context o f circular arrays of closely spaced sensors, we show in section 4.2 that uncorrelated equi-variance contributions from ambient noise fields to the phase-mode signals will affect the rms sidelobe level o f mode-space beams in a very similar manner. Uncorrelated equi-variance aperture errors, on the other hand, generally lead to an angle-dependent rms sidelobe pattern, unless the array elem ents are omnidirectional. This is shown for amplitude, phase, element displacement and pattern rotation errors in sections 4.3 and 4.4.
Various schemes for monitoring, correcting and maintaining the accuracy of antenna and sonar arrays, have appeared in the calibration literature. Those relevant to arrays with digital beamforming, ordinarily comprise initial pre-deploym ent calibration measurements and a re-alignment procedure based on a near-field source or on internally injected test signals [Bar80], [War89], [Lon85a]. The latter procedure is executed prior to, or interlaced with the normal operational deployment of the array, using test equipment attached to or built into the system. Sections 4.5 and 4,6 of this chapter examine ways of re-aligning a digitally-beamformed circular array of a known geometry with given element patterns and channel responses. The idea behind these calibrating algorithms has been inspired by [Lon85b] which deals with the correction of site effects on a phased array radiation pattern. We start by introducing the concept of ‘least squares’ pattern correction for a single co-phased beam at a single frequency, where a set of correction weights is applied to the array channels. A narrowband correction algorithm for the special case of a single phase mode pattern is then considered, and later extended to include a multimode scheme which involving two sets o f correction weights. A similar formulation for a multibeam excitation is then followed, and is shown to be equivalently implementable as a two-stage multimode correction algorithm. Sections 4.7, 4.8 and 4.9 conclude this chapter with wideband versions of the foregoing correction algorithms, a set of simulation plots and a summary of the main results.
4 . 2 M o d e - s o a c e ex cita tio n errors_________________________________________ iiilllllillllll 7 0
4.2
MODE-SPACE EXCITATION ERRORS
Let us consider an M-element circular array excited to form a nominal (i.e. error-free) mode-space beam pattern whose peak is ‘scanned’ to direction Çm = il7ülM)m. The beam is synthesised by the linear combination o f (2A +1) of the circular array phase modes O.Aj ^-A+h • ■ • » » A<M/2, Each of the processed phase modes is assumed to have undergone appropriate alignment, in addition to which it is linearly phased and weighted, as represented (for the | t ’th phase mode) by the phasing operator and the (possibly complex) weight If (for an even M) A = M/2, we assume the alignment of phase mode fx = ±M/2 to be such that:
■■ - - cos(M<p/2) + distortion terms C(MP,)0 C(MI2)0
The sensitivity of the array to uncorrelated mode-space errors pertains, as mentioned in section 4.1, to the effect of a circumferentially-isotropic and elevationwise- impulsive ambient noise field. Its analysis, being similar to that relating to linear arrays, is based on the inclusion o f a zero-mean phase error and a fractional amplitude error with each of the summed phase modes in the expression (323) for FmiçX where the errors are uncorrelated and their variances independent of fx. The perturbed far-field mode-space beam pattern, denoted here by j/m (A is given by:
A
Jm {(p) = “ V X CP/1 + K ^ e j^ e - M 9 - 2 n m /M ) . . . (42J )
and the expected value of \%i((p)\^ is clearly given by
A A
* M ^ ^ ^ ') ( 1 +
//=-A fjr=-A
■■■(422)
Now, for small errors one can approximately write:
5 [ ( l + Ky)(l+K-^)e/(V-*y^ = d - o i ) [ l + o 2 5 ( / i '- / / ') ] . . . (423)
71 Iiilllllillllll __________________________________________________________P attern correction -A < ^< A . . . (42.4) P
<j2 = {oI +gI)K I-gI) .. • (425)
and 5(n) is the Kronecker delta function. Expression (4.22) may consequently be rewritten as:
S\:Fm{(pt * + 1^1^) • ■ • (42.6)
where the nominal power pattern \Fm{(p)9’ has been augmented by an additional constant term IzlFP which is given by:
\AF? = (cf/M ) y l a / = — ^ — IF„(2ron/Ai)|2 • • • {42.7) fi=-A ( 2 A + 1 ) ^ with^ A A a ^ P /( 2 y l+ l) X l a / ■■■(42.8) fJ^-A fjb=-A
The constant factor a^/(2A+ \)Q multiplying WmQ.mnlM)?- on the right hand side of (421), constitutes the rms sidelobe level of the mode-space beam, and is seen to be directly proportional to the sum of the error variances, and inversely proportional to the number of phase modes and to the gain factor of the tapered array. A very similar result has been obtained for an M-element linear array under uncorrelated zero-mean amplitude and phase aperture errors, with M substituted for (2A +1) in the expressions (42.7) and (42.8) above.
In the next two sections we shall be examining the effect of uncorrelated equi- variance amplitude and phase errors at the circular array elements, as well as element displacement and pattern rotation errors, on the sidelobe level of mode-space beams.
Q is commonly referred to in the array literature as the gain factor^ denoting the relative power per unit solid angle directed, under the given weighting taper, by the array in the direction of its main lobe peak, as compared to the power per unit solid angle directed in that direction by the same array under a uniform taper.
4 .3 A perture excitation errors_____________________________________________ iiilllllillllll 7 2
4.3
APERTURE EXCITATION ERRORS
In order to analyse the effect of aperture errors on mode-space beams, we assume as before the presence o f uncorrelated zero-mean equi-variance phase and fractional amplitude errors, but this time at the element channels. The receive channel connected to the m ’th array element thus includes a random error signal whose phase and fractional amplitude are given by em and Km respectively. The expression for the /t’th perturbed phase mode on the azimuth (9 = nH)plane becomes:
/ M-1
0 = - L V Y ( l 4 - K ) y i ') e / G " 'e V M M ) ( / z + 0 / R 'g / ( G ) f y c ) c o s i<p-2nm'IM) . . . f4J jj
and the m ’th mode-space beam generated from a set of (2A +1) modes, is given by:
A
!Fm (<P) = - \ z r X [ (G), 7Ü/2)]e/(2?r/M)m/i<p^(;j/2, <p, CO)
M-1
Y ( 1 + V(2;:/M)(ff+;)/»'g/(û)A/c)cos(y-2;rm'/M) . . . ( 4 3 2 )
m'=0
where the (finite) Fourier series representation (22.4) for the element patterns has been used. The expected value of lj^(^)l^ is given for small errors by the expression:
5Um(ç>)l^ = * ^t^Afir=.AC^o(CD,nl2)C*^Q(co,7il2) M-1 M-1 m'=0 m"=0 = ( l - 0 # ) ( l f m ( # + l ^ m ( # ) ■ • • (43.4) with X m -i.G ) R lc ) c o s {(p - 2 n m /M ) , 0 < m < M - l • • • ( 4 3 5 )