Flexible paired-orthogonal codes for multi-user MIMO-OFDM

The Journal of China Universities of Posts and Telecommunications February 2011, 18(1): 36–41 www.sciencedirect.com/science/journal/10058885

http://www.jcupt.com

Flexible paired-orthogonal codes for multi-user MIMO-OFDM WANG Ya-chen ( ), FAN Yan-hong, MA Xiao-feng, LI Ye 1. Shandong Computer Science Center, Jinan 250014, China 2. Shandong Provincial Key Laboratory of Computer Network, Jinan 250014, China

Abstract

The work proposes two novel spreading codes, called extensive double-orthogonal code (EDOC) and flexible paired-orthogonal code (FPOC), for multi-user downlink multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The goal of code designs is to obtain an improved bit error rate (BER) performance without loss of bandwidth efficiency. The code designs achieve space diversity by employing multiple antennas at both transmit ends and receive ends. Codes are reasonably spread into time, frequency and space domain. The proposed codes are attractively characterized with length flexibility, which breaks the “power of two” constraint. The simulation results indicate that the proposed codes improve overall BER performance, by exploiting space diversity gains. The flexibility in code length also results in high spectral efficiency. Keywords flexible paired-orthogonality, MIMO, OFDM, multi-user, space diversity

1

Introduction

Future wireless systems are expected to offer much higher data rates and to improve the quality of transmission. As a very promising approach to physical layer, MIMO-OFDM [1] is a very effective solution to both frequency selective fading and significant capacity requirement. To take advantage of multiple antennas, a large amount of research has been made, which started from space-time coding schemes [2] to space-frequency coding [3] and even space-time-frequency coding [4]. Unfortunately, all these literatures focus on single-user scenarios. Due to the existence of multi-user interference (MUI), we cannot expect excellent performance by applying current space-time-frequency coding for single-user scenarios directly to multi-user ones. Code division multiple access (CDMA) technology [5] has been integrated with MIMO to suppress MUI while achieving space diversity in multi-user systems. In Ref. [6], a space-time spreading scheme was proposed and analyzed, where spreading sequences with user symbols are transmitted as the Alamouti scheme. Yang in Ref. [7] Received date: 10-05-2010 Corresponding author: WANG Ya-chen, E-mail: [email protected] DOI: 10.1016/S1005-8885(10)60025-8

proposed an effective MIMO space-CDMA system obtaining an improved capacity with space-time spreading. High efficiency is an urgent requirement for code designs. Current code lengths, constrained to a limited range, are expected to be loosed. Take Walsh-Hadamard (WH) codes for example. The spreading factor is set as multiples of power of two, which is called ‘power of two’ constraint in the paper. Hence the chip rate is limited to two, four, eight, …, times of transmission rate. If a chip rate of 24 units is demanded, 32 units have to be assigned, which incurs a waste of 25 % of the total bandwidth occupied by both sides. Our interest in the paper lies on the spreading code designs that achieve space diversity and are flexible in code length. On the one side, we will investigate the design criterion to obtain orthogonal codes among transmit antennas. On the other side, the ‘power of two’ constraint will be loosed in the code design. The rest of this paper is organized as follows. Firstly we introduce the spreading scheme and propose the extensive double-orthogonal codes (EDOC) in Sect. 2. In Sect. 3, we put forward the flexible paired-orthogonality, according to which FPOC is proposed. Space diversity of FPOC is also investigated. In Sect. 4, code lengths of EDOC and FPOC are analyzed, and BER performances of EDOC and FPOC with conventional spreading codes are compared,

Issue 1

WANG Ya-chen, et al. / Flexible paired-orthogonal codes for multi-user MIMO-OFDM

which is finally followed by the conclusions drawn in Sect. 5.

2 Spreading scheme and extensive double-orthogonal code design 2.1

Space-time-frequency spreading scheme

We consider downlink multi-user MIMO-OFDM using nT transmit antennas and nR

receive antennas. Orthogonal

37

S = N ⊗S .

The first three steps are sufficient to construct EDOC if L = 1 . Clearly the spreading fact of EDOC is multiples of nT2 . Constructing matrices for EDOC are not limited to WH. DOC is obtained when WH codes are used, indicating that the design above is a generalization of double-orthogonal codes.

3

Flexible paired-orthogonal codes

codes with length P are spread in time, frequency and space domains, with space factor nT and time/frequency factor

3.1

Q = P nT . The scheme with nT = 4 is given with bold

We are inspired by Ref. [9] that keeping orthogonal spreading sequences among transmit antennas helps achieving space diversity. Considering the length constraint for spreading codes mentioned in Sect. 1, we put forward the flexible paired-orthogonality as ⎫⎪ SS H = I P (1) ⎬ H S k S k = α I nT ; k = 1,2,..., P ⎪⎭

rectangles in Fig. 1. Multiple transmit antennas here serve to realize space diversity, which is quite different from those in traditional ones that realize multiplexing. Due to abundant researches on time/frequency diversity, we mainly investigate space diversity in the following of the paper.

Flexible paired-orthogonality

where Sk = ⎡⎣ sk ,1

sk ,2 " sk ,Q ⎤⎦ ∈ CnT ×Q , Q≥nT ,

for the conjugate transpose, and α nT = tr ( Sk S

H k

H

)=s

stands H k k

s =1 .

The orthogonality in Eq. (1) is in a similar form with that in Ref. [8], but has different code lengths. The spreading factor in Ref. [8] has to be multiples of nT2 while that in our paper Fig. 1 The spreading ways for double-orthogonal codes and flexible paired-orthogonal codes

2.2

Extensive double-orthogonal code design

The double-orthogonal code (DOC) is firstly proposed in Ref. [8]. Here we put forward an extension of DOC, named EDOC, which pursues a broader application in doubleorthogonality. The code design of EDOC is described as below: Step 1 Generate normalized orthogonal matrices E , F , M ∈ C nT × nT , the norms of whose entries are equivalent to nT−1 .

Step 2 Stack the columns M

vector m, i.e., m = ⎡⎣ m1T

into a nT2 × 1 column T

m2T " mnTT ⎤⎦ , mi denotes the ith column of M, and superscript T denotes the transpose

of a vector or a matrix. Step 3 Compute S = nT m D ( E ⊗ F ) , where ‘ D ’ means the element-by-row multiplication of a vector and a matrix [8]. Step 4 Generate a L × L unitary matrix N, and compute

is multiples of nT , which enables a wider length range of orthogonal codes. The spreading ways for codes with flexible paired-orthogonality is shown by the rectangle with broken lines in Fig. 1, where P = 24 and nT = 4 . Spreading sequences in rows of S k are assigned to corresponding transmit antennas. It is easily seen that as long as the second equation in Eq. (1) is satisfied, the orthogonality among transmit antennas could be obtained. 3.2

Code design

A novel code, named flexible paired-orthogonal code (FPOC), is proposed to undertake a more flexible spreading factor. Before introducing the design steps, we give two definitions as follows. Definition 1 For a matrix A ∈ C N × N , let an denote the nth column of A , n = 1,2,..., N . Matrix An ∈ C N × N is

denoted by An = [ an matrix ⎡⎣ A1T

A ∈ CN

2

an +1 " a N

×N

is

a1 a2 " an −1 ] , and

denoted

by

A = tr ( A ) =

T

A2T " ANT ⎤⎦ .

Definition 2

For matrices A ∈ C( MN )× M and B ∈ C N × N ,

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The Journal of China Universities of Posts and Telecommunications

( a )m be A ( n ) ∈ CM ×M

let

the mth row of A , m = 1,2,..., MN . Denote as a matrix composed from the

M + 1) th row to the

( nM ) th

( ( n − 1) ⋅

row of A , n = 1,2,..., N , i.e. T

T T T A ( n ) = ⎡( a )( n −1) M +1 ( a )( n −1) M + 2 " ( a )nM ⎤ . Let ( b )n be ⎣ ⎦ the nth row of B , and denote a MN × MN matrix by A ⊗M ×1 B =

⎡( A (1) ⊗ ( b ) ) 1 ⎢⎣

T

( A( 2) ⊗ ( b) )

T

"

2

( A( N ) ⊗ ( b) ) N

T

T

⎤ . ⎥⎦

The encoding steps of the proposed FPOC are: Step 1 Construct orthogonal matrices F ∈ CQ×Q and M ∈ C nT × nT , Q≥nT . The norms of each element are Q −1 and nT−1 respectively. Step 2

Construct a Q 2 × Q matrix F by F = Trans ( F ) .

Take the first

nTQ

rows of

F

and compute

S=

( F )1:n Q ⊗Q×1 M . T

Step 3

Q ×1

Partition each column of S to nT vectors with

dimensions, i.e.,

{s

u,n

n = 1,2,..., nT } = par ( su , nT ) ,

u = 1,2,..., nTQ . Construct an nT × Q matrix Su by Su =

⎡⎣ su ,1

T

su ,2 " su , nT ⎤⎦ . Stack the rows of Su into an

nTQ × 1 vector su by su = vec ( Su ) . Step 4

⎡⎣ s1

Construct an nTQ × nTQ matrix S by S =

s2 " snT Q ⎤⎦ .

Matrices F and M are anticipated to be arbitrary in dimensions, so they are not subject to the length constraint. Two typical kinds of orthogonal matrices whose dimensions are not under the ‘power of two’ constraint are discrete Fourier transform (DFT) and HouseHolder matrices, which have been investigated as options for spreading codes [10–11]. Therefore nT and Q can be set any positive integers, as long as link quality is satisfied. Take nT = 4 and Q = 5 for example. Provided that the link quality is satisfied, FPOC with nTQ = 20 preserves 12 units on band for each user, compared to WH code which has to deploy 32 units. Actually FPOC has all properties of DOC in Ref. [8]. While DOC can only be designed by WH matrices, FPOC can be designed by all kinds of orthogonal codes, merely with the requirement that the norms of constructing matrices are Q −1 or nT−1 . For length limitation, we skip the proof of flexible paired-orthogonality of the code design and readers who are interested in it may refer to [12] for details. We take an example to show a clear process of above steps.

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Example Constructing Steps 1–3 of FPOC is shown below with Q = 3 and nT = 2 . Step 1 ⎡ f1,1 ⎢ F = ⎢ f 2,1 ⎢ f 3,1 ⎣ ⎡ m1,1 M =⎢ ⎣ m2,1

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

f1,3 ⎤ ⎥ f 2,3 ⎥ f 3,3 ⎥⎦

f1,2 f 2,2 f 3,2 m1,2 ⎤ m2,2 ⎥⎦

Let fi = ⎡⎣ f1,i j = 1,2 .

f 2,i ⎤⎦

T

(2)

and

( m ) j = ⎡⎣ m j ,1

m j ,2 ⎤⎦ , i = 1,2,3 ,

Step 2

⎡ f1 F = trans ( F ) = ⎢⎢ f 2 ⎣⎢ f3

f3 f1 f2

⎡ [ f1 S = ( F )1:6 ⊗3×1 M = ⎢ ⎢⎣[ f 2 [ s1 s2 " s6 ]

f2 ⎤ f 3 ⎥⎥ f1 ⎦⎥ 9×3 f 2 ] ⊗ ( m )1 ⎤ ⎥ f 3 ] ⊗ ( m )2 ⎥⎦ 6× 6

f3 f1

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ = ⎪ ⎪ ⎪ ⎪⎭

(3)

Step 3

⎡ sk ,1 sk → S k = ⎢ ⎣ sk ,4

sk ,2 sk ,5

sk ,3 ⎤ sk ,6 ⎥⎦ 3× 2

sk = ⎡⎣ sk ,1

sk ,2

sk ,5

3.3

sk ,4

sk ,3

⎫ ⎪⎪ ⎬ T⎪ sk ,6 ⎤⎦ ⎭⎪

(4)

Analysis of space diversity

Set K as the number of active users and xk as the data symbol of the kth user, whose spreading sequence is denoted by sk = ⎡⎣ sk ,1

T

sk ,2 " sk , P ⎤⎦ . Invoke the assumption that

channel fading keeps constant during each OFDM symbol. The channel matrix for user k is Hk =

diag ( H k ,1

H k ,2 " H k ,Q ) ∈ CnR Q× nT Q . Let ε ∈ C P×1 denote

additive white Gaussian noise vector. We have K

yk = H k Sx + ε = H k ∑ sk xk + ε = k =1

⎡ H k ⎢ ∑ skT,1 xk ⎣ k =1 K

where

S = [ s1

and x = [ x1

K

∑ skT,2 xk " k =1

s2 " s K ] ,

T

⎤ skT,Q xk ⎥ + ε ∑ k =1 ⎦ K

sk = ⎡⎣ skT,1

(5)

skT,2 " skT,Q ⎤⎦

T

x2 " xK ] . To investigate the diversity gain T

by the method of pair-wise error probability (PEP) analysis, we reconstruct the system model in Eq. (5) as yk = Chk + ε = diag ( C1 , C 2 ,", CQ ) ⎣⎡ hkT,1 hkT,2 " hkT,Q ⎦⎤ + ε T

(6)

Issue 1

WANG Ya-chen, et al. / Flexible paired-orthogonal codes for multi-user MIMO-OFDM

⎛ K ⎞ where C q = I R ⊗ cqT , cqT = ⎜ ∑ skT, q xk ⎟ , ⎝ k =1 ⎠

included in pair of parentheses on diagonal. The PEP between C and Cˆ (C ≠ Cˆ ) is upper bounded as −1

⎛ 2r − 1⎞⎛ ⎞ ⎛ρ ⎞ p C → Cˆ hk ≤⎜ ⎟⎜ ∏ γ i ⎟ ⎜ ⎟ r ⎝ ⎠ ⎝ i =1 ⎠ ⎝ nT ⎠

(

)

r

( C − Cˆ ) E ( h h ) ( C − Cˆ ) nonzero eigenvalues of ( C − Cˆ ) E ( h h ) ( C − Cˆ )

where r is the rank of

hkT, q = ⎡⎣ hk1,1, q " hk1,,nqT hk2,1, q " hk2,, qnT " hkn,Rq,1 " hkn,Rq, nT ⎤⎦ and diag ( ⋅) indicates a diagonal matrix with entries

−r

(7)

39

k

k

H k

H k

H

H

, γ i is the .

Easily we arrive at E ( hk hkH ) = ( RT ⊗ RF ) ⊗ I nT nR where RT ∈ C M × M

RF ∈ C

Q M ×Q M

[13],

is the temporal correlation matrix,

is the frequency correlation matrix and ‘ ⊗ ’

denotes Kronecker product. Denoting the elements of RT ⊗ RF as rq , q ′ , we have

⎡ r1,1 ( c1T − cˆ1T )( c1* − cˆ1* ) r1,2 ( c1T − cˆ1T )( c2* − cˆ2* ) " r1,Q ( c1T − cˆ1T )( cQ* − cˆQ* ) ⎤ ⎢ ⎥ ⎢ r2,1 ( c2T − cˆ2T )( c1* − cˆ1* ) r2,2 ( c2T − cˆ2T )( c2* − cˆ2* ) " ⎥ H # H ˆ ˆ R = C − C E ( hk hk ) C − C = ⎢ ⎥ ⊗ I nR ⎢ ⎥ # # # ⎢ ⎥ T T * * T T * * T T * * ⎢⎣ rQ ,1 ( cQ − cˆQ )( c1 − cˆ1 ) rQ ,2 ( cQ − cˆQ )( c2 − cˆ2 ) " rQ ,Q ( cQ − cˆQ )( cQ − cˆQ ) ⎦⎥ and r≤ min {rank ( RT ) LnR , QnR } . Here L is the number of high probability of full rank with K > 1 .

(

)

(

)

(8)

path. To achieve the full space diversity at the transmitter, Q should not be less than nT . 1) The case with Q = nT and K = 1 2 −1 2 T * ⎪⎧ x1 − xˆ1 nT ; q = q′ T T * * ˆ ˆ ˆ c c c c s s − − = x − x = ( q q )( q′ q′ ) 1 1 1,q 1,q′ ⎨ ⎪⎩0; q ≠ q′

(9) and the full space diversity achieves with r = nT nR . 2) The case with Q = nT and K > 1 To achieve full space diversity, maximally nT users are served at a moment. Thus K ≤nT and users are assigned

Fig. 2

with spread codes that satisfies skT, q sk*′, q ′ = mq , j mq*′, j ′ f qT f q*′ = 0

The statistic distribution of the rank R ( Q = nT = 4 )

when q ≠ q′ . We have

(c

T q

− cˆqT )( cq*′ − cˆq*′ ) = [ x1 − xˆ1 " xK − xˆ K ] ⋅ ⎡ s1,Tq s1,* q ′ ⎢ T * ⎢ s2, q s1, q ′ ⎢ # ⎢ T * ⎣⎢ sK , q s1, q ′

s1,Tq s2,* q ′ " s1,Tq sK* , q ′ ⎤ ⎡ x1* − xˆ1* ⎤ ⎥⎢ ⎥ s2,T q s2,* q ′ " s2,T q sK* , q ′ ⎥ ⎢ x2* − xˆ2* ⎥ = # # ⎥⎢ # ⎥ ⎥ ⎢ ⎥ sKT , q s2,* q ′ " s2,T q sK* , q ′ ⎦⎥ ⎣⎢ xK* − xˆ K* ⎦⎥

⎧⎛ K T ⎞⎛ K * * ⎞ ⎪⎜ ∑ sk , q xk ⎟⎜ ∑ sk , q ′ xk ⎟ ; q = q′ ⎨⎝ k =1 ⎠⎝ k =1 ⎠ ⎪0; q ≠ q′ ⎩

(10)

Finally R has a full rank of r = nT nR . When more than nT users are served at a moment, full space diversity cannot be always obtained. However, it can be obtained with a very high probability (see Fig. 2). 3) The case with Q > nT R does not have an obvious form of full rank with Q > nT , thus we investigate the statistic distribution of its rank instead. Fig. 3 shows a full rank with K = 1 and a very

Fig. 3

The statistic distribution of the rank R ( Q = 6, nT = 4 )

Above all, the flexible paired-orthogonality has an outstanding property of space diversity gain.

4

Code length analysis and simulation results

4.1

Code length analysis

EDOC and FPOC have an outstanding property in

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The Journal of China Universities of Posts and Telecommunications

flexibility of code length, which saves precious band resource when necessary. Table 1 shows the code lengths that can be obtained by WH code, DOC, EDOC and FPOC within 32. In Table 1, the numbers in the cells denote the code lengths that

2011

can be obtained; ‘X’ stands for the lengths that cannot be obtained; ‘-’ means the lengths that are out of consideration. Table 1 gives a clear description that EDOC and FPOC achieve a wider range of code lengths.

Table 1 Code lengths comparison for DOC, EDOC and FPOC within 32 Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

4.2

WH

DOC

EDOC

FPOC

nT = 2

nT = 3

nT = 4

nT = 2

nT = 3

nT = 4

nT = 2

nT = 3

nT = 4

nT = 2

nT = 3

nT = 4

4 8 X 16 X X X 32 -

X X X X X X X X X X X X X X X X

16 32 -

4 8 X 16 X X X 32 -

X X X X X X X X X X X X X X X X

16 32 -

4 8 12 16 20 24 28 32 -

9 18 27 -

16 32 -

X 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

X X 9 12 15 18 21 24 27 30 -

X X X 16 20 24 28 32 -

Simulation results

In the simulations, the carrier frequency is 3.5 GHz, the channel bandwidth is 5 MHz, and the multipath channel model is six-path typical urban channel [14]. We assume the MIMO channel is spatially independent and identically distributed. The radio bandwidth is divided into 256 OFDM subcarriers, with 54-sample’s guard interval for an OFDM symbol. The vehicular speed is 50 km/h and the maximum Doppler frequency is approximately 162 Hz. Quadrature phase-shift keying (QPSK) constellation and maximum mean square error (MMSE) are used at transmit end and receive end respectively. Chip synchronization and perfect knowledge of channel information at the receive end are both assumed. Figs. 4 and 5 show BER comparison between the novel codes and conventional DFT/HouseHolder codes. Here nT = nR = 4 . In both figures, EDOC and FPOC exhibit

Fig. 4 BER performance among conventional codes, EDOC and FPOC (24≤P≤48)

excellent performance priorities to conventional ones. From the setting of code lengths, we also know that EDOC and FPOC have a wider flexibility in usage. The performance of FPOC with P = K = 24 is quite close to that of EDOC with P = K = 32 ; however, if BER performance is satisfied and spectral efficiency is more cared about, FPOC is preferred.

Fig. 5 BER performance among conventional codes, EDOC and FPOC (40≤P≤80)

5

Conclusions In this work, we put forward two code designs, called

Issue 1

WANG Ya-chen, et al. / Flexible paired-orthogonal codes for multi-user MIMO-OFDM

EDOC and FPOC, for multi-user MIMO-OFDM systems. EDOC is an extension of DOC, while FPOC is a kind of fully novel codes. We have proved that FPOC promises great advantages of exploiting space diversity. Furthermore, it makes an effective contribution in a broader variety of spreading factor. By setting FPOC with a proper length, radio resource is able to be adequately utilized, without losing link quality. Acknowledgements This work was supported by the Shandong Province Natural Science Foundation (ZR2010FQ008).

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(Editor: WANG Xu-ying)