ADAPTIVE ANTENNAS. SPATIAL BF

Transcription

1 ADAPTIVE ANTENNAS SPATIAL BF 1

2 1-Spatial reference BF -Spatial reference beamforming may not use of embedded training sequences. Instead, the directions of arrival (DoA) of the impinging waves are used to synthesize beams steered at the wanted signal with nulls directed to other co channel users (interference) -DoA algorithms do their work on the signal received at the array sensor output and computes the DoA of all the incoming signals -Once the AoA is known, it is fed into the beamforming network to compute the complex weight vectors required for beam steering 2

3 -All DoA estimation algorithms need information about the number of source signals. If the information is not provided, it has to be estimated from the data ( measured or formulated ) -This information can then be used to localize the signal sources, form high gain for SOI or to steer nulls to SNOI -The physical measurements collected by a sensor array contain noise. When deriving the array signal processing algorithms, the noise is conventionally modeled as White Gaussian random process with zero mean and variance σ 2. Under this condition the derived optimal estimation is different from the actual model (measurements) in which the noise is non Gaussian -Most DoA estimation algorithms and methods, for estimating the number of source signals are based on the senor array covariance matrix or its eigenvalues and eigenvectors - DoA estimation algorithms and methods should have high resolution, i.e., they should be able to distinguish between one source and two sources with close DoAs -igh resolution DoA estimation is important in many systems such as radar, sonar, electronic surveillance and seismic exploration 3

4 Some terminologies Array manifold s 1 (t) s 2 (t) s D (t) The array input vector X(t) can be written as X ( t) A( ) s( t) n( t) θ D where A(θ) is the MxD matrix of array steering vectors or array response for directions θ i A ) [ a( ),..., a( )] ( 1 D s t) [ s ( t),..., s ( t)] D ( 1 T x 1 (t) x 2 (t) x M (t) a(θ i ) is Mx1 vector represents the steering vector for directions θ i the snap shot at time t from the sources s 1,,s D Array manifold is a set composed of all array response (steering) vectors over the entire parameter space The columns of A(θ) elements are elements of that set This set is completely determined by the sensor directivity pattern and the array geometry. For complex array geometry this set can be determined by calibrations ( i.e., physical measurement )

5 Signal subspace The observed data vectors X(t) for the D signal sources is called the D dimensional observed signal subspace spanned by the D vectors a(θ i ) [columns of A(θ)] -In the absence of noise measurements X(t) =A(θ)s(t); the outputs of the sensor array lie in the D dimensional observed signal subspace spanned by the columns of A(θ) i.e., once the D independent vectors has been observed, the observed signal subspace is known, The intersection between that observed signal subspace and the array manifold yield the set of vectors from the array manifold that span the observed signal subspace It is clear that in absence of noise, the parameter estimates can be obtained by finding the intersections of the array manifold with the signal subspace Or equivalently finding the elements of the manifold that that are orthogonal to the noise subspace X(t 4 ) X(t 3 ) a(θ 2 ) X(t 1 ) a(θ 1 ) X(t 2 ) Two sources Observed signal subspace Array manifold 5

6 -In noisy measurements, X(t) =A(θ)s(t)+n(t); the outputs of the sensor array are available but the D dimensional signal subspace spanned by the columns of A(θ) must be estimated such that the estimated signal subspace be spanned from the manifold and assuming unknown deterministic signal and Gaussian noise -As a conclusion: With perfect knowledge of the signal subspace, searching for the array manifold for D intersections with the signal subspace can be quite costly, especially for multidimensional parameters ( azimuth, elevation and range ). The problem is further complicated in the presence of noise which is hardly to find any intersection A potential solution is to find the elements of A(θ) that are closest to the signal subspace 6

7 2- Conventional techniques of DoA beamformers The conventional beamformer is one of the older techniques for localizing signal sources. The idea is to steer the array in one direction at a time and measure the output power The steering directions which result in maximum power at the output provide the DOA estimates That is to say conventional methods are based on using beamforming and nullsteering to scan through the spatial power spectrum to identify power peaks that correspond to valid signal direction of arrivals y( k) W X ( k) 2 P cbf E[ y( k) ] E[ W X ( k) ] W W RW 2 E[ X ( k) X ( k)] W R is the spatial correlation matrix of the sensor array output data Array output x 1 (k) x 2 (k) x M (k) 7

8 Fourier method (delay-and-sum method) -The concept is to form a narrow beam at each angle over the angular region of interest in discrete steps by forming weights W = a(θ), where a(θ) is the steering vector associated with DoA θ ( like what we did in main beam steering) -Then determine the output power E { y(θ) 2 } for different DoA i.e., for different steering vector associated with DoA θ 2 E { y ( t ) } P cbf ( ) w Rw a ( ) R a ( ) Array output x 1 (k) x 2 (k) x M (k) 8

9 STEPS -Estimating the input autocorrelation matrix R X from the source R ss, noise R nn and the steering array vectors A -Knowing the steering vectors a(θ) for all θ's, we can estimate the output power as a function of the DoAs -The angle of arrival can be estimated by finding the angles that correspond to the peaks in the output power P CBF θ 2 θ θ 1 -It performs well under the presence of a single signal In case if the signal beam is arrived from multiple sources & directions, the width of the beam and the size of the side lobes limit the effectiveness leading to poor resolution since it contains contributions from the SOI and also from SNOI 9

10 Example: An array of M elements have beamwidth 8.5 o For angle of arrivals +10,-10 the array can resolve the two angles 10

11 For angle of arrivals +5,-5 the array can not resolve the two angles The peak power at angle 0 11

12 CAPON S MINIMUM VARIANCE -Capon's method is similar to Fourier but it attempts to overcome the contribution of the undesired interferences by minimizing the total output power of y(k) = W X(k) while maintaining a constant gain in the look direction -i.e., the weight vector is chosen according to the minimum variance distortionless response criteria for optimum beamforming P Ccapon W R xx W Rxx a( ) W ( ) 1 a ( ) R a( ) ( ) R ( ) R 1 xx 1 xx -The DoA can be estimated by locating peaks in the spatial spectrum P(θ) -Better resolution compared with the Fourier method a a When other signals present are correlated with the SOI, the correlated components may be combined destructively and method fails 1 R a( ) xx xx a 1 Rxx 1 Rxx a( ) a( ) a 1 ( ) R 1 xx a( ) Requires a computation of a matrix inversion which can increase the computational cost for large arrays 12

13 Example: An array of M elements have beamwidth 8.5 o For angle of arrivals +5,-5 the array can resolve the two angles 13

14 db Example: For comparison of another conventional beamformer and Capon's method in the situation where two independent random 4-QAM signals of equal power (SNR is 20 db) from directions 81 o and 99 o arrive to a 6-element ULA with inter-element spacing equal to half a wavelength In this example the number of snapshots is K = 300 and the noise is complex Gaussian 14

15 Regardless of the available data quality or amount, conventional beamforming can not resolve two signals with close angles of arrival, i.e. its resolution is limited. It can be shown that for a ULA of M sensors, the beamforming resolution limit is approximately λ / Md. Note that the low resolution also limits the number of DOAs that can be estimated Example: We can summarize that, the conventional beamforming drawbacks & limitations in resolution cause the difficulties in forming structure of data input at the sensor array outputs ULA of 6 sensors of half-wavelength inter-element spacing, the approximate resolution limit equals 1/3 rad = 19 o The advantage of the Fourier and Capon estimation methods is that these are nonparametric solutions and one does not need an a priori knowledge of the specific statistical properties of the signal and the array structure 15

16 3- Subspace techniques of DoA MUltiple SIgnal Classification (MUSIC) -It is based on exploiting the eigen structure of the input spatial covariance matrix so it is called a subspace method -It provides information about the number of incident signals, DoA of each signal, strength and cross correlations between incident signals, noise power If the number of signals is D, the number of signal eigenvalues and eigenvectors is D and the number of noise eigenvalues and eigenvectors is M D (M is the number elements) -It makes the assumption that the noise in each channel is uncorrelated making the noise correlation matrix diagonal -It requires very precise and accurate array calibration -STEPS -Collect input samples (snapshots) X ( k)...where k 0,1,, K 1 and the input covariance matrix is estimated by or calculate the array correlation matrix from Rˆ xx R xx 1 K AR K k0 ss A X k X 2 n k I 16

17 - Find the eigenvalues and eigenvectors for R xx -Produce D eigenvectors associated with the signals and M D eigenvectors associated with the noise (the eigenvectors associated with the smallest eigenvalues ) -Construct the M (M D) dimensional subspace spanned by the noise eigenvectors V q q N 1 2 q M D -This noise subspace eigenvectors are orthogonal to the array steering vectors at the angles of arrival θ 1, θ 2,..., θ D Because of this orthogonality, one can show that the Euclidean distance d 2 = 0 for each and every arrival angle θ 1, θ 2,..., θ D N a( ) V V a( ) 0 N Placing this distance expression in the denominator creates sharp peaks at the angles of arrival. The MUSIC pseudospectrum is now: P MUSIC ( ) a ( ) V 1 N V N a( ) 17

18 Example: M = 6 element array. With element spacing d = λ/2 uncorrelated, equal amplitude sources, (s 1, s 2 ), and σ 2 n = 0.1, and the pair of arrival angles given by ±5 -The eigenvalues are given by λ1 = λ2 = λ3 = λ4 = σ n 2 =.1, λ5 = 2.95, and λ6 = The subspace created by the M D = 4 noise eigenvectors again is given as V N This results for estimated Rxx from the source Rss and noise Rnn P MUSIC θ 2 θ θ 1 18

19 If we calculate the averaging Rxx from the K snapshots we get P MUSIC i.e., the more practical application we must collect several time samples of the received signal plus noise, assume ergodicity, and estimate the correlation matrices via time averaging When the source correlation matrix is not diagonal (correlated ), or the noise variances vary, the plots of P MUSIC can change dramatically and the resolution will diminish, so we should use the time averaging correlation matrix Rxx θ 2 θ θ 1 MUSIC fails when impinging signals s(t ) are highly correlated 19

20 Root MUSIC algorithm - Root-MUSIC implies that the MUSIC algorithm is reduced to finding roots of a polynomial instead of searching for peaks in the pseudospectrum -It is based on polynomial rooting and applied only for uniformly spaced linear array, whose the m th element steering vector a(θ) is jmd sin am e... m 1,2,..., M β=2π/λ is the phase propagation coefficient, d is the spacing between the array elements and θ is the angle from the normal to the array -The MUSIC spatial spectrum can be expressed as ( 1 P MUSIC ) a ( ) Qa( ) Q V V N N -The denominator of MUSIC spatial spectrum can be expressed as 1 M M ^ P MUSIC ( ) e m1 n1 jmd sin Q mn e jndsin M 1 lm 1 Q l e jdlsin where Q l is the sum of the diagonal elements of Q along the l th diagonal such that Q l Q l mnl 20

21 -We can recall the denominator equation and simplify it to be in the form of a polynomial whose coefficients are Q l, thus P ( z ) M 1 pm 1 Q l z l z e jd sin The roots of P(z) that lie closest to the unit circle correspond to the poles of the MUSIC pseudospectrum P MUSIC -This polynomial is of order 2(M 1) and thus has roots of z 1, z 2,..., z 2(M 1). Each root can be complex and using polar notation can be written as z i z i e jangle( z i Exact zeros in P(z) exist when the root magnitudes z i = 1 ence, we can calculate the DOA by comparing e j angle(zi ) to e jkdsinθi sin 1 i 2d ) angle( z i ) 21

22 Disadvantages of MUSIC algorithm -Complete knowledge of the array manifolds is required -It is very sensitive with respect to array imperfections -The search parameter space is computationally very expensive -Under high signal correlation the traditional MUSIC algorithm breaks down and other methods must be implemented to correct this weakness

23 Example: M = 4 element array. With element spacing d = λ/2 uncorrelated, equal amplitude sources, (s 1, s 2 ), and σ n 2 = 0.3, and the arrival angles given by -4, +8 and 300 snapshots Imaginary Real 23

24 P root MUSIC θ 24

25 Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) -It exploits the rotational invariance in the signal subspace which is created by two arrays with a translational invariance structure i.e., a linear phase shift along the array is assumed (is well suited to uniform linear arrays) -It assumes multiple identical arrays called doublets. These can be separate arrays or can be composed of subarrays of one larger array like M elements of the receiving array divided into two identical overlapping sub-arrays, each of which consists of the M - 1 element sensors Ex: For a four elements ULA the array is divided into two sub-arrays of three elements or doublets These two subarrays are translationally displaced by the distance d d Sub array 1 Sub array 2 The structure looks like two identical sub arrays displaced from each other by a known displacement vector of magnitude d The sub array displacement vector is considered as the reference direction for the DoA estimates and is also considered as the scale of the structure The sources can be either random or deterministic and the noise is assumed to be random with zero-mean

26 -The output of each sub-array is denoted by X 0 (t) and X 1 (t). Using matrix and vector notation these two outputs can be written as X 0( t) A0 s( t) n0 ( t) X1( t) A0s ( t) n1 ( t) where s(t) denotes the Dx1 vector of source signals as observed at a reference element of the first sub array n o (t), and n 1 (t) denote the noise present on the elements of the two sub-arrays A 0 denotes a MxD matrix, with its columns denoting the D steering vectors corresponding to D directional sources associated with the first sub-array A 0 Ф is the steering vectors corresponding to D directional sources associated with the second sub-array Ф is an D x D diagonal matrix whose diagonal elements represent the phase delays between the doublet sensors for the D signals and is given by diag{ e, e jkd sin jkd sin2,..., e jkd sin 1 D } 26

27 27 -The complete received signal considering the contributions of both subarrays is ) ( N N S A A X X t X o N S A X Dx MxD M The correlation matrix is estimated for the two array outputs K k o o oo k X k X K R 1 ) ( ) ( 1 ˆ K k X k k X K R ) ( ) ( 1 ˆ or calculated from the source and noise correlation matrices as I A A R R n o ss o oo 2 I A R A R n o ss o 2 11 or the correlation matrix of the entire X(t) I A R A R n ss MxD XX Mx M 2 or from the time averaged correlation of the entire array output X(t)

28 -Using the total least-squares (TLS) criterion we can estimate the rotation operator Ψ Construct the signal subspaces U o,u 1 from the entire array signal subspace U (M D matrix composed of the signal eigenvectors) such that U o is the first M/2 + 1 rows ((M + 1)/2 + 1 for odd arrays) of U U 1 is the last M/2+1 rows ((M+ 1)/2 + 1 for odd arrays) of U -Form a 2D 2D matrix using the signal subspaces such that C U U o 1 U -Form the eigenvalues decomposition (EVD)of C we can get U C from such that λ 1 λ 2 λ 2D and ᴧ= diag {λ 1, λ 2,..., λ 2D } -Partition U C into four D D submatrices such that U U 21 U U 1 C -Calculate the eigenvalues of Ψ λ 1, λ 2,..., λ D U o U 1 U 12 U 22 C U U C C i sin 1 (arg i / d), i 1,2,.., D It is less sensitive with respect to array imperfections than MUSIC It does not require an exhaustive search through all possible steering vectors to estimate the DoA of the incoming signal 28

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