Transmission code optimization method for incoherent scatter radar

0.15 MB·English

by  Juha Vierinen

ManuscriptpreparedforAnn. Geophys.
withversion1.3oftheLATEXclasscopernicus.cls.
Date: 3February2008
Transmission code optimization method for incoherent scatter radar
8
0
0 JuhaVierinen1,MarkkuS.Lehtinen1,MikkoOrispa¨a¨1,andIlkkaI.Virtanen2
2
1Sodankyla¨ geophysicalobservatory

n 2UniversityofOulu
a
J

9
Abstract. Whenstatisticalinversionofalagprofileisused Wehavepreviouslystudiedthetargetestimationvariance

to determine an incoherent scatter target, the posterior vari- ofacoherenttargetwherethetargetbackscatterisassumed
]
n ance of the estimated target can be used to determine how to stay constant while the transmission travels through the
a
wellacertainsetoftransmissioncodesperform.Inthiswork target (Vierinen et al., 2006). We found using an optimiza-
-
a wepresentanincoherentscatterradartransmissioncodeop- tionalgorithmthatacombinationofamplitudeandarbitrary
t
a timization search method suitable for different modulation phase modulation can achieve very close to optimum cod-
d types,includingbinaryphase,polyphaseandamplitudemod- ing(theorder10−4 lessthanoptimalintermsofnormalized
s. ulation.Wefindthatthecombinationofamplitudeandphase variance). In this study we apply an optimization method,
c modulation provides better performance than traditional bi- similartotheoneusedforcoherenttargets,tofindtransmis-
i
s naryphasecoding,insomecasesgivingbetteraccuracythan sion codes that minimize the variance of incoherent target
y
alternatingcodes. autocorrelation function estimates. We compare results of
h
the optimization algorithm for several different modulation
p
[ methods.

All formulas in this paper use discrete time, unless oth-
1
1 Introduction erwisestated. Allwaveformsdiscussedarecomplexvalued
v
3 baseband signals. The ranges will be defined as round-trip
4 Incoherentscatterradarlagprofilemeasurementscanbede- timeforthesakeofsimplicity.
3
convolvedusingstatisticalinversionwitharbitraryrangeand
1
time resolution as shown by Virtanen et al. (2007a). In this
.
1 case, the transmission does not need to be completely free
2 Generaltransmissioncode
0
ofambiguities. Theonlyimportantfactoristhevarianceof
8
the estimated target autocorrelation function. Even though
0 A code with length L can be described as an infinite length
: alternating codes are transmission sequences that are opti-
v sequencewithafinitenumberofnonzerobaudswithphases
mal in terms of posterior variance when integrated over the
i andamplitudesdefinedbyparametersφ anda . Thesepa-
X code transmission cycle (Lehtinen, 1986), shorter and only k k
rameters obtain values φ ∈ [0,2π] and a ∈ [a ,a ],
k k min max
ar slightlylessoptimalcodegroupsarebeneficialinmanycases where k ∈ [1,...,L] : k ∈ N. The reason why one might
where an alternating code sequence is too long. Also, a
want to restrict the amplitudes to some range stems from
shorter code group offers much more flexibility when de-
practicalconstraintsintransmissionequipment. Usually,the
signingradarexperiments,e.g.,makingiteasiertocombine
maximumpeakamplitudeisrestrictedinadditiontoaverage
multiple different experiments in the same frequency chan-
duty cycle. Also, many systems only allow a small number
nelandsimplifyinggroundclutterremoval. Theuseofshort
of phases. The commonly used binary phase coding allows
transmissioncodesisdescribedinmoredetailinthecompan-
onlytwophases: φ ∈{0,π}.
k
ion paper by Virtanen et al. (2007b) submitted to this same
Byfirstdefiningδ(t)witht∈Zas
issue.
Correspondenceto: JuhaVierinen (cid:26)1 when t=0
δ(t)= (1)
(Juha.Vierinen@iki.fi) 0 otherwise,
2 Vierinenet.al.: Transmissioncodeoptimizationmethodforincoherentscatterradar
wecandescribeanarbitrarybasebandradartransmission wheres2isthevarianceofthemeasurementnoise.
envelope(cid:15)(t)as In this case, the normalized measurement “noise power”
oflagτ canthenbeapproximatedinfrequencydomainas
L
(cid:88)
(cid:15)(t)= akeiφkδ(t−k+1). (2) (cid:90) 2π N (L−τ)
P ≈ c dω, (8)
k=1 τ 0 (cid:80)Nc=c1|(cid:15)ˆcτ(ω)|2
We restrict the total transmission code power to be con-
stantforallcodesofsimilarlength. Withoutanylossofgen- where(cid:15)ˆc(ω)=FM{(cid:15)c(t)}isazeropaddeddiscreteFourier
τ D τ
erality,wesetcodepowerequaltocodelength transformofthetransmissionenvelopewithtransformlength
M (cid:29) L. N is the number of codes in the transmission
c
L
(cid:88) groupandListhenumberofbaudsinacode. Eachcodein
L= |(cid:15)(t)|2. (3)
agroupisassumedtobethesamelength.
t=1
willnot
Thiswillmakeitpossibletocompareestimatorvariancesof For alternating codes of both Lehtinen (1986) and Sulzer
codes with different lengths. It is also possible to compare (1993)type,P =1forallpossiblevaluesofτ.Forconstant
τ
codes of the same length and different transmission powers amplitude codes, this is the lower limit. On the other hand,
byreplacingLwiththerelativetransmissionpower. if amplitude modulation is used, this is not the lower limit
anymore, because in some cases more radar power can be
used on certain lags, even though the average transmission
3 Lagestimationvariance
poweristhesame.
We will only discuss estimates of the target autocorrelation To give an idea of how phase codes perform in general,
functionσ (r)withlagsτ thatareshorterthanthelengthof Fig. 1 shows the mean lag noise power for random code
τ
a transmission code (here r is the range in round-trip time, groups at several different code and code group lengths. It
and it is discretized by the baud length). The lags are as- is evident that when the code group is short and the code
sumed to be non-zero multiples of the baud length of the length is large, the average behaviour is not very good. On
transmissioncode. Autocorrelationfunctionestimationvari- theotherhand,whenthereisasufficientnumberofcodesin
ance is presented more rigorously in the companion paper a group, the performance is fairly good even for randomly
byLehtinenetal.(2007)submittedtothesamespecialissue. chosencodegroups. Thus,weonlyneedtoworryaboutper-
Thevariancepresentedtherealsoincludespulse-to-pulseand formanceofshortcodegroupsthataresufficientlylong.
fractionallags,takingintoaccounttargetpost-integrationas
well.
4 Codeoptimizationcriteria
Lag profile inversion is conducted using lagged products
forthemeasuredreceivervoltage,definedforlagτ as
Nearly all practical transmission code groups result in such
m (t)≡m(t)m(t+τ). (4) avastsearchspacethatthereisnopossibilityforanexhaus-
τ
tive search. As we cannot yet analytically derive the most
As more than one code is used to perform the measure-
optimal codes, except in a few selected situations, we must
ment, we index the codes with c as (cid:15)c(t). For convenience,
resort to numerical means. The problem of finding a trans-
wedefinealaggedproductofthecodeas
mission code with minimal estimation variance is an opti-
(cid:15)c(t)≡(cid:15)c(t)(cid:15)c(t+τ). (5) mizationproblemandthereexistanumberofalgorithmsfor
τ
approachingthisproblemnumerically.
Withthehelpofthesetwodefinitions,thelaggedproduct
A typical approach is to define an optimization criteria
measurement can be stated as a convolution of the lagged
f(x)withaparametervectorx. Theoptimizationalgorithm
product of the transmission with the target autocorrelation
thenfindsx thatminimizesf(x). Inthecaseoftransmis-
min
function
sioncodegroups,xwillcontainthephaseφc andamplitude
k
mc(t)=((cid:15)c ∗σ )(t)+ξ (t), (6) ac parametersofeachcodeinthecodegroup
τ τ τ τ k
Theequationalsocontainsanoisetermξτ(t),whichisrather x={a1,...,aNc,φ1,...,φNc}. (9)
1 L 1 L
complicated, as it also includes the unknown target σ (r).
τ
This term is discussed in detail, e.g., by Huuskonen and Therearemanydifferentwaystodefinef(x)inthecaseof
Lehtinen (1996). In the case of low SNR, which is typical transmissioncodegroups,butatrivialoneisaweightedsum
forincoherentscattermeasurements,thethermalnoisedom- ofthenormalizedlagpowerP ,withweightsw selectedin
τ τ
inates and ξ (t) can be approximated as a Gaussian white suchawaythattheyreflecttheimportanceofthatlag
τ
noiseprocessdefinedas
(cid:88)
f(x)= w P . (10)
Eξ (i)ξ (j)=δ(i−j)s2, (7) τ τ
τ τ τ
Vierinenet.al.: Transmissioncodeoptimizationmethodforincoherentscatterradar 3
Inthispaper,wesetw = 1foralllags. Thisgiveseach modulation, which we shall refer to as general modulation.
τ
laganequalimportance. Thisisasomewhatarbitrarychoice Inthisexample,weuseda = 1fortheconstantamplitude
k
of weights, in reality they should be selected in a way the modulationsandallowedamplitudesintherangea ∈[0,2]
k
reflectstheimportanceofthelagintheexperiment. forgeneralmodulationcodes,whilestillconstrainingtheto-
taltransmissioncodepowerinbothcasestobethesame.
The results are shown in Fig. 1. In this case the re-
5 Optimizationalgorithm
sults are shown in terms of mean lag noise power P =
(L−1)−1(cid:80) P . Itisevidentthatsignificantimprovement
Asoursearchmethodwillalsohavetoworkwithcodesthat τ τ
can be achieved when the code group length is short. For
haveafinitenumberofphases,weneededanalgorithmthat
longer code groups, the optimized groups do not different
could also work with situations were an analytic or numer-
thatmuchfromrandomcodegroups. Also,onecanseethat
ical derivative of f(x) cannot be defined. We developed an
optimizedpolyphasecodesaresomewhatbetterthanbinary
algorithmforthisspecifictask,whichbelongstotheclassof
phase codes; ultimately general phase codes are better than
random local algorithms (Lewis and Papadimitriou, 1997).
polyphasecodes–insomecasesthemeanlagnoisepoweris
Another well known algorithm belonging to the same class
actually better than unity. The reason for this is that ampli-
is simulated annealing (Kirkpatrick et al., 1983), which has
tudemodulationallowstheuseofmorepowerformeasuring
somesimilaritiestotheoptimizationalgorithmthatweused.
some lags, in addition to allowing more freedom in remov-
Therandomlocaloptimizationalgorithmisfairlyefficient
ing range ambiguities. It should also be noted, that when
at converging to a minima of f(x) and it can also to some
thecodeorcodegrouplengthisincreased,thedifferencebe-
extent jump out of local minima. In practice, it is faster to
tweenmodulationmethodsalsobecomeslesssignificant.
restarttheoptimizationsearchwithadifferentrandominitial
parameter set, in order to efficiently locate the minima of
f(x) that can then be compared the different optimization
7 Conclusions
runs.
Asimplifieddescriptionofourcodesearchalgorithmthat
We have introduced an optimization method suitable for
searchesforlocalminimaoff(x)isasfollows:
searching optimal transmission codes when performing lag
profile inversion. General radar tranmission coding, i.e.,
1. Randomizeparametersinx.
modulationthatallowsamplitudeandarbitraryphaseshifts,
2. Forasufficientnumberofsteps,randomizeanewvalue is shown to perform better than plain binary phase modula-
for one of the elements of x and accept the change if tion. Amplitude modulation is shown to be even more ef-
f(x)isimproved. fective than alternating codes, as the amplitude modulation
allowstheuseofmoreradarpowerinasubsetofthelags.
3. Randomizeallparametersx,acceptthechangeiff(x)
For sake of simplicity, we have only dealt with estima-
isimproved.
tion variances for lags that are non-zero multiples of the
baud length, with the additional condition that the lags are
4. Ifsufficientconvergencetoalocalminimaoff(x)has
shorterthanthetransmissionpulselength. Itisfairlyeasyto
been achieved, save x and goto step 1. Otherwise go
extend this same methodology for more complex situations
to step 2. The location of the minima can be further
that,e.g.,takeintoaccounttargetpost-integration,fractional
finetunedusinggradient-basedmethods,ifagradientis
or pulse-to-pulse lags. This is done by modifying the opti-
definedforf(x).
mizationcriterionf(x).
In practice, our algorithm also included several tunable Inthecasesthatweinvestigated,theroleofthemodulation
variables that were used in determining the convergence of method is important when the code length is short. When
f(x)toalocalminima. Also,thenumberoflocalminimato usinglongercodesorcodegroups,themodulationbecomes
searchfordependsalotonthenumberofparametersinthe lessimportant. Also,theneedforoptimizingcodesbecomes
problem. Inmanycaseswearesurethattheglobalminima smallerwhenthecodegrouplengthisincreased.
was not even found as the number of local minima was so FurtherinvestigationofthehighSNRcasewouldbeben-
vast. eficial and the derivation of variance in this case would be
interesting, albeitmaybenotasrelevantinthecaseofinco-
herentscatterradar.
6 Optimizationresults
Acknowledgements. ThisworkhasbeensupportedbytheAcademy
In order to demonstrate the usefulness of the optimization
of Finland (application number 213476, Finnish Programme for
method,wesearchedforcodegroupsthatusethreedifferent CentresofExcellenceinResearch2006-2011). TheEISCATmea-
types of modulation: binary phase modulation, polyphase surementsweremadewithspecialprogrammetimegrantedforFin-
modulation,andthecombinationofamplitudeandpolyphase land. EISCAT is an international assosiation supported by China
4 Vierinenet.al.: Transmissioncodeoptimizationmethodforincoherentscatterradar
Random Binary−phase
1.4 1.4
1.3 1.3
M M
e e
an1.2 an1.2
n n
o o
is is
e e
p1.1 p1.1
o o
w w
e e
r r
1.0 1.0
0.9 0.9
2.0 2.0
5 5
C1od0e gr1o5up le2ng0th 25 1lo.0g10(Co1de. l5ength) C1od0e gr1o5up le2ng0th 25 1lo.0g10(Co1de. l5ength)
30 30
0.5 0.5
Polyphase General
1.4 1.4
1.3 1.3
M M
e e
an1.2 an1.2
n n
o o
is is
e e
p1.1 p1.1
o o
w w
e e
r r
1.0 1.0
0.9 0.9
2.0 2.0
5 5
C1od0e gr1o5up le2ng0th 25 1lo.0g10(Co1de. l5ength) C1od0e gr1o5up le2ng0th 25 1lo.0g10(Co1de. l5ength)
30 30
0.5 0.5
Fig.1. Themeanlagnoisepowerforrandomcodes,optimizedbinaryphasecodes,optimizedpolyphasecodesandoptimizedamplitude
a ∈[0,2]andarbitraryphasemodulated(generalmodulation)codes. Thelargestimprovementsareachievedforshortcodegroups. Also,
k
itisclearthatthecombinationofamplitudeandphasemodulationprovidesthebestlagvariance.
(CRIRP),Finland(SA),Germany(DFG),Japan(STELandNIPR), els,JournalofAtmosphericandTerrestrialPhysics,58,453–463,
Norway(NFR),Sweden(VR)andUnitedKingdom(PPARC). 1996.
Kirkpatrick,S.,Gelatt,C.D.,andVecchi,M.P.: Optimizationby
SimulatedAnnealing,Science,220,671–680,1983.
Lehtinen,M.:Statisticaltheoryofincoherentscattermeasurements,
References EISCATTech.Note86/45,1986.
Lehtinen,M.S.,Virtanen,I.I.,andVierinen,J.: FastComparison
Huuskonen, A. and Lehtinen, M. S.: The accuracy of incoherent ofISradarcodesequencesforlagprofileinversion,Submittedto
scattermeasurements: errorestimatesvalidforhighsignallev-
Vierinenet.al.: Transmissioncodeoptimizationmethodforincoherentscatterradar 5
AnnalesGeophysicae,2007.
Lewis, H. R. and Papadimitriou, C. H.: Elements of the Theory
of Computation, Prentice Hall, Upper Saddle River, NJ, USA,
1997.
Sulzer,M.: Aradartechniqueforhighrangeresolutionincoherent
scatter autocorrelation function measurements utilizing the full
averagepowerofklystronradars,RadioScience,21,1033–1040,
1986.
Sulzer,M.P.:Anewtypeofalternatingcodeforincoherentscatter
measurements,RadioScience,28,1993.
Vierinen,J.,Lehtinen,M.S.,Orispa¨a¨,M.,andDamtie,B.:General
radartransmissioncodesthatminimizemeasurementerrorofa
statictarget,http://aps.arxiv.org/abs/physics/0612040v1,2006.
Virtanen,I.I.,Lehtinen,M.S.,Nygren,T.,Orispaa,M.,andVieri-
nen,J.: LagprofileinversionmethodforEISCATdataanalysis,
AcceptedforpublicationinAnnalesGeophysicae,2007a.
Virtanen, I. I., Lehtinen, M. S., and Vierinen, J.: Towards multi-
purposeISRradarexperiments,SubmittedtoAnnalesGeophys-
icae,2007b.