The transmit subspace for MIMO systems

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.

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

i

t j i

i i

e a t

x

1

) 2 (

)

The correlation of this function at times of t and t  kis defined as [1]:

k f j i xx

i

e a k t x t x E k

The correlation matrix for K times of observation is expressed as:









































] 0 [

] 2 [ ] 1 [

)]

2 ( [

] 0 [ ]

1 [

)]

1 ( [(

] 1 [ ]

0 [

xx xx

xx

xx xx

xx

xx xx

xx xx

r K

r K r

K r r

r

K r r

r

R

It can be rewritten to emphasise the influence of subspace:

H

where S is defined as: Ss1 s2 sL in which si,i 1L that is defined as:

]

1

[ j2 f j2 (K 1)f

s

and [ , 2, , 2]

2 2

a diag



Therefore, the subspace of a signal consists of linear combinations of all vectors

L

i

i,  1

s of S Rxx can be then rewritten so as to emphasize the influence of the SVD (Singular Value Decomposition) is defined as:Rxx UΣ VH,where U,V are unitary matrices and Σ is the diagonal matrix where Uu1 u2 uL

When the correlation matrix Rxxis known, the change of the direction of the component signal x i t of the signal x t can be given by the eigenvectors ui,i 1Lof matrix,

U 1 2 extracted from the SVD of Rxx

(1)

(2)

(3)

(4) (5)

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

Figure 1 MIMO model including moving mobile

The channel matrix in the MIMO model in the discrete physical model stated as:

h nmNxM

H  , where h is the connection coefficient between the m th element at the nm

transmit antenna and the nth element at the receive antenna where:

 

L

l

s n s m j j l nm

l R l T

l

e















1

sin ) 1 ( sin







where l is the magnitude of path l,





  2 where  is wavelength of signal, vt  where z

receiver moves

The important relationship between the correlation matrices: rhh,g(p), rhh,q(p) and the

corresponding columns of channel matrix h ,g hqis:

H g g

g

hh, (p) E h (t p)h (t)

r   ; rhh,q(p)E hq(tp)hq(t)H

In the context of the MIMO model in the discrete physical model, rhh,g(p)is equivalent to

)

(

hh

r This indicates that the correlation matrices Rhh,g, Rhh,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

…

Path L

…

T

s

1

sin

T

s

1

z

L



Moving of the receiver

…

R

s

… Path 1

N elements

M elements

L



(6)

(7)

transmit dimensions Given the covariance matrix after K times of observation at the receiver,Rhh, the subspace of the channel vector extracted from this matrix is rewritten as:

S 1 2

where

The information of the phases of the component entriesejl ej( m1)s Tsinl e ju l vt in the

L

l

s m j j l m

l l T

e t











1

sin ) 1 (

subspace at the pth time of observation at the receiver in which the vectors used for giving this information are extracted from sl,l  1Ldefined as:

where f l u l v/2,u l cosl

For the case where l2 0,l1L, these magnitudes of the lth 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 vectorsslp,p1K,l 1L Hence, the optimum weight vectors at thepth time of observation can be rewritten as:

L l

e e

e e

e

e e

e e e e

e e

e

M s j f K

s j f j K

f j K

M s j f j

s j f j

f j M s j

s j

s j

j l

T l l

T l l

l

T l l

T l l

l T l

T l

T l



















































































































1 ,

1

) 1 ( sin 2

) 1 (

sin 2

) 1 (

2 ) 1 (

) 1 ( sin 2

sin 2

2 ) 1 ( sin

2 sin sin











































s

(9)

 

 

L l

K p

e e

e e

e e

p l j T s l M j

p l j T s l j

p l j l

j













































1 , 1 ,

1 2 sin 1

1 2 sin

1 2

















s

L l

K p

e e

e e

e e

T

p f j s M jk

p f j s jk

p f j

j H lp lp

l T l

l T l l















































1 , 1 ,

) 1 ( 2 sin ) 1 (

) 1 ( 2 sin

) 1 ( 2













s w

The Lvectors wlp,p1K,l 1Lare known from the covariance matrix Rhh at

thepth time of observation at the receiver in which the transmitted power is allocated to these

vectors In terms of the Lvectors wlp,p1K,l 1Loffered by the covariance matrix

at the receiver, the array factor (beam patterns) of the vector wlp,p1K,l 1L as

defined in [2]:











M

m

s m j lp lp

T

e m M

AF

1

) sin ) 1 ( (

) (

1 )

where wlp(m),m 1M is the m th entry of vector wlp, lp

1

is the normalized vector of

lp

w

Applying SVD at the receiver to decompose the covariance matrixRhh, i.e Rhh UΣ VH

(when slp,p1K,l 1L are not available at the receiver) leads to the vectors

L

l

l,  1

u of matrix Uu1 u2 uL The productive transmit vector at the pth

observation wlp,l 1L are then uH lp,l  1L, where ulp,l  1L consists of the

1

)

1

(p 

M th to theMpth entries of vector ul,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 150,2 3150,1 1350,2 2250at 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

Velocity of the receiver v 40(km/h)

The spacing between the transmit elements

) ( 5

The number of elements at M 4,N 4

(10)

(11)

The simulation of the beam pattern with different transmit angles1 150,300,450, ,1200,

is shown in figure 2

Figure 2 Simulations of beam patterns when moving the receiver at different transmit

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:

M

h h

 h

(c)1 450

1 30



(d)1  600

(e)1  750 (f)1 900

(12)

where ju vt

L

l

s m jk j l m

l l T

e t











1

sin ) 1 (

For the optimum weight transmit vector for the conventional method, [3] presented it, as;

H

w w



 h w

L

l

s m j j l m

m

l l T

e t

h t











1

sin ) 1 ( 1

At the first observation at the receiver,t 0, this vector can be rewritten as:

H

w w



 h w









L

l

s m j j l m

m

l T

l e e t

h

w

1

sin ) 1 ( 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

2 2 0 1

of signal: 0,1 Distance between two element antennas: s T s R 0,1 Velocity of mobile:

h

km

v40 / Number of observation: K 100 Signal to noise ratio: S/N 5dB

The higher capacity (bit/s/Hz) can be seen in Fig.3

5 10 15 20

30

210

60

240 90

270 120

300

150

330

Beam pattern

transmit angle

5 10 15 20

30

210

60

240 90

270 120

300

150

330

Beam pattern

transmit angle

5 10 15 20 25

30

210

60

240 90

270 120

300

150

330

Beam pattern

transmit angle

(13)

(14)

Beams for two paths

Strongest beam

0 10 20 30 40 50 60 70 80 90 100 5.5

6 6.5 7 7.5 8 8.5 9 9.5 10

Times of observation

CAPACITY IN CASE OF USING BEAMS AND STRONGEST BEAM

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 KHÔNG GIAN CON PHáT BứC Xạ TRONG MIMO

Không gian con tín hiệu là một khái niệm cơ bản và được ứng dụng nhiều trong hệ thống thông tin hiện đại Trong MIMO, khái niệm này có thể được sử dụng để đưa ra các không gian phát bức xạ, dựa trên ma trận các hệ số pha đinh hiện có tại máy phát Bài báo làm rõ các không gian con dành cho bức xạ phát của một mô hình MIMO điển hình Hơn nữa, tác giả còn đưa ra được khả năng tăng dung lượng khi sử dụng các không gian con cho bức xạ phát so với bức xạ truyền thống thông qua mô phỏng

Từ khúa: Thụng tin vụ tuyến, hệ thống MIMO, khụng gian con phỏt

Nhận bài ngày 10 tháng 4 năm 2014 Hoàn thiện ngày 15 tháng 9 năm 2014 Chấp nhận đăng ngày 25 tháng 9 năm 2014

Địa chỉ: Khoa Điện- Điện tử, Trường Đại học Giao thụng Vận tải Hà nội,

Email: hoaitrunggt@yahoo.com ,

Strongest beam Beams baed on subspace