following document object identiﬁer (doi): 10.1109/jstars

IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING 1

SNR degradation in GNSS-R measurements underthe effects of Radio-Frequency Interference

Jorge Querol Student Member IEEE Alberto Alonso-Arroyo Student Member IEEERaul Onrubia Student Member IEEE Daniel Pascual Student Member IEEE Huyk Park Member IEEE

and Adriano Camps Fellow IEEE

ccopy2017 IEEE Personal use of this material is permitted Permission from IEEE must be obtained for all other uses in any current or future media includingreprintingrepublishing this material for advertising or promotional purposes creating new collective works for resale or redistribution to servers or lists orreuse of any copyrighted component of this work in other works The official published version of this paper may be obtained via IEEE Xplore ccopyusing thefollowing Document Object Identifier (DOI) 101109JSTARS20162597438

AbstractmdashRadio-Frequency Interference (RFI) is a seriousthreat for systems working with low power signals such as thosecoming from the Global Navigation Satellite Systems (GNSS) TheSpectral Separation Coefficient (SSC) is the standard figure ofmerit to evaluate the Signal-to-Noise Ratio (SNR) degradationdue to RFI However an in-depth assessment in the field ofGNSS-Reflectometry (GNSS-R) has not been performed yet andparticularly about which is the influence of RFI on the so-calledDelay-Doppler Map (DDM) This work develops a model thatevaluates the contribution of intra-inter-GNSS and external RFIeffects to the degradation of the SNR in the DDM for bothconventional and interferometric GNSS-R techniques Moreovera Generalized SSC (GSSC) is defined to account for the effects ofnon-stationary RFI signals The results show that highly directiveantennas are necessary to avoid interference from other GNSSsatellites whereas mitigation techniques are essential to keepGNSS-R instruments working under external RFI degradation

Index TermsmdashRFI interference GNSS-R reflectometry SNRDDM degradation WAF ambiguity SSC GSSC Wigner-Ville

I INTRODUCTION

NOWADAYS the consolidation of mass-market wirelesscommunication applications together with the increasing

demand of wider bandwidths of operation have fostered theproblem of Radio-Frequency Interference (RFI) in an over-crowded frequency spectrum RFI are undesirable signals thatdegrade or even disrupt the performance of a receiver andthey are particularly threatening for devices that use very lowpower signals such as Global Navigation Satellite Systems(GNSS) receivers Despite their inherent protection to RFIdue to the spread-spectrum codes used in navigation signalsin order to achieve Code Division Multiple Access (CDMA)GNSS-based instruments are prone to suffer from RFI effectsIn the last years the number of reported RFI occurrencesin GNSS has been increasing [1] even though legal policiesare established to protect GNSS bands Furthermore severalstudies such as [2] and [3] have shown that in addition tojamming signals even unintentional RFI signals can degradethe performance of GNSS receivers In GNSS RFI signalsinclude aeronavigation signals such as Distance MeasurementEquipment (DME) at the GPS L5Galileo E5 bands spuriousor harmonics of lower frequency and even contributions of

This work has received funding by the project ldquoAGORA Tecnicas Avan-zadas en Teledeteccion Aplicada Usando Senales GNSS y Otras Senalesde Oportunidadrdquo of the Spanish Ministerio de Economıa y Competitividad(MINECOFEDER) ESP2015-70014-C2-1-R and by the grant for the recruit-ment of early-stage research staff FI-DGR 2016 of the AGAURGeneralitatde Catalunya 2016FI-B00738

the same GNSS satellites sharing the same band of operationamong others Although anthropogenic RFI signals are themost common source natural emissions may also be consid-ered as RFI signals such as Sun L-band surface glints whichcan also affect GNSS-R measurements

The impact of RFI signal in GNSS receivers has beenstudied in detail over the last decade The Spectral SeparationCoefficient (SSC) was defined as a figure of merit to quantifythe degree of interference that a GNSS signal suffers due toother signals sharing the band of operation [4] in terms ofa reduction of the Signal-to-Noise Ratio (SNR) Indeed theSSC allows to characterize also GNSS intra-system and inter-system SNR degradation [5] That is a measurement of therejection ratio between navigation codes taken into account intheir design process in order to guarantee the co-existence ofmultiple GNSS

The RFI problem in GNSS is of special concern dueto the high number of existing GNSS-enabled applicationsOne of these applications is GNSS-Reflectometry (GNSS-R)a promising technique in the field of remote sensing thatwas first proposed in 1988 for scatterometry applications [6]Later in 1993 GNSS-R was first suggested for mesoscaleocean altimetry [7] and since then this technique has beenused in the retrieval of many geophysical parameters such asmeasuring sea surface state [8] sea surface salinity [9] soilmoisture [10] and ocean altimetry [11] among others

GNSS-R devices receive simultaneously the signals trans-mitted by multiple GNSS satellites that have been scatteredby the surface of interest This configuration is known asmultistatic in radar theory [12] There are two main GNSS-Rapproaches [13] The first is the so-called conventional GNSS-R (cGNSS-R) in which the reflected signal is cross-correlatedwith a locally generated replica of the transmitted one Onlysignals with open codes can be used in this approach (egGPS L1 CA) The second one is the so-called interferometricGNSS-R (iGNSS-R) in which the reflected signal is cross-correlated with a signal captured directly from the satelliteIn this case both open and restricted codes are used simul-taneously in a specific band (eg CA C P and M codes atGPS L1 band) Figure 1 illustrates the scenario correspondingto each approach Regarding their resilience in front of RFIcGNSS-R is not immune to RFI due to the finite rejectionratio of the spread-spectrum signal whereas iGNSS-R is evenmore prone to RFI since an undesired signal captured by bothantennas will produce a non-zero cross-correlation

The observable that contains all the information of GNSS-

2 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING

(a) (b)

Fig 1 Illustrative diagram representing satellite RFI and ther-mal noise signals involved in the study of the SNR degradationfor a) cGNSS-R scenario and b) iGNSS-R scenario

R measurements is the so-called Delay-Doppler Map (DDM)which shows how the scattering process over the surfaceof interest spreads the energy of the transmitted signal inthe delay-Doppler space and how this spreading changes intime The generally accepted model for incoherent GNSS-R reflections assuming fully diffuse scattering from roughsurfaces and originally stated for sea-surface observations isthe Zavorotny-Voronovich (Z-V) model [14] The Z-V modelcan be expressed in a simplified version as a two dimensionalconvolution [15] and is equivalent to the so-called radarmapping equation [12] The model can be expressed as in[16] as lang

|Y (ν τ)|2rang

= |χ(ν τ)|2 lowast lowast Σ(ν τ) (1)

where 〈|Y (ν τ)|2〉 is the expected DDM ν and τ are theDoppler and delay variables respectively the lowastlowast operatorindicates a two-dimensional convolution in ν and τ domains|χ(ν τ)|2 is the Woodwardrsquos Ambiguity Function (WAF) ofthe navigation code and Σ(ν τ) accounts for the scatteringat the surface propagation losses antenna patterns and therest of terms in the bistatic radar equation Moreover if thereflection process is coherent because it takes place overa planar surface (eg calm water ice floes or flat lands)the measured DDM will be an attenuated version of thesignal WAF Furthermore any combination of coherent andincoherent reflection will led to a DDM that can be expressedas an attenuated and distorted version of the signal WAF

The WAF is the squared modulus of the ambiguity functionfirst proposed in [17] and it was introduced in radar theory[18] as the appropriate function to evaluate the degree ofambiguity in the measurements when the number of radarechoes is sufficiently large This approach is translated in anon-coherent detection process after the cross-correlation asused in most of GNSS receivers [19] in order to increasethe SNR and necessary in GNSS-R applications due to theeffect of speckle noise introduced in the scattering process[12] which is caused by the addition of a large number ofrandom phased scatterers over the surface of interest

According to the experiments carried out in [2] the effectof RFI signals in cGNSS-R is translated into an equivalent

rise of the noise floor and thus into a desensitization of thereceiver However these effects have not been studied in depthto the best of authorsrsquo knowledge This work aims at providinga general model that accounts for the effects of RFI signals inGNSS-R instruments operating with CDMA spread-spectrumGNSS signals including intra- and inter-GNSS interferenceand other external RFI signals

The results derived along this work are completely generalso they can be applied to any GNSS-R scenario using theproper geometry parameters However for the sake of claritythese results are illustrated using specific cases that take intoaccount no spatial filtering (isotropic antennas) perfect specu-lar reflection (no scattering) and equal received power for allsatellites (same transmission power propagation attenuationand surface reflectivity)

The outline of this paper is as follows In Section II thegeneral model of SNR degradation is developed for bothcGNSS-R and iGNSS-R techniques In Section III and Sec-tion IV the particular effects on the SNR of cross-correlationbetween GNSS signals and external RFI signals respectivelyare described in detail The main conclusions of this paper arestated in Section V

II GENERAL SNR MODEL FOR GNSS-R

A Signal model and detection

After the scattering process GNSS-R signals are collectedby the instrument antennas conditioned by the front-end stagewith bandwidth Br and finally delivered to the processingstage The complex base-band signal at this stage can beexpressed as the addition of three terms

yr(t) = sr(t)+ ir(t)+nr(t) =

Lminus1suml=0

srl(t)+ ir(t)+nr(t) (2)

where the subscript r stands for ldquoreflectedrdquo signals L is thenumber of ldquoin-viewrdquo GNSS signals srl corresponds to the lthsignal ir is the aggregate of RFI signals and nr is band-limited thermal noise (see Fig 1a)

GNSS signals are detected after the delay-Doppler corre-lator or ldquomatched filterrdquo that performs the spread-spectrumdemodulation [19] In cGNSS-R the signal after the correlatorfor a particular kth tracked signal (eg GPS L1 CA GalileoE5 Beidou B2) is obtained as

χyrck(ν τ) =1

Tc

intTc

yr(t) clowastk(tminus τ) eminusj2πνt dt (3)

where yr(t) is the received complex base-band signal definedin (2) ck isin C is the clean replica of the kth navigation codeassumed known deterministic and with normalized powerwithin the bandwidth Br ν and τ are the Doppler anddelay variables respectively and Tc is known as the coherentintegration time

As aforementioned the detection process in GNSS-R mustbe non-coherent as in radar systems so that the new observableafter the detector becomes the expected power expressed as

|χ|2yrck (ν τ) = E|χyrck(ν τ)|2

(4)

QUEROL et al SNR DEGRADATION IN GNSS-R MEASUREMENTS UNDER THE EFFECTS OF RADIO-FREQUENCY INTERFERENCE 3

The expected operator E middot is introduced since the signal yris a random process Furthermore it is often replaced by theaverage operator 〈middot〉 as an unbiased estimator of the expectedvalue and usually referred as incoherent averaging [14] Inthis work |χ|2yrck is introduced as the Cross-WAF (CWAF)between yr and ck analogously to the definition of the (Auto-)WAF for random processes in radar theory [18] Then |χ|2yrckturns to be equal to the measured DDM for the kth satellitesignal using the cGNSS-R approach and equivalent to theexpected DDM 〈|Y (ν τ)|2〉 defined in [14] In other wordsthe CWAF is the mathematical operation used to compute themeasured DDM

In iGNSS-R the clean replica of the code ck is replacedby the signal acquired directly from the ldquoin-viewrdquo satellitesdefined as

yd(t) = sd(t)+id(t)+nd(t) =

Lminus1suml=0

sdl(t)+id(t)+nd(t) (5)

where the subscript d stands for ldquodirectrdquo signals (see Fig 1b)and it is analogous to (2) Therefore the output of thecorrelator becomes χyryd(ν τ) and the DDM in iGNSS-R canbe defined as |χ|2yryd (ν τ) Furthermore signal yd is treated asa unit-less signal since it becomes the impulse response of theldquomatchedrdquo filter and its power represents a non-dimensionalscale factor in the iGNSS-R DDM

B DDM decomposition

In order to study the contribution of RFI signals to themeasured DDM the CWAF is decomposed into the termscorresponding to each signal in (2) Assuming that sr ir andnr have zero mean and are statistically independent from eachother |χ|2yrck (ν τ) can be expressed using the SussmanrsquosFormula [20] as follows (see proof in Appendix A)

|χ|2yrck (ν τ) = |χ|2srkck (ν τ)+

Lminus1sumlge0l 6=k

|χ|2srlck+|χ|2irck+|χ|2nrck

(6)where the first term corresponds to the useful tracked signalDDM the second one is the cross-sat term the third one isthe RFI term and the last one is the thermal noise term Fig-ure 2 depicts an hypothetical example of DDM decompositionshowing each one of the terms corresponding to a cGNSS-RDDM using GPS L1 CA with Br = 2046 MHz

The CWAF between the reflected signal of the kth trackedsatellite srk and its clean replica crk |χ|2srkck (ν τ) iscoined as tracked or useful signal DDM in this work Thetracked signal DDM contains all the information about the sur-face of interest and hence the rest of the terms in (6) representundesired contributions which degrade the SNR Besides incGNSS-R all of them are translated into a rise of a uniformpower floor as it can be appreciated in Fig 2 Among them thecross-sat term include all intra- and inter-GNSS interferencewhereas the RFI term represents the additive contribution ofexternal RFI signals to the received DDM Both terms will beanalyzed in detail in next sections Eventually assuming thatthe thermal noise after the front-end is a complex Gaussian

(a) (b)

(c) (d)

(e)

Fig 2 Sample decomposition of a cGNSS-R DDM Thefollowing conditions have been assumed GPS L1 CA codeperfect specular reflection (no scattering) Br = 2046 MHzTc = 1 ms and equal unitary received power for all signalsEach subfigure corresponds to (a) the received DDM obtainedfrom (6) (b) the tracked signal DDM (c) the cross-sat term(for 10 ldquoin-viewrdquo additional satellites 5 GPS L1 + 5 GalileoE1) (d) the RFI term (jammer chirp RFI) and (e) the noiseterm

stochastic process with flat Power Spectral Density (PSD)within Br and equal to N0r the noise term in cGNSS-R canbe approximated as

|χ|2nrck=N0r

Tc

int ν+Br2

νminusBr2

Sck(f) df N0r

Tc (7)

where Sck(f) is the PSD function of ck and it has beenconsidered that the range of the variations of ν is very smallas compared to Br

On the other hand the DDM in iGNSS-R can be obtainedfrom |χ|2yryd (ν τ) Taking into account the same assumptionsas in cGNSS-R and that sd id and nd are statisticallyindependent between them the iGNSS-R DDM can be ex-pressed as in (8) (see proof in Appendix B) where eachterm represents from left to right the useful tracked signalDDM as in (6) but for a GNSS composite signal the cross-talk term the cross-sat term the RFI term and the last threeones are the interferometric thermal noise terms (ie |χ|2noise)


|χ|2yryd (ν τ) = |χ|2srksdk (ν τ) + |χ|2cross-talk + |χ|2cross-sat + |χ|2RFI (ν τ) +(|χ|2nrsdk

+ |χ|2srknd+ |χ|2nrnd

) (8)

(a) (b)

(c) (d)

(e) (f)

Fig 3 Sample decomposition of a iGNSS-R DDM Thefollowing conditions have been assumed GPS composite L1(CA C P(Y) and M codes) perfect specular reflection (noscattering) Br = 3069 MHz Tc = 1 ms and equal unitaryreceived power for all signals Each subfigure corresponds to(a) the received DDM obtained from (8) (b) the tracked signalDDM (c) the cross-talk term (for 10 ldquoin-viewrdquo additionalsatellites 5 GPS L1 + 5 Galileo E1) (d) the cross-sat term(e) the RFI term (jammer chirp RFI) and (f) the noise terms

Besides in iGNSS-R the subscript k is used to indicate allthose codes that belong to a GNSS composite signal receivedfrom a particular satellite (eg GPS L2 composed by C P(Y)and M codes Galileo E6 composed by CS and PRS codesBeidou B3 with a single code) whereas in cGNSS-R thesubscript k refers to the particular clean code replica used inthe delay-Doppler correlator (eg GPS L1 CA)

An example showing each one of the terms correspondingto an iGNSS-R DDM using GPS composite L1 signal with Br= 3069 MHz is depicted in Fig 3 Note that the comparisonbetween Figures 2 and 3 is not direct since the assumed front-end bandwidth Br differs in each case

In (8) the cross-talk term appears as an exclusive effect of

the iGNSS-R technique Cross-talk is composed by the DDMscorresponding to the ldquoin-viewrdquo non-tracked satellite signalsand it can have an overwhelming effect on the SNR (muchhigher than cross-sat) if it overlaps the tracked signal DDMThe overlapping between DDMs depends on the relativeposition and velocity between the satellites and the receiveron the WAF of the involved signals and on the surface ofreflection A detailed study of cross-talk effect is beyond thescope of this paper (see [21])

Moreover interferometric cross-sat and RFI terms areequivalent to the ones in the cGNSS-R case Eventuallyinterferometric noise terms can be approximated as follows

|χ|2noise N0r

TcPsdk + Psrk

N0d

Tc+N0rN0d

BrTc (9)

where Tc is the coherent integration time N0r and N0d arethe PSD of thermal noise after the respective front-end afterdown-looking and up-looking antennas with equal bandwidthBr and considering the same approximation applied in (7)This expression of the interferometric noise matches with theone developed in [22]

C SNR definition and degradation

The SNR is the figure of merit that has been chosen toevaluate the impact of RFI on GNSS-R measurements and it istypically defined at the maximum of the tracked signal DDMIn this paper in order to maintain the whole information of theDDM a point-to-point relation will be considered similarly tothe one used in [22] Therefore a general SNR definition forGNSS-R measurements considering the kth tracked satellitesignal can be written as

ρk(ν τ) =|χ|2kth-sat (ν τ)

|χ|2cross-talk (ν τ) + |χ|2cross-sat + |χ|2RFI (ν τ) + |χ|2noise

(10)with every undesired term in the denominator defined accord-ing to the involved technique and evaluated in the correspond-ing delay-Doppler point (ν τ) Two examples of ρk(ν τ) areshown in Fig 4 taking the values obtained from each one ofthe simulations performed in Fig 2 and Fig 3 for cGNSS-Rand iGNSS-R respectively

Finally the SNR degradation as a function of the undesiredterms is defined with respect to the thermal SNR (ie ρkthermal )The latter is the quotient between the useful signal and thecorresponding thermal noise Then the SNR can also beexpressed in the following way

ρk(ν τ) = ρkthermal(ν τ) ∆k(ν τ) (11)

where ∆k represents SNR degradation corresponding to thecombined effect of cross-talk cross-sat and RFI terms Fur-thermore SNR degradation introduced by cross-sat and RFIterms are further studied in subsequent sections


(a) (b)

Fig 4 Point-to-point SNR in the delay-Doppler plane obtainedfrom (10) Each subfigure corresponds to (a) cGNSS-R tech-nique taking the terms depicted in Fig 2 and (b) iGNSS-Rtechnique according to the terms in Fig 3

III CROSS-SAT SNR DEGRADATION

The cross-sat effect appears when one or more GNSSsignals apart from the one that is used to perform the GNSS-Rmeasurements are captured by the instrument antennas andthey are not completely rejected by the front-end So thatit represents the contribution of all the cross-correlationsbetween these satellite signals that share the same band ofoperation or at least their PSD are partially overlappedMoreover cross-sat determines the level of intra- and inter-GNSS interference that is fixed by design and it is evaluatedwith the SSC parameter first used in [4] The SSC betweentwo signals x and y is defined as

κxy(ν) =

intBr

Sx(f)Sy(f minus ν) df = Sx(ν) lowast Sy(ν) (12)

where Br is the bandwidth of the receiver and Sx and Syare the normalized PSD of x and y respectively The SSC ismeasured in dBHz and it reveals the rejection or attenuationper unit of bandwidth suffered by other GNSS signals ratherthan the one under correlation

As stated previously GNSS codes have been designed fornavigation purposes and considering that GNSS-R takes anadvantage of the properties of these signals in an opportunisticway it seems reasonable to study how much this overlappingaffects to GNSS-R measurements and in particular to theSNR Furthermore cross-talk effect also takes place under thesame conditions for the iGNSS-R technique and its impact onthe SNR may be even worse Thus the following evaluationwill be considered valid if cross-talk effect is not present

A SSC values

The magnitude of the cross-sat degradation depends onthe addition of the residual power of the undesired GNSSsignals at the output of the front-end Psrl weighted by theSSC between each the received code and the tracked oneAccording to the definition the SSC depends on the Dopplerof the receiver however the dependence can be neglectedsince the Doppler variations are much smaller as comparedto Br Then SSC is taken as constant with ν similarly tothe approximation applied with the noise term SSC valuesbetween the main GNSS codes are shown in Table I for upper

L-band (1559-1610 MHz) and in Table II for lower L-band(1164-1300 MHz) These values have been obtained using(12) and taking Br equal to the main lobe bandwidth as in[23]

Since cross-sat degradation is highly dependent on thescenario number type position velocity and emitted powerof satellites signal attenuation and scattering at the reflec-tion surface and antennas and front-end of the receiver anassessment of the SNR degradation induced by the cross-satphenomenon must be performed in each particular case Forthe sake of simplicity a general case is discussed for cGNSS-Rand iGNSS-R in the following subsections

B Cross-sat in cGNSS-R

The cross-sat term in cGNSS-R can be expressed as (seeproof in Appendix C)

Lminus1sumlge0l 6=k

|χ|2srlck 1

Tc

Lminus1sumlge0l 6=k

Psrl κsrlck (13)

and the SNR degradation when only this term is present as

∆kcross-sat

(1 +

sumLminus1l 6=k Psrl κsrlck

N0r

)minus1 (14)

An example of SNR degradation induced by cross-sat effectin cGNSS-R is depicted in Fig 5 GNSS satellites from thesame constellation have antenna patterns shaped to producenearly constant power density at surface level regardless ofsatellite elevation Transmitted power is also monitored andkept constant with a limited margin However small differencesare expected due to aging For the sake of simplicity equalreceived power after the front-end for all interfering satelliteshas been assumed as well as the different transmitted powerfor different codes Nevertheless the received power from eachsatellite is not a priori known but a minimum received powerfor each service is guaranteed (see Table I and II) Giventhis the assumption of equal received power (and isotropicantennas) allows to calculate approximate values of the impactof cross-sat signals on the SNR Further results about cross-sat effect in GNSS-R must include particular geometry andantenna pattern case by case

According to results depicted in Fig 5 ∆kcross-sat mayrepresent a degradation of several dB in case a non-directiveantenna is used However highly directive antennas of theorder of 30 dB can reduce the cross-sat effect to smallervalues although the effect produced by non-used codes fromthe tracked satellite will still remain

C Cross-sat in iGNSS-R

The cross-sat term in iGNSS-R can be expressed as

(15)

|χ|2cross-sat Lminus1sumlge0l 6=k

(1

TcPsrl

Lminus1sumk=0

Psdk κsrlsdk +

+N0r

TcPsdl + Psrl

N0d

Tc

)


TABLE I SSC values between main codes at upper GNSS band obtained from (12) in dBHz SSC values depend on whichis the GNSS-R receiver bandwidth Br the spectrum of tracked GNSS signal used in cross-correlation the spectrum of thecross-sat signals and the bandwidth where they are transmitted in Bt Infinite negative values indicate no overlapping betweenGNSS signal spectra In addition the last row shows additional information about minimum received power for each GNSSservice

SSC [dBHz] Crossed GPS Galileo BeidouUpper Band Bt[MHz] 3069 25552 4092 1600

Tracked Br[MHz]HHHH

L1CA L1C L1P L1M E1OS E1PRS B1

GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068

Minimum Received Power [dBW] -1615 -1600 -1645 -1600 -1570 -1570 -1570

TABLE II SSC between main codes at lower GNSS band The structure is equivalent to that presented in Table I

SSC [dBHz] Crossed GPS Galileo Beidou GLONASSLower Band Bt[MHz] 3069 2400 5115 4092 3600 2046

Tracked Br[MHz]H

HHHL2C L2P L2M L5 E5 E6CS E6PRS B2OS B2AS B3 L3OC

GPS

2046 L2C -6181 -7021 -9241 -infin -infin -infin -infin -infin -infin -infin -infin2046 L2P -6990 -7125 -7997 -infin -infin -infin -infin -9958 -9079 -infin -infin3069 L2M -8706 -7979 -7155 -infin -9573 -infin -infin -9340 -8957 -infin -infin2046 L5 -infin -infin -infin -7100 -7417 -infin -infin -infin -infin -infin -infin

Galileo5115 E5 -11038 -9814 -9573 -7417 -7430 -infin -infin -7312 -7435 -infin -76181023 E6CS -infin -infin -infin -infin -infin -6863 -9656 -infin -infin -8582 -infin3069 E6PRS -infin -infin -infin -infin -infin -8588 -7134 -infin -infin -7348 -infin

Beidou4092 B2OS -infin -infin -infin -infin -7333 -infin -infin -6478 -7032 -infin -73932046 B2AS -10743 -9518 -9284 -infin -7436 -infin -infin -7011 -7140 -infin -73362046 B3 -infin -infin -infin -infin -infin -8339 -7345 -infin -infin -7140 -infin

GLONASS 2046 L3OC -infin -infin -infin -infin -7618 -infin -infin -7353 -7335 -infin -7098

Minimum Received Power [dBW] -1615 -1645 -1600 -1570 -1520 -1550 -1550 -1630 -1630 -1600 -

according to Appendices B and C In this case the corre-sponding cross-correlations between nr and sdl and betweensrl and nd must be considered in addition to the combinationbetween all non-tracked codes The degradation ∆kcross-sat isdefined analogously to the previous case Besides SSC valuesin Tables I and II must be recalculated using Br equal to thecomposite signal bandwidth in each case

Figure 6 illustrates the same example of Fig 5 but con-sidering the iGNSS-R technique SNR degradation levels iniGNSS-R are similar to the ones seen in cGNSS-R except inGPS composite L1 which are substantially worse as comparedto L1 CA This is mainly given by the difference in receiverbandwidth Br used in each case In cGNSS-R using GPS L1CA Br = 2046 MHz whereas Br = 3069 MHz in iGNSS-Rwith GPS L1 band Larger bandwidth implies more cross-satpower is received In addition when comparing the iGNSS-Rcross-sat degradation in L1E1 band with L5E5 band L1E1band is worse because it is more ldquocrowdedrdquo than L5E5 bandiGNSS-R L1 has slightly better performance than E1 becausereceiver bandwidth is also slightly smaller (3069 MHz vs35805 MHz) The same happens when comparing L5 and

E5 (2046 MHz vs 51315 MHz)

IV RFI SNR DEGRADATION

RFI signals other than interfering GNSS codes can degradeor totally disrupt the performance of GNSS-R devices Thedegradation of the SNR produced by RFI signals have alsobeen studied in GNSS using the SSC and the level ofinterference is highly dependent on the kind of RFI presentHowever the approach followed in the state of the art is onlylimited to stationary RFI signals

A Generalized SSC

In order to evaluate the impact of any RFI (stationary ornon-stationary) in GNSS-R DDMs an extension of the SSCconcept named Generalized SSC (GSSC) is introduced in thiswork The GSSC between two signals x and y after the front-end is defined as

γxy(ν τ) =1

Tc

intBr

intTc

Wx(t f)Wy(tminusτ fminusν) dt df (16)


(a) (b)

(c) (d)

Fig 5 Sample SNR degradation induced by cross-sat effectin cGNSS-R and obtained from (13) and (14) The followingassumptions are considered equal received power after thefront-end for all satellites isotropic antennas and SSC valuesand minimum received power stated in Tables I and II Eachsubfigure corresponds to cGNSS-R using (a) GPS L1 CA(b) Galileo E1 OS (c) GPS L5 and (d) Galileo E5 The totalCross-Sat-to-Noise ratio refers to the sum of power receivedfrom all interfering GPS and Galileo satellites divided by thethermal noise power with N0r = -204 dBWHz

where Wx and Wy are the normalized Wigner-Ville Spectrum(WVS) [24] or non-stationary spectrum of x and y respec-tively The WVS is used instead of the PSD defined only forstationary random processes and its use accounts for the non-stationary nature of RFI whose spectrum may change overthe time

B RFI in cGNSS-R

In cGNSS-R the RFI term can be expressed using the GSSCas follows (see proof in Appendix C)

|χ|2irck 1

TcPir γirck (17)

where Pir is the RFI power at the reflected antenna Tc isthe coherent integration time and γirck is the GSSC obtainedfrom (16) between the RFI at the reflected antenna ir andthe clean code replica ck γirck has been approximated as aconstant taking into account the same assumptions as for theSSC case Moreover the SNR degradation when this is thedominant term is

∆kRFI (

1 +Pir γirckN0r

)minus1 (18)

In order to validate this model a commercial chirp jammer thatemits a 15 MHz frequency modulated signal centered at L1band has been used Figure 7 shows a comparison between thetheoretical SNR degradation obtained using (18) and the one

(a) (b)

(c) (d)

Fig 6 Sample SNR degradation induced by cross-sat effectin iGNSS-R and obtained from (15) and (14) The followingassumptions are considered equal received power after thefront-end for all satellites isotropic antennas and SSC valuesand minimum received power stated in Tables I and II Eachsubfigure corresponds to iGNSS-R using (a) GPS compositeL1 (CA C P(Y) and M codes) (b) Galileo composite E1(OS and PRS codes) (c) GPS L5 and (d) Galileo E5 The totalCross-Sat-to-Noise ratio refers to the sum of power receivedfrom all interfering GPS and Galileo satellites divided by theinterferometric noise power with N0r = N0d = -204 dBWHz

measured at the laboratory using the jammer with three com-mercial GPS L1 CA receivers Theoretical SNR degradationpredicted by the model shows that from 25 dB of Interference-to-Noise Ratio (INR) on SNR degradation changes has -10dB per decade Comparing this with the curves obtained fromthe three GPS receivers they show a trend similar to thetheoretical degradation but their results are between 5 and 15dB better mainly because they incorporate some mitigationtechniques Furthermore the improved performance using acustom RFI mitigation system described in [25] also showsthe same trend but the resilience against the RFI signal hasbeen increased 30 dB more This result is twice valuable Onthe one hand the theoretical model has been validated withreal measurements obtained from commercial systems On theother hand it demonstrates that mitigation techniques are thesolution to reduce the impact of external RFI signals in GNSS-R measurements


Fig 7 Sample SNR degradation induced by RFI effect incGNSS-R The simulated RFI signal is a 15 MHz sweep chirpjammer cross-correlated with GPS L1 CA code Theoreticalresult is compared to real measurements obtained from severalcommercial GPS receivers with and without a custom RFImitigation system implemented at UPC-BarcelonaTech [25]

C RFI in iGNSS-R

In iGNSS-R the RFI term can be expressed as follows (seeAppendices B and C)

|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl

+ PirPid γirid(ν τ) + PirN0d

Tcγirnd

+N0r

TcPid γnrid

)

(19)

and the SNR degradation can be defined in an analogous wayto the cGNSS-R case In iGNSS-R four sub-terms contributeconstantly to the SNR degradation whereas there is only onethat depends on ν and τ γirid(ν τ) corresponds to the cross-correlation between RFI signals captured by both direct andreflected antennas and it can reach magnitudes much higherthan the useful signal even if high directive antennas are used

In order to illustrate the effects of RFI signals in iGNSS-R GSSC values in (19) are depicted in Fig 8 for the caseof a DME RFI signal interfering GPS L5 and Galileo E5iGNSS-R measurements It can be appreciated that the GSSCterms γsrlid γirsdl γirnd

and γnrid are of the order of theSSC values obtained in Section III for cross-sat terms butthey may depend on the DME frequency channel However ifthe received RFI power represented by Pid and Pir is muchhigher than the received noise and satellite signal power theeffect of RFI SNR degradation will be worse than in thecross-sat case Moreover the GSSC corresponding to a cross-correlation between direct and reflected DME signals γirid is concentrated in few delay lags rather that spread over thewhole delay-Doppler space This fact implies that the SNRdegradation around this peak will be much higher as comparedto other terms Further information about the impact of DMEsignals on iGNSS-R measurements can be found in [26]

(a) (b)

Fig 8 Sample GSSC values obtained using (16) This exampleconsiders an iGNSS-R instrument interfered with a DME RFIsignal DME spectrum is overlapped with GPS L5 and GalileoE5 spectra Each subfigure corresponds to (a) the GSSC valuesfor crossed combinations DME to Noise and DME to GNSSsignals and (b) GSSC values for DME signals present at bothdirect and reflected antennas

(a) (b)

(c) (d)

Fig 9 Sample GSSC values obtained using (16) consideringGPS L1 iGNSS-R with Br = 3069 MHz under different RFIsignals Each subfigure corresponds to (a) a CW signal (b)a narrow-band chirp with Bchirp = 15 MHz and 1024 microsrepetition period (c) a wide-band chirp with Bchirp = 60 MHzand 8192 micros repetition period and (d) a Pseudo-RandomNoise (PRN) signal with 25 MHz of bandwidth Results areshown for GPS L1 iGNSS-R with Br = 3069 MHz

As stated above the GSSC between direct and reflectedsignals in iGNSS-R may increase the SNR degradation con-siderably For the sake of completeness the GSSC valuescorresponding to four typical RFI signal shapes [27] areshown in Fig 9 The chosen RFI signals are a sinusoidalor Continuous Wave (CW) signal a narrow-band chirp withBchirp = 15 MHz and 1024 micros repetition period a wide-bandchirp with Bchirp = 60 MHz and 8192 micros repetition periodand a Pseudo-Random Noise (PRN) signal with 25 MHz ofbandwidth Results are shown for GPS L1 iGNSS-R with Br= 3069 MHz


Figure 9 depicts how RFI signals whose power is moreconcentrated in frequency such as CW and narrow-band chirpare more spread in the delay domain after the correlationprocess On the contrary wide-band chirp and PRN RFIsignals are more concentrated in delay domain This behavioris explained taking into account that delay and frequency arecomplementary domains [28] Moreover the behavior in theDoppler domain will depend on the RFI signal shape itself

CW and in general narrow-band signals are in principlemore troublesome for GNSS-R measurements since they aremore spread in the delay-Doppler space Conversely wide-band signals will only degrade the SNR at particular delaybins but they can be fatal if the tracked signal DDM informa-tion is also in these bins Therefore the probability of SNRdegradation will be higher for narrow-band signals than forwide-band signals However mitigation techniques are moreeffective for narrow-band than for than wide-ban signals

V CONCLUSIONS

RFI signals can overwhelm the performance of GNSS-Rinstruments as it was demonstrated in in the past (eg [2]) Inthis paper a model to evaluate the degradation of the SNR inGNSS-R measurements has been proposed for both cGNSS-R and iGNSS-R techniques and it has been evaluated with ageneral case for each technique

Regarding cGNSS-R the degradation of the thermal SNRcan be produced by either cross-sat effect or external RFIeffect or both Moreover they have a constant degradationover the delay-Doppler space The cross-sat effect may rep-resent degradation of more than 10 dB considering severalsatellites with all existing and forthcoming GNSS signalsNevertheless the cross-sat effect can be neglected when usinghighly directive antennas In particular antenna arrays withbeam-steering capabilities are able to track a specific satellitewhile they attenuate signals from the rest of them On theother hand the external RFI effect may represent a corruptionof the measurements since it is known that they can disruptthe performance of GNSS receivers In this case mitigationtechniques must be used in order to reduce the error producedby the RFI effect [27]

The iGNSS-R technique has been proven to be much moresensitive to RFI than cGNSS-R Moreover in this case anew phenomenon is introduced the cross-talk effect thatmay represent a total corruption of the measurements Cross-talk together with cross-sat effect may also be attenuatedwith directive antennas Finally external RFI effect can havea catastrophic effect in the measurements even worse ascompared to cGNSS-R if the interference is captured by bothantennas reflected and direct In this case the use of highlydirective antennas can help to decorrelate the signals capturedby them However the RFI effect will still remain because ofthe multiple residual cross-correlations between RFI signalsand thermal noise captured by both antennas Therefore miti-gation techniques will be essential to improve the performanceof iGNSS-R instruments under RFI conditions

A specific mention must be given to former GLONASSwhich uses spread-spectrum signals with a single code com-bined with Frequency Division Multiple Access (FDMA) Due

to the use of different frequencies for different satellites cross-talk and cross-sat effects will disappear in both cGNSS-Rand iGNSS-R techniques even if non-directive antennas areused Nevertheless external RFI signals are still troublesomeWith the conventional approach if the RFI power is wellconcentrated in frequency (for instance a CW RFI signal) theSNR degradation could be much higher for some GLONASSsatellites (those where the RFI signal overlaps their respectivespectra) as compared to GPS or Galileo satellites The op-posite would occur if the RFI signal is wide-band Moreoverin the interferometric approach SNR degradation would beof similar order since the cross-correlated term between directand reflected signal does not depend on the used GNSS

APPENDIX ADERIVATION OF DDM DECOMPOSITION IN CGNSS-R

The CWAF has been defined as the operation needed tocalculate the measured DDM in GNSS-R and for two genericsignals x and y it is computed as

|χ|2xy (ν τ) = E|χxy(ν τ)|2

(20)

In the state of the art the Sussmanrsquos formula was introducedin [20] as a general proof of several properties of ambiguityfunctions such as the Siebertrsquos theorem [29] and the Stuttrsquosinvariant relation [30] and states that if a Cross-AmbiguityFunction (CAF) is defined as

χxy(ν τ) =

intx(t) ylowast(tminus τ) eminusj2πνtdt (21)

the following identity is fulfilled

(22)χxy(ν τ)χlowastwz(ν τ)

=

intintχxw(νprime τ prime)χlowastyz(ν

prime τ prime) ej2π(νprimeτminusντ prime)dνprime dτ prime

so that the modulus squared of any CAF can be expressed as

(23)

|χxy(ν τ)|2

= χxy(ν τ)χlowastxy(ν τ)

=

intintχxx(νprime τ prime)χlowastyy(νprime τ prime) ej2π(ν

primeτminusντ prime)dνprime dτ prime

This expression was derived for infinite coherent integrationtime however this is always finite In order to obtain anexpression equivalent to (23) for finite Tc the equation definedfor the delay-Doppler correlator in (3) is modulus squared asfollows

|χxy(ν τ)|2 = χxy(ν τ)χlowastxy(ν τ)

=

[1

Tc

intTc

x(t1) ylowast(t1 minus τ) eminusj2πνt1dt1

][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc

)x(t1) ylowast(t1 minus τ)

xlowast(t2) y(t2 minus τ) eminusj2πν(t1minust2)dt1 dt2

(24)


Applying the change of variables t1 = t and t2 = t minus τ prime

with dt1 dt2 = |J(t τ prime)| dt dτ prime = dt dτ prime being J the Jacobianmatrix of the transformation (24) yields

|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc

)x(t)xlowast(tminus τ prime)

1

TcΠ

(tminus τ prime

Tc

)ylowast(tminus τ) y(tminus τ minus τ prime) eminusj2πντ

primedt dτ prime

(25)

Then using the inverse of the identities

χxx(νprime τ prime) =

int1

TcΠ

(t

Tc

)x(t)xlowast(tminus τ prime) eminusj2πν

primet dt

= sinc(Tc νprime) lowastνprimeχxx(νprime τ prime)

(26a)

and

χyy(νprimeprime τ prime) =

int1

TcΠ

(tminus τ prime

Tc

)ylowast(tminus τ) y(tminus τ minus τ prime)

eminusj2πνprimeprime(tminusτ) dt

=[sinc(Tc ν

primeprime) eminusj2πνprimeprime(τ primeminusτ)

]lowastνprimeprimeχyy(νprimeprime τ prime)

(26b)

(25) can be expressed as

|χxy(ν τ)|2

=

intint [intχxx(νprime τ prime) ej2πν

primetdνprime]

[intχyy(νprimeprime τ prime) ej2πν

primeprime(tminusτ)dνprimeprime]lowasteminusj2πντ

primedt dτ prime

=

intintχxx(νprime τ prime)

intχlowastyy(νprimeprime τ prime)

[inteminusj2π(ν

primeprimeminusνprime)tdt

]ej2πν

primeprimeτeminusj2πντprimedνprimeprime dνprime dτ prime

=


intχlowastyy(νprimeprime τ prime) δ(νprimeprime minus νprime)

ej2π(νprimeprimeτminusντ prime)dνprimeprime dνprime dτ prime

=

intintχxx(νprime τ prime) χlowastyy(νprime τ prime) ej2π(ν


(27)

where χxx and χyy are windowed and delayed versions of χxxand χyy respectively Indeed the last ones correspond to theauto-correlation functions of the input signals using correlatordefined in (3) as was done in [18] Therefore if x and y arealready taken windowed by Tc and delayed by τ χxx = χxxχyy = χyy and (27) is equivalent to (23) Henceforth thislast statement will be assumed

Eventually substituting (27) (or equivalent (23)) in (20)and considering statistical independence between x and y theCWAF yields

|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=

intintE χxx(νprime τ prime)E

χlowastyy(νprime τ prime)

ej2π(ν

primeτminusντ prime)dνprimedτ prime

=

intintAx(νprime τ prime)Alowasty(νprime τ prime) ej2π(ν


(28)

where Ax and Ay are the Expected Ambiguity Function (EAF)of signals x and y respectively This expression will be usedto prove the DDM decomposition for cGNSS-R stated in (6)

Therefore using (28) the CWAF in (4) can be expressedas a function of Ayr and Ack Ack is the EAF of the trackedcode whereas Ayr is the EAF of the received signal after thefront-end obtained from

Ayr (ν τ) =1

Tc

intTc

E yr(t) ylowastr (tminus τ) eminusj2πνt dt (29)

E yr(t) ylowastr (tminus τ) can be expanded using the terms of yr in


(2) as follows

E yr(t) ylowastr (tminus τ) = E

[Lminus1suml=0

srl(t) + ir(t) + nr(t)

][Lminus1sumlprime=0

slowastrlprime (tminus τ)+ilowastr(tminusτ)+nlowastr(tminusτ)

]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+

Lminus1sumlge0l 6=lprime

Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0

E srl(t) ilowastr(tminus τ)

+

Lminus1suml=0

E srl(t)nlowastr(tminus τ)

+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)

+ E ir(t) ilowastr(tminus τ)+ E ir(t)nlowastr(tminus τ)

+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)

+ E nr(t) ilowastr(tminus τ)+ E nr(t)nlowastr(tminus τ)

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+ E ir(t) ilowastr(tminus τ)+ E nr(t)nlowastr(tminus τ)

(30)

assuming that sr ir and nr are statistically independentbetween them that E

srl(t)s

lowastrlprime

(tminus τ)

= 0 because of codeorthogonality and that E ir(t) = E nr(t) = 0 withoutloss of generality Then (29) can be expressed as

Ayr (ν τ) =

Lminus1suml=0

Asrl(ν τ) +Air (ν τ) +Anr(ν τ) (31)

and eventually the DDM decomposition stated in (6) isobtained by substituting (31) in (28) yielding

(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)

+ |χ|2irck (ν τ) + |χ|2nrck(ν τ)

= |χ|2srkck (ν τ) +

Lminus1sumlge0l 6=k

|χ|2srlck

+ |χ|2irck + |χ|2nrck

where the dependence with ν and τ in the last terms has beenomitted since it does not apply This result may be derived

intuitively taking into account the statistical independencebetween the additive terms of the input signal

APPENDIX BDERIVATION OF DDM DECOMPOSITION IN IGNSS-R

The derivation of the DDM in the case of iGNSS-R can beobtained from expressions in Appendix A First using (28)the CWAF can be expressed as

(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=

intintAyr (νprime τ prime)Alowastyd(νprime τ prime) ej2π(ν


since yr and yd are assumed statistically independent Besidesthe EAF of yd can be written as

Ayd(ν τ) =

Lminus1sumlprime=0

Asdlprime (ν τ) +Aid(ν τ) +And(ν τ) (34)

considering the same assumptions and derivations taken in(29) (30) and (31) but for the direct received signals in (5)Eventually substituting (31) and (34) in (33) the interfero-


metric CWAF yields

(35)|χ|2yryd (ν τ)

=

intint [Lminus1suml=0

Asrl(νprimeτ prime) +Air (νprimeτ prime) +Anr

(νprimeτ prime)


Alowastsdlprime (νprimeτ prime)+Alowastid(νprimeτ prime)+Alowastnd

(νprimeτ prime)

]ej2π(ν


=

intint [ Lminus1suml=0

Lminus1sumlprime=0

Asrl(νprimeτ prime)Alowastsdlprime (ν

primeτ prime)

+

Lminus1suml=0

Asrl(νprimeτ prime)Alowastid(νprimeτ prime)

+

Lminus1suml=0

Asrl(νprimeτ prime)Alowastnd

(νprimeτ prime)

+Air (νprimeτ prime)

Lminus1sumlprime=0

Alowastsdlprime (νprimeτ prime)

+Air (νprimeτ prime)Alowastid(νprimeτ prime) +Air (νprimeτ prime)Alowastnd(νprimeτ prime)

+Anr(νprimeτ prime)

Lminus1sumlprime=0

Alowastsdlprime (νprimeτ prime)+Anr

(νprimeτ prime)Alowastid(νprimeτ prime)

+Anr (νprimeτ prime)Alowastnd(νprimeτ prime)

]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0

|χ|2srlsdlprime (ν τ) +

Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)

+ |χ|2irid (ν τ) + |χ|2irnd(ν τ) +

Lminus1suml=0

|χ|2nrsdl(ν τ)

+ |χ|2nrid(ν τ) + |χ|2nrnd

(ν τ)

The last formula can be simplified in order to obtain thefollowing expression for DDM decomposition for the kthcomposite satellite signal using iGNSS-R

(36a)|χ|2yryd (ν τ) = |χ|2srksdk (ν τ) + |χ|2cross-talk (ν τ)

+ |χ|2cross-sat (ν τ)

+ |χ|2RFI (ν τ) + |χ|2noise (ν τ)

where the last four terms are defined as follows

|χ|2cross-talk (ν τ) =

Lminus1sumlge0l 6=k

|χ|2srlsdl (ν τ) (36b)

|χ|2cross-sat (ν τ) =


Lminus1sumlprime=0

|χ|2srlsdlprime (ν τ)

+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0

|χ|2srlid (ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)

+ |χ|2irid (ν τ) + |χ|2irnd(ν τ) + |χ|2nrid

(ν τ)

(36d)

and

|χ|2noise (ν τ) = |χ|2srknd(ν τ)+|χ|2nrsdk

(ν τ)+|χ|2nrnd(ν τ)

(36e)Eventually (8) has been obtained from the combination of(36a) and (36e) and omitting the dependence with ν and τ inthe terms that it does not apply

APPENDIX CDERIVATION OF THE GSSC AND REDUCTION TO SSC

A CWAF and particularly the result obtained in (27) canbe expressed in an alternative way as a function of the WVSof each one of the involved signals which are the Fouriertransform of the their respective EAF which are related asfollows [28]

Ax(ν τ) =1

Tc

intintTc

Wx(t f) eminusj2π(νtminusfτ) dt df (37)

with the WVS defined as

Wx(t f) =

intE x(t)xlowast(tminus τ) eminusj2πfτ dτ (38)

taking into account the time window Tc as in Appendix AThen substituting (37) into (28) |χ|2xy can also be written as

(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc

Wx(t f) eminusj2π(νprimetminusfτ prime)dt df

][

1

Tc

intintTc

Wy(tprime f prime) eminusj2π(νprimetprimeminusf primeτ prime)dtprime df prime

]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν

prime(tprimeminust+τ)dνprime]

[inteminusj2π(f

primeminusf+ν)τ primedτ prime]dtprime df prime dt df

=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime) δ(tprime minus t+ τ)

δ(f prime minus f + ν) dtprime df prime dt df

=1

T 2c

intintTc

Wx(t f)Wy(tminus τ f minus ν) dt df

=1

T 2c

Wx(τ ν) lowast lowastWy(τ ν)

where the reality and commutative properties of the WVS havebeen applied [28] This result resembles the definition of theSSC in [31] but in this case it involves the convolution of non-stationary spectrum functions For this reason a more generalfigure of merit is defined in this paper which is the GSSCThe GSSC between two generic signals x and y is defined as

γxy(ν τ) =1

TcWx(τ ν) lowast lowastWy(τ ν) (40)


where Wx and Wy have been normalized dividing by theirrespective power Px and Py in analogy to the classical SSCTherefore using (40) (39) yields

|χ|2xy (ν τ) =1

TcPx Py γxy(ν τ) (41)

The use of the WVS accounts for non-stationary randomprocesses whose spectra may change over time However ifthe random process x is stationary (ie its statistical momentsare constant over time) or at least quasi-stationary (ie itsstatistical moments are constant over a fraction of time Tc)the WVS becomes the stationary PSD [24]

Wx(t f) = Sx(f) (42)

and in this case the GSSC yields

(43)

γxy(ν τ) =1

TcWx(τ ν) lowast lowastWy(τ ν)

=1

Tc

intintTc


=1

Tc

intintTc

Sx(f)Sy(f minus ν) dt df

=

intSx(f)Sy(f minus ν) df

Therefore in this case the GSSC is equivalent to the classicaldefinition of SSC in [31]

γxy(ν τ) = κxy(ν) = Sx(ν) lowast Sy(ν) (44)

and a CWAF can be expressed as

|χ|2xy (ν τ) =1

TcPx Py κxy(ν) (45)

Furthermore considering an ideal transfer function for thefront-end with bandwidth Br the GSSC can be written asin (16) and the classical SSC turns to be equal to the one in(12)

REFERENCES

[1] ldquoPower Utilities Mitigating GPS Vulnerabilities and Protecting PowerUtility Network Timingrdquo Symmetricom Inc San Jose CaliforniaTechnical Report 2013 [Online] Available httpswwwaventasinccomwhitepapersWP Power Utilitiespdf[Lastaccessed]2016-07-26

[2] J Querol G F Forte and A Camps ldquoStudy of RFI signals inprotected GNSS bands generated by common electronic devices Effectson GNSS-R measurementsrdquo in 2014 IEEE Geoscience and RemoteSensing Symposium Quebec Canada 2014 pp 4050ndash4053

[3] M Wildemeersch A Rabbachin E Cano and J Fortuny ldquoInterferenceassessment of DVB-T within the GPS L1 and Galileo E1 bandrdquoin Programme and Abstract Book - 5th ESA Workshop on SatelliteNavigation Technologies and European Workshop on GNSS Signals andSignal Processing NAVITEC 2010 2010

[4] J W Betz ldquoEffect of Partial-Band Interference on Receiver Estimationof CN0 Theoryrdquo The MITRE Corporation Technical Report pp 716ndash723 2001

[5] J Betz and B Titus ldquoIntersystem and intrasystem interferencewith signal imperfectionsrdquo in PLANS 2004 Position Location andNavigation Symposium (IEEE Cat No04CH37556) 2004 pp 558ndash565

[6] C Hall and R Cordey ldquoMultistatic Scatterometryrdquo in InternationalGeoscience and Remote Sensing Symposium rsquoRemote Sensing MovingToward the 21st Centuryrsquo 1988 pp 561ndash562

[7] M Martin-Neira ldquoA Passive Reflectometry and Interferometry System(PARIS) - Application to ocean altimetryrdquo ESA journal vol 17 no 4pp 331ndash355 1993

[8] A Alonso-Arroyo A Camps H Park D Pascual R Onrubia andF Martin ldquoRetrieval of Significant Wave Height and Mean Sea SurfaceLevel Using the GNSS-R Interference Pattern Technique Results Froma Three-Month Field Campaignrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 53 no 6 pp 3198ndash3209 Jun 2015

[9] R Sabia M Caparrini and G Ruffini ldquoPotential Synergetic Useof GNSS-R Signals to Improve the Sea-State Correction in the SeaSurface Salinity Estimation Application to the SMOS Missionrdquo IEEETransactions on Geoscience and Remote Sensing vol 45 no 7 pp2088ndash2097 Jul 2007

[10] N Rodriguez-Alvarez X Bosch-Lluis A Camps M Vall-llosseraE Valencia J Marchan-Hernandez and I Ramos-Perez ldquoSoil MoistureRetrieval Using GNSS-R Techniques Experimental Results Over aBare Soil Fieldrdquo IEEE Transactions on Geoscience and RemoteSensing vol 47 no 11 pp 3616ndash3624 Nov 2009

[11] E Cardellach A Rius M Martin-Neira F Fabra O Nogues-CorreigS Ribo J Kainulainen A Camps and S DrsquoAddio ldquoConsolidating thePrecision of Interferometric GNSS-R Ocean Altimetry Using AirborneExperimental Datardquo IEEE Transactions on Geoscience and RemoteSensing vol 52 no 8 pp 4992ndash5004 Aug 2014

[12] A W Rihaczek Principles of high-resolution radar McGraw-Hill1969

[13] V U Zavorotny S Gleason E Cardellach and A Camps ldquoTutorialon Remote Sensing Using GNSS Bistatic Radar of Opportunityrdquo IEEEGeoscience and Remote Sensing Magazine vol 2 no 4 pp 8ndash45Dec 2014

[14] V U Zavorotny and A G Voronovich ldquoScattering of GPS signals fromthe ocean with wind remote sensing applicationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 38 no 2 pp 951ndash964 2000

[15] T Elfouhaily D R Thompson and L Linstrom ldquoDelay-Doppleranalysis of bistatically reflected signals from the ocean surface Theoryand applicationrdquo IEEE Transactions on Geoscience and Remote Sensingvol 40 no 3 pp 560ndash573 2002

[16] J F Marchan-Hernandez A Camps N Rodriguez-Alvarez E ValenciaX Bosch-Lluis and I Ramos-Perez ldquoAn efficient algorithm to thesimulation of delay-Doppler maps of reflected global navigation satellitesystem signalsrdquo IEEE Transactions on Geoscience and Remote Sensingvol 47 no 8 pp 2733ndash2740 2009

[17] J Ville ldquoTheorie et applications de la notion de signal analytiquerdquoCables et Transmissions vol 2A no 1 pp 61ndash74 1948

[18] P Woodward ldquoRadar Ambiguity Analysisrdquo RRE Technical Note No731 vol February 1967

[19] E Kaplan and C Hegarty Understanding GPS Principles andApplications Second Edition Artech House 2005

[20] S Sussman ldquoLeast-square synthesis of radar ambiguity functionsrdquo IRETransactions on Information Theory vol 8 no 3 1962

[21] D Pascual H Park R Onrubia A A Arroyo J Querol andA Camps ldquoCrosstalk Statistics and Impact in Interferometric GNSS-Rrdquo IEEE Journal of Selected Topics in Applied Earth Observationsand Remote Sensing pp 1ndash10 2016

[22] M Martın-Neira S DrsquoAddio C Buck N Floury and R Prieto-Cerdeira ldquoThe PARIS ocean altimeter in-Orbit demonstratorrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6 PART2 pp 2209ndash2237 2011

[23] D Pascual A Camps F Martin H Park A A Arroyo and R On-rubia ldquoPrecision bounds in GNSS-R ocean altimetryrdquo IEEE Journalof Selected Topics in Applied Earth Observations and Remote Sensingvol 7 no 5 pp 1416ndash1423 2014

[24] W Martin and P Flandrin ldquoWigner-Ville spectral analysis of nonsta-tionary processesrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 6 1985

[25] J Querol and A Camps ldquoSistema y metodo para la deteccion yeliminacion en tiempo real de interferencias de radiofrecuenciardquo ESPatent Application Submitted 2015

[26] R Onrubia J Querol D Pascual A Alonso-Arroyo H Park andA Camps ldquoDMETACAN Impact Analysis on GNSS ReflectometryrdquoIEEE Journal of Selected Topics in Applied Earth Observations andRemote Sensing pp 1ndash10 2016

[27] J Querol A Alonso-Arroyo R Onrubia D Pascual and A CampsldquoAssessment of back-end RFI mitigation techniques in passive remotesensingrdquo in 2015 IEEE International Geoscience and Remote SensingSymposium (IGARSS) Milan Italy IEEE Jul 2015 pp 4746ndash4749

[28] L Cohen Time-frequency Analysis ser Electrical engineering signalprocessing Prentice Hall PTR 1995

[29] W M Siebert ldquoStudies of Woodwardrsquos uncertainty functionrdquo MITQuarterly Progress Report pp 90ndash94 1958


[30] C Stutt ldquoA note on invariant relations for ambiguity and distancefunctionsrdquo IEEE Transactions on Information Theory vol 5 no 4pp 164ndash167 Dec 1959

[31] J A Avila-Rodrıguez ldquoOn Generalized Signal Waveforms for SatelliteNavigationrdquo PhD dissertation University FAF Munich 2008

Introduction

General SNR model for GNSS-R

Signal model and detection

DDM decomposition

SNR definition and degradation

Cross-sat SNR degradation

SSC values

Cross-sat in cGNSS-R

Cross-sat in iGNSS-R

RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions

Appendix A Derivation of DDM decomposition in cGNSS-R

Appendix B Derivation of DDM decomposition in iGNSS-R

Appendix C Derivation of the GSSC and reduction to SSC

References


(a) (b)

Fig 1 Illustrative diagram representing satellite RFI and ther-mal noise signals involved in the study of the SNR degradationfor a) cGNSS-R scenario and b) iGNSS-R scenario

R measurements is the so-called Delay-Doppler Map (DDM)which shows how the scattering process over the surfaceof interest spreads the energy of the transmitted signal inthe delay-Doppler space and how this spreading changes intime The generally accepted model for incoherent GNSS-R reflections assuming fully diffuse scattering from roughsurfaces and originally stated for sea-surface observations isthe Zavorotny-Voronovich (Z-V) model [14] The Z-V modelcan be expressed in a simplified version as a two dimensionalconvolution [15] and is equivalent to the so-called radarmapping equation [12] The model can be expressed as in[16] as lang

|Y (ν τ)|2rang

= |χ(ν τ)|2 lowast lowast Σ(ν τ) (1)

where 〈|Y (ν τ)|2〉 is the expected DDM ν and τ are theDoppler and delay variables respectively the lowastlowast operatorindicates a two-dimensional convolution in ν and τ domains|χ(ν τ)|2 is the Woodwardrsquos Ambiguity Function (WAF) ofthe navigation code and Σ(ν τ) accounts for the scatteringat the surface propagation losses antenna patterns and therest of terms in the bistatic radar equation Moreover if thereflection process is coherent because it takes place overa planar surface (eg calm water ice floes or flat lands)the measured DDM will be an attenuated version of thesignal WAF Furthermore any combination of coherent andincoherent reflection will led to a DDM that can be expressedas an attenuated and distorted version of the signal WAF

The WAF is the squared modulus of the ambiguity functionfirst proposed in [17] and it was introduced in radar theory[18] as the appropriate function to evaluate the degree ofambiguity in the measurements when the number of radarechoes is sufficiently large This approach is translated in anon-coherent detection process after the cross-correlation asused in most of GNSS receivers [19] in order to increasethe SNR and necessary in GNSS-R applications due to theeffect of speckle noise introduced in the scattering process[12] which is caused by the addition of a large number ofrandom phased scatterers over the surface of interest

According to the experiments carried out in [2] the effectof RFI signals in cGNSS-R is translated into an equivalent

rise of the noise floor and thus into a desensitization of thereceiver However these effects have not been studied in depthto the best of authorsrsquo knowledge This work aims at providinga general model that accounts for the effects of RFI signals inGNSS-R instruments operating with CDMA spread-spectrumGNSS signals including intra- and inter-GNSS interferenceand other external RFI signals

The results derived along this work are completely generalso they can be applied to any GNSS-R scenario using theproper geometry parameters However for the sake of claritythese results are illustrated using specific cases that take intoaccount no spatial filtering (isotropic antennas) perfect specu-lar reflection (no scattering) and equal received power for allsatellites (same transmission power propagation attenuationand surface reflectivity)

The outline of this paper is as follows In Section II thegeneral model of SNR degradation is developed for bothcGNSS-R and iGNSS-R techniques In Section III and Sec-tion IV the particular effects on the SNR of cross-correlationbetween GNSS signals and external RFI signals respectivelyare described in detail The main conclusions of this paper arestated in Section V

II GENERAL SNR MODEL FOR GNSS-R

A Signal model and detection

After the scattering process GNSS-R signals are collectedby the instrument antennas conditioned by the front-end stagewith bandwidth Br and finally delivered to the processingstage The complex base-band signal at this stage can beexpressed as the addition of three terms

yr(t) = sr(t)+ ir(t)+nr(t) =

Lminus1suml=0

srl(t)+ ir(t)+nr(t) (2)

where the subscript r stands for ldquoreflectedrdquo signals L is thenumber of ldquoin-viewrdquo GNSS signals srl corresponds to the lthsignal ir is the aggregate of RFI signals and nr is band-limited thermal noise (see Fig 1a)

GNSS signals are detected after the delay-Doppler corre-lator or ldquomatched filterrdquo that performs the spread-spectrumdemodulation [19] In cGNSS-R the signal after the correlatorfor a particular kth tracked signal (eg GPS L1 CA GalileoE5 Beidou B2) is obtained as

χyrck(ν τ) =1

Tc

intTc

yr(t) clowastk(tminus τ) eminusj2πνt dt (3)

where yr(t) is the received complex base-band signal definedin (2) ck isin C is the clean replica of the kth navigation codeassumed known deterministic and with normalized powerwithin the bandwidth Br ν and τ are the Doppler anddelay variables respectively and Tc is known as the coherentintegration time

As aforementioned the detection process in GNSS-R mustbe non-coherent as in radar systems so that the new observableafter the detector becomes the expected power expressed as

|χ|2yrck (ν τ) = E|χyrck(ν τ)|2

(4)





Lminus1suml=0



B DDM decomposition



Lminus1sumlge0l 6=k




(a) (b)

(c) (d)

(e)



|χ|2nrck=N0r

Tc

int ν+Br2

νminusBr2

Sck(f) df N0r

Tc (7)






) (8)

(a) (b)

(c) (d)

(e) (f)







|χ|2noise N0r

TcPsdk + Psrk

N0d

Tc+N0rN0d

BrTc (9)











(a) (b)




κxy(ν) =

intBr




A SSC values






Lminus1sumlge0l 6=k

|χ|2srlck 1

Tc

Lminus1sumlge0l 6=k

Psrl κsrlck (13)


∆kcross-sat

(1 +


N0r

)minus1 (14)





(15)


(1

TcPsrl

Lminus1sumk=0

Psdk κsrlsdk +

+N0r

TcPsdl + Psrl

N0d

Tc

)




Tracked Br[MHz]HHHH


GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068




Tracked Br[MHz]H


GPS








E5 (2046 MHz vs 51315 MHz)



A Generalized SSC


γxy(ν τ) =1

Tc

intBr

intTc



(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References





Lminus1suml=0



B DDM decomposition



Lminus1sumlge0l 6=k




(a) (b)

(c) (d)

(e)



|χ|2nrck=N0r

Tc

int ν+Br2

νminusBr2

Sck(f) df N0r

Tc (7)






) (8)

(a) (b)

(c) (d)

(e) (f)







|χ|2noise N0r

TcPsdk + Psrk

N0d

Tc+N0rN0d

BrTc (9)











(a) (b)




κxy(ν) =

intBr




A SSC values






Lminus1sumlge0l 6=k

|χ|2srlck 1

Tc

Lminus1sumlge0l 6=k

Psrl κsrlck (13)


∆kcross-sat

(1 +


N0r

)minus1 (14)





(15)


(1

TcPsrl

Lminus1sumk=0

Psdk κsrlsdk +

+N0r

TcPsdl + Psrl

N0d

Tc

)




Tracked Br[MHz]HHHH


GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068




Tracked Br[MHz]H


GPS








E5 (2046 MHz vs 51315 MHz)



A Generalized SSC


γxy(ν τ) =1

Tc

intBr

intTc



(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References




) (8)

(a) (b)

(c) (d)

(e) (f)







|χ|2noise N0r

TcPsdk + Psrk

N0d

Tc+N0rN0d

BrTc (9)











(a) (b)




κxy(ν) =

intBr




A SSC values






Lminus1sumlge0l 6=k

|χ|2srlck 1

Tc

Lminus1sumlge0l 6=k

Psrl κsrlck (13)


∆kcross-sat

(1 +


N0r

)minus1 (14)





(15)


(1

TcPsrl

Lminus1sumk=0

Psdk κsrlsdk +

+N0r

TcPsdl + Psrl

N0d

Tc

)




Tracked Br[MHz]HHHH


GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068




Tracked Br[MHz]H


GPS








E5 (2046 MHz vs 51315 MHz)



A Generalized SSC


γxy(ν τ) =1

Tc

intBr

intTc



(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References


(a) (b)




κxy(ν) =

intBr




A SSC values






Lminus1sumlge0l 6=k

|χ|2srlck 1

Tc

Lminus1sumlge0l 6=k

Psrl κsrlck (13)


∆kcross-sat

(1 +


N0r

)minus1 (14)





(15)


(1

TcPsrl

Lminus1sumk=0

Psdk κsrlsdk +

+N0r

TcPsdl + Psrl

N0d

Tc

)




Tracked Br[MHz]HHHH


GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068




Tracked Br[MHz]H


GPS








E5 (2046 MHz vs 51315 MHz)



A Generalized SSC


γxy(ν τ) =1

Tc

intBr

intTc



(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References




Tracked Br[MHz]HHHH


GPS

2046 L1CA -6181 -6828 -7021 -9241 -6826 -12450 -infin14322 L1C -6816 -6546 -7044 -8294 -6544 -10374 -106372046 L1P -6990 -7043 -7125 -7997 -7041 -10147 -101483069 L1M -8706 -8186 -7979 -7155 -8186 -8672 -8210

Galileo 14322 E1OS -6813 -6544 -7042 -8293 -6542 -10373 -1063635805 E1PRS -9908 -9424 -8644 -8672 -10115 -6845 -6998

Beidou 4092 B1 -9947 -9563 -8694 -8749 -12039 -7302 -7068




Tracked Br[MHz]H


GPS








E5 (2046 MHz vs 51315 MHz)



A Generalized SSC


γxy(ν τ) =1

Tc

intBr

intTc



(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References


(a) (b)

(c) (d)



B RFI in cGNSS-R


|χ|2irck 1

TcPir γirck (17)


∆kRFI (

1 +Pir γirckN0r

)minus1 (18)


(a) (b)

(c) (d)





C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References



C RFI in iGNSS-R


|χ|2RFI (ν τ) 1

Tc

(Lminus1suml=0

PsrlPid γsrlid +

Lminus1suml=0

PirPsdl γirsdl


Tcγirnd

+N0r

TcPid γnrid

)

(19)




(a) (b)


(a) (b)

(c) (d)






V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References




V CONCLUSIONS









(20)


χxy(ν τ) =




=




(23)

|χxy(ν τ)|2


=





=

[1

Tc

intTc


][

1

Tc

intTc


]lowast=

intint1

TcΠ

(t1Tc

)1

TcΠ

(t2Tc



(24)




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References




|χxy(ν τ)|2

=

intint1

TcΠ

(t

Tc


1

TcΠ

(tminus τ prime

Tc


primedt dτ prime

(25)



int1

TcΠ

(t

Tc


primet dt


(26a)

and


int1

TcΠ

(tminus τ prime

Tc



=[sinc(Tc ν



(26b)


|χxy(ν τ)|2

=


primetdνprime]



primedt dτ prime

=



[inteminusj2π(ν


]ej2πν


=




=



(27)



|χ|2xy (ν τ)

= E|χxy(ν τ)|2

= E



=



ej2π(ν


=



(28)



Ayr (ν τ) =1

Tc

intTc




(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References


(2) as follows


[Lminus1suml=0




]

=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)

+


Lminus1sumlprime=0

Esrl(t)s

lowastrlprime

(tminus τ)

+

Lminus1suml=0


+

Lminus1suml=0


+

Lminus1sumlprime=0

Eir(t)s

lowastrlprime

(tminus τ)


+

Lminus1sumlprime=0

Enr(t) s

lowastrlprime

(tminus τ)


=

Lminus1suml=0

Esrl(t)s

lowastrl

(tminus τ)


(30)


srl(t)s

lowastrlprime

(tminus τ)


Ayr (ν τ) =

Lminus1suml=0



(32)

|χ|2yrck (ν τ) =

Lminus1suml=0

|χ|2srlck (ν τ)



Lminus1sumlge0l 6=k

|χ|2srlck






(33)|χ|2yryd (ν τ)

= E|χyryd(ν τ)|2

=




Ayd(ν τ) =

Lminus1sumlprime=0




metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References


metric CWAF yields

(35)|χ|2yryd (ν τ)

=



(νprimeτ prime)



(νprimeτ prime)

]ej2π(ν


=


Lminus1sumlprime=0


primeτ prime)

+

Lminus1suml=0


+

Lminus1suml=0


(νprimeτ prime)


Lminus1sumlprime=0




Lminus1sumlprime=0




]ej2π(ν


=

Lminus1suml=0

Lminus1sumlprime=0


Lminus1suml=0

|χ|2srlid (ν τ)

+

Lminus1suml=0

|χ|2srlnd(ν τ) +

Lminus1suml=0

|χ|2irsdl (ν τ)


Lminus1suml=0

|χ|2nrsdl(ν τ)


(ν τ)







Lminus1sumlge0l 6=k




Lminus1sumlprime=0


+

Lminus1sumlge0l 6=k

|χ|2srlnd(ν τ) +

Lminus1sumlge0l 6=k

|χ|2nrsdl(ν τ)

(36c)

|χ|2RFI (ν τ) =

Lminus1suml=0


Lminus1suml=0

|χ|2irsdl (ν τ)


(ν τ)

(36d)

and






Ax(ν τ) =1

Tc

intintTc



Wx(t f) =



(39)|χ|2xy (ν τ)

=

intint [1

Tc

intintTc


][

1

Tc

intintTc


]lowastej2π(ν


=1

T 2c

intintTc

Wx(t f)

intintTc

Wy(tprime f prime)

[intej2πν


[inteminusj2π(f


=1

T 2c

intintTc

Wx(t f)

intintTc



=1

T 2c

intintTc


=1

T 2c



γxy(ν τ) =1




|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References



|χ|2xy (ν τ) =1



Wx(t f) = Sx(f) (42)


(43)

γxy(ν τ) =1


=1

Tc

intintTc


=1

Tc

intintTc


=





|χ|2xy (ν τ) =1



REFERENCES

































Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References




Introduction



DDM decomposition



SSC values



RFI SNR degradation

Generalized SSC

RFI in cGNSS-R

RFI in iGNSS-R

Conclusions




References

following document object identiﬁer (doi): 10.1109/jstars

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