PROCEEDINGS OF THE 1989 INTERNATIONAL SYMPOSILJM
ON NOIS& AND CLLn%R RWECIION IN RADARS AND
IMAOINO SENSORS, cdilod
7.Suuki, H Ogun and S. Flrjimun
@ IEICE, 1969
227
DESIGN OF CLUTTER-OPTIMUM DISCRETE PHASE CODED RADAR SIGNALS
VIA INTEGER PROGRAMMING
T.K. Bhattacharya, P.R. Mahapatra and N. Balakrishnan
Department of Aerospace Engineering
Indian Institute of Science
Bangalore 560 012, INDIA
Abstract
T h e p a p e r p r e s e n t s a m e t h o d t o t a c k l e t h e p r o b l e m of d e s i g n i n g d i s c r e t e p h a s e
codes f o r t h e r e j e c t l o n of d e l a y - D o p p l e r c l u t t e r . S i g n a l - t o - I n t e r f e r e n c e R a t i o ( S I R )
has been used as t h e performance criterion and t h e maximization of SIR, for t h e case of
c o d e d r e c t a n g u l a r p u l s e b u r s t s , h a s b e e n r e d u c e d t o t h e m i n i m i z a t i o n of a c l u t t e r
integral. To solve t h e associated nonlinear optimization problem a n integer programming
algorithm has been adopted, using which it has been possible to obtain optimum signals
f o r a v a r i e t y of c l u t t e r d i s t r i b u t i o n s .
T h e c a p a b i l i t y o f the a l g o r i t h m t o p r o v i d e
o p t i m u m s o l u t i o n s i n s p i t e of t h e l a r g e n u m b e r of l o c a l m i n i m a , m a k e s it p o s s i b l e t o
design optimum d i s c r e t e coded signals t o r e j e c t a r b i t r a r y delay- Doppler c l u t t e r with
m o d e r a t e computational efforrs.
I. Introduction
Signal design is known t o be a very robust method t o c o m b a t c l u t t e r i n t e r f e r e n c e
i n r a d a r s . A n u m b e r of s i g n a l f a m i l i e s h a v e b e e n c o n s i d e r e d i n t h e p a s t t o r e j e c t
c l u t t e r effects.
O n e of t h e most studied (and advantageous) family of signals h a s been
p h a s e c o d e d p u l s e b u r s t s . T h e a u t h o r s of t h i s p a p e r h a v e e l s e w h e r e [ l ] t a c k l e d t h e
problem o f designing analog phase coded pulse bursts, t o optimally r e j e c t arbitr,ary
c l u t t e r d i s t r i b u t i o n s i n t h e d e l a y - D o p p l e r p l a n e of t h e r a d a r . In t h e a b o v e p a p e r , t h e
subpulse phases w e r e taken t o lie anywhere between 0-2n. A scheme, which would be more
a t t r a c t i v e from t h e point of view of implementation is one in which t h e phase of e a c h
s u b p u l s e is c h o s e n f r o m a f i n i t e set of p h a s e s , g i v i n g r i s e t o d i s c r e t e p h a s e c o d e d
p u l s e b u r s t s . T h e r e is no m e t h o d , a s y e t , w h i c h c a n b e u s e d f o r d e s i g n i n g o p t i m u m
In t h i s
d i s c r e t e coded signals against a r b i t r a r y delay-Doppler c l u t t e r distribution.
paper, we present a n integer programming methodology t o t a c k l e t h e problem and results
a r e presented to confirm t h e workability of t h e proposed method.
1
T h e k n o w n p h a s e c o d e s , s u c h a s t h e B a r k e r C o d e [ 2 ] a n d t h e F r a n k c o d e [3] are
designed t o obtain good aperiodic autocorrelation 'function behavior. The optimization
of t h e a u t o c o r r e l a t i o n b e h a v i o r i m p l i e s r e j e c t i o n of c l u t t e r w h i c h i s s t a t i o n a r y w i t h
respect to t h e t a r g e t . However, i n p r a c t i c e t h e c l u t t e r c a n and does have a finite
v e l o c i t y d i f f e r e n c e as w e l l a s s p r e a d , a p a r t f r o m t h e r a n g e d i f f e r e n c e w i t h r e s p e c t t o
the target.
W h a t i s n e e d e d t o r e j e c t s u c h d e l a y - D o p p l e r d i s t r i b u t e d c l u t t e r , is t o
d e s i g n a r a d a r s i g n a l w i t h good a m b i g u i t y f u n c t i o n r a t h e r t h a n good a u t o c o r r e l a t i o n
function behavior.
T h e f u n d a m e n t a l p r o b l e m in c o d e s d e s i g n e d on t h e b a s i s of
a u t o c o r r e l a t i o n f u n c t i o n , i s t h e p r e s e n c e of high s i d e l o b e s in t h e a m b i g u i t y f u n c t i o n
of r h e s i g n a l , w h i c h leads t o d e g r a d e d p e r f o r m a n c e of t h e c o d e s i n p r e s e n c e of a n y
Doppler shift of t h e c l u t t e r with respect t o t h e target.
R u m m l e r [4] a n d M i t c h e l l a n d Rihaczek [ 5 ] h a v e c o n s i d e r e d s p e c i f i c a s p e c t s of
T.X. Bhanachntya
228
et a1
signal d e s i g n a g a i n s t 2-D c l u t t e r .
T h e y h a v e t a k e n f o r c o n s i d e r a t i o n only t h e p u l s e
This assurnption leads to a simple expresbion f o r
bursts with 50% o r lesber duty cycle.
t h e c l u t t e r i n t e g r a l . S u c h a s i g n a l , h o w e v e r , r e d u c e s t h e a v e r a g e p o w e r of t h e
transmitter.
Since t h e minimum pulsewidth is not a design criterion, being decided by
t h e system l i m i t a t i o n s s u c h a s t h e p e a k p o w e r of t h e t r a n s m i t t e r a n d o p e r a t i o n a l
requirements such a s t h e t o t a l energy of the transmitted signal,
t h e sparse burst also
increases the minimum operating range of t h e radar for a given number of subpulses.
2. Formulation
Signal- to-Interference R a t i o (SIR) h a s been used as t h e performance c r i t e r i o n a n d
t h e m a x i m i z a t i o n of SIR, u n d e r m a t c h e d f i l t e r a s s u m p t i o n , f o r t h e case of c o d e d
rectangular pulse bursts, has been shown [ I ] t o reduce to the minimization of a c l u t t e r
integral (CI), defined as
p.;
00
The complex envelope of t h e constant-amplitude phase coded c o m p a c t pulse burst i s
described as
...,
w h e r e pi i = (k,
N] r e f e r s t o t h e p h a s e of t h e ith s u b p u l s e a n d g i ( t ) is t h e g a t e
function given in t e r m s of a unit s t e p function u(t) and t h e subpulse duration T as
g i ( t ) = u{t
-
(i-I)T} - u { t
-
(3)
iT}
T h e parametiers of optimization in this
o n l y f r o m p u l s e to p u l s e i n t h e b u r s t .
significant simplification in t h e evaluation
e v a l u a t i o n of t h e c l u t t e r i n t e g r a l . T o a i d
as,
c a s e a r e t h e subpulse phases, which c h a n g e
T h i s p r o p e r t y of t h e s i g n a l l e a d s t o a
of t h e a m b i g u i t y f u n c t i o n , a n d h e n c e t h e
t h e s i m p l i f i c a t i o n , d e l a y Z can be w r i t t e n
t =( I + F ) T
(4)
Thus, I
W h e r e , I is t h e i n t e g e r p a r t of a n y , s h i f t a n d F i s t h e f r a c t i o n a l p a r t .
b e l o n g s t o t h e set of i n t e g e r s [0,1,2,.,,(N-l)],
and F belongs t o t h e set of f r a c t i o n s
[O < F < I].
T h e c l u t t e r i n t e g r a l C , g i v e n b y e q . ( I ) , can b e e v a l u a t e d for t h e p h a s e c o d e d
compact pulse bursts for a given c l u t t e r distribution function P(.c,v).
We now consider the g e n e r a l 2-dimensional uniform c l u t t e r distribution given as
Design of Clutter-Oprimwn Dkmtt Phase Codcd Rodnr Sign&
229
The constant K is determined by t h e normalization condition
In o r d e r t o s i m p l i f y t h e e x p r e s s i o n s , t h e e x t e n t of t h e c l u t t e r d i s t r i b u t i o n is
assurned t o be rriultiples of t h e subpulse width, i.e. F,in
and Fmax a r e identically zero.
F o r a r e a s o n a b l y l a r g e n u m b e r of s u b p u l s e s i n t h e t r a i n , t h i s d o e s n o t i m p o s e a n y
serious limitation. The resulting c l u t t e r integral is, then,
vWk3X
K
1
C.= -
NK*
I-U(V)
'
*mx
~ , ~ ~ I , v ~ , ~ + ~ . ( I ,+ v (U-Cos(V))
~ I ' )
J
l"Imin
'mi n
I=lmin
r:
....( 7 )
Were
and,
2vi)tpi+pi-I-11
for 0
<
I
< (N-2)
(8)
3. Description of t h e Algorithm
T h e bulk of w o r k d o n e on s o l u t i o n s of d i s c r e t e v a r i a b l e p r o b l e m s d e a l w i t h
g e n e r a l a p p l i c a t i o n s i n l i n e a r p r o g r a m m i n g f i e l d . In t h e f i e l d of n o n l i n e a r i n t e g e r
a n d m i x e d - i n t e g e r p r o g r a m m i n g , s u b s t a n t i a l l y less a t t e n t i o n s e e m s t o h a v e b e e n paid.
F o r t h e case of a q u a d r a t i c , p o s i t i v e s e m i - d e f i n i t e f u n c t i o n w i t h l i n e a r c o n s t r a i n t s ,
Kunzi et al. [61 suggested a method which makes use of transformations t o convert t h e
p r o b l e m i n t o a s e q u e n c e of i n t e g e r l i n e a r p r o b l e m s .
W i t z g a l l [71, a l s o s u g g e s t e d a
i
X Bhanachaiya a aL
230
s i m i l a r m e t h o d f o r the c a s e o f a l i n e a r o b j e c t i v e f u n c t i o n w i t h p a r a b o l i c c o n s t r a i n t s ,
in i n t e g e r variables.
Again, a t r a n s f o r m a t i o n , t h i s t i m e of t h e c o n s t r a i n t s , i n t o a
pseudo linear form is made.
A large number of other papers, as reviewed in [8], have
a l s o suggested such methods, which c o n v e r t t h e nonlinear problem into a linear
problems, and hence a r e applicable t o only a restrictive class of problems.
A m o r e g e n e r a l a p p r o a c h s u g g e s t e d by R e i t e r e t al. 191, h a n d l e s t h e p r o b l e m s ,
A modified
where both t h e objective function and t h e constraint a r e quadratic.
The m e t h o d is also claimed to b e applicable to wider
gradient- type method, is applied.
r a n g e of n o n l i n e a r problems. A n o t h e r i n t e r e s t i n g a p p r o a c h , u t i l i z i n g t h e c o n c e p t of
penalty functions, was suggested by Cellatly and Marcal [lo].
T h e p r o b l e m of PSK s i g n a l d e s i g n , f o r c l u t t e r r e j e c t i o n , i n v o l v e s t h e c l u t t e r
i n t e g r a l , w h i c h is a highly n o n l i n e a r f u n c t i o n of t h e s i g n a l p h a s e s .
H e n c e , m o s t of
t h e available methods which transforms t h e nonlinear problem into a sequence of linear
o n e s a r e n o t d i r e c t l y a p p l i c a b l e . Also, t h e m e t h o d s w h i c h r e q u i r e t h e e x p l i c i t use of
g r a d i e n t i n f o r m a t i o n a r e not a t t r a c t i v e .
F o r t h i s r e a s o n , out of a l l t h e a v a i l a b l e
methods, t h e penalty function technique, SUMT (Sequential Unconstrained Minimization
T e c h n i q u e s ) [ I I], w h i c h h a s been"' used q u i t e e f f e c t i v e l y i n m a n y c o n t i n u o u s v a r i a b l e
p r o b l e m s , i s a d a p t e d t o handle i n t e g e r p r o b l e m s in a m a n n e r s i m i l a r t o t h a t s u g g e s t e d
by G e l l a t l y a n d M a r c a l [lo]. T h e a p p l i c a b i l i t y of t h e m e t h o d i s a l s o d e m o n s t r a t e d b y
m e a n s of numerical examples.
T h e problem of integer phase coding, t o b e solved, c a n b e written as
min C(p), where p ={pi,p2,...,p
P
'
Nl is
t h e vector of subpulse phases
subject to
p E R d , t h e f e a s i b l e set of d i s c r e t e v a r i a b l e s .
w h e r e C is the c l u t t e r integral for t h e PSK c o d e s given by eq. (7).
The c o n s t r a i n e d p r o b l e m s t a t e d a b o v e c a n b e t r a n s f o r m e d i n t o a s e r i e s o f
unconstrained problems.
In t h i s t e c h n i q u e , t h e p e n a l t y f o r n o n c o n f o r m i t y t o t h e
The s e q u e n c e of
d i s c r e t e s e t is s u c c e s s i v e l y i n c r e a s e d a f t e r e v e r y i t e r a t i o n .
w e i g h t i n g f a c t o r s was c h o s e n w e r e c h o s e n b a s e d on an h e u r i s t i c m e t h o d .
The
unconstrained minimization at each iteration is tackled by t h e Simplex method of Nelder
a n d M e a d [12]. T h i s cornbination of t h e S i m p l e x m e t h o d a n d t h e i t e r a t i v e i n c r e a s e i n
t h e p e n a l t y has been found to ensure t h a t t h e minimum is reached with t h e constraints
satisfied within 5-10 iterations.
4. Numerical Results and Discussions
T h e a l g o r i t h m d e s c r i b e d a b o v e w a s u s e d t o d e s i g n i n t e g e r PSK b u r s t s f o r t h e
r e j e c t i o n of d e l a y - D o p p l e r c l u t t e r d i s t r i b u t i o n s .
A v a r i e t y of c l u t t e r d i s t r i b u t i o n s
w e r e c o n s i d e r e d . For i l l u s t r a t i v e p u r p o s e s , w e p r e s e n t t h r e e s a m p l e ' r e s u l t s f o r t h e ' '
design of optimum 13-pulse biphase codes against given delay-Duppler clutter.
Fig. 1-3
P r e s e n t t h e r e s u l t i n t e r m s of t h e p h a s e s t r u c t u r e a n d t h e a m b i g u i t y f u n c t i o n o f the
o p t i m u m biphase c o d e s obtained. T h e r e l e v a n t c l u t t e r distributions a r e a l s o indicated
in t h e P l o t s O f t h e a m b i g u i t y f u n c t i o n i t s e l f , b y t h e d a r k e r m e s h . T h e o p t i m a l i t y of
h i p of Cher-Optitnum Dircreic Phase W
d Radar Sip&
23 1
t h e s i g n a l is c l e a r l y e v i d e n t by t h e a b s e n c e of any s i g n i f i c a n t s i d e l o b e l e v e l s i n t h e
c l u t t e r occupied regions in t h e delay Doppler plane of t h e radar.
T h u s , the e f f e c t i v e n e s s of d i s c r e t e p h a s e c o d e s t o deal w i t h a v a r i e t y of
d e l a y - D ~ p p l e rc l u t t e r d i s t r i b u t i o n s i s v i n d i c a t e d . T h e r e s u l t s a l s o e s t a b l i s h t h e
usefulness of t h e novel integer programming technique adapted in t h i s work.
Since t h e
hardware requirement t o field program t h e parameters of t h e biphase code using digital
implementation a r e easy t o realize, t h e signal design methodology of this paper has t h e
potential t o introduce t h e adaptive c l u t t e r rejection capability in radars.
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[I21 J.A.
Fig. I
1:.
ti
Fig. 2
1.1
-.
'
,.I
I.,
t-
in n
l.w..lla
Fig. 3
ni
Phase structure and Ambiguity f u n c t i o n of o p t i m u m 13- pulse biphase c o d e , f o r t h e
clutter distributions indicated by darker mesh in (b).
111. ..I11
w.
PlDPll"
N
w
h)