Phase rotation precoding for Vertical-BLAST

Tong Xia Modern Education Technical Department

Minzu University of China Beijing, China

tongxt@sohu.com

Abstract—In order to improve the reliability of VBLAST in block fading channel, phase rotation precoding is proposed. Through phase rotation of the modulated signals at the transmitter, the burst decoding error is changed into random error at the receiver, which can be corrected easily. So the system performance is enhanced. Simulation results demonstrate the validity of the proposed scheme.

Keywords- vblast; phase rotation precoding; block fading; burst error; random error

I. INTRODUCTION The reliability and system capacity of wireless

communication can be significantly improved by MIMO technology [1-5]. However, with the spatial fading correlation at the transmitter, the transmitted signals would interfere with each other, which leads to the degradation of system performance. In this case, the precoding at the transmitter can change the channel characteristic and enhance the system performance [6-10].

[9] have proposed power allocation and phase rotation precoding for 2 2× MIMO system, in which the optimal power and phase are obtained through exhaustive searching. The precoding in [9] can remarkably improve the system performance. However, the computational complexity is so high that it cann’t apply to pratical system. In order to reduced the computational complexity, phase rotation precoding is proposed in [10], in which each transmit antenna has the same power. The scheme is simple and effective.

However, only 2 2× MIMO system is considered in [10] and computational complexity is still very high with large number of transmit antennas. In addition, the channel characteristic has not taken into consideration. In the same fading block, the minimize distance between received signals keeps constant. There possibly exists burst decoding error at the receiver in deep fading channel. So, phase rotation precoding for VBLAST is proposed in this paper. The signals at the transmitter are rotated a certain angle according a sequence. Through phase rotation precoding of the transmitted signals, the burst decoding error is changed into random error, which can be corrected easily. So the system performance is enhanced. Simulation results demonstrate that at the BER of

410− the gain of the proposed code is 1dB-2dB.

II. SYSTEM MODEL

Consider a system with TM transmit antennas and RM receiver antennas. The system model of the proposed scheme is shown in Figure 1.

...s

( )S θ

Figure 1. model of the proposed scheme

Each m bits of information sequence is grouped as vector 1 , m

k k kv v=v , {0,1}k ∈v . Then the modulator maps kv to complex valued signal ( )k ku μ= v in 2m dimension constellation according to the mapping rule.

Next, every TM symbols of sequence ku is grouped as a

vector 1[ , ]T

TMu u=u . Then, u is linearly transformed

according to the phase rotation precoding matrix F to get the vector x with dimension of TM , F and x are expressed as

1

11 1

0

0

[ , ] [ , ]

MT

MT

T T

j

j

jj TM M

e

e

x x u e u e

θ

θ

θθ

=

= = =

F

x Fu

(1)

where iθ denotes the rotation angle of the i th transmitted antenna. iθ is drawn from sequence ( )S θ , which has the feature of pseudo random white noise.

Then, x the is encoded as VBLAST code. The symbols of x are transmitted after series-parallel conversion.

Assuming Rayleigh block fading channel, the received signals can be expressed as

/ 2HT= +r H R x n (2)

where r is 1RM × received signals, H is R TM M× channel matrix, the elements of H are obtained from an independent and standard normal distribution; TR denotes the transmit correlation matrix of dimension R TM M× and / 2H

TR is an upper triangular matrix, obtained by Cholesky decomposition

1480978-1-61284-459-6/11/$26.00 ©2011 IEEE

of TR . n is the noise vector, ( ) 0

2 R R

HM M

NE ×=nn I , ( )E ⋅

denotes mathematics expectation, 0N is the noise power,

R RM M×I presents R RM M× unit matrix.

Let / 2HT=H H R . By substituting (1) into (2), we have

= +r HFu n (3) We know from (3) that the channel matrix is changed owing to the introduction of F and the effective channel matrix is HF .

With all the channel state information and maximum Likelihood (ML) decoder at the receiver, the decoded signal

′u can be expressed as 2arg min′ = −u r HFu (4)

III. RELIABILITY ANALYSIS

Lemma: let [ ]1 2 mA ,A , AA = is a m m× matrix, Ai is the i th column of A . The elements of G are vectors of length m and each element of these vectors is QPSK modulated signals. GE denotes the set of subtraction of any element of G . [ ]11 12 1, T

mw w w= ∈w1 GE and at least two elements of w1 is not equal to zero, such as 11 0w ≠ . If w1 satisfies (5)

{ }arg min ,= ∈w1 Aw w GE (5)

we can get that the angle between 1 11wA and 12

m

i ii

w=

A ,

denoted by 0β , is in the area of 3 5,4 4π π

Proof: to simplify analysis, we assume

1 11 12

m

i ii

w w=

>A A and 1 11 12

m

i ii

w w=

∠ < ∠A A . If 0β is in

the area of 3,4 4π π , the angle between / 2

1 11jw e π−A and

12

m

i ii

w=

A , denoted by 0β ′ , is 0 2πβ + . So, 0β ′ satisfies

03 54 4π πβ ′≤ ≤ . Let / 2

11 12 1,jmw e w wπ−=w2 ∈w2 GE ,

in the following Aw1 and Aw2 are compared.

1 11 12

1 11 1 02

1 11 12

cos

22

m

i ii

m

i ii

m

i ii

w w

w w

w w

β

=

=

=

= +

= −

≥ −

Aw1 A A

A A

A A

(6)

/ 21 11 1

2

1 11 12

22

mj

i ii

m

i ii

w e w

w w

π−

=

=

= +

< −

Aw2 A A

A A (7)

We know from (6) and (7) that

>Aw1 Aw2 (8) (8) can be obtained using the similar method when

034 4π πβ− < < . However, (8) is contradictive to the known

condition. So, 03

4 4π πβ< ≤ cann’t be tenable. Thus, we have

03 54 4π πβ≤ ≤ .

The reliability analysis of the proposed scheme on a case of QPSK is presented in this section.

Given ML decoding, the Pair Error Probability (PEP) that the receiver mistakes u for ( )′ ′≠u u u , 1[ , ]

T

TMu u′ ′′ =u , is

( ) 2

( )4

Qρ ′−

′→ =HF u u

P u u (9)

where ( ) 2 / 212

t

x

Q x e dtπ

∞−=

Let 1[ , ]T

TMe e′ =e = u - u i i ie u u′= − (9) can be

rewritten as 2

( )4

Qρ

=HFe

P e (10)

When the other conditions are the same, 2HFe is the main factor for determining the system PEP. Since ( )Q x is a

decreasing function of x , the less 2HFe is, the higher PEP

is. In what follows, 2HFe for these two codes is analyzed.

A. The analysis of 2HFe with no precoding

With no phase rotation precoding, 2 2F = I × , in the same

fading block, 2HFe is given by 2

21 2 1

2

1 12

,T T

T

T

M M

M

i ii

e e

e e=

=

= +

HFe h h h

h h (11)

where 1 2 TM=H h h h , ih is the i th row of H , 1 Ti M≤ ≤ .

Assuming at least two elements in the set are not equal to zero, there exists 01 0,

T

T

Me e=0e , ∈0e , 01 0e ≠ , which satisfies (12),

{ }2arg min ,i i= ∈0e

e HFe e (12)

When 0e = e , (11) reduces to 2

201 1 0

2

TM

i ii

e e=

= +0HFe h h (13)

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Let 01 01 1e=h h and 02 02

TM

i ii

e=

=h h . We know from the lemma

that the angle between 01h and 02h , denote by β , is

,04πβ π α α= ± ≤ ≤ . To simplify analysis, we assume

01 02h > h . Then, we have

( )22

01 02

22

01 02

cos

1

α

α

= −

≤ − −

0HFe h h

h h (14)

If α is small, (14) can be approximate to

( )2201 02≈ −0HFe h h (15)

the PEP that the receiver mistakes u for 0u + e , is

( )201 02( )

4Q

ρ −=

h hP e (16)

B. The analysis of 2HFe with proposed precodings

In the proposed scheme, 2HFe is given by

1

1

22

1 2 1

2

1 12

, MT

T T

T

i

TjjM M

Mjj

i ii

e e e e

e e e e

θθ

θθ

=

=

= +

HFe h h h

h h (17)

Let 11 1 1

je e θ ′=h h and 22

T

i

Mj

i ii

e e θ

=

′=h h . The angle

between 1′h and 2

′h is not a constant, but changes in the area

of 3 5,4 4π π due to the randomness of the elements of F .

The probability of (16) indeed exists, but very small. We know from Figure 2 that without phase rotation

precoding, the mimimize of 2He keeps constant. However,

the minimize of 2HFe is random variation. In the same

fading block, the average of 2HFe is increased by 0.4

compared with 2He . Thus, the average PEP of the proposed scheme is lower than that of precoding. So, the system performance is enhanced by phase rotation precoding.

From the perspective of channel, channel keeps the same in a fading block. For deep fading, there probably exists burst decoding errors. However, with phase rotation precoding, the fading is accelerated, and the system have the ability to resistance to the effect of deep fading.

0 100 200 3001.5

2

2.5

3

3.5

no precoding

proposed

average of the proposed

Figure . 2HFe and 2He

IV. SIMULATION RESULTS In this section, the performance of the proposed scheme

and the traditional scheme is compared through simulation. Figure 3 and Figure 4 compare the BER curves of the two

schemes for 2 2× and 4 4× MIMO system, perspectively. Random sequence, chaos sequence and m sequence are adopted. We know from these Figure that the less the correlation of these sequences, the better the system performance. The gain of the proposed scheme is 1dB-2dB for 4 4× MIMO system at the BER of 410− .

5 10 15 2010

-4

10-3

10-2

10-1

SNR/dB

BE

R

proposed(charos)

proposed(random)proposed(m)

VBLAST

Figure . of the two schemes in 2 2× MIMO

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0 5 10 15 20

10-4

10-3

10-2

10-1

100

SNR/dB

BE

R

proposed (random)

proposed (chaos) proposed(m)

VBLAST

Figure . of the two schemes in 4 4× MIMO

V. CONCLUSION The proposed scheme is especially suitable for block

fading channel. Analysis and simulation show that the system performance of the proposed scheme is significantly improved in block fading channel. Moreover, the proposed scheme can be applied to system with any number of transmit antennas. So the application range is extended compared to the scheme in [10].

ACKNOWLEDGEMENT The authors would like to thank anonymous reviewers for

their constructive comments.

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