Research Article Coherent RAKE Receiver for CPM...

Research ArticleCoherent RAKE Receiver for CPM-Based Direct SequenceSpread Spectrum

Ke Zhou12 Shilian Wang1 and Eryang Zhang1

1Department of Electronic Science and Engineering National University of Defense Technology Changsha 410073 China2Institute of Electronic Engineering China Academy of Engineering Physics Mianyang 621900 China

Correspondence should be addressed to Shilian Wang wangslnudteducn

Received 6 March 2016 Revised 20 May 2016 Accepted 29 May 2016

Academic Editor Leonid Shaikhet

Copyright copy 2016 Ke Zhou et alThis 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

Direct sequence spread spectrum (DSSS) using continuous phase modulation (CPM) inherits the techniquesrsquo benefits constantenvelope anti-interference and spectral efficiency To get diversity gains over a Rayleigh-fading multipath channel as inconventional direct sequence spread-spectrum binary phase shift keying (DSSS-BPSK) system a new class of coherent RAKEreceivers is proposed in this work By introducing chip branch metric to the receiver scheme despreading and data detection canbe done meanwhile based on Maximum Likelihood Sequence Detection (MLSD) Compared to the conventional RAKE receiverwhich sums decision metrics symbol-by-symbol the proposed DSSS-CPM RAKE receiver accumulates symbol branch metricincrements over every phase state of multiple paths after chip phase synchronization Numerical results show that DSSS-CPMusing the synchronous despreading and demodulation algorithm has no performance loss compared to CPM system that employsMLSD algorithm under the same test conditions Moreover the proposed RAKE receiver outperforms conventional RAKE receiverand achieves a remarkable diversity gain of bit error rate (BER) under the Rayleigh-fading multipath channel

1 Introduction

The state-of-the-art continuous phase modulation (CPM)known for its efficient spectral properties has the meritsof continuous phase and constant envelop [1] while DirectSequence Spread Spectrum (DSSS) is a mature techniqueand benefits from narrowband interference suppression lowprobability of intercept and multiple-access communication[2]The so-called DSSS-CPM signals inherit both of the tech-nical merits specifically Firstly the constant envelope allowsthe usage of the nonlinear amplifiers which are more power-efficient and cheaper Moreover the narrow power spectraldensity (PSD) will increase the processing gain in band-limitconditions and the spread-spectrum techniqueswillmake thesignal format to have a low probability of intercept (LPI) [3]Finally based on the correlation properties of pseudorandomspreading sequences a remarkable diversity gain effect can beobtained usingRAKE receiver under amultipath channel anda code division multiple-access (CDMA) system is availablefor multiuser scenarios [4]

Historically several types of DSSS-CPM scheme havebeen proposed but most of the studies focused on modula-tion schemes signal format design multiaccess interference(MAI) and CDMA under the AWGN channel Lok andLehnert presented a DSSS-CPM format with continuousphase in symbol transitions as well as chip intervals [5]which is the basic scheme discussed in this paper Becausespreading sequences and data symbols are not separable theconventional receiver structure ofDSSS is not available in thisDSSS-CPM system To solve this problem Hsu and Lehnertproposed a CDMA system In this work the transmitterfirstly generates CPM signal of pseudorandom spreadingsequence using a continuous phase modulator and then theCPM signal is multiplied by data signal [6] Obviously thiskind of signal is not efficient in PSD due to discontinuitiesat symbol intervals Thus McDowell proposed a dual-phaseDSSS-CPM signal format [7 8] which is unique from theaspect of spreading sequences and data symbols effectingthe carrier phase separately Even though this signal formatis phase-continuous and despreading and dada detection

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6971083 7 pageshttpdxdoiorg10115520166971083

2 Mathematical Problems in Engineering

s(t 120572 c h)

Multi-hCPM

generator

120572i

cij f(t)hi+j

Figure 1 DSSS-CPM transmitter

are separable the data signal is limited in a minimum-shift keying (MSK) format and the scheme is a compromisebetween performance and complexity There are also someother researches that concentrate on spreading sequencesdesign to eliminate the multiaccess interference (MAI) andachieve multiuser communication [9ndash13] However all ofthese studies did not consider RAKE receiver for DSSS-CPMsystem

Diversity combining techniques were proposed becausethe received multipath signal will cause errors in a multipathfading channel [14] Specifically if the same messages overdifferent paths are collected and recombined after phase chipsynchronization the receiver will overcome this problem andget diversity gain in performance Conventional DSSS-PSKRAKE receiver takes advantage of pseudorandom spreadingsequences to despread multipath signals and recombinesthe decision metrics symbol-by-symbol This kind of RAKEreceiver cannot be used in DSSS-CPM system since CPMsignal is nonlinear and sequence detection is required due tothe phase memory feature

In this paper we propose and analyze a coherent RAKEreceiver for DSSS-CPM system under a Rayleigh-fadingmultipath channel By introducing chip branch metric tothe receiver scheme synchronous despreading and datadetection can be done based on Maximum LikelihoodSequence Detection (MLSD) The proposed RAKE receiveraccumulates the symbol branchmetric increments over everyphase state ofmultiple paths after chip phase synchronizationConsequently large diversity gains as well as desirable spec-tral properties can be achieved

This paper is organized as follows In Section 2 wedescribe the transmitter the signal format and the Rayleigh-fading multipath channel model Section 3 presents thecoherent RAKE receiver techniques Some evaluations andcomparisons illustrate the result in Section 4

2 System Model

We describe the DSSS-CPM communication system in thissection The transmitter for this DSSS-CPM signal is pre-sented in Section 21 The signal format and state trellisstructure are defined in Section 22 Finally in Section 23 weprovide a model for the Rayleigh-fading multipath channel

21 Transmitter The conceptual transmitter structure of theDSSS-CPM signal is shown in Figure 1 Each informa-tion symbol is multiplied by a finite-length pseudorandomspreading sequence to form baseband spreading signalswhich are then used as input to the CPMmodulator Since the

signal phase is continuous throughout the symbol and chiptransmissions it inherits the desirable properties of typicalCPM signals

In the DSSS-CPM scheme the low pass equivalent signalis defined as

119904 (119905120572 c h) = radic

2119864119888

119879119888

exp 119895120595 (119905120572 c h) (1)

where 119864119888is the chip energy and 119879

119888is the chip period The

information data 120572 = (1205720 1205721 ) is included in the phase

function which can be expressed as

120595 (119905120572 c h) = 2120587

infin

sum

119894=0

119873119888minus1

sum

119895=0

120572119894119888119894119895ℎ119894+119895

119902 (119905 minus (119895 + 119894119873119888) 119879119888) (2)

where 120572119899is the 119872-ary information symbol with 120572

119899isin

plusmn1 plusmn3 plusmn(119872 minus 1) and c = (1198881198940 1198881198941 119888

119894119873119888

) 119888119894119895is the

pseudorandom spreading sequence with 119888119894119895

= plusmn1 119879119904is the

symbol duration such that 119879119904= 119873119888119879119888 The shape of the phase

response can be defined as 119902(119905) = int

119905

0

119892(120591)119889120591 The frequencysmoothing pulse 119892(119905) is limited in (0 119871119879

119888) for positive integer

119871 The scheme is called full-response CPM for 119871 = 1 andpartial-response CPM for 119871 gt 1 [9] The modulation indexℎ(119894+119895)mod119873

ℎ

is from Π = ℎ0 ℎ1 ℎ

119873ℎminus1 of cyclically

varying with a fixed frequency period of119873ℎ The modulation

index is fixed in the chip duration The signal format isreferred to Spread-Spectrum Single-ℎ CPM (SSSH-CPM) for119873ℎ= 1 and Spread-Spectrum Multi-ℎ CPM (SSMH-CPM)

for 119873ℎ

gt 1 The spreading sequence modulation indexsymbol period and chip duration are assumed to be known apriori by both the transmitter and the intended receiver

22 State Trellis Structure (Signal Format) As shown in [2 4]during the 119897th information symbol and the 119899th chip intervalthe phase function can be expressed as

120595 (119905120572 c h) = 2120587

119899

sum

119895=119899minus119871+1

120572119897119888119897119895ℎ119897+119895

119902 (119905 minus (119895 + 119897119873119888) 119879119888)

+ 120579119897119899

= 120579 (119905 120572119897 119888119897119899 ℎ119897+119899

) + 120579119897119899

(3)

where ℎ119897+119899

= ℎ119897+119899 mod 119873

ℎ

= 119896119901 and 119896 and 119901 are fixed integers120579119897119899 called chip accumulation phase state can be expressed as

120579119897119899

= (120587

119897

sum

119894=0

119899minus119871

sum

119895=0

119886119894119888119894119895ℎ119894+119895

) mod 2120587 (4)

Obviously 120579119897119899is aMarkov chain with119901 or 2119901 phase states

and the chip accumulated phase state 120579119897119899exhibits a periodic

phase state trellis structure If the numerator 119896 of ℎ119897+119899

is aneven number 120579

119897119899will come from a limited set as

Θ119878= 0

120587119896

119901

2120587119896

119901

(119901 minus 1) 120587119896

119901

(5)

Otherwise when 119896 is an odd number then

Θ119878= 0

120587119896

119901

2120587119896

119901

2 (119901 minus 1) 120587119896

119901

(6)

Mathematical Problems in Engineering 3

r(t)

n(t)sum

120573V(t)1205732(t)1205731(t)

120591V12059121205911 middot middot middot

s(t 120572 c h)

Figure 2 Tapped delay line channel model

In (3) 120579(119905 120572119897 119888119897119899 ℎ119897+119899

) can be expressed as

120579 (119905 120572119897 119888119897119899 ℎ119897+119899

)

= 2120587120572119897119888119897119899ℎ119897+119899

119902 (119905 minus (119899 + 119897119873119888) 119879119888)

+ 2120587

119899minus1

sum

119895=119899minus119871+1

120572119897119888119897119895ℎ119897+119895

119902 (119905 minus (119895 + 119897119873119888) 119879119888)

(7)

The first term of (7) expresses the phase increment causedby the current 119899th chip of 119897th symbol And the second termindicates the phase change caused by the 119871minus1 chips sequence120572119897119862119897(119899minus119871+1)

120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)

of the 119897th symbol 120572119897 and it

is called chip related phase state vector with 2119871minus1

119872 statesIn conclusion at 119905 = ln119879

119888interval of the transmission

the signaling state trellis depends on the accumulation phasestate and the chip related phase state vector as

s119899= 120579119897119899 120572119897119862119897(119899minus119871+1)

120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)

(8)

At the next interval 119905 = 119897(119899 + 1)119879119888 the state trellis can be

expressed as

s119899+1

= 120579119897(119899+1)

120572119897119888119897(119899minus119871+2)

120572119897119888119897(119899minus1)

120572119897119888119897(119899)

(9)

where the accumulation phase state is recursive as

120579119897(119899+1)

= 120579119897119899+ 120587ℎ119897+119899minus119871+1

120572119897119888119897(119899minus119871+1)

(10)

These expressions describe the phase state trellis struc-tures which can be used for MLSD and Viterbi detectionThetotal states of the trellis structure are 119901119872 sdot 2

119871minus1 or 119901119872 sdot 2119871

23 Channel Model The typical features of mobile channelare multiple paths and fading due to the electromagneticwaversquos reflection and scattering on buildings trees andother obstacles Meanwhile because of the movement of thereceiver and obstacles the characteristics of channel becometime-varying This section gives a Rayleigh-fading multipathchannel model for the following analysis and evaluationsThis tapped delay line channelmodel is addressed in Figure 2

We assume that there are total 119881 paths over this channelmodelHence the channel impulse response can be expressedas

ℎ (120591 119905) =

119881

sum

119896=1

120573119896(119905) 120575 (119905 minus 120591

119896(119905)) (11)

where 120573119896(119905) is the attenuation of the 119896th path and 120591

119896(119905) is the

relative time delay of the 119896th pathFor transmitted signal 119904(119905120572 c h) under this channel we

will receive signal

119903 (119905) =

119881

sum

119896=1

120573119896(119905) 119904 (119905 minus 120591

119896(119905) 120572 c h) + 119899 (119905) (12)

where 119899(119905) is white Gauss noise with a single sided spectraldensity of119873

0

The channel impulse response will be a zero mean Gaussprocess if the fading channel has large number of multiplepaths In this situation the channel response will obeythe Rayleigh distribution and the signal phase will obey auniform distribution within [0 2120587]

3 Coherent RAKE Receiver

In a synchronous DSSS-CPM system MLSD can be usedfor optimum performance [2] However since spreadingsequences and data symbols are not separable the conven-tional receiver structure of the DSSS is not available in thissystem To solve this problem a coherent RAKE receiverwith synchronous despreading and demodulation algorithmis derived in this section

For frequency selective slow fadingmultipath channel weassume that the attenuations and time delays are not timing-varying at least in a symbol duration which implies 120573

119896(119905) =

120573119896and 120591119896(119905) = 120591

119896 The slow fading makes the accurate phase

shift estimation possible and consequentlywe can implementcoherent signal detection


Thus the received DSSS-CPM signal under a frequencyselective slow fading multipath channel in (12) can beexpressed as

119903 (119905) =

119881

sum

119896=1

120573119896119904 (119905 minus 120591

119896120572 c h) + 119899 (119905) (13)

We assume that the received multipath signals are ideal-synchronized and independent with each other For conve-nience of expression we use and to indicate assumedvalues and estimated values

The received signal over the 119896th path can be expressed as

119903119896

(119905) = 120573119896119904119896

(119905 minus 120591119896120572 c h) + 119899

119896

(119905) (14)

From the MLSD theory the receiver makes a symboldecision of

120580with the principle that the received single-path

signal 119903119896(119905) and the assumed waveform 119904119896

(119905 120580 c h) have the

minimum squared distance as

120582119896

() = int

infin

minusinfin

10038161003816100381610038161003816119903119896

(119905) minus 119904119896

(119905 c h)10038161003816100381610038161003816

2

119889119905 (15)

Due to the constant envelope of DSSS-CPM signalminimum equation (15) is equivalent to the maximum cross-correlation given by

120582119896

() = Re [intinfin

minusinfin

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905] (16)

Using Viterbi algorithm [2] the right side of (16) can beexpressed as

120582119896

119894(119899) = 120582

119896

119894(119899 minus 1)

+ Re[int(119899+1)119879

119904

119899119879119904

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Δ120582119896

119894(119899)

(17)

where the first term 120582119896119894(119899 minus 1) is the branch metric of the

119894th survive path of state trellis over the 119896th multipath at 119905 =

(119899 minus 1)119879119904and the second term which is called symbol branch

metric increment presents the metric increment caused bythe 119899th possible symbol

119899

With the assumed symbol 119899and the spreading sequence

119899119888119899(1)

119899119888119899(119873119888minus1)

119899119888119899(119873119888) the symbol branch metric

increment can be written in the following manner

Δ120582119896

119894(119899) = Re[int

(119898+119873119888)119879119888

119898119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]

=

119873119888minus1

sum

119895=0

Re[int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Λ119896

119899(119895)

(18)

where Λ119896119899(119895) called chip branch metric increment indicates

the chip branchmetric caused by the 119895th possible chip 119899119888119899(119895)

Thus we have the following equationΔ120582119896

119894(119899) = sum

119873119888minus1

119895=0Λ119896

119899(119895)

As presented in Section 2 the chip accumulated phasestate 120579

119897119899is from a limited setΘ

119878Thus the chip branchmetric

increment can be expressed as

Λ119896

119899(119895)

= Re[119890minus119895120579119899119895 int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119890minus119895120579(119905

119899119888119899119895ℎ119899+119895)

119889119905]

(19)

where 119890minus119895120579119899119895 forms a limited phase rotation networkWith (18)and (19) we can now rewrite (17) as

120582119896

119894(119899) = 120582

119896

119894(119899 minus 1) +

119873119888minus1

sum

119895=0

Λ119896

119899(119895) (20)

These expressions indicate that the decision metrics ofthe 119896th path form a Markov chain and a Viterbi decodercan be used for demodulating the information symbolsof independent paths under a Rayleigh-fading multipathchannel

Thus we have the decision metrics and the next stepis to implement a RAKE receiver to get diversity gain overmultipath channel A normal idea is combining the metricsbefore symbol decision but it will not work Since DSSS-CPM signal format is nonlinear and MLSD algorithm is asequence decision method different from the conventionalRAKE receiver computing the decision metrics symbol-by-symbol we proposed a coherent RAKE receiver for DSSS-CPM system which accumulates the symbol branch metricincrement in every phase state of the trellis structure overmultiple paths instead Based on this idea the RAKE receiversynchronizes the chip phase of the main separated 119866 pathsand combines the symbol branchmetric increments where119866is the number of main separated paths that we choose basedon principle of minimizing the BER and acceptable hardwarecomplexityThe total symbol branchmetric increment can beexpressed as

Δ120582119894(119899) =

119866

sum

119896=1

120574119896Δ120582119896

119894(119899) =

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (21)

where 120574119896is the metric weighting coefficient of the 119896th path

which is defined as

120574119896=

120573119896

sum119866

119904=1120573119904

(22)

This combination method is called maximum ratio com-bining (MRC) The larger the path attenuation is the lesscontribution is made to the total symbol branch metricincrement

The total symbol branch metric of the 119894th survive path inphase state trellis structure can be expressed as

120582119894(119899) = 120582

119894(119899 minus 1) +

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (23)

Equation (23) illuminates that decision metrics are theweighted sum of chip branch metric increments over every


r(t)

Path (1)

Path (k)

Path (G)

GM2Lminus1

Tc

Ts

n

Waveformmatchingdelay and

attenuationestimate Matched

filters

Matchedfilters

Phaserotate

Phaserotate

GpM2LGpM2Lminus1

detectionViterbi

N119888minus1

sumj=0

Λn(j)

N119888minus1

sumj=0

Λn(j)

Chip phasessynchronization

MRCG

sumk=1

N119888minus1

sumj=0

120574kΛkn(j)

Figure 3 Coherent RAKE receiver for DSSS-CPM

phase state Using the branch metric and Viterbi decoderalgorithm the conceptual structure of coherent RAKEreceiver architecture for DSSS-CPM is shown in Figure 3

StandardDSSS-BPSK system takes advantage of orthogo-nal spreading sequences to synchronize with local spreadingcodes and obtain multipath time delay and attenuationHowever the DSSS-CPM signal is such that the transmittedsignals have continuous phase Hence it is difficult to takeuse of the orthogonality of spreading sequences to distinguishthe multipath signals as in DSSS-BPSK system As a possiblesolution the waveform-matching block as shown in Figure 3is used to distinguish multipaths and estimating the multi-path time delay and attenuation The received signal is firstlymoved to match with the CPM waveform generated by thechosen spreading sequence using the same CPMmodulationparameters as the transmitter According to the waveform-matching correlation peaks the receiver can obtain the delaysand relative attenuations of main paths

The matched filters block calculates chip branch metricincrement according to (19) which calculate the correlationof the 119896th path signal 119903119896(119905) with the assumed local waveform119904119896

(119905 120580 c h) The correlation is then multiplied by the phase

rotation network 119890minus119895120579119899119895 in phase rotate block

We have calculated that the total phase state of thetrellis structure is 119901119872 sdot 2

119871minus1 or 119901119872 sdot 2119871 over one path of

the channel In order to obtain the complete chip branchmetric increments Λ

119896

119899(119895) for 119896 = 1 2 119866 the RAKE

receiver needs a bank of 119866119901119872 sdot 2119871minus1 or 119866119901119872 sdot 2

119871 matchedcorrelation filters to match with local waveforms 119904119896(119905

120580 c h)

for 119896 = 1 2 119866 as is shown in Figure 3 Obviouslythe hardware complexity increases with diversity level 119866modulation level 119872 and memory length 119871 Hence thecompromise between complexity and performance should betaken into consideration in an actual DSSS-CPM system

4 Numerical Examples

In this section we numerically present the simulation resultsof synchronous despreading and demodulation and coherentRAKE receiver for DSSS-CPM system In Section 41 wevalidate the feasibility of the coherent receiver algorithm forSSSH-CPM and SSMH-CPM BER performance results ofcoherent RAKE receiver for DSSS-CPM under a Rayleigh-fading multipath channel are presented in Section 42

SH SH

M = 4

M = 2

2 4 6 80 10

EbN0 (dB)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

= 127SSSH Nc

= 255SSSH Nc

= 255SSSH Nc

= 127SSSH Nc

Figure 4 BER performance of SSSH-CPM in an AWGN channel(119871 = 1 ℎ = 12)

Moreover a comparison is made to DSSS-BPSK whichemploys a conventional RAKE receiver In the followingexamples 119898-sequences generated by a linear shift registerand raised cosine pulses are used to determine the per-formance We assume that the symbol the carrier andthe pseudorandom spreading sequence are synchronizedperfectly The parameters of the multipath channel are nottiming-varying at least in a symbol interval The followingwork is completed in MATLAB

41 BER of Synchronous Despreading and DemodulationWhile significant study has been done on general CPMsignaling format a BER performance comparison is made tothe optimal receiver based on MLSD in an AWGN channel

In Figures 4-5 we show BER results for SSSH-CPMsystem and SSMH-CPM system with different modulationparameter values ldquoSHrdquo and ldquoMHrdquo indicate single-ℎ CPMand multi-ℎ CPM The curves reflect that the BER perfor-mances of DSSS-CPM system with 119873

119888= 127 and 119873

119888=

255 are coincident when other parameter values are thesame Moreover there is almost no performance loss in


M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc

Figure 5 BER performance of SSMH-CPM in an AWGN channel(ℎ = [14 24])

SSSH-CPM and SSMH-CPM compared to SH-CPM andMH-CPM which employ optimal receiver based on MLSD

These examples and analysis reflect that the synchronousdespreading and demodulation receiver based on MLSDpresented in the last section is effective for all kinds of DSSS-CPM including SSSH-CPM SSMH-CPM full-responseDSSS-CPM partial-response DSSS-CPM and two- or four-level DSSS-CPM

42 Performance of Coherent RAKE Receiver There are alarge number of multipath components in an actual mul-tipath scattering channel (eg ionosphere and tropospherescattering) but we can obtain appropriate diversity gainand suitable hardware complexity with 119866 = 2sim3 in themost multipath conditions [14] According to typical cellu-lar and microwave environment the following simulationsamples use relative attenuations [0 minus5 dB minus10 dB] whichare the maximum energy paths of the totally 119881 paths overthe Rayleigh-fading multipath channel model presented inSection 23 The relative delay time of the three paths is[0 16119879stp 32119879stp] where the temporal resolution is definedas 119879stp = 119879

119904(119873119888lowast 119873samp) and the sampling rate is set

to 119873samp = 4 Hence the relative delay time of the 3paths can be rewritten as [0 4119879

119888 8119879119888] The multipath signal

phases obey a uniform distribution within [0 2120587] Set thenormalizedDoppler frequency as119891

119889= 001Hz whichmeans

the multipath channel is slow fadingThe received signal is firstly moved to match with the

CPM waveform generated by the spreading sequence asshown in the first block of Figure 3 We can then obtain thetime delays and the relative attenuations of the paths thathave larger energy according to these waveform-matchingcorrelation peaks as shown in Figure 6 Moreover it isconcluded that the multipath time delay temporal resolutionis less than a chip interval So the receiver can add up thechip branch metric increments of main paths within the

Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion

Figure 6 Normalized waveform-matching correlation peaks

G = 3 DSSS-BPSKG = 2 DSSS-BPSKG = 1 DSSS-BPSK

G = 3 SSSH-CPMG = 2 SSSH-CPMG = 1 SSSH-CPM

5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

Figure 7 BER curves of the SSSH-CPM and DSSS-BPSK usingRAKE receiver (119872 = 2 119871 = 1 ℎ = 12 and119873

119888= 63)

chip duration according to the estimation of time delays andattenuations

The probability curves of BER for DSSS-CPM using theproposed RAKE receiver in a Rayleigh-fading multipathchannel are shown in Figures 7 and 8 Besides Figure 7also shows the BER performance curves of a standard DSSS-BPSK system using a conventional RAKE receiver with thesame symbol duration 119879

119904and the same 119898-sequence for

comparison 119866 = 1 indicates the BER performance withoutdiversity gain and the receiver uses only one path to makesymbol decisions 119866 = 2 3 means two- or three-leveldiversity Due to the memory characteristic of DSSS-CPMsignal format SSSH-CPM outperforms the conventionalDSSS-BPSK system at high signal-to-noise ratio There areabout 4 dB and 3 dB improvement of performance at BER =

10minus5 for two- and three-level diversity compared to DSSS-

BPSK in the same test conditions It is also observed from


G = 3 SSMH-CPMG = 2 SSMH-CPMG = 1 SSMH-CPM

5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

Figure 8 BER curves of the SSMH-CPMusingRAKE receiver (119872 =

4 119871 = 2 ℎ = [14 24] and119873119888= 63)

Figure 7 that the proposed RAKE receiver has remarkableperformance gainwhile achieving 12 dB and 16 dB of diversitygain at BER = 10

minus5 for 119866 = 2 3 considered in this paperSimilar diversity gain for SSMH-CPM is shown in Figure 8

These numerical results show that performance improve-ment can be achieved over conventional DSSS-BPSK RAKEreceiver Furthermore the proposed RAKE receiver canobtain a remarkable diversity gain while the DSSS-CPMsignaling format maintains constant envelope and spectralefficiency

5 Conclusions

Motivated by the excellent properties of DSSS-CPM com-munication system a new class of coherent RAKE receiverbased on MLSD has been proposed in this paper By intro-ducing chip branch metric increment to the receiver schemethe synchronous despreading and demodulation algorithmand the coherent RAKE receiver are presented for DSSS-CPM system We have provided numerical examples tovalidate the efficiency of the algorithm It shows that there isalmost no performance loss in SSSH-CPM and SSMH-CPMcompared with simple CPM system employing an optimalreceiver Moreover SSSH-CPM significantly outperformsconventional standard DSSS-BPSK under a Rayleigh-fadingmultipath channel and remarkable diversity gains can beachieved in DSSS-CPM system using the proposed RAKEreceiver

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] C-E Sundberg ldquoContinuous phase modulationrdquo IEEE Com-munications Magazine vol 24 no 4 pp 25ndash38 1986

[2] W D Lane and A M Bush ldquoSpread-spectrum multi-h modu-lationrdquo IEEE Journal on Selected Areas in Communications vol8 no 5 pp 728ndash742 1990

[3] D K Asano T Hayashi and R Kohno ldquoModulation and pro-cessing gain tradeoffs in DS-CDMA spread spectrum systemsrdquoin Proceedings of the IEEE 5th International Symposium onSpread Spectrum Techniques and Applications vol 1 pp 9ndash131998

[4] R T Hsu and J S Lehnert ldquoContinuous phase-coded direct-sequence spread-spectrum multiple-access communicationsrdquoin Proceedings of the 10th Annual International Phoenix Confer-ence on Computers and Communications pp 441ndash447 March1991

[5] T M Lok and J S Lehnert ldquoDSSSMA communication systemwith trellis coding and CPMrdquo IEEE Journal on Selected Areas inCommunications vol 12 no 4 pp 716ndash722 1994

[6] R T Hsu and J S Lehnert ldquoThe performance of continuous-phase-coded DSSSMA communicationsrdquo IEEE Transactionson Communications vol 46 no 4 pp 533ndash543 1998

[7] A T McDowell J S Lehnert and Y K Jeong ldquoDual-phasecontinuous phase modulation for spread-spectrum multiple-access communicationrdquo IEEETransactions onCommunicationsvol 52 no 5 pp 823ndash833 2004

[8] A T McDowell and J S Lehnert ldquoPhase-independent con-tinuous phase modulation for bandwidth efficient multiple-access communicationrdquo in Proceedings of the IEEE MilitaryCommunications Conference (MILCOM rsquo92) CommunicationsFusing Command Control and Intelligence vol 1 pp 104ndash107IEEE San Diego Calif USA 1992

[9] R R Muller and A Lampe ldquoSpectral efficiency of ran-dom CDMA with constant envelope modulationrdquo AEUmdashInternational Journal of Electronics and Communications vol65 no 8 pp 701ndash706 2011

[10] J S Lehnert ldquoSerial MSK spread-spectrum multiple-accesscommunicationsrdquo IEEE Transactions on Communications vol40 no 6 pp 1119ndash1127 1992

[11] R R Muller ldquoOn random CDMA with constant enveloperdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo11) pp 1663ndash1667 St Petersburg Russia August2011

[12] N Mazzali G Colavolpe and S Buzzi ldquoCPM-based spreadspectrum systems formulti-user communicationsrdquo IEEETrans-actions on Wireless Communications vol 12 no 1 pp 358ndash3672013

[13] Y Fengfan H Leung B Guangguo and Y Ming ldquoThedesign criterion of novel phase spreading sequences for mobileDSSSMA communicationsrdquo in Proceedings of the IEEE SixthInternational Symposium on Spread Spectrum Techniques andApplications vol 1 pp 207ndash211 ParsippanyNJUSA September2000

[14] G T Chyi J G Proakis and C M Keller ldquoDiversity selec-tioncombining schemeswith excess noise-only diversity recep-tion over a rayleigh-fadingmultipath channelrdquo in Proceedings ofthe Conference on Information Sciences and Systems (CISS rsquo88)Princeton University March 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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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


s(t 120572 c h)

Multi-hCPM

generator

120572i

cij f(t)hi+j

Figure 1 DSSS-CPM transmitter

are separable the data signal is limited in a minimum-shift keying (MSK) format and the scheme is a compromisebetween performance and complexity There are also someother researches that concentrate on spreading sequencesdesign to eliminate the multiaccess interference (MAI) andachieve multiuser communication [9ndash13] However all ofthese studies did not consider RAKE receiver for DSSS-CPMsystem

Diversity combining techniques were proposed becausethe received multipath signal will cause errors in a multipathfading channel [14] Specifically if the same messages overdifferent paths are collected and recombined after phase chipsynchronization the receiver will overcome this problem andget diversity gain in performance Conventional DSSS-PSKRAKE receiver takes advantage of pseudorandom spreadingsequences to despread multipath signals and recombinesthe decision metrics symbol-by-symbol This kind of RAKEreceiver cannot be used in DSSS-CPM system since CPMsignal is nonlinear and sequence detection is required due tothe phase memory feature

In this paper we propose and analyze a coherent RAKEreceiver for DSSS-CPM system under a Rayleigh-fadingmultipath channel By introducing chip branch metric tothe receiver scheme synchronous despreading and datadetection can be done based on Maximum LikelihoodSequence Detection (MLSD) The proposed RAKE receiveraccumulates the symbol branchmetric increments over everyphase state ofmultiple paths after chip phase synchronizationConsequently large diversity gains as well as desirable spec-tral properties can be achieved

This paper is organized as follows In Section 2 wedescribe the transmitter the signal format and the Rayleigh-fading multipath channel model Section 3 presents thecoherent RAKE receiver techniques Some evaluations andcomparisons illustrate the result in Section 4

2 System Model

We describe the DSSS-CPM communication system in thissection The transmitter for this DSSS-CPM signal is pre-sented in Section 21 The signal format and state trellisstructure are defined in Section 22 Finally in Section 23 weprovide a model for the Rayleigh-fading multipath channel

21 Transmitter The conceptual transmitter structure of theDSSS-CPM signal is shown in Figure 1 Each informa-tion symbol is multiplied by a finite-length pseudorandomspreading sequence to form baseband spreading signalswhich are then used as input to the CPMmodulator Since the

signal phase is continuous throughout the symbol and chiptransmissions it inherits the desirable properties of typicalCPM signals

In the DSSS-CPM scheme the low pass equivalent signalis defined as

119904 (119905120572 c h) = radic

2119864119888

119879119888

exp 119895120595 (119905120572 c h) (1)

where 119864119888is the chip energy and 119879

119888is the chip period The

information data 120572 = (1205720 1205721 ) is included in the phase

function which can be expressed as

120595 (119905120572 c h) = 2120587

infin

sum

119894=0

119873119888minus1

sum

119895=0

120572119894119888119894119895ℎ119894+119895

119902 (119905 minus (119895 + 119894119873119888) 119879119888) (2)

where 120572119899is the 119872-ary information symbol with 120572

119899isin

plusmn1 plusmn3 plusmn(119872 minus 1) and c = (1198881198940 1198881198941 119888

119894119873119888

) 119888119894119895is the

pseudorandom spreading sequence with 119888119894119895

= plusmn1 119879119904is the

symbol duration such that 119879119904= 119873119888119879119888 The shape of the phase

response can be defined as 119902(119905) = int

119905

0

119892(120591)119889120591 The frequencysmoothing pulse 119892(119905) is limited in (0 119871119879

119888) for positive integer

119871 The scheme is called full-response CPM for 119871 = 1 andpartial-response CPM for 119871 gt 1 [9] The modulation indexℎ(119894+119895)mod119873

ℎ

is from Π = ℎ0 ℎ1 ℎ

119873ℎminus1 of cyclically

varying with a fixed frequency period of119873ℎ The modulation

index is fixed in the chip duration The signal format isreferred to Spread-Spectrum Single-ℎ CPM (SSSH-CPM) for119873ℎ= 1 and Spread-Spectrum Multi-ℎ CPM (SSMH-CPM)

for 119873ℎ

gt 1 The spreading sequence modulation indexsymbol period and chip duration are assumed to be known apriori by both the transmitter and the intended receiver

22 State Trellis Structure (Signal Format) As shown in [2 4]during the 119897th information symbol and the 119899th chip intervalthe phase function can be expressed as

120595 (119905120572 c h) = 2120587

119899

sum

119895=119899minus119871+1

120572119897119888119897119895ℎ119897+119895

119902 (119905 minus (119895 + 119897119873119888) 119879119888)

+ 120579119897119899

= 120579 (119905 120572119897 119888119897119899 ℎ119897+119899

) + 120579119897119899

(3)

where ℎ119897+119899

= ℎ119897+119899 mod 119873

ℎ

= 119896119901 and 119896 and 119901 are fixed integers120579119897119899 called chip accumulation phase state can be expressed as

120579119897119899

= (120587

119897

sum

119894=0

119899minus119871

sum

119895=0

119886119894119888119894119895ℎ119894+119895

) mod 2120587 (4)

Obviously 120579119897119899is aMarkov chain with119901 or 2119901 phase states

and the chip accumulated phase state 120579119897119899exhibits a periodic

phase state trellis structure If the numerator 119896 of ℎ119897+119899

is aneven number 120579

119897119899will come from a limited set as

Θ119878= 0

120587119896

119901

2120587119896

119901

(119901 minus 1) 120587119896

119901

(5)

Otherwise when 119896 is an odd number then

Θ119878= 0

120587119896

119901

2120587119896

119901

2 (119901 minus 1) 120587119896

119901

(6)


r(t)

n(t)sum

120573V(t)1205732(t)1205731(t)


s(t 120572 c h)


In (3) 120579(119905 120572119897 119888119897119899 ℎ119897+119899


120579 (119905 120572119897 119888119897119899 ℎ119897+119899

)

= 2120587120572119897119888119897119899ℎ119897+119899

119902 (119905 minus (119899 + 119897119873119888) 119879119888)

+ 2120587

119899minus1

sum

119895=119899minus119871+1

120572119897119888119897119895ℎ119897+119895

119902 (119905 minus (119895 + 119897119873119888) 119879119888)

(7)


120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)






s119899= 120579119897119899 120572119897119862119897(119899minus119871+1)

120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)

(8)


expressed as

s119899+1

= 120579119897(119899+1)

120572119897119888119897(119899minus119871+2)

120572119897119888119897(119899minus1)

120572119897119888119897(119899)

(9)


120579119897(119899+1)

= 120579119897119899+ 120587ℎ119897+119899minus119871+1

120572119897119888119897(119899minus119871+1)

(10)


119871minus1 or 119901119872 sdot 2119871



ℎ (120591 119905) =

119881

sum

119896=1

120573119896(119905) 120575 (119905 minus 120591

119896(119905)) (11)


119896(119905) is the


will receive signal

119903 (119905) =

119881

sum

119896=1

120573119896(119905) 119904 (119905 minus 120591

119896(119905) 120572 c h) + 119899 (119905) (12)


0





119896(119905) =

120573119896and 120591119896(119905) = 120591





119903 (119905) =

119881

sum

119896=1

120573119896119904 (119905 minus 120591

119896120572 c h) + 119899 (119905) (13)



119903119896

(119905) = 120573119896119904119896

(119905 minus 120591119896120572 c h) + 119899

119896

(119905) (14)




(119905 120580 c h) have the


120582119896

() = int

infin

minusinfin

10038161003816100381610038161003816119903119896

(119905) minus 119904119896

(119905 c h)10038161003816100381610038161003816

2

119889119905 (15)


120582119896

() = Re [intinfin

minusinfin

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905] (16)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1)

+ Re[int(119899+1)119879

119904

119899119879119904

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Δ120582119896

119894(119899)

(17)





119899


119899119888119899(1)

119899119888119899(119873119888minus1)



Δ120582119896

119894(119899) = Re[int

(119898+119873119888)119879119888

119898119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]

=

119873119888minus1

sum

119895=0

Re[int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Λ119896

119899(119895)

(18)




119894(119899) = sum

119873119888minus1

119895=0Λ119896

119899(119895)





Λ119896

119899(119895)

= Re[119890minus119895120579119899119895 int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119890minus119895120579(119905

119899119888119899119895ℎ119899+119895)

119889119905]

(19)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1) +

119873119888minus1

sum

119895=0

Λ119896

119899(119895) (20)



Δ120582119894(119899) =

119866

sum

119896=1

120574119896Δ120582119896

119894(119899) =

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (21)


which is defined as

120574119896=

120573119896

sum119866

119904=1120573119904

(22)



120582119894(119899) = 120582

119894(119899 minus 1) +

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (23)



r(t)

Path (1)

Path (k)

Path (G)

GM2Lminus1

Tc

Ts

n



filters

Matchedfilters

Phaserotate

Phaserotate

GpM2LGpM2Lminus1

detectionViterbi

N119888minus1

sumj=0

Λn(j)

N119888minus1

sumj=0

Λn(j)


MRCG

sumk=1

N119888minus1

sumj=0

120574kΛkn(j)










119896

119899(119895) for 119896 = 1 2 119866 the RAKE



120580 c h)




SH SH

M = 4

M = 2

2 4 6 80 10

EbN0 (dB)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

= 127SSSH Nc

= 255SSSH Nc

= 255SSSH Nc

= 127SSSH Nc





119888= 127 and 119873

119888=



M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc












Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion




5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


119888= 63)









5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r(t)

n(t)sum

120573V(t)1205732(t)1205731(t)


s(t 120572 c h)


In (3) 120579(119905 120572119897 119888119897119899 ℎ119897+119899


120579 (119905 120572119897 119888119897119899 ℎ119897+119899

)

= 2120587120572119897119888119897119899ℎ119897+119899

119902 (119905 minus (119899 + 119897119873119888) 119879119888)

+ 2120587

119899minus1

sum

119895=119899minus119871+1

120572119897119888119897119895ℎ119897+119895

119902 (119905 minus (119895 + 119897119873119888) 119879119888)

(7)


120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)






s119899= 120579119897119899 120572119897119862119897(119899minus119871+1)

120572119897119888119897(119899minus2)

120572119897119888119897(119899minus1)

(8)


expressed as

s119899+1

= 120579119897(119899+1)

120572119897119888119897(119899minus119871+2)

120572119897119888119897(119899minus1)

120572119897119888119897(119899)

(9)


120579119897(119899+1)

= 120579119897119899+ 120587ℎ119897+119899minus119871+1

120572119897119888119897(119899minus119871+1)

(10)


119871minus1 or 119901119872 sdot 2119871



ℎ (120591 119905) =

119881

sum

119896=1

120573119896(119905) 120575 (119905 minus 120591

119896(119905)) (11)


119896(119905) is the


will receive signal

119903 (119905) =

119881

sum

119896=1

120573119896(119905) 119904 (119905 minus 120591

119896(119905) 120572 c h) + 119899 (119905) (12)


0





119896(119905) =

120573119896and 120591119896(119905) = 120591





119903 (119905) =

119881

sum

119896=1

120573119896119904 (119905 minus 120591

119896120572 c h) + 119899 (119905) (13)



119903119896

(119905) = 120573119896119904119896

(119905 minus 120591119896120572 c h) + 119899

119896

(119905) (14)




(119905 120580 c h) have the


120582119896

() = int

infin

minusinfin

10038161003816100381610038161003816119903119896

(119905) minus 119904119896

(119905 c h)10038161003816100381610038161003816

2

119889119905 (15)


120582119896

() = Re [intinfin

minusinfin

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905] (16)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1)

+ Re[int(119899+1)119879

119904

119899119879119904

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Δ120582119896

119894(119899)

(17)





119899


119899119888119899(1)

119899119888119899(119873119888minus1)



Δ120582119896

119894(119899) = Re[int

(119898+119873119888)119879119888

119898119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]

=

119873119888minus1

sum

119895=0

Re[int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Λ119896

119899(119895)

(18)




119894(119899) = sum

119873119888minus1

119895=0Λ119896

119899(119895)





Λ119896

119899(119895)

= Re[119890minus119895120579119899119895 int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119890minus119895120579(119905

119899119888119899119895ℎ119899+119895)

119889119905]

(19)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1) +

119873119888minus1

sum

119895=0

Λ119896

119899(119895) (20)



Δ120582119894(119899) =

119866

sum

119896=1

120574119896Δ120582119896

119894(119899) =

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (21)


which is defined as

120574119896=

120573119896

sum119866

119904=1120573119904

(22)



120582119894(119899) = 120582

119894(119899 minus 1) +

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (23)



r(t)

Path (1)

Path (k)

Path (G)

GM2Lminus1

Tc

Ts

n



filters

Matchedfilters

Phaserotate

Phaserotate

GpM2LGpM2Lminus1

detectionViterbi

N119888minus1

sumj=0

Λn(j)

N119888minus1

sumj=0

Λn(j)


MRCG

sumk=1

N119888minus1

sumj=0

120574kΛkn(j)










119896

119899(119895) for 119896 = 1 2 119866 the RAKE



120580 c h)




SH SH

M = 4

M = 2

2 4 6 80 10

EbN0 (dB)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

= 127SSSH Nc

= 255SSSH Nc

= 255SSSH Nc

= 127SSSH Nc





119888= 127 and 119873

119888=



M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc












Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion




5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


119888= 63)









5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903 (119905) =

119881

sum

119896=1

120573119896119904 (119905 minus 120591

119896120572 c h) + 119899 (119905) (13)



119903119896

(119905) = 120573119896119904119896

(119905 minus 120591119896120572 c h) + 119899

119896

(119905) (14)




(119905 120580 c h) have the


120582119896

() = int

infin

minusinfin

10038161003816100381610038161003816119903119896

(119905) minus 119904119896

(119905 c h)10038161003816100381610038161003816

2

119889119905 (15)


120582119896

() = Re [intinfin

minusinfin

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905] (16)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1)

+ Re[int(119899+1)119879

119904

119899119879119904

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Δ120582119896

119894(119899)

(17)





119899


119899119888119899(1)

119899119888119899(119873119888minus1)



Δ120582119896

119894(119899) = Re[int

(119898+119873119888)119879119888

119898119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]

=

119873119888minus1

sum

119895=0

Re[int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119904119896

(119905 c h)lowast 119889119905]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Λ119896

119899(119895)

(18)




119894(119899) = sum

119873119888minus1

119895=0Λ119896

119899(119895)





Λ119896

119899(119895)

= Re[119890minus119895120579119899119895 int119899119879119904+(119895+1)119879

119888

119899119879119904+119895119879119888

119903119896

(119905) 119890minus119895120579(119905

119899119888119899119895ℎ119899+119895)

119889119905]

(19)


120582119896

119894(119899) = 120582

119896

119894(119899 minus 1) +

119873119888minus1

sum

119895=0

Λ119896

119899(119895) (20)



Δ120582119894(119899) =

119866

sum

119896=1

120574119896Δ120582119896

119894(119899) =

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (21)


which is defined as

120574119896=

120573119896

sum119866

119904=1120573119904

(22)



120582119894(119899) = 120582

119894(119899 minus 1) +

119866

sum

119896=1

119873119888minus1

sum

119895=0

120574119896Λ119896

119899(119895) (23)



r(t)

Path (1)

Path (k)

Path (G)

GM2Lminus1

Tc

Ts

n



filters

Matchedfilters

Phaserotate

Phaserotate

GpM2LGpM2Lminus1

detectionViterbi

N119888minus1

sumj=0

Λn(j)

N119888minus1

sumj=0

Λn(j)


MRCG

sumk=1

N119888minus1

sumj=0

120574kΛkn(j)










119896

119899(119895) for 119896 = 1 2 119866 the RAKE



120580 c h)




SH SH

M = 4

M = 2

2 4 6 80 10

EbN0 (dB)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

= 127SSSH Nc

= 255SSSH Nc

= 255SSSH Nc

= 127SSSH Nc





119888= 127 and 119873

119888=



M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc












Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion




5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


119888= 63)









5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r(t)

Path (1)

Path (k)

Path (G)

GM2Lminus1

Tc

Ts

n



filters

Matchedfilters

Phaserotate

Phaserotate

GpM2LGpM2Lminus1

detectionViterbi

N119888minus1

sumj=0

Λn(j)

N119888minus1

sumj=0

Λn(j)


MRCG

sumk=1

N119888minus1

sumj=0

120574kΛkn(j)










119896

119899(119895) for 119896 = 1 2 119866 the RAKE



120580 c h)




SH SH

M = 4

M = 2

2 4 6 80 10

EbN0 (dB)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

= 127SSSH Nc

= 255SSSH Nc

= 255SSSH Nc

= 127SSSH Nc





119888= 127 and 119873

119888=



M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc












Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion




5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


119888= 63)









5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M = 4 L = 3

M = 2 L = 1

MH MH

2 4 6 8 10 120

EbN0 (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER

SSMH = 127Nc

SSMH = 255Nc

SSMH = 127Nc

SSMH = 255Nc












Path 2

Path 3

Path 1

50 100 150 200 250 300 350 4000Sampling point

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

wav

efor

m-m

atch

ing

corr

elat

ion




5 10 15 20 25 30 350

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


119888= 63)









5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 10 15 20 25 30 35 400

EbN0 (dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

BER


4 119871 = 2 ℎ = [14 24] and119873119888= 63)




5 Conclusions


Competing Interests


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Coherent RAKE Receiver for CPM...

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