PERFORMANCE OF MC-CDMA SYSTEMS WITH SPACE-TIME BLOCK CODES IN FREQUENCY SELECTIVE FADING CHANNELS M.Valipour, H. Shafiee Department of Electrical and Computer Engineering University of Tehran, Tehran, Iran ABSTRACT In this paper, the performance of a space time coded multi-user MC-CDMA wireless communication system in frequency selective fading channels is analyzed. The space time code is a block code with a 4× 4 transition matrix. Analytical and simulation symbol error rate results are obtained for the system with two transmit antennas. The analysis also shows that a more relaxed set of orthogonality conditions are sufficient for the spreading sequences. 1. INTRODUCTION Multi-carrier code-division-multiple-access (MCCDMA) is currently being considered as a suitable modulation technique for high rate communication in wideband wireless networks. MC-CDMA combines the benefits of CDMA in a multi-user environment with that of multi-carrier modulation for high rate data transmission over frequency selective fading channels. In principle, the modulated signal is spread in the frequency domain using a spreading sequence and then transmitted over orthogonal sub-carriers. With spreading codes being orthogonal, MC-CDMA allows the interference from other users to be completely canceled in an ideal, i.e., additive white Gaussian noise, (AWGN), channel as well as in a flat fading channel. Modulation over orthogonal sub-carriers, on the other hand, provides for robust and convenient demodulation of spread symbols, as the fading for each sub-channel can be considered to be nonselective.

analyze the performance of a ST block code with a 4× 4 transition matrix which is applied to a system with two transmit antennas. We derive relations for the bit error rate (BER) performance of a multi-user system in frequency selective fading channels. The analysis also shows that a more relaxed set of orthogonality conditions are sufficient for the spreading sequences. The organization of the rest of this paper as follows: in Section II and III, the mathematical model for the STcoded multi-user CDMA system of interest is presented and relations for the transmitted and received signals in a frequency selective fading wireless channel are obtained. The performance of the system is analyzed in Section IV and relations for the system error rate are derived. In Section V, the results of BER simulations are presented. 2. SYSTEM MODEL Figure 1 shows a generic block diagram for the ST coded system considered. The number of transmit and receive antennas are taken to be 2 and N RX , respectively. The number of users in the system is dented by N u . We employ the space-time block code given in [2]. Here, if the block of three input symbols are denoted as [d1 d 2 d 3 ]T , the code produces four 4-tuple set of symbols as shown in the matrix below: ⎡ ⎢ d1 ⎢ ⎢− d 2* ⎢ ⎢ d3* ⎢ ⎢ 2* ⎢ d3 ⎢ 2 ⎣

d2 d1*

d3 2 d3 2 − d1 − d1* + d 2 − d 2* 2 d 2 + d 2* + d1 − d1* 2

d3 ⎤ ⎥ 2 ⎥ d3 ⎥ − ⎥ 2 − d 2 − d 2* + d1 − d1* ⎥ ⎥ 2 ⎥ * * − d1 − d1 − d 2 + d 2 ⎥ ⎥ 2 ⎦

To provide diversity and coding gains, the application of space-time (ST) codes in multiple-antenna-array structure transmitters has been under intensive investigation in recent years. Both space-time block codes (STBC) and trellis codes (STTC) have been used in multi-input-multi-output (MIMO) wireless structures. A simple and effective ST block code was introduced in [1] for a 2 × 2 MIMO structure, i.e., one with two transmit and two receive antennas. This so-called Alamouti code improves error rate performance without any power or bandwidth penalties. Space time block codes with four transmission antennas have also been developed and shown to increase the diversity gain [2].

In a multi-user environment, the modulated symbols for the mth user are grouped into 3-symbol blocks and applied to the ST encoder. If dm[n] denotes the modulated

The application of ST codes to MC-CDMA systems has also been considered although this has largely been limited to the case of Alamouti code [3]. In this paper, we

symbol for the mth user at time n, vector d [n] is formed by three input symbols to the encoder as defined below, i.e.,

d3* 2 d3* − 2

The first column in the above matrix corresponds to symbols generated at four spaces in the first time-interval. The second column denotes symbols transmitted in second time interval and so on.

m

transform (IDFT) operation is performed on the two resulting 2P-sample blocks. The time-domain block of samples (i.e., those at the output of the two IDFT blocks) are denoted as x im,1 [ n] and x im, 2 [ n] . Combining all the operation performed above on the ST-coded symbols, we can write:

[d [d

]

[3n] d m [3n + 1] d m [3n + 2] T m d 2m [n] d 3m [n] 1 [ n] m

]

T

m

1 ≤ i, l ≤ 4 where i and l correspond to time and space,

respectively. For each l , the set of (time-domain) symbols generated are grouped together as:

[

s 2m,l

s 3m,l

s 4m,l

]

T

1≤ l ≤ 4

,

x im, 2 [ n] = FoH C 2m sim, 2 [ n ] + FeH C m4 sim, 4 [ n]

(2)

Fu ,v =

We will denote each encoded symbol by s i ,l [ n]

s lm = s1m,l

(1)

where matrix Fe and Fo are matrices obtained from the odd and even rows of the Fourier matrix F , respectively. F is a matrix of size 2 P × 2 P for which the element at the uth row and vth column is given by

Figure 1: Block diagram for the transmitter

d m [ n] = =

xim,1[n] = FoH C1m sim,1[n] + FeH C3m sim,3[n]

⎛ 2π (u − 1)(v − 1) ⎞ exp⎜ − ⎟ 2P 2P ⎝ ⎠ 1

A complete multi-carrier modulated block is formed by adding a cyclic prefix at the beginning which is utilized so that the effect of inter-symbol interference is removed. After this operation and parallel to serial conversion, the two generated sequences are upconverted to the desired carrier frequency and transmitted by antennas 1 and 2, respectively.

Each encoded symbol is now multiplied by a spreading sequence. The spreading code (of length P) for th

[

]

± 1 / 4 P . Recall that in a typical CDMA system, the spreading codes for all users are ideally orthogonal, so that the interferences from all transmitters other than the intended user can be perfectly canceled when the channel is ideal.

In general, in an MC-CDMA system, the number of transmit antennas may or may not be the same as the number of spaces allocated by the space-time code. In the following, with the 4 × 4 ST code used, the number of transmit antennas is chosen to be two. However, the methodology presented is general so that it could be applied to other cases as well. With two antennas available, the four spread sequences at time n, are grouped into two set of samples organized in the two vectors shown below:

[C

] [ T

]

mT m mT m s [n] C 3mT sim,3 [n] , C 2 s i , 2 [n] C 4 s i , 4 [n]

mT m 1 i ,1

3. RECEIVER STRUCTURE

th

the l space for the m user is given by C lm = c lm,1 K c lm, P T , where each term has a value of

T

Note that we have opted to place the first and third sequences in one group, and the other two sequences in the other group. An interleaver operates on each of the vectors above by sequentially taking one sample from each of its two sub-blocks. This operation helps to mitigate the impact of a deep null in the channel fading response, as fewer adjacent symbols are affected after despreading in the receiver. To modulate the resulting block of symbols on orthogonal sub-carriers, an inverse discrete Fourier

(3)

When the signal is transmitted through a frequency selective fading channel, each receiver antenna picks up the signal from the desired user as well as the interference from the other N u − 1 users. Assuming that the cyclic prefix is removed, a DFT operation is performed on each set of received symbols. The output samples at the output of the DFT block for the jth antenna which corresponding to the ith time for the nth ST-coded block can be written as: y i , j [n] =

Nu

2

∑ ∑ F H mk , j x im,k [n] + w i , j [n]

(4)

m =1 k =1

where index k shows the summation due to two transmit antennas for each user and w i , j [ n ] represents noise. Here, H k , j is a matrix whose elements represent channel m

coefficients from the kth transmitter antenna of the mth user to the jth receiver antenna. Specifically, if we take the length of each of the frequency-selective fading channel responses to be L, H mk , j x im, k [n] produces the timedomain samples at the output of the related channel which is due to the (circular) convolution of the input samples with the channel response. Furthermore, matrix F is again the Fourier matrix which when applied on the resulting vector above, the block of received samples in the frequency domain is obtained. By combining samples from the four time instants of the nth ST-coded block, we form the complete set of samples for the jth receiver antenna, i.e.,

[

]

Y j [ n] = y 1T, j [n] y T2, j [n] y T3, j [n] y T4, j [ n]

T

To make the notation more compact, we combine the inverse Fourier transform at the transmitter, channel response and the Fourier transform operation at the receiver, such that Φ mk , j = F H mk , j F H . With the effect

from all antennas need to be added together. Thus,

of cyclic prefix taken into account, it can be readily verified that Φ mk , j is indeed a diagonal matrix which can

where Ψ v [n] = Θ v ,1

ϕ

be written as Φ mk , j = Diag (

ϕ

the vector

m k,j

m k,j

⎛ ⎝

φkm, j (v) = ∑ hkm, j [r ] exp⎜ − J r =0

Y[n] = Ψ1[n]D[n] + Ψ 2 [n]D *[n] + W[n],

[

) , where each element of

is given by

L −1

N Rx receiver antennas, the contributions

If there are

2πvr ⎞ ⎟ 2P ⎠

(5)

m

where 0 ≤ v ≤ 2 p − 1 , J = − 1 and hk , j [ r ] corresponds

T

Θ v,2

T

... Θ v , N Rx

(9)

].

T T

4. PERFORMANCE ANALYSIS Let us first consider the case of one receiver antenna. Without loss of generality, we assume that the first user is the desired user. Starting with Equation (7), it is shown in Appendix B that the estimate of d [ n] denoted by 1

dˆ 1[n] is given by

to the time-domain channel fading response coefficients. By forming new matrices Φ mk ,,oj and Φ mk ,,ej using odd and

H T dˆ 1 [n] = θ11, j Y j [n] + θ12, j Y*j [n] .

(10)

even columns of matrix Φ mk , j , Equation (4) can be written

The above relation can be written as: 1 ˆ1 dˆ 1[n] = Q1 d1[n] + Wmu , INT [n] + W j [n]

(11)

using the space-time coded symbols, i.e., M

where Q1 =

2

y i , j [n] = ∑∑ Φ mk ,,oj C mk sim, k [n] + Φ mk ,,ej C mk+ 2 sim, k + 2 m =1 k =1

(6)

+ w i , j [ n] To be able to derive relations for the performance of the receiver, we further need to write the expression above for the input symbols to the ST block encoder using the ST code transition matrix. It can be shown that the vector, Y j [n] , which as defined above contains samples due to all four time intervals of the nth ST-coded block can be written as: Nu

m =1

(

m

m 2, j

m

ˆ 1 [ n] = θ 1 H W [ n ] + θ 1 T W * [ n] . W j j j 1 2

(12)

For BPSK modulation, the decisions are made based ˆ 1[ n]) on Re( dˆ 1[ n ]) . The covariance matrix of Re( W j can be shown to be Q 1σ n2 I 3 / 2 . As shown in Appendix C, the interference terms in Wmu , INT [n] are 1

shown to be approximately Gaussian, with the variances given by [β mu ,1 β mu , 2 β mu ,3 ]T . These terms, which are 1

)

Y j [ n] = ∑ θ d [ n] + θ d * [ n] + W j [ n] m 1, j

1 ⎛ 1 H 1 1 H 1 ⎞ ⎜ ϕ1, j ϕ1, j + ϕ 2, j ϕ 2, j ⎟ , and 4P ⎝ ⎠

(7)

where θ1m, j and θ m2, j are matrices of size 8 P × 3 which clearly depend on the structure of the ST encoder, but additionally on the spreading codes and the channel responses. For the particular ST block encoder used in this paper, the two matrices are given in Appendix A. W j [n] is also a vector which, similar to Y j [n] , is constructed by combining the noise vectors w i , j [ n] ,

1≤ i ≤ 4.

dependent on Q are computed in Appendix C. In addition, the probability distribute function of Q1 can be shown to be L −1 L −1 ⎛ 2q ⎞⎤ 2Π Π ⎡ ⎛ 2q ⎞ PQ1 (q1 ) = ∑∑ 2 n m2 ⎢exp⎜⎜ − 21 ⎟⎟ − exp⎜⎜ − 21 ⎟⎟⎥ n =0 m=0 σ n − σ m ⎣ ⎢ ⎝ σn ⎠ ⎝ σ m ⎠⎦⎥ (13) n≠ m ⎛Π + ∑ 4⎜⎜ 2n n =0 ⎝ σ n L −1

where Π n =

2

⎞ ⎛ 2q ⎞ ⎟ q1 exp⎜ − 21 ⎟ ⎟ ⎜ σ ⎟ n ⎠ ⎠ ⎝

σ n2 ∑ 2 2 m =0 σ n − σ m L −1

for n = 0,..., L − 1

m≠ n

By combining θ mk , j matrices as well as d m [n] vectors together, the above relation can be equivalently written as follows: Yj [n] = Θ1, j D[n] + Θ 2, j D *[n] + Wj [n]

(8)

where Θ v, j and D [ n] are given by

[

]

θ v2, j K θ vN,uj , v = 1, 2 ,

Θ v , j = θ1v , j

D [n] = ⎡d 1 [n] d 2 [n] K d ⎢⎣ T

T

Nu T

[n]⎤ ⎥⎦

T

With the estimates of the transmitted symbols and the noise variances available, the system bit error rate can be obtained as given below: ⎡σ 2 q ⎤ ⎞⎞ 1 ∞⎛ 3 ⎛ Pe = ∫ ⎜⎜ ∑ Q⎜ q12 / ⎢ n 1 + β mu ,i|q ⎥ ⎟ ⎟⎟ PQ ( q1 ) dq1 (14) ⎟ 3 0 ⎜ i =1 ⎜ ⎣ 2 ⎦ ⎠ ⎟⎠ ⎝ ⎝ where Q denotes the complementary error function. We can arrive at Equation (11), if we assume that all spreading sequences from all users are orthogonal. But this is not necessary as the orthogonality conditions can be relaxed without any adverse impact on performance. Specifically, the following orthogonality requirements for 1

1

the spreading codes for the two users m1 and m2 are indeed sufficient:

C lm1H C lm1 = 0.25 ,

l = 1, K ,4 , ∀ m1

(15a)

C lm1H C lm2 = 0 , C1m1H C m2 2 = 0 , C m1H C m2 = 0 ,

l = 1,K,4 , m1 ≠ m2

(15b)

∀ m1 , m 2

(15c)

∀ m1 , m 2

(15d)

4

1E+0

In the case of N RX receiver antennas, Equation (10) will be generalized to the following relation: H T dˆ 1[n] = Λ11 Y[n] + Λ12 Y * [n]

[

where Λ = θ 1 v

1 T v ,1

θ

1 T v,2

(16)

... θ

T 1 v , N Rx

], T

v = 1, 2 .

One user Tw o users

1E-1 Bit Error Rate

3

antennas. Relations for the bit error rate performance of the system in frequency selective fading channels were derived. Simulation results obtained for different cases showed excellent agreement with analytical relations obtained. The analysis also showed that a relaxed set of orthogonality conditions for the spreading codes can be considered.

The

Four users Flat Channel (Theory)

1E-2

1E-3

1E-4

relation for the BER can be similarly obtained. 1E-5

RESULTS AND DISCUSSION

Figure 3 shows analytical and simulation bit error rate plots vs. SNR for a flat fading channel as well as for frequency selective channels of length two and three. Channel coefficients are normalized so that the total power is all channels are the same. Excellent agreement between analytical and simulation results is observed. Notice that as the channel length is increased, the resulting additional diversity improves the error rate performance. For the 3-tap frequency selective fading channel, Figure 4 shows simulation as well as analytical results for different number of users. Note that the channel now disrupts the orthogonality of the codes, and hence, the performance is degraded as the number of users is increased. Note that, for this channel response, the performance with one or two users is almost the same. 6. CONCLUSIONS The performance of a space time coded MC-CDMA communication system was analyzed for the case of a ST block code with a 4x4 transition matrix and two transmit

5

10 SER

15

20

Figure 2: BER vs. SNR for different number of users in Raleigh flat fading channel

1E+0 Flat channel 2-tap channel

1E-1

Bit Error Rate

In this section, we present the results of the error rate simulations of an MC-CDMA system, which uses the ST block code described in Section II and the structure shown in Figure 1. The binary bit sequence is BPSK modulated. Walsh-Hadamard sequences of length 16 with the orthogonality conditions as given in Equations (15a) to (15d) are used for spreading of the encoded symbols. With the length of the spreading sequence set at 16, the input blocks to the IDFT and DFT are therefore, of size 32. Figure 2 shows plots of bit error rate (BER) vs. signal to noise ratio (SNR) for flat fading Rayleigh channel when the number of users in the system is taken to be one, two and four. It is observed that regardless of the number of users, the system produces the same bit error rate. This indeed confirms that orthogonality conditions stated previously result in complete cancellation of interference from other users in the flat fading channel.

0

3-tap channel

1E-2 1E-3 1E-4 1E-5 0

5

10

15

SER

20

Figure 3: BER vs. SNR for a single user for different frequency selective channels. Solid lines with filled markers correspond to analytical results; dotted lines with hollow markers correspond to simulation results 1E+0 One user Tw o users

1E-1

Bit Error Rate

5.

Three users Four users

1E-2 1E-3 1E-4 1E-5 0

5

10

SER

15

20

25

Figure 4: BER vs. SNR for a frequency selective channel for different number of users. Solid lines with filled markers correspond to analytical results; dotted lines with hollow markers correspond to simulation results

Appendix A m m For the ST block code of Section I, θ1, j and θ 2, j are

where Re( a11,,1m ) = Re( a 12,,m2 ) = Re( a 31,,m3 ) , and

⎡ 0 ⎢ * = ⎢− b11,,2m ⎢ 0 ⎣

given as follows.

θ1m, j

θm 2, j

Φ1m, ,jo C1m ⎡ ⎢ ⎢ ⎢ 0 ⎢ = ⎢ m,e m m ,e m ⎢ −Φ1, j C3 + Φ 2, j C 4 2 ⎢ ⎢ ⎢ Φ1m, ,jeC3m −Φ 2m,,jeC 4m ⎢⎣ 2

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ,o m Φ1m, ,jo C1m + Φ m 2, j C2 ⎥ 2 ⎥ ⎥ Φ1m, ,jo C1m − Φ 2m,,jo C 2m ⎥ ⎥⎦ 2

0

0

Φ m , o C1m 1, j

,e m Φ1m, ,je C 3m + Φ m 2, j C 4 2 − Φ1m, ,je C 3m − Φ 2m,,je C 4m 2

⎡ 0 ⎢ ⎢ ⎢ ⎢ Φ m2,,joC m2 ⎢ ⎢ =⎢ ⎢ − Φ1m, ,jeC3m − Φ m2,,je C m4 ⎢ 2 ⎢ ⎢ m ,e m ⎢ − Φ1, j C3 − Φ m2,,je C m4 ⎢ 2 ⎣

Φ C +Φ C ⎤ ⎥ 2 ⎥ ⎥ m ,e m m ,e m Φ1, j C3 − Φ 2, j C 4 ⎥ ⎥ 2 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎦ m ,e 1, j

− Φ m2,,joC m2

0 − Φ1m, ,jeC3m + Φ m2,,jeC m4 2 −Φ C +Φ C m ,e 1, j

m 3

m ,e 2, j

m 4

2

m 3

m ,e 2, j

) )D *[n] + Wˆ [n]

Θ 2, j + θ

1 2, j

Θ

* 1, j

[ [B

dˆ 1 [n] = A1j,1 1,1 j

] ]D *[n] + Wˆ [n]

Δ

Δ

[ ]

B1j,m = θ11, j θ 2m, j + θ11, j θ 2m,*j = bi1,,mj H

T

. (B.2)

1 j

[ ]

where A1j,m = θ11, j θ1m, j + θ12, j θ 2m,*j = ai1,,mj T

Re( w

(C.1) Using Appendix A and Equation (B.3), we can write: L −1 L −1

var(Re(a11,,1m ) | q1 ) = ∑∑

[(α

3×3

(B.3)

(

)

* ˆ 1 [n] dˆ 1 [ n] = Q 1d1 [n] + ∑ A 1j,m d m [n] + B 1j,m d m [n] + W j

Then,

A

1, m j

a

1, m 2, 2

0

)

]

(C.2)

2 2 σ t2 ⎛ E ⎜ X = h1m (k ) + h2m (k ) Q1 = q1 ⎞⎟

⎝

2

) + (α smt −, nk ,1,1 ) 2

m , n ,1,1 2 ck −t

Δ 2P

here α cm , n,v , w = ∑ C sm,v C sn , w cos( l

where

⎠

s =1

⎠

]

(C.3)

2π ls ) , m, n = 1,..., Nu 2P

2π ls ) v, w = 1,2 2P

[

Cm ,1 = c1m,1 , c3m,1 , c1m, 2 , c3m, 2 , K clm, P c3m, P

[

] = [C T

m ,1 1

] [

,..., C2mP,1

]

T

Cm, 2 = c2m,1 , c4m,1 , c2m, 2 , c4m, 2 , K c2m, P c4m, P = C1m , 2 ,..., C2mP, 2 From Equation (C.1), the variance of components of, 1 can be found to be: Wmu , INT [ n]

m=2

.

(B.4)

a11,,2m

m , n ,1, 2 2 st − k

Nu

(B.2), we can show:

⎡ a11,,1m ⎢ * = ⎢− a11,,2m ⎢ 0 ⎣

) + (α

[(α

β mu ,3 = ∑ [var(Re(a11,,1m )] Nu

m=2

θ1m, j and θ m2, j given in Appendix A. using Equation

m=2

⎝

2

m , n ,1, 2 2 ck − t

β mu , 2 = β mu ,1 = ∑ [var(Re(a11,,1m ) + var(Re(a11,,2m + b11,,2m ))]

3×3

For the expression above, we can now use the matrices

Nu

k =0 t =0

2 2 σ t2 ⎛ E ⎜ X = h1m (k ) + h2m (k ) Q1 = q1 ⎞⎟

k =0 t =0

l

A1j, 2 K A1j, N Rx D[ n] +

H

⎡ Re(a11,,1m ) Re(a11,,2m + b11,,2m ) 0 ⎤ ⎡d1m [ n]⎤ ⎢ ⎥⎢ ⎥ 1, m 1, m 1, m ) = ∑ ⎢− Re(a1, 2 + b1, 2 ) Re(a1,1 ) 0 ⎥ ⎢d 2m [ n]⎥ m=2 1 , m m ⎢ 0 0 Re(a1,1 ) ⎥⎦ ⎢⎣d3 [ n]⎥⎦ ⎣ Nu

1 mu , ISI

Δ 2P

1 j

B1j, 2 K B1j, N Rx

From Equation (B.4), we have:

α sm,n ,v ,w = ∑ C sm,v C sn ,w sin(

(B.1)

(B.6)

Appendix C

s =1

Using the definition of Θ v, j , this can further be written as:

m = 1,K, N u

1,1 ˆ 1 [n] then follows. and B j = 0 .The relation for d

var(Re(a11,,2m + b11,,2m ) | q1 ) = ∑∑

H T dˆ 1 [n] = θ11, j Θ1, j + θ12, j Θ *2, j D[n] + T

0 0

L −1 L −1

In this appendix, the derivation of the expression for dˆ 1 [ n] is outlined. Let us use the expression given for Y j [n] in Equation (8) in Equation (10). This gives:

1 H 1, j

0⎤ ⎥ 0⎥ , 0⎥⎦

Using the orthogonality requirements of the codes given in Equations (15a) to (15d), we can show A 1j,1 = q 1I 3

m 4

Appendix B

( (θ

B

1, m j

b11,,2m

0 ⎤ ⎥ 0 ⎥ m = 1,K, Nu a 31,,m3 ⎥⎦

(B.5)

REFERENCES S. M. Alamouti, “A simple transmit diversiy technique for wireless communications”, IEEE J. Seclt. Areas Comunin., vol. 16, pp. 1451-1458, Oct. 1998 [2] V. Tarokh, H. Jafarkhani, A. R. Calderbank, "Space-time block codes from orthogonal designs" IEEE Trans. Inform. Theory, vol. 45, pp. 1456-1467, July 1999. [3] X. Hu, Y. H. Chew, "Performance of space-time block coded MC-CDMA system over frequency selective fading channel using semi-blind channel estimation technique," IEEE WCNC 2003, volume 1, pp. 411-415. [1]

]