Over-the-horizon Radar Array Calibration Using Echoes ... - IEEE Xplore

Over-the-horizon radar array calibration using echoes frolm ionised meteor trails I.S.D.Solomon D.A. Gray Yu.l.Abramovich S.J.Anderson

zdexing terms: Radar, Array calibration, Weiss-Friedlander method, Ionised meteor trail echoes, Coupling matrices

Abstract: The Weiss-Friedlander MUSIC-type approach for estimating sensor positions and mutual coupling are combined and extended for over-the-horizon radar applications. The proposed method exploits backscattered echoes from ionised meteor trails, which can be considered as disjoint sources of opportunity from an array calibration perspective. This method uses a more accurate model for the coupling matrix, with more unknowns, and estimates the coupling matrix and sensor positions using disjoint sources of unknown direction-of-arrival. Simulations have been conducted, using an experimentally measured coupling matrix, to illustrate the good performance of this inethod and to show that it is statistically efficient.

1

Introduction

Over-the-horizon (OTH) radars currently being developed for coastal surveillance incorporate antenna arrays that can be erected quickly with minimal site preparation. Errors in sensor positions, mutual coupling and receiver gainlphase variations are known to degrade array performance [l-31. For OTH radar, receiver gain/phase errors can be calibrated relatively easily by injecting signals internally at the receiver inputs, but sensor posii:ion errors and mutual coupling need more sophisticated calibration techniques. Weiss and Friedlander have developed a method for estimating sensor positions for an array with no mutual coupling [4], and a method for estimating a Toeplitz coupling matrix for an array with known sensor positions [5]. These methods are based on the fact that the noise subspace is (for an estimate of the covariance matrix) almost orthogonal to the true signal steering vector, the cost function used being a minimum when 0IEE, 1998 IEE Proceedings online no. 19%1899 Paper first received 20th October 1997 and in revised form 15th January 1998 I.S.D. Solomon and S.J. Anderson are with the Defence Science and Technology Organisation, Building 200 Labs, PO Box 1500, Salisbury 5108, Australia

I.S.D. Solomon is additionally with, and D.A. Gray is with the CRC for Sensor Signal and Information Processing (CSSIP),Technology Park, The Levels 5095, Australia and the University of Adelaide Yu.1. Abramovich is with C S S P and the University of South Australia IEE Proc.-Radar. Sonar Navig., Vol. 145, No. 3, June 1998

the signal steering vector is almost orthogonal to the estimated noise subspace. The parameter values at this minimum are the estimated values. In this paper we try to estimate the coupling matrix and sensor position errors using disjoint sources of opportunity, which are present in OTH radar in the form of echoes from ionised meteor trails. By disjoint we mean that they do not occupy both the same time snapshots and the same radar range cells. The WeissFriedlander method is modified for the case of disjoint sources, and results in fewer sources being required for a given number of unknowns. Further, instead of employing a Toeplitz coupling matrix we use a more accurate model for the coupling matrix. An experimentally measured coupling matrix is used in simulations to assess the performance of this method, and to compare the performance of the method with the CramerRao lower bound. The Weiss-Friedlander method [5] performed poorly for this coupling matrix, since the Toeplitz approximation is not applicable in this case. 2

Background

2. I Properties of echoes from ionised meteor trails In [6] we discussed sources for calibrating OTH radars, and mentioned that while noise sources and beacons may be used for calibrating OTH radar arrays, echoes from ionised meteor trails have several properties which make them superior from an array calibration perspective. Backscattered echoes from ionised meteor trails [6-101 in general: (a) have planar wavefronts at the ranges of interest (b) are sources of opportunity (c) are received in large numbers over the entire highfrequency band (d) do not occupy the same time snapshots and also the same range cells, i.e. they are disjoint sources (e) have high signal-to-noise ratios (SNR) cf> are of adequate duration for sufficient snapshots to be obtained for array calibration. In this paper echoes from ionised meteor trails will often be referred to as sources or signals.

2.2 Problem formula tion For a narrowband signal impinging an A 4 element array, in the absence of mutual coupling, the output of the mth sensor is Zml(t) =

+a

(1

m )e-34ms1 ( t ) e - 3 w T m 1

+ nm(t) (1) 173

where a, and 4, are the gain and phase errors, sl(tj is the received signal and n,(t) is additive receiver noise. The radar operating frequency is w,zml= (x, sin O1 + y , cos O1)lv, x, and y , define the position of the mth sensor, el is the direction-of-arrival (DOA) (azimuth) of the signal (with respect to broadside) and v is the speed of light in free space. The vector of M sensor outputs, from the array, is

zl(t) = ra(ol)sl(t)+ n(t)

(2) where q ( t j = [zll(tj, zzl(t), ..., ~,,,,~(t)]~, I? = diag((1 + al)e-J@l, ..., (1 + a M ) e - J @ M } , a(@,) = [e-JWtln, e-JWT2n > '", e-JwzMnITand n(t) = [nl(t),nz(t), ..., n&t)lT. In the presence of mutual coupling 1111

+

q ( t )= CTa(&)s,(t) n(t) (3) where C = (IM+ Z0/ZL)-l is called the coupling matrix. Matrix I, is the A4 x A4 identity matrix, Zo is the array mutual coupling matrix and Z, is the scalar load impedance. The covariance matrix for this signal, assuming zero mean noise, is

R1 = E{zl(t)zl(t)H} (4) Now consider N disjoint echoes. Most meteor trail echoes we have observed are resolvable in time and range, indicating different underlying physical mechanisms, and hence we represent these as statistically disjoint sources. The covariance matrix for the nth disjoint echo is R, = E{.,(t)Zn(t)H} (5) where z,(t) is the vector of M sensor outputs for the nth disjoint echo. In practice, the covariance matrices are estimated as 1 ln Rn = - C z n ( t ) z n ( t ) H

3

Parameter estimation

The Weiss-Friedlander method is based on the eigendecomposition of the received data sample covariance matrix. The cost function used is based on the MUSIC algorithm, which is very sensitive to model errors. Hence for the exact covariance matrix or for an estimated covariance matrix with sufficient number of integrations, this cost function is a good choice. The modified cost function we use is N

(7) n=l

where U(nj is the matrix whose columns are the eigenvectors of R, which correspond to the noise eigenvalues of R,. Note in Weiss-Friedlander's method U is independent of n, and is created from the single covariance matrix of the nondisjoint sources. Note that the cost function (eqn. 7) is a least-squares formulation of complex equations as in [4]; in this case there are N(M - 1) complex equations. There are 2M 3 unknown sensor positions (see assumptions below), N unknown directions-of-arrival and M ( M + 1) unknown coupling parameters (since we estimate a complex symmetric coupling matrix). Hence a solution can exist only if 2M - 3 + N + M(A4 + 1) 5 2N(M - 1). If nondisjoint signals are used, the number of complex equations would be N(M - N) [4]. Clearly, by using disjoint sources, one can estimate many more parameters for a given number of signals. The proposed iterative procedure for minimising eqn. 7 contains three steps. The first step uses the best estimates of sensor positions and coupling values available, and estimates the N signal directions-of-arrival using the MUSIC algorithm.

Tn t=l

where T, is the number of snapshots obtainable from the nth echo. The problem is then to estimate the sensor positions and coupling matrix, given the N covariance matrices. We have assumed here that the data zn(t) have been corrected for receiver gain/phase errors by injecting signals internally at the receiver inputs, hence r = IM

2.3 Coupling matrix Mutual coupling is caused by energy being received by an antenna element from other elements. The voltage received by an element is then the sum of the direct voltage due to a signal, plus the sum of all voltages arising from reradiation by other elements in the array. Mutual coupling causes elements in an array of identical elements to have nonequal radiation patterns, i.e. not all elements 'see' the same environment. The coupling matrix C in eqn. 3 is independent of incident signal powers and their directions. The matrix

C is equal to its transpose due to the 'reciprocity' theorem, which states that the energy received by element A due to the reradiation of element B is equal to the energy received by B due to the reradiation of element A. We found this to be valid in experimentally measured C . In our model for the coupling matrix we hence assume C is symmetric. Note that in [5], for nominally uniform linear arrays, a Toeplitz coupling matrix was used; hence assuming equal coupling between any two equally spaced sensors in the array. 174

DOAs corresponding to peak values of P,(O) are the estimated DOAs for the N signals. The DOAs estimated, and the best available values of the coupling parameters, are then used in the second step to estimate the sensor positions. Weiss-Friedlander [4] showed that the sensor positions can be estimated (using a closed-form solution that is related to the Gauss-Newton technique) as follows: vZy= [Re{BHB}]-lRe{BHZ} (9) where vxy = [vxT, vyTT, v, and vy are vectors of Ax, and Ay,, respectively, and for this problem we have B = [ W I T , B(2IT, ..., B(N>TT, B(n) = -[U(n)Hdiag(al(O,)), U(nIHdiag(a2(en>>l,z = [ Z ( l ) TZ(2IT, , Z(N)TT, Z(n> = a(On), a l ( Q = ja(O,)(w/v) sin(O,), az(Bn) = ja(O,)(w/v) cos(0,). For the third step, estimation of the coupling matrix, if one can rewrite Ca(0,) as A(0,)c (i.e. the unknowns are placed in the vector c and the known values in the matrix A), then the cost function can be rewritten as "1,

N

n=l

N n=l

The quadratic minimum for estimating c, under the constraint CI1= 1 (to exclude the trivial solution), is IEE Proc.-Radar, Sonar Navig , Vol. 145, No. 3, June 1998

r

*O

where here G = Z g ,

U ( T Z ) U (A(@,), ~ ) ~ e = [l, 0,

..) O]T.

It can be shown that for the symmetric coupling matrix we employ, the matrix A is given by

II P(m + 1) 1 k p(m I ) + I Q151 {imp, otherwise am+k--p(m)

Amk =

p(m) =

-

-



T

=T

trace{R,’(dR,/dSk)R,l

(aR,/dQl)}

n=l

(22)

IO.1 DOA and DOA terms Since we have disjoint sources, dRn/dO, is a zero matrix for n z 1 and hence Jek,eris zero for k # 1. Now J8, ,ek

= Ttrace{R,l(dRk/dBk)Ri’ (dRk/d&)} (23)

= TcJ:~Tczc~{RC’CAQ, (ok)a(B~,)~C~ H

H

R;’CAe, (&)a(&) c + R , l C ~ e , ( B k ) a ( B k ) H C H R ; l C a ( B k ) ~ ~ b ( B k ) H CH +R,lCa(Bk)&, (Qk)HCHR;’C&,( Q k ) a ( B k ) H C H +R;lCa(Bk)Ao, (Bk)HCHR;lCa(Bk)A.e, (&)“C”} (24) = 2To$iR{tr~ce{R~~CA~, (Bk)a(Bk)H CH H

H

R;lc~o,(Qk)a(@k)c H H +R;lC5Qk(ek)a(ek) c R i ’ Ca(h)as, CH}}(25) JQk,Q,

=

2To:R2(tra~e{R;~CA~~ (Bk)a(Bk)HCHR;’

C(Ao,

+ a(Qk)&,(Bk)H)CH}} (26)

where a(,@), = da(@,)/d&= D(O,)a(Ok) (with D(0,) = (-2q7A) diag{x cos(&) y sin(&)}) A is the radar wavelength and the property trace{zH} = conj(trace{z}) has been used in eqn. 25. ~

179

10.2 Sensor position and sensor position terms JZ,,Zl

+Ri1CCk a(8n)a(0,)HCHR;1Ca(0,)a(0,)HC: +R,1Ca(0,)a(8n)H Cz RL1CCL a(0,)a(8n)HC H

=

+R;1Ca(0,)a(8n)Hk~R;1Ca(B,)a(Bn)HC~}

N

T

trace{Ril (dR,/dzk)R,l(dR,/dzl)}

(27)

(31)

n=l

N

= To,"

N

trace{R,lC~,,(O,)a(O,)HCH

R i l CAZ,(0,)a( 8,) CH +RL1C8,, (Bn)a(0n)HCHR,1Ca(Bn)8,1 (B,)HCH +R;1Ca(8,)AZ, (0,) H CH RilCAZl(On)a(O,)HCH +Ril Ca(0,) AZk(8,)

CHRi1Ca(0,) AZl(8,) H C H } (28)

= N

2Tat

R{trace{R;lCAZ, (0,)a(8n)HCH n=l

R;lc(Azl

R{ trace{ R;' C,

2To;

n=l

JZ,,Zl

=

Jc,,cI

+ a(Bn)& (&JH)CH}}

(Qn)a(Qn)H

(29) Note axk(On) = d a ( 0 , ) i d x k = dxk 0 a(0,), where 0 is the Hadamard product and dxk is an A4 element row vector with all but the kth element zero; the kth element is (-2q7A) sin(On). Similarly for the y co-ordinate and x-y co-ordinate terms.

e,)

a(8,) a(

CHR;

'

n=l

(CQ48, )a(@,) CH

+ Ca(&)a(&) 6: )1

(32) where matrix = d c / d c k , c k being the amplitude/ phase of an element in the symmetric coupling matrix. Note cckhas only one nonzero element if ck is a diagonal element, and has only two nonzero elements (of equal value) if ck is an off-diagonal element.

cck

10.4 Cross terms The cross terms can be similarly derived as JSk,zl =

Tattruce{RI,'C(Ao, ( B k ) a ( O k ) H

+ a(Qk)&,

cH~,$(&, ( e k ) a ( o d H+ +wZ, (eP)cH} (33) =

JSk,cl

Tottrace{ R i l C (AS, ( H / c ) a ( B k )

" + a(

O h )A

Q ( ~0 k ) H ,

CHR,1(Ccla(0k)a(0~)HCH+Ca(8k)a(0k)HC~)} 10.3 Coupling and coupling terms Jc,,cl

(34)

= JCk,Zl

N

T

trace{R,l(dR,/dck)R,l

(dR,/dcl)}

(30)

n=l

= N

trace{RL1(C,, a(Q,)a(

To:

CH

n= 1

N

= To,"

RLIC(& ( 4 L ) a ( Q 7 J H Rc1Cc1 a(On)a(O,)"CH

180

+ca(&)a(0,)"Ct-',

trace{R,1Cc,a(0,)a(8,)HCH

n=l

+ a(OnPzl( & J H ) C H } (35)

IEE Pro,.-Radar, Sonar Navig.,Vol. 145,No. 3, June 1998