Research Article Fast Maximum-Likelihood Decoder for Quasi...

Research ArticleFast Maximum-Likelihood Decoder for Quasi-OrthogonalSpace-Time Block Code

Adel Ahmadi1 and Siamak Talebi12

1Department of Electrical Engineering Shahid Bahonar University of Kerman PO Box 76169-133 Kerman Iran2Advanced Communications Research Institute Sharif University of Technology PO Box 11365-9363 Tehran Iran

Correspondence should be addressed to Adel Ahmadi adelahmadiengukacir

Received 8 February 2015 Accepted 13 April 2015

Academic Editor Sergio Preidikman

Copyright copy 2015 A Ahmadi and S Talebi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Motivated by the decompositions of sphere andQR-basedmethods in this paper we present an extremely fastmaximum-likelihood(ML) detection approach for quasi-orthogonal space-time block code (QOSTBC)The proposed algorithm with a relatively simpledesign exploits structure of quadrature amplitude modulation (QAM) constellations to achieve its goal and can be extended toany arbitrary constellation Our decoder utilizes a new decomposition technique for ML metric which divides the metric intoindependent positive parts and a positive interference part Search spaces of symbols are substantially reduced by employing theindependent parts and statistics of noise Symbols within the search spaces are successively evaluated until the metric is minimizedSimulation results confirm that the proposed decoderrsquos performance is superior to many of the recently published state-of-the-art solutions in terms of complexity level More specifically it was possible to verify that application of the new algorithms with1024-QAM would decrease the computational complexity compared to state-of-the-art solution with 16-QAM

1 Introduction

Quasi-orthogonal space-time block codes (QOSTBCs) [1 2]have attracted much attention lately due to their desired per-formances and pairwise detections Given these advantagesin [3ndash5] investigation is carried out to optimize diversityand coding gain performances One drawback of thesecodes however is that complexity of pairwise optimummaximum-likelihood (ML) decoder rises drastically whensize of constellation is increased A survey of related literatureshows that much research has been underway to develop anapproach that reduces complexity of QOSTBC decoder In[6 7] lattice reduction aided (LRA) method is combinedwith suboptimal detectors in order to improve performanceof the suboptimal decoders inmultiple-inputmultiple-output(MIMO) communications systems In [8] the QOSTBC isemployed to reduce gap between error ratio of ML andLRA methods This work also compares the performancesof spatial multiplexing and QOSTBC with those of LRAzero-forcing and ML decoders Authors of [9] employ QRdecomposition and sorting to simplify detection of QOSTBC

with four transmit antennas This method achieves ML errorperformance and offers low complexity For rotatedQOSTBCwith four transmit antennas and quadrature amplitude mod-ulation (QAM) constellation [10] explores a fast schemewith ML error performance and reduced complexity In [11]a different pairwise structure is proposed for QOSTBCsdecodingwith three or four transmit antennasThe algorithmin [11] initializes the first symbol by rounding its zero-forcing estimation to the nearest constellation point Thenthe other symbol of the pair is detected byminimizing its costfunction and the first symbol is discovered by minimizingits cost function The quality of the detected symbols isdependent upon the initial value which means that thedetection procedure may have to be repeated a number oftimes if the distance between the selected value and equalizedsymbol is higher than a threshold This method achieves anear-ML performance with low complexity decoder A new4 times 4 QOSTBC is reported in [12] that employs precoderand rotated symbols The complexity of the decoder is lowand a reasonable suboptimum error performance is obtainedwhen rotation angles are optimized The optimization

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 654865 6 pageshttpdxdoiorg1011552015654865

2 Mathematical Problems in Engineering

procedure is related to signal-to-noise ratio (SNR) andutilized constellation In [13] a suboptimum fast decodingalgorithm is investigated for block diagonal QOSTBC witharbitrary transmit antennas This decoder employs precom-puted look-up tables and statistics of phases in order toprepare a limited search area Authors of [14] introduce a 4times 4nonorthogonal space-time block code (STBC)with improvedcoding gain and a low complexity decoder In [15] a STBCis proposed for four transmit antennas based on mapping ofPAM and QAM constellationsThis code offers a high codinggain and authors design a low complexity decoder for it Anew structure for fast group decodable codes is presented in[16] whose decoder offers the lowest worst-case complexityamong STBCs

The innovative method developed in this letter decom-poses the received vector into two pairs of symbols whichare detected independently To do this the ML metricminimization of each pair is transformed into a sum ofindependent positive parts and an interference part It shouldbe noted that independency of the positive parts facilitatesdetection by allowing us to considerably limit the searchspaces of the symbols The candidates placed in relevantpartial search areas are gradually evaluated and then thetransmitted symbols are estimated by computing interferencebetween them If the search areas are small and no symbol isdetected then they are extended and evaluation is repeatedtill the transmitted symbols are detected The ML metricdecomposition studied in this letter boasts two importantfeatures namely decomposition is not a highly complexprocess and more significantly most of the decomposedparts are independent of each other and therefore the searchspace can be efficiently limited Based on these features theproposed method offers the desired ML performance at verylow complexity

Compared with the proposed method Sphere [17] andQR [9] decoders are more complex because their searchspaces cannot effectively be reduced and they both requiresignificant precomputation steps As for the fast ML decoderin [10] authors utilize a simplified quadratic ML decodingstatistic for QAM constellations and comparison of the sim-ulation results indicates that the average complexity of thiswork is higher than that of the proposed method confirmingthat the latter can be extended to any arbitrary constellationsOn the other hand the nonoptimum detectors in [12 13]both require rotation angle optimization and significantmemory locations for look-up tables respectively whereasthe proposed solution does not require any optimization orlook-up tables

2 System Model

QOSTBC for 119872119879

= 4 transmit antennas has the followingform

X =

[[[[[

[

1199041

1199042

1199043

1199044

minus119904lowast

2119904lowast

1minus119904lowast

4119904lowast

3

1199043

1199044

1199041

1199042

minus119904lowast

4119904lowast

3minus119904lowast

2119904lowast

1

]]]]]

]

(1)

where 1199043

= 119890j1205791199043 1199044

= 119890j1205791199044 j = radicminus1 and the data symbols

1199041 119904

4belong to constellationC with unit average energy

The rotation angle 120579 can be selected such that full diversityand high coding gain are attained For example 120579 = 1205874

is optimum for QAM [10] A space-time communicationsystem with four transmit antennas and119872

119877receive antennas

that transmits four symbols over four time slots can berepresented by an equivalent receive antenna 119895 as

y119895= radic

120588

119872119879

H119895s + w119895 (2)

where SNR is indicated by 120588 and s = [1199041 119904

4]119879 is the trans-

mitted vector The equivalent received vector and equivalentadditive white Gaussian noise (AWGN) vector are denotedby y119895

= [1199101119895

119910119872119879119895

]119879 and w

119895= [119908

1119895 119908

119872119879119895]119879

respectively Also 119910119894119895and119908

119894119895denote the equivalent received

signal and its relative noise at the 119894th time slot of the119895th receive antenna respectively The AWGN has complexGaussian distribution with zero mean and unit varianceThe equivalent channel matrix for the 119895th receive antenna isdefined as

H119895≜ [

[

B (ℎ1119895

ℎ2119895

) 119890j120579B (ℎ

3119895 ℎ4119895

)

B (ℎ3119895

ℎ4119895

) 119890j120579B (ℎ

1119895 ℎ2119895

)

]

]

(3)

where ℎ119894119895

stands for channel fade between the 119894th transmitantenna and the 119895th receive antenna such that

B (ℎ119894119895 ℎ119894+1119895

) ≜ [

ℎ119894119895

ℎ119894+1119895

ℎlowast

119894+1119895minusℎlowast

119894119895

] (4)

The complete received vectors can be concatenated as

y = radic

120588

119872119879

Hs + w (5)

where y ≜ [y1198791 y119879

119872119877]119879 w ≜ [w119879

1 w119879

119872119877]119879 and H ≜

[H1198791 H119879

119872119877]119879

3 Fast Maximum-Likelihood Decoder

31 Proposed ML Detection Method TheML decoder shouldminimize the following norm in order to estimate thetransmitted symbols

s = argminsisinC119872119879

1003817100381710038171003817100381710038171003817100381710038171003817

y minus radic

120588

119872119879

Hs1003817100381710038171003817100381710038171003817100381710038171003817

(6)

where norm x is defined as x = radicx119867x and x119867 indicatesconjugate transpose of x The above minimization can berewritten as


H (s minus z) (7)

where z = [1199111 119911

119872119879]119879 is defined as

z ≜ radic

119872119879

120588

(H119867H)

minus1

H119867y (8)

Mathematical Problems in Engineering 3

The matrixH119895can be decomposed into

H119895=

1

2

UH1015840119895UV (9)

where V ≜ diag(I2 119890

j120579I2) U ≜ [

I2 I2I2 minusI2 ] I2 ≜ [

1 0

0 1] and

H1015840119895

≜ [

[

B (ℎ1119895

+ ℎ3119895

ℎ2119895

+ ℎ4119895

) 02times2

02times2

B (ℎ1119895

minus ℎ3119895

ℎ2119895

+ ℎ4119895

)

]

]

(10)

By employing (9) and doing some math operations weare able to compute z through fewer mathematic operationsas

z = radic

119872119879

4120588

V119867UDminus1119873

sum

119895=1

H1015840119895

119867Uy119895 (11)

whereD ≜ diag(120572I2 120573I2) and

120572 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

+ ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

+ ℎ4119895

10038161003816100381610038161003816

2

)

120573 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

minus ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

minus ℎ4119895

10038161003816100381610038161003816

2

)

(12)

In addition H(s minus z)2 can be decomposed into twoindependent parts by utilizing (9) which results in

H (s minus z)2 =2

sum

119897=1

(s119897minus z119897)119867T (s

119897minus z119897) (13)

where z119897≜ [119911119897 119911119897+2

]119879 and s

119896≜ [119904119897 119904119897+2

]119879 for 119897 = 1 and 2 By

defining 1199050≜ (120572 + 120573)2 and 119905

1≜ (120572 minus 120573)2 the matrix T can

be represented as

T ≜ [

1199050

119890j1205791199051

119890minusj120579

1199051

1199050

] (14)

Based on (13) for 119897 = 1 and 2 the ML decoder canindependently detect s

119897≜ [119904119897 119904119897+2

]119879 by

(119904119897 119904119897+2

) = argmin119904119897 119904119897+2isinC

(s119897minus z119897)119867T (s

119897minus z119897) (15)

For the remaining part of this section we focus on detectionof (1199041 1199043) noting that the other pair (119904

2 1199044) can be detectable

by applying a similar approach Furthermore for the sakeof simplicity and clarity let us consider a square 119872

2-QAMconstellation although the solution can also be straightfor-wardly extended to other constellations For 119896 = 1 and 3119904119896and 119911

119896can be represented by their real and imaginary

parts as 119904119896

= s119896+ js119896+1

and 119911119896

= z119896+ jz119896+1

respectivelyUnder this scenario s

1198974

119897=1are independent and belong to

the setR = 1198861 1198862 119886

119872 with real members The decoder

searches within R to find the best choices for the real andimaginary parts of symbols which minimizes the ML metric

The ML minimization (15) can be reformulated as

120574 ≜

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+ 12059111205811+ 12059121205812 (16)

where 1205911

≜ 2120582 cos 120579 1205912

≜ 2120582 sin 120579 120582 ≜ 11990511199050 1205811

≜ 12057511205753+

12057521205754 1205812≜ 12057521205753minus 12057511205754 and 120575

119897≜ s119897minusz119897for 119897 = 1 4 Based

on (12) 1199050

ge |1199051| and we can rewrite minimization of (16) as

minimization of 120576 ≜ 120574(1199050minus |1199051|) which is decomposable into

sum of positive parts

120576 =

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+

|120582|

1 minus |120582|

1003816100381610038161003816(1205751+ j1205752) + 120583 (120575

3+ j1205754)1003816100381610038161003816

2 (17)

where 120583 ≜ sign(1199051)119890

j120579 and sign(sdot) stands for the signumfunction In (17) the parts 120575

1198974

119897=1are independent of each

other which allows us to reduce the search space effectivelyand the purpose of the third part is to represent interferencebetween the other parts

By assuming that the minimum of (17) is smaller than 1199032

we deduce that |120575119897| lt 119903 for 119897 = 1 4 Therefore the output

of the ML decoder has to be located within correspondingintervals [z

119897minus 119903z

119897+ 119903] which reduces the search spaces

effectively The ML metric (16) for the remaining membersis incrementally evaluated piece by piece to remove moreinappropriate members and finally we obtain symbols 119904

1=

s1+ js2and 1199043

= s3+ js4which minimize the ML metric

These steps for an 1198722-QAM constellation are demonstrated

as a pseudocode in Pseudocode 1 In this pseudocodecardinality of setR is indicated by |R|

119888 the119898th member of

this set is denoted byR(119898) andR119897≜ 119886 isin R |119886 minusz

119897| lt 119903

for 119897 = 1 4The proposed method can be extended to other constel-

lations with arbitrary distribution of signal points Under thiscondition the real part of a symbol is related to its imaginarypart and therefore they are examined together

32 Algorithm Initialization The proposed method needs tobegin with an appropriate initial value for 119903 which shouldbe suitably selected to avoid unnecessary complexity It isworth mentioning that in the proposed method the initialvalue selection is approximately similar to the initial radiusselection in the sphere method If the initial value of 119903 is toohigh size of search spaces and thus the detection complexitywill be raised Conversely if value of 119903 is initialized with avery small value the search spaces include no candidate andno symbol will be detected Under this situation we haveto increase the value of 119903 and apply our method again tillsymbols are detected In this letter we set the initial valueof 119903 to 119903initial = 119889min2 where 119889min indicates the minimumdistance between two distinct constellation points In thiscase search region of each symbol contains just one candidatepoint which is almost the best possible choice for moderateto high SNRs As will be shown in Section 4 this type ofinitialization ensures that the proposed decoder achieves lowcomplexity


120574th = (1 minus |120582|)1199032

] = 1(1 minus |120582|)FOR 119898

1= 1 2 |R

1|119888

1205751= R1(1198981) minus z1

1205761= 1205752

1

FOR 1198982= 1 2 |R

2|119888

1205752= R2(1198982) minus z2

1205762= 1205761+ 1205752

2

IF 1205762gt 1199032

CONTINUEENDFOR 119898

3= 1 2 |R

3|119888

1205753= R3(1198983) minus z3

1205763= 1205762+ 1205752

3

IF 1205763gt 1199032


4= 1 2 |R

4|119888

1205754= R4(1198984) minus z4

1205764= 1205763+ 1205752

4

IF 1205764gt 1199032

CONTINUEEND1205811= 12057511205753+ 12057521205754

1205812= 12057521205753minus 12057511205754

120574 = 1205764+ 12059111205811+ 12059121205812

IF 120574 gt 120574th120576 = ]120574119903 = radic120576120574th = 1205741199041= R1(1198981) + jR

2(1198982)

1199043= R3(1198983) + jR

4(1198984)

ENDEND

ENDEND

END

Pseudocode 1 Pseudocode of search stage of proposed method

33 Complexity Analysis The decoder operation can bedivided into precomputation and search stages The initialvalue of variables and candidate points are obtained byperforming about 64119872

119877+ 29 real additions 40119872

119877+ 31

multiplications and four divisions during the first stage Inaddition the proposed decoder on average employs about8log2119872 and 8119872 relational operators for an119872-ary QAM and

other constellations respectivelyThe complexity of the search stage depends on complex

Gaussian noises From (8) we infer that 1199111

= 1199041+ 1205901198991and

1199113= 1199043+ (984858120590)119890

minusj1205791198991+ 12059010158401198993 where 119904

1and 1199043are transmitted

symbols 1198991and 1198993are independent complex Gaussian noises

with zero means and unit variances 1205902 = 119872119879(120572 + 120573)2120588120572120573

984858 = 119872119879(120572 minus 120573)2120588120572120573 and 120590

10158402

= 2119872119879120588(120572 + 120573) Based on

the complex Gaussian distribution the real and imaginaryparts of 119911

1minus 1199041and 1199113minus 1199043are located within [minus120596 +120596] with

probability 119875(120596) This probability is given by

119875 (120596) =

1

2120587

sdot int

120596120590

minus120596120590

int

120596120590

minus120596120590

[Q(minus

120596

1205901015840minus 1205781) minus Q(

120596

1205901015840minus 1205781)]

sdot [Q(minus

120596

1205901015840minus 1205782) minus Q(

120596

1205901015840minus 1205782)] 119890minus(12)(119899

2+11989910158402

)119889119899 119889119899

1015840

(18)

where 1205781= (984858120590120590

1015840)(119899 cos 120579+119899

1015840 sin 120579) 1205782= (984858120590120590

1015840)(1198991015840 cos 120579minus

119899 sin 120579) and Q(119906) = (1radic2120587) int

infin

119906exp(minus120577

22)119889120577 When 120596 ≫

1205901015840 the value of 119875(120596) can be bounded by

119875 (120596) gt [1 minus 2Q(

120596

1205901015840)]

2

[1 minus 2Q (4)]2 (19)

By assuming that the decoder can truly detect 1199041and 1199043

the complexity of our pseudocode can be bounded by

C13

le 119888M (120596)N (120596) (20)

where 119888 is a constant value andM(120596) andN(120596) are functionsthat indicate the number of constellation points which arelocated within the search intervals of 119904

1and 1199043 respectively

The inequality (19) implies that 119875(120596) gets very close to onefor 119889min2 le 120596 le 119889min and acceptable values of SNRswhich results in 120590

1015840≪ 119889min Under this scenario the real and

imaginary parts of 119911119896approach those of 119904

119896 such that very few

signal points take place inside the created search regionsThisstatistic feature reduces the complexity of search stage andmakes it become independent of constellation size when SNRis reasonable

The detection complexity of 1199041198964

119896=1in the search stage

depends on the power of noises which practically becomesmall for reasonable values of SNRs Under this conditionon average either one or two candidates are selected forevaluation per each dimension hence requiring about 22ndash44additions and 22ndash44multiplications for aQAMconstellationThis means that complexity of the search stage is low andtherefore that of precomputation is taken as the dominantfactor

Compared against the sphere detection (SD) methodwhere the precomputation stage requires about 424119872

119877+ 28

additions 432119872119877

+ 120 multiplications with real equivalentchannel having 8119872

119877rows and 8 columns [17] or against the

QR-based methods involving about 104119872119877minus8 additions and

136119872119877+ 112 multiplications for the preparation stage [9] it

is obvious that the overall complexity of the proposed modelenjoys a clear advantage

34 Comparison with Sphere The SD method reformulatesthe complex ML metric of (6) in a real form as y minus Hswhere y isin R8119872119877times1 H isin R8119872119877times8 and s isin R8times1 Thenafter applying Cholesky factorization and solving two linearsystems a new metric is obtained in form of R(z minus s)where R isin R8times8 is an upper triangular matrix and z isthe real zero-force equalized vector Therefore the algorithm


should minimize the following metric for the real entries ofs isin R8times1

8

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=119894

119903119894119895

(z119895minus s119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=100381610038161003816100381611990388

(z8minus s8)1003816100381610038161003816

2

+100381610038161003816100381611990377

(z7minus s7) + 11990378

(z8minus s8)1003816100381610038161003816

2+ sdot sdot sdot

+100381610038161003816100381611990311

(z1minus s1) + 11990312

(z2minus s2) + sdot sdot sdot

+ 11990318

(z8minus s8)1003816100381610038161003816

2

(21)

where 119903119894119895 z119895 and s

119895are entries of r z and s respectively

This equation consists of eight positive terms which aredependent on each other This means that the search spacesof s119894rsquos (for 119894 = 1 8) are dependent on the values of

s119894+1

s119894+2

and s8while in the decomposed metric (17)

of the proposed method all terms are independent of eachother except the last term This fact helps us to preparethe search spaces individually and reduce their sizes moreefficiently which is not possible with the SDmethod Also theSD algorithm requires Cholesky factorization and two linearsystems solving which results in higher complexity

4 Simulation Results

In order to evaluate performance of the proposed decoderexperimentally authors simulated a QOSTBC system withRayleigh channel model under different constellations usingfour transmit antennas and one receive antenna As men-tioned in Section 3 the complexity of the algorithm dependson the initial value Figure 1 clearly illustrates the impactsof initial value selection on average complexity for differenttypes of constellation when the BER is approximately 10

minus4

and the constellations have unit average energy FromFigure 1it is evident that complexity is reduced when initial value of119903 is selected to be about 119889min2 For the remainder of thissection we set the initial value of 119903 as 119903initial = 119889min2 whichresults in a reasonably acceptable complexity

The required number of real additions and multiplica-tions by the proposed method and those of several state-of-the-art works are tabulated in Tables 1 and 2 These resultsshow that the numbers by the proposed method are lowerthan those by the competitors and that they remain almostunaltered for different constellations For example when biterror rate (BER) is about 10minus4 complexity of our methodfor the given QAMs reduces to about 118 addition and 96multiplication operations respectively which is compara-tively lower than that of the other solutions even with smallerconstellation size This is a good indication of the superiorityof the new algorithm Moreover Figure 2 charts BERs ofthe proposed and ML decoders which illustrates that it isquite possible to arrive at a solution capable of optimumperformance at very low complexity

5 Conclusion

In this letter an innovative fast maximum-likelihood (ML)decoder for QOSTBCwas introducedThe proposedmethod

0 05 1 1510

2

103

104

rinitial

Avg

num

ber o

f add

ition

s and

mul

tiplic

atio

ns

1024-QAM256-QAM

64-QAM16-APSK

Figure 1The relation between average complexity and initial value

5 10 15 20 25 30 35 40 4510

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

Bit e

rror

rate

ML of (6) QPSKML of (6) 8-PSKML of (6) 16-APSKML of (6) 64-QAMML of (6) 256-QAMML of (6) 1024-QAM

Prop ML QPSKProp ML 8-PSKProp ML 16-APSKProp ML 64-QAMProp ML 256-QAMProp ML 1024-QAM

Figure 2 Error performance of ML and proposed method

exploited structure ofQAMconstellations in order to developan algorithm at low computational complexity By using newalgorithm the ML metrics was decomposed into a sum ofindependent positive parts and an interference part Giventhat metrics within the positive parts are independent of eachother the proposed technique substantially limited the searcharea and hence complexity of detection decreased exponen-tially It was demonstrated that for bit error rate of 10minus4 theresulting complexity remained almost unaltered for differentconstellations and was less than those of state-of-the-art


Table 1 Average number of additions and multiplications for the proposed method and several BERs

Const 16-QAM 64-QAM 256-QAM 1024-QAMBER + times + times + times + times

10minus2 1436 1199 1515 1266 1597 1338 1636 137110minus3 1222 998 1243 1017 1269 1040 1277 104710minus4 1162 941 1169 947 1176 954 1180 957Const QPSK 8-PSK 8-APSK 16-APSKBER + times + times + times + times

10minus2 1357 1117 1506 1249 1397 1142 1542 127110minus3 1203 978 1217 991 1203 975 1242 101210minus4 1156 936 1159 937 1154 932 1163 941

Table 2 Average number of additions and multiplications

of 16-QAM 64-QAM 256-QAM+ times + times + times

[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364

schemes Also the proposed scheme can be extended to workwith any constellation type

Conflict of Interests

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

References

[1] H Jafarkhani ldquoA quasi-orthogonal space-time block coderdquoIEEE Transactions on Communications vol 49 no 1 pp 1ndash42001

[2] F Fazel and H Jafarkhani ldquoQuasi-orthogonal space-frequencyand space-time-frequency block codes for MIMO-OFDMchannelsrdquo IEEE Transactions on Wireless Communications vol7 no 1 pp 184ndash192 2008

[3] W Su andX-G Xia ldquoSignal constellations for quasi-orthogonalspace-time block codeswith full diversityrdquo IEEETransactions onInformation Theory vol 50 no 10 pp 2331ndash2347 2004

[4] A Sezgin and E A Jorswieck ldquoOn optimal constellations forquasi-orthogonal space-time codesrdquo in Proceedings of the IEEEInternational Conference on Accoustics Speech and Signal Pro-cessing (ICASSP rsquo03) vol 4 pp 345ndash348 IEEE Hong KongApril 2003

[5] A Sezgin E A Jorswieck and H Boche ldquoPerformance criteriaanalysis and further performance results for quasi-orthogonalspace-time block codesrdquo in Proceedings of the 3rd IEEE Interna-tional Symposium on Signal Processing and Information Technol-ogy (ISSPIT rsquo03) pp 102ndash105 IEEE December 2003

[6] H Yao and G W Wornell ldquoLattice-reduction-aided detectorsfor MIMO communication systemsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference (GLOBECOM rsquo02) pp424ndash428 Taipei Taiwan November 2002

[7] D Wubben R Bohnke V Kuhn and K-D KammeyerldquoNear-maximum-likelihood detection of MIMO systems usingMMSE-based lattice-reductionrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC 04) pp798ndash802 Paris France June 2004

[8] A Sezgin E A Jorswieck and E Costa ldquoLattice-reductionaided detection spatial multiplexing versus quasi-orthogonalSTBCrdquo in Proceedings of the IEEE 60th Vehicular TechnologyConference (VTC-Fall rsquo04) pp 2389ndash2393 September 2004

[9] M-T Le V-S Pham L Mai and G Yoon ldquoLow-complexitymaximum-likelihooddecoder for four-transmit-antenna quasi-orthogonal space-time block coderdquo IEEE Transactions on Com-munications vol 53 no 11 pp 1817ndash1821 2005

[10] S KunduW SuDA Pados andM JMedley ldquoFastmaximum-likelihood decoding of 4 x 4 full-diversity quasi-orthogonalSTBCs with QAM signalsrdquo in Proceedings of the 53rd IEEEGlobal Communications Conference (GLOBECOM rsquo10) pp 1ndash5December 2010

[11] S J Alabed J M Paredes and A B Gershman ldquoA low com-plexity decoder for quasi-orthogonal space time block codesrdquoIEEE Transactions on Wireless Communications vol 10 no 3pp 988ndash994 2011

[12] S Kundu D A Pados W Su and R Grover ldquoToward a pre-ferred 4times4 space-time block code a performance-versus-com-plexity sweet spot with linear-filter decodingrdquo IEEE Transac-tions on Communications vol 61 no 5 pp 1847ndash1855 2013

[13] A Ahmadi S Talebi and M Shahabinejad ldquoA new approachto fast decode Quasi-orthogonal space-time block codesrdquo IEEETransactions on Wireless Communications vol 14 no 1 p 1232015

[14] N SharmaM R Bhatnagar andM Agrawal ldquoNon-orthogonalSTBC for four transmit antennas with high coding gain andlowdecoding complexityrdquo inProceedings of the 5th InternationalConference on Signal Processing and Communication Systems(ICSPCS rsquo11) pp 1ndash5 IEEE Honolulu Hawaii USA December2011

[15] N Sharma M R Bhatnagar and M Agrawal ldquoA high codinggain and low decoding complexity STBC for four transmitantennasrdquo in Proceedings of the 18th National Conference onCommunications (NCC rsquo12) pp 3ndash5 February 2012

[16] A Ismail J Fiorina and H Sari ldquoA new family of low-com-plexity STBCs for four transmit antennasrdquo IEEE Transactionson Wireless Communications vol 12 no 3 pp 1208ndash1219 2013

[17] Z Safar W Su and K J Liu ldquoA fast sphere decoding algo-rithm for space-frequency block codesrdquo EURASIP Journal onAdvances in Signal Processing vol 2006 Article ID 097676 14pages 2006

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


procedure is related to signal-to-noise ratio (SNR) andutilized constellation In [13] a suboptimum fast decodingalgorithm is investigated for block diagonal QOSTBC witharbitrary transmit antennas This decoder employs precom-puted look-up tables and statistics of phases in order toprepare a limited search area Authors of [14] introduce a 4times 4nonorthogonal space-time block code (STBC)with improvedcoding gain and a low complexity decoder In [15] a STBCis proposed for four transmit antennas based on mapping ofPAM and QAM constellationsThis code offers a high codinggain and authors design a low complexity decoder for it Anew structure for fast group decodable codes is presented in[16] whose decoder offers the lowest worst-case complexityamong STBCs

The innovative method developed in this letter decom-poses the received vector into two pairs of symbols whichare detected independently To do this the ML metricminimization of each pair is transformed into a sum ofindependent positive parts and an interference part It shouldbe noted that independency of the positive parts facilitatesdetection by allowing us to considerably limit the searchspaces of the symbols The candidates placed in relevantpartial search areas are gradually evaluated and then thetransmitted symbols are estimated by computing interferencebetween them If the search areas are small and no symbol isdetected then they are extended and evaluation is repeatedtill the transmitted symbols are detected The ML metricdecomposition studied in this letter boasts two importantfeatures namely decomposition is not a highly complexprocess and more significantly most of the decomposedparts are independent of each other and therefore the searchspace can be efficiently limited Based on these features theproposed method offers the desired ML performance at verylow complexity

Compared with the proposed method Sphere [17] andQR [9] decoders are more complex because their searchspaces cannot effectively be reduced and they both requiresignificant precomputation steps As for the fast ML decoderin [10] authors utilize a simplified quadratic ML decodingstatistic for QAM constellations and comparison of the sim-ulation results indicates that the average complexity of thiswork is higher than that of the proposed method confirmingthat the latter can be extended to any arbitrary constellationsOn the other hand the nonoptimum detectors in [12 13]both require rotation angle optimization and significantmemory locations for look-up tables respectively whereasthe proposed solution does not require any optimization orlook-up tables

2 System Model

QOSTBC for 119872119879

= 4 transmit antennas has the followingform

X =

[[[[[

[

1199041

1199042

1199043

1199044

minus119904lowast

2119904lowast

1minus119904lowast

4119904lowast

3

1199043

1199044

1199041

1199042

minus119904lowast

4119904lowast

3minus119904lowast

2119904lowast

1

]]]]]

]

(1)

where 1199043

= 119890j1205791199043 1199044

= 119890j1205791199044 j = radicminus1 and the data symbols

1199041 119904

4belong to constellationC with unit average energy

The rotation angle 120579 can be selected such that full diversityand high coding gain are attained For example 120579 = 1205874

is optimum for QAM [10] A space-time communicationsystem with four transmit antennas and119872

119877receive antennas

that transmits four symbols over four time slots can berepresented by an equivalent receive antenna 119895 as

y119895= radic

120588

119872119879

H119895s + w119895 (2)

where SNR is indicated by 120588 and s = [1199041 119904

4]119879 is the trans-

mitted vector The equivalent received vector and equivalentadditive white Gaussian noise (AWGN) vector are denotedby y119895

= [1199101119895

119910119872119879119895

]119879 and w

119895= [119908

1119895 119908

119872119879119895]119879

respectively Also 119910119894119895and119908

119894119895denote the equivalent received

signal and its relative noise at the 119894th time slot of the119895th receive antenna respectively The AWGN has complexGaussian distribution with zero mean and unit varianceThe equivalent channel matrix for the 119895th receive antenna isdefined as

H119895≜ [

[

B (ℎ1119895

ℎ2119895

) 119890j120579B (ℎ

3119895 ℎ4119895

)

B (ℎ3119895

ℎ4119895

) 119890j120579B (ℎ

1119895 ℎ2119895

)

]

]

(3)

where ℎ119894119895

stands for channel fade between the 119894th transmitantenna and the 119895th receive antenna such that

B (ℎ119894119895 ℎ119894+1119895

) ≜ [

ℎ119894119895

ℎ119894+1119895

ℎlowast

119894+1119895minusℎlowast

119894119895

] (4)

The complete received vectors can be concatenated as

y = radic

120588

119872119879

Hs + w (5)

where y ≜ [y1198791 y119879

119872119877]119879 w ≜ [w119879

1 w119879

119872119877]119879 and H ≜

[H1198791 H119879

119872119877]119879

3 Fast Maximum-Likelihood Decoder

31 Proposed ML Detection Method TheML decoder shouldminimize the following norm in order to estimate thetransmitted symbols


1003817100381710038171003817100381710038171003817100381710038171003817

y minus radic

120588

119872119879

Hs1003817100381710038171003817100381710038171003817100381710038171003817

(6)

where norm x is defined as x = radicx119867x and x119867 indicatesconjugate transpose of x The above minimization can berewritten as


H (s minus z) (7)

where z = [1199111 119911

119872119879]119879 is defined as

z ≜ radic

119872119879

120588

(H119867H)

minus1

H119867y (8)



H119895=

1

2

UH1015840119895UV (9)


j120579I2) U ≜ [


1 0

0 1] and

H1015840119895

≜ [

[

B (ℎ1119895

+ ℎ3119895

ℎ2119895

+ ℎ4119895

) 02times2

02times2

B (ℎ1119895

minus ℎ3119895

ℎ2119895

+ ℎ4119895

)

]

]

(10)


z = radic

119872119879

4120588

V119867UDminus1119873

sum

119895=1

H1015840119895

119867Uy119895 (11)


120572 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

+ ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

+ ℎ4119895

10038161003816100381610038161003816

2

)

120573 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

minus ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

minus ℎ4119895

10038161003816100381610038161003816

2

)

(12)


H (s minus z)2 =2

sum

119897=1

(s119897minus z119897)119867T (s

119897minus z119897) (13)

where z119897≜ [119911119897 119911119897+2

]119879 and s

119896≜ [119904119897 119904119897+2

]119879 for 119897 = 1 and 2 By

defining 1199050≜ (120572 + 120573)2 and 119905


be represented as

T ≜ [

1199050

119890j1205791199051

119890minusj120579

1199051

1199050

] (14)


119897≜ [119904119897 119904119897+2

]119879 by

(119904119897 119904119897+2

) = argmin119904119897 119904119897+2isinC

(s119897minus z119897)119867T (s

119897minus z119897) (15)






parts as 119904119896

= s119896+ js119896+1

and 119911119896

= z119896+ jz119896+1


1198974


the setR = 1198861 1198862 119886




120574 ≜

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+ 12059111205811+ 12059121205812 (16)

where 1205911

≜ 2120582 cos 120579 1205912

≜ 2120582 sin 120579 120582 ≜ 11990511199050 1205811

≜ 12057511205753+

12057521205754 1205812≜ 12057521205753minus 12057511205754 and 120575


on (12) 1199050




120576 =

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+

|120582|

1 minus |120582|

1003816100381610038161003816(1205751+ j1205752) + 120583 (120575

3+ j1205754)1003816100381610038161003816

2 (17)

where 120583 ≜ sign(1199051)119890


1198974






119897minus 119903z



1=

s1+ js2and 1199043






119897| lt 119903





120574th = (1 minus |120582|)1199032

] = 1(1 minus |120582|)FOR 119898

1= 1 2 |R

1|119888

1205751= R1(1198981) minus z1

1205761= 1205752

1

FOR 1198982= 1 2 |R

2|119888

1205752= R2(1198982) minus z2

1205762= 1205761+ 1205752

2

IF 1205762gt 1199032


3= 1 2 |R

3|119888

1205753= R3(1198983) minus z3

1205763= 1205762+ 1205752

3

IF 1205763gt 1199032


4= 1 2 |R

4|119888

1205754= R4(1198984) minus z4

1205764= 1205763+ 1205752

4

IF 1205764gt 1199032

CONTINUEEND1205811= 12057511205753+ 12057521205754

1205812= 12057521205753minus 12057511205754

120574 = 1205764+ 12059111205811+ 12059121205812


2(1198982)

1199043= R3(1198983) + jR

4(1198984)

ENDEND

ENDEND

END



119877+ 29 real additions 40119872

119877+ 31




= 1199041+ 1205901198991and

1199113= 1199043+ (984858120590)119890

minusj1205791198991+ 12059010158401198993 where 119904




984858 = 119872119879(120572 minus 120573)2120588120572120573 and 120590

10158402

= 2119872119879120588(120572 + 120573) Based on




119875 (120596) =

1

2120587

sdot int

120596120590

minus120596120590

int

120596120590

minus120596120590

[Q(minus

120596

1205901015840minus 1205781) minus Q(

120596

1205901015840minus 1205781)]

sdot [Q(minus

120596

1205901015840minus 1205782) minus Q(

120596

1205901015840minus 1205782)] 119890minus(12)(119899

2+11989910158402

)119889119899 119889119899

1015840

(18)

where 1205781= (984858120590120590

1015840)(119899 cos 120579+119899

1015840 sin 120579) 1205782= (984858120590120590

1015840)(1198991015840 cos 120579minus


infin

119906exp(minus120577

22)119889120577 When 120596 ≫


119875 (120596) gt [1 minus 2Q(

120596

1205901015840)]

2

[1 minus 2Q (4)]2 (19)



C13

le 119888M (120596)N (120596) (20)












119877+ 28

additions 432119872119877









8

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=119894

119903119894119895

(z119895minus s119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=100381610038161003816100381611990388

(z8minus s8)1003816100381610038161003816

2

+100381610038161003816100381611990377

(z7minus s7) + 11990378

(z8minus s8)1003816100381610038161003816

2+ sdot sdot sdot

+100381610038161003816100381611990311

(z1minus s1) + 11990312


+ 11990318

(z8minus s8)1003816100381610038161003816

2

(21)

where 119903119894119895 z119895 and s



s119894+1

s119894+2





minus4



5 Conclusion


0 05 1 1510

2

103

104

rinitial

Avg

num

ber o

f add

ition

s and

mul

tiplic

atio

ns

1024-QAM256-QAM

64-QAM16-APSK


5 10 15 20 25 30 35 40 4510

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

Bit e

rror

rate












[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











H119895=

1

2

UH1015840119895UV (9)


j120579I2) U ≜ [


1 0

0 1] and

H1015840119895

≜ [

[

B (ℎ1119895

+ ℎ3119895

ℎ2119895

+ ℎ4119895

) 02times2

02times2

B (ℎ1119895

minus ℎ3119895

ℎ2119895

+ ℎ4119895

)

]

]

(10)


z = radic

119872119879

4120588

V119867UDminus1119873

sum

119895=1

H1015840119895

119867Uy119895 (11)


120572 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

+ ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

+ ℎ4119895

10038161003816100381610038161003816

2

)

120573 ≜

119872119877

sum

119895=1

(

10038161003816100381610038161003816ℎ1119895

minus ℎ3119895

10038161003816100381610038161003816

2

+

10038161003816100381610038161003816ℎ2119895

minus ℎ4119895

10038161003816100381610038161003816

2

)

(12)


H (s minus z)2 =2

sum

119897=1

(s119897minus z119897)119867T (s

119897minus z119897) (13)

where z119897≜ [119911119897 119911119897+2

]119879 and s

119896≜ [119904119897 119904119897+2

]119879 for 119897 = 1 and 2 By

defining 1199050≜ (120572 + 120573)2 and 119905


be represented as

T ≜ [

1199050

119890j1205791199051

119890minusj120579

1199051

1199050

] (14)


119897≜ [119904119897 119904119897+2

]119879 by

(119904119897 119904119897+2

) = argmin119904119897 119904119897+2isinC

(s119897minus z119897)119867T (s

119897minus z119897) (15)






parts as 119904119896

= s119896+ js119896+1

and 119911119896

= z119896+ jz119896+1


1198974


the setR = 1198861 1198862 119886




120574 ≜

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+ 12059111205811+ 12059121205812 (16)

where 1205911

≜ 2120582 cos 120579 1205912

≜ 2120582 sin 120579 120582 ≜ 11990511199050 1205811

≜ 12057511205753+

12057521205754 1205812≜ 12057521205753minus 12057511205754 and 120575


on (12) 1199050




120576 =

4

sum

119897=1

1003816100381610038161003816120575119897

1003816100381610038161003816

2+

|120582|

1 minus |120582|

1003816100381610038161003816(1205751+ j1205752) + 120583 (120575

3+ j1205754)1003816100381610038161003816

2 (17)

where 120583 ≜ sign(1199051)119890


1198974






119897minus 119903z



1=

s1+ js2and 1199043






119897| lt 119903





120574th = (1 minus |120582|)1199032

] = 1(1 minus |120582|)FOR 119898

1= 1 2 |R

1|119888

1205751= R1(1198981) minus z1

1205761= 1205752

1

FOR 1198982= 1 2 |R

2|119888

1205752= R2(1198982) minus z2

1205762= 1205761+ 1205752

2

IF 1205762gt 1199032


3= 1 2 |R

3|119888

1205753= R3(1198983) minus z3

1205763= 1205762+ 1205752

3

IF 1205763gt 1199032


4= 1 2 |R

4|119888

1205754= R4(1198984) minus z4

1205764= 1205763+ 1205752

4

IF 1205764gt 1199032

CONTINUEEND1205811= 12057511205753+ 12057521205754

1205812= 12057521205753minus 12057511205754

120574 = 1205764+ 12059111205811+ 12059121205812


2(1198982)

1199043= R3(1198983) + jR

4(1198984)

ENDEND

ENDEND

END



119877+ 29 real additions 40119872

119877+ 31




= 1199041+ 1205901198991and

1199113= 1199043+ (984858120590)119890

minusj1205791198991+ 12059010158401198993 where 119904




984858 = 119872119879(120572 minus 120573)2120588120572120573 and 120590

10158402

= 2119872119879120588(120572 + 120573) Based on




119875 (120596) =

1

2120587

sdot int

120596120590

minus120596120590

int

120596120590

minus120596120590

[Q(minus

120596

1205901015840minus 1205781) minus Q(

120596

1205901015840minus 1205781)]

sdot [Q(minus

120596

1205901015840minus 1205782) minus Q(

120596

1205901015840minus 1205782)] 119890minus(12)(119899

2+11989910158402

)119889119899 119889119899

1015840

(18)

where 1205781= (984858120590120590

1015840)(119899 cos 120579+119899

1015840 sin 120579) 1205782= (984858120590120590

1015840)(1198991015840 cos 120579minus


infin

119906exp(minus120577

22)119889120577 When 120596 ≫


119875 (120596) gt [1 minus 2Q(

120596

1205901015840)]

2

[1 minus 2Q (4)]2 (19)



C13

le 119888M (120596)N (120596) (20)












119877+ 28

additions 432119872119877









8

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=119894

119903119894119895

(z119895minus s119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=100381610038161003816100381611990388

(z8minus s8)1003816100381610038161003816

2

+100381610038161003816100381611990377

(z7minus s7) + 11990378

(z8minus s8)1003816100381610038161003816

2+ sdot sdot sdot

+100381610038161003816100381611990311

(z1minus s1) + 11990312


+ 11990318

(z8minus s8)1003816100381610038161003816

2

(21)

where 119903119894119895 z119895 and s



s119894+1

s119894+2





minus4



5 Conclusion


0 05 1 1510

2

103

104

rinitial

Avg

num

ber o

f add

ition

s and

mul

tiplic

atio

ns

1024-QAM256-QAM

64-QAM16-APSK


5 10 15 20 25 30 35 40 4510

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

Bit e

rror

rate












[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120574th = (1 minus |120582|)1199032

] = 1(1 minus |120582|)FOR 119898

1= 1 2 |R

1|119888

1205751= R1(1198981) minus z1

1205761= 1205752

1

FOR 1198982= 1 2 |R

2|119888

1205752= R2(1198982) minus z2

1205762= 1205761+ 1205752

2

IF 1205762gt 1199032


3= 1 2 |R

3|119888

1205753= R3(1198983) minus z3

1205763= 1205762+ 1205752

3

IF 1205763gt 1199032


4= 1 2 |R

4|119888

1205754= R4(1198984) minus z4

1205764= 1205763+ 1205752

4

IF 1205764gt 1199032

CONTINUEEND1205811= 12057511205753+ 12057521205754

1205812= 12057521205753minus 12057511205754

120574 = 1205764+ 12059111205811+ 12059121205812


2(1198982)

1199043= R3(1198983) + jR

4(1198984)

ENDEND

ENDEND

END



119877+ 29 real additions 40119872

119877+ 31




= 1199041+ 1205901198991and

1199113= 1199043+ (984858120590)119890

minusj1205791198991+ 12059010158401198993 where 119904




984858 = 119872119879(120572 minus 120573)2120588120572120573 and 120590

10158402

= 2119872119879120588(120572 + 120573) Based on




119875 (120596) =

1

2120587

sdot int

120596120590

minus120596120590

int

120596120590

minus120596120590

[Q(minus

120596

1205901015840minus 1205781) minus Q(

120596

1205901015840minus 1205781)]

sdot [Q(minus

120596

1205901015840minus 1205782) minus Q(

120596

1205901015840minus 1205782)] 119890minus(12)(119899

2+11989910158402

)119889119899 119889119899

1015840

(18)

where 1205781= (984858120590120590

1015840)(119899 cos 120579+119899

1015840 sin 120579) 1205782= (984858120590120590

1015840)(1198991015840 cos 120579minus


infin

119906exp(minus120577

22)119889120577 When 120596 ≫


119875 (120596) gt [1 minus 2Q(

120596

1205901015840)]

2

[1 minus 2Q (4)]2 (19)



C13

le 119888M (120596)N (120596) (20)












119877+ 28

additions 432119872119877









8

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=119894

119903119894119895

(z119895minus s119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=100381610038161003816100381611990388

(z8minus s8)1003816100381610038161003816

2

+100381610038161003816100381611990377

(z7minus s7) + 11990378

(z8minus s8)1003816100381610038161003816

2+ sdot sdot sdot

+100381610038161003816100381611990311

(z1minus s1) + 11990312


+ 11990318

(z8minus s8)1003816100381610038161003816

2

(21)

where 119903119894119895 z119895 and s



s119894+1

s119894+2





minus4



5 Conclusion


0 05 1 1510

2

103

104

rinitial

Avg

num

ber o

f add

ition

s and

mul

tiplic

atio

ns

1024-QAM256-QAM

64-QAM16-APSK


5 10 15 20 25 30 35 40 4510

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

Bit e

rror

rate












[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











8

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

8

sum

119895=119894

119903119894119895

(z119895minus s119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=100381610038161003816100381611990388

(z8minus s8)1003816100381610038161003816

2

+100381610038161003816100381611990377

(z7minus s7) + 11990378

(z8minus s8)1003816100381610038161003816

2+ sdot sdot sdot

+100381610038161003816100381611990311

(z1minus s1) + 11990312


+ 11990318

(z8minus s8)1003816100381610038161003816

2

(21)

where 119903119894119895 z119895 and s



s119894+1

s119894+2





minus4



5 Conclusion


0 05 1 1510

2

103

104

rinitial

Avg

num

ber o

f add

ition

s and

mul

tiplic

atio

ns

1024-QAM256-QAM

64-QAM16-APSK


5 10 15 20 25 30 35 40 4510

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

Bit e

rror

rate












[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















[9] 373 469 1311 1179 5637 4302[10] 327 437 647 443 1479 1445[11] 232 308 808 1076 3112 4148[12] 225 332 225 348 225 364




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Fast Maximum-Likelihood Decoder for Quasi...

Documents

Transcript of Research Article Fast Maximum-Likelihood Decoder for Quasi...