A POWERFUL RANGE-DOPPLER CLUTTER REJECTION STRATEGY
FOR NAVIGATIONAL RADARS
T.K. Bhattacharya and P.R.Mahapatra
D e p a r t m e n t of Aerospace Engineering
Indian I n s t i t u t e of Science
Bangalore-5600 12, INDIA
ABSTRACT
mainframes of t h e t y p e available at typical
navigational facilities.
Navigational radars a r e used t o d e t e c t and
In
c o n t i n u o u s l y t r a c k a n u m b e r of a i r c r a f t .
strongly cluttered environment, this c a n lead t o
broken and interrupted tracks. This could b e
hazardous specially in t h e terminal a r e a s a n d
approach corridors.
H e n c e t h e optimum r e j e c t i o n
of clutter is a primary requirement in
navigational radars.
This paper describes a
method t o c o m b a t a r b i t r a r y delay-Doppler c l u t t e r .
The problem is approached via t h e signal
ambiguity function.
Matched filter receiver is
assumed a n d transmitted signal p a r a m e t e r s a r e
optimized t o achieve the clutter rejection
c a p a b i l i t y . T h e s i g n a l set c h o s e n i s t h e f a m i l y
of frequency coded constant amplitude pulse
bursts, a n d a c r i t e r i o n f u n c t i o n is d e r i v e d i n
t e r m s of t h e f r e q u e n c i e s of i n d i v i d u a l s u b p u l s e s
which h a s been optimized f o r a v a r i e t y of c l u t t e r
distributions.
The results obtained clearly
bring o u t t h e e f f e c t i v e n e s s of t h e method.
The problem has been formulated using
signals t e r m e d Frequency-Shift Keyed (FSK),
c o m p a c t pulse bursts.
This f a m i l y h a s c e r t a i n
h i g h l y d e s i r a b l e p r o p e r t i e s as d e s c r i b e d l a t e r .
Using t h i s signal family, s p e c i f i c results a r e
derived and presented for specific interference
distributions.
BACKGROUND
The interference t o a radar return consists
of a s u p e r p o s i t i o n of n o i s e a n d c l u t t e r i.e. t h e
combined returns received from unwanted
scatterers. Most o f t e n t h e most severe problem
is t o d e t e c t t h e t a r g e t ( s ) i n p r e s e n c e o f t h e s e
scatterers; it may not b e necessary t o resolve
t h e individual o b j e c t s c a u s i n g t h i s i n t e r f e r e n c e .
Also, m o s t o f t e n , c l u t t e r r e t u r n s a r e s t r o n g
enough to "bury" t h e e c h o power received f r o m t h e
t a r g e t(s)
.
INTRODUCTION
In g e n e r a l , a n i n d i v i d u a l s c a t t e r e r w o u l d
have both a range displacement and a radial
velocity r e l a t i v e t o t h e t a r g e t of i n t e r e s t a n d
t h u s c a n b e r e p r e s e n t e d as a s t r e n g t h w e i g h t e d
point on a delay-Doppler plane c e n t e r e d on t h e
t a r g e t delay and Doppler.
The most general
c l u t t e r distribution may b e represented by a
c o n t i n u o u s d i s t r i b u t i o n of s u c h p o i n t s i n t h e
delay-Doppler plane.
W e r e f e r t o i t as 2dimensional c l u t t e r or delay-Doppler c l u t t e r .
Navigational radars traditionally provide
i n f o r m a t i o n r e g a r d i n g a i r c r a f t . In m o r e m o d e r n
a p p l i c a t i o n s i n v o l v i n g l a r g e s c a l e a u t o m a t i o n of
t h e navigational and air traffic control
functions, t h e radars a r e required to provide
precise and continuous track information on
a i r c r a f t in a multiple t a r g e t environment.
However, i n strongly cluttered regions, t h e
d e t e c t i o n o f t h e a i r c r a f t by t h e r a d a r b e c o m e s
uncertain, resulting in broken and interrupted
t r a c k s . In d e n s e t r a f f i c a n d h i g h l y d y n a m i c
navigational situations such a s terminal a r e a s
a n d a p p r o a c h c o r r i d o r s , a n y s i g n i f i c a n t loss o f
t r a c k would b e hazardous.
Thus t h e need for
a g i l e a n d s i t u a t i o n - s p e c i f i c r e j e c t i o n of 2dimensional (range and Doppler) clutter is
extremely important for modern navigational
radars. This paper presents a method which
optimizes t h e d e t e c t i o n p e r f o r m a n c e of t h e r a d a r
for any given c l u t t e r distribution in t h e delayD o p p l e r p l a n e of t h e r a d a r .
The method is
v e r s a t i l e and powerful enough to handle a r b i t r a r y
d i s t r i b u t i o n s , a n d y e t is c o m p u t a t i o n a l l y
tractable even on minicomputers and lower end
I t is well known t h a t noise effects c a n b e
minimized t h r o u g h t h e use of a m a t c h e d f i l t e r
receiver. For reducing c l u t t e r effects, however,
s p e c i f i c d e s i g n of t h e r a d i a t e d s i g n a l e n v e l o p e
and/or receiver filter characteristics is
n e c e s s a r y . Ideally, t h e u s e of a m i s m a t c h e d
f i l t e r a n d t h e s i m u l t a n e o u s o p t i m i z a t i o n of t h e
signal-filter combination should lend more
versatility for clutter rejection than an optimum
signal-matched-filter pair.
However, Rihaczek
[I] has shown that a properly designed
transmitted waveform, along with i t s matched
f i l t e r , p e r f o r m s a l m o s t as w e l l as a n o p t i m u m
signal-mismatched-filter pair in most realistic
situations, barring some very special clutter
132
CH2759-918910000-0132$1 .OO 0 1989 IEEE
distributions.
On t h e other hand, matched
filters are, in general, simple to realize and
h a v e t h e a d d e d a d v a n t a g e of b e i n g i n h e r e n t l y
o p t i m u m against t h e white-noise c o m p o n e n t of t h e
interference.
For these reasons, in this paper
t h e matched filter assumption is made and a
t e c h n i q u e f o r t h e d e s i g n of o p t i m u m t r a n s m i t t e d
waveforms is presented.
For discrete coded
signals t h e m a t c h e d f i l t e r assumption reduces t h e
p a r a m e t e r s o f o p t i m i z a t i o n b y a f a c t o r o f 2.
Since computational efforts for multivariable
optimization grows exponentially with t h e problem
d i r n e n s i o n a l i t y , t h i s r e d u c t i o n i n t h e n u m b e r of
optimization parameters greatly improves t h e
tractability of t h e optimization, especially for
long pulse codes.
D e l o n g a n d H o f s t e t t e r [61 h a v e a l s o s t u d i e d
t h e g e n e r a l problem b u t confining t h e signals and
receiver impulse responses t o uniformly spaced,
phase- a n d magnitude-tapered, pulse trains.
They
have e v a l u a t e d t h e p e r f o r m a n c e of both m a t c h e d
and "clutter" filters for general waveforms and
have devised a n iterative technique for
optimizing t h e clutter filter performance for
pulse t r a i n signals.
In a l a t e r w o r k [ 7 ] t h e
same authors have evolved a n algorithm for
designing signal waveforms under a dynamic r a n g e
c o n s t r a i n t i.e., w i t h a l i m i t o n t h e r a t i o of t h e
maximum to t h e minimum a m p l i t u d e of t h e t r a n s m i t
signal.
Their approach resulted in a nonlinear
p r o g r a m m i n g f o r m u l a t i o n f o r t h e o p t i m i z a t i o n of
SIR.
If n o a priori information i s available
about t h e target scenario, all signals a r e
equally good o r bad since i t i s known t h a t n o
signal c a n h a v e good overall behaviour throughout
t h e delay-Doppler plane.
This conclusion follows
f r o m t h e c o n s t a n t volume p r o p e r t y of t h e signal
ambiguity function.
T h e range-Doppler ambiguity
f u n c t i o n s e r v e s as a n a t u r a l t o o l f o r s i g n a l
d e s i g n a g a i n s t 2-D c l u t t e r .
The ambiguity
f u n c t i o n h a s b e e n used f r e q u e n t l y in t h e p a s t t o
understand c l u t t e r rejection behaviour of
transmitted signal-receiver filter combinations
and t o design clutter-resistant signals and/or
filters.
T h e design of a n o p t i m u m mismatched f i l t e r
response for d e t e c t i n g a given signal in c l u t t e r
h a s b e e n c o n s i d e r e d b y S t u t t a n d S p a f f o r d [SI.
T h e y h a v e u s e d t h e t e c h n i q u e s of v a r i a t i o n a l
c a l c u l u s to minimize t h e o u t p u t c l u t t e r power
s u b j e c t to c e r t a i n c o n s t r a i n t s on t h e signal a n d
filter responses.
They obtained a linear
i n t e g r a l e q u a t i o n , t h e s o l u t i o n of w h i c h l e a d s t o
t h e optimum receiver filter, and they have
suggested a method for solving t h e equation.
Spafford's work [91, h a s e x t e n d e d t h i s t h e o r y a n d
outlined a n iterative procedure to obtain t h e
optimum signal-filter pair.
Recently Ziomek a n d Sibul [ l o ] have e x t e n d e d
t h e a p p r o a c h i n [61 t o f o r m u l a t e t h e p r o b l e m of
o p t i m a l signal design f o r t h e d e t e c t i o n of a
doubly-spread target against general
c l u t t e r / r e v e r b e r a t i o n . The problem h a s been posed
as a n u m b e r o f n o n l i n e a r o p t i m i z a t i o n p r o b l e m s .
T h e problem of point t a r g e t d e t e c t i o n against 2-D
c l u t t e r w o u l d b e a s p e c i a l c a s e of t h e i r
formulation.
However, solutions to t h e
optimization problem h a v e not been a t t e m p t e d by
t h e m , a n d no results a r e available.
O n e of t h e e a r l i e s t a t t e m p t s t o d e s i g n a
f i l t e r to o b t a i n o p t i m u m p e r f o r m a n c e a g a i n s t
c l u t t e r w a s by U r k o w i t z [2] w h o c o n s i d e r e d the
c l u t t e r a s t h e o n l y f o r m of i n t e r f e r e n c e .
he
showed t h a t t h e filter which maximizes t h e
Signal-to-Interference R a t i o (SIil) f o r c l u t t e r
which is stationary with respect to t h e t a r g e t
h a s a f r e q u e n c y r e s p o n s e which is t h e i n v e r s e of
t h e s p e c t r u m of t h e t r a n s m i t t e d signal.
Since
t h e f r e q u e n c y s p e c t r u m of a n y p h y s i c a l l y
realizable signal has to approach z e r o on both
s i d e s of t h e s p e c t r u m , t h e i d e a l c l u t t e r f i l t e r ,
as suggested by him, b e c o m e s unrealizable, s i n c e
i t w i l l h a v e t o h a y e i n f i n i t e ba,ndwidth.
The
d r a w b a c k s of t h e I n v e r s e f i l t e r a r e d i s c u s s e d
by Turin [31.
Since radar detection is more frequently
clutter-limited rather than noise-limited,
mitigation of c l u t t e r effects i s a strong
r e q u i r e m e n t in r a d a r s y s t e m design.
When t h e
c l u t t e r is one-dimensional, essentially spread
a l o n g t h e r a n g e a x i s o n l y ( e.g. g r o u n d c l u t t e r ) ,
t h e m e t h o d s of c o m b a t i n g i t s effect a r e f a i r l y
s t r a i g h t f o r w a r d a n d u s u a l l y c o n s i s t of u s i n g a
simple waveform with t h e receiver f i l t e r tuned in
t e r m s of i t s frequency c h a r a c t e r i s t i c s .
However,
when t h e c l u t t e r i s two-dimensional, d i s t r i b u t e d
along t h e range and velocity (Doppler)
dimensions, t h e clutter alleviation strategy
becomes more complex. A potent method of
c o m b a t i n g 2-D c l u t t e r e f f e c t s i s t o c h o o s e
transmitted signal with a large time-bandwidth
product (TB), t h e received signal being processed
through a m a t c h e d o r a m i s m a t c h e d filter.
T h e c o m b i n e d i n t e r f e r e n c e by n o i s e a n d
c l u t t e r ( s t a t i o n a r y ) w a s f i r s t c o n s i d e r e d by
Manasse [41.
H e showed t h a t when t h e c l u t t e r i s
uniformly distributed in range, t h e optimum
c l u t t e r r e j e c t i n g signal is a n impulse of
i n f i n i t e s i m a l d u r a t i o n , l e a d i n g t o a s i g n a l of
infinite bandwidth and peak-power.
When a n y
reasonable restriction is put on t h e signal
bandwidth, h e showed, t h a t t h e o p t i m u m signal is
o n e w h i c h h a s a f l a t s p e c t r u m o v e r t h e b a n d of
interest.
Ares [ 5 ] m a d e a physical approach t o t h e
g e n e r a l p r o b l e m of i n d e p e n d e n t l y o p t i m i z i n g t h e
s i g n a l a n d t h e f i l t e r a n d c o n s i d e r e d t h e case of
c l u t t e r extended in range, but confined in
Doppler t o a narrow s t r i p p a r a l l e l to t h e d e l a y
axis.
For specific c l u t t e r distributions, wellk n o w n l a r g e T B s i g n a l s , s u c h as t h e c h i r p s i g n a l ,
have been used along with their matched filters.
However, when the cluttyr distyibution is
arbitrary, such relatively simple signals d o
133
H e n c e f r o m eq (4),
n o t n e c e s s a r i l y p e r f o r m w e l l i n a l l cases. T h i s
f o l l o w s f r o m t h e l i m i t e d n u m b e r of c o n t r o l
p a r a m e t e r s f o r such signals, which r e s t r i c t s t h e
a b i l i t y of t h e w a v e f o r m t o a d a p t t o t h e c l u t t e r
distribution in t h e delay-Doppler plane. A m o r e
modern a p p r o a c h i s t o h a v e discrete-coded signals
which offer m o r e control p a r a m e t e r s for finer
c o n t r o l of t h e ambiguity surface.
C(0,O)= E:.CO(O,O)
(6)
Where, CO r e f e r s to t h e c l u t t e r o u t p u t f o r a unit
energy t r a n s m i t t e d signal.
A l s o , if t h e n o i s e s p e c t r a l d e n s i t y ( o n e s i d e d ) a t t h e i n p u t of t h e r e c e i v e r i s No, t h e n
t h e n o i s e a t t h e o u t p u t of t h e r e c e i v e r i s g i v e n
by
FORMULATION
N = NOES
Dense c l u t t e r may be described as a
distribution p ( ~ , v )o v e r t h e delay-Doppler plane.
A m e a s u r e of t h e d e t e c t a b i l i t y of t h e t a r g e t i s
t h e 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 (SIR), w h i c h i s
t h e ratio between t h e target signal and t h e
i n t e r f e r e n c e (clutter+noise), both observed at
t h e d e l a y a n d D o p p l e r c o r r e s p o n d i n g t o t h o s e of
t h e target.
(7)
T h e r e f o r e , t h e SIR a t t h e o u t p u t o f t h e
receiver is, from e q s (3)-(5),
The output of t h e receiver filter f o r a
given t r a n s m i t t e d signal i s given by t h e crossc o r r e l a t i o n function
m
(X12(7,v)(2 = ( S s I ( t ) s ; ( t - + ) e - j 2 ~ v t d t / 2 (1)
m
where sI(t) is t h e t r a n s m i t t e d signal and
s2(t) is t h e signal t o which t h e receiver is
matched. Under a m a t c h e d f i l t e r assumption,
Sl(t)
I
(8)
= s2(t) = s(t)
T h e d e n o m i n a t o r of e q (8) c o n s i s t s of t w o
terms.
The f i r s t t e r m (NO/Es) depends on t h e
e n e r g y of t h e s i g n a l a n d c a n b e d e c r e a s e d , f o r a
g i v e n r e c e i v e r n o i s e l e v e l , o n l y by i n c r e a s i n g
t h e e n e r g y of t h e t r a n s m i t t e d signal.
The second
t e r m is t h e contribution due t o t h e clutter and
i s i n d e p e n d e n t of t h e e n e r g y of t h e t r a n s m i t t e d
signal.
However, since t h e normalized ambiguity
f u n c t i o n i s s t r o n g l y d e p e n d e n t o n t h e s h a p e (i.e.
c o m p l e x e n v e l o p e ) of t h e t r a n s m i t t e d s i g n a l , i t s
weighted i n t e g r a l C O will also, in general, b e
d e p e n d e n t o n t h e w a v e s h a p e . Any r e d u c t i o n i n
t h i s f a c t o r c a n t h e r e f o r e be brought about only
by d e s i g n i n g a s u i t a b l e s i g n a l w a v e f o r m t o
minimize Co.
a n d t h e o u t p u t of t h e r e c e i v e r i s g i v e n by
t h e ambiguity function of t h e t r a n s m i t t e d signal
m
W i t h o u t loss of g e n e r a l i t y , t h e o r i g i n of
t h e (z,v) plane may b e l o c a t e d at t h e delay a n d
D o p p l e r c o r r e s p o n d i n g t o t h o s e of t h e t a r g e t .
T h e n t h e o u t p u t S of t h e r e c e i v e r f i l t e r a t (0,O)
d u e to t h e t a r g e t is
An e f f i c i e n t optimization procedure would b e
to e x p r e s s t h e signal in p a r a m e t r i c f o r m and t h e n
o p t i m i z e t h e p a r a m e t e r s minimizing C O ,f o r a given
c l u t t e r d i s t r i b u t i o n p(r,v). To f a c i l i t a t e t h i s
p r o c e d u r e , s i g n a l s of s p e c i f i e d f a m i l i e s m a y b e
c h o s e n . In t h i s p a p e r w e c o n s i d e r c o n s t a n t amplitude compact frequency coded pulse bursts
and f o r m u l a t e t h e problem of optimization against
arbitrary delay-Doppler clutter distributions.
Where E is t h e energy c o n t e n t of t h e signal
s(t). Also, the o u t p u t of t h e r e c e i v e r d u e t o
c l u t t e r at t h e t a r g e t delay and Doppler is given
by t h e i n t e g r a l
m
If lXOl2
f u n c t i o n , (i.e.
normalized to
property of t h e
Frequency coded rectangular compact pulse
bursts f o r m a f r e q u e n t l y used c l a s s of signals in
m o d e r n h i g h p e r f o r m a n c e r a d a r s b e c a u s e of i t s
o b v i o u s a d v a n t a g e s o v e r o t h e r c l a s s e s of c o d e d
bursts. An FSK burst of N subpulses have a TB of
N 2 as c o m p a r e d t o N f o r a PSK b u r s t of s a m e
is the normalized ambiguity
ambiguity function for t h e signal
unit energy), t h e n a well-known
ambiguity function [ I l l is
(5)
134
of t h e FSK burst as
length.
A l s o t h i s o f f e r s l a r g e n u m b e r of
optimization p a r a m e t e r s (N for a n N pulse long
b u r s t ) f o r b e t t e r c o n t r o l of t h e a m b i g u i t y
behavior.
S o m e l i m i t e d e f f o r t s h a v e b e e n m a d e by a
n u m b e r of o t h e r a u t h o r s [12-151, t o s t u d y t h e
p e r f o r m a n c e of FSK b u r s t s f o r c l u t t e r r e j e c t i o n .
The a p p r o a c h in m o s t cases, however, has been t o
choose a certain coding structure and then study
t h e p e r f o r m a n c e of t h e c o d e i n p r e s e n c e of
interference.
Such a n approach severely limits
t h e a d a p t a b i l i t y of t h e r e s u l t a n t w a v e f o r m t o
c a t e r t o different t a r g e t scenarios.
An e f f o r t
h a s b e e n m a d e , in t h i s p a p e r , t o f o r m u l a t e t h e
p e r f o r m a n c e c r i t e r i o n in t e r m s of t h e FSK
p a r a m e t e r s ( n a m e l y t h e f r e q u e n c y of s u b p u l s e s )
and to o p t i m i z e t h e p e r f o r m a n c e with r e s p e c t to
t h e p a r a m e t e r s of t h e signal.
N
+Ee
k = I +2
where
U
=TIN
xi = v
A s d i s c u s s e d p r e v i o u s l y , t h e p r o b l e m of
maximizing SIR r e d u c e s to t h e minimization of t h e
n o r m a l i z e d c l u t t e r i n t e g r a l , C , i n e q (4).
T h e e v a l u a t i o n of CO f o r a n FSK g u r s t i s h i g h l y
computation intensive and t o effectively use t h e
f o r m u l a t i o n t h e e v a l u a t i o n of t h e c l u t t e r
integral must b e simplified.
The piecewise
invariance of t h e p a r a m e t e r s of t h e FSK burst c a n
b e used to effect this simplification.
t
fi
y k = V t fk
+ f ,. - I
+
and
fk-I-l
Eq ( 1 1 ) c a n b e u s e d t o o b t a i n a f a i r l y
t r a c t a b l e e x p r e s s i o n f o r t h e c l u t t e r i n t e g r a l C,
which c a n b e used as t h e p e r f o r m a n c e c r i t e r i o n
a n d c a n b e e v a l u a t e d f o r a g i v e n d i s t r i b u t i o n of
p ( z , v ) a n d o p t i m i z e d w . r . t fi's t o o b t a i n t h e
o p t i m u m signal f o r t h a t case.
OPTIMIZATION CRITERION
However, f r o m t h e approach i t is c l e a r t h a t
t h i s m o d i f i e d f o r m u l a t i o n c a n n o t t a k e c a r e of
cases where 1 < 0. This c a n be handled by using
t h e s y m m e t r y property of t h e ambiguity function
and defining a modified distribution function
T h e c o m p l e x e n v e l o p e of t h e c o n s t a n t a m p l i t u d e frequency coded c o m p a c t pulse bursts is
described as
N
k d ( - c , v ) = p ( r , v ) + p(-r,-v) for
(9)
= o
z>
0
(12)
e I sewher e
RESULTS
where f .
i = { I ,...,NI r e f e r s t o t h e f r e q u e n c y
o f t h e 'ith s u b p u l s e a n d g i ( t ) i s t h e g a t e
f u n c t i o n g i v e n i n t e r m s of a u n i t s t e p f u n c t i o n
u(t) as
T h e f o r m u l a t i o n of t h e p r e v i o u s s e c t i o n
p r o v i d e s us w i t h a t o o l f o r d e s i g n i n g s i g n a l s of
s p e c i f i e d f a m i l i e s , t h e c o m p a c t FSK b u r s t s i n
t h i s case, f o r o p t i m a l r e j e c t i o n of a r b i t r a r y 2-D
clutter distributions.
In t h i s s e c t i o n t h e
u s e f u l n e s s of t h e f o r m u l a t i o n i s d e m o n s t r a t e d by
application against assumed representative
clutter distributions.
The effectiveness can
e a s i l y b e judged by e x a m i n i n g t h e a m b i g u i t y
behaviour.
Let
f = ( I + F)T
(10)
W e p r e s e n t s o m e r e p r e s e n t a t i v e r e s u l t s in
Fig. 1-4 w h i c h s h o w t h e a m b i g u i t y f u n c t i o n f o r
t h e o p t i m u m FSK b u r s t s f o r d i f f e r e n t b u r s t
l e n g t h s a n d c l u t t e r d i s t r i b u t i o n s . F o r ease i n
v i s u a l i z a t i o n t h e p l o t s s h o w o n l y t h e r e g i o n of
interest.
T h e a b s e n c e of a n y s i g n i f i c a n t
s i d e l o b e s in t h e r e g i o n of t h e d e l a y - D o p p l e r
p l a n e o c c u p i e d by t h e c l u t t e r i s q u i t e e v i d e n t
a n d c l e a r l y d e m o n s t r a t e t h e e f f e c t i v e n e s s of t h e
m e t hod.
i.e. I i s t h e i n t e g e r p a r t of a n y s h i f t i n d e l a y
and F t h e f r a c t i o n a l part. Therefore,
I = 0,1,2,..,(N-I)
and
0
<
F
<
1
S u b s t i t u t i n g t h e v a l u e of s ( t ) as g i v e n b y
e q (9) and using e q (IO) w e g e t , a f t e r
s i m p l i f i c a t i o n of e q (21, t h e a m b i g u i t y f u n c t i o n
135
The clutter distribution for t h e optimum
s i g n a l , s h o w n i n F i g . 1 a n d 2, a r e s i m i l a r i n
d e s c r i p t i o n in t h e normalized signal d u r a t i o n
f r a m e and t h e i n c r e a s e in t h e b u r s t l e n g t h m u s t
lead to a c o m m e n s u r a t e improvement. This c a n b e
e a s i l y s e e n in t h e i r a m b i g u i t y p l o t s . T h i s i s
a l s o c o n f i r m e d by t h e i m p r o v e m e n t i n S I R ,
normalized with reference t o a n uncoded burst,
which has a value of
10.6 f o r t h e 5-pulse burst
and irrrproves to 45.5 for t h e 13-pulse case.
I X Ir, vl I
F i g . 3 a n d 4 s h o w t h e a m b i g u i t y f u n c t i o n of
t h e optimum 13-pulse burst against different
c l u t t e r d i s t r i b u t i o n s as m e n t i o n e d .
The
distribution h a v e been so chosen so as to c o v e r
different delay-Doppler c l u t t e r occupations, and
t h u s i n d i c a t e s t h e e f f e c t i v e n e s s of t h e s i g n a l
f o r m a t a n d t h e o p t i m i z a t i o n algorithm to a d a p t to
any general c l u t t e r distribution. The workability
of t h e m e t h o d c a n b e easily judged by t h e
ambiguity behavior obtained.
2
I x Ir,V I I
t
Fig.1
-
2
t
Fig.3
The ambiguity function of t h e o p t i m a l 5p u l s e FSK b u r s t , a g a i n s t u n i f o r m c l u t t e r
d i s t r i b u t e d in a r e c t a n g u l a r region given
by
1 < Z: < 5 a n d - I / N T < v < 1/NT.
-
The ambiguity function of t h e o p t i m a l 13p u l s e FSK b u r s t , a g a i n s t u n i f o r m c l u t t e r
d i s t r i b u t e d in a r e c t a n g u l a r region given
by
1 < Z: < 5 and - I / N T < v < I/NT.
I X lr, vl I
2
t
V’
Fig.2
-
_
I
I
The ambiguity function of t h e o p t i m a l 13p u l s e FSK b u r s t , a g a i n s t u n i f o r m c l u t t e r
distributed in a r e c t a n g u l a r region given
by
1 < t < 1 3 aqd - I / N T < v < l/NT.
Fig.4
136
-
The ambiguity function of t h e o p t i m a l 13p u l s e FSK b u r s t , a g a i n s t u n i f o r m c l u t t e r
d i s t r i b u t e d in a r e c t a n g u l a r region given
by
0 < T < 1 and -2/NT < v < 2/NT.
CONCLUSIONS
[61
D.F. D e l o n g a n d E.M. H o f s t e t t e r , "On t h e
d e s i g n of O p t i m u m R a d a r W a v e f o r m s f o r
C 1u t t e r R e j ec t i on", IE E E T r a ns I n f o r m a t i o n
Theory, IT-I3 (July 1967), pp. 454-463.
.
In summary, t h e problem of o p t i m u m c l u t t e r
r e j e c t i o n i n a g i v e n r e g i o n of t h e d e l a y - D o p p l e r
plane has been solved.
The approach is based on
optimizing frequency-coded pulse train
t r a n s m i t t e d signals, keeping t h e r e c e i v e r f i l t e r
matched t o this signal all the time.
This
reduces t h e dimensionality of t h e problem leading
t o l a r g e saving in cornputational e f f o r t s .
This
factor together with t h e simplification achieved
using t h e p i e c e w i s e c o n s t a n t p r o p e r t y of t h e
signal holds t h e potential for adaptive signal
optimization in time-varying c l u t t e r situations.
The a c t u a l design of t h e signal depends t o a
large extent on t h e numerical optimization
a l g o r i t h m . U s e of a f a s t e r c o n v e r g i n g a l g o r i t h m
will b e a n added a d v a n t a g e and will c o n t r i b u t e
much t o t h e a p p l i c a t i o n in t i m e varying c l u t t e r
environment.
i i o w e v e r , t h e t y p e s of f u n c t i o n s
e n c o u n t e r e d h e r e a r e highly non-linear and h a v e a
l a r g e n o . of l o c a l m i n i m a .
T h e d e s i g n of a n
efficient optimization algorithm, for such
f u n c t i o n s , i s q u i t e d i f f i c u l t a n d t h e a u t h o r s of
this paper a r e currently working towards
developing more efficient algorithms for t h e
c l a s s of f u n c t i o n s e n c o u n t e r e d i n t h i s t y p e s of
signal design problems.
[7]
D.F. Delong and E.M. H o f s t e t t e r , "The design
of C l u t t e r - R e s i s t a n t R a d a r W a v e f o r m s w i t h
limited Dynamic Range", IEEE Trans.
I n f o r m a t i o n T h e o r y , I T - I 5 ( M a y 1 9 6 9 ) , pp.
376-385.
[8]
C . A . S t u t t a n d L . J . S p a f f o r d , "A ' b e s t '
Mismatched F i l t e r Response for Radar C l u t t e r
D i s c r i m i n a t ion" , I E E E T r a n s . I n f o r m a t i o n
Theory, IT-I4 (Mar. 19681, pp. 280-287.
191
L.J. S p a f f o r d , " O p t i m u m R a d a r S i g n a l
Processing in Clutter", IEEE Trans.
I n f o r m a t i o n T h e o r y , I T - I 4 ( S e p . 1 9 6 8 ) , pp.
734-743.
[IO]
L.J. Ziomek and L.H. Sibul, "Maximization of
Signal to I n t e r f e r e n c e R a t i o for a Doublyspread target: Problems in nonlinear
Programming", Signal Processing, 5 (July
19831, pp, 355-368.
[ I l l C.E. Cook and Ni. Bernfeld, Radar Signals: An
Introduction to Theory and Application. New
York: A c a d e m i c Press, 1967.
[I21 R.M. M e r s e r e a u a n d T.S. S e a y , " M u l t i p l e
Access Frequency h o p p i n g P a t t e r n s with Low
Ambiguity", IEEE Trans. A e r o s p a c e and
E l e c t r o n i c S y s t e m s , AES-17 ( J u l y 1 9 8 1 ) , pp.
57 1-578.
REFERENCES
A.W. Rihaczek, Principles of High-Resolution
Radar. New York: McGraw Hill, 1969.
H. Urkowitz, "Filters f o r d e t e c t i o n of small
i i a d a r s i g n a l s in C l u t t e r " , J o u r n a l Appl.
Phys., Vol. 24 (Aug. I953), pp. 1024-1031.
[ 131 E.L. Titlebaum, "Time-Frequency Hop Signals
P a r t I: C o d i n g b a s e d u p o n t h e T h e o r y o f
Linear Congruences", IEEE Trans. Aerospace
and Electronic Systems, AES-17 (July 1981),
pp. 490-493.
G.L. T u r i n , " A n I n t r o d u c t i o n t o & l a t c h e d
f i l t e r s " , I R E T r a n s . I n f o r m a t i o n T h e o r y , IT6 ( J u n e 1960), pp. 311-329.
[I41 E.L. T i t l e b a u m a n d L.H. S i b u l , " T i m e Frequency Hop Signals P a r t 11: Coding based
upon Q u a d r a t i c C o n g r u e n c e s " , IEEE T r a n s .
A e r o s p a c e a n d E l e c t r o n i c S y s t e m s , AES-17
(July 1981), pp. 494-500.
R. M a n a s s e , " T h e u s e o f P u l s e C o d i n g t o
D i s c r i m i n a t e a g a i n s t c l u t t e r " , MIT L i n c o l n
Lab., Group Rept. 312-12(Rev.l), J u n e 1961.
M. Ares, "Radar Waveforms f o r Suppression of
[I51 C . N . O l i g b o h a n d M.H. A c k r o y d , " L i n e a r
frequency coded sequences", IEE Proc. P a r t G, Vol. 127 (Aug. 1980), pp. 191-198.
Extended C l u t t e r " , IEEE Trans. Aerospace and
E l e c t r o n i c S y s t e m s , A E S - 3 ( J a n . 1 9 6 7 ) , pp.
138-1 4 1.
137