Design of Spatial Decoupling Scheme

Design of Spatial Decoupling Scheme Design of Spatial Decoupling Scheme using Singular Value Decomposition for Multi-User Systems Abstract In this paper, we present the use of a polynomial singular value decomposition (PSVD) algorithm to examine a spatial decoupling based block transmission design for multiuser systems. This algorithm facilitates joint and optimal decomposition of matrices arising inherently in multiuser systems. Spatial decoupling allows complex multichannel problems of suitable dimensionality to be spectrally diagonalized by computing a reduced-order memoryless matrix through the use of the coordinated transmit precoding and 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 PSVD for the proposed algorithm. Index Terms-polynomial singular value decomposition, paraunitary systems, MIMO system. INTRODUCTION Block transmission based systems allows parallel, ideally noninterfering, virtual communication channels between multiuser channels. Minimally spatial decoupling channels are needed whenever more than two transmitting channels are communicate simultaneously. The channel of our interest here, is the multiple input multiple output channels, consisting of multiple MIMO capable source terminals and multiple capable destinations. This scenario arises, obviously, in multi-user channels. Since certain phases of relaying involves broadcasting, it also appears in MIMO relaying contexts. The phrase MIMO broadcast 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 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 common 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 common multipoint-to-multipoint MIMO channels is not much attractive, given 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 have 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 power; (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 signal coding but had its own demerits.   Amongst many, it shown in cite{Ta2005,Ta2007} that an appropriate additional amount of additive samples  still 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 zero-forcing at the destination has been examined [12], [13] as a mean of coordinated 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]. Despite 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; 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: Section 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 vector v, (i) A(i, j)  gives the ith element on the jth column of A; (ii) v(i)  {ˆ y1 }R(r+1,r+s) = ˜Π£{x }R(r+1,r+s) + _ UHn1 _ R(r+1,r+s) ,   {ˆ y2 }R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+1,pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s) = ˜Άº{x }R(r+1,r+s) + _ VHn2 _ R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+1,pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s) , {ˆ y1 }R(1,r) = {x }R(1,r) + _ UHn1 _ R(1,r) , {ˆ y2 }R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s+1,p) = {x }R(r+s+1,t) + _ VHn2 _ R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s+1,p) . (1)  gives the element of v at the ith position. {A}R(n) and  {A}C(n) denote the sub-matrices consisting respectively of the  first n rows, and the first n columns of A. Let {A}R(m,n)  denote the sub-matrix consisting of the rows m through n  of A. The expression A = diag (a1, . . . , an) indicates that  A is rectangular diagonal; and that first n elements on its  main diagonal are a1, . . . , an. rank (A) gives the rank of  A. The operators ( à £Ã†â€™Ã‚ » )H, and ( à £Ã†â€™Ã‚ »)à ¢Ã‹â€ Ã¢â‚¬â„¢1 denote respectively the  conjugate transpose and the matrix inversion. C mÃÆ'-n is the  space spanned by mÃÆ'-n matrices containing possibly complex  elements. The channel between the wireless terminals T1 and  T2 in a MIMO system is designated T1 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢T2.   II. MATHEMATICAL FRAMEWORK Let us examine GSVD to see how it can be used for  beamforming. There are two major variants of GSVD in the  literature (e.g. [20] vs. [21]). We use them both here to  elaborate the notion of GSVD-based beamforming. A. GSVD Van Loan definition Let us first look at GSVD as initially proposed by Van Loan [20, Thm. 2]. Definition 1: Consider two matrices, H à ¢Ã‹â€ Ã‹â€ C mÃÆ'-n with  m à ¢Ã¢â‚¬ °Ã‚ ¥n, and G à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, having the same number n of  columns. Let q = min (p, n). H and G can be jointly  decomposed as H = UÃŽÂ £Q, G = VΆºQ (2) where (i) U à ¢Ã‹â€ Ã‹â€ C mÃÆ'-m,V à ¢Ã‹â€ Ã‹â€ C pÃÆ'-p are unitary, (ii) Q à ¢Ã‹â€ Ã‹â€  C nÃÆ'-n non-singular, and (iii) ÃŽÂ £= diag (à Ã†â€™1, . . . , à Ã†â€™n) à ¢Ã‹â€ Ã‹â€  C mÃÆ'-n, à Ã†â€™i à ¢Ã¢â‚¬ °Ã‚ ¥0; Άº= diag (ÃŽÂ »1, . . . , ÃŽÂ »q) à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, ÃŽÂ »i à ¢Ã¢â‚¬ °Ã‚ ¥0. As a crude example, suppose that G and H above represent  channel matrices of MIMO subsystems S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 and S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2  having a common source S. Assume perfect channel-stateinformation  (CSI) on G and H at all S,D1, and D2. With  a transmit precoding matrix Qà ¢Ã‹â€ Ã¢â‚¬â„¢1, and receiver reconstruction  matrices UH,VH we get q non-interfering virtual broadcast channels. The invertible factor Q in (2) facilitates jointprecoding  for the MIMO subsystems; while the factors U,V   allow receiver reconstruction without noise enhancement. Diagonal  elements 1 through q of ÃŽÂ £,Άºrepresent the gains  of these virtual channels. Since Q is non-unitary, precoding  would 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 average  total transmit power at the desired level. This is the essence of GSVD-based beamforming for  a single source and two destinations. As would be shown  in Section III, this three-terminal configuration appears in  various MIMO subsystems making GSVD-based beamforming  applicable. B. GSVD Paige and Saunders definition Before moving on to applications, let us appreciate GSVDbased  beamforming in a more general sense, through another  form of GSVD proposed by Paige and Saunders [21, (3.1)]. This version of GSVD relaxes the constraint m à ¢Ã¢â‚¬ °Ã‚ ¥n present  in (2). Definition 2: Consider two matrices, H à ¢Ã‹â€ Ã‹â€ C mÃÆ'-n and  G à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, having the same number n of columns. Let CH = _ HH,GH _ à ¢Ã‹â€ Ã‹â€ C nÃÆ'-(m+p), t = rank(C), r = t à ¢Ã‹â€ Ã¢â‚¬â„¢rank (G) and s = rank(H) + rank (G) à ¢Ã‹â€ Ã¢â‚¬â„¢t. H and G can be jointly decomposed as H = U (ÃŽÂ £ 01 )Q = UÃŽÂ £{Q}R(t) , G = V (Άº 02 )Q = VΆº{Q}R(t) , (3) where (i) U à ¢Ã‹â€ Ã‹â€ C mÃÆ'-m,V à ¢Ã‹â€ Ã‹â€ C pÃÆ'-p are unitary, (ii) Q à ¢Ã‹â€ Ã‹â€ C nÃÆ'-n non-singular, (iii) 01 à ¢Ã‹â€ Ã‹â€ C mÃÆ'-(nà ¢Ã‹â€ Ã¢â‚¬â„¢t), 02 à ¢Ã‹â€ Ã‹â€  C pÃÆ'-(nà ¢Ã‹â€ Ã¢â‚¬â„¢t) zero matrices, and (iv) ÃŽÂ £Ãƒ ¢Ã‹â€ Ã‹â€ C mÃÆ'-t,ΆºÃƒ ¢Ã‹â€ Ã‹â€  C pÃÆ'-t have structures ÃŽÂ £_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  IH ˜Π£ 0H à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚   and Άº_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  0G ˜Άº IG à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚  . IH à ¢Ã‹â€ Ã‹â€ C rÃÆ'-r and IG à ¢Ã‹â€ Ã‹â€ C (tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s)ÃÆ'-(tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s) are identity  matrices. 0H à ¢Ã‹â€ Ã‹â€ C (mà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s)ÃÆ'-(tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s), and 0G à ¢Ã‹â€ Ã‹â€  C (pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r)ÃÆ'-r are zero matrices possibly having no  rows or no columns. ˜Π£= diag (à Ã†â€™1, . . . , à Ã†â€™s) ,˜Άº= diag (ÃŽÂ »1, . . . , ÃŽÂ »s) à ¢Ã‹â€ Ã‹â€ C sÃÆ'-s such that 1 > à Ã†â€™1 à ¢Ã¢â‚¬ °Ã‚ ¥. . . à ¢Ã¢â‚¬ °Ã‚ ¥ à Ã†â€™s > 0, and à Ã†â€™2 i + ÃŽÂ »2i = 1 for i à ¢Ã‹â€ Ã‹â€  {1, . . . , s}. Let us examine (3) in the MIMO context. It is not difficult  to see that a common transmit precoding matrix _ Qà ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(t) and receiver reconstruction matrices UH,VH would jointly  diagonalize the channels represented by H and G.  For broadcasting, only the columns (r+1) through (r +s)  of ÃŽÂ £and Άºare of interest. Nevertheless, other (t à ¢Ã‹â€ Ã¢â‚¬â„¢s)  columns, when they are present, may be used by the source  S to privately communicate with the destinations D1 and configuration # common channels # private channels S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ {D1,D2} S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2 m > n,p à ¢Ã¢â‚¬ °Ã‚ ¤n p n à ¢Ã‹â€ Ã¢â‚¬â„¢p 0 m à ¢Ã¢â‚¬ °Ã‚ ¤n, p > n m 0 n à ¢Ã‹â€ Ã¢â‚¬â„¢m m à ¢Ã¢â‚¬ °Ã‚ ¥n, p à ¢Ã¢â‚¬ °Ã‚ ¥n n 0 0 m + p à ¢Ã‹â€ Ã¢â‚¬â„¢n n à ¢Ã‹â€ Ã¢â‚¬â„¢p n à ¢Ã‹â€ Ã¢â‚¬â„¢m (m + p) > n n à ¢Ã¢â‚¬ °Ã‚ ¥(m + p) 0 m p TABLE I NUMBERS OF COMMON CHANNELS AND PRIVATE CHANNELS FOR  DIFFERENT CONFIGURATIONS D2. It is worthwhile to compare this fact with [22], and  appreciate the similarity and the conflicting objectives GSVDbased  beamforming for broadcasting has with MIMO secrecy  communication. Thus we can get ˆ y1 à ¢Ã‹â€ Ã‹â€ C mÃÆ'-1, ˆ y2 à ¢Ã‹â€ Ã‹â€ C pÃÆ'-1 as in (1) at  the detector input, when x à ¢Ã‹â€ Ã‹â€ C tÃÆ'-1 is the symbol vector  transmitted. It can also be observed from (1) that the private  channels always have unit gains; while the gains of common  channels are smaller. Since, à Ã†â€™is are in descending order, while the ÃŽÂ »is ascend  with i, selecting a subset of the available s broadcast channels  (say k à ¢Ã¢â‚¬ °Ã‚ ¤s channels) is somewhat challenging. This highlights  the need to further our intuition on GSVD. C. GSVD-based beamforming Any two MIMO subsystems having a common source  and channel matrices H and G can be effectively reduced,  depending on their ranks, to a set of common (broadcast) and  private (unicast) virtual channels. The requirement for having  common channels is rank (H) + rank (G) > rank (C) where C = _ HH,GH _ H. When the matrices have full rank, which is the case with  most MIMO channels (key-hole channels being an exception),  this requirement boils down to having m +p > n . Table I  indicates how the numbers of common channels and private  channels vary in full-rank MIMO channels. It can be noted  that the cases (m > n,p à ¢Ã¢â‚¬ °Ã‚ ¤n) and (m à ¢Ã¢â‚¬ °Ã‚ ¥n, p à ¢Ã¢â‚¬ °Ã‚ ¥n)  correspond to the form of GSVD discussed in the Subsection II-A. Further, the case n à ¢Ã¢â‚¬ °Ã‚ ¥(m + p) which produces only  private channels with unit gains, can be seen identical to zero  forcing at the transmitter. Thus, GSVD-based beamforming is  also a generalization of zero-forcing. Based on Table I, it can be concluded that the full-rank  min (n,m + p) of the combined channel always gets split  between the common and private channels. D. MATLAB implementation A general discussion on the computation of GSVD is found  in [23]. Let us focus here on what it needs for simulation:  namely its implementation in the MATLAB computational  environment, which extends [14, Thm. 8.7.4] and appears as  less restrictive as [21]. The command [V, U, X, Lambda, Sigma] = gsvd(G, H);  gives1 a decomposition similar to (3). Its main deviations  from (3) are,   1Reverse order of arguments in and out of gsvd function should be noted. ) ) D1 y1 , r1 S x ,w ( ( ) ) D2 y2 , r2 _ H1 __ n1 _ __ H2 n2 Fig. 1. Source-to-2 destination MIMO broadcast system  Ãƒ ¢Ã¢â€š ¬Ã‚ ¢ QH = X à ¢Ã‹â€ Ã‹â€ C nÃÆ'-t is not square when t . Precoding  for such cases would require the use of the pseudo-inverse  operator. à ¢Ã¢â€š ¬Ã‚ ¢ ÃŽÂ £has the same block structure as in (3). But the structure  of Άºhas the block 0G shifted to its bottom as follows: Άº_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  ˜Άº IG 0G à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚  . This can be remedied by appropriately interchanging the  rows of Άºand the columns of V. However, restructuring  ÃƒÅ½Ã¢â‚¬ ºis not a necessity, since the column position of the  block ˜Άºwithin Άºis what matters in joint precoding.   Following MATLAB code snippet for example jointly  diagonalizes H,G to obtain the s common channels (3)  would have given. MATLAB code % channel matrices H = (randn(m,n)+i*randn(m,n))/sqrt(2); G = (randn(p,n)+i*randn(p,n))/sqrt(2); % D1, D2: diagonalized channels [V,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)-t; D1 = U(:,r+1:r+s)*H*w(:,r+1:r+s); D2 = V(:,1:s)*G*w(:,r+1:r+s); III. APPLICATIONS Let us look at some of the possible applications of GSVDbased beamforming. We assume the Van Loan form of GSVD  for simplicity, having taken for granted that the dimensions  are such that the constraints hold true. Nevertheless, the Paige  and Saunders form should be usable as well. A. Source-to-2 destination MIMO broadcast system   Consider the MIMO broadcast system shown in Fig. 1,  where the source S broadcasts to destinations D1 and D2.  MIMO subsystems S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 and S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2 are modeled  to have channel matrices H1 ,H2 and additive complex   Gaussian noise vectors n1 , n2. Let x = [x1, . . . , xn]T ) ) R1 y1 , F1 ( ( S x ,w ( ( ) ) D y3 ,r1 y4 ,r2 ) ) R2 y2 , F2 ( ( _ ___ H3 _ n3 H1 ___ n1 _ ___ H2 n2 _ H4 ___ n4 Fig. 2. MIMO relay system with two 2-hop-branches  be the signal vector desired to be transmitted over n à ¢Ã¢â‚¬ °Ã‚ ¤Ã‚  min (rank (H1 ) , rank (H2 )) virtual-channels. The source  employs a precoding matrix w. The input y1 , y2 and output ˆ y1 , ˆ y2 at the receiver filters   r1 , r2 at D1 and D2 are given by y1 = H1wx + n1 ; ˆ y1 = r1 y1 , y2 = H2wx + n2 ; ˆ y2 = r2 y2 . Applying GSVD we get H1 = U1 ÃŽÂ £1 V and H2 = U2 ÃŽÂ £2V. Choose the precoding matrix w = ÃŽÂ ± _ Và ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(n) ; and receiver reconstruction matrices r1 = _ U1 H _ R(n) _ , r2 = U2 H _ R(n) . The constant ÃŽÂ ± normalizes the total average transmit power. Then we get, ˆ y1(i) = ÃŽÂ ±ÃƒÅ½Ã‚ £1(i, i) x(i) + Ëœn1(i) , ˆ y2(i) = ÃŽÂ ±ÃƒÅ½Ã‚ £2(i, i) x(i) + Ëœn2(i), ià ¢Ã‹â€ Ã‹â€  {1 . . . n}, where Ëœn1 , Ëœn2 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 a  source S communicates a symbol vector x with a destination  D via two relays R1 and R2. MIMO channels S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢R1, S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ R2, R1 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D and R2 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D are denoted: Hi , i à ¢Ã‹â€ Ã‹â€  {1, 2, 3, 4}. Corresponding channel outputs and additive complex Gaussian  noise vectors are yi , ni for i à ¢Ã‹â€ Ã‹â€  {1, 2, 3, 4}. Assume relay  operations to be linear, and modeled as matrices F1 and F2 . Assume orthogonal time-slots for transmission. The source  S uses w as the precoding matrix. Destination D uses  different reconstruction matrices r1 , r2 during the time slots  2 and 3. Then we have: Time slot 1: y1 = H1wx + n1 , y2 = H2wx + n2 Time slot 2: y3 = H3 F1 y1 + n3 Time slot 3: y4 = H4 F2 y2 + n4 Let ˆ y = r1 y3 +r2 y4 be the input to the detector. Suppose n à ¢Ã¢â‚¬ °Ã‚ ¤min i (rank (Hi )) virtual-channels are in use. ) ) R y1 , F ( ( S x ,w ( ( ) ) D y2 ,r1 y3 ,r2 _ ___ H3 _ n3 H1 ___ n1 H2 _ n2 Fig. 3. MIMO relay system having a direct path and a relayed path  Applying GSVD on the broadcast channel matrices we get H1 = U1 ÃŽÂ £1 Q and H2 = U2 ÃŽÂ £2 Q. Through SVD we  obtain H3 = V1 Άº1 R1 H and H4 = V2 Άº2 R2 H. Choose w = ÃŽÂ ± _ Qà ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(n) ; F1 = R1U1 H; F2 = R2U2 H; r1 = _ V1 H _ R(n) ; r2 = _ V2 H _ R(n) . The constant ÃŽÂ ± normalizes  the total average transmit power. Then we get