where in general L > M , the required set of weights is found by solving, in the least squares sense, the following matrix equation:
G K v = F
where:
V = [vo vi • • • V;n • • • v m-iY • • • (453)
P = [F((Po) F((f>i) ■ • ■ P((p^ ' • • ^ • • • (^3.4)
K is a diagonal Mx-M co-phase weighting matrix whose mm^ih element is given in the case of a circular array (for 6 = %H) by (223), and G is an L xM element pattern matrix available from calibration measurements, whose /m ’th element is given by:
[G\/rn = eJ(<oRlc)cos [2^r(//L.m/M)]^[;r/2,2;r(//L-m/M), w]
0 < /< L - l , 0< m < M -l --■(43.5)
The optimal vector v is obtained from (452) by minimising the following cost function expression:
£ = [v " K " G " -F ^ W [G K V -F ] • • • (45.6)
where W is an LxL real diagonal weighting matrix whose elements may be chosen, for satisfactory sidelobe performance down to a level of 77 dB below the peak of the main lobe, as:
[TV]//=l/m ax[10-’>"MF((p^)/F(ç)^i)P] , ■■■(45.7)
IF(W I^ = max[lF(vH))P, lf(% )F , . . . IF(%.i)P] . . . (45.8)
Taking the ‘complex gradient* with respect to v ” and equating to zero‘d we obtain the standard least-squares solution to the above problem:
V = [K "G "W G K ]-i K^'G^W F • • • (45.9)
Note that the evaluated weight vector v is only optimal with respect to the target pattern F . In section 4.6 we shall be looking at ways of handling the simultaneous
4 . 5 S in g le-p a tte rn correction 7 8 correction o f a number of beams. But before we do that, let us first apply the same concept for the correction of a single circular-airay phase mode. The /I’th uncorrected phase mode d>^(;r/2, <p, co) and the target mode pattern will be represented by the L x l vectors:
= [O^(Jt/2,0,® )• • ■ CO)--- û))]’' • • • {4520}
Tw = ['F'^CO)... ' ¥ y ! ? ^ l ) ) f
= [1 e-K 2nlL )n/ g-j(27t/L)fi(l^l)Y (4.5.11)
and our aim is to find a ‘pre-DFT’ correction vector v o f complex weights, which we shall apply, as illustrated in Fig. 45.2., to the outputs o f the array channels prior to the formation of modes such that the expression
£ = IGE^v - V P / = [ V % G " - [G E > - ï y {4522}
is minimised. In (45.12) above, is an AfxM diagonal mode forming matrix for the j f i h phase mode, whose m m ’th element is given by:
(4.5.13) 0 1 A r r a y E l e m e n t s M-2 M-1 P h a s e m o d e s , , . C o r r e c t i o n u n i t D i g i t a l D F T A M -A M-2 M-1
7 9 P a t t e r n c o r r e c t i o n
Note that since the target pattern is omnidirectional in amplitude, there is no need for a weighting matrix, v is obtained as before by differentiating £ with respect to v " and equating to zero. The mode-dependent result is:
V = E ^ [ G " G ] - ' G " « V
and the single corrected phase mode is given by:
(45.14)
(45.15)
One rather complex scheme for the simultaneous correction of more than one phase mode is suggested in Fig 4 5 3 . It is based on (45.15), rewritten as
L-l M-1
¥>/, m) = £ { £ [ G ] / „ [ r U t ) , 0 ^ / ^ L - l • • • (45.16) k=0 m=0
with [r\mk denoting the mk^xh element of F = [G”G ]'^G ”. The above expression indeed constitutes an L-point DFT as implemented in Fig. 4 3 .3 . A more elegant multimode scheme based on a two-stage correction is the subject of section 4.6.
0 A r r a y m El e m e n t s M - 1 N e t w o r k D i g i t i s i n g [■Tioo L - l D i g i t a l DFT P h a s e m o d e s
4 . 6 Multi-pattern correction 8 0
4.6
MULTI-PATTERN CORRECTION
Although the pre-DFT weight vector v as given by (4.5.14) is phase-mode dependent and may not provide simultaneous correction for more than one phase mode, we may try and find another ‘global’ weight vector that will minimise the deviation of a prescribed set of phase modes from their respective ideal patterns. Noting that for an error-free circular array:
it appears reasonable to add a second set
u = Ufj,--' • • • (4.6.1)
of complex weights at the output of the DFT unit of Fig. 4 .5 2 , which we shall refer to as post-DFT correction weights. This scheme, which is illustrated in Fig. 4.6.1, is more easily implemented that the one suggested in Fig. 4 5 .3 , and moreover a post- DFT weighting unit would in any case be needed for the purpose of mode alignment and for the possible application o f an additional (amplitude) taper in the implementation of low-sidelobe mode-space radiation beams.
0 M-2 M-1 A r r a y E l e m e n t s D i g i t i s i n g N e t w o r k P r e - D F T W e i g h t i n g P o s t - D F T W e i g h t i n g C o r r e c t e d p h a s e m o d e s UM-l M -A M-2 M-1 M-2 M-1