PHASE-MODE EXCITATION

Phase mode excitation refers to the excitation of circular-array elements with uniform amplitude but with a linear phase sequence which obeys the periodicity condition, allowing only an integer number of 360° phase cycles around the array - see Fig. 2.4.1. For a radially-symmetric ring array of M elements whose frequency- independent patterns are symmetrically identical and denoted by (2.2.1), the ^ ’th far- field phase mode pattern (p, (û) is expressible as the following sum:

M-l

(p, CO) = S ( p -2 7tm lM ) e - j [ i' ^ ^ i ^ ) ^ ^ ' n - i c û R / c ) s m ecos { ( p - l n m l M ) ]

m=0

which, under the Fourier series representation (22.4) for the element patterns, may be rewritten as: M - l I (p, co) = — }i.(^Q^^j((p-27Cm/M)i-^g-j[i27C/M)fim-icûR/c)smecos (<jp - 2 n m /M )] ^ m = 0 i=-I I M - l _ _ 1_X h (^Q)eji 9 X £ - j ( 2 n f M ) ( j i + i ) m gj((ûRlc)s'm 6 cos ( ç - 27cmlM) ^ i = - I ‘ (2.42) with I denoting the maximum non-vanishing element-pattem coefficient in (2.2.4). A simpler though quite useful representation for the element pattern is the following:

J

^-j(27tlM)(M-l)ii

g-j{2nlM)(M-2)gL

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= . p > 0 ■■■(2.43)

with ho (6) denoting an arbitrary elevation dependence, for which the corresponding array pattern is -

co) = —^ ^ Q-j{27zlM)^m^[{(ùRlc)+j\np\sin ©cos {(p-lnmlM) . . . (2.4.4)

^ m=0

The term sin 6cos(9)-2wn/M) which appears in (2.4.2) is expressible as the following infinite Bessel series:

^jicùRc) sinflcos {<p-2nmlM) = ^ jVJ^ Q'^^-jv{<p-2nmlM) . . . (2.45) \^-eo

Substituting in (2.42) and changing the order of summations we have:

(p, co) =

/ ~ M-l

2 X y''-^v(^sin0)e/O -'')9’[ ^ ] £ g-;(2;r/M)(;x+i-v)m] ...(2.4.6)

i—~î y=-oo t?i—0

The bracketed term in (2.4.6) is equal to zero or to 1 according to the values of m, i and V.

M-l

e-K2n/M)(fi+i-v)m^'^ S (^+ i-v+ qM ) (2.4.7)

^m=0

The expression for 0 ^ (0 , cp, co) therefore takes the form:

0^(e,(p,w ) = ' Z A,(e) 2 f q=-oo / _ =

E [Ê

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where 6) is the frequency-dependent phase mode coefficient of order q for the ji'th phase mode, and is given by:

/

e) = s (0) V,>^M [(û)R'c)sin0] ■.. (2.4.9) i=-I

The far-field pattern of a phase mode is thus expressible as a sum of the form

<p, <a) = X 6)e-i<M+‘!tf)<i> . . . (2.4.10)

q=-oo

which may be viewed as the azimuthally-omnidirectional linear-phase pattern CpiQ{cû,0)e-iy-^^ distorted by an infinite series of higher order terms^'^. Note that (2A.10) still applies when element patterns of the form (2A3) are assumed, only each phase mode coefficient is given by (see appendix B.2 for details):

C^î(û). m = p A o (e );(^ -^ » * « V ,M [(^ + ;ln p )sin 0 ] ■ • • (2.4.11)

For array elements that are omnidirectional in azimuth, we have /z/ = 0 , zVO in {22 A), or equivalently p = 1 in {2A3}, both of which lead to:

CMi<o, e) v , M ( ^ s i n e ) ■ • • (2.4.12)

Since (2A.12) also applies to the dominant ^ = 0 mode, whenever the frequency and the array radius are such th a t/^ (-^ ^ ) hits one of its zeros, there will effectively be a ‘hole’ in the far-field circumferential coverage of that mode around elevation zero (0 = n fl). In the vicinity of the zero, the far-field azimuth pattern is not completely cut out, but ripple from higher order terms (especially those characterised by <7 = ± 1) will dominate, and thus limit the practical usefulness of that mode. If on the other hand the element patterns are of the form:

The original motivation for considering the zero-order term as dominant stems from the corresponding analysis of a continuously-distributed circular-array source under linear-phase excitation [Day 83], [Jon 90] where the far-field pattern for the /i’th mode is seen to be given by In the case of a circular array of discrete but closely spaced elements (an inter-element spacing of less than half a wavelength), higher order modes (though not necessarily the first one) wül indeed die out by virtue of the asymptotic properties of a Bessel function whose order is much greater than its argument.

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g(G, ç) = ^ e (0)cos2^ = ^g(0) [ l+ lc o s ç ) ]

then.

where Jy (• ) signifies the derivative of the Bessel function of the first kind with respect to its argument. Since the zeros of /v(* ) and of Jy (• ) do not coincide for any V, it follows that for this type of element pattern the phase mode coefficients never fall to zero. In fact noting that the zeros of a Bessel function of integer order are always real it follows that any element pattern of the form (2.4,3) which is either outward-directional (p < 1) or even inward-directional (p > 1) leads to phase mode coefficients (2.4.11) that never fall to zero.

One last result relates to symmetric element patterns given either by (2.4.3) or by (22.4) with:

hi(e) = h.i(e) , i > o

The phase mode coefficients of symmetric modes are then related by:

/

= E . . . (2.4.13)

* = - /

as can also be directly deduced from (2.4.11), where use has been made of the Bessel identity (22 J5). Consequently if 0^(0, ç,co) is given by (2.4.10) then the corresponding expression for (p, co) is:

0 .^(6 , «).£»)= X . . . (2.4.14)

q=-eo

In the next chapter we shall show how the above symmetry may be used in phase- comparison direction finding without resorting to mode alignment. Another important point to note is that by symmetry, the phase-mode coefficients for modes fi = M U and

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