research article a flexible modulation scheme design for c...

Research ArticleA Flexible Modulation Scheme Design for C-Band GNSS Signals

Rui Xue Qing-ming Cao and Qiang Wei

College of Information amp Communication Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Rui Xue xuerui0216hotmailcom

Received 9 February 2015 Revised 10 June 2015 Accepted 10 June 2015

Academic Editor Emiliano Mucchi

Copyright copy 2015 Rui Xue et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the spectrum congestion of current navigation signals in L-band C-band has been taken into consideration as a candidatefrequency band for global navigation satellite system (GNSS) As is known modulation scheme is the core part of signal structureand how to design a modulation waveform that could make full use of narrow bandwidth 20MHz and satisfy the constraintcondition of frequency compatibility in C-band is the main research content of this paper In view of transmission characteristicsand constraint condition of compatibility in C-band multi-h continuous phase modulation (CPM) is proposed as a candidatemodulation schemeThen the classical channel capacity estimation and a comprehensive evaluation criterion for GNSSmodulationsignals are employed to assess the proposed scheme in the aspects of the capacity over additive white Gaussian noise (AWGN)tracking accuracy multipath mitigation antijamming and so on Simulation results reveal that through optimizing the numberand size of modulation indexes the flexible scheme could offer better performance in terms of code tracking multipath mitigationand antijamming compared with other candidates such as MSK and GMSK while maintaining high band efficiency and moderateimplementation complexity of receiver Moreover this paper also provides a reference for next generation modulation signals inC-band

1 Introduction

With the rapid development of global navigation satellitesystems (GNSS) namely GPS GLONASS Galileo andCom-pass together with various regional navigation satellite sys-tems and space-based augmentation systems the number ofnavigation signals in space is anticipated to be over 400 by2030 [1] which will further aggravate an already crowdedradio spectrum in L-band (1164sim1610MHz) Consequentlyimproving the signal spectral efficiency [2] and providingnew bands for radio navigation satellite service (RNSS) arethe main means to solve the above problem and the fre-quency band between 5010MHz and 5030MHz offering abandwidth of 20MHz has been already allocated as C-bandportion by International Telecommunication Union (ITU)for RNSS applications Unfortunately the use of C-band navi-gation signals offers both advantages and drawbacks [3]However technological progress in electronics and spacecraftmight balance some of the disadvantages and C-band hascaused great interests as a candidate for the future GNSSservices [4]

Although the performance of signals inC-band is difficultto surpass L-band [5] signal combination of L- and C-band could improve positioning accuracy and timing per-formance and promote the comprehensive performance ofRNSS [6 7] Predictably the multiband combined navigationand compatibility among different navigation systems willbecome research hotspots in the future As is known to allmodulation scheme whose quality completely determinesthe upper bound of GNSS performance is the core partof signal structure and power spectrum envelopes decidedby modulation waveforms could play a dominant role intracking accuracymultipathmitigation antijamming and soforthAt present how to design amodulation scheme that canmake full use of bandwidth 20MHz and satisfy the constraintcondition of frequency compatibility is the main challengesubject of C-band signal system

The compatibility between newly introduced signalsin C-band and the existing signals in adjacent frequencybands such as radio astronomy (RA) band and microwavelanding system (MLS) must be carefully analyzed beforeany new band for satellite navigation is put into use [8]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 165097 8 pageshttpdxdoiorg1011552015165097

2 Mathematical Problems in Engineering

Therefore band-limited signals or signals with continuousphase should be the first choice for C-band signal system Inaddition the signals are easily distorted due to the nonlineareffect of the high power amplifier (HPA) [9] and a constantenvelope modulation reveals the excellent characteristic thatHPA can run in saturation or close to saturation which couldincrease the total efficiency of amplification By the aboveanalysis a modulation scheme with constant envelope andcontinuous phase will be a top priority for C-band signal

Binary offset carrier (BOC) modulation and BOC-derived modulations such as multiplexed BOC (MBOC) andalternative BOC (AltBOC) have been alreadywidely acceptedby next generation GNSS Nevertheless the BOC modulatedsignals have larger spectral side lobes and are more prone toproduce larger interferences with signals of other coexistingnavigation systems [10] Reference [11] has pointed out thatminimum shift keying (MSK) can offer better ranging pre-cision within a certain bandwidth and introduce less interfer-ence to other signals Subsequently MSK MSK-BOC andGaussian-filtered MSK (GMSK) have been proposed aspotential candidates for C-band signals by Galileo systemHowever the tracking bias of MSK-BOC modulated signalshas not been solved effectively so far [12] and hardwareimplementation ofGMSK is complicated and its tracking per-formance is hard to optimize Moreover MSK and GMSK arethe special cases in continuous phasemodulation (CPM) butthey do not possess the optimal performance in CPM family

Based on the above problems a special subclass of CPMthat is multi-ℎ CPM is presented as a modulation waveformfor C-band signal due to its many excellent characteristicssuch as flexible parameter adjusting high power and spec-trum efficiency a large number of alternative waveformsand being easily compatible with existing signals The restof this paper is organized as follows Section 2 describesmathematical model and an improved calculation of powerspectra for multi-ℎ CPM signals The capacity of multi-ℎ CPM signals is derived in Section 3 Section 4 providesperformance evaluation criterion forGNSSmodulation Sim-ulation results are discussed in Section 5 Finally we concludethe paper in Section 6

2 Multi-ℎ CPM Signal

21 Mathematical Model The complex-baseband multi-ℎCPM signal is given by

119904 (119905120572) = exp 119895120601 (119905120572) (1)

where 120601(119905120572) is the information-carrying phase denoted as

120601 (119905120572) = 2120587infin

sum

119894=minusinfin

120572

119894ℎ

119894119902 (119905 minus 119894119879) (2)

where 119879 is the symbol duration 120572 = 120572119894 are the information

symbols in the 119872-ary alphabet plusmn1 plusmn3 plusmn(119872 minus 1) 119902(119905)is the phase pulse and ℎ

119894

119873ℎminus1119894=0 is a set of 119873

ℎmodulation

index where ℎ119894is constant in symbol duration 119879 In this

paper the underlined subscript notation in (2) is defined asmodulo-119873

ℎ that is 119899 ≜ 119899 mod 119873

ℎ We always assume ℎ

119894le 1

because power spectra of CPM will behave like BOC signalswith spectrum splitting whenmodulation index is more thanone which goes against compatibility with other signals inrelatively narrow bandwidth

The phase pulse 119902(119905) and the frequency pulse 119891(119905) arerelated by

119902 (119905) = int

119905

minusinfin

119891 (120591) 119889120591 (3)

The frequency pulse is supported over the time interval (0119871119879) and is subject to the constraints

int

119871119879

0119891 (120591) 119889120591 = 119902 (119871119879) =

12

(4)

When 119871 = 1 the signal is called full-response formats andwhen 119871 gt 1 the signal is called partial-response formatsSome general pulse shapes are length-119871119879 rectangular(119871REC) length-119871119879 raised-cosine (119871RC) and Gaussian

In light of the constraints on 119891(119905) and 119902(119905) (2) can berewritten as

120601 (119905120572) = 120579 (119905120572119899) + 120579

119899minus119871

= 2120587119899

sum

119894=119899minus119871+1120572

119894ℎ

119894119902 (119905 minus 119894119879)

+(120587

119899minus119871

sum

119894=minusinfin

120572

119894ℎ

119894) mod 2120587

(5)

where the term 120579(119905120572119899) is a function of the 119871 symbols being

modulated by the phase pulse For ℎ119894= 2119896119894119901 (119896119894 119901 integers)

the phase state 120579119899minus119871

takes on 119901 distinct values The signalis described by a trellis containing 119901119872

119871minus1 states with 119872

branches at each state Each branch is defined by the (119871 + 1)-tuple 120590

119899= (120579

119899minus119871 120572

119899minus119871+1 120572119899minus119871+2 120572119899)

22 Easy Calculation of Power Spectra Reference [13] pro-vides an easy method which allows a fast calculation of thespectra of 119872-ary multi-ℎ CPM signals under the generalhypothesis of (1) 119872-ary symbols (119872 a power of 2) (2)arbitrary pulse shaping (3) partial response signaling (4) anarbitrary set ofmodulation indexesTheprocedure of the easymethod is as follows

Let 119909(119905) = exp119895120601(119905) be the complex envelope of thetransmitted signal 119909(119905) To find the baseband equivalent 119878(119891)of the power spectrum 119878(119891) of the signal 119909(119905) we use the so-called ldquoautocorrelation-basedrdquo approach

The autocorrelation function of 119909(119905) is defined as

120588 (1199051 1199052)

≃ 119864 119909 (1199051) 119909lowast

(1199052)

= 119864exp[119895+infin

sum

119894=minusinfin

2120587120572119894ℎ

119894(119902 (1199051 minus 119894119879) minus 119902 (1199052 minus 119894119879))]

(6)

Mathematical Problems in Engineering 3

As we know ℎ119894+119870

= ℎ

119894and 119870119879 = superbaud period and the

independence of the data symbols we are able to put 120588(1199051 1199052)in the form that is

120588 (1199051 1199052)

=

+infin

prod

119894=minusinfin

119872minus1sum

119903=0

1119872

exp 1198952120587 (2119903 minus119872 + 1) ℎ119894times [119902 (1199051 minus 119894119879) minus 119902 (1199052 minus 119894119879)]

=

+infin

prod

119894=minusinfin

119901 (1199051 minus 119894119870119879 1199052 minus 119894119870119879)

(7)

where

119901 (119886 119887) ≃

119870minus1prod

119898=0

sin (2120587119872ℎ

119898(119902 (119886) minus 119902 (119887)))

119872 sin (2120587ℎ119898(119902 (119886) minus 119902 (119887)))

(8)

The process 119909(119905) is cyclostationary in the wide sense withperiod 119870119879 and average autocorrelation is expressed as

119877 (120591) ≃

1119870119879

int

119870119879

0120588 (119905 119905 + 120591) 119889119905

(9)

It is seen that 119901(119886 119887) = 1 when 119886 119887 ge (119871 + 119870 minus 1)119879 or119886 119887 le 0 The number of factors in (7) can be reduced and (9)is rewritten as

119877 (120591) =

1119870119879

int

119870119879

0

119898+1prod

119894=lceil(1minus119871)119870rceil119901 (119905 minus 119894119870119879 119905 + 120591

1015840

minus (119894 minus 119898)119870119879) 119889119905

(10)

where 119898 ≃ lfloor120591119870119879rfloor 1205911015840 ≃ 120591 minus 119898119870119879 (0 le 120591

1015840

lt 119870119879) lfloor119910rfloor ≃maximum integer le 119910 and lceil119910rceil ≃minimum integer ge 119910

The power spectral density (PSD) 119878(119891) can be derivedfrom

119877(120591) by Fourier transform (FT) that is

119878 (119891) = 2

int

(1minus119904)119870119879

0

119877 (120591) cos 2120587119891120591119889120591

+R[[

int

(2minus119904)119870119879(1minus119904)119870119879

119877 (120591) exp (minus1198952120587119891120591) 119889120591

(1 minus 1198621119870 exp (minus1198952120587119891119870119879))]

]

(11)

where 119904 ≃ lfloor(1 minus 119871)119870rfloor 1198621119870 ≃ prod

119870minus1119898=0[sin(119872120587ℎ

119898)

(119872sin(120587ℎ119898))] and R(sdot)denotes real part of complex

In this paper we always assume the spread spectrumcode rate and 119872 in all schemes are equal to 5 times 1023MHzand 2 respectively The PSD of CPM signals with differentparameters are shown in Figure 1 As seen in Figure 1 theCPM signals using RC pulse can effectively decrease PSDamplitude of side lobes and concentrate more energy intomain lobe compared to REC pulse when the modulationindexes are the same Moreover it is noteworthy that energyin main lobe tends to be more centralized and decay rateof side lobes is almost the same when the average of allelements in set ℎ1 ℎ2 is getting larger In terms of mainlobe energy the CPM signals with RC or REC are moreconcentrated than GMSK Also power fluctuation of side

0 10 20 30Frequency (MHz)

Pow

er sp

ectr

al d

ensit

y (d

BWH

z)

MSKGMSK

minus30 minus20 minus10

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

minus70

minus60

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

Figure 1 The PSD of different CPM signals

lobes in GMSK decays faster than CPM signals with RECinferior to CPM adopting RC pulse

The PSD of modulated signals have a direct effect ontracking performance ability of multipath mitigation andantijamming Particularly the PSD of modulated signalsshould have the following characteristics in order to meet thestrict compatibility condition constraints of the C-band (1)the majority of power should be concentrated into the mainlobe (2) a stronger spectrum roll-off in side lobes should beprovided in order to significantly cut down spectrum leakageor interference to RA and MLS services

In the CPM family multi-ℎ CPM signals as a special sub-class still maintain all of excellent properties just as a generalCPM In addition a properly chosen cyclic set of modula-tion indexes results in delayed merging of neighboring phasetrellis paths and therefore provides a larger minimumEuclidean distance than conventional single-ℎ CPM whichleads to improving error performance and making its spec-trummore compact and out-of-band rejection better Underthe condition of limited bandwidth and power just assatellite navigation systems multi-ℎ CPM has more excellenttransmission performance than single-ℎ CPM [14] By theabove analysis ldquoflexiblerdquo means that a particular multi-ℎCPM can be designed according to some restrictions in somepractical systems through optimizing the number and size ofmodulation indexes

3 Capacity of CPM over AWGN Channel

Modern satellite navigation systems employ high efficientchannel codes which operate very close to capacity Thismakes the channel capacity an effective parameter in evalu-ating the performance of a modulation scheme It has beenshown that a CPM modulator can be decomposed into acontinuous phase encoder (CPE) followed by a memorylessmodulator (MM) Since the CPE is Markov the CPM mod-ulator can be described by a finite-state machine (FSM) with


Continuousphase modulator

AWGN channelFSMC

XNminus1

0

VNminus1

0 WNminus1

0

YNminus1

0

Figure 2 FSMC model for CPM over AWGN channel

119901119872

119871minus1 states which is ergodic and stationaryThus the CPMmodulator together with AWGNchannel can bemodeled as aFSM channel (FSMC) shown in Figure 2 and the capacity ofCPM is themutual information between the input and outputsequences of FSMC [15]

Suppose (1198830 1198831 119883119873minus1) is the transmitted symbolsequence and (1198840 1198841 119884119873minus1) is the received sequenceof symbols Let 119883119895

119894denote the sequence of symbols 119883

119894

119883

119894+1 119883119895 Let 119868(119883119873minus10 119884

119873minus10 ) denote the mutual informa-

tion between the transmitted and the received sequencesThen the channel capacity can be calculated as

119862iud = lim119873rarrinfin

1119873

119868 (119883

119873minus10 119884

119873minus10 ) (12)

From the chain rule of entropy

119868 (119883

119873minus10 119884

119873minus10 ) = 119867(119883

119873minus10 ) minus119867(119883

119873minus10 | 119884

119873minus10 )

=

119873minus1sum

119894=0119867(119883

119894)

minus

119873minus1sum

119894=0119867(119883

119894| 119883

119894minus10 119884

119873minus10 )

(13)

where 119867(119883119894| 119883

119894minus10 ) = 119867(119883

119894) is used Because input 119883

119894is

independent and uniformly distributed (iud) over 119872 con-stellation points 119867(119883

119894) = log2119872 and all that remains to be

calculated is119867(119883119894| 119883

119894minus10 119884

119873minus10 ) From the definition of con-

ditional entropy

119867(119883

119894| 119883

119894minus10 119884

119873minus10 ) = minus119864 [log2119901 (119883119894 | 119883

119894minus10 119884

119873minus10 )] (14)

The above expectation can be found using Monte Carlo inte-gration To compute the probability 119901(119883

119894| 119883

119894minus10 119884

119873minus10 ) we

first apply Bayesrsquo rule to obtain the following equation

119901 (119883

119894| 119883

119894minus10 119884

119873minus10 ) =

119901 (119883

119894 119883

119873minus10 119884

119873minus10 )

sum

119883119894119901 (119883

119894 119883

119894minus10 119884

119873minus10 )

(15)

Similar to [16] a BCJR-like method can be used to compute119901(119883

119894 119883

119873minus10 119884

119873minus10 ) Rather than explicitly calculating the

denominator in (15) its value is found such that sum119883119894119901(119883

119894|

119883

119894minus10 119884

119873minus10 ) = 1 More details of calculation process for 119901(119883

119894

119883

119873minus10 119884

119873minus10 ) are shown in [17]

It is obvious that approximate estimation of channelcapacity is derived from (12)sim(14) and computational com-plexity of (14) is reduced from exponential order of 119873to linear due to BCJR introduced into Arnold algorithm

0 5 10 15 200

05

1

15

2

25

3

Chan

nel c

apac

ity (b

itsy

mbo

l)

M = 8 h = 12

M = 8 h = 14

M = 8 h = 15

M = 8 h = 17

M = 8 h = 16

M = 8 h = 12 14

M = 4 h = 12

M = 4 h = 14

M = 4 h = 12 14

EsN0 (dB)minus5minus10minus15

Figure 3 The channel capacity of different CPM signals

According to Monte Carlo theory the approximate estima-tion of channel capacity is infinite approaching to true valuewhen length of119873 is closing to infinite In fact the deviation ofchannel capacity descends with the increasing of119873 and tendsto be stable when119873 is larger than 5000

Figure 3 shows the channel capacity of different CPMsignals with 2RC We can see from Figure 3 that capacity isgradually improved with the increasing of 119872 when mod-ulation index is fixed at a certain value Under the samecondition of119872 the greater average of all elements in set ℎ

119894

is the earlier channel capacity convergesThat is to say119864119904119873

0

required to achieve the same capacity is getting smaller withthe increasing of average of all elements in set ℎ

119894 if other

conditions are the same For instance the channel capacityof signal with h = 12 14 has to be intermediate betweenℎ = 12 and ℎ = 14 when119872 is the same

4 Performance Evaluation forModulation Signal

Generally an excellent performance evaluation criterion formodulation signal is characterized by objectivity accuracycomprehensiveness adaptability and so forth A rather com-prehensive performance evaluation method for GNSS mod-ulation has been proposed by [18] and the tracking accuracymultipathmitigation and antijamming performance are usedas performance evaluation standard which provides sig-nificant references on satellite navigation signal design Nextwe will expound the above performance indexes one by one

41 Tracking Accuracy Gabor bandwidth and code trackingerrors can be used as the reference indexes for evaluating thetracking performance Based on a coherent early-minus-late


(EML) code tracking loop the code tracking errors in AWGNchannel are defined as follows [19]

120590

2120576=

119861

119871(1 minus 05119861

119871119879

119894) int

1198612minus1198612 119866 (119891) sin

2(120587119891119889) 119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 119891119866 (119891) sin (120587119891119889) 119889119891]

2 (16)

where 119861119871denotes the tracking loop bandwidth 119866(119891) is the

PSD of the signal that is normalized to unit power overinfinite transmission bandwidth and symmetric to the carrierfrequency119889denotes the correlation time spacing between theearly and late reference signals 1198621198730 is the carrier-to-noiseratio (CNR) 119861 is the receiver prefiltering bandwidth and 119879

119894

is the coherent integration timeAssuming that the signal is ideal and 119861

119871119879

119894is small

enough (16) in the limit defined as 119889 is vanishingly small andbecomes

lim119889rarr 0

120590

2120576cong 120590

2CRB

=

119861

119871int

1198612minus1198612 119866 (119891) (120587119891119889)

2119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 120587119891

2119889119866 (119891) 119889119891]

2

=

119861

119871120587

2119889

2int

1198612minus1198612 119891

2119866 (119891) 119889119891

(2120587)2 (1198621198730) 1205872119889

2[int

1198612minus1198612 119891

2119866 (119891) 119889119891]

2

=

119861

119871

(2120587)2 (1198621198730) int1198612minus1198612 119891

2119866 (119891) 119889119891

(17)

with

Δ119891Gabor = radicint1198612

minus1198612119891

2119866 (119891) 119889119891

(18)

where 1205902CRB and Δ119891Gabor are referred to as the Cramer-Raolower bound and Gabor bandwidth respectively

From (17) it is obvious that the Gabor bandwidth can beapproximately interpreted as Cramer-Rao lower bound andthe greater the Gabor bandwidth the better the code trackingaccuracy

42 Multipath Error Envelopes Themultipath errors are oneof the dominant error sources in GNSS The multipath errorenvelopes and average multipath errors are valuable indexesto evaluate the multipath mitigation ability The receivedbaseband signals disturbed by other reflected signals can beexpressed as follows [20]

119903 (119905) = 11988601198901198951205950119909 (119905 minus 1205910) +

119873

sum

119899minus1119886

119899119890

119895120595119899119909 (119905 minus 120591

119899) (19)

where 1198860 1205910 and1205950 are the amplitude delay and phase of thedirect signal Likewise 119886

119899 120591119899 and120595

119899are the amplitude delay

and phase of reflected signals and119873 denotes the number of

reflected signals If we only consider one reflected path anduse a coherent EML discriminator the discriminator outputcan be described as follows [21]

119863 (120576) = 1198860 [R(120576 minus

119889

2)minusR(120576 +

119889

2)]

+ 1198861 [R(120576 minusΔ120591minus

119889

2)minusR(120576 minusΔ120591+

119889

2)]

times cos (Δ120595) equiv 0

(20)

where Δ120591 and Δ120595 are the delay and carrier phase differencebetween the multipath and direct signals with Δ120591 = 1205911 minus 1205910and Δ120595 = 1205951 minus 1205950 separately and R(sdot) and 120576 denote auto-correlation function of the signal and the multipath errorsrespectively To explore the theoretical lower bound of themultipath errors the cases Δ120595 = 0 and Δ120595 = 120587 correspond-ing to the worst multipath errors are considered By the defi-nition of the Maclaurin series (20) can be simplified as

119863 (120576) asymp 119863 (0) +1198631015840 (0) 120576 (21)

According to the Wiener-Khintchine theorem 119863(0) and119863

1015840

(0) can be obtained by substituting ldquo0rdquo into (20) and thecorresponding first-order derivative that is

119863 (0) = plusmn 21198861 int1198612

minus1198612119866 (119891) sin (minus2120587119891Δ120591) sin (120587119891119889) 119889119891

119863

1015840

(0)

= 41205871198860 int1198612

minus1198612119891119866 (119891) sin (120587119891119889) 119889119891

plusmn 41205871198861 int1198612

minus1198612119891119866 (119891) cos (minus2120587119891Δ120591) sin (120587119891119889) 119889119891

(22)

By combining (21)sim(22) the multipath error envelopescan be estimated eventually by

120576 (Δ120591)

asymp

plusmn119886 int

1198612minus1198612 119866 (119891) sin (2120587119891Δ120591) sin (120587119891119889) 119889119891

2120587int1198612minus1198612 119891119866 (119891) sin (120587119891119889) [1 plusmn 119886 cos (2120587119891Δ120591)] 119889119891

(23)

with 119886multipath to direct ratio (MDR) namely 119886 = 11988611198860The corresponding average multipath errors can be given

by

120576av (Δ1205911015840

) =

1Δ120591

1015840int

Δ1205911015840

0

1003817

1003817

1003817

1003817

1205760 (Δ120591)1003817

1003817

1003817

1003817

+

1003817

1003817

1003817

1003817

120576

120587(Δ120591)

1003817

1003817

1003817

1003817

2119889Δ120591

(24)

where 1205760(Δ120591) and 120576120587(Δ120591) are the multipath errors under theconditions Δ120595 = 0 and Δ120595 = 120587

43 Antijamming The narrowband-jamming and matched-spectrum-jamming are the main threats to the pseudocodeand carrier tracking as well as the demodulation processIn order to effectively evaluate the antijamming ability of


Table 1 The parameters in evaluation test

Transmitted bandwidthMHz BMHz C1198730dBsdotHz MDRdB Δ120591m dchip 119861

119871Hz

2046 2046 20sim40 minus6 0sim300 01 1

navigation signals against the above interferences the paperintroduces four parameters including anti-narrowband-jamming merit factor 119876DemAJNW and anti-matched-spec-trum-jamming merit factor 119876DemAJMS for the demodulationprocess and the anti-narrowband-jamming merit factors119876CTAJNW and anti-matched-spectrum-jamming merit factor119876CTAJMS for the code tracking process [22] They are definedas

119876DemAJNW = 10times log10 [1

119877 timesmax [119866119904(119891)]

] (dB)

119876DemAJMS = 10times log10 [

[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)

119876CTAJNW = 10times log10 [

[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)

119876CTAJMS = 10times log10 [

[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)

where 119866119904(119891) and 119877 are the PSD of the desired signals and

symbol rate separately Also the greater the merit factors thestronger the antijamming abilities

5 Simulation Results and Analysis

In order to test the validity of multi-ℎ CPM as a candidatemodulation for C-band signals we employ Monte Carlosimulations to evaluate the candidatersquos performance in theaspects of tracking accuracy multipath mitigation and anti-jamming The parameters in all simulations are listed inTable 1 as is mentioned above 119861 is the receiver prefilteringbandwidth and is equal to 2046MHz 1198621198730 is the carrier-to-noise ratio (CNR) in AWGN channel and limited in scope[20 40] dBsdotHz multipath to direct ratio (MDR) is set asminus6 dB Δ120591 is the delay between the multipath and directsignals and is within 0sim300m the correlation time spacingbetween the early and late reference signals 119889 is 01 chip andtracking loop bandwidth 119861

119871is 1 Hz

Figure 4 displays the Gabor bandwidth of CPM signalswith different parameters As we see fromFigure 4 the Gaborbandwidth of CPM signals with RC is larger than others withREC in the same condition of modulation index when 119861 isvaried from 0 to 25MHz If other parameters of CPM signalsare identical the large average of all elements in set ℎ

119894 leads

to a great Gabor bandwidth The Gabor bandwidth order isas follows when 119861 is fixed at 2046MHz RCh = 12 34 gtRECh = 12 34 gt RCℎ = 12 gtGMSK gtMSK gt RCh =14 12 gt RECh = 14 12

0 5 10 15 20 25Receiver prefiltering bandwidth (MHz)

Gab

or b

andw

idth

(MH

z)MSKGMSK

prefilteringbandwidth

0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

Figure 4 The Gabor bandwidth of different CPM signals

20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4

Carrier-to-noise ratio (dBmiddotHz)

Figure 5 The code tracking error of different CPM signals

The relation curves of code tracking error and 1198621198730 areshown in Figure 5 Obviously the code tracking error of allCPM schemes is descending with the increasing of1198621198730 andtends to be stable when 1198621198730 is more than 32 dBsdotHz and allother parameters are the same as the previous simulationThe code tracking error of CPM signals with RC is relativelysmall compared to other CPM signals with REC in the samecondition of modulation index In addition the large averageof all elements in set ℎ

119894 results in a small code tracking error


0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

Figure 6 The multipath error envelopes of different CPM signals

if other parameters of CPM signals are identical The codetracking error order is as follows when 1198621198730 is varied from20 to 40 dBsdotHz RCh = 12 34 lt RECh = 12 34 ltRCℎ = 12 ltGMSK ltMSK lt RCh = 14 12 lt RECh =14 12

Figure 6 shows the relation curves of multipath errorenvelopes of different CPM schemes and path difference thatis product of Δ120591 and velocity of light In Figure 6 we observethat the multipath error of CPM signals with RC is smallerthan other CPM signals with REC in the same condition ofmodulation index when path difference is in the region of1 to 150m Similarly a great average of all elements in setℎ

119894 tends to be a small multipath error if other parameters

of CPM signals are the same When path difference is variedfrom 1 to 60m the CPM signal with RCh = 12 34 hasthe best performance of multipath mitigation in the aboveschemes andGMSK is superior to other scheme in the regionof 60 to 150m

Figure 7 shows the comparison of antijamming of CPMsignals with different parameters As we see from Figure 7in the aspect of 119876DemAJNW GMSK has the best performancein all CPM schemes and the CPM signal with RCh =12 34 has almost the same performance as that withRECh = 12 34 but both of them are inferior to GMSKIn the aspect of 119876CTAJNW the CPM signal employed RCh= 12 34 has the best performance in all CPM schemesIn the aspect of 119876DemAJMS the CPM scheme adopted RCh= 12 34 parameter has better performance than othersinferior to GMSK In the aspect of119876CTAJMS the CPM schemewith RCh = 12 34 has the equivalent performance toGMSK and both of them are superior to others After theabove analysis we can obtain that the CPM scheme withRCh = 12 34 is relatively close to GMSK in termsof antijamming ability If we continue to rationally adjustmodulation index set there must be an optimized multi-ℎCPM signal whose antijamming performance is superior toGMSK

DemAJNW CTAJNW DemAJMS CTAJMS0

10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

Figure 7The comparison of antijamming of different CPM signals

6 Conclusions

Multi-ℎ CPM signals as a special subclass in CPM familystill maintain all of excellent characteristics just as a generalCPM such as constant envelope high power and bandwidthefficiency suitable for adopting HPA Besides they also havesome unique properties that is flexible parameter adjustinga large number of alternative waveforms being easily com-patible with existing signals and deployment capabilities ofmultiband and multifrequency and so on Therefore multi-ℎ CPM will bring extensive application prospects for futureGNSS signals

According to transmission properties and the constraintcondition of compatibility in C-band multi-ℎ CPM is pro-posed as a candidate modulation waveform for C-band sig-nals and a proper tradeoff among channel capacity band effi-ciency navigation performance and implementation com-plexity of receiver can be realized by optimizing modulationindexes Simulation results show that multi-ℎ CPM signalsnot only provide better performance in terms of trackingaccuracy multipath mitigation and antijamming comparedto MSK GMSK and other single-ℎ CPM signals under theconstraint on transmitted bandwidth in C-band but alsohave relatively high channel capacity and band efficiencywhile taking into account receiver complexity Furthermorethe proposed modulation scheme could offer new ideas andfeasibility demonstration for signal system design of nextgeneration GNSS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (Grant no 61403093) Science


Foundation of Heilongjiang Province of China for ReturnedScholars (Grant no LC2013C22) the Assisted Project byHeilongjiang Province of China Postdoctoral Funds for Sci-entific Research Initiation (Grant no LBH-Q14048) andChina Fundamental Research Funds for Central Universities(Grant no HEUCF1508)

References

[1] J W Betz ldquoSignal structures for satellite-based navigation pastpresent and futurerdquo Inside GNSS vol 8 pp 34ndash42 2013

[2] X Liu M Liang Y Morton P Closas T Zhang and Z HongldquoPerformance evaluation of MSK and OFDM modulations forfuture GNSS signalsrdquo GPS Solutions vol 18 no 2 pp 163ndash1752014

[3] E Colzi G Lopez-Risueno J Samson et al ldquoAssessment ofthe feasibility of GNSS in C-bandrdquo in Proceedings of the 26thAIAA International Communications Satellite Systems Confer-ence (ICSSC rsquo08) pp 1ndash15 June 2008

[4] M Liu X Zhan W Li and M Chen ldquoMSK-BCS modulationfor GNSS radio frequency compatibility in C bandrdquo Journal ofNetworks vol 9 no 10 pp 2713ndash2720 2014

[5] I Mateu M Paonni and J L Issler ldquoA search for spectrum-GNSS signals in S bandrdquo Inside GNSS vol 5 no 7 pp 46ndash532010

[6] J-L Issler M Paonni and B Eissfeller ldquoToward centimetricpositioning thanks to L- and S-band GNSS and to meta-GNSSsignalsrdquo in Proceedings of the 5th ESA Workshop on SatelliteNavigation Technologies and European Workshop on GNSSSignals and Signal Processing (NAVITEC rsquo10) pp 1ndash8 December2010

[7] P Qing ldquoThe research of the signal in the S frequency bandrdquo inProceedings of the China Satellite Navigation Conference (CSNCrsquo13) vol S2 pp 1ndash5 May 2013

[8] M Liu X Zhan W Li and M Chen ldquoA compatibility analysisbetweenGNSS and radio astronomymicrowave landing systemin C bandrdquo Journal of Aeronautics Astronautics and Aviationvol 46 no 2 pp 102ndash107 2014

[9] X-L Hu Y-H Ran Y-Q Liu and T Ke ldquoNew option for mod-ulation in GNSS signal designrdquo Systems Engineering and Elec-tronics vol 32 no 9 pp 1962ndash1967 2010

[10] S Attia K Rouabah D Chikouche and M Flissi ldquoSide peakcancellation method for sine-BOC(mn)-modulated GNSS sig-nalsrdquo EURASIP Journal on Wireless Communications and Net-working vol 2014 article 34 2014

[11] J A Avila-Rodriguez S Wallner J H Won and B EissfellerldquoStudy on a Galileo signal and service plan for C-bandrdquo inProceedings of the 21th International Technical Meeting of theSatellite Division pp 2515ndash2530 September 2008

[12] X Gu R Li D Zeng and D Yao ldquoStudy on MSK modulationand tracking technique for satellite navigation systemsrdquo inProceedings of the IET International Radar Conference April2013

[13] M Luise ldquoEasy calculation of power spectra for multi-h phase-coded signalsrdquo Electronics Letters vol 21 no 14 pp 608ndash6091985

[14] S Zhong C Yang and J Zhang ldquoSymbol timing estimationwith multi-h CPM signalsrdquo Journal of Networks vol 9 no 4pp 921ndash926 2014

[15] R Chen F Wei M Huang B Li and B Bai ldquoOn the capacityof CPM over Rayleigh fading channelsrdquo in Proceedings of

the International Conference on Wireless Communications andSignal Processing (WCSP 12) pp 1ndash15 Huangshan ChinaOctober 2012

[16] D M Arnold H-A Loeliger P O Vontobel A Kavcic andW Zeng ldquoSimulation-based computation of information ratesfor channels with memoryrdquo IEEE Transactions on InformationTheory vol 52 no 8 pp 3498ndash3508 2006

[17] S ChengM C Valenti andD Torrieri ldquoCoherent continuous-phase frequency-shift keying parameter optimization and codedesignrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 1792ndash1802 2009

[18] Z Tang H Zhou and X Hu ldquoResearch on performance evalu-ation of COMPASS signalrdquo Scientia Sinica Physica Mechanicaamp Astronomica vol 5 pp 592ndash602 2010

[19] J W Betz and K R Kolodziejski ldquoGeneralized theory of codetracking with an early-late discriminator Part I Lower boundand coherent processingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 45 no 4 pp 1538ndash1556 2009

[20] M Sahmoudi and R J Landry ldquoMultipath mitigation tech-niques using maximum likelihood principlerdquo Inside GNSS vol8 pp 24ndash29 2008

[21] E D Kaplan and C J Hegarty Understanding GPS Principleand Application Artech House Boston Mass USA 2006

[22] R Xue Y Sun and D Zhao ldquoCPM signals for satellite navi-gation in the S and C bandsrdquo Sensors vol 15 no 6 pp 13184ndash13200 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Therefore band-limited signals or signals with continuousphase should be the first choice for C-band signal system Inaddition the signals are easily distorted due to the nonlineareffect of the high power amplifier (HPA) [9] and a constantenvelope modulation reveals the excellent characteristic thatHPA can run in saturation or close to saturation which couldincrease the total efficiency of amplification By the aboveanalysis a modulation scheme with constant envelope andcontinuous phase will be a top priority for C-band signal

Binary offset carrier (BOC) modulation and BOC-derived modulations such as multiplexed BOC (MBOC) andalternative BOC (AltBOC) have been alreadywidely acceptedby next generation GNSS Nevertheless the BOC modulatedsignals have larger spectral side lobes and are more prone toproduce larger interferences with signals of other coexistingnavigation systems [10] Reference [11] has pointed out thatminimum shift keying (MSK) can offer better ranging pre-cision within a certain bandwidth and introduce less interfer-ence to other signals Subsequently MSK MSK-BOC andGaussian-filtered MSK (GMSK) have been proposed aspotential candidates for C-band signals by Galileo systemHowever the tracking bias of MSK-BOC modulated signalshas not been solved effectively so far [12] and hardwareimplementation ofGMSK is complicated and its tracking per-formance is hard to optimize Moreover MSK and GMSK arethe special cases in continuous phasemodulation (CPM) butthey do not possess the optimal performance in CPM family

Based on the above problems a special subclass of CPMthat is multi-ℎ CPM is presented as a modulation waveformfor C-band signal due to its many excellent characteristicssuch as flexible parameter adjusting high power and spec-trum efficiency a large number of alternative waveformsand being easily compatible with existing signals The restof this paper is organized as follows Section 2 describesmathematical model and an improved calculation of powerspectra for multi-ℎ CPM signals The capacity of multi-ℎ CPM signals is derived in Section 3 Section 4 providesperformance evaluation criterion forGNSSmodulation Sim-ulation results are discussed in Section 5 Finally we concludethe paper in Section 6

2 Multi-ℎ CPM Signal

21 Mathematical Model The complex-baseband multi-ℎCPM signal is given by

119904 (119905120572) = exp 119895120601 (119905120572) (1)

where 120601(119905120572) is the information-carrying phase denoted as

120601 (119905120572) = 2120587infin

sum

119894=minusinfin

120572

119894ℎ

119894119902 (119905 minus 119894119879) (2)

where 119879 is the symbol duration 120572 = 120572119894 are the information

symbols in the 119872-ary alphabet plusmn1 plusmn3 plusmn(119872 minus 1) 119902(119905)is the phase pulse and ℎ

119894

119873ℎminus1119894=0 is a set of 119873

ℎmodulation

index where ℎ119894is constant in symbol duration 119879 In this

paper the underlined subscript notation in (2) is defined asmodulo-119873

ℎ that is 119899 ≜ 119899 mod 119873

ℎ We always assume ℎ

119894le 1

because power spectra of CPM will behave like BOC signalswith spectrum splitting whenmodulation index is more thanone which goes against compatibility with other signals inrelatively narrow bandwidth

The phase pulse 119902(119905) and the frequency pulse 119891(119905) arerelated by

119902 (119905) = int

119905

minusinfin

119891 (120591) 119889120591 (3)

The frequency pulse is supported over the time interval (0119871119879) and is subject to the constraints

int

119871119879

0119891 (120591) 119889120591 = 119902 (119871119879) =

12

(4)

When 119871 = 1 the signal is called full-response formats andwhen 119871 gt 1 the signal is called partial-response formatsSome general pulse shapes are length-119871119879 rectangular(119871REC) length-119871119879 raised-cosine (119871RC) and Gaussian

In light of the constraints on 119891(119905) and 119902(119905) (2) can berewritten as

120601 (119905120572) = 120579 (119905120572119899) + 120579

119899minus119871

= 2120587119899

sum

119894=119899minus119871+1120572

119894ℎ

119894119902 (119905 minus 119894119879)

+(120587

119899minus119871

sum

119894=minusinfin

120572

119894ℎ

119894) mod 2120587

(5)

where the term 120579(119905120572119899) is a function of the 119871 symbols being

modulated by the phase pulse For ℎ119894= 2119896119894119901 (119896119894 119901 integers)

the phase state 120579119899minus119871

takes on 119901 distinct values The signalis described by a trellis containing 119901119872

119871minus1 states with 119872

branches at each state Each branch is defined by the (119871 + 1)-tuple 120590

119899= (120579

119899minus119871 120572

119899minus119871+1 120572119899minus119871+2 120572119899)

22 Easy Calculation of Power Spectra Reference [13] pro-vides an easy method which allows a fast calculation of thespectra of 119872-ary multi-ℎ CPM signals under the generalhypothesis of (1) 119872-ary symbols (119872 a power of 2) (2)arbitrary pulse shaping (3) partial response signaling (4) anarbitrary set ofmodulation indexesTheprocedure of the easymethod is as follows

Let 119909(119905) = exp119895120601(119905) be the complex envelope of thetransmitted signal 119909(119905) To find the baseband equivalent 119878(119891)of the power spectrum 119878(119891) of the signal 119909(119905) we use the so-called ldquoautocorrelation-basedrdquo approach

The autocorrelation function of 119909(119905) is defined as

120588 (1199051 1199052)

≃ 119864 119909 (1199051) 119909lowast

(1199052)

= 119864exp[119895+infin

sum

119894=minusinfin

2120587120572119894ℎ

119894(119902 (1199051 minus 119894119879) minus 119902 (1199052 minus 119894119879))]

(6)


As we know ℎ119894+119870

= ℎ



120588 (1199051 1199052)

=

+infin

prod

119894=minusinfin

119872minus1sum

119903=0

1119872


=

+infin

prod

119894=minusinfin

119901 (1199051 minus 119894119870119879 1199052 minus 119894119870119879)

(7)

where

119901 (119886 119887) ≃

119870minus1prod

119898=0

sin (2120587119872ℎ

119898(119902 (119886) minus 119902 (119887)))

119872 sin (2120587ℎ119898(119902 (119886) minus 119902 (119887)))

(8)


119877 (120591) ≃

1119870119879

int

119870119879

0120588 (119905 119905 + 120591) 119889119905

(9)


119877 (120591) =

1119870119879

int

119870119879

0

119898+1prod


1015840

minus (119894 minus 119898)119870119879) 119889119905

(10)


1015840




119878 (119891) = 2

int

(1minus119904)119870119879

0

119877 (120591) cos 2120587119891120591119889120591

+R[[

int

(2minus119904)119870119879(1minus119904)119870119879

119877 (120591) exp (minus1198952120587119891120591) 119889120591

(1 minus 1198621119870 exp (minus1198952120587119891119870119879))]

]

(11)


119870minus1119898=0[sin(119872120587ℎ

119898)




Pow

er sp

ectr

al d

ensit

y (d

BWH

z)

MSKGMSK


minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

minus70

minus60

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34









AWGN channelFSMC

XNminus1

0

VNminus1

0 WNminus1

0

YNminus1

0


119901119872




119894

119883

119894+1 119883119895 Let 119868(119883119873minus10 119884




1119873

119868 (119883

119873minus10 119884

119873minus10 ) (12)


119868 (119883

119873minus10 119884

119873minus10 ) = 119867(119883

119873minus10 ) minus119867(119883

119873minus10 | 119884

119873minus10 )

=

119873minus1sum

119894=0119867(119883

119894)

minus

119873minus1sum

119894=0119867(119883

119894| 119883

119894minus10 119884

119873minus10 )

(13)

where 119867(119883119894| 119883

119894minus10 ) = 119867(119883


119894is



calculated is119867(119883119894| 119883

119894minus10 119884


ditional entropy

119867(119883

119894| 119883

119894minus10 119884

119873minus10 ) = minus119864 [log2119901 (119883119894 | 119883

119894minus10 119884

119873minus10 )] (14)


119894| 119883

119894minus10 119884

119873minus10 ) we


119901 (119883

119894| 119883

119894minus10 119884

119873minus10 ) =

119901 (119883

119894 119883

119873minus10 119884

119873minus10 )

sum

119883119894119901 (119883

119894 119883

119894minus10 119884

119873minus10 )

(15)


119894 119883

119873minus10 119884



119894|

119883

119894minus10 119884


119894

119883

119873minus10 119884



0 5 10 15 200

05

1

15

2

25

3

Chan

nel c

apac

ity (b

itsy

mbo

l)

M = 8 h = 12

M = 8 h = 14

M = 8 h = 15

M = 8 h = 17

M = 8 h = 16

M = 8 h = 12 14

M = 4 h = 12

M = 4 h = 14

M = 4 h = 12 14





119894


0


119894 if other







120590

2120576=

119861

119871(1 minus 05119861

119871119879

119894) int

1198612minus1198612 119866 (119891) sin

2(120587119891119889) 119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 119891119866 (119891) sin (120587119891119889) 119889119891]

2 (16)



119894


119871119879

119894is small


lim119889rarr 0

120590

2120576cong 120590

2CRB

=

119861

119871int

1198612minus1198612 119866 (119891) (120587119891119889)

2119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 120587119891

2119889119866 (119891) 119889119891]

2

=

119861

119871120587

2119889

2int

1198612minus1198612 119891

2119866 (119891) 119889119891

(2120587)2 (1198621198730) 1205872119889

2[int

1198612minus1198612 119891

2119866 (119891) 119889119891]

2

=

119861

119871

(2120587)2 (1198621198730) int1198612minus1198612 119891

2119866 (119891) 119889119891

(17)

with


minus1198612119891

2119866 (119891) 119889119891

(18)




119903 (119905) = 11988601198901198951205950119909 (119905 minus 1205910) +

119873

sum

119899minus1119886

119899119890

119895120595119899119909 (119905 minus 120591

119899) (19)


119899 120591119899 and120595




119863 (120576) = 1198860 [R(120576 minus

119889

2)minusR(120576 +

119889

2)]

+ 1198861 [R(120576 minusΔ120591minus

119889


119889

2)]


(20)


119863 (120576) asymp 119863 (0) +1198631015840 (0) 120576 (21)


1015840


119863 (0) = plusmn 21198861 int1198612


119863

1015840

(0)

= 41205871198860 int1198612

minus1198612119891119866 (119891) sin (120587119891119889) 119889119891

plusmn 41205871198861 int1198612


(22)


120576 (Δ120591)

asymp

plusmn119886 int

1198612minus1198612 119866 (119891) sin (2120587119891Δ120591) sin (120587119891119889) 119889119891


(23)


by

120576av (Δ1205911015840

) =

1Δ120591

1015840int

Δ1205911015840

0

1003817

1003817

1003817

1003817

1205760 (Δ120591)1003817

1003817

1003817

1003817

+

1003817

1003817

1003817

1003817

120576

120587(Δ120591)

1003817

1003817

1003817

1003817

2119889Δ120591

(24)






119871Hz

2046 2046 20sim40 minus6 0sim300 01 1



119877 timesmax [119866119904(119891)]

] (dB)


[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)





119871is 1 Hz


119894 leads



Gab

or b

andw

idth

(MH

z)MSKGMSK


0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4






0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










As we know ℎ119894+119870

= ℎ



120588 (1199051 1199052)

=

+infin

prod

119894=minusinfin

119872minus1sum

119903=0

1119872


=

+infin

prod

119894=minusinfin

119901 (1199051 minus 119894119870119879 1199052 minus 119894119870119879)

(7)

where

119901 (119886 119887) ≃

119870minus1prod

119898=0

sin (2120587119872ℎ

119898(119902 (119886) minus 119902 (119887)))

119872 sin (2120587ℎ119898(119902 (119886) minus 119902 (119887)))

(8)


119877 (120591) ≃

1119870119879

int

119870119879

0120588 (119905 119905 + 120591) 119889119905

(9)


119877 (120591) =

1119870119879

int

119870119879

0

119898+1prod


1015840

minus (119894 minus 119898)119870119879) 119889119905

(10)


1015840




119878 (119891) = 2

int

(1minus119904)119870119879

0

119877 (120591) cos 2120587119891120591119889120591

+R[[

int

(2minus119904)119870119879(1minus119904)119870119879

119877 (120591) exp (minus1198952120587119891120591) 119889120591

(1 minus 1198621119870 exp (minus1198952120587119891119870119879))]

]

(11)


119870minus1119898=0[sin(119872120587ℎ

119898)




Pow

er sp

ectr

al d

ensit

y (d

BWH

z)

MSKGMSK


minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

minus70

minus60

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34









AWGN channelFSMC

XNminus1

0

VNminus1

0 WNminus1

0

YNminus1

0


119901119872




119894

119883

119894+1 119883119895 Let 119868(119883119873minus10 119884




1119873

119868 (119883

119873minus10 119884

119873minus10 ) (12)


119868 (119883

119873minus10 119884

119873minus10 ) = 119867(119883

119873minus10 ) minus119867(119883

119873minus10 | 119884

119873minus10 )

=

119873minus1sum

119894=0119867(119883

119894)

minus

119873minus1sum

119894=0119867(119883

119894| 119883

119894minus10 119884

119873minus10 )

(13)

where 119867(119883119894| 119883

119894minus10 ) = 119867(119883


119894is



calculated is119867(119883119894| 119883

119894minus10 119884


ditional entropy

119867(119883

119894| 119883

119894minus10 119884

119873minus10 ) = minus119864 [log2119901 (119883119894 | 119883

119894minus10 119884

119873minus10 )] (14)


119894| 119883

119894minus10 119884

119873minus10 ) we


119901 (119883

119894| 119883

119894minus10 119884

119873minus10 ) =

119901 (119883

119894 119883

119873minus10 119884

119873minus10 )

sum

119883119894119901 (119883

119894 119883

119894minus10 119884

119873minus10 )

(15)


119894 119883

119873minus10 119884



119894|

119883

119894minus10 119884


119894

119883

119873minus10 119884



0 5 10 15 200

05

1

15

2

25

3

Chan

nel c

apac

ity (b

itsy

mbo

l)

M = 8 h = 12

M = 8 h = 14

M = 8 h = 15

M = 8 h = 17

M = 8 h = 16

M = 8 h = 12 14

M = 4 h = 12

M = 4 h = 14

M = 4 h = 12 14





119894


0


119894 if other







120590

2120576=

119861

119871(1 minus 05119861

119871119879

119894) int

1198612minus1198612 119866 (119891) sin

2(120587119891119889) 119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 119891119866 (119891) sin (120587119891119889) 119889119891]

2 (16)



119894


119871119879

119894is small


lim119889rarr 0

120590

2120576cong 120590

2CRB

=

119861

119871int

1198612minus1198612 119866 (119891) (120587119891119889)

2119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 120587119891

2119889119866 (119891) 119889119891]

2

=

119861

119871120587

2119889

2int

1198612minus1198612 119891

2119866 (119891) 119889119891

(2120587)2 (1198621198730) 1205872119889

2[int

1198612minus1198612 119891

2119866 (119891) 119889119891]

2

=

119861

119871

(2120587)2 (1198621198730) int1198612minus1198612 119891

2119866 (119891) 119889119891

(17)

with


minus1198612119891

2119866 (119891) 119889119891

(18)




119903 (119905) = 11988601198901198951205950119909 (119905 minus 1205910) +

119873

sum

119899minus1119886

119899119890

119895120595119899119909 (119905 minus 120591

119899) (19)


119899 120591119899 and120595




119863 (120576) = 1198860 [R(120576 minus

119889

2)minusR(120576 +

119889

2)]

+ 1198861 [R(120576 minusΔ120591minus

119889


119889

2)]


(20)


119863 (120576) asymp 119863 (0) +1198631015840 (0) 120576 (21)


1015840


119863 (0) = plusmn 21198861 int1198612


119863

1015840

(0)

= 41205871198860 int1198612

minus1198612119891119866 (119891) sin (120587119891119889) 119889119891

plusmn 41205871198861 int1198612


(22)


120576 (Δ120591)

asymp

plusmn119886 int

1198612minus1198612 119866 (119891) sin (2120587119891Δ120591) sin (120587119891119889) 119889119891


(23)


by

120576av (Δ1205911015840

) =

1Δ120591

1015840int

Δ1205911015840

0

1003817

1003817

1003817

1003817

1205760 (Δ120591)1003817

1003817

1003817

1003817

+

1003817

1003817

1003817

1003817

120576

120587(Δ120591)

1003817

1003817

1003817

1003817

2119889Δ120591

(24)






119871Hz

2046 2046 20sim40 minus6 0sim300 01 1



119877 timesmax [119866119904(119891)]

] (dB)


[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)





119871is 1 Hz


119894 leads



Gab

or b

andw

idth

(MH

z)MSKGMSK


0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4






0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











AWGN channelFSMC

XNminus1

0

VNminus1

0 WNminus1

0

YNminus1

0


119901119872




119894

119883

119894+1 119883119895 Let 119868(119883119873minus10 119884




1119873

119868 (119883

119873minus10 119884

119873minus10 ) (12)


119868 (119883

119873minus10 119884

119873minus10 ) = 119867(119883

119873minus10 ) minus119867(119883

119873minus10 | 119884

119873minus10 )

=

119873minus1sum

119894=0119867(119883

119894)

minus

119873minus1sum

119894=0119867(119883

119894| 119883

119894minus10 119884

119873minus10 )

(13)

where 119867(119883119894| 119883

119894minus10 ) = 119867(119883


119894is



calculated is119867(119883119894| 119883

119894minus10 119884


ditional entropy

119867(119883

119894| 119883

119894minus10 119884

119873minus10 ) = minus119864 [log2119901 (119883119894 | 119883

119894minus10 119884

119873minus10 )] (14)


119894| 119883

119894minus10 119884

119873minus10 ) we


119901 (119883

119894| 119883

119894minus10 119884

119873minus10 ) =

119901 (119883

119894 119883

119873minus10 119884

119873minus10 )

sum

119883119894119901 (119883

119894 119883

119894minus10 119884

119873minus10 )

(15)


119894 119883

119873minus10 119884



119894|

119883

119894minus10 119884


119894

119883

119873minus10 119884



0 5 10 15 200

05

1

15

2

25

3

Chan

nel c

apac

ity (b

itsy

mbo

l)

M = 8 h = 12

M = 8 h = 14

M = 8 h = 15

M = 8 h = 17

M = 8 h = 16

M = 8 h = 12 14

M = 4 h = 12

M = 4 h = 14

M = 4 h = 12 14





119894


0


119894 if other







120590

2120576=

119861

119871(1 minus 05119861

119871119879

119894) int

1198612minus1198612 119866 (119891) sin

2(120587119891119889) 119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 119891119866 (119891) sin (120587119891119889) 119889119891]

2 (16)



119894


119871119879

119894is small


lim119889rarr 0

120590

2120576cong 120590

2CRB

=

119861

119871int

1198612minus1198612 119866 (119891) (120587119891119889)

2119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 120587119891

2119889119866 (119891) 119889119891]

2

=

119861

119871120587

2119889

2int

1198612minus1198612 119891

2119866 (119891) 119889119891

(2120587)2 (1198621198730) 1205872119889

2[int

1198612minus1198612 119891

2119866 (119891) 119889119891]

2

=

119861

119871

(2120587)2 (1198621198730) int1198612minus1198612 119891

2119866 (119891) 119889119891

(17)

with


minus1198612119891

2119866 (119891) 119889119891

(18)




119903 (119905) = 11988601198901198951205950119909 (119905 minus 1205910) +

119873

sum

119899minus1119886

119899119890

119895120595119899119909 (119905 minus 120591

119899) (19)


119899 120591119899 and120595




119863 (120576) = 1198860 [R(120576 minus

119889

2)minusR(120576 +

119889

2)]

+ 1198861 [R(120576 minusΔ120591minus

119889


119889

2)]


(20)


119863 (120576) asymp 119863 (0) +1198631015840 (0) 120576 (21)


1015840


119863 (0) = plusmn 21198861 int1198612


119863

1015840

(0)

= 41205871198860 int1198612

minus1198612119891119866 (119891) sin (120587119891119889) 119889119891

plusmn 41205871198861 int1198612


(22)


120576 (Δ120591)

asymp

plusmn119886 int

1198612minus1198612 119866 (119891) sin (2120587119891Δ120591) sin (120587119891119889) 119889119891


(23)


by

120576av (Δ1205911015840

) =

1Δ120591

1015840int

Δ1205911015840

0

1003817

1003817

1003817

1003817

1205760 (Δ120591)1003817

1003817

1003817

1003817

+

1003817

1003817

1003817

1003817

120576

120587(Δ120591)

1003817

1003817

1003817

1003817

2119889Δ120591

(24)






119871Hz

2046 2046 20sim40 minus6 0sim300 01 1



119877 timesmax [119866119904(119891)]

] (dB)


[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)





119871is 1 Hz


119894 leads



Gab

or b

andw

idth

(MH

z)MSKGMSK


0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4






0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120590

2120576=

119861

119871(1 minus 05119861

119871119879

119894) int

1198612minus1198612 119866 (119891) sin

2(120587119891119889) 119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 119891119866 (119891) sin (120587119891119889) 119889119891]

2 (16)



119894


119871119879

119894is small


lim119889rarr 0

120590

2120576cong 120590

2CRB

=

119861

119871int

1198612minus1198612 119866 (119891) (120587119891119889)

2119889119891

(2120587)2 (1198621198730) [int1198612minus1198612 120587119891

2119889119866 (119891) 119889119891]

2

=

119861

119871120587

2119889

2int

1198612minus1198612 119891

2119866 (119891) 119889119891

(2120587)2 (1198621198730) 1205872119889

2[int

1198612minus1198612 119891

2119866 (119891) 119889119891]

2

=

119861

119871

(2120587)2 (1198621198730) int1198612minus1198612 119891

2119866 (119891) 119889119891

(17)

with


minus1198612119891

2119866 (119891) 119889119891

(18)




119903 (119905) = 11988601198901198951205950119909 (119905 minus 1205910) +

119873

sum

119899minus1119886

119899119890

119895120595119899119909 (119905 minus 120591

119899) (19)


119899 120591119899 and120595




119863 (120576) = 1198860 [R(120576 minus

119889

2)minusR(120576 +

119889

2)]

+ 1198861 [R(120576 minusΔ120591minus

119889


119889

2)]


(20)


119863 (120576) asymp 119863 (0) +1198631015840 (0) 120576 (21)


1015840


119863 (0) = plusmn 21198861 int1198612


119863

1015840

(0)

= 41205871198860 int1198612

minus1198612119891119866 (119891) sin (120587119891119889) 119889119891

plusmn 41205871198861 int1198612


(22)


120576 (Δ120591)

asymp

plusmn119886 int

1198612minus1198612 119866 (119891) sin (2120587119891Δ120591) sin (120587119891119889) 119889119891


(23)


by

120576av (Δ1205911015840

) =

1Δ120591

1015840int

Δ1205911015840

0

1003817

1003817

1003817

1003817

1205760 (Δ120591)1003817

1003817

1003817

1003817

+

1003817

1003817

1003817

1003817

120576

120587(Δ120591)

1003817

1003817

1003817

1003817

2119889Δ120591

(24)






119871Hz

2046 2046 20sim40 minus6 0sim300 01 1



119877 timesmax [119866119904(119891)]

] (dB)


[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)





119871is 1 Hz


119894 leads



Gab

or b

andw

idth

(MH

z)MSKGMSK


0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4






0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119871Hz

2046 2046 20sim40 minus6 0sim300 01 1



119877 timesmax [119866119904(119891)]

] (dB)


[

1

119877 times int

1198612minus1198612 119866

2119904(119891) 119889119891

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

max [1198912119866

119904(119891)]

]

]

(dB)


[

int

1198612minus1198612 119891

2119866

119904(119891) 119889119891

int

1198612minus1198612 119891

2119866

2119904(119891) 119889119891

]

]

(dB)

(25)





119871is 1 Hz


119894 leads



Gab

or b

andw

idth

(MH

z)MSKGMSK


0

02

04

06

08

1

12

14

16

18

2

2046MHz

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


20 22 24 26 28 30 32 34 36 38 40

35

Cod

e tra

ckin

g er

ror (

m)

0

05

1

15

2

25

3

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34

4






0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160 180 200

0

5

10

15

Path difference (m)

Mul

tipat

h er

ror e

nvelo

pes (

m)

minus5

minus10

minus15

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34








10

20

30

40

50

60

70

80

Ant

ijam

min

g m

erit

fact

ors (

dB)

MSKGMSK

RCh = 14 12

RCh = 12

RCh = 12 34

RECh = 14 12

RECh = 12 34


6 Conclusions





Acknowledgments




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a flexible modulation scheme design for c...

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