poorer performance Note that curvature is independent of the array radius and is primarily defined by the angular sector occupied by the array elements Varying the radius of a circular array represents a change in interelernent spacing as a function of wavelength and that does effect performance The difference between a small radius (oversampled) array and a large radius (undersampled array) is significant Separation performance as a function of radius is a sampling issue that does not seem to be enhanced by interpolation An alternative interpolation algorithm that relies on phase-mode excitation has been used by many researchers for direction-of-arrival estimation, spatial smoothing pre-processing, etc [1, 7] The basic concept is that the discrete Fourier transform of the output of a circular array can be reordered to match that of a uniform linear array except for a diagonal mapping of Bessel functions [7] The output of a uniform linear virtual array can thus be estimated using a reverse operation This method uses approximations that are valid with large number of sensors and relies on a theoretic depiction of the output of a uniform circular array The concept relies on applying a series of linear mappings that result in a lower dimension uniform linear array with a manifold that exhibits a Vandermode structure and is thus amenable to array pre-processing techniques such as spatial smoothing and spatial filtering A third interpolation algorithm [8] generates the Vandermode signal subspace by the null space of a constructed Toeplitz Induced Mapping matrix TIM The algorithm maps the manifold of a circular array into that of a uniform linear array using a real valued Teoplitz matrix constructed from the eigenvectors of covariance matrix By assuming that the observation vector represents the coefficients of a polynomial with some roots on the unit circle and others inside or outside that circle, this algorithm generates the array manifold with the Vandermode structure by exploiting those roots that are on the unit circle Although mathematically involved, the implementation of this algorithm is a simple sequence of algebraic manipulations 5 CONCLUSIONS Circular antenna arrays at the base station of a mobile communication system are attractive alternative to uniform linear arrays mainly because of their 360 degrees field of view However, array curvature results in signal nonstationarity and leads to poor source separation performance Interpolation techniques that map a circular array into a linear array have been successfully implemented and tested for the purpose of blind source separation at the base station The performance of the interpolated circular array with a 360 degrees field of view is comparable to that of a uniform linear array with a 180 degrees field of view [1] [2] [3] [4] 6 REFERENCES T Rouphael and JR Cruz, Enhanced CDMA Cellular System Using Interpolated Circular Arrays, IEEE Vehicular Technology Conference, pp , 1996 J-F Cardoso and A Souloumiac, Blind beamforming for non-gaussian signals, IEE Proceedings-F, Vol 140, No 6, pp , December 1993 P Zetterberg and B Otterson, The spectrum efficiency of a base station antenna array system for spatially selective transmission,ieee Trarasactions on Vehicular Technology, Vol 44, No 3, pp , August 1995 S Anderson, M Millnert, M Vlberg, and B Wahlberg, ~~An adaptive array for mobile communication system,ieee Transactions on Vehicular Technology, Vol 40, No 1, pp , February 1991

4 [5] S Swales, M Beach, D Edwards, and J McGeehan, The Performance Enhancement of Multibeam Adaptive Base-Station Antennas for Cellular Land Mobile Radio Systems, IEEE Transactions on Vehicular Technology, Vol 39, No 1, pp 56-67, February 1990 [6] [7] [8] [9] Zatman, M A, Circular Array STAP, National Radar Conference 99 Tewfik A H and W Hong, On the application of uniform linear array bearing estimation techniques to uniform circular arrays, IEEE Transactions on Signal Processing, Vol 40, No 4, April 1992 Lu, M A Toeplitz-Induced Mapping Technique in Sensor Array Processing, IEEE Transactions on Signal Processing, Vol 43,N0 5, pp , May 1995 Wax, M and Sheinvald, J, Direction Finding of Coherent Signals via Spatial Smoothhg for Uniform Circular Arrays, IEEE Transactions on Antennas and Pro~a~ation, VO1 42, No 5, PP , May [10] Friedlander B, The root-music algorithm for direction finding with interpolated arrays, Signal Processing, Vol 30, pp 15-29, , I 1 1,,,! I /, A- / +, + / + \ / 087- =, {0=- T z B ~ ow -E3E!ii t 1 1!! Noise Power (db) Figure 3 shows a comparison between the normalized rejection levels for linear, circular, and circular (with interpolation) arrays

5 -%> 1I I / : L----_l <* *,,;~~ *! :, - +{ o ; > * r?, :*,> $$, -, : :: -: -,* * t; J ! A? c :, 1 ;:;?:,: -:? 0 :----- \ : > - - ; : ; : /?,,: --- : 2 ~ I I 1 -, :, 0,1 : :, ;:: % -2 ~ : :: -,: y - $: -:>: 0 : : : ::% Z* ~ ;/ ;;,: -,<: :: 0 :,: > -?*- % : -:i : ;:, 2r ) 1 * : ~,, ~: & :, : 0 ;: :: ; - $ {;::~f + 2 ~ Figures 1 and 2 show (from top to bottom) one of the original signals, the recovered signal using a linear array, the recovered signal using a circular a,rray without interpolation, and the recovered signal using a circular array with interpolation