505
1 INTRODUCTION
Nowadaysthenecessityofmarineradardesignwith
usingpulse compressionwaveformsexists, allowing
to reduce significantly the peak power of radiation
and thus to improve the working conditions of
seafarers and electromagnetic compatibility with
other vessel’s devices. There are two main tasks of
radar: detecting a target and det
ermining its range.
Fairlyearlyrangehas expendedtoinclude direction
to the target and radial velocity between the radar
and the target. Application of pulse compression
waveforms gives the possibility remove many of
constraints,providingexecutivenewtasks:increasing
therangeofradaroperationunderlimitedtransmitter
peak power; improving det
ectability small‐size
targets on the background sea surface by mean
increasing Doppler selectivity. Extraction of signals
frominterferingreflectionsisveryimportantforsuch
kindofradarwaveformandfilterdesign.Thequality
of suchextraction significantlydepends from range‐
velocity distribution of interfering reflections [1,2,3].
Theproblemofthemisma
tchedfilterandwaveform
designthatmaximizesthesignal‐to‐noise‐plus‐clutter
ratioatthereceiverfilteroutputhasbeenformulated
and addressed in [4], [5], [6], [7], [8], [9], [10], [11].
Mismatched filtering may causes degradations in
signal‐to‐white noise ratio. In [10] the method of a
filt
er optimization which maximizes the signal‐to‐
noiseratiounderadditionalquadraticconstraintswas
developed.In [12]the methodsof jointoptimization
signalandfilterforinterferingreflectionssuppression
under additional constraints on range resolving
performance, signal‐to‐noise ratio loses and given
amplitude modulation of signal with different
limit
ationsonthememoryandthewidthofthepass
bandofthefilterweredeveloped.
Thesignalsandfiltersdesigntechniquepresented
in [13] is extension of methods [12] to the case of
waveforms and filters sets with group‐
complementary properties, which are optimized
simultaneously. In this work we consider discrete
signalsandtheoptimizingdiscretefilt
ersforcases
electronically scanning antenna and mechanical
scanning rotating antenna. The method of filter
optimization,considering Doppler shift ofsignal for
bothcases,issuggested.
We consider discrete signals and the optimizing
discretefilterswithcomplexenvelopes[10,11]:
Sets of Waveform and Mismatched Filter Pairs for
Clutter Suppression in Marine Radar Application
V.Koshevyy&V.Popova
NationalUniversity“OdessaMaritimeAcademy”,Odessa,Ukraine
ABSTRACT: Setsof waveformand mismatched filter pairs are used.Onthe contrary with Golays matched
waveformfilterpairthemismatchedwaveformfilterpairdoesexistforallN(numberpulsesinwaveform).
UsingcorrespondingshapesoffiltergoodDopplertolerancemaybeprovided.Thispropertytogetherwitha
goodrangeside‐lobslevelsuppressionma
kesit’sattractableforuseinmarineradar.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 3
September 2017
DOI:10.12716/1001.11.03.17
506
P
p
N
n
pn
nTTptuStS
1
1
0
0
))1(()(
(1)
P
p
M
m
fpm
TmqTptuWtW
1
1
0
0
))1((()(
(2)
where
1, (m-1)Z
0, of t
tmcZ
ut mZ
for other values
;
0
Т –iselementarypulseu(t)duration;T–repetition
period of signals;
S
f
W
F
q
F
– is a parameter which
characterizes the pass‐band of the filter
W
F
in
respecttothespectralwidthofthesignal
S
F ;
pn
S
;
pm
W
– are complex amplitudes and weighting
coefficientsofwaveformS
pandmismatchedfilterWp
pairs.
P‐numberofwaveformandmismatchedfilterpairs
inset.
Considering optimization reduces to a choice of
thesignalS(t)andofthefilterW(t)whichmaximizes
theratio[9]:
dfdI)f,(X)f,(dt)t(W
0,0)(XA
2
SW0
2
2
SW
2
(3)
where:
*2ft
(, ) () ( ) dt
i
SW
Xf WtSte
(4)
istheCrossambiguityfunctionofsignal(1)andfilter
(2);
)f(f),()f,(
;
)f,(
istherange–
velocity distribution of the interfering reflections;
)f(
is delta‐function; A,
0
are para‐meters which
characterizes the reflecting properties ofthe target
and the interfering reflections;
,
are the
parameters for controlling signal‐to‐noise ratio loses
and range resolving properties correspond‐dingly.
More over, for the considered below problem of
multiplejointwaveformsandfiltersoptimization,by
a proper parameter selection, the ideal correlation
properties(norangeside‐lobes)forthesumofcross‐
correlation functions
(complementary property) may
beenforced.
2 FILTEROPTIMIZATION
Weconsiderthecasewhen,
00 0
,,21
ff
MNsoNTNT Mq T T if q T
(5)
In the case (5) expression (4) can be written as
follows(see[13],[14],[15],[16])
21
1
,,
pp
p
ifpT
SW S W
p
Xf X fe
(6)
Formula(6)simplifiesthetaskofmaximizing(3).
Sowe havegotexpression (3)for thesets ofsignals
andfiltersatthesameformaswehaveforthesingle
signaland filterand we canuse iteration processof
joint optimization signal and filter which was
described
in[12]. Theiterationprocess isfollows: at
first for the given sets of signals we are getting
optimal setsof filters,then for thissets of filterswe
are getting optimal sets of signals and so on. The
convergence of the iteration process was proved in
[17].
Ratio(3)for
signal(1)andfilter(2)withusing(6)
mayberewriteinthematrixform
2
2
0
*
*
AW S
WvI DW
(7)
whereS, W arethevectors ofcomplexamplitudeof
signals set and filters set; I – identity matrix;
D
is
correlation matrix of interfering reflections with
range‐velocitydistribution
)f,(
.
Atthefirststepofiterationprocesswechooseany
initial set of signals vector
0
S
and find for it
optimum vector sets of filters
0
W
according to
expression[12]
0
1
00
0
S
WvID S
(8)
Atthenextstepforthe
)(
W
0
wefind
)(
S
1
[12]
0
1
10
0
S
SvID W
(9)
Forthegiven amplitudeofthe signalatthis step
we choose only phases of the signal according to
algorithmwhich wasdescribed in [12].In this work
onlyfirststepofoptimizationwillbeconsidered.
IfconsiderthecaseM=Nand
f
q
=1in(7),(8),(9)
thendimensionofsetofsignalsvectoranddimension
of set of filters vector are the same and equal PN,
dimensionofmatrixisPN
PN.
3 NUMERICALRESULTS
We consider the case only filter optimization
according to (8). As an example we calculate
)(
W
0
from (8) for the case N=M=3, P=2,
(f))f,(
[14]vector
t)0(
S
[1‐1‐1111]andcalculatedfilter
t)0(
W
[1‐0,5‐0,51‐0,50,5].
So, we have set of signals
111
0
1
t)(
S
,
111
)0(
2
t
S
and set of filters
50501
0
1
,,W
t)(
,
55001
0
2
,,W
t)(
.
Crosscorrelationfunctions
507
11
WS
R
[‐0,5;0;2;‐0,5;‐1]
22
WS
R
=[0,5;0;1;0,5;1].
Aswecanseethispairsofsignalsandfiltersare
complementary (the sum of cross correlation
functions has zero side‐lobes). This example is
interesting because the classical Golay
complementary pair for N=3 doesn’t exist, but for
mismatchedcaseitdoes[13,14].
In this example N=M=3, P=2 signal‐to‐noise ratio
loses [5]
=0,5. But if we increase memory of filter
N=3,M=5weget
=0,7[14].Sosignal‐to‐noiseratio
losesaredecreased.Forthiscasewehave
t)(
S
0
1
=[1‐1‐1];
t)(
S
0
2
=[111];
t)(
W
0
1
=[‐0,5;1,0;‐0,9;‐0,4;0,2];
t
W
)0(
2
=[‐0,5;1,0;0,1;0,8;‐0,2]
Crosscorrelationfunctions
11
WS
k
R
=[0;0,2;‐0,6;‐0,7;2,3;‐0,6;‐0,5;0,5;0];
22
WS
k
R
=[0;‐0,2;0,6;0,7;1,9;0,6;0,5;‐0,5;0].
AnotherexampleN=5,M=5,P=2[11]
S
1=[1111‐1];S2=[111‐11]
W
1=[1416‐4];W2=[14‐1‐64].
Crosscorrelationfunctions
RSW
1=[‐4;2;3;7;16;0;4;‐3;‐1];
RSW
2=[4;‐2;‐3;‐7;14;0;‐4;3;1];
=0,64.
For increased filter memory N=5, M=7 we have
[11]
W
1=[‐1,5;3,5;2;2;6;‐6,5;‐1,5];
W
2=[‐1,5;3,5;5;1;‐6;6,5;1,5];
=0,79.
So,signal‐to‐noiseratiolosesarealsodecrease.
Considerafewexampleselse:N=6,M=6,P=2
S
1=[111111];S2=[11111‐1];
W
1=[‐1;7;‐1;5;11;‐7];W2=[‐1;7;1;‐5;‐11;7];
=0,39.
N=7,M=7,P=2
S
1=[11‐1‐11‐11];S2=[111‐1‐11‐1];
W
1=[1;0,75;1;1;2,5;0,75;0,75];
W
2=[1;0,75;1;‐1;‐1;‐0,75;‐0,75];
=0,21,
butforincreasedmemoryN=7,M=9,weget
=0,24.
S
1=[1111111];S2=[111‐1‐11‐1];
W
1=[0,11;‐0,11;0,11;‐0,14;0,14;‐0,11;0,14];
W
2=[0,11;0,11;0,11;‐0,14;‐0,14;0,11;‐0,14];
=0,33.
N=5,M=5,P=6
S
1=[111‐11];S2=[11‐111];S3=[1‐1111];
S
4=[11111];S5=[1111‐1];S6=[11111];W1=S1;
W
2=S2;W3=S3;W4=S4;W5=S5;W6=‐S6;
=0,44
N=5,M=5,P=4
S
1=[111‐11];S2=[1‐111‐1];S3=[1‐1‐1‐11];
S
4=[1‐1‐1‐1‐1];W1=S1;W2=S2;W3=S3;W4=S4.
N=6,M=6,P=4
S
1=[1‐111‐11];S2=[1‐1111‐1];S3=[111‐1‐1‐1];
S
4=[111‐111];
W
1=S1;W2=S2;W3=S3;W4=S4.
N=7,M=7,P=8[13]
S
1=[111‐1‐11‐1]; S2,S3,S4,S5,S6,S7– are cyclic shifts of
signalS
1;S8=[1111111];
Wi=Si(i=1,2,…P).
N=11,M=11,P=4
S
1=[111‐111‐11‐1‐11];S2=[111‐11‐1‐1‐1‐111];
S
3=[111‐1111‐111‐1];S4=[111‐11‐1111‐1‐1];
Wi=Si(I=1,2,…P);
=1.
Cross correlation functions are represented on
fig.1‐5.
Figure1.forS1Figure2.forS2
Figure3.forS3 Figure4forS4
Figure5.forthesumofthecorrelationfunctions
Inlastfourexampleswe getthe maximumvalue
ofp=1,whichcorrespondstomatchedfilters(without
signal‐to‐noiseratioloses).
For considered cases
)f()f,(
matrix
D
,
S*W
in(7)isformedbythenextway:
1
1
**
1
1
... ,...,
N
k
kpk
kN
pk
S
DSS
S
(10)
508
were
pk
S ‐vector of signal with number p in set,
whichisshiftedonkposition:
0
2
0
.
,0
pk
p
p
Sifk
S
S
3
2
1
,0
.
0
Np
Np
pk
Np
S
S
Sifk
S
1
****
121 2
1
...
*...
PPPP
P
P
S
WS W W W W S
S
S
(11)
Considered method of optimization set of filters
for given set of waveforms, as a first step of joint
waveform‐filter sets optimization, suggested in [13],
givesthepossibilitytogetthesolutionsforarbitrary
discretesignalsandarbitrarynumberofsignalsinset.
The minimum loses in signal‐to‐noise ratio are
provided for given sets of signals. For existing non
trivial solutions only no singularity of matrix (10)
should be provided. For example in the case of
identicalsignalsinsetthematrix(10)issingularand
no trivialsolutions forset offilters, whichprovided
complementary property, doesn’t
exist. Increasing
memory of filters leads to decreasing of signal‐to‐
noise ratio loses. It is very important to note that
gettingsolutionsallowgettingnewsetsofsignalsand
filters without any additional calculations. Really, if
we have set of two signals with equal N and set of
two
corresponding mismatched filters, which are
complementary, we may create new complementary
sets of two signals and two filters, but with twice
lengthsbythenextway[11]:
S
1(2N)=[S1(N);S2(N)];S2(2N)=[S1(N);‐S2(N)];
W
1(2N)=[W1(N);W2(N)];W2(2N)=[W1(N);‐W2(N)].
ThevalueofpisreservedasforthelengthN.
We can demonstrate it for example which are
consideredaboveforN=3,M=3,P=2
S
1(6)=[1111‐1‐1];S2(6)=[111‐111];
W
1(6)=[2‐112‐1‐1];W2(6)=[2‐11‐211]; =0,5.
If we have set of two signals with different N
(N
1;N2) and set of two corresponding mismatched
filters,whicharecomplementary,wemaycreatenew
complementary sets of four signals and four filters
withlengthsN
1+N2bythenextway[11]:
S
1(N1+N2)=[S1(N1);S1(N2)];
W
1(N1+N2)=[W1(N1);W1(N2)];
S
2(N1+N2)=[S1(N1);‐S1(N2)];
W
2(N1+N2)=[W1(N1);‐W1(N2)];
S
3(N1+N2)=[S2(N1);S2(N2)];
W
3(N1+N2)=[W2(N1);W2(N2)];
S
4(N1+N2)=[S2(N1);‐S2(N2)];
W
4(N1+N2)=[W2(N1);‐W2(N2)].
We can demonstrate it for examples which were
calculated above for N=M=6=N
1=6, P=2 and
N=M=3=N
2,P=2.
S
1(9)=[11111‐11‐1‐1];
W
1(9)=[‐1;7;‐1;5;11;‐7;1;‐0,5;‐0,5]
S
2(9)=[11111‐1‐111];
W
2(9)=[‐1;7;‐1;5;11;‐7;‐1;0,5;0,5];
S
3(9)=[1111‐11111];
W
3(9)=[‐1;7;1;‐5;‐11;7;1;‐0,5;0,5];
S
4(9)=[1111‐11‐1‐1‐1];
W
4(9)=[‐1;7;1;‐5;‐11;7;‐1;0,5;‐0,5]; =0,37
This suggested approach to construction new
complementary sets of filters and signals is an
extension of known approach for the mismatched
case.
Resembling task is considered in [17] where the
filterdesigntechniquewasworkedoutforgivenset
of signalsand considered only signal‐to‐noiseloses
and complementary properties and doesn’t consi‐
dered another type of interference. The task of
maximizingsignal‐to‐noiseratiounderconstraintson
fulfilling complementary properties was solved.
Number of equationswhich must be solved for
thatisequaltoPN+2N‐1.Thisismuchmorethanwe
haveinourcase.
Althoughoursapproachesarealso
maximizing signal‐to‐noise ratio and besides
suppressinganothertypesofinterference.
In[18]alsoisconsideringthetaskofmaximization
signal‐to‐noise ratio under constraints on
complimentaryproperty, butonlyfor twosignals in
set.Approachissimilarto[17].
Beside in [17]
a few questions of using some
properties of shifting m‐sequences for designing the
array of Golay’s sequences are considered, but
without any background of it. In [13] the back
groundingofthispropertyonthebaseofanalyzesthe
fundamental properties of Cross ambiguity function
andextensionthispropertyon
anotherwideclassof
signalshavebeendone.
Allsignalsandfiltersconsideredmaybeusedfor
as group‐complementary sets of waveforms and
filters for the case of antenna with electronically
scanning.
In the case of rotating antenna the group‐
complementary properties of sets of waveform and
filters (figure
a) are destroyed due to amplitude
modulation. So the construction waveforms and
filtersmayberealiseinotherway(figureb),which
guarantees zero side‐lobes level in nearby peak of
correlation function zone independently of rotating
antennaeffect.
Figurea Diagram of signals for the case of an electronic
scanningantenna
509
Figureb. Diagram of signals for the case of the rotating
antenna
WedemonstratesitontheexampleofN=15onthe
baseofwaveformsandfilterssetforN
1=5,M1=5,P=2,
whichwewereconsideredabove:S=[1111‐100000111‐
11];W=[1;4;1;6;‐4;00000;1;4;‐1;‐6;4]. Cross correlation
functions for these signal and filter (filter tuned on
different Doppler frequencies L=0; L=1; L=2) are
shown on Fig. 6. On this Fig. we can see zero side‐
lobeslevelinthenearbyzoneofthe
crosscorrelation
function central peak.
= 0,64. L=
0
4
F
wNT
(Fw –
frequencyoffiltertuned).
L=0L=1
L=2
Figure6.WhentheDopplershiftofsignall
1=0forN=15
We alsomay considerthe signal andfilter N=77,
whichareconstructedonthebaseofwaveformsand
filterssetforN
1=11;M1=11;P=4.
S=[111‐111‐11‐1‐1100000000000111‐11‐1‐1‐1‐
11100000000000111‐1111‐111‐100000000000111‐11‐
1111‐1‐1];W=S.Crosscorrelationfunctionsareshown
onFig.7(L=0;L=1;L=2).
= 1.
L=0L=1
L=2
Figure7WhentheDopplershiftofsignall
1=0forN=77
ThetoleranceforDopplershiftofsignalshouldbe
provided for both cases of antenna scanning
(electronicallyandmechanicallyrotating)bymeansof
special filter counting [16]. Results of such counting
fortheDopplershiftofsignall
1=1( )are
shown for last two examples on Fig.8 and Fig.9
correspondently.TolerancetoDopplershiftofsignal
canbeseenfromcomparisonofthecrosssectionsl
1=0,
L=0(picturesonFig.6,Fig.7)andl
1=1,L=1(pictureson
Fig.8,Fig.9).
L=0L=1
L=2
Figure8WhentheDopplershiftofsignall
1=1forN=15
510
L=0L=1
L=2
Figure9.WhentheDopplershiftofsignall
1=1forN=77
4 CONCLUSION
This paper demonstrated the efficiency of filter
synthesis under additional constraints with group‐
complementaryproperties.Itwasshownthatsignal‐
to‐noise ratio loses is decrease with increasing
memoryofoptimizingfiltersinset.
Approaches for the construction of new sets of
signalsandfiltersonthebaseof
knownsetsofsignals
and filters with complementary properties were
suggestedfordifferentkindofantennascanning.
The counting of filters, which provides the
tolerance for Doppler shifts of signal are also
suggested.
REFERENCES
[1]Marineradar.EditedbyV.I.Vinokurov.(InRussian)‐L.
Sudoctroenie,1986.
[2]RadarHandbook.Editor‐In‐ChiefM.I.Skolnik.McGraw‐
HillBookCompany,1970.
[3]N. Levanon, E. Mozeson. Radar Signals. –Wiley‐
Interscience,2004.
[4]W.D.Rummler. ’Clutter Suppression by Complex
Weighting of Coherent Pulse Trains.”IEEE Trans. on
AES.Vol.
AES‐2.No.6.pp.810‐818.Nov.1966.
[5]W.D.Rummler. “A Technique for Improvingthe
Clutter Performance of Coherent Pulse Train
Signals.”IEEETrans.onAES.vol.AES‐3.No.6.pp.898‐
906.Nov.1967.
[6]D.F.Delong.Jr.andE.M.Hofstetter.‘OntheDesignof
OptimumRadarWaveformsforClutter
rejection.”IEEE
Trans.onIT.Vol.IT‐13.No.3.pp.454‐463.July1967.
[7]L.J. Spafford. ”Optimum Radar Signal Processing in
Clutter.”IEEETrans.onIT.Vol.I14.No.5.pp.734‐743.
Sept.1968.
[8]C.A.StattandL.J.Spafford.“A“Best”MismatchedFilter
Response for Radar Clutter Discrimination .”
IEEE
Trans.onIT.vol.IT‐14.No.2.pp.280‐287.Mar.1968.
[9]V.T. Dolgochub and M.B. Sverdlik. “ Generalized v‐
filters.” Radio Engeneering and ElectronicPhysics.
Vol.15.pp.147‐150.January1970.
[10]Y.I. Abramovich and Sverdlik. “Synthesis of a filter
which maximizes the signsl‐to‐noise
ratio under
additionalquadraticconstraints.’RadioEngineeringand
ElectronicPhysics.vol.15.pp.1977‐1984.Nov.1970.
[11]P. Stoica,J. Li,M. Xue. “Transmit Codes and Receive
Filters for Pulse Compression Radar Systems.” IEEE
SignalProcessingMag. vol. 25.no. 6. pp.815‐845. Nov.
2008.
[12]V.M. Koshevyy, M.B Sverdlik
.’Synthesis of Signal –
Filer pair under additional constraints’. Radio
EngineeringandElectronics.vol21.no.6.pp.1227‐1234.
June1976.
[13]V.M. Koshevyy. “Synthesis of Waveform‐Filter pairs
under Additional Constraints with Group‐
Complementary Properties” IEEE, Radar Conference
2015,May2015,Arlington,VA(USA),pp.0616‐0621.
[14]V.M. Koshevyy. “Efficiency
of Filter Synthesis under
Additional Constaints with Group‐Complementary
Properties” IEEE, 2016 International Conference
“Radioelectronics & InfoCommunications” (UkrMiCo),
September11‐16,2016,Kiev,Ukraine,pp.978‐982
[15]A.W. Rihachek.Principlesof High Resolution
Radar.NewYork:McGraw‐Hill.1969.pp.153.
[16]V.M. Koshevyy. Synthesisofmultifrequencycodes
under additional constraints Radio Engineering and
Electronics.Vol29.N11.1984.
[17]V.M.Koshevyy,M.B.Sverdlik.Jointptimizationof
SignalandFilterintheProblemsofExtractionofSignals
from Interfering Reflections. Radio Engineering and
ElectronicPhysics.Vol.20.N10pp.48‐55.
[18]Glenn Weathers, Edvard M. Holiday.Group‐
ComplementaryArrayCoding for Radar Clutter
Rejection. IEEE Trans. On Aerospace and Electronic
System.Vol.AES‐19.N3May1983.
[19]Bi,J.Rohling,H.ComplementaryBinaryCodeDesign
basedonMismatchedFilter.IEEETrans.onAerospace
andElectronicSystem,Vol.48,N2,January2012.
[20]V.M.Koshevyy,M.B.
Sverdlik.“Ononepropertyofan
optimum signal” Radio Engeneering and Electronic
Physics.1977,No.10,Vol.22,pp.73‐75.
