research article comparison of semidefinite relaxation ...research article comparison of...

Research ArticleComparison of Semidefinite Relaxation Detectors forHigh-Order Modulation MIMO Systems

Z Y Shao1 S W Cheung2 and T I Yuk2

1 School of Mechanical amp Electric Engineering Guangzhou University Guangzhou Higher Education Mega CenterGuangzhou 510006 China

2Department of Electrical amp Electronic Engineering University of Hong Kong Pokfulam Road Hong Kong

Correspondence should be addressed to Z Y Shao zyshaoeeehkuhk

Received 30 June 2014 Revised 26 October 2014 Accepted 30 October 2014 Published 27 November 2014

Academic Editor Stefano Selleri

Copyright copy 2014 Z Y Shao et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multiple-input multiple-output (MIMO) system is considered to be one of the key technologies of LTE since it achievesrequirements of high throughput and spectral efficiency The semidefinite relaxation (SDR) detection for MIMO systems is anattractive alternative to the optimummaximum likelihood (ML) decoding because it is very computationally efficient We proposea new SDR detector for 256-QAM MIMO system and compare its performance with two other SDR detectors namely BC-SDRdetector and VA-SDR detector The tightness and complexity of these three SDR detectors are analyzed Both theoretical analysisand simulation results demonstrate that the proposed SDR can provide the best BLER performance among the three detectorswhile the BC-SDR detector and the VA-SDR detector provide identical BLER performance Moreover the BC-SDR has the lowestcomputational complexity and the VA-SDR has the highest computational complexity while the proposed SDR is in between

1 Introduction

Multiple-input multiple-output (MIMO) system has beenconsidered as a promising solution to provide high datarate and good quality of future wireless communicationsIn MIMO systems detection algorithm is one of the majorchallenges due to its limitations of either unsatisfactoryperformance or high complexity The maximum likelihood(ML) detection can provide the best block-error-rate (BLER)performance but its computational complexity is extremelyhigh since it searches the vectors in the entire latticespace of the transmitted signals Although equalization-baseddetectors such as zero-forcing (ZF) decoding have very lowcomplexity they suffer from unacceptable degradations inBLER performance Sphere decoding (SD) is able to providethe BLER performance of ML detection with less complexityby searching only a subset of the entire lattice space Never-theless it has been proven that its expected complexity is stillexponential [1]Thus it becomes impractical when the systemorder is high and the signal-to-noise ratio (SNR) is low

The decoding algorithms based on semidefinite relax-ation (SDR) approach have becomemore andmore attractive

simply because of the fact that semidefinite programming(SDP) problems can be efficiently solved in polynomial time[2ndash5] The SDR approach was firstly applied to detect binaryphase-shift keying (BPSK) and quadrature amplitude modu-lation (4-QAM) signals [6 7]Then the extensions to differentSDR techniques for 16-QAM signals had been proposedsuch as polynomial-inspired SDR (PI-SDR) [8] bound-constrained SDR (BC-SDR) [9] and virtually antipodal SDR(VA-SDR) [10] all exhibiting acceptable BLER performanceand relatively low complexity In [11] it has been proved thatthere exists an equivalence among PI-SDR BC-SDR andVA-SDR for 16-QAM The extension of BC-SDR and VA-SDRto 256-QAM has also been investigated and both of themprovide the same BLER performance However due to itshigh complexity the PI-SDR is not suitable for extension to256-QAM system

In this paper a new SDR detector is proposed for256-QAM system [12] Then a comprehensive comparisonbetween the proposed method and the previous SDR detec-tors is made The results show that the proposed SDR canprovide better BLER performance than both the BC-SDRand VA-SDR Moreover the complexity of the proposed

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 614876 8 pageshttpdxdoiorg1011552014614876

2 International Journal of Antennas and Propagation

method is higher than that of the BC-SDR but lower thanthat of the VA-SDR This paper is organized as follows InSection 2 the BC-SDR and VA-SDR are reviewed and theproposed SDR is introduced In Sections 3 and 4 the tightnessand the complexity of these methods are analyzed Thenthe simulation results are shown in Section 5 Finally weconclude the paper in Section 6

2 Semidefinite Relaxation Detectors for256-QAM MIMO System

21 System Model The system model for the MIMO trans-mission using 119872-QAM is considered in this paper TheMIMO system is modeled as

r = Hx + n (1)

where r is the119873119877-dimensional received signal vector and x is

the 119873119879-dimensional signal vector in the transmit lattice H

denotes the channel matrix with elements ℎ119894119895representing

the transfer function from the 119895th transmit antenna to 119894threceive antenna n is the119873

119877-dimensional additive noise The

signal vector x is assumed to be a statistically independentvariable with zero mean and variance 1205902 = 119873

02 where 119873

0

is noise power spectral density Perfect channel knowledgeis assumed to be known to the receiver In addition thechannel matrix is assumed to be a flat fading channel and allits entries are complex Gaussian and independent The noiseis an independently and identically distributed (iid) zero-mean Gaussian noise vector with elements having a fixedvariance

The complex transmission in (1) can be equivalentlyrepresented in real matrix form as

[Re (r)Im (r)] = [

Re (H) minus Im (H)Im (H) Re (H)

] [Re (x)Im (x)] + [

Re (n)Im (n)]

r = Hx + n(2)

with Re(∙) and Im(∙) being the real and imaginary partsof (∙) respectively The dimension of x and r is 119872

119877=

2119873119877and 119872

119879= 2119873

119879 respectively H becomes an 119872

119877times

119872119879matrix And the noise n is an 119872

119877-dimensional vector

ML decoding calculates the Euclidean distances betweenthe possible transmit signal vectors and the received signalvector and then chooses the one with shortest distanceas the solution For MIMO system using 256-QAM theML detection aims at finding the solution of the followingoptimization problem

min r minusHx2 (3a)

st x isin plusmn1 plusmn3 plusmn5 plusmn7 plusmn9 plusmn11 plusmn13 plusmn15119872119879 (3b)

where sdot represents the vector 2-norm It is well knownthat ML detection can provide the best BLER performanceHowever its computation requirement is too complicated tobe implemented especially for the cases with large number ofantennas or high level modulation

22 Review of BC-SDR Define a rank-1 semidefinite matrixΩ which is given by

Ω = [x119879 1]119879

[x119879 1] (4)

It is easy to find that the ML detection problem given in(3a) and (3b) can be rewritten as

min TrΩ [H119879H minusH119879r

minusr119879H r119879r ] (5a)

st Ω = Ω119879 isin R(119872119879+1)times(119872119879+1) (5b)

Ω11= Ω12Ω21isin R119872119879times119872119879 (5c)

Ω12isin plusmn1 plusmn3 plusmn5 plusmn7 plusmn9 plusmn11 plusmn13 plusmn15

119872119879 (5d)

Ω22= 1 (5e)

It can be observed that the high complexity of the MLdetection is due to the presence of the two nonconvexconstraints (5c) and (5d)Thus relaxation of these constraintswill be engaged to transform the original problem into asemidefinite problem which can then be efficiently solvedin polynomial time First constraint (5d) implies 1 le 1199092

119894le

225 where 119909119894denotes the 119894th component of x Then (5b)

along with (5c) can be relaxed into Y≻0 A new symbol Yis introduced here to distinguish Y from the aforementionedΩ since they are actually different matrixes after relaxationConsequently the BC-SDR problem is obtained as

min TrY [H119879H minusH119879r

minusr119879H r119879r ] (6a)

st Y isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (6b)

I≺Diag Y11 ≺ 255I I isin R119872119879times119872119879 (6c)

Y22= 1 (6d)

The BC-SDR problem (6a) (6b) (6c) and (6d) can thenbe solved by any of the SDP solvers such as Sedumi [3] basedon interior point methods Although (6b) could be deducedfrom (5b) and (5c) and (6c) can also be deduced from (5d)problem (6a) (6b) (6c) and (6d) is however not exactlyequivalent to problem (5a) (5b) (5c) (5d) and (5e) Thusthe solution obtained by solving (6a) (6b) (6c) and (6d) hasmore errors than ML solution in (5a) (5b) (5c) (5d) and(5e) Once the solution is found the randomization approachis applied to quantize the resulting Y

12till constraint (3b) is

satisfied

23 Review of VA-SDR It is worth noting that when con-straint (3b) is expected to be satisfied the signal x could beexpressed as

x = Up119879 (7)

where U = [I 2I 4I 8I] p = [p1 p2p3p4] I isin R119872119879times119872119879

and p119894isin plusmn1

119872119879 119894 = 1 2 3 4

International Journal of Antennas and Propagation 3

Table 1 The value of x

q2

q1

minus3 minus1 1 3

minus3 minus15 minus7 1 9


1 minus11 minus3 5 13


Substituting (7) into (4) and defining a matrix Z which isgiven by

Z = [p 1]119879 [p 1] (8)

the objective function (5a) can be equivalently transformedinto (9a) and constraints (5d) and (5e) are equal to (9b)Similarly (5b) along with (5c) can be relaxed into (9c) Thuswe obtain the VA-SDR problem given by

min TrZ [U119879H119879HU minusU119879H119879rminusr119879HU r119879r ] (9a)

st Z isin R(4119872119879+1)times(4119872119879+1)≻ 0 (9b)

Diag (Z) = 1I (9c)

Since the first 4119872119879elements of the last row in the solution

Z can be considered as p = [p1 p2p3p4] thus the

optimum solution x is reconstructed by using (7) Finally thesolution is quantized by using randomization

24 Proposed SDR Considering constraint (3b) the signal xcould also be expressed as

x = Vq119879 (10)

where V = [I 4I] q = [q1 q2] I isin R119872119879times119872119879 and q

1 q2isin

plusmn1 plusmn3119872119879

Table 1 gives the value of 119909119895for the possible combinations

of 1199021119895and 119902

2119895 where 119909

119895 1199021119895 and 119902

2119895denote the 119895th element

of x q1 and q

2 respectively

By substituting (10) into (4) and defining a matrix Wwhich is given by

W = [q 1]119879 [q 1] (11)

the objective function (5a) can be equivalently transformedinto (12a) Moreover it can be known from (10) that con-straint (5d) is equivalent to q

1 q2isin plusmn1 plusmn3

119872119879 which areessentially thewell-known indices used to characterize the 16-QAM constellation Herein the set operation method [13] isengaged to formulate the alphabet constraint (5d) into (12b)and (12c) Similarly (5b) along with (5c) can be relaxed into

(12d) Also (5e) can be reformulated as (12e) Thus we obtainthe proposed SDR problem given by

min TrW [V119879H119879HV minusV119879H119879rminusr119879HV r119879r ] (12a)

st 1I≺Diag W11 ≺ 9I I isin R2119872119879times2119872119879 (12b)

Diag W11 plusmn 4Diag W

12 + 3I≻ 0 I isin R2119872119879times2119872119879

(12c)

W isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (12d)

W22= 1 (12e)

Similar to the VA-SDR method the first 2119872119879elements

of the last row in the solution W can be considered asq = [q1 q

2] thus the optimum solution x is reconstructedby using (10) Finally the solution is quantized by usingrandomization

3 Comparison of Tightness

As mentioned in Section 2 these three SDR problems areall relaxed from the original problem (3a) and (3b) andtheir objective functions are equivalent which are actuallycalculating the Euclidean distance given by (3a) Thus thetightness of the constraints of each SDR algorithm implieshow close it is to the ML decoding In what follows we willcompare their tightness

31 Equivalence of BC-SDR and VA-SDR Firstly we willdemonstrate that the constraints of the BC-SDR problem areequivalent to those of the VA-SDR problemThis equivalencehas been considered in [11] and for completeness of under-standing we will reorganize the proof in the following twosteps

Step 1 For each matrix Z isin R(4119872119879+1)times(4119872119879+1) that satisfiesconstraints (9b) and (9c) there must be a matrix Y isin

R(119872119879+1)times(119872119879+1) which satisfies constraints (6b)ndash(6d)

Proof For any matrix Z which satisfies constraints (9b) and(9c) it has the following form

Z = [Z11 isin R4119872119879times4119872119879 Z

12isin R4119872119879times1

Z21isin R1times4119872119879 1

] (13)

Since Z≻ 0 there should be a reversible matrix

Λ = [12058211205822 120582

119872119905120582119872119905+1

120582119872119905+2

12058221198721199051205822119872119905+1

1205822119872119905+2

12058231198721199051205823119872119905+1

1205823119872119905+2

12058241198721199051205824119872119905+1

]

(14)


that satisfies

Z = Λ119879Λ (15)

where 120582119898isin R(4119872119879+1)times1119898 = 1 2 4119872

119879+ 1

Now we construct a semidefinite matrix Y isin

R(119872119879+1)times(119872119879+1) which is defined as

Y = GZG119879 = (ΛG119879)119879

(ΛG119879) (16)

where

G = [U 00 1

] = [I isin R119872119879times119872119879 2I isin R119872119879times119872119879 4I isin R119872119879times119872119879 8I isin R119872119879times119872119879 0 isin R119872119879times10 isin R1times119872119879 0 isin R1times119872119879 0 isin R1times119872119879 0 isin R1times119872119879 1

] (17)

From (13)ndash(15) and (9c) we get

10038171003817100381710038171205821198991003817100381710038171003817 = (120582

119879

119899120582119899)12

= 1 119899 = 1 2 4119872119879 (18)

120582119879

4119872119879+11205824119872119879+1

= 1 (19)

Substituting (14) into (16) gives

Y = [Y11 Y12

Y21

Y22

]

= [(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4) (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)1198791205824119872119879+1

120582119879

4119872119879+1(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879

120582119879

4119872119879+11205824119872119879+1

]

(20)

where Λ1= [12058211205822 120582

119873119905] isin R(4119872119879+1)times119872119879 Λ

2= [120582119872119879+1

120582119872119879+2

1205822119872119879] isin R(4119872119879+1)times119872119879 Λ

3= [1205822119873119905+1

1205822119873119905+2

1205823119873119905] isin R(4119872119879+1)times119872119879 and Λ

4= [1205823119873119905+1

1205823119873119905+2

1205824119873119905] isin

R(4119872119879+1)times119872119879 From (19) and (20) it can be known thatY

22= 1 and (6d)

is satisfied Moreover we have

Y11= (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)

(21)

The element located in the 119895th row and the 119895th column ofY11

is

119910119895119895=10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

2

(22)

From the perspective of geometry and (18) it is easy to knowthat

1 =1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120582119895

10038171003817100381710038171003817minus100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817minus1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895

10038171003817100381710038171003817+100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817= 15

(23)

Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(24)

The proof is complete

Step 2 For each matrix Y isin R(119872119879+1)times(119872119879+1) that satis-fies constraints (6b)ndash(6d) there must be a matrix Z isin

R(4119872119879+1)times(4119872119879+1) which satisfies constraints (9b) and (9c)

Proof For anymatrixY that satisfies constraints (6b)ndash(6d) ithas the following form

Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (25)

Since Y≻0 there must be a reversible matrix Γ = [1205741 1205742

1205743 120574

119872119879 120574119872119879+1

] that satisfies

Y = Γ119879Γ (26)

where 120574119897isin R(119872119879+1)times1 119897 = 1 2 119872

119879+ 1

From (25) and (26) and (6c) it can be obtained that10038171003817100381710038171003817120574119872119879+1

10038171003817100381710038171003817= (120574119879

119872119879+1120574119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120574119895

10038171003817100381710038171003817= (120574119879

119895120574119895)12

le 15 119895 = 1 2 119872119879

(27)

From the perspective of geometry it is easy to know thatthere should be vectors 120574

1198951 1205741198952 1205741198953 1205741198954isin R(119872119879+1)times1 which

satisfy

120574119895= 1205741198951+ 21205741198952+ 41205741198953+ 81205741198954 (28)

100381710038171003817100381710038171205741198951

10038171003817100381710038171003817=100381710038171003817100381710038171205741198952

10038171003817100381710038171003817=100381710038171003817100381710038171205741198953

10038171003817100381710038171003817=100381710038171003817100381710038171205741198954

10038171003817100381710038171003817= 1 (29)

Now we construct another matrix on the basis of (26) and(28) which is given by

Γ1015840= [12057411 12057412 12057413 12057414 12057421 12057422 12057423 12057424 120574

1198721199051

1205741198721199052 1205741198721199053 1205741198721199054 120574119872119905+1

]

isin R(119872119879+1)times(4119872119879+1)

(30)


Next we construct a semidefinite matrix Z isin


Z = (Γ1015840)119879

Γ1015840 (31)

Substituting (30) into (31) the element located in the (119895 times 119894)throw and the (119895 times 119894)th column of Z is given as

119911(119895119894)(119895119894)

= (120574119879

119895119894120574119895119894) =10038171003817100381710038171003817120574119895119894

10038171003817100381710038171003817

2

= 1 (32)

where 119895 = 1 2 119872119879 119894 = 1 2 3 4 Moreover the (4119872

119879+

1)th row and the (4119872119879+ 1)th column of Z are

119911(4119872119879+1)(4119872119879+1)

= (120574119879

119872119879+1120574119872119879+1

) = 1 (33)

From (32) and (33) it can be obtained that

Diag (Z) = 1I (34)


From both Steps 1 and 2 the equivalence of BC-SDR andVA-SDR can be concluded

32 Proposed SDR Tighter Than BC-SDR and VA-SDR Sec-ondly we will demonstrate that the constraints of proposedSDR problem are tighter than those of the BC-SDR problemand also tighter than those of the VA-SDR due to theaforementioned equivalence For this purpose a new SDRproblem is constructed given by

min TrW1015840 [V119879H119879HV minusV119879H119879rminusr119879HV r119879r ] (35a)


W1015840 isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (35c)

W101584022= 1 (35d)

By comparing (35a) (35b) (35c) and (35d) and (12a)(12b) (12c) (12d) and (12e) it is apparent that the proposedSDR is tighter than the new SDR since the proposed SDR isnearly the same as the new SDR except that the proposed SDRhas an extra constraint (12c) In what follows we will provethat the new SDR given by (35a) (35b) (35c) and (35d) isequivalent to BC-SDR given by (6a) (6b) (6c) and (6d) andtherefore the proposed SDR being tighter than BC-SDR canbe proven

Step 1 For each matrix W1015840 isin R(2119872119879+1)times(2119872119879+1) that satis-fies constraints (35b)ndash(35d) there must be a matrix Y isin


Proof For any matrix W1015840 that satisfies constraints (35b)ndash(35d) it has the following form

W1015840 = [W1015840

11isin R2119872119879times2119872119879 W1015840

12isin R2119872119879times1

W101584021isin R1times2119872119879 1

] (36)

SinceW1015840≻0 there should be a reversible matrix

Ψ

= [12059511205952 120595

119872119879120595119872119879+1

120595119872119879+2

12059521198721198791205952119872119879+1

]

(37)

that satisfies

W1015840 = Ψ119879Ψ (38)


119879+ 1

From (36)ndash(38) (35b) and (35d) give

1 le10038171003817100381710038171205951198961003817100381710038171003817 = (120595

119879

119896120595119896)12

le 3 119896 = 1 2 2119872119879 (39)

120595119879

2119872119879+11205952119872119879+1

= 1 (40)

Now we construct a semidefinite matrix Y isin R(119872119879+1)times(119872119879+1)

which is defined as

Y = EW1015840E119879 = (ΨE119879)119879

(ΨE119879) (41)

where

E = [V 00 1

]

= [I isin R119872119879times119872119879 4I isin R119872119879times119872119879 0 isin R119872119879times10 isin R1times119872119879 0 isin R1times119872119879 1

]

(42)


Y = [Y11 Y12

Y21

Y22

]

= [(Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (Ψ1+ 4Ψ2)1198791205952119872119879+1

120595119879

2119872119879+1(Ψ1+ 4Ψ2)119879120595119879

2119872119879+11205952119872119879+1

]

(43)

where

Ψ1= [12059511205952 120595

119873119905] isin R

(2119872119879+1)times119872119879

Ψ2= [120595119872119879+1

120595119872119879+2

1205952119872119879] isin R

(2119872119879+1)times119872119879

(44)

From (40) and (43) it can be known that Y22= 1 and

(6d) is satisfied Moreover we have

Y11= (Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (45)

and the element located in the119895th row and the 119895th column ofY11

is

119910119895119895=10038171003817100381710038171003817120595119895+ 4120595119873119905+119895

10038171003817100381710038171003817

2

119895 = 1 2 119872119879 (46)


1 le100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120595119895

10038171003817100381710038171003817le10038171003817100381710038171003817120595119895+ 4120595119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120595119895

10038171003817100381710038171003817+100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817le 15

(47)


Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)


Step 2 For each matrix Y isin R(119872119879+1)times(119872119879+1) that satis-fies constraints (6b)ndash(6d) there must be a matrix W1015840 isinR(2119872119879+1)times(2119872119879+1) which satisfies constraints (35b)ndash(35d)


Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)

Since Y≻0 there must be a reversible matrix Δ =

[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1

From (49) (50) and (6c) it can be obtained that10038171003817100381710038171003817120590119872119879+1

10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)

From the perspective of geometry it is easy to know that thereshould be vectors 120590

11989511205901198952isin R(119872119879+1)times1 which satisfy

120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)

Next we construct a semidefinite matrix W1015840 isin


W1015840 = (Δ1015840)119879

Δ1015840 (55)

Substituting (54) into (55) the element located in the (119895times 119894)throw and the (119895 times 119894)th column ofW1015840 is given as

1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2

Then from (53) and (56) it can be obtained that

1I≺Diag W101584011 ≺ 9I I isin R2119872119879times2119872119879 (57)

Moreover the (2119872119879+ 1)th row and the (2119872

119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)


From both Steps 1 and 2 the equivalence of BC-SDRand new SDR given by (35a) (35b) (35c) and (35d) can beconcluded Therefore the constraints of the proposed SDRare tighter than those of the BC-SDR and also tighter thanthose of the VA-SDR due to the aforementioned equivalence

4 Comparison of Complexity

Firstly the BC-SDR given in (6a)ndash(6d) consists of a (119872119879+1)times

(119872119879+ 1)matrix variable Y and119874(119872

119879+ 1) linear constraints

Since constraints (6c) and (6d) are separable the complexityof the BC-SDR detector is 119874((119872

119879+ 1)35) [9] Secondly the

VA-SDR given in (9a)ndash(9c) involves a (4119872119879+ 1) times (4119872

119879+ 1)

matrix variable Z and 119874(4119872119879+ 1) The constraint (9c) is

also separable thus the complexity of the VA-SDR detectoris119874((4119872

119879+1)35) Finally the proposed SDR shown in (12a)ndash

(12e) consists of a (2119872119879+ 1) times (2119872

119879+ 1) matrix variable

W and two119874(4119872119879) linear constraints According to [13 14]

it can be known that the complexity of the proposed SDRdetector is 119874(4(2119872

119879+ 1)45+ 8(2119872

119879+ 1)35)

5 Simulation Results

Computer simulations were conducted to evaluate the per-formance of these three SDR detectors An uncoded MIMOsystem with independent Rayleigh fading channel was takeninto account and the Sedumi toolbox within Matlab softwarewas used to implement the SDRdetection algorithms Figures1 and 2 show the BLER performances of the three SDRdetectors for 4 times 4 and 8 times 8 256-QAM systems respectivelyIt can be observed that the BC-SDR detector and the VA-SDR detector provide exactly the same BLER performancewhile the proposed SDR detector can provide the best BLERperformance among these three detectors This agrees wellwith the analysis presented in Section 3 What is more itcan be found that the improvement concerning the BLERperformance provided by the proposed SDR detector in thecase of 8 times 8 MIMO system is larger than that in the caseof 4 times 4 MIMO system The relaxation of the alphabet setengaged in theBC-SDRand theVA-SDRwill cause increasingerrors with the increase of the number of the antennas Thatis because the inaccurate detection of any dimension of thetransmitted signal will result in the failure of the detection ofthe whole signal vector and the higher the dimension of theproblem is the bigger the possibility of failure will be Themain merit of the proposed SDR over the BC-SDR and theVA-SDR is that it can offer more accurate relaxation of thealphabet set Moreover the computational times for solvingone signal vector of these SDR detectors are illustrated inFigures 3 and 4 These figures demonstrate that the BC-SDRhas the lowest computational complexity and the VA-SDRhas the highest computational complexity while the proposedSDR is in between the two detection methods It is alsofound that the complexity of SDR detectors is independentof SNR This is a distinct advantage over the SD whosecomplexity varies as a function of SNR making SD difficultto be implemented in practice


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR

Proposed SDRML decoding

EbN0 (dB)

Figure 1The BLER performance of the SDR detectors for 4times4 256-QAMMIMO systems

0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)

Figure 2 The BLER performance of the SDR detectors for 8 times 8256-QAMMIMO systems

6 Conclusion

In this paper we proposed a SDR detector for 256-QAMMIMO system and also reviewed two other SDR detectorsnamely BC-SDR detector and VA-SDR detector Then weanalyzed the tightness and the complexity of these three SDRdetectors Both theoretical analysis and simulation resultsdemonstrate that the proposed SDR can provide the bestBLER performance among these three detectors while theBC-SDR detector and the VA-SDR detector provide exactlythe same BLER performance Moreover the BC-SDR has thelowest computational complexity and the VA-SDR has the

0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1

BC-SDRVA-SDRProposed SDR

EbN0 (dB)

Figure 3 The computational time of the SDR detectors for 4 times 4256-QAMMIMO systems

0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)


highest computational complexity while the proposed SDRis in between

Conflict of Interests

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


References

[1] J Jalden and B Ottersten ldquoAn exponential lower bound on theexpected complexity of sphere decodingrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo04) vol 4 pp iv-393ndashiv-396 MontrealCanada May 2004

[2] A Ben-Tal and A Nemirovski Lectures on Modern ConvexOptimization MPS-SIAM Series on Optimization 2001

[3] J F Sturm ldquoUsing SEDUMI 102 A Matlab toolbox for opti-mization over symmetric conesrdquo Optimization Methods andSoftware vol 11-12 pp 1ndash30 1999

[4] B Steingrimsson Z Q Luo and K M Wong ldquoSoft quasi-maximum-likelihood detection for multiple-antenna wirelesschannelsrdquo IEEE Transactions on Signal Processing vol 51 no11 pp 2710ndash2719 2003

[5] Z Y Shao SWCheung andT I Yuk ldquoLattice-reduction-aidedsemidefinite relaxation detection algorithms for multiple-inputmultiple-output systemsrdquo IETCommunications vol 8 no 4 pp448ndash454 2014

[6] W K Ma T N Davidson K M Wong Z Q Luo andP C Ching ldquoQuasi-maximum-likelihood multiuser detectionusing semi-definite relaxation with application to synchronousCDMArdquo IEEE Transactions on Signal Processing vol 50 no 4pp 912ndash922 2002

[7] A Wiesel Y C Eldar and S Shamai ldquoSemidefinite relaxationfor detection of 16-QAM signaling in MIMO channelsrdquo IEEESignal Processing Letters vol 12 no 9 pp 653ndash656 2005

[8] P H Tan and L K Rasmussen ldquoThe application of semidefiniteprogramming for detection inCDMArdquo IEEE Journal on SelectedAreas in Communications vol 19 no 8 pp 1442ndash1449 2001

[9] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[10] Z Mao X Wang and X Wang ldquoSemidefinite programmingrelaxation approach for multiuser detection of QAM signalsrdquoIEEE Transactions on Wireless Communications vol 6 no 12pp 4275ndash4279 2007

[11] W-K Ma C-C Su J Jalden T-H Chang and C-Y ChildquoThe equivalence of semidefinite relaxation MIMO detectorsfor higher-order QAMrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1038ndash1052 2009

[12] Z Y Shao SWCheung andT I Yuk ldquoSemi-definite relaxationdecoder for 256-QAM MIMO systemrdquo Electronics Letters vol46 no 11 pp 796ndash797 2010

[13] Y Yang C Zhao P Zhou and W Xu ldquoMIMO detection of 16-QAM signaling based on semidefinite relaxationrdquo IEEE SignalProcessing Letters vol 14 no 11 pp 797ndash800 2007

[14] C Helmberg F Rendl R J Vanderbei and H Wolkowicz ldquoAninterior-point method for semidefinite programmingrdquo SIAMJournal on Optimization vol 6 no 2 pp 342ndash361 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



method is higher than that of the BC-SDR but lower thanthat of the VA-SDR This paper is organized as follows InSection 2 the BC-SDR and VA-SDR are reviewed and theproposed SDR is introduced In Sections 3 and 4 the tightnessand the complexity of these methods are analyzed Thenthe simulation results are shown in Section 5 Finally weconclude the paper in Section 6

2 Semidefinite Relaxation Detectors for256-QAM MIMO System

21 System Model The system model for the MIMO trans-mission using 119872-QAM is considered in this paper TheMIMO system is modeled as

r = Hx + n (1)

where r is the119873119877-dimensional received signal vector and x is

the 119873119879-dimensional signal vector in the transmit lattice H

denotes the channel matrix with elements ℎ119894119895representing

the transfer function from the 119895th transmit antenna to 119894threceive antenna n is the119873

119877-dimensional additive noise The

signal vector x is assumed to be a statistically independentvariable with zero mean and variance 1205902 = 119873

02 where 119873

0

is noise power spectral density Perfect channel knowledgeis assumed to be known to the receiver In addition thechannel matrix is assumed to be a flat fading channel and allits entries are complex Gaussian and independent The noiseis an independently and identically distributed (iid) zero-mean Gaussian noise vector with elements having a fixedvariance

The complex transmission in (1) can be equivalentlyrepresented in real matrix form as

[Re (r)Im (r)] = [

Re (H) minus Im (H)Im (H) Re (H)

] [Re (x)Im (x)] + [

Re (n)Im (n)]

r = Hx + n(2)

with Re(∙) and Im(∙) being the real and imaginary partsof (∙) respectively The dimension of x and r is 119872

119877=

2119873119877and 119872

119879= 2119873

119879 respectively H becomes an 119872

119877times

119872119879matrix And the noise n is an 119872

119877-dimensional vector

ML decoding calculates the Euclidean distances betweenthe possible transmit signal vectors and the received signalvector and then chooses the one with shortest distanceas the solution For MIMO system using 256-QAM theML detection aims at finding the solution of the followingoptimization problem

min r minusHx2 (3a)

st x isin plusmn1 plusmn3 plusmn5 plusmn7 plusmn9 plusmn11 plusmn13 plusmn15119872119879 (3b)

where sdot represents the vector 2-norm It is well knownthat ML detection can provide the best BLER performanceHowever its computation requirement is too complicated tobe implemented especially for the cases with large number ofantennas or high level modulation

22 Review of BC-SDR Define a rank-1 semidefinite matrixΩ which is given by

Ω = [x119879 1]119879

[x119879 1] (4)

It is easy to find that the ML detection problem given in(3a) and (3b) can be rewritten as

min TrΩ [H119879H minusH119879r

minusr119879H r119879r ] (5a)

st Ω = Ω119879 isin R(119872119879+1)times(119872119879+1) (5b)

Ω11= Ω12Ω21isin R119872119879times119872119879 (5c)

Ω12isin plusmn1 plusmn3 plusmn5 plusmn7 plusmn9 plusmn11 plusmn13 plusmn15

119872119879 (5d)

Ω22= 1 (5e)

It can be observed that the high complexity of the MLdetection is due to the presence of the two nonconvexconstraints (5c) and (5d)Thus relaxation of these constraintswill be engaged to transform the original problem into asemidefinite problem which can then be efficiently solvedin polynomial time First constraint (5d) implies 1 le 1199092

119894le

225 where 119909119894denotes the 119894th component of x Then (5b)

along with (5c) can be relaxed into Y≻0 A new symbol Yis introduced here to distinguish Y from the aforementionedΩ since they are actually different matrixes after relaxationConsequently the BC-SDR problem is obtained as

min TrY [H119879H minusH119879r

minusr119879H r119879r ] (6a)

st Y isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (6b)

I≺Diag Y11 ≺ 255I I isin R119872119879times119872119879 (6c)

Y22= 1 (6d)

The BC-SDR problem (6a) (6b) (6c) and (6d) can thenbe solved by any of the SDP solvers such as Sedumi [3] basedon interior point methods Although (6b) could be deducedfrom (5b) and (5c) and (6c) can also be deduced from (5d)problem (6a) (6b) (6c) and (6d) is however not exactlyequivalent to problem (5a) (5b) (5c) (5d) and (5e) Thusthe solution obtained by solving (6a) (6b) (6c) and (6d) hasmore errors than ML solution in (5a) (5b) (5c) (5d) and(5e) Once the solution is found the randomization approachis applied to quantize the resulting Y

12till constraint (3b) is

satisfied

23 Review of VA-SDR It is worth noting that when con-straint (3b) is expected to be satisfied the signal x could beexpressed as

x = Up119879 (7)

where U = [I 2I 4I 8I] p = [p1 p2p3p4] I isin R119872119879times119872119879

and p119894isin plusmn1

119872119879 119894 = 1 2 3 4



q2

q1

minus3 minus1 1 3






Z = [p 1]119879 [p 1] (8)



st Z isin R(4119872119879+1)times(4119872119879+1)≻ 0 (9b)

Diag (Z) = 1I (9c)





x = Vq119879 (10)


1 q2isin



of 1199021119895and 119902

2119895 where 119909

119895 1199021119895 and 119902


of x q1 and q

2 respectively


W = [q 1]119879 [q 1] (11)








12 + 3I≻ 0 I isin R2119872119879times2119872119879

(12c)

W isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (12d)

W22= 1 (12e)










Z = [Z11 isin R4119872119879times4119872119879 Z

12isin R4119872119879times1

Z21isin R1times4119872119879 1

] (13)


Λ = [12058211205822 120582

119872119905120582119872119905+1

120582119872119905+2

12058221198721199051205822119872119905+1

1205822119872119905+2

12058231198721199051205823119872119905+1

1205823119872119905+2

12058241198721199051205824119872119905+1

]

(14)


that satisfies

Z = Λ119879Λ (15)


119879+ 1



Y = GZG119879 = (ΛG119879)119879

(ΛG119879) (16)

where

G = [U 00 1


] (17)


10038171003817100381710038171205821198991003817100381710038171003817 = (120582

119879

119899120582119899)12

= 1 119899 = 1 2 4119872119879 (18)

120582119879

4119872119879+11205824119872119879+1

= 1 (19)


Y = [Y11 Y12

Y21

Y22

]

= [(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4) (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)1198791205824119872119879+1

120582119879

4119872119879+1(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879

120582119879

4119872119879+11205824119872119879+1

]

(20)

where Λ1= [12058211205822 120582

119873119905] isin R(4119872119879+1)times119872119879 Λ

2= [120582119872119879+1

120582119872119879+2

1205822119872119879] isin R(4119872119879+1)times119872119879 Λ

3= [1205822119873119905+1

1205822119873119905+2

1205823119873119905] isin R(4119872119879+1)times119872119879 and Λ

4= [1205823119873119905+1

1205823119873119905+2

1205824119873119905] isin


22= 1 and (6d)


Y11= (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)

(21)


is

119910119895119895=10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

2

(22)


1 =1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120582119895

10038171003817100381710038171003817minus100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817minus1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895

10038171003817100381710038171003817+100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817= 15

(23)

Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(24)





Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (25)


1205743 120574

119872119879 120574119872119879+1

] that satisfies

Y = Γ119879Γ (26)


119879+ 1


10038171003817100381710038171003817= (120574119879

119872119879+1120574119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120574119895

10038171003817100381710038171003817= (120574119879

119895120574119895)12

le 15 119895 = 1 2 119872119879

(27)



satisfy

120574119895= 1205741198951+ 21205741198952+ 41205741198953+ 81205741198954 (28)

100381710038171003817100381710038171205741198951

10038171003817100381710038171003817=100381710038171003817100381710038171205741198952

10038171003817100381710038171003817=100381710038171003817100381710038171205741198953

10038171003817100381710038171003817=100381710038171003817100381710038171205741198954

10038171003817100381710038171003817= 1 (29)


Γ1015840= [12057411 12057412 12057413 12057414 12057421 12057422 12057423 12057424 120574

1198721199051

1205741198721199052 1205741198721199053 1205741198721199054 120574119872119905+1

]

isin R(119872119879+1)times(4119872119879+1)

(30)




Z = (Γ1015840)119879

Γ1015840 (31)


119911(119895119894)(119895119894)

= (120574119879

119895119894120574119895119894) =10038171003817100381710038171003817120574119895119894

10038171003817100381710038171003817

2

= 1 (32)


119879+


119911(4119872119879+1)(4119872119879+1)

= (120574119879

119872119879+1120574119872119879+1

) = 1 (33)


Diag (Z) = 1I (34)






W1015840 isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (35c)

W101584022= 1 (35d)





W1015840 = [W1015840

11isin R2119872119879times2119872119879 W1015840

12isin R2119872119879times1

W101584021isin R1times2119872119879 1

] (36)


Ψ

= [12059511205952 120595

119872119879120595119872119879+1

120595119872119879+2

12059521198721198791205952119872119879+1

]

(37)

that satisfies

W1015840 = Ψ119879Ψ (38)


119879+ 1


1 le10038171003817100381710038171205951198961003817100381710038171003817 = (120595

119879

119896120595119896)12

le 3 119896 = 1 2 2119872119879 (39)

120595119879

2119872119879+11205952119872119879+1

= 1 (40)


which is defined as

Y = EW1015840E119879 = (ΨE119879)119879

(ΨE119879) (41)

where

E = [V 00 1

]


]

(42)


Y = [Y11 Y12

Y21

Y22

]

= [(Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (Ψ1+ 4Ψ2)1198791205952119872119879+1

120595119879

2119872119879+1(Ψ1+ 4Ψ2)119879120595119879

2119872119879+11205952119872119879+1

]

(43)

where

Ψ1= [12059511205952 120595

119873119905] isin R

(2119872119879+1)times119872119879

Ψ2= [120595119872119879+1

120595119872119879+2

1205952119872119879] isin R

(2119872119879+1)times119872119879

(44)



Y11= (Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (45)


is

119910119895119895=10038171003817100381710038171003817120595119895+ 4120595119873119905+119895

10038171003817100381710038171003817

2

119895 = 1 2 119872119879 (46)


1 le100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120595119895

10038171003817100381710038171003817le10038171003817100381710038171003817120595119895+ 4120595119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120595119895

10038171003817100381710038171003817+100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817le 15

(47)


Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)




Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)


[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1


10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)



120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)



W1015840 = (Δ1015840)119879

Δ1015840 (55)


1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2




119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)








119879+ 1)35) [9] Secondly the


119879+ 1)








119879+ 1)45+ 8(2119872

119879+ 1)35)




0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











q2

q1

minus3 minus1 1 3






Z = [p 1]119879 [p 1] (8)



st Z isin R(4119872119879+1)times(4119872119879+1)≻ 0 (9b)

Diag (Z) = 1I (9c)





x = Vq119879 (10)


1 q2isin



of 1199021119895and 119902

2119895 where 119909

119895 1199021119895 and 119902


of x q1 and q

2 respectively


W = [q 1]119879 [q 1] (11)








12 + 3I≻ 0 I isin R2119872119879times2119872119879

(12c)

W isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (12d)

W22= 1 (12e)










Z = [Z11 isin R4119872119879times4119872119879 Z

12isin R4119872119879times1

Z21isin R1times4119872119879 1

] (13)


Λ = [12058211205822 120582

119872119905120582119872119905+1

120582119872119905+2

12058221198721199051205822119872119905+1

1205822119872119905+2

12058231198721199051205823119872119905+1

1205823119872119905+2

12058241198721199051205824119872119905+1

]

(14)


that satisfies

Z = Λ119879Λ (15)


119879+ 1



Y = GZG119879 = (ΛG119879)119879

(ΛG119879) (16)

where

G = [U 00 1


] (17)


10038171003817100381710038171205821198991003817100381710038171003817 = (120582

119879

119899120582119899)12

= 1 119899 = 1 2 4119872119879 (18)

120582119879

4119872119879+11205824119872119879+1

= 1 (19)


Y = [Y11 Y12

Y21

Y22

]

= [(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4) (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)1198791205824119872119879+1

120582119879

4119872119879+1(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879

120582119879

4119872119879+11205824119872119879+1

]

(20)

where Λ1= [12058211205822 120582

119873119905] isin R(4119872119879+1)times119872119879 Λ

2= [120582119872119879+1

120582119872119879+2

1205822119872119879] isin R(4119872119879+1)times119872119879 Λ

3= [1205822119873119905+1

1205822119873119905+2

1205823119873119905] isin R(4119872119879+1)times119872119879 and Λ

4= [1205823119873119905+1

1205823119873119905+2

1205824119873119905] isin


22= 1 and (6d)


Y11= (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)

(21)


is

119910119895119895=10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

2

(22)


1 =1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120582119895

10038171003817100381710038171003817minus100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817minus1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895

10038171003817100381710038171003817+100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817= 15

(23)

Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(24)





Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (25)


1205743 120574

119872119879 120574119872119879+1

] that satisfies

Y = Γ119879Γ (26)


119879+ 1


10038171003817100381710038171003817= (120574119879

119872119879+1120574119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120574119895

10038171003817100381710038171003817= (120574119879

119895120574119895)12

le 15 119895 = 1 2 119872119879

(27)



satisfy

120574119895= 1205741198951+ 21205741198952+ 41205741198953+ 81205741198954 (28)

100381710038171003817100381710038171205741198951

10038171003817100381710038171003817=100381710038171003817100381710038171205741198952

10038171003817100381710038171003817=100381710038171003817100381710038171205741198953

10038171003817100381710038171003817=100381710038171003817100381710038171205741198954

10038171003817100381710038171003817= 1 (29)


Γ1015840= [12057411 12057412 12057413 12057414 12057421 12057422 12057423 12057424 120574

1198721199051

1205741198721199052 1205741198721199053 1205741198721199054 120574119872119905+1

]

isin R(119872119879+1)times(4119872119879+1)

(30)




Z = (Γ1015840)119879

Γ1015840 (31)


119911(119895119894)(119895119894)

= (120574119879

119895119894120574119895119894) =10038171003817100381710038171003817120574119895119894

10038171003817100381710038171003817

2

= 1 (32)


119879+


119911(4119872119879+1)(4119872119879+1)

= (120574119879

119872119879+1120574119872119879+1

) = 1 (33)


Diag (Z) = 1I (34)






W1015840 isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (35c)

W101584022= 1 (35d)





W1015840 = [W1015840

11isin R2119872119879times2119872119879 W1015840

12isin R2119872119879times1

W101584021isin R1times2119872119879 1

] (36)


Ψ

= [12059511205952 120595

119872119879120595119872119879+1

120595119872119879+2

12059521198721198791205952119872119879+1

]

(37)

that satisfies

W1015840 = Ψ119879Ψ (38)


119879+ 1


1 le10038171003817100381710038171205951198961003817100381710038171003817 = (120595

119879

119896120595119896)12

le 3 119896 = 1 2 2119872119879 (39)

120595119879

2119872119879+11205952119872119879+1

= 1 (40)


which is defined as

Y = EW1015840E119879 = (ΨE119879)119879

(ΨE119879) (41)

where

E = [V 00 1

]


]

(42)


Y = [Y11 Y12

Y21

Y22

]

= [(Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (Ψ1+ 4Ψ2)1198791205952119872119879+1

120595119879

2119872119879+1(Ψ1+ 4Ψ2)119879120595119879

2119872119879+11205952119872119879+1

]

(43)

where

Ψ1= [12059511205952 120595

119873119905] isin R

(2119872119879+1)times119872119879

Ψ2= [120595119872119879+1

120595119872119879+2

1205952119872119879] isin R

(2119872119879+1)times119872119879

(44)



Y11= (Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (45)


is

119910119895119895=10038171003817100381710038171003817120595119895+ 4120595119873119905+119895

10038171003817100381710038171003817

2

119895 = 1 2 119872119879 (46)


1 le100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120595119895

10038171003817100381710038171003817le10038171003817100381710038171003817120595119895+ 4120595119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120595119895

10038171003817100381710038171003817+100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817le 15

(47)


Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)




Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)


[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1


10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)



120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)



W1015840 = (Δ1015840)119879

Δ1015840 (55)


1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2




119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)








119879+ 1)35) [9] Secondly the


119879+ 1)








119879+ 1)45+ 8(2119872

119879+ 1)35)




0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










that satisfies

Z = Λ119879Λ (15)


119879+ 1



Y = GZG119879 = (ΛG119879)119879

(ΛG119879) (16)

where

G = [U 00 1


] (17)


10038171003817100381710038171205821198991003817100381710038171003817 = (120582

119879

119899120582119899)12

= 1 119899 = 1 2 4119872119879 (18)

120582119879

4119872119879+11205824119872119879+1

= 1 (19)


Y = [Y11 Y12

Y21

Y22

]

= [(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4) (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)1198791205824119872119879+1

120582119879

4119872119879+1(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879

120582119879

4119872119879+11205824119872119879+1

]

(20)

where Λ1= [12058211205822 120582

119873119905] isin R(4119872119879+1)times119872119879 Λ

2= [120582119872119879+1

120582119872119879+2

1205822119872119879] isin R(4119872119879+1)times119872119879 Λ

3= [1205822119873119905+1

1205822119873119905+2

1205823119873119905] isin R(4119872119879+1)times119872119879 and Λ

4= [1205823119873119905+1

1205823119873119905+2

1205824119873119905] isin


22= 1 and (6d)


Y11= (Λ1+ 2Λ2+ 4Λ3+ 8Λ4)119879(Λ1+ 2Λ2+ 4Λ3+ 8Λ4)

(21)


is

119910119895119895=10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

2

(22)


1 =1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120582119895

10038171003817100381710038171003817minus100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817minus1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895+ 2120582119872119879+119895

+ 41205822119872119879+119895

+ 81205823119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120582119895

10038171003817100381710038171003817+100381710038171003817100381710038172120582119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381741205822119872119879+119895

10038171003817100381710038171003817+1003817100381710038171003817100381781205823119872119879+119895

10038171003817100381710038171003817= 15

(23)

Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(24)





Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (25)


1205743 120574

119872119879 120574119872119879+1

] that satisfies

Y = Γ119879Γ (26)


119879+ 1


10038171003817100381710038171003817= (120574119879

119872119879+1120574119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120574119895

10038171003817100381710038171003817= (120574119879

119895120574119895)12

le 15 119895 = 1 2 119872119879

(27)



satisfy

120574119895= 1205741198951+ 21205741198952+ 41205741198953+ 81205741198954 (28)

100381710038171003817100381710038171205741198951

10038171003817100381710038171003817=100381710038171003817100381710038171205741198952

10038171003817100381710038171003817=100381710038171003817100381710038171205741198953

10038171003817100381710038171003817=100381710038171003817100381710038171205741198954

10038171003817100381710038171003817= 1 (29)


Γ1015840= [12057411 12057412 12057413 12057414 12057421 12057422 12057423 12057424 120574

1198721199051

1205741198721199052 1205741198721199053 1205741198721199054 120574119872119905+1

]

isin R(119872119879+1)times(4119872119879+1)

(30)




Z = (Γ1015840)119879

Γ1015840 (31)


119911(119895119894)(119895119894)

= (120574119879

119895119894120574119895119894) =10038171003817100381710038171003817120574119895119894

10038171003817100381710038171003817

2

= 1 (32)


119879+


119911(4119872119879+1)(4119872119879+1)

= (120574119879

119872119879+1120574119872119879+1

) = 1 (33)


Diag (Z) = 1I (34)






W1015840 isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (35c)

W101584022= 1 (35d)





W1015840 = [W1015840

11isin R2119872119879times2119872119879 W1015840

12isin R2119872119879times1

W101584021isin R1times2119872119879 1

] (36)


Ψ

= [12059511205952 120595

119872119879120595119872119879+1

120595119872119879+2

12059521198721198791205952119872119879+1

]

(37)

that satisfies

W1015840 = Ψ119879Ψ (38)


119879+ 1


1 le10038171003817100381710038171205951198961003817100381710038171003817 = (120595

119879

119896120595119896)12

le 3 119896 = 1 2 2119872119879 (39)

120595119879

2119872119879+11205952119872119879+1

= 1 (40)


which is defined as

Y = EW1015840E119879 = (ΨE119879)119879

(ΨE119879) (41)

where

E = [V 00 1

]


]

(42)


Y = [Y11 Y12

Y21

Y22

]

= [(Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (Ψ1+ 4Ψ2)1198791205952119872119879+1

120595119879

2119872119879+1(Ψ1+ 4Ψ2)119879120595119879

2119872119879+11205952119872119879+1

]

(43)

where

Ψ1= [12059511205952 120595

119873119905] isin R

(2119872119879+1)times119872119879

Ψ2= [120595119872119879+1

120595119872119879+2

1205952119872119879] isin R

(2119872119879+1)times119872119879

(44)



Y11= (Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (45)


is

119910119895119895=10038171003817100381710038171003817120595119895+ 4120595119873119905+119895

10038171003817100381710038171003817

2

119895 = 1 2 119872119879 (46)


1 le100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120595119895

10038171003817100381710038171003817le10038171003817100381710038171003817120595119895+ 4120595119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120595119895

10038171003817100381710038171003817+100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817le 15

(47)


Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)




Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)


[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1


10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)



120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)



W1015840 = (Δ1015840)119879

Δ1015840 (55)


1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2




119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)








119879+ 1)35) [9] Secondly the


119879+ 1)








119879+ 1)45+ 8(2119872

119879+ 1)35)




0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Z = (Γ1015840)119879

Γ1015840 (31)


119911(119895119894)(119895119894)

= (120574119879

119895119894120574119895119894) =10038171003817100381710038171003817120574119895119894

10038171003817100381710038171003817

2

= 1 (32)


119879+


119911(4119872119879+1)(4119872119879+1)

= (120574119879

119872119879+1120574119872119879+1

) = 1 (33)


Diag (Z) = 1I (34)






W1015840 isin R(2119872119879+1)times(2119872119879+1) ≻ 0 (35c)

W101584022= 1 (35d)





W1015840 = [W1015840

11isin R2119872119879times2119872119879 W1015840

12isin R2119872119879times1

W101584021isin R1times2119872119879 1

] (36)


Ψ

= [12059511205952 120595

119872119879120595119872119879+1

120595119872119879+2

12059521198721198791205952119872119879+1

]

(37)

that satisfies

W1015840 = Ψ119879Ψ (38)


119879+ 1


1 le10038171003817100381710038171205951198961003817100381710038171003817 = (120595

119879

119896120595119896)12

le 3 119896 = 1 2 2119872119879 (39)

120595119879

2119872119879+11205952119872119879+1

= 1 (40)


which is defined as

Y = EW1015840E119879 = (ΨE119879)119879

(ΨE119879) (41)

where

E = [V 00 1

]


]

(42)


Y = [Y11 Y12

Y21

Y22

]

= [(Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (Ψ1+ 4Ψ2)1198791205952119872119879+1

120595119879

2119872119879+1(Ψ1+ 4Ψ2)119879120595119879

2119872119879+11205952119872119879+1

]

(43)

where

Ψ1= [12059511205952 120595

119873119905] isin R

(2119872119879+1)times119872119879

Ψ2= [120595119872119879+1

120595119872119879+2

1205952119872119879] isin R

(2119872119879+1)times119872119879

(44)



Y11= (Ψ1+ 4Ψ2)119879(Ψ1+ 4Ψ2) (45)


is

119910119895119895=10038171003817100381710038171003817120595119895+ 4120595119873119905+119895

10038171003817100381710038171003817

2

119895 = 1 2 119872119879 (46)


1 le100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817minus10038171003817100381710038171003817120595119895

10038171003817100381710038171003817le10038171003817100381710038171003817120595119895+ 4120595119872119879+119895

10038171003817100381710038171003817

le10038171003817100381710038171003817120595119895

10038171003817100381710038171003817+100381710038171003817100381710038174120595119872119879+119895

10038171003817100381710038171003817le 15

(47)


Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)




Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)


[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1


10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)



120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)



W1015840 = (Δ1015840)119879

Δ1015840 (55)


1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2




119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)








119879+ 1)35) [9] Secondly the


119879+ 1)








119879+ 1)45+ 8(2119872

119879+ 1)35)




0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Thus we have1 le 119910119895119895le 225

1I≺Diag Y11 ≺ 225I

(48)




Y = [Y11 isin R119872119879times119872119879 Y

12isin R119872119879times1

Y21isin R1times119872119879 1

] (49)


[120590112059021205903 120590

119872119879120590119872119879+1

] that satisfies

Y = Δ119879Δ (50)


119879+ 1


10038171003817100381710038171003817= (120590119879

119872119879+1120590119872119879+1

)12

= 1

1 le10038171003817100381710038171003817120590119895

10038171003817100381710038171003817= (120590119879

119895120590119895)12

le 15 119895 = 1 2 119872119879

(51)



120590119895= 1205901198951+ 41205901198952 (52)

1 le10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817= (120590119879

119895119894120590119895119894)12

le 3 119894 = 1 2 (53)


Δ1015840= [12059011120590121205902112059022 120590

11987211990511205901198721199052120590119872119905+1

]

isin R(119872119879+1)times(2119872119879+1)

(54)



W1015840 = (Δ1015840)119879

Δ1015840 (55)


1199081015840

(119895119894)(119895119894)= (120590119879

119895119894120590119895119894) =10038171003817100381710038171003817120590119895119894

10038171003817100381710038171003817

2

(56)

where 119895 = 1 2 119872119879 119894 = 1 2




119879+ 1)th column

ofW1015840 are

W101584022= 1199081015840

(2119872119879+1)(2119872119879+1)= (120590119879

119872119879+1120590119872119879+1

) = 1 (58)








119879+ 1)35) [9] Secondly the


119879+ 1)








119879+ 1)45+ 8(2119872

119879+ 1)35)




0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


0 5 10 15 20 25

BLER

100

10minus1

10minus2

10minus3

BC-SDRVA-SDR


EbN0 (dB)


6 Conclusion


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

10minus1


EbN0 (dB)


0 5 10 15 20 25

Com

puta

tiona

l tim

e (s)

100

101

102

10minus1


EbN0 (dB)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article comparison of semidefinite relaxation ...research article comparison of...

Documents