Tuesday, June 4, 2019

Design of Spatial Decoupling Scheme

Design of Spatial Decoupling arrangementDesign of Spatial Decoupling Scheme using Singular Value Decomposition for Multi-User SystemsAbstract In this paper, we front the use of a polynomial singular value decomposition (PSVD) algorithm to study a spacial decoupling based occlusion communicateting design for multiuser agreements. This algorithm facilitates joint and optimal decomposition of matrices arising inherently in multiuser systems. Spatial decoupling solelyows complex multichannel problems of fitted dimensionality to be spectrally diagonalized by computing a reduced-order memoryless matrix through the use of the coordinated transmit precoding and pass receiver equalization matrices.A primary application of spatial decoupling based system can be useful in discrete multitone (DMT) systems to combat the induced crosstalk interference, as well as in OFDM with intersymbol interference. We present here simulation-based performance analysis results to justify the use of PSV D for the proposed algorithm.Index Terms-polynomial singular value decomposition, paraunitary systems, MIMO system.INTRODUCTIONBlock infection based systems allows parallel, ideally noninterfering, virtual communicating convey between multiuser channels. Minimally spatial decoupling channels ar needed whenever more than two transmitting channels are communicate simultaneously. The channel of our worry here, is the multiple input multiple output channels, consisting of multiple MIMO capable quotation terminals and multiple capable destinations.This scenario arises, obviously, in multi-user channels. Since certain phases of relaying involves hand outing, it also appears in MIMO relaying contexts. The phrase MIMO course of study channel is frequently used in a loose sense in the literature, to include point-to-multipoint unicast (i.e. private) channels carrying different messages from a single source to individually of the multiple destinations (e.g. in multi-user MIMO). Its us e in this paper is more specific, and denotes the presence of at least one joint 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 requires parkland multipoint-to-multipoint MIMO channels is not much attractive, prone the fact that the total number of private and common channels is limited by the number of antennas the source has.Wherever each receiver of a broadcast channel conveys what it receives orthogonally to the same destination, as in the case of pre-and post-processing block transmission, the whole system can be envisaged as a single point-to-point MIMO channel.Block transmission techniques stool been demonstrated for point-to-point MIMO channels to benefit the system complexities. Other advantages includes (i) channel interference is removed by creating $K$ independent subchannels (ii) paraunitarity of precoder allows to control transmit former (iii) paraunitarity of equalizer does not amplify the channel noise (iv) spatial redundancy can be achieved by discarding the weakest subchannels.Though the technique outperform the conventional mansion coding but had its own demerits. Amongst many, it shown in citeTa2005,Ta2007 that an appropriate additional amount of additive samplesstill require individual processing, e.g. per- tone equalisation, to remove ISI, and the receiver does not exploit the case of structured noise.However, the choice of optimal relay gains, although known for certain cases (e.g. 10, 11), is not straightforward with this approach. 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 nought-forcing at the destination has been raised 12, 13 as a mean of coordinate d beamforming, since it does not require transmitter 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. in spite of GSVD being the natural generalization of SVD for two matrices, we are yet to see in the literature, a generalization of SVD-based beamforming to GSVD-based beamforming. Although the purpose and the use is somewhat different, the reference 17, p.1 appears to be the first 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 then 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 Section II presents the mathematical framework, highlighting how and infra which constraints GSVD can be used for beamforming. Section III examines how GSVD-based beamforming can be applied in certain dewy-eyed MIMO and MIMO relaying configurations. Performance analysis is conducted in character IV on one of these applications. Section V concludes with some final remarks.Notations Given a matrix A and a vector v, (i) A(i, j)gives the ith element on the jth column 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. rank (A) gives the rank ofA. The operators ( )H, and ( )1 denote respectively theconjugate transpose and the matrix inversion. C m-n is the office 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. MATHEMATICAL 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 vox populi of GSVD-based beamforming.A. GSVD Van Loan definitionLet us first look at GSVD as init ially proposed by Van Loan20, Thm. 2.Definition 1 Consider 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 example, suppose 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 reconstructive memorymatrices UH,VH we get q non-interfering virtual broadcast channels. The invertible factor Q in (2) facilitates jointprecodingfor the MIMO subsystems magic spell the factors U,Vallow receiver reconstruction without noise enhancement. Diagonalelements 1 through q of ,represent the gainsof these virtual channels. Since Q is non-unitary, precodingwould cause the ins tantaneous transmit power to fluctuate.This is a drawback not present in SVD-based beamforming. shine signal should be normalized to maintain the reasonabletotal 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 rate GSVDbasedbeamforming in a more general sense, through an separateform 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 = rank(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 jointlydiagonalize 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 CHANNELS AN D PRIVATE CHANNELS FORDIFFERENT CONFIGURATIONSD2. It is worthwhile to compare this fact with 22, andappreciate the similarity and the conflicting objectives GSVDbasedbeamforming for broadcasting has with MIMO secrecycommunication.Thus we can get y1 C m-1, y2 C p-1 as in (1) atthe sensor input, when x C t-1 is the symbol vector communicable. 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 beamformingAny 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 channels 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 . plank 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)correspond 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 splitbetween the common and private channels.D. MATLAB implementationA general intelligence on the computation of GSVD is foundin 23. Let us focus here on what it needs for simulationnamely its implementation in the MATLAB computationalenvironment, which extends 14, Thm. 8.7.4 and appea rs 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 function 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). But 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 cha nnelsV,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 grow 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 (rank (H1 ) , rank (H2 )) virtual-channels. The sourc eemploys 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.Then 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- slots)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. speech D usesdifferent reconstruction matrices r1 , r2 during the time slots2 and 3. Then 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 with(predicate) 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

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