The practical usefulness of the non-adaptive mode-space beamforming, null steering and sectorally-confined direction finding schemes that have been discussed in this chapter, depends, to a large extent, on our ability to equalise the intrinsic frequency response variations between the excited phase modes which constitute the building blocks in the implementation of the above techniques. One possible solution to the problem is that of narrowband mode alignment, which refers to the calibration of each phase mode over the relevant frequency band, so that a look-up table may be set up and used (in conjunction with possible interpolation) for its alignment at each frequency. This method is instantaneously narrowband in the sense that it is limited to the reception of narrowband signals that must all have the same carrier frequency, which is either known or pre-detected.
Of course if element patterns were controllable (alas, they rarely are) one could try and achieve wideband mode alignment by searching for an ideal element pattern that will flatten, or at least linearise the frequency responses of all phase mode coefficients. A hint to the solution for this somewhat academic pattern control problem has been provided in section 2.4 o f chapter 2, where an azimuth element pattern of (l+cosç>) was shown to lead to phase mode coefficients that caimot fall to zero. In fact by examining the asymptotic expression for the relevant Bessel functions of large arguments^®, expression (2.4.13) for the | t ’th phase mode coefficient takes the form:
C^oio), 6) ~ ge(.e)e-j”/^ (2c/jia>Rnn . . . ( jg j;
which is indeed linear in its phase response, but requires amplitude equalisation for broadband operation. A little thought will reveal the ‘optimal’ solution for which all phase mode coefficients become linear in phase and invariant in amplitude, only it turns out to be impulsive in qr.
g(6,V>) = g e ( 6 ) 'Z e i i f ■■■{3.62} (=-oo
Under (3.62) all phase mode coefficients become (see (2.4.9)):
When the order and argument of Jfi+i[(ci}R/c)smd] are such that: \ii+i\ «(coR /c)sin 6, or equivalently l/i+zl « Af/2 (for an inter-element spacing smaller than XJl and 9 = n /l) then:
5 9 Iiilllllillllll ___________________C o n v e n tio n a l m o d e - s o a c e t e c h n iq u e s an d a p p lica tio n s
C^f(co,e ) ^ X 0] = e
...
but this affects all coefficients regardless o f order, with the result that each phase mode pattern is given by a series of M impulses at angles (p = litm lM , 0<m <A f-l, which, after all, is to be expected from M impulsive element patterns. The pursuit after the ideal element pattern is further discussed in appendix C.3.
A more pragmatic approach for the broadband alignment of circular-array phase modes to which we devote the rest of this section, involves the separate deconvolution of their dominant coefficients {C^o(û>» ^o))/i for a given azimuth cut The required analogue transfer function for the /I’th phase mode filter is accordingly given by:^i
Hii(s) \s=j(o=^'-^ ^IC ^(co , n il) , coijo^ co< cùhi
where the (assumed finite) time delay r is needed to make the impulse response of H^(jœ) causal, and cdlo and œni are, respectively, the lower and upper frequency in
the operating band. The stability of such a filter would depend on the radiation properties of the element patterns. Assuming the latter to be of the form (2.43) then from (2.4.11) we have:
c^o(to, TcfZ) = + j\np) - ■ • (3.6.4)
and the filter’s Laplace-domain transfer function is
= e-^Vpj^J^[-j(sR/c)+jlnp] • ■ • (3.6.5)
Noting that the zeros of a Bessel function o f integer order are always real, it follows that for outward directional element patterns characterised by 0 < p < l, the poles of H^(^s) are located on the left half of the complex 5-plane, rendering it stable. In contrast, omnidirectional elements (p = 1) as well as inward-directional elements (p > 1) both lead to unstable filter designs.
In the context of discrete time-sampled signals, a corresponding causal digital filter WÜ1 be stable if and only if the poles of its transfer function y^(z) all lie inside the unit circle on the complex z-plane. A stable realisation of as l/(D^(z), where 2^(z) is the digital implementation of (3.6.4), therefore depends on îD^(z) having all
3 . 6 B an dw id th c o n sid e r a tio n s 60
its zeros confined to within the unit circle, which in turn, as we shall shortly see, is again a function of the shape of the element patterns. For the implementation of a causal digital filtering unit (D^(z) having a finite impulse response of length N, and a non-aliased frequency response which, over the bandwidth 0<Q )<cohj, is equal (to within a constant group delay) to C^q(co, we must have:
^ C^o(û)"2^pM^^/2)5(û)2ir-2;r/7)evXû>-2;r/?/4r)(N-i)z\f/2
p=-e
(3.6,6)
where S (H) is a sampling window function which is zero for values of £2 outside the interval [-2;r+ 2;r- and A t is the temporal sampling interval (with a>s = 2nlAt denoting the sampling f r e q u e n c y ) ^ ^ _ pig^ 3.6.1. The corresponding
transfer function for this filter is then given by:
2^(z) = îD^(lzle/^ =
pjfi2-m)/2 ^ S(a-2xp)ei’V(.N-i)j^[(R/cAtXa-27cp)+j]n ^
pyMz (N -W g ( ^ ) / [A S + yin P. ] , -cO H iA t^C i^m i^t
cAt
(3.6.7)
S (Q) Cpo(n/At, Kl2)e-J^(N.1)P.
Fig. 3.6.1 Digital implementation of phase mode coefficient response
^2 Under the finite-impulse-response assumption (which is later proven), the addition of the delay term e'^^'^xo Cfioio), nH) with t = (N-\)Atl2., renders the filter causal.
61 C o n v e n tio n a l m o d e - s p a c e t e c h n iq u e s an d a p p lica tio n s
and it is evident from (3.6.7) that the zeros o f iD^(z) (and therefore the poles of lie on a circle o f radius in the complex z-plane. A stable design once more depends on the array elements being outward directional with p < 1 - see Fig. 3.6.2.
In order to adequately describe a single Bessel function J^(coR/c) multiplied by a window S(O)At) o f approximate temporal duration^ ^ Stu/cOs from its frequency
samples, the frequency-sampling period Act) - 2%!Zs must be such that:
% < nlAo) (3.6.8)
where rjji is the maximum effective extent in the time domain of the convolved expression/v(0*5 (f), where: Jv(t) dcoeJ^^JvicoR/c) — (jYTyict/Fl) Id < R/c 7c[(R/cf - 0 Id > R/c (3.6.9)
S (t) is the Fourier transform of S (oA t) and Ty(x) is the Chebyshev polynomial of the first kind of order v. Since Jy(t) and S (t) are bounded in the time domain by Id < R/c and Id < Ak/cOs respectively, it follows that:
z-plane
cAt/R.
Fig. 3.62 Poles of l/^D^iz) on the complex z-plane
This is the null-to-null width of mainlobe + first sidelobes in the Fourier transform of a rectangular window whose cutoff frequencies are 2 1 (0 At = ± n
3 . 6 B andw idth c o n sid e r a tio n s ___________________________________________ iiilllllillllll 6 2 lllllll
tHi = (Fi/c) + 47t/û)s • • • (3.6,10)
which in view of (3.6.8) also means that:
Tj > 2R /C + Sk/o)s • • • (3.6.11 )
Now, exactly the same argument holds for a Bessel function J^(coRlc) of any order. Since each phase mode coefficient is expressible as a sum o f such Bessel functions^^. It follows that Aco is also the frequency sampling period required for the digital representation of C^xo(û), 7r/2)S((oAt), and the fact that it is effectively bounded both in frequency and in time justifies our attempts to implement it as a finite impulse response (FIR) filter. The number of required frequency samples N which defines the order of the filter (Dfi{z) is given by the inequality:
N = zJA t > 2(2+ RIcAt) • • • (3.6.12)
whereas the frequency sampling period must not exceed:
A a = 27clt, = ^ < — % — ■■■Q.6J3) N 4+cOsRlnc Denoting Kx = (ùsl2(ûiii > 1 > 1 (3.6.U) we can write: N = [2 Ko)i2 + KrCOHiR/nc)] • • • (3.6.15)
where Fxl the ceiling of x is defined as-
r%l = 1 +Int(x) - Int[Int(x+1 ) - x ] - • • (3.6.16)
Noting that the arc wise inter-element spacing must be kept smaller than 0.5 or even
This clearly follows from (2.4.9) when element patterns are represented by (2.2.4), but it is also the case under the representation (2.43) by virtue of the identity:
lllllll 6 3 Iiilllllillllll ___________________ C o n v e n tio n a l m o d e - s p a c e t e c h n iq u e s an d a p p lica tio n s