The transmit subspace for MIMO systems

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Subspace of signal is convenional concept and very useful for applying to communication theory. In MIMO, the transmit beams can be created based on this concept, that can be predicted channel fading matrix. Here, the paper considers good subspace for transmitter can form for these beams. Moreover, the author using simulation to show higher capacity given by these beams than conventional method of creating transmit beam.

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Nội dung Text: The transmit subspace for MIMO systems

Nghiên cứu khoa học công nghệ THE TRANSMIT SUBSPACE FOR MIMO SYSTEMS TRAN HOAI TRUNG Abtract: Subspace of signal is convenional concept and very useful for applying to communication theory. In MIMO, the transmit beams can be created based on this concept, that can be predicted channel fading matrix. Here, the paper considers good subspace for transmitter can form for these beams. Moreover, the author using simulation to show higher capacity given by these beams than conventional method of creating transmit beam. Keywords: Wireless communication, MIMO system, Transmit subspace 1. PROBLEM The subspace method, obtained from the covariance of the channel matrix, represents the productive transmit dimensions and the power allocation at the receiver. Simulations of the productive dimensions are used to investigate the invariance of these dimensions at the transmitter. 2. THE SUBSPACE OF A SIGNAL When a signal can be expressed in terms of its phase and time parameters [1]: L x(t )   a i e j ( 2fi t  i ) (1) i 1 The correlation of this function at times of t and t  k is defined as [1]: rxx (k )  E x(t ) x(t  k )  ai2 e j 2fi k (2) The correlation matrix for K times of observation is expressed as:  rxx [0] rxx [ 1] ... rxx [(( K  1)]  r [1] rxx [0] ... rxx [( K  2)]  (3) R xx   xx  ... ... ... ...    rxx [ K  1] rxx [ K  2] ... rxx [0]  It can be rewritten to emphasise the influence of subspace: R xx  SPS H , (4) where S is defined as: S  s1 s2 ... s L  in which s i , i  1  L that is defined as: s i  [1 e j 2f ... e j 2 ( K 1) f ] (5) and P  diag[a12 , a 22 ,..., a L2 ] . Therefore, the subspace of a signal consists of linear combinations of all vectors s i , i  1  L of S . R xx can be then rewritten so as to emphasize the influence of the SVD (Singular Value Decomposition) is defined as: R xx  UΣ V H ,where U, V are unitary matrices and Σ is the diagonal matrix where U  u1 u 2 ... u L  . When the correlation matrix R xx is known, the change of the direction of the component signal xi t  of the signal xt  can be given by the eigenvectors u i , i  1  L of matrix, U  u1 u 2 ... u L  extracted from the SVD of R xx . Tạp chí Nghiên cứu KH&CN quân sự, Số 33, 10 - 2014 51 Kỹ thuật điện tử & Khoa học máy tính 3. MIMO MODEL The discrete physical MIMO model in the discrete physical model is defined through this paper as a multi-path radio link with multiple elements at the transmit antenna and multiple elements at the receive antenna as pictured in Fig 1. Path 1 Moving of the receiver … 1 1 L z sR sT sT sin 1 … … … L N elements M elements Path L Figure 1. MIMO model including moving mobile. The channel matrix in the MIMO model in the discrete physical model stated as: H  hnm NxM , where hnm is the connection coefficient between the m th element at the transmit antenna and the n th element at the receive antenna where: L hnm    l e  j l e  j m 1 sin l sT  ( n 1) sin  l s R e ju l vt (6) l 1 2 where  l is the magnitude of path l ,   where  is wavelength of signal, vt  z where  v is the velocity of the receiver, t is the time of moving the receiver and z is the distance the receiver moves. The important relationship between the correlation matrices: rhh , g ( p ) , rhh ,q ( p ) and the corresponding columns of channel matrix h g , h q is: rhh , g ( p )  E h g (t  p )h g (t ) H ; rhh ,q ( p )  E h q (t  p )h q (t ) H (7) In the context of the MIMO model in the discrete physical model, rhh , g ( p ) is equivalent to rhh ,q ( p ) . This indicates that the correlation matrices R hh , g , R hh ,q are the same. Therefore, in the MIMO model, the correlation matrix of any column of the channel matrix is referred to as the correlation matrix of the first column as defined in the MISO model when it can be interpreted as the correlation matrix of other columns of the channel matrix. 4. TRANSMIT BEAMS BASED ON THE SUBSPACE OF MISO When the subspace of the channel vector is known (i.e., when the covariance matrix is available at the receiver), it is possible to use this subspace to determine the productive 52 (8) T. H. Trung, “The transmit subspace for MIMO systems.” Nghiên cứu khoa học công nghệ transmit dimensions. Given the covariance matrix after K times of observation at the receiver, R hh , the subspace of the channel vector extracted from this matrix is rewritten as: S  s1 s 2 ... s L  where  1   e  j sin l sT     e  j sin l sT 2     ...   e  j sin l sT ( M 1)   e j 2f l     j l  e j 2f l e  j sin l sT  sl  e  , l  1  L ...  j 2f l  j sin l sT ( M 1)   e e   ...   ( K 1) j 2f l   e   e ( K 1) j 2f l e  j sin l sT     ...  e ( K 1) 2f l e  j sin l sT ( M 1)  The information of the phases of the component entries e  j l e  j ( m 1) sT sin l e jul vt in the L channel vector h  hm1  where hm1 (t )    l e  j l e  j ( m 1) sT sin l e jul vt is known from this l 1 subspace at the p th time of observation at the receiver in which the vectors used for giving this information are extracted from s l , l  1  L defined as:  e j 2f l  p 1    j sin l sT j 2f l  p 1  e e s l  e  j l  , p  1  K , l  1  L (9)  ...    j M 1sin  s j 2f  p 1  l Te l e  where f l  u l v / 2 , u l   cos l . For the case where  l2  0, l  1  L , these magnitudes of the l th path of the discrete physical environment can be given by the matrix, P  diag[ 12  22 ...  L2 ] . They were extracted by the covariance matrix, given that maximum gains of these physical paths are achievable when the weight vectors are the conjugate transpose of vectors s lp , p  1  K , l  1  L . Hence, the optimum weight vectors at the p th time of observation can be rewritten as: Tạp chí Nghiên cứu KH&CN quân sự, Số 33, 10 - 2014 53 Kỹ thuật điện tử & Khoa học máy tính T  e  j 2f l ( p 1)   jk sin l sT  j 2f l ( p 1)  e e w lp  s lpH  e j l   , p  1  K, l 1  L (10)  ...   jk ( M 1) sin l sT  j 2f l ( p 1)  e e  The L vectors w lp , p  1  K , l  1  L are known from the covariance matrix R hh at the p th time of observation at the receiver in which the transmitted power is allocated to these vectors. In terms of the L vectors w lp , p  1  K , l  1  L offered by the covariance matrix at the receiver, the array factor (beam patterns) of the vector w lp , p  1  K , l  1  L as defined in [2]: 1 M AFlp ( )   w lp (m)e  j ( ( m1) sT sin  ) M m1 (11) 1 where w lp (m), m  1  M is the m th entry of vector w lp , w lp is the normalized vector of M w lp . Applying SVD at the receiver to decompose the covariance matrix R hh , i.e. R hh  UΣ V H (when s lp , p  1  K , l  1  L are not available at the receiver) leads to the vectors u l , l  1  L of matrix U  u1 u2 ... u L  . The productive transmit vector at the p th observation w lp , l  1  L are then u lpH , l  1  L , where u lp , l  1  L consists of the M ( p  1)  1 th to the Mp th entries of vector u l , l  1  L . 5. TRANSMIT BEAM IN CASE MOVING RECEIVER The observation of the beam pattern using the strongest dimension is given. When implementing this beam pattern, the parameters that have to be considered in the discrete physical environments are: the AoD,  l , l  1  L and the AoA,  l , l  1  L . In beam patterns, the directions of physical paths are basically related to the AoD. A method to validate the changes of these directions as the receiver moves is to choose the different AoD and observe the changes of directions of the physical paths when moving the receiver. At first, a two-path environment is assumed with  1  15 0 ,  2  315 0 ,  1  135 0 , 2  225 0 at the beginning of receiver movement. Other parameters are illustrated in table 1. Table 1. The parameters in a two-path environment excluding transmit and receive angles. Wavelength   1(m) Velocity of the receiver v  40(km / h) The spacing between the sT  0.5(m) transmit elements The number of elements at M  4, N  4 transmit and receive antennas Magnitudes of paths 1   2  1 The number of observation K  200 54 T. H. Trung, “The transmit subspace for MIMO systems.” Nghiên cứu khoa học công nghệ The simulation of the beam pattern with different transmit angles  1  15 0 ,30 0 ,45 0 ,...,120 0 , is shown in figure 2. (a)  1  15 0 (b) 1  30 0 (c)  1  45 0 (d) 1  60 0 0 0 (e) 1  75 (f) 1  90 Figure 2. Simulations of beam patterns when moving the receiver at different transmit angles  1  15 0 ,30 0 ,45 0 ,...,120 0 , dotted bold lines describe the transmit angles at the beginning of the moving receiver where  1  15 0 ,30 0 ,45 0 ,...,120 0 and  2  315 0 . The directions of physical paths at the beginning of receiver movement are illustrated as dotted lines in figure 2. This figure also presents change of these paths as straight lines when receiver is moving. The figure indicates that these paths changes slowly when receiver moves (the receiver’s velocity is: v  40(km / h) ). 6. COMPARISON The subspace method permits the higher theoretical channel capacity compared to the conventional method that uses only the strongest dimension. This section defines this conventional method based on the first column of channel matrix as defined in [3], [4] and [5]. The first column of channel matrix in MIMO discrete physical model can be written as: h  h11 h21 ... hM 1  T (12) Tạp chí Nghiên cứu KH&CN quân sự, Số 33, 10 - 2014 55 Kỹ thuật điện tử & Khoa học máy tính L where hm1 (t )    l e  j l e  jk ( m 1) sT sin l e jul vt l 1 For the optimum weight transmit vector for the conventional method, [3] presented it, as; w  h H  w11 w12 ... w1M  (13) L where w1m (t )  hm1 (t )*    l e j l e j ( m 1) sT sin l e  jul vt l 1 At the first observation at the receiver, t  0 , this vector can be rewritten as: w  h H  w11 w12 ... w1M  (14) L where w1m  hm1 (t )*    l e j l e j ( m1) sT sin l l 1 The author uses simulation to show the advantage of subspace method. In this simulation, the author uses some parameters such as: number of path L  2 , gains for two paths:  1  1;  2  1 , angles of arrivals and destinations: 1  1  45 0 ;  2   2  315 0 . Wavelength of signal:   0,1 . Distance between two element antennas: sT  s R  0,1 . Velocity of mobile: v  40km / h . Number of observation: K  100 . Signal to noise ratio: S / N  5dB . The higher capacity (bit/s/Hz) can be seen in Fig.3. Beam pattern 90 Beams for Beam pattern 90 20 20 120 15 60 two paths 120 15 60 150 10 30 150 10 30 5 5 array factors array factors 180 0 180 0 210 330 210 330 240 300 240 300 270 270 transmit angle transmit angle Beam pattern 90 25 120 60 20 15 150 30 10 5 array factor 180 0 Strongest beam 210 330 240 300 270 transmit angle 56 T. H. Trung, “The transmit subspace for MIMO systems.” Nghiên cứu khoa học công nghệ CAPACITY IN CASE OF USING BEAMS AND STRONGEST BEAM 10 9.5 9 Beams baed on subspace 8.5 Capacity C(bits/Hz/s) 8 7.5 Strongest beam 7 6.5 6 5.5 0 10 20 30 40 50 60 70 80 90 100 Times of observation Figure 3. Beam types and capacity comparison CONCLUSION The author uses the generalized correlation matrix to find how to form beams in physical multipath environment. Moreover, the author also gives the advantage to increase capacity of the proposed method compared to conventional method using only one beam. REFERENCES [1]. T. K. Moon, W. C. Stirling, Mathematical methods and algorithms for signal processing, Prentice Hall, 2000. [2]. J. Litva, T. K-Y. Lo, Digital beamforming in wireless communications, Artech House, 1996. [3]. C. Brunner, Efficient space-time processing schemes for WCDMA, PhD thesis, Institute for Circuit Theory and Signal Processing, Munich University of Technology, 2000.. [4]. S. A. Jafar, A. Goldsmith "On optimality of beamforming for Multiple Antenna Systems with Imperfect Feedback," IEEE International Symposium, 2001. [5]. G. Jongren, M. Skoglund and B. Ottersten "Combining beamforming and orthogonal space-time block coding," IEEE Transactions on Information Theory, vol.48, issue. 3, pp.611-627, 2002. CÁC KH¤NG GIAN CON PH¸T BøC X¹ TRONG MIMO Kh«ng gian con tÝn hiÖu lµ mét kh¸i niÖm c¬ b¶n vµ ®îc øng dông nhiÒu trong hÖ thèng th«ng tin hiÖn ®¹i. Trong MIMO, kh¸i niÖm nµy cã thÓ ®îc sö dông ®Ó ®a ra c¸c kh«ng gian ph¸t bøc x¹, dùa trªn ma trËn c¸c hÖ sè pha ®inh hiÖn cã t¹i m¸y ph¸t. Bµi b¸o lµm râ c¸c kh«ng gian con dµnh cho bøc x¹ ph¸t cña mét m« h×nh MIMO ®iÓn h×nh. H¬n n÷a, t¸c gi¶ cßn ®a ra ®îc kh¶ n¨ng t¨ng dung lîng khi sö dông c¸c kh«ng gian con cho bøc x¹ ph¸t so víi bøc x¹ truyÒn thèng th«ng qua m« pháng. Từ khóa: Thông tin vô tuyến, hệ thống MIMO, không gian con phát NhËn bµi ngµy 10 th¸ng 4 n¨m 2014 Hoµn thiÖn ngµy 15 th¸ng 9 n¨m 2014 ChÊp nhËn ®¨ng ngµy 25 th¸ng 9 n¨m 2014 Địa chỉ: Khoa Điện- Điện tử, Trường Đại học Giao thông Vận tải Hà nội, Email: hoaitrunggt@yahoo.com, Điện thoại: 0982341176. Tạp chí Nghiên cứu KH&CN quân sự, Số 33, 10 - 2014 57