Expressions for the Autocorrelation Function and Power...

Research ArticleExpressions for the Autocorrelation Function and Power SpectralDensity of BOC Modulation Based on Convolution Operation

Jiangang Ma1 Yikang Yang 1 Hengnian Li2 and Jisheng Li12

1School of Electronic and Information Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China2State Key Laboratory of Astronautic Dynamics Xirsquoan Satellite Control Center Xirsquoan 710049 China

Correspondence should be addressed to Yikang Yang yangyk74mailxjtueducn

Received 5 September 2019 Revised 10 April 2020 Accepted 17 April 2020 Published 5 June 2020

Academic Editor Javier Martinez Torres

Copyright copy 2020 Jiangang Ma et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present universal expressions for the autocorrelation functions (ACFs) of the Binary Offset Carrier (BOC) Multiplexed BOC(MBOC) and Alternative BOC (AltBOC) modulations based on convolution operations We also derive the expressions for thepower spectrum densities (PSDs) of these modulations using the Fourier transform of their ACFs -e results obtained in thiscontribution are useful for Global Navigation Satellite System (GNSS) signal simulation performance evaluation and high-performance acquisition and tracking algorithm design-e derivation methods of the expressions for the ACFs are common andcan be used to derive expressions for the ACFs of other BOC-based modulations

1 Introduction

-e initial modulation scheme for the Global NavigationSatellite System (GNSS) signal is binary phase shift keying(BPSK) modulation which enjoys considerable success asthe first-generation signal scheme in the Global PositioningSystem (GPS) With the development of GNSS applicationsthere are continuing expectations for improved accuracyBPSK modulation with a faster-spreading code rate im-proves accuracy but also requires larger bandwidth and haslimited improvement in multipath performance [1] Tobetter satisfy increasing application demands Binary OffsetCarrier (BOC) modulation was proposed because of itshigher accuracy better spectral isolation from heritagesignals and better multipath interference resistance andflexibility in implementation compared with BPSK modu-lation [2] -e BOC modulation has been adopted in themodernized GPS [3] European Galileo System [4] andBeiDou Navigation System [5] -e GPS L1M and L2Msignals adopt sine-phased BOC BOCsin(105) while theGalileo E6 Public Regulated Service (PRS) signal uses cosine-phased BOC BOCcos(105) and E1 PRS uses BOC-cos(1525) Some new BOC-based modulations have alsobeen adopted by GNSS signals A Time Multiplexed BOC

(TMBOC) modulation TMBOC(61111) was selected forthe GPS L1C signal and a Composite BOC (CBOC)CBOC(61111) was selected for the Galileo E1 OpenService (OS) signal [6] An Alternative BOC (AltBOC)modulation AltBOC(1510) was used to transmit GalileoE5a and E5b signals [7]

-e BOC modulation uses a square-wave subcarrier tocreate separated spectra on each side of the transmittedspectrum which provides spectral isolation from heritagesignals and leads to significant improvements in terms oftracking interference and multipath mitigation Funda-mentally these excellent characteristics of the BOC mod-ulation are determined by its autocorrelation function(ACF) and power spectrum density (PSD) -erefore in-vestigations into the ACF and PSD properties of the BOCmodulations are important Knowing the analytical ex-pressions for their ACFs it is possible in principle toquantitatively calculate the potential code tracking accuracyand estimate the signal resolution under multipath propa-gation and interference conditions For example the analyticexpressions for the ACFs of the BOC modulations are usefulfor GNSS signal simulation [8 9] and performance evalu-ation [10 11] -ey can also enable designers to developnear-optimal receiver discriminators which would ensure

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2063563 12 pageshttpsdoiorg10115520202063563

unambiguous tracking of the main peak of the ACFswherever possible andminimize the probability of capture oftheir false peaks [12ndash16] -e deduced explicit formulas forthe ACFs of the BOC modulations play an important role inGNSS signal research However there are currently nouniversal expressions for the ACFs of the BOC and BOC-based modulations -e expressions for the ACFs ofBOCsin(pnn) BOCcos(pnn) MBOC(6 1 111) andconstant and nonconstant envelope AltBOC(15 10) arepresented in [8 16ndash18] -ose expressions available for theACFs are typically given for particular cases -us there iscontinuing expectation for universal expressions for theACFs of these modulations -e PSDs of the BOC modu-lations determine the bandwidth consumption and thespectral separation with legacy signals sharing the samefrequency band which can be used to evaluate the trackingand demodulation performance of the signal in thermalnoise and interference environments -e expressions forthe PSDs of the BOC and BOC-based signals have beenpresented in [19]

In this paper we present universal expressions for theACFs of the BOCMBOC andAltBOCmodulations based onconvolution operations In [8] Sousa and Nunes derived theexpressions for the ACFs of BOCsin(pnn) and BOCcos(pnn)by using the characteristic of piecewise linearity of the ACFsWe utilize the conversion relationship between the convo-lution operation and the correlation function calculationcombined with the characteristics of randomness and sym-metry of these modulations to derive the expressions for theACFs of the BOC MBOC and AltBOC modulations In thenext section we first establish the mathematical models ofthese modulations and then present details about the ex-pressions for the ACFs of these modulations We finallyderive the expressions for the PSDs of these modulationsusing the Fourier transform of their ACFs

11 Signal Model Considering a BOC signal as the productof a pseudorandom noise (PRN) spreading code modulateddata with a square-wave subcarrier we assume that the PRNspreading code is random infinite aperiodic identically dis-tributed and independent and the signal bandwidth is infinite

111 BOC Signal -e BOC signal can be expressed as thefollowing equation

s(t) 1113944+infin

kminus infinckp t minus kTc( 1113857 (1)

where ck denotes the data-modulated PRN spreading codechip and p(middot) denotes the square-wave subcarrier symbolwith support time Tc which is defined in either sine-phasedor cosine-phased form as follows

psin(t) sign sin 2πfst( 1113857( 1113857 0le tleTc

0 otherwise1113896

pcos(t) sign cos 2πfst( 1113857( 1113857 0le tleTc

0 otherwise1113896

(2)

where fs denotes the subcarrier frequency and sign(middot) is thesign function -e BOC signal is typically denoted byBOC(fs fc) fc which equals to 1Tc is the PRN spreadingcode chip rate -e designation BOC(α β) with referencefrequency fref which equals to 1023MHz is used as theabbreviation of BOC(fs fc) the subcarrier frequencyfs αtimes fref and the PRN spreading code rate fc β times fref

-ere is also another model defining the BOC signal withp(middot) broken up into rectangular pulses with amplitudes ofplusmn1 For the sine-phased BOC signal psin(t) is broken upinto N 2αβ rectangular pulses of duration ts TcNwhich can be expressed as

psin(t) μts(t)otimes 1113944

Nminus 1

m0(minus 1)

mδ t minus mts( 1113857 (3)

where otimes denotes the convolution operation δ(middot) denotesthe impulse function and μT(t) is a rectangular pulse withsupport T defined as

μT(t) 1 0le tleT

0 otherwise1113896 (4)

Figure 1 shows the sine-phased BOC subcarrier waveFor the cosine-phased BOC signal pcos(t) is broken up

into 2N rectangular pulses of duration ts2 so pcos(t) canalso be expressed as

pcos(t) μts2(t)otimes 11139442Nminus 1

m0(minus 1)

lfloor(m+1)2rfloorδ t minusmts

21113874 1113875 (5)

where lfloormiddotrfloor denotes the floor function Figure 2 shows thecosine-phased BOC subcarrier wave

112 MBOC Signal -e MBOC signal is defined in thefrequency domain and a specific case of MBOC modula-tions denoted MBOC(61111) is recommended based onextensive work to meet technical constraints [6] whose PSD(pilot and data components together) is given by

GMBOC(61111)(f) 1011

GBOC(11)(f) +111

GBOC(61)(f)

(6)

where GBOC(αβ)(f) is the PSD of the BOC signal -ere are avariety of methods producing MBOC(61111) signalswhich are typically produced by two different methodsCBOC and TMBOC

(1) CBOC the subcarrier of a CBOC signal comprisesfour-level symbols formed by the weighted sum ofdifferent BOC subcarrier symbols and the model ofthe CBOC signal can be defined as

sCBOC(t) 1113936+infin

kminus infinckpCBOC t minus kTc( 1113857

pCBOC(t) w1pBOC(11)(t) + w2pBOC(61)(t)

⎧⎪⎪⎨

⎪⎪⎩(7)

where w1 and w2 denote amplitude weighting factorssatisfying w2

1 + w22 1 Furthermore pBOC(11)(t)

and pBOC(61)(t) can also be expressed as

2 Mathematical Problems in Engineering

pBOC(11)(t) μTc12(t)otimes 111394411

m0(minus 1)

lfloorm6rfloorδ t minusmTc

121113874 1113875


11

m0(minus 1)

mδ t minusmTc

121113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(8)

Figure 3 shows the CBOC subcarrier waves weassume that the signal power is split into 5050between data and pilot components and the CBOCsubcarrier is used on both pilot and data compo-nents -en the MBOC implementation using theCBOC method denoted CBOC(61111) can beexpressed as

pCBOC(t) μTc12(t)otimes 111394411

m0δ t minus

mTc

121113874 1113875

middot

1011

1113970

(minus 1)lfloorm6rfloor

+

111

1113970

(minus 1)m

1113890 1113891

(9)

(2) TMBOC the model of a TMBOC signal is definedas

sTMBOC(t) 1113944+infin

kminus infinckpk t minus kTc( 1113857 (10)

where different BOC subcarrier symbols are usedfor different values of k in a TMBOC time se-quence we assume that the signal power is splitinto 2575 between data and pilot componentsand the TMBOC subcarrier is used on pilotcomponents -en the MBOC signal produced byusing the TMBOC method denoted asTMBOC(61433) can be implemented by placingthe BOC(61) subcarrier symbol in locations 1 5 7and 30 of each 33 BOC(11) subcarrier symbollocation

113 AltBOC Signal AltBOC can transmit four signalcomponents at most its spectrum has two sidebands andeach sideband can be processed independently as a

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)

Figure 1 Sine-phased BOC subcarrier wave (a) sine-phased BOC subcarrier wave when N 2 6 10 (b) sine-phased BOC subcarrierwave when N 4 8 12 (c) sine-phased BOC subcarrier wave when N 3 7 11 and (d) sine-phased BOC subcarrier wave whenN 5 9 13

Mathematical Problems in Engineering 3

quadrature phase shift keying (QPSK) signal -e BOCsignal is a particular case of AltBOC when the four PRNspreading codes are made identical -e AltBOC subcarrierbased on the BOC subcarrier is complex so the spectrum ofthe signal component is not split up but only shifted to upperor lower sidebands [7] We assume that both pilot and datacomponents are introduced and four PRN spreading codesare needed -e signal model of AltBOC can be expressed as

sAltBOC 1113944+infin

kminus infinc

DL + jc

PL1113872 1113873plowastAltBOC t minus kTc( 1113857

+ cDU + jc

PU1113872 1113873pAltBOC t minus kTc( 1113857

(11)

where pAltBOC(t) pcos(t) + jpsin(t) denotes the subcarriersymbol cD

L and cDU are the data codes of lower and upper

sidebands respectively cPL and cP

U are the pilot codes of lower

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

Wav

efor

m

0 Time t (s) Tc

ts

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)

Figure 2 Cosine-phased BOC subcarrier wave (a) cosine-phased BOC subcarrier wave when N 2 6 10 (b) cosine-phased BOCsubcarrier wave whenN 4 8 12 (c) cosine-phased BOC subcarrier wave whenN 3 7 11 and (d) cosine-phased BOC subcarrierwave when N 5 9 13

0 Time t (s)

Wav

efor

m

w1 + w2

w1 ndash w2

ndashw1 ndash w2

ndashw1 + w2

CBOC (61w22rsquo+rsquo)

Tc

ndash1

0

1

(a)

Wav

efor

m

w1 ndash w2ndashw1 + w2

ndashw1 ndash w2

w1 + w2CBOC (61w2

2rsquondashrsquo)

0 Time t (s) Tc

ndash1

0

1

(b)

Figure 3 CBOC subcarrier waves


and upper sidebands respectively Furthermore pAltBOC(t)

can also be expressed as

pAltBOC(t) 1113944+infin

kminus infinμts2(t)otimes 1113944

2Nminus 1

m0(minus 1)


2minus kTc1113874 1113875 + j 1113944

+infin

kminus infinμts

(t)otimes 1113944Nminus 1

m0(minus 1)

mδ t minus mts minus kTc( 1113857 (12)

However the signal defined in (11) may be distortedwithin the high-power amplifier of the satellite payload dueto nonlinear amplification because it does not have a con-stant envelope -us the constant envelope AltBOC withfour codes is derived in [7] and defined as follows

sAltBOC 1113944

+infin

kminus infinc

DL + jc

PL1113872 1113873plowastd t minus kTc( 1113857

+ cDU + jc

PU1113872 1113873pd t minus kTc( 1113857 + c

DL + jc

PL1113872 1113873plowastp t minus kTc( 1113857

+ cDU + jc

PU1113872 1113873pp t minus kTc( 1113857

(13)

withcD

L cPUcD

UcPL

cPL cD

UcPUcD

L

cDU cD

L cPUcP

L

cPU cD

UcDL cP

L

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

and pd(t) scd(t) + jscd(t minus Ts4) and pp(t) scp(t) +

jscp(t minus Ts4) scd(t) and scp(t) are waves defined as

scd(t)

2

radic

4sign cos 2πfst minus

π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 +

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

scp(t)

minus

2

radic


π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 minus

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)

Furthermore scd(t) and scp(t) can also be expressed as

scd(t) μts4(t)otimes 11139444Nminus 1

m0

2

radic

4(minus 1)

lfloor(m+1)4rfloor+12(minus 1)

lfloor(m+2)4rfloor+

2

radic

4(minus 1)

lfloor(m+3)4rfloor1113890 1113891δ t minus

mts

41113874 1113875

scp(t) μts4(t)otimes 11139444Nminus 1

m0

2

radic

4(minus 1)

1+lfloor(m+1)4rfloor+12(minus 1)

lfloor(m+2)4rfloor+

2

radic

4(minus 1)

1+lfloor(m+3)4rfloor1113890 1113891δ t minus

mts

41113874 1113875

(16)

-e top panel of Figure 4 shows the wave of scd(t) andthe bottom panel shows the wave of scp(t) Similar to theBOC signal the AltBOC signal is generally denoted asAltBOC(p q) with the reference frequency fref whichequals to 1023MHz

12 Autocorrelation Functions -e ACF of the BOC signalcan be expressed as

R(t t + τ) E s(t)slowast(t + τ)( 1113857

E 1113944+infin

kminus infinckpt minus kTc( 1113857 1113944

+infin

lminus infinclowastl plowast

t + τ minus lTc( 1113857⎛⎝ ⎞⎠

1113944+infin

kminus infin1113944

+infin

lminus infinRc(l)E p t minus kTc( 1113857(

middot plowast

t + τ minus kTc minus lTc( 11138571113857

(17)


where Rc(l) E(ckclowastk+l) denotes the ACF of the PRNspreading code For an ideal spreading code Rc(l) becomes

Rc(l) 1 l 0

0 lne 01113896 (18)

Substituting (18) into (17) R(t t + τ) can be simplified to

R(t t + τ) 1113944+infin

kminus infinE p t minus kTc( 1113857p

lowastt + τ minus kTc( 1113857( 1113857 (19)

-e BOC signal is not a wide-sense stationary processbut rather has the following characteristics

E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)

-us the BOC signal is cyclostationary and its ACF canbe obtained by averaging the ACF over the intervalt isin [0 Tc]

R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)

Substituting (19) into (21) the ACF of the BOC signalcan be written as

R(τ) 1Tc

1113946Tc

01113944

+infin

kminus infinp t minus kTc( 1113857p

lowastt + τ minus kTc( 1113857dt

1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc

p(t)plowast(t + τ)dt

1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc

p(τ)otimesplowast(minus τ)

(22)

-erefore the ACF of the BOC signal can be obtained bythe convolution of the subcarrier symbol and the conjugateof the mirror function of the subcarrier symbol

121 BOC Signal For the sine-phased BOC subcarriersymbol according to Figure 1 we found the followingproperty

psin(t) minus psin Tc minus t( 1113857 N is even

psin Tc minus t( 1113857 N is odd1113896 (23)

-en substituting (23) into (22) the ACF of the sine-phasedBOC signal can be expressed as

RBOC sin(τ) 1Tc

(minus 1)Nminus 1

psin(τ)otimespsin(τ)otimes δ τ + Tc( 1113857

(24)

Substituting (3) into (24) RBOCsin(τ) can be derived

RBOCsin(τ) 1Tc

(minus 1)Nminus 1μts

(τ)otimes 1113944Nminus 1

m0(minus 1)

mδ τ minus mts( 1113857otimes μts(τ)

middot 1113944Nminus 1

n0(minus 1)

nδ τ minus nts( 1113857otimes δ τ + Tc( 1113857

1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts

middot τ minus (m + n + 1 minus N)ts( 1113857

(25)

where ΛT(t) is the triangle function with support 2T de-fined as

ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)

For the cosine-phased BOC subcarrier symbolaccording to Figure 2 we found the following property

pcos(t) pcos Tc minus t( 1113857 N is even

minus pcos Tc minus t( 1113857 N is odd1113896 (27)

-en substituting (27) into (22) the ACF of the cosine-phased BOC signal can be expressed as

RBOCcos(τ) 1Tc

(minus 1)N

pcos(τ)otimespcos(τ)otimes δ τ + Tc( 1113857 (28)

Substituting (5) into (28) RBOCcos(τ) can be derived

Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)

Figure 4 scd(t) and scp(t) waves


RBOCcos(τ) 1Tc

(minus 1)Nμts2(τ)otimes 1113944

2Nminus 1

m0(minus 1)

lfloor(m+1)2rfloorδ τ minusmts

21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)

lfloor(n+1)2rfloorδ τ minusnts

21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)

lfloor(m+1)2rfloor+lfloor(n+1)2rfloor+N ts

Tc

times Λts2 τ minus (m + n + 1 minus 2N)ts

21113874 1113875

(29)

-e ACFs of the sine-phased and cosine-phased BOCsignals when N 2 3 4 and 5 which are constructedaccording to (25) and (29) are presented in Figure 5

122 MBOC Signal -e CBOC subcarrier symbolaccording to the definition has the following property

pCBOC(t) minus pCBOC Tc minus t( 1113857 (30)

-en substituting (30) into (22) the ACF of the CBOCsignal can be expressed as

RCBOC(τ) minus1Tc

pCBOC(τ)otimespCBOC(τ)otimes δ τ + Tc( 1113857 (31)

Substituting (7) and (8) into (31) the ACF of the CBOCsignal can be derived

RCBOC(τ) minus1Tc

μTc12(τ)otimes 111394411

m0w1(minus 1)

lfloorm6rfloor+ w2(minus 1)

m1113872 1113873δ τ minus

mTc

121113874 1113875otimesμTc12(τ)otimes1113944

11

n0w1(minus 1)

lfloorn6rfloor+ w2(minus 1)

n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)

lfloorm6rfloor+lfloorn6rfloor+1+ w

22(minus 1)

m+n+1+2w1w2(minus 1)

lfloorm6rfloor+n+11113872 1113873 timesΛTc12 τ minus (m + n minus 11)

Tc

121113874 1113875

(32)

For the TMBOC signal its ACF can be expressed as

RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)

lfloorm6rfloor+lfloorn6rfloor+11113874 1113875

times ΛTc12 τ minus (m + n minus 11)Tc

121113874 1113875

(33)

with 0lewle 1-e ACFs of CBOC(61111rsquo+rsquo) CBOC(61111rsquominus rsquo)

and TMBOC(61433) which are constructed according to(32) and (33) are presented in Figure 6

123 AltBOC Signal -e ACF of the nonconstant envelopeAltBOC can be expressed as

RNCEminus AltBOC(τ) 2Tc

1113946Tc

0pAltBOC(t)p

lowastAltBOC(t + τ)(

+ plowastAltBOC(t)pAltBOC(t + τ)1113857dt

4 RBOCsin(τ) + RBOCcos(τ)( 1113857

(34)

Substituting (25) and (29) into (34) the ACF can bederived as follows

RNCEminus AltBOC(τ) 4 1113944Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc

times Λtsτ minus (m + n + 1 minus N)ts( 1113857⎛⎝

+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)

For the constant envelope AltBOC signal its ACF can beexpressed as

RCEminus AltBOC(τ) 2Tc

1113946Tc

0pd(t)p

lowastd (t + τ) + p

lowastd (t)pd(t + τ) + pp(t)p

lowastp (t + τ) + p

lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc

0scd(t)scd(t + τ) + scd t minus Ts4( 1113857scd t minus Ts4 + τ( 1113857 + scp(t)scp(t + τ)1113872 +scp t minus Ts4( 1113857scp t minus Ts4 + τ( 11138571113873dt

(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos

ndashTc ndash05Tc 0 05Tc TcTime t (s)

(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)

Figure 5 Normalized ACFs of BOC signals when N 2 3 4 and 5

ACF

CBOC(61111rsquo+)TMBOC(61 433)

CBOC(61111rsquondashrsquo)

ndash05

0

05

1


Figure 6 Normalized ACFs of CBOC(61111rsquo+rsquo) CBOC(61111rsquominus rsquo) and TMBOC(61433)


Let scd1(t) scd(t) scd2(t) scd(t minus Ts4) scp1(t)

scp(t) and scp2(t) scp(t minus Ts4) then the ACF of theconstant envelope AltBOC signal can be expressed as


scd1(τ)otimes scd1(minus τ) + scd2(τ)otimes scd2(minus τ)(

+ scp1(τ)otimes scp1(minus τ) + scp2(τ)otimes scp2(minus τ)1113873

4Tc

Rscd1(τ) + Rscd2

(τ) + Rscp1(τ) + Rscp2

(τ)1113874 1113875

(37)

According to the definition scd1(t) scd2(t) scp1(t) andscp2(t) have the following property

scd1(t) (minus 1)Nscd1 Tc minus t( 1113857

scd2(t) (minus 1)Nminus 1scd2 Tc minus t( 1113857

scp1(t) (minus 1)Nscp1 Tc minus t( 1113857

scp2(t) (minus 1)Nminus 1scp2 Tc minus t( 1113857

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)

-us substituting (38) into (37) the ACFs of scd1(t)scd2(t) scp1(t) and scp2(t) can be derived

Rscd1(τ) (minus 1)Nscd1(τ)otimes scd1(τ)otimes δ τ + Tc( 1113857

Rsc2(τ) (minus 1)Nminus 1scd2(τ)otimes scd2(τ)otimes δ τ + Tc( 1113857

Rscp1(τ) (minus 1)Nscp1(τ)otimes scp1(τ)otimes δ τ + Tc( 1113857

Rscp2(τ) (minus 1)Nminus 1scp2(τ)otimes scp2(τ)otimes δ τ + Tc( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)

Substituting (16) and (39) into (37) the ACF of theconstant envelope AltBOC signal can be obtained

RCEminus AltBOC(τ) 11139444Nminus 1

m01113944

4Nminus 1

n0

12(minus 1)

lfloorm4rfloor+lfloorn4rfloor+Nminus 11113874

+12(minus 1)

lfloor(m+2)4rfloor+lfloor(n+2)4rfloor+N

+(minus 1)lfloor(m+1)4rfloor+lfloor(n+3)4rfloor+N

1113873ts

Tc


41113874 1113875

(40)

-e ACFs of nonconstant and constant envelope Alt-BOC(1510) which are constructed according to (35) and(40) are presented in Figure 7

13 Power Spectral Density According to the WienerndashKhinchin theorem [20] the PSD of the BOC signal is theFourier transform of its ACF

G(f) FT[R(τ)] (41)

131 BOC Signal Considering that N is either even or oddthe PSD of the sine-phased BOC signal can be derived asfollows

GBOCsin(f)

1Tc

sin πfts( 1113857sin πfTc( 1113857

πf cos πfts( 11138571113888 1113889

2

N is even

1Tc

sin πfts( 1113857cos πfTc( 1113857

πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)

Considering that N is either even or odd the PSD of thecosine-phased BOC signal can be derived as follows

GBOCcos(f)

1Tc

1 minus cos πfts( 1113857( 1113857sin πfTc( 1113857

πfcos πfts( 11138571113888 1113889

2

N is even

1Tc

1 minus cos πfts( 1113857( 1113857cos πfTc( 1113857

πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

-e PSDs of the sine-phased and cosine-phased BOCsignals when N 2 3 4 and 5 which are constructedaccording to (42) and (43) are presented in Figure 8

132 MBOC Signal For the CBOC signal its PSD can beexpressed as

GCBOC(f) 1Tc

times w211 minus cos πfTc( 1113857( 1113857

2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888

+ 2w1w21 minus cos πfTc( 1113857( 1113857sin πfTc( 1113857

sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)

For the TMBOC signal its PSD can be expressed as

GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)

-e PSDs of CBOC(61111rsquo+rsquo) CBOC(61111rsquominus rsquo)and TMBOC(61433) which are constructed according to(44) and (45) are presented in Figure 9

133 AltBOC Signal Considering that N is either even orodd the PSD of the nonconstant envelope AltBOC signalcan be derived as follows


Nonconstant envelopeConstant envelope

ACF


ndash05

0

05

1

Figure 7 Normalized ACFs of nonconstant and constant envelope AltBOC(1510)

PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

80ndash4ndash8 4Frequency (MHz)

(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos

ndash4 0 4 8ndash8Frequency (MHz)

(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)

Figure 8 PSDs of BOC signals when N 2 3 4 and 5


GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+

1 minus cos πfts( 1113857( 11138572sin2 πfTc( 1113857

π2f2cos2 πfts( 11138571113888 1113889 N is even

4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+

1 minus cos πfts( 1113857( 11138572cos2 πfTc( 1113857

π2f2cos2 πfts( 11138571113888 1113889 N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Considering that N is either even or odd the PSD of theconstant envelope AltBOC signal can be derived as follows

GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857

π2f2cos2 πfts( 1113857times 32cos2 πf ts4( 1113857( 1113857 + 16 sin πfts( 1113857sin πf ts2( 1113857( 1113857 + 161113872 1113873 N is even

1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857

π2f2cos2 πfts( 1113857times 32cos2 πf ts4( 1113857( 1113857 + 16 sin πfts( 1113857sin πf ts2( 1113857( 1113857 + 161113872 1113873 N is odd

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)

-e PSDs of nonconstant and constant envelope Alt-BOC(1510) which are constructed according to (46) and(47) are presented in Figure 10

2 Conclusion

-is paper derives explicit analytical expressions for theACFs of the BOC MBOC and AltBOC modulations Byexpressing the ACFs as the sum of triangle functions it ispossible to determine expressions for the PSDs additionally-e derive method uses the conversion relationship betweenthe convolution operation and the calculation of the cor-relation function-emethod is common and can be used toderive analytical expressions for the ACFs of other BOC-based modulations With the knowledge of the analyticalexpressions for the ACFs for a satellite navigation system itis possible to calculate the potential code tracking accuracy

quantitatively and to estimate the signal resolution undermultipath propagation and interference conditions De-signers can consciously overcome difficulties when devel-oping a discriminator for a receiver to ensure unambiguoustracking of the main peak of ACFs and minimize theprobability of capture of their false peaks Moreover theanalytical expressions for the ACFs are useful for GNSSsignal simulation and performance evaluation

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

CBOC(61111rsquo+rsquo)TMBOC(61 433)CBOC(61111rsquondashrsquo)

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


Figure 9 PSDs of CBOC(61111rsquo+rsquo) CBOC(61111rsquominus rsquo) andTMBOC(61433)

PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

Figure 10 PSDs of nonconstant and constant envelopeAltBOC(1510)


Acknowledgments

-is study was funded by the National Key Research andDevelopment Program of China (no 2017YFC1500904) theNational Key Research and Development Program of China(no 2016YFB0501301) and the National 973 Program ofChina (no 613237201506)

References

[1] J W Betz ldquoBinary offset carrier modulations for radio-navigationrdquo Navigation vol 48 no 4 pp 227ndash246 2001

[2] J W Betz ldquo-e offset carrier modulation for GPS modern-izationrdquo in Proceedings of ION NTM 1999 pp 639ndash648 SanDiego CA USA 1999

[3] GPS ldquoGPS space segmentuser segment L1C interface IS-GPS-800B technical reportrdquo in Global Positioning SystemDirectorate Systems Engineering and Integration SpringerBerlin Germany 2011

[4] I Galileo Galileo Open Service Signal in Space InterfaceControl Document European Space AgencyEuropean GNSSSupervisory Authority New York NY USA 2015

[5] BeiDou BeiDou Navigation Satellite System Signal in SpaceInterface Control Document Open Service Signal B1C (Version10) China Satellite Navigation Office Beijing China 2017

[6] G W Hein J A Avila-Rodriguez S Wallner et al ldquoMBOCthe new optimized spreading modulation recommended forGALILEO L1 OS and GPS L1Crdquo in Proceedings of IEEEIONPLANS 2006 pp 883ndash892 San Diego CA USA 2006

[7] L Lestarquit G Artaud and J L Issler ldquoAltBOC forDummies or Everything You Always Wanted to Know aboutAltBOCrdquo in Proceedings of the ION GNSS 2008 pp 961ndash970Savannah GA USA September 2008

[8] F M G Sousa and F D Nunes ldquoNew expressions for theautocorrelation function of BOC GNSS signalsrdquo Navigationvol 60 no 1 pp 1ndash9 2013

[9] J L Garrison ldquoA statistical model and simulator for ocean-reflected GNSS signalsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 54 no 10 pp 6007ndash6019 2016

[10] R Luo Y Xu and H Yuan ldquoPerformance evaluation of thenew compound-carrier-modulated signal for future naviga-tion signalsrdquo Sensors vol 16 no 2 p 142 2016

[11] J Zhang Z Yao and M Lu ldquoGeneralized theory anddecoupled evaluation criteria for unmatched despreading ofmodernized GNSS signalsrdquo Sensors vol 16 no 7 p 11282016

[12] K Rouabah and D Chikouche ldquoGPSGalileo multipath de-tection and mitigation using closed-form solutionsrdquo Math-ematical Problems in Engineering vol 2009 Article ID1068702009 2009

[13] F Liu and Y Feng ldquoA new acquisition algorithm withelimination side peak for all BOC signalsrdquo MathematicalProblems in Engineering vol 2015 Article ID 140345 2015

[14] Y Feng F Liu X Yao and X Zhang ldquoAn acquisition al-gorithm with NCCFR for BOC modulated signalsrdquo Journal ofElectrical and Computer Engineering vol 2017 Article ID4241297 2017

[15] Z Yao Y Gao Y Gao and M Lu ldquoGeneralized theory ofBOC signal unambiguous tracking with two-dimensionalloopsrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 53 no 6 pp 3056ndash3069 2017

[16] M S Yarlykov ldquoCorrelation functions of BOC and AltBOCsignals as the inverse Fourier transforms of energy spectrardquo

Journal of Communications Technology and Electronicsvol 61 no 8 pp 857ndash876 2016

[17] F D Nunes F M G Sousa and J M N Leitao ldquoGatingfunctions for multipath mitigation in GNSS BOC signalsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 43 no 3 pp 951ndash964 2007

[18] E S Lohan A Lakhzouri and M Renfors ldquoComplex double-binary-offset-carrier modulation for a unitary character-isation of Galileo and GPS signalsrdquo IEE Proceedings - RadarSonar and Navigation vol 153 no 5 pp 403ndash408 2006

[19] E Rebeyrol C Macabiau L Lestarquit et al ldquoBOC PowerSpectrum Densitiesrdquo BOC in Proceedings of ION NTM 2005pp 24ndash26 San Diego CA USA January 2005

[20] L W Couch M Kulkarni and U S Acharya Digital andAnalog Communication Systems Prentice-Hall Upper SaddleRiver NJ USA 1997


unambiguous tracking of the main peak of the ACFswherever possible andminimize the probability of capture oftheir false peaks [12ndash16] -e deduced explicit formulas forthe ACFs of the BOC modulations play an important role inGNSS signal research However there are currently nouniversal expressions for the ACFs of the BOC and BOC-based modulations -e expressions for the ACFs ofBOCsin(pnn) BOCcos(pnn) MBOC(6 1 111) andconstant and nonconstant envelope AltBOC(15 10) arepresented in [8 16ndash18] -ose expressions available for theACFs are typically given for particular cases -us there iscontinuing expectation for universal expressions for theACFs of these modulations -e PSDs of the BOC modu-lations determine the bandwidth consumption and thespectral separation with legacy signals sharing the samefrequency band which can be used to evaluate the trackingand demodulation performance of the signal in thermalnoise and interference environments -e expressions forthe PSDs of the BOC and BOC-based signals have beenpresented in [19]

In this paper we present universal expressions for theACFs of the BOCMBOC andAltBOCmodulations based onconvolution operations In [8] Sousa and Nunes derived theexpressions for the ACFs of BOCsin(pnn) and BOCcos(pnn)by using the characteristic of piecewise linearity of the ACFsWe utilize the conversion relationship between the convo-lution operation and the correlation function calculationcombined with the characteristics of randomness and sym-metry of these modulations to derive the expressions for theACFs of the BOC MBOC and AltBOC modulations In thenext section we first establish the mathematical models ofthese modulations and then present details about the ex-pressions for the ACFs of these modulations We finallyderive the expressions for the PSDs of these modulationsusing the Fourier transform of their ACFs

11 Signal Model Considering a BOC signal as the productof a pseudorandom noise (PRN) spreading code modulateddata with a square-wave subcarrier we assume that the PRNspreading code is random infinite aperiodic identically dis-tributed and independent and the signal bandwidth is infinite

111 BOC Signal -e BOC signal can be expressed as thefollowing equation

s(t) 1113944+infin

kminus infinckp t minus kTc( 1113857 (1)

where ck denotes the data-modulated PRN spreading codechip and p(middot) denotes the square-wave subcarrier symbolwith support time Tc which is defined in either sine-phasedor cosine-phased form as follows

psin(t) sign sin 2πfst( 1113857( 1113857 0le tleTc

0 otherwise1113896

pcos(t) sign cos 2πfst( 1113857( 1113857 0le tleTc

0 otherwise1113896

(2)

where fs denotes the subcarrier frequency and sign(middot) is thesign function -e BOC signal is typically denoted byBOC(fs fc) fc which equals to 1Tc is the PRN spreadingcode chip rate -e designation BOC(α β) with referencefrequency fref which equals to 1023MHz is used as theabbreviation of BOC(fs fc) the subcarrier frequencyfs αtimes fref and the PRN spreading code rate fc β times fref

-ere is also another model defining the BOC signal withp(middot) broken up into rectangular pulses with amplitudes ofplusmn1 For the sine-phased BOC signal psin(t) is broken upinto N 2αβ rectangular pulses of duration ts TcNwhich can be expressed as

psin(t) μts(t)otimes 1113944

Nminus 1

m0(minus 1)

mδ t minus mts( 1113857 (3)

where otimes denotes the convolution operation δ(middot) denotesthe impulse function and μT(t) is a rectangular pulse withsupport T defined as

μT(t) 1 0le tleT

0 otherwise1113896 (4)

Figure 1 shows the sine-phased BOC subcarrier waveFor the cosine-phased BOC signal pcos(t) is broken up

into 2N rectangular pulses of duration ts2 so pcos(t) canalso be expressed as

pcos(t) μts2(t)otimes 11139442Nminus 1

m0(minus 1)


21113874 1113875 (5)

where lfloormiddotrfloor denotes the floor function Figure 2 shows thecosine-phased BOC subcarrier wave

112 MBOC Signal -e MBOC signal is defined in thefrequency domain and a specific case of MBOC modula-tions denoted MBOC(61111) is recommended based onextensive work to meet technical constraints [6] whose PSD(pilot and data components together) is given by

GMBOC(61111)(f) 1011

GBOC(11)(f) +111

GBOC(61)(f)

(6)

where GBOC(αβ)(f) is the PSD of the BOC signal -ere are avariety of methods producing MBOC(61111) signalswhich are typically produced by two different methodsCBOC and TMBOC

(1) CBOC the subcarrier of a CBOC signal comprisesfour-level symbols formed by the weighted sum ofdifferent BOC subcarrier symbols and the model ofthe CBOC signal can be defined as

sCBOC(t) 1113936+infin

kminus infinckpCBOC t minus kTc( 1113857

pCBOC(t) w1pBOC(11)(t) + w2pBOC(61)(t)

⎧⎪⎪⎨

⎪⎪⎩(7)

where w1 and w2 denote amplitude weighting factorssatisfying w2

1 + w22 1 Furthermore pBOC(11)(t)

and pBOC(61)(t) can also be expressed as



m0(minus 1)


121113874 1113875


11

m0(minus 1)

mδ t minusmTc

121113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(8)



m0δ t minus

mTc

121113874 1113875

middot

1011

1113970


+

111

1113970

(minus 1)m

1113890 1113891

(9)






0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)





kminus infinc

DL + jc


+ cDU + jc


(11)





0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

Wav

efor

m

0 Time t (s) Tc

ts

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)


0 Time t (s)

Wav

efor

m

w1 + w2

w1 ndash w2

ndashw1 ndash w2

ndashw1 + w2


Tc

ndash1

0

1

(a)

Wav

efor

m


ndashw1 ndash w2

w1 + w2CBOC (61w2

2rsquondashrsquo)

0 Time t (s) Tc

ndash1

0

1

(b)







2Nminus 1

m0(minus 1)


2minus kTc1113874 1113875 + j 1113944

+infin

kminus infinμts


m0(minus 1)



sAltBOC 1113944

+infin

kminus infinc

DL + jc


+ cDU + jc

PU1113872 1113873pd t minus kTc( 1113857 + c

DL + jc


+ cDU + jc

PU1113872 1113873pp t minus kTc( 1113857

(13)

withcD

L cPUcD

UcPL

cPL cD

UcPUcD

L

cDU cD

L cPUcP

L

cPU cD

UcDL cP

L

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



scd(t)

2

radic


π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 +

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

scp(t)

minus

2

radic


π4


2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)



m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875


m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875

(16)




E 1113944+infin


+infin


t + τ minus lTc( 1113857⎛⎝ ⎞⎠

1113944+infin

kminus infin1113944

+infin


middot plowast


(17)



Rc(l) 1 l 0

0 lne 01113896 (18)


R(t t + τ) 1113944+infin




E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)


R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)


R(τ) 1Tc

1113946Tc

01113944

+infin



1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc


1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc


(22)






RBOC sin(τ) 1Tc

(minus 1)Nminus 1


(24)


RBOCsin(τ) 1Tc



m0(minus 1)



n0(minus 1)


1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts


(25)


ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)





RBOCcos(τ) 1Tc

(minus 1)N



Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)



RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References
























m0(minus 1)


121113874 1113875


11

m0(minus 1)

mδ t minusmTc

121113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(8)



m0δ t minus

mTc

121113874 1113875

middot

1011

1113970


+

111

1113970

(minus 1)m

1113890 1113891

(9)






0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)





kminus infinc

DL + jc


+ cDU + jc


(11)





0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

Wav

efor

m

0 Time t (s) Tc

ts

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)


0 Time t (s)

Wav

efor

m

w1 + w2

w1 ndash w2

ndashw1 ndash w2

ndashw1 + w2


Tc

ndash1

0

1

(a)

Wav

efor

m


ndashw1 ndash w2

w1 + w2CBOC (61w2

2rsquondashrsquo)

0 Time t (s) Tc

ndash1

0

1

(b)







2Nminus 1

m0(minus 1)


2minus kTc1113874 1113875 + j 1113944

+infin

kminus infinμts


m0(minus 1)



sAltBOC 1113944

+infin

kminus infinc

DL + jc


+ cDU + jc

PU1113872 1113873pd t minus kTc( 1113857 + c

DL + jc


+ cDU + jc

PU1113872 1113873pp t minus kTc( 1113857

(13)

withcD

L cPUcD

UcPL

cPL cD

UcPUcD

L

cDU cD

L cPUcP

L

cPU cD

UcDL cP

L

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



scd(t)

2

radic


π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 +

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

scp(t)

minus

2

radic


π4


2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)



m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875


m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875

(16)




E 1113944+infin


+infin


t + τ minus lTc( 1113857⎛⎝ ⎞⎠

1113944+infin

kminus infin1113944

+infin


middot plowast


(17)



Rc(l) 1 l 0

0 lne 01113896 (18)


R(t t + τ) 1113944+infin




E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)


R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)


R(τ) 1Tc

1113946Tc

01113944

+infin



1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc


1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc


(22)






RBOC sin(τ) 1Tc

(minus 1)Nminus 1


(24)


RBOCsin(τ) 1Tc



m0(minus 1)



n0(minus 1)


1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts


(25)


ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)





RBOCcos(τ) 1Tc

(minus 1)N



Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)



RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References

























kminus infinc

DL + jc


+ cDU + jc


(11)





0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(a)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(b)

Wav

efor

m

0 Time t (s) Tc

ts

ndash1

0

1

(c)

0 Time t (s) Tc

ts

Wav

efor

m

ndash1

0

1

(d)


0 Time t (s)

Wav

efor

m

w1 + w2

w1 ndash w2

ndashw1 ndash w2

ndashw1 + w2


Tc

ndash1

0

1

(a)

Wav

efor

m


ndashw1 ndash w2

w1 + w2CBOC (61w2

2rsquondashrsquo)

0 Time t (s) Tc

ndash1

0

1

(b)







2Nminus 1

m0(minus 1)


2minus kTc1113874 1113875 + j 1113944

+infin

kminus infinμts


m0(minus 1)



sAltBOC 1113944

+infin

kminus infinc

DL + jc


+ cDU + jc

PU1113872 1113873pd t minus kTc( 1113857 + c

DL + jc


+ cDU + jc

PU1113872 1113873pp t minus kTc( 1113857

(13)

withcD

L cPUcD

UcPL

cPL cD

UcPUcD

L

cDU cD

L cPUcP

L

cPU cD

UcDL cP

L

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



scd(t)

2

radic


π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 +

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

scp(t)

minus

2

radic


π4


2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)



m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875


m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875

(16)




E 1113944+infin


+infin


t + τ minus lTc( 1113857⎛⎝ ⎞⎠

1113944+infin

kminus infin1113944

+infin


middot plowast


(17)



Rc(l) 1 l 0

0 lne 01113896 (18)


R(t t + τ) 1113944+infin




E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)


R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)


R(τ) 1Tc

1113946Tc

01113944

+infin



1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc


1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc


(22)






RBOC sin(τ) 1Tc

(minus 1)Nminus 1


(24)


RBOCsin(τ) 1Tc



m0(minus 1)



n0(minus 1)


1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts


(25)


ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)





RBOCcos(τ) 1Tc

(minus 1)N



Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)



RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References



























2Nminus 1

m0(minus 1)


2minus kTc1113874 1113875 + j 1113944

+infin

kminus infinμts


m0(minus 1)



sAltBOC 1113944

+infin

kminus infinc

DL + jc


+ cDU + jc

PU1113872 1113873pd t minus kTc( 1113857 + c

DL + jc


+ cDU + jc

PU1113872 1113873pp t minus kTc( 1113857

(13)

withcD

L cPUcD

UcPL

cPL cD

UcPUcD

L

cDU cD

L cPUcP

L

cPU cD

UcDL cP

L

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



scd(t)

2

radic


π4

1113874 11138751113876 1113877 +12sign cos 2πfst( 11138571113858 1113859 +

2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

scp(t)

minus

2

radic


π4


2

radic

4sign cos 2πfst +

π4

1113874 11138751113876 1113877 0le tleTc

0 others

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(15)



m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875


m0

2

radic

4(minus 1)


lfloor(m+2)4rfloor+

2

radic

4(minus 1)


mts

41113874 1113875

(16)




E 1113944+infin


+infin


t + τ minus lTc( 1113857⎛⎝ ⎞⎠

1113944+infin

kminus infin1113944

+infin


middot plowast


(17)



Rc(l) 1 l 0

0 lne 01113896 (18)


R(t t + τ) 1113944+infin




E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)


R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)


R(τ) 1Tc

1113946Tc

01113944

+infin



1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc


1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc


(22)






RBOC sin(τ) 1Tc

(minus 1)Nminus 1


(24)


RBOCsin(τ) 1Tc



m0(minus 1)



n0(minus 1)


1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts


(25)


ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)





RBOCcos(τ) 1Tc

(minus 1)N



Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)



RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References
























Rc(l) 1 l 0

0 lne 01113896 (18)


R(t t + τ) 1113944+infin




E s t + Tc( 1113857( 1113857 E(s(t))

R t + Tc t + τ + Tc( 1113857 R(t t + τ)(20)


R(τ) 1Tc

1113946Tc

0R(t t + τ)dt (21)


R(τ) 1Tc

1113946Tc

01113944

+infin



1Tc

1113944

+infin

kminus infin1113946

(1minus k)Tc

minus kTc


1Tc

1113946+infin

minus infinp(t)p

lowast(t + τ)dt

1Tc


(22)






RBOC sin(τ) 1Tc

(minus 1)Nminus 1


(24)


RBOCsin(τ) 1Tc



m0(minus 1)



n0(minus 1)


1113944

Nminus 1

m01113944

Nminus 1

n0(minus 1)

m+n+Nminus 1 ts

Tc

times Λts


(25)


ΛT(t)

1 minus|t|

T |t|ltT

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(26)





RBOCcos(τ) 1Tc

(minus 1)N



Wav

efor

m

05

ndash05

( 2 + 1)2

ndash( 2 + 1)2

scd (t)

0 Time t (s) Tc

ndash1

0

1

(a)

Wav

efor

m

0 Time t (s)

05

ndash05

( 2 ndash 1)2

ndash( 2 ndash 1)2

scp (t)

Tc

ndash1

0

1

(b)



RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References























RBOCcos(τ) 1Tc


2Nminus 1

m0(minus 1)


21113874 1113875otimes μts2(τ) 1113944

2Nminus 1

n0(minus 1)


21113874 1113875otimes δ τ + Tc( 1113857

12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875

(29)





RCBOC(τ) minus1Tc



RCBOC(τ) minus1Tc


m0w1(minus 1)


m1113872 1113873δ τ minus

mTc


11

n0w1(minus 1)


n1113872 1113873δ τ minus

nTc

121113874 1113875otimesδ τ+ Tc( 1113857

112

1113944

11

m01113944

11

n0w

21(minus 1)


22(minus 1)



Tc

121113874 1113875

(32)


RTMBOC(τ) 112

1113944

11

m01113944

11

n0

3w

4(minus 1)

m+n+1+ 1 minus

3w

41113874 1113875(minus 1)



121113874 1113875

(33)





1113946Tc

0pAltBOC(t)p




(34)



m01113944

Nminus 1

n0(minus 1)

m+n+(Nminus 1) ts

Tc


+12

1113944

2Nminus 1

m01113944

2Nminus 1

n0(minus 1)


Tc


21113874 1113875⎞⎠

(35)



1113946Tc

0pd(t)p




lowastp (t)pp(t + τ)1113872 1113873dt

4Tc

1113946Tc


(36)


ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References























ACF

N = 2

ndash1

05

0

05

1

BOCsinBOCcos


(a)

N = 3

ACF

ndash1

05

0

05

1

BOCsinBOCcos


(b)

BOCsinBOCcos

N = 4

ACF

ndash1

05

0

05

1


(c)

N = 5A

CF

ndash1

05

0

05

1

BOCsinBOCcos


(d)


ACF



ndash05

0

05

1









4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References




























4Tc

Rscd1(τ) + Rscd2


(τ)1113874 1113875

(37)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(38)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(39)



m01113944

4Nminus 1

n0

12(minus 1)


+12(minus 1)



1113873ts

Tc


41113874 1113875

(40)



G(f) FT[R(τ)] (41)


GBOCsin(f)

1Tc


πf cos πfts( 11138571113888 1113889

2

N is even

1Tc


πf cos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)


GBOCcos(f)

1Tc


πfcos πfts( 11138571113888 1113889

2

N is even

1Tc


πfcos πfts( 11138571113888 1113889

2

N is odd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



GCBOC(f) 1Tc


2

sin2 πf Tc12( 1113857( 1113857+ w

22

sin2 πfTc( 1113857

cos2 πf Tc12( 1113857( 11138571113888


sin πf Tc12( 1113857( 1113857cos πf Tc12( 1113857( 11138571113889

middotsin πf Tc12( 1113857( 1113857

πf1113888 1113889

2

(44)


GTMBOC(f) 1Tc

3w

4sin2 πf Tc12( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc12( 1113857( 11138571113888

+ 1 minus3w

41113874 1113875

sin2 πf Tc2( 1113857( 1113857sin2 πfTc( 1113857

π2f2cos2 πf Tc2( 1113857( 11138571113889

(45)





ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References
























ACF


ndash05

0

05

1


PSD

(dB)

N = 2

BOCsinBOCcos

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(a)

N = 3

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

BOCsinBOCcos


(b)

N = 4

PSD

(dB)

BOCsinBOCcos


ndash110

ndash100

ndash90

ndash80

ndash70

ndash60

(c)

N = 5

BOCsinBOCcos

PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60


(d)



GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References























GNCEminus AltBOC(f)

4Tc

sin2 πfts( 1113857sin2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



4Tc

sin2 πfts( 1113857cos2 πfTc( 1113857

π2f2cos2 πfts( 1113857+



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)


GCEminus AltBOC(f)

1Tc

sin2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


1Tc

cos2 πfTc( 1113857sin2 πf ts4( 1113857( 1113857


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(47)


2 Conclusion



Data Availability





PSD

(dB)

ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



PSD

(dB)



ndash110

ndash100

ndash90

ndash80

ndash70

ndash60



Acknowledgments


References























Acknowledgments


References























Expressions for the Autocorrelation Function and Power...

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Transcript of Expressions for the Autocorrelation Function and Power...