Research Article Design of Simplified Maximum-Likelihood ...downloads.hindawi.com/journals/tswj/2014/174294.pdf · Multiuser receiver 1 K K s1 (t, 1) s2 (t, 2) s(t, ) fK, K sK (t,

Research ArticleDesign of Simplified Maximum-Likelihood Receivers forMultiuser CPM Systems

Li Bing and Baoming Bai

State Key Laboratory of ISN Xidian University Xirsquoan 710071 China

Correspondence should be addressed to Li Bing libingprcgmailcom

Received 26 August 2013 Accepted 4 November 2013 Published 27 January 2014

Academic Editors N Aydin and X Fan

Copyright copy 2014 L Bing and B Bai This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A class of simplified maximum-likelihood receivers designed for continuous phase modulation based multiuser systems isproposed The presented receiver is built upon a front end employing mismatched filters and a maximum-likelihood detectordefined in a low-dimensional signal space The performance of the proposed receivers is analyzed and compared to some existingreceivers Some schemes are designed to implement the proposed receivers and to reveal the roles of different system parametersAnalysis and numerical results show that the proposed receivers can approach the optimum multiuser receivers with significantly(even exponentially in some cases) reduced complexity and marginal performance degradation

1 Introduction

As an efficient constant envelopemodulation scheme contin-uous phasemodulation (CPM) has gained extensive attentionsince it was developed in 1980s [1] and has been proven tohave outstanding performance in single user [2ndash5] and mul-tiuser systems [6ndash9] Compared with conventional multiusersystems based on linearmodulations [10 11] an extraordinaryproperty of CPM is the constant envelope and thus the abilityto overcome the nonlinear distortion introduced by a class-Camplifier This property makes CPM an attractive scheme inmodern wireless systems

While increasing the efficiency the optimum multiuserreceiver consisting of a front-end followed by a detector hasa considerable complexity Generally the optimum receiversuch as maximum-likelihood (ML) multiuser receiver suf-fers from the exponentially increased complexity (with thenumber of users) and is considered too complicated tobe practical To reduce the complexity some suboptimumreceivers were proposed [7] The main strategy is simplifyingthe detector at the expense of acceptable performance lossin terms of bit error rate (BER) With properly designedparameters such as increasing the frequency separationeven a suboptimal receiver can successfully narrow the gapbetween suboptimum and optimum receivers [7] As amatter

of fact the design of front-end for CPM-based multiusersystems gains less attention The front-end is generallyinterpreted as an independent module generating sufficientstatistics The most existing designs focus on the techniquesof oversampling [1] and generalized Laurent decomposition(LD) (see [12] and references therein) Since they are notparticularly suited for multiuser systems the complexity ofsuch a front-end is linear to the number of users whichprobably finally exceeds the capacity of receivers

In this paper the joint design of simplified ML receiversis considered A simplified front-end is designed with pur-posely incorporated performance loss The detector followedis a ML detector defined in a low-dimensional signal spaceand thus complexity reduction can be obtained instantlyTheparameters of the receiver are then optimized such that theperformance loss is minimized for a given complexity reduc-tion The main tools employed are the principal componentanalysis [13] andmismatched filters [14 15]The performanceof the proposed receiver is measured by the average energyloss the minimum achievable Euclidean distance and thetotal complexity reduction

This paper is organized as follows The system modelis presented in Section 2 Section 3 presents the simplifiedreceivers and compares its complexity with existing receivers

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 174294 6 pageshttpdxdoiorg1011552014174294

2 The Scientific World Journal

User 1

User 2

User K

CPMmodulator

CPMmodulator

CPMmodulator

times

times

times

Equivalent multiple access channel

AWGNLow-dimensional

front endsum +r(t)

f1 1206011

f2 1206012

ML detector Multiuser receiver

1198341

1198342

119834K

119834K

s1(t 1198341)

s2(t 1198342)s(t 119834) 119834119955

fK 120601K

sK(t )

Figure 1 System model

Section 4 gives the numerical results and Section 5 concludesthe paper

2 System Model

A multiple-access-channel (MAC) type multiuser CPM sys-tem consisting of 119870 users is shown in Figure 1 It is assumedthat all users employ an identical CPM scheme where themodulation parameters such as the transmitted energy 119864modulation level119872 modulation index ℎ and phase response119902(119905) are the same The 119896th (1 le 119896 le 119870) user maps its infor-mation sequence a119896 = 1198861198961 1198861198962 119886119896119873 independently tothe transmitted signal whose equivalent complex basebandsignal reads [1]

119904119896 (119905 a119896) = radic2119864

119879exp 119895 120593 (119905 a119896) + 2120587119891119896119905 + 120601119896 (1)

where 119879 119864 and 119902(119905) are the symbol duration signal energyand phase response respectively The information-bearingphase is

120593 (119905 a119896) = 2120587ℎsum

119894

119886119896119894119902 (119905 minus 119894119879) (2)

where ℎ = 119896119901 is the modulation index and the transmittedsymbol 119886119896119894 isin plusmn1 plusmn3 plusmn(119872 minus 1) The phase responsefunction 119902(119905) is defined as

119902 (119905) = int

119905

minusinfin

119892 (120591) 119889120591 (3)

where 119892(120591) is the frequency response assumed to be causaland of duration 119871 symbols The phase response has thefollowing property

119902 (119905) =

1

2119905 ge 119871119879

0 119905 lt 0

(4)

The frequency separation 119891119896 and phase separation 120601119896 are

119891119896 = Δ119891(119896 minus119870 + 1

2)

120601119896 = Δ120601(119896 minus119870 + 1

2)

(5)

where Δ119891 and Δ120601 are frequency spacing and phase spacing[6] respectively According to the model the superimposedsignal (or the equivalent MAC signal) reads

119904 (119905 a) =119896=119870

sum

119896=1

119904119896 (119905 a119896) (6)

3 Design of Simplified Receivers

As it was mentioned above the basic idea is to employ areceiver defined in a low-dimensional signal space whichis optimized such that the minimum achievable distance ismaximized The low-dimensional receiver here is a general-ized form of the one presented in [14 15] where the receiveris built upon a shortened frequency response scheme Thedetails are presented below

31 Principles of the Proposed Receivers The transmittedsignal of the 119896th user is defined as 119904119896(119905 a119896) which reads

119904119896 (119905 a119896) = radic2119864119896

119879exp 120593119896 (119905 minus 120591 a119896) + 2120587119891119896 (119905 minus 120591) + 120601119896

(7)

and the corresponding alternative in received signal space iswritten as

119904119877119896 (119905 a119896) = radic2119864119896

119879exp 120595119896 (119905 a119896) + 2120587119891119896119905 + 120601119896 + 120579 (8)

where 120595(119905 a119896) is based on a shorter frequency response 119892119877(119905)whose duration is 119871119877 (119871119877 le 119871) The quantities 120579 and correcttiming 120591 are incorporated to achieve the best fitness betweenthe transmitted and received phase trajectories [14 15] Thealternative superimposed signal is 119904119877(119905 a) = sum

119896=119870

119896=1119904119877119896(119905 a119896)

and thus the received signal space is 119904119877(119905 a) Obviously thesize of the signal set is reduced from119872

119870119871 down to119872119870119871119877 andresults in a119872119870(119871minus119871119877)-fold complexity reduction As a specialcase when 119871119877 = 119871 there is no complexity reduction in thedetector However it is still possible to simplify the front-endas we will see below

Unfortunately the number of filters required is 119872119870119871119877which still might be unacceptable Therefore the principalcomponents analysis is introduced into the front-end to

The Scientific World Journal 3

further reduce the complexity As we shall see later thenumber of the filters can now be reduced significantly

The proposed scheme is summarized below

(1) find the 120579 the optimum 119902119877(119905) and the correctingtiming 120591

(2) calculate the orthogonal basis of 119904119877(119905 a) and thosewith nonzero eigenvalues are considered effective(this is equivalent to determine the effective dimen-sions and effective basis) the number of which isdesignated as119873119864

(3) the front-end that is a bank ofmatched filters is builtupon the effective basis 120573(119905) = 1205731(119905) 120573119873119864

(119905)(4) the sufficient statistics r(wrt 119904119877(119905 a)) are gener-

ated and sent to the detector followed(5) the ML detector delivers the detected a of a

For more details of the calculations and parameter opti-mization see [14 15] It should be pointed out that theresulting receiver is rather versatile It can be optimum (119871 =

119871119877) or suboptimum (119871 lt 119871119877) depending on the signalspace 119904119877(119905 a) being considered In the rest of this paperwe focus on the ML detector (wrt 119904119877(119905 a)) to evaluate theasymptotic performance

32 Performance Measurements Three measurements areconsidered in this paper the minimum achievable distancethe average energy loss and the number of effective dimen-sions The minimum distance is principally the same as insingle user systems [14 15] which reads

1198892= min

a = b

1

2119864119887

[

1003816100381610038161003816119904 (119905 a) minus 119904119877 (119905 b)1003816100381610038161003816

2minus1003816100381610038161003816119904 (119905 a) minus 119904119877 (119905 a)

1003816100381610038161003816

2

1003816100381610038161003816119904119877 (119905 a) minus 119904119877 (119905 b)1003816100381610038161003816

]

2

(9)

where 119864119887 is the average transmitted energy per informationbit It is noticed that 1198892 is positive by definition but is notadditiveTherefore no efficientmethod but exhaustive searchis employed to find this quantity in most cases

The average energy loss is defined as

120576 =1

119872119870119871[sum

a1 minus

1003816100381610038161003816⟨119904 (119905 a) 120573 (119905)⟩1003816100381610038161003816

2

|119904 (119905 a)|2] (10)

where ⟨ ⟩ designates the inner product operation The quan-tity 120576 is essentially the energy loss projecting 119904(119905 a) to thelow-dimensional signal space averaged over the transmittedsignal set

The number of effective dimensions implies the numberof complex filters required by the front-end This quantity isusually defined as the number of nonzero eigenvalues [13]It is now redefined as the number of eigenvalues greater than10minus4 for practical purpose It is obvious that such an operation

does not undermine the accuracy of the front-end On theother hand LD-based receivers also exploited a similar ideabut the real-valued filters required usually are defined overdurations of several symbol intervals [12 16]

33 Complexity It should be evident from the discussionabove that the proposed receiver can reduce the complexityexponentially from 119872

119870119871 down to 119872119870119871119877 This results in twokinds of complexity reduction the number of sates in atrellis and the computational effort of the branch metricsThere exist other receivers based on oversampling [7] orLD among which only those based on LD can achievesimultaneously simplified front-end and state reduction inthe detector In the case of single user systems it is observedthat the LD-based receiver and the proposed receiver havethe same performance in terms of BER and complexity aswas stated in [12] If those most significant components areused in LD a simpler trellis can be constructedwith negligibleperformance loss in single user systems However this doesnot work in multiuser systems where a degradation up to15 dB (around BER 10

minus3) is observed [9]Actually LD is rather a decomposition of the phase

trajectories than the CPM signal itself Therefore LD-basedfront ends for CPM multiuser systems must consider thesignals of individuals [9] This implies a linearly increasednumber of filters in LD receivers To conclude we say thatthe LD-based receiver and proposed receiver are roughlycomparable in single user systems [12] but may differ whenproceeding to multiuser systems

As to the proposed receiver there are three ways tosimplify the complexity by (1) reducing the number ofeffective filters 119873119864 (2) reducing the size of 119904119877(119905 a) by letting119871119877 lt 119871 or combing (1) and (2) together These methods areevaluated numerically in the next section

4 Numerical Results

The performance of the proposed receivers is evaluated indifferent scenarios Different multiuser schemes are designedto demonstrate the impact on the performance by differentmodulation parameters such as Δ119891 120579 119871119877 and the number ofeffective filters being used1198731015840

119864

In Table 1 for ℎ = 05 two systems based on araised-cosine frequency response of duration 2 (ie binaryCPM2RC) or a rectangular shape of duration 2 (ie binaryCPM2REC) are detected by MSK-based receivers Differentsystems are evaluated according to the measurements inSection 3 It is evident that the proposed receiver can reducethe complexity significantly while the average energy lossis marginal with much less required matched filters Thenumber of filters required by the LD-based receiver is 119870 sdot

2119875(119871minus1)

(2119875minus 1) [12] where 119875 is defined as 2119875minus1 lt 119872 le 2

119875To make a fair comparison we should take into accountthat the filters of LD are usually real valued whose durationsare several symbol intervals while the filters of proposedreceivers are complex valued and their duration is onesymbol interval Since the proposed receivers are designed toprocess the superimposed signals the signals of new usersdo not always increase the dimensions due to the strongcorrelation between CPM signals This is also observed inTable 1 where119873119864 increases slowly or even remains the samewhile increasing119870

4 The Scientific World Journal

Table 1 Comparison and performance analysis of some MSK-based receivers

Transmitter 119870 Δ119891Number of effective filters119873

119864 LD front end Energy loss Complexity reduction119904(119905 a) 119904119877(119905 a)

Binary CPM 2REC

1 000 3 2 2 0010 2-fold

2000 3 2

40015 4-fold

025 4 3 0086 4-fold05 4 3 0100 4-fold000 3 2

100021 32-fold

5 025 6 5 0098 32-fold

Binary CPM 2RC

05 8 6 0026 32-fold1 000 3 2 2 0002 2-fold

2000 4 2

40002 4-fold

025 5 3 0058 4-fold05 5 3 0098 4-fold

5000 4 2

100003 32-fold

025 8 5 0101 32-fold05 9 6 0031 32-fold

The optimum (119871119877 = 119871) receivers for different two-userbinary CPM2RC systems are considered in Figure 2 In thesesystems the main concern is to examine the impact of 119873119864The conventional front-end consisting of119872119870119871 = 16 filters isalso shown as a reference To reduce the complexity furthersome (ie 1198731015840

119864) most significant dimensions are employed

For a given observation length 119873 the minimum achievabledistance1198892 versusmodulation index ℎ is shown It is observedthat119873119864 effective filters are sufficient to reconstruct the signalsand no degradation is made When1198731015840

119864= 119873119864 minus 1 a marginal

but negligible degradation is observed When 1198731015840119864= 119873119864 minus 2

the gap is up to 06 dB (ℎ isin [0 05]) Therefore1198731015840119864= 119873119864 minus 1

would be a good choice It is also observed that CPM-basedmultiuser systems also suffer from weak index [1] It shouldbe pointed out that Δ119891 = 0 which makes this multiusersystem the most band-efficient scheme The parameter Δ120601 isoptimized to maximize 1198892These designedmultiuser systemscan approach the single user systems asymptotically with nosacrifice of bandwidth efficiency The use of Δ120601 is justified

The proposed suboptimum receivers based on 1REC areconsidered in Figure 3 with optimized Δ120601 and Δ119891 = 0 Thisfrequency response was particularly suited for single userbinary 2RC systems [14] The performances of the subopti-mum and the optimum receivers are compared It is seen thatthese suboptimum receivers have a performance loss nomorethan 1 dB However due to the severe degradation caused bythe dimension reduction observed in single user systems thistechnique is not adopted in these multiuser receivers

Presented in Figure 4 is a comparison of two suboptimumreceivers based on 1REC and 1RC respectively It is seenthat the 1RC based receiver is 15 dB worse than the 1RECbased receiverThis figure implies that the frequency responseparticularly suited for single user system is probably abetter choice than shortening the frequency response of thetransmitter directly

Based on the results and discussion above it can be seenthat the proposed receivers are successfully implemented in

3

25

2

15

1

05

0

0 01 02 03 04 05 06 07 08 09 1

d2

Modulation index h

One user N = 4

Two users N = 4 N998400

E= NE minus 1

Two users N = 4 N998400

E= 16

Two users N = 4 N998400

E= NE

Two users N = 4 N998400

E= NE minus 2

One user Nrarr +infin

Figure 2 The minimum achievable distance 1198892 versus the modula-tion index ℎ optimum (119871119877 = 119871) receivers

CPM-based systems The designed multiuser systems withoptimized parameters almost have an identical BER as thecorresponding single user systems It can be expected thatthe performance can be further improved using the methodpresented in [15] Another issue is the choice of differentparameters especially Δ120601 and Δ119891 In our case there is noneed to use Δ119891 gt 0 However for different systems theconclusions may differ It is also noticed that for some mod-ulation indices such as ℎ isin [05 08] a severe degradation

The Scientific World Journal 5

3

25

2

15

1

05

0

0 01 02 03 04 05 06 07 08 09 1

d2

Modulation index h

One user N = +infin optimum N998400

E= NE

One user N = 5 suboptimum N998400

E= NE

One user N = 5 suboptimum N998400

E= NE minus 1

Two users N = 5 optimum N998400

E= NE

Two users N = 5 suboptimum N998400

E= NE

Figure 3 The minimum achievable distance 1198892 versus the modula-tion index ℎ for suboptimum (119871119877 lt 119871) receivers Δ119891 = 0 and Δ120601 isoptimized

3

25

2

15

1

05

0

0 01 02 03 04 05 06 07 08 09 1

d2

Modulation index h

One user N = 4 suboptimumTwo users N = 4 suboptimum based on 1RECTwo users N = 4 suboptimum based on 1RC

One user Nrarr +infin optimum

Figure 4 The minimum achievable distance 1198892 versus the modula-tion index ℎ Δ119891 = 0 and Δ120601 is optimized

is observed This is due to the fact that a longer observationlength (ie 119873) is required Anyhow it is evident that aproperly designed CPM-based system has an asymptoticallyidentical BER with the corresponding single user systems

5 Conclusion

A class of simplified maximum-likelihood receivers is pro-posed for CPM-based multiuser systems The basic idea is

to perform detection over a low-dimensional signal spacesuch that the computational effort is reduced significantly(even exponentially in some cases) The performance of theproposed receiver is evaluated by means of analysis andjustified by the minimum achievable Euclidean distance Theimpact ofmodulation parameters is examined in detail for thedesigned schemes which reveal that the proposed receiverrequires less filters than some existing schemes and can befurther reducedwith negligible performance lossThough themain concern is designing maximum-likelihood receiversit should be obvious that the presented principles can begeneralized to other suboptimum receivers (such as [17]) withfew modifications

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor Tor Aulin for theuseful discussions and proofreading of this paper This workwas supported in part by the 973 Program of China underGrant 2012CB316100 NSFC under Grant 61372074 and theOpen Research Fund from the Science and Technology onInformation Transmission and Dissemination in Communi-cation Networks Laboratory (ITD-U12006)

References

[1] T Aulin N Rydbeck and C -E Sundberg ldquoContinuousphase modulation Part I and Part IIrdquo IEEE Transactions onCommunications Systems vol 29 pp 196ndash225 1981

[2] P Moqvist and T M Aulin ldquoSerially concatenated continuousphase modulation with iterative decodingrdquo IEEE Transactionson Communications vol 49 no 11 pp 1901ndash1915 2001

[3] M Xiao and T M Aulin ldquoSerially concatenated continuousphase modulation with convolutional codes over ringsrdquo IEEETransactions on Communications vol 54 no 8 pp 1387ndash13962006

[4] A Graell I Amat C Abdel Nour and C Douillard ldquoSeriallyconcatenated continuous phase modulation for satellite com-municationsrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3260ndash3269 2009

[5] A Perotti A Tarable S Benedetto andGMontorsi ldquoCapacity-achieving CPM schemesrdquo IEEE Transactions on InformationTheory vol 56 no 4 pp 1521ndash1541 2010

[6] P Moqvist Multiuser serially concatenated continuousphase modulation [PhD thesis] Chalmers University ofTechnology Goteborg Sweden 2002 httpwwwchalmerssecseENresearchresearch-groupstelecommunication-theorypublicationsphdtheses

[7] A Piemontese and G Colavolpe ldquoA novel graph-based subop-timal multiuser detector for FDM-CPM transmissionsrdquo IEEETransactions onWireless Communications vol 9 no 9 pp 2812ndash2819 2010

[8] P Moqvist and T Aulin ldquoMultiuser serially concatenatedcontinuous phase modulationrdquo in International Symposium onTurbo Codes pp 211ndash214 Brest France January 2013

6 The Scientific World Journal

[9] P A Murphy M Golanbari G E Ford and M J ReadyldquoOptimum and reduced complexity multiuser detectors forasynchronous CPM signalingrdquo IEEE Transactions on WirelessCommunications vol 5 no 8 pp 1959ndash1965 2006

[10] A J Viterbi CDMA Principles of Spread-Spectrum Communi-cation Addison-Wesley Wireless Communication 1995

[11] A D Wyner ldquoMulti-tone multiple access for cellular systemsrdquoATampT Bell Labs Technical Memorandum BL011217-920812-12TM 1992

[12] E Perrins and M Rice ldquoPAM decomposition of M-ary multi-h CPMrdquo IEEE Transactions on Communications vol 53 no 12pp 2065ndash2075 2005

[13] P Moqvist and T M Aulin ldquoOrthogonalization by principalcomponents applied to CPMrdquo IEEE Transactions on Commu-nications vol 51 no 11 pp 1838ndash1845 2003

[14] T Aulin C-E Sundberg and A Svensson ldquoSimple Viterbidetectors for partial response continuous phase modulatedsignalsrdquo inNational Telecommunications Conference Record ppA761ndashA767 New Orleans La USA 1981

[15] A Svensson C-E Sundberg and T Aulin ldquoA class of reduced-complexity Viterbi detectors for partial response continuousphase modulationrdquo IEEE Transactions on Communications vol32 no 10 pp 1079ndash1087 1984

[16] P A Laurent ldquoExact and approximate construction of digitalphase modulations by superposition of amplitude modulatedpulses (AMP)rdquo IEEE Transactions on Communications vol 34no 2 pp 150ndash160 1986

[17] XWang andHV Poor ldquoIterative (Turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999

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