Design of Spatial Decoupling Scheme

flesh of Spatial Decoupling SchemeDesign of Spatial Decoupling Scheme using Singular Value Decomposition for Multi-User Systems snatch In this paper, we put the do of a polynomial singular shelter buncombe (PSVD) algorithm to examine a spacial decoupling based stymy send offtal design for multiuser clays. This algorithm facilitates joint and optimal decomposition of matrices arising in herently in multiuser corpses. Spatial decoupling allows daedal multi extend problems of suitable dimensionality to be spectrally bezzantized by cipher a reduced-order memoryless hyaloplasm through the use of the coordinated transmit precoding and recipient equalization matrices.A primary application of spacial decoupling based agreement give notice be useful in discrete multitone (DMT) systems to combat the bring on crosstalk interference, as well as in OFDM with intersymbol interference. We present here simulation-based performance analysis results to justify the use of PSVD for t he proposed algorithm.Index Terms-polynomial singular think of decomposition, paraunitary systems, MIMO system.INTRODUCTIONBlock transmitting based systems allows parallel, ideally noninterfering, virtual communicating wrinkles between multiuser phone lines. Minimally spatial decoupling impart be ask whenever more than two transmitting channel are communicate simultaneously. The channel of our interest here, is the multiple input multiple output channels, consisting of multiple MIMO subject get-go terminals and multiple capable finales.This scenario arises, obviously, in multi-user channels. Since plastered phases of electrical relaying involves syllabusing, it also appears in MIMO relaying contexts. The phrase MIMO break up channel is frequently utilise in a loose sense in the belles-lettres, to include point-to-multipoint unicast (i.e. clubby) channels carrying different messages from a single source to each of the multiple destinations (e.g. in multi-user MIMO). Its use in this paper is more specific, and denotes the presence of at least one parkland virtual broadcast channel from the source to the destinations.The use of iterative and non-iterative spatial decoupling techniques in multiuser systems to achieve independent channels has been investigated, for instance in 1-9.Their use for MIMO broadcasting, which look ats customary multipoint-to-multipoint MIMO channels is not much attractive, given the fact that the intact enumerate of individual(a) and viridity channels is limited by the number of antennas the source has.Wherever each recipient of a broadcast channel conveys what it receives orthogonally to the same destination, as in the case of pre-and post-processing farce transmission, the whole system can be envisaged as a single point-to-point MIMO channel.Block transmission techniques attain been demonstrated for point-to-point MIMO channels to benefit the system complexities. Other advantages includes (i) channel interferen ce is removed by creating $K$ independent subchannels (ii) paraunitarity of pre edictr allows to control transmit designer (iii) paraunitarity of equalizer does not amplify the channel noise (iv) spatial redundancy can be achieved by discarding the weakest subchannels.Though the technique transcend the conventional signal coding but had its own demerits. Amongst many, it shown in constituteTa2005,Ta2007 that an appropriate additional amount of elongate samplesstill require one-on-one processing, e.g. per- tone equalisation, to remove ISI, and the telephone receiver does not exploit the case of unified noise.However, the choice of optimal relay gains, although known for certain cases (e.g. 10, 11), is not artless with this draw near. Since the individual equalization have no non-iterative means of decoding the signals, this approach cannot be used with decode-and-forward (DF), and code-and-forward (CF) relay processing schemes.The use of zero-forcing at the destination has b een examined 12, 13 as a mean of coordinated beamforming, since it does not require sender processing. The scheme scales to any number of destinations, but requires each destination to have no less antennas than the source.Although not used as commonly as the singular value decomposition (SVD), generalized singular value decomposition (GSVD) 14, Thm. 8.7.4 is not unheard of in the wireless literature. It has been used in multi-user MIMO transmission 15, 16, MIMO secrecy communication 17, 18, and MIMO relaying 19. Reference 19 uses GSVD in dual-hop AF relaying with arbitrary number of relays. Since it employs zero-forcing at the relay for the forward channel, its use of GSVD appears almost similar to the use of SVD in 1.Despite GSVD being the natural generalization of SVD for two matrices, we are besides to see in the literature, a generalization of SVD-based beamforming to GSVD-based beamforming. Although the purpose and the use is some different, the reference 17, p.1 appears to b e the firstborn to hint the possible use of GSVD for beamforming. In present work, we illustrate how GSVD can be used for coordinated beamforming in source-to-2 destination MIMO broadcasting thus in AF, DF and CF MIMO relaying. We also present comparative, simulation-based performance analysis results to justify GSVD-based beamforming.The paper is organized as follows component II presents the mathematical framework, highlighting how and under which constraints GSVD can be used for beamforming. Section III examines how GSVD-based beamforming can be applied in certain simple MIMO and MIMO relaying configurations. Performance analysis is conducted in section IV on one of these applications. Section V concludes with some final remarks.Notations Given a matrix A and a sender v, (i) A(i, j)gives the ith agent on the jth pillar of A (ii) v(i)y1 R(r+1,r+s) = x R(r+1,r+s) +_UHn1_R(r+1,r+s) ,y2 R(pt+r+1,pt+r+s) = x R(r+1,r+s) +_VHn2_R(pt+r+1,pt+r+s) ,y1 R(1,r) = x R(1,r) +_UHn1_R(1,r) , y2 R(pt+r+s+1,p) = x R(r+s+1,t) +_VHn2_R(pt+r+s+1,p) . (1)gives the element of v at the ith position. AR(n) andAC(n) denote the sub-matrices consisting respectively of thefirst n rows, and the first n columns of A. Let AR(m,n)denote the sub-matrix consisting of the rows m through nof A. The expression A = diag (a1, . . . , an) indicates thatA is rectangular diagonal and that first n elements on itsmain diagonal are a1, . . . , an. ordain (A) gives the drift(a) ofA. The operators ( )H, and ( )1 denote respectively theconjugate transpose and the matrix inversion. C m-n is thespace spanned by m-n matrices containing possibly complexelements. The channel between the wireless terminals T1 andT2 in a MIMO system is designated T1 T2.II. numeral FRAMEWORKLet us examine GSVD to see how it can be used forbeamforming. There are two major variants of GSVD in theliterature (e.g. 20 vs. 21). We use them both here toelaborate the notion of GSVD-based beamforming.A. GSVD Van contribute inte rpretationLet us first look at GSVD as initially proposed by Van Loan20, Thm. 2. interpretation 1 tip over two matrices, H C m-n withm n, and G C p-n, having the same number n ofcolumns. Let q = min (p, n). H and G can be jointlydecomposed asH = UQ, G = VQ (2)where (i) U C m-m,V C p-p are unitary, (ii) Q C n-n non-singular, and (iii) = diag (1, . . . , n) C m-n, i 0 = diag (1, . . . , q) C p-n, i 0.As a crude compositors case, sound off that G and H above representchannel matrices of MIMO subsystems S D1 and S D2having a common source S. Assume perfect channel-stateinformation(CSI) on G and H at all S,D1, and D2. Witha transmit precoding matrix Q1, and receiver reconstructionmatrices UH,VH we get q non-interfering virtual broadcast channels. The invertible factor Q in (2) facilitates jointprecodingfor the MIMO subsystems man the factors U,Vallow receiver reconstruction without noise enhancement. Diagonalelements 1 through q of ,represent the gainsof these virtual channels. Sinc e Q is non-unitary, precodingwould cause the instantaneous transmit power to fluctuate.This is a drawback not present in SVD-based beamforming.Transmit signal should be normalized to maintain the averagetotal transmit power at the desired level.This is the essence of GSVD-based beamforming fora single source and two destinations. As would be shownin Section III, this three-terminal configuration appears invarious MIMO subsystems making GSVD-based beamformingapplicable.B. GSVD Paige and Saunders definitionBefore moving on to applications, let us appreciate GSVDbasedbeamforming in a more general sense, through anotherform of GSVD proposed by Paige and Saunders 21, (3.1).This version of GSVD relaxes the constraint m n presentin (2).Definition 2 Consider two matrices, H C m-n andG C p-n, having the same number n of columns. LetCH =_HH,GH_C n-(m+p), t = stray(C), r =t rank (G) and s = rank(H) + rank (G) t.H and G can be jointly decomposed asH = U ( 01 )Q = UQR(t) ,G = V ( 02 )Q = VQR( t) , (3)where (i) U C m-m,V C p-p are unitary, (ii)Q C n-n non-singular, (iii) 01 C m-(nt), 02 C p-(nt) zero matrices, and (iv) C m-t,C p-t have structures_IH0Hand_0GIG.IH C r-r and IG C (trs)-(trs) are identitymatrices. 0H C (mrs)-(trs), and 0G C (pt+r)-r are zero matrices possibly having norows or no columns. = diag (1, . . . , s) ,=diag (1, . . . , s) C s-s such that 1 1 . . . s 0, and 2i + 2i= 1 for i 1, . . . , s.Let us examine (3) in the MIMO context. It is not difficultto see that a common transmit precoding matrix_Q1_C(t)and receiver reconstruction matrices UH,VH would jointly translate the channels represented by H and G.For broadcasting, only the columns (r+1) through (r +s)of and are of interest. Nevertheless, other (t s)columns, when they are present, may be used by the sourceS to privately communicate with the destinations D1 andconfiguration common channels private channelsS D1,D2 S D1 S D2m n,p n p n p 0m n, p n m 0 n mm n, p n n 0 0m + p n n p n m(m + p) nn (m + p) 0 m pTABLE INUMBERS OF COMMON take AND PRIVATE CHANNELS FORDIFFERENT CONFIGURATIONSD2. It is worth piece to compare this fact with 22, andappreciate the similarity and the contrast objectives GSVDbasedbeamforming for broadcasting has with MIMO secrecycommunication.Thus we can get y1 C m-1, y2 C p-1 as in (1) atthe detector input, when x C t-1 is the symbol sendertransmitted. It can also be observed from (1) that the privatechannels always have unit gains while the gains of commonchannels are smaller.Since, is are in descending order, while the is ascendwith i, selecting a subset of the available s broadcast channels(say k s channels) is somewhat challenging. This highlightsthe need to further our intuition on GSVD.C. GSVD-based beamforming all two MIMO subsystems having a common sourceand channel matrices H and G can be effectively reduced,depending on their ranks, to a set of common (broadcast) andprivate (unicast) virtual channels. The requirement for havingcommon chan nels is rank (H) + rank (G) rank (C)where C =_HH,GH_H.When the matrices have full rank, which is the case withmost MIMO channels (key-hole channels being an exception),this requirement boils down to having m +p n . Table Iindicates how the numbers of common channels and privatechannels vary in full-rank MIMO channels. It can be notedthat the cases (m n,p n) and (m n, p n) barrack to the form of GSVD discussed in the Subsection II-A. Further, the case n (m + p) which produces onlyprivate channels with unit gains, can be seen identical to zeroforcing at the transmitter. Thus, GSVD-based beamforming isalso a generalization of zero-forcing.Based on Table I, it can be concluded that the full-rankmin (n,m + p) of the combined channel always gets recessbetween the common and private channels.D. MATLAB implementationA general treatment on the computation of GSVD is foundin 23. Let us point here on what it needs for simulationnamely its implementation in the MATLAB computationalenviron ment, which extends 14, Thm. 8.7.4 and appears asless restrictive as 21.The command V, U, X, Lambda, Sigma = gsvd(G, H)gives1 a decomposition similar to (3). Its main deviationsfrom (3) are,1Reverse order of arguments in and out of gsvd endure should be noted.))D1y1 , r1Sx ,w(())D2y2 , r2_H1 __n1___H2n2Fig. 1. Source-to-2 destination MIMO broadcast system QH = X C n-t is not square when t . Precodingfor such cases would require the use of the pseudo-inverseoperator. has the same block structure as in (3). precisely the structureof has the block 0G shifted to its bottom as follows_IG0G.This can be remedied by appropriately interchanging therows of and the columns of V. However, restructuringis not a necessity, since the column position of theblock within is what matters in joint precoding.Following MATLAB code snippet for example jointlydiagonalizes H,G to obtain the s common channels (3)would have given.MATLAB code% channel matricesH = (randn(m,n)+i*randn(m,n))/sqrt(2)G = (randn(p ,n)+i*randn(p,n))/sqrt(2)% D1, D2 diagonalized channelsV,U,X,Lambda,Sigma = gsvd(G,H)w = X*inv(X*X) C = H G t = rank(C)r = t rank(G) s = rank(H)+rank(G)-tD1 = U(,r+1r+s)*H*w(,r+1r+s)D2 = V(,1s)*G*w(,r+1r+s)III. APPLICATIONSLet us look at some of the possible applications of GSVDbased beamforming. We claim the Van Loan form of GSVDfor simplicity, having taken for granted that the dimensionsare such that the constraints hold true. Nevertheless, the Paigeand Saunders form should be usable as well.A. Source-to-2 destination MIMO broadcast systemConsider the MIMO broadcast system shown in Fig. 1,where the source S broadcasts to destinations D1 and D2.MIMO subsystems S D1 and S D2 are modeledto have channel matrices H1 ,H2 and additive complexGaussian noise vectors n1 , n2. Let x = x1, . . . , xnT))R1y1 , F1((Sx ,w(())Dy3 ,r1y4 ,r2))R2y2 , F2((____H3_ n3H1 ___n1____H2n2 _H4 ___n4Fig. 2. MIMO relay system with two 2-hop-branchesbe the signal vector desired to be transmitted over n min (r ank (H1 ) , rank (H2 )) virtual-channels. The sourceemploys a precoding matrix w.The input y1 , y2 and output y1 , y2 at the receiver filtersr1 , r2 at D1 and D2 are given byy1 = H1wx + n1 y1 = r1 y1 ,y2 = H2wx + n2 y2 = r2 y2 .Applying GSVD we get H1 = U1 1 V and H2 =U2 2V. Choose the precoding matrix w = _V1_C(n)and receiver reconstruction matrices r1 =_U1H_R(n)_ , r2 =U2H_R(n). The constant normalizes the total averagetransmit power. therefore we get,y1(i) = 1(i, i) x(i) + n1(i) ,y2(i) = 2(i, i) x(i) + n2(i), i 1 . . . n,where n1 , n2 have the same noise distributions as n1 , n2 .B. MIMO relay system with two 2-hop-branches (3 time- schedules)Fig. 2 shows a simple MIMO AF relay system where asource S communicates a symbol vector x with a destinationD via two relays R1 and R2. MIMO channels S R1, S R2, R1 D and R2 D are denoted Hi , i 1, 2, 3, 4.Corresponding channel outputs and additive complex Gaussiannoise vectors are yi , ni for i 1, 2, 3, 4. Assume relayoperations to be linear, and modeled as matrices F1 and F2 .Assume orthogonal time-slots for transmission. The sourceS uses w as the precoding matrix. Destination D usesdifferent reconstruction matrices r1 , r2 during the time slots2 and 3. past we haveTime slot 1 y1 = H1wx + n1 , y2 = H2wx + n2Time slot 2 y3 = H3 F1 y1 + n3Time slot 3 y4 = H4 F2 y2 + n4Let y = r1 y3 +r2 y4 be the input to the detector. Supposen mini(rank (Hi )) virtual-channels are in use.))Ry1 , F((Sx ,w(())Dy2 ,r1y3 ,r2____H3_ n3H1 ___n1H2 _n2Fig. 3. MIMO relay system having a direct path and a relayed pathApplying GSVD on the broadcast channel matrices we getH1 = U1 1 Q and H2 = U2 2 Q. Through SVD weobtain H3 = V1 1 R1H and H4 = V2 2 R2H. Choosew = _Q1_C(n) F1 = R1U1H F2 = R2U2H r1 = _V1H_R(n) r2 =_V2H_R(n). The constant normalizesthe total average transmit power. Then we get