AP04-7
AN ALGORIMM FOR
PATlERN COMPUTATION OF TRIANGULAR LA!LTICE
PHASED ARRAYS
S.A. Bokhari, N. Balakrishnan and P.R. Mahapatra
Department of Aerospace Ehgineering
Indian Institute of Science
Bangalore
India
012,
N u m e r i c a lm e t h o d sf o ra r r a yp a t t e r no p t i m i z a t i o nr e q u i r e
efficient techniques for the pattern computation.
Among t h e known
geometries, the e q u i l a t e r a l t r i a n g u l a r l a t t i c e is a widely used configuration for large phased a r r a y s i n view of its substantial savings in
e t a1 [I]havesuggesteduseofMerserau's[2]
hardware.Corey
hexagonal FFT a l g o r i t h mf o rt h ee f f i c i e n tp a t t e r nc o m p u t a t i o n .
i s known t o b e r a t h e r
H o w e v e r ,t h ec o d i n go ft h i sa l g o r i t h m
complicated. h e to the large volume of literature on rectangular FFT
a l g o r i t h m s , a v a i l a b i l i t y o f computer programs, and the ease in coding
some versions such as the row-column algorithm, it is advantageous to
reduce the hexagonal FFT to a rectangular FFT. Otneil e t a1 [3] have
developedanalgorithm
on these grounds. More recently, Geussom and
Merserau[4]have
shown t h a tm u l t id i m e n s i o n a l
DFTs defined on
arbitrary but periodic sampling lattices can be reduced to rectangular
DFTs through a permutationof the input sequence.
I n t h i s p a p e r , a hexagonal DFT c o n v e n i e n t f o r t r i a n g u l a r g r i d
phased array computations is described. Based on thegeneraltheory
fast computation has ,been developed. Further[4],analgorithmfor
more, a n a l g o r i t h m f o r c a r r y i n g o u t t h e p e r m u t a t i o n s
of t h e i n p u t
sequence Itin placett has also been developed. This is o f p a r t i c u l a r
importance i n d e a l i n g w i t h FFTs o f l a r g e o r d e r as it s i g n i f i c a n t l y
r e d u c e s t h e - a u x i l a r y s t o r a g e r e q u i r e d . The a l g o r i t h m d i f f e r s f r o m
that of O'neil e t a1 ,[3] i n t h a t t h e p e r i o d i c i t y l a t t i c e s i n b o t h
s p a t i a l and s p a t i a l frequency domains are identical. A main advantage
of this algorithm is that it is remarkably simple to code and requires
-about the same number of operations as the vector radix algorithm [2].
A p p l i c a t i o n o f t h e FFT a l g o r i t h m f o r pattern computation often
adequateresolution
requiresexcessive"zero
paddingtt to r e s u l ti n
i n t h e s p a t i a l f r e q u e n c y domain. When computer s t o r a g e becomes a
l i m i t a t i o n , some f o r moifn t e r p o l a t i o n
becomes necessary.
An
a l g o r i t h m similar t o t h e Itpseudosamplingmethodtt
[5] hasbeen
described i n this paper f o r use with the hexagonal FFT
CH2435-6/87/0000-0137$01.00 @ 1987 IEEE
137
!Ike Hexagonal
FFT algorithu
While performing DFTs of a function of spatial coordinates, it i s
more c o n v e n i e n t t o a d o p t t h e d e f i n i t i o n g i v e n b e l o w i n s t e a d o f t h e
more widely used definition in signal processing. This allows direct
correspondance w i t h the array factor. 'The hexagonal DFT of order
( N x N) f o r N = 2m, m
being an integer, can be written
as
(W21-1
Fh(kl,k;~)=
(N/2)-1
z
2
n1 = -N/2
n2
- k2)
-(*I
+
=
fh(nlYn2)
exp[-jfl((a1
- n2)
-N/2
n&)/N],
-N/2
5 kq ,k2 <
N/2
(1)
where fh(nl,n ) d e n o t e st h ea r r a ye x c i t a t i o nc o e f f i c i e n t sw i t h
r e f e r e n c e t o &e c o o r d i n a t e a x e s shown i n Fig. 1, kl , k2 d e n o t e t h e
frequency v a r i a b l e s i n the s p a t i a l frequency domain and Fh denotes the
hexagonal DFT of f h w i t h i d e n t i c a l p e r i o d i c i t y .
The fundamental
in both domains is a rhombus and is convenient for
region of support
s t o r a g e a s two dimensionalarrays.Anotheradvantageofthis
d e f i n i t i o n o v e r t h e more g e n e r a l o n e [ 2 ] is t h a t it r e d u c e s t o a
s i n g l es q u a r e DFT. The i n v e r s e hex DFT i s i d e n t i c a l t o Eqn. (1)
e x c e p t f o r 3 change i n s i g n o f t h e e x p o n e n t , a n d
a normalization
f a c t o r (1/N ). The f o l l o w i n g d e f i n i t i o n o f t h e s q u a r e DFT has been
employed since most a v a i l a b l e computer programs are based upon it.
<
0 5 k!,,
k$
(2)
N
Equation. (1) can be reduced to Eqn. (2) by a nonlineartransformation
of variables.
The procedure w i t h d e t a i l s omitted is given below.
( a ) Form the perturbed -array f r ( n j ,n$) = fh(nl ,n2)
where n1 = (n!,
+
nS)\t N
andn2
= (n!,
+ 2nb) \ N
The modulo operation i s denoted by *\', and i s assumed t o wrap
around and return a nmber between -N/2 and N/2-1.
(b) Compute the rectangular FFT Fr(Y ,k5) of
(C)
Set Fh(kA ,k2) = Fr(k{ \N,k$\N)
138
fr(nj ,nj)
The computation i n s t e p (a) can be done "in-placeff by exploiting the
periodicity property of the i n d i c e s g e n e r a t e d by t h e a p p l i c a t i o n o f
be grouped i n t o sets
theformula i n (b). The inputsequencescanthen
of different periodicities and the interchange of memory locations is
t h e nc a r r i e do u t .
The a l g o r i t h m ,i l l u s t r a t e db yt h ef o l l o w i n g
four nested loops is v a l i d for N > 2.
rFor
a
=
1 to
(m-1)
instepsof
I
For b = 0 to (N - t) i ns t e p s of t
c = 2 : If b = O then c = l
For
to +I i n steps of
d = -1
c
I
nl=b
For
: n2=b+d*s
e
=
1 to (p
n{
=
(nl
fh(n1,n2)
- 1)
+ n2)\N
n 2 < 0 then n 2 = N - s
: If
i ns t e p s of 1
: n$ = (nl
<=> fh(n!,,ni)
+ 2n2)\N
: n, = n!,
: n i = n$
Next
The symbol <=> denotesaninterchangeof
memory l o c a t i o n s .
be done by a simple block interchange.
operation .in step (c) can
The
The a l g o r i t h m f o r i n c r e a s i n g t h e s p a t i a l f r e q u e n c y r e s o l u t i o n
a chosen
consists of resampling the aperture distribution in steps of
resolution factor, Fourter transforming the individual sets, phase
compensation of each s e t f o l l o w e d by a summing operation.
Nuuerical Illustration and Conclusion
As an example, a 2437 e l e m e n t a r r a y l o c a t e d w i t h i n a c i r c l e of
radius 15 1 i n the Y-Z plane of the Cartesian coordinate system, with
Az = 0.5 ), is considered. The aperturedistribution
i s &ken as
[ 1 - ( ~ - / a ) ~ ]The
~ . computed b r o a d s i d e p a t t e r n u s i n g a 64 x 6 4 p o i n t
hex FFT with a r e s o l u t i o n f a c t o r o f 4 a l o n g b o t h a x e s i s shown i n
Fig. 2. The distorted appearanceof the main beam is due to the rectangular plot of the
hexagonal grid samples. Patterns a t other angles
can be readily obtained using the complex s h i f t i n g theorem of DFTs.
The present algorithm provides a s i m p l i f i e d , y e t e f f i c i e n t method
f o r t h ep a t t e r nc o m p u t a t i o no ft r i a n g u l a r l y
packed a r r a y s . In
s i t u a t i o n s where there is a storage limitation,theabovealgorithm
combined w i t h the hex WT becomes convenient, -however a t the expense
ofcomputation time. The high resolution algorithm has theadvantage
of being exact and not involving the complexity of f i l t e r design as i n
other high resolution algorithms.
139
References
[ I ] L.E. Corey, J.C. Weed and T.C. Speake, IIModeling of t r i a n g u l a r l y
packed a r r a y s u s i n g h e x a g o n a l FFTI, IEEE AP-S I n t . Symp. D i g e s t ,
pp. 507-510, 1984.
[2] R.M. M e r s e r a u , ? l T h ep r o c e s s i n go fh e x a g o n a l l ys a m p l e d
two
dimensionalsignals11, Proc. IEEE, vol. 67, pp. 930-949, 1979.
[3] D.R.Otneil, L.E. Coreyand E.A. Nelson, trEvaluating the Fourier
DW',
transform of a hexagonally
sampled signal using rectangular
Proc. IEEE SouthEastcon, pp. 282-285, 1985.
[4] A. Guessom and R.M. Merserau,Wxt algorithm f o rt h ec a l c u l a t i o n
o f t h e m u l t i d i m e n s i o n a l DFTI, IEEE Trans Acoustics, Speech and
SignalProcessing,vol.
ESP-34, pp.937-944,1986.
[ 51 O.M. Eucei, G. D'Eliaand G. Franceschetti, Vfficient computation
o f t h e far f i e l d of a parabolic reflector by pseudo sampling
algorithmt1, IEEE Trans. Antennas Propagat., vol. AP-31, pp. 12591272, 1983.
'
0..
0
' .
.
'
.
o
".
I
........
.......
r',.......
0
.
0
~
Y
.
Fig.1 Coordinate axes for
the hexagonal geometry
1.L I
Fig. 2 Radiation
pattern
Sin 8 Sin 0
140