BER analysis of MPSK space-time code with differential detection over correlated block-fading Rayleigh channel

THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volume 15, Issue 2, June 2008

ZOU Yu-long, ZHENG Bao-yu

BER analysis of MPSK space-time code with differential detection over correlated block-fading Rayleigh channel CLC number

TN929.5

Document

A

Abstract MIMO technology proposed in recent years can effectively combat the multipath fading of wireless channel and can considerably enlarge the channel capacity, which has been investigated widely by researchers. However, its performance analysis over correlated block-fading Rayleigh channel is still an open and challenging objective. In this article, an analytic expression of bit error rate (BER) is presented for multiple phase shift keying (MPSK) space-time code, with differential detection over correlated block-fading Rayleigh channel. Through theoretical analysis of BER, it can be found that the differential space-time scheme without the need for channel state information (CSI) at receiver achieves distinct performance gain compared with the traditional nonspace-time system. And then, the system simulation is complimented to verify the above result, showing that the diversity system based on the differential space-time block coding (DSTBC) outperforms the traditional nonspace- time system with diversity gain in terms of BER. Furthermore, the numerical results also demonstrate that the error floor of the differential space-time system is much lower than that of the differential nonspace-time system. Keywords differential space-time block code, correlated blockfading Rayleigh channel, CSI, error floor

1

Introduction

As already well-known to all, the multipath effect, which is the intrinsic property of wireless channel, is a major factor of hindering the improvement of quality of service (QoS). Moreover, the destination cannot correctly demodulate the received signal transmitted from the source when the channel fading is deep. Consequently, many diversity techniques are proposed to enhance the reliability of communication system, among which the space-time diversity method is applied broadly to current wireless communication systems. The reason why the multiple antenna system usually employs the

Article ID 1005-8885 (2008) 02-0018-08 space-time code is that the space-time code can considerably enhance the transmission rate by making full use of space and time resources. Thus far, the space-time code can be classified into space-time trellis code (STTC) and space-time block code (STBC). STTC can provide both diversity gain and coding gain as possible, regardless of the cost of the transmitted bandwidth. Meanwhile, its decoding complexity increases exponentially with the increment of transmission rate when the number of receiving antenna is fixed. For the purpose of resolving this shortcoming of STTC, Alamouti [1] presented STBC, which was applied to the double-antenna system, and then Tarokh [2] explored a space-time block coding scheme from the orthogonal principle, which can be applied to the arbitrary antenna number. Although the performance of STBC is inferior to STTC [3], the decoding complexity of STBC is much lower than that of STTC. Thus, STBC is adopted gradually by next-generation wireless networks (e.g., WSN, MANET, and Beyond 3G). All the above works assume that the receivers have perfect knowledge of the CSI through some channel estimation method. This leads to a very complex system when many network nodes are involved since the channel estimation is needed for each pair of communication nodes. So the consideration of system design without the knowledge of CSI at receiver becomes more realistic in some situations. Differential modulation and detection technique compliment the communication without CSI. On common differential binary phase shift keying (BPSK) modulation scheme, such a simple differential detection rule incurs 3 dB performance loss compared with the coherent detection over AWGN and independent Rayleigh flat-fading channel [4]. With regards to temporal correlated Rayleigh fading channel, the result is not the case. There exists an irreducible error floor in the high SNR region as mentioned in Ref. [5]. For DBPSK, § E · 1 U lim Pe ¨ s2 ¸ Es 2 ©V ¹ of 2 V

Received date: 2007-09-06 ZOU Yu-long ( ), ZHENG Bao-yu Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003, China E-mail: [email protected]

where U denotes correlation coefficient between the fading gains at two consecutive symbol intervals. In Ref. [6], the authors illustrated the difference between the independent

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ZOU Yu-long, et al.: BER analysis of MPSK space-time code with differential detection over… where {hk , id }i

19

denotes the fading coefficient from the

Rayleigh fading channels and correlated Rayleigh fading channels and gave an approximate BER performance of FSK modulation over correlated Rayleigh fading channel. Then a closed-form expression for bit error probability of MDPSK over the nonselective correlated Rayleigh channel was given by Kam in Ref. [7]. As to differential space-time modulation, a differential detection scheme was proposed by Tarokh and Jafarkhani [8] for exploiting diversity gain. Soon later, Hughes [9] put forth a general approach for differential modulation for multiple transmit antennas based on unit space-time group code. Then, the BER performance of MPSK space-time block code with differential detection was given by Gao and Haimovich over the independent block-fading Rayleigh channel in Ref. [10]. In practice, however, the channels are temporally correlated, which can be described by Jakes’ model given in Ref. [11]. Therefore, BER analysis of MPSK space-time block code with differential detection over the correlated block-fading Rayleigh channel is still an open challenge. This article will research the differential space-time code over the temporal correlated block-fading Rayleigh channel. The remainder of this article is organized as follows: Section 2 describes the system model and illustrates in detail the implementation of the MPSK-based DSTBC scheme. Section 3 deduces the closed-form expression of BER for MPSK space-time code with differential detection over correlated block-fading Rayleigh channel. In Sect. 4, computer simulation is conducted to show that our proposed differential scheme achieves distinct diversity gain. Finally, the concluding remarks are given in Sect. 5.

between the fading gains at two consecutive space-time block intervals is given by U b E  hk , id hk*1, id J 0  4ʌBdTs , where

2 Implementation of space-time block code with differential detection

bk , i

2.1

System model

Here, we consider a wireless communication system with one source node that is equipped with double antenna ( S1 and S 2 ) and one destination node with single antenna (d), whereas

the antenna at any node can be used for both transmission and reception. And the system model we use is shown in Fig. 1.

Fig. 1 Diversity system model-based space-time code

At the destination node, the received signal at the first slot of block k can be expressed as (1) rk , 1 hk , 1d sk , 1  hk , 2 d sk , 2  nk , 1

1, 2

transmitting antenna i to the receiving antenna d at the block k. Also, the fading coefficients are regarded as constant during a correlation time. Since the correlation time is far more than a space-time block interval including two symbol periods, the block-fading channel can be utilized to analyze the system performance. Throughout the article, we make the following assumptions: 1) All the wireless channels are molded as correlated block-fading channel, which are temporally correlated in the time domain. In detail, the fading gains are invariable during a space-time block interval. But the fading gains vary with the different block intervals, and the correlation coefficient Ub

BdTs is the symbol-rate normalized Doppler shift. 2) nkˈ1 is the complex white Gaussian noise with zero mean and power density N 0 . 3) All the fading channels are independent in space, namely, all hk ,id are independent for each i. 4) Both antenna S1 and S2 transmit signal with unit power, i.e., Es 1 . In addition, the italic characters in this article represent scalar variables and bold italic characters denote vector variables or matrix variables. 2.2 Implementation of the differential space-time scheme

Let the MPSK symbol at the time slot i of block interval k be e

j

2 mʌ M

where j

; m

0, 1,..., M  1

(2)

1 . The differential block code ( sk , 1 , sk , 2 ) is

recursively defined as bellow § sk 1, 1 sk 1, 2 · §1 1· ( sk , 1 sk , 2 ) (bk , 1 bk , 2 ) ¨ * ¸ ; S0 = ¨ ¸ (3) * ©1 1 ¹ ©  sk 1, 2 sk 1, 1 ¹ After the Alamouti space-time coding to the differential block code ( sk , 1 , sk , 2 ), the sending matrix S k can be written as following expression §s  sk*, 2 · S k ¨¨ k , 1 ¸¸ * © sk , 2 s k , 1 ¹ Combining Eqs. (3) and (4), we can have §s  sk* 1, 2 ·§ bk , 1 bk*, 2 · Sk S k 1Bk ¨¨ k 1, 1 ¸¨ ¸¸ * * ¸¨ © sk 1, 2 sk 1, 1 ¹© bk , 2 bk , 1 ¹

(4)

(5)

To simplify the expression, the time slot assignment of differential transmitting scheme is illustrated specifically in Table 1.

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The Journal of CHUPT Table 1

Time-slot assignment of transmission scheme

Time slot

Slot 2

S1 Ј d

Slot 1 sk , 1

S2 Ј d

sk , 2

sk*, 1

I1

 sk*, 2

II1

As shown in Table 1, the whole transmitting process is separated into two consecutive symbol periods (phases), i.e., time Slot 1 and time Slot 2. During the first symbol period, sk , 1 and sk , 2 are transmitted in the broadcasting manner through the transmitting antenna S1 and S2, where sk ,1 and sk ,2 are the differential results of bk ,1 and bk ,2 defined in

Eq. (3). Then, during the second symbol period,  sk*, 2 and s

* k, 1

are transmitted in the broadcasting manner by the

transmitting antenna S1 and S2. At receiver d, the received signals at the two consecutive symbol periods can be given by ­°rk , 1 hk , 1d sk , 1  hk , 2 d sk , 2  nk , 1 (6) ® * * °¯rk , 2  hk , 1d sk , 2  hk , 2 d sk , 1  nk , 2 Form Eq. (6), it is easy to obtain §r  rk*, 2 · § sk , 1 sk , 2 · § hk , 1d  hk*, 2 d · Rk ¨¨ k , 1 ¸¸ ¨ * ¸¸  * ¸¨ * * ¨ © rk , 2 rk , 1 ¹ ©  sk , 2 sk , 1 ¹ © hk , 2 d hk , 1d ¹ § nk , 1 nk*, 2 · (7) ¨¨ ¸¸ * © nk , 2 nk , 1 ¹ For convenience, channel matrix H k and noise matrix

N k is defined as follows, respectively ­ § hk , 1d  hk*, 2 d · ° H k ¨¨ ¸¸ * ° © hk , 2 d hk , 1d ¹ (8) ® § nk , 1  nk*, 2 · ° * ° N k ¨¨ n ¸¸ © k , 2 nk , 1 ¹ ¯ Substituting H k and N k from Eq. (8) into Eq. (7), we can easily give Rk SkH H k  N k

(9)

According to the principle of differential decoding, it is not difficult to obtain the differential detection result as †

§r r * ·  rk , 1 rk*, 2 ¨¨ rk , 1 r *k , 2 ¸¸ k, 1 ¹ © k, 2 * § r r* · (10)  rk , 1 rk*, 2 ¨¨ rk 1, 1 rk 1, 2 ¸¸ © k 1, 2 k 1, 1 ¹ where ‘ † ’ denotes the Hermitian operation. From Eq. (10), the detection results of bk , 1 and bk , 2 are calculated by



bk , 1 bk , 2



­°bk , 1 rk , 1rk*1, 1 +rk*, 2 rk 1, 2 I1  II1  III1 (11) ® * * °¯bk , 2 rk , 1rk 1, 2  rk , 2 rk 1, 1 =I 2  II 2  III 2 Here, item Ii , IIi and IIIi (i 1, 2) represent the desired , item, interference item, and noise item, respectively, as below

2008

h h

* k , 1d k 1, 1d

h

 hk*, 2 d hk 1, 2 d sk , 1sk* 1, 1 

* k , 1d k 1, 1d

 hk*, 2 d hk 1, 2 d sk , 2 sk* 1, 2

h h h h

h

*

h

* k , 1d k 1, 2 d

 hk 1, 1d hk*, 2 d sk , 1sk* 1, 2   hk 1, 2 d hk*, 1d sk , 2 sk* 1, 1

* k , 2 d k 1, 1d

III1

h s

k , 1d k , 1

 hk , 2 d sk , 2 nk* 1, 1   hk 1, 2 d sk* 1, 1  hk 1, 1d sk* 1, 2 nk*, 2 

* k , 2d k , 1

s

 hk*, 1d sk , 2 nk 1, 2   hk*1, 2 d sk* 1, 2 +hk*1, 1d sk* 1, 1 nk , 1

(11a) and I 2  hk , 2 d hk*1, 2 d  hk 1, 1d hk*, 1d sk , 2 sk 1, 1 

h h h h

* k , 1d k 1, 1d

II 2

III 2

h

 hk 1, 2 d hk*, 2 d sk , 1sk 1, 2

* k , 1d k 1, 2 d

 hk*, 2 d hk 1, 1d sk , 1sk 1, 1 

* k , 1d k 1, 2 d

 hk , 2 d hk*1, 1d sk , 2 sk 1, 2

h

h

* k , 1d k , 2

s

*

 hk*, 2 d sk , 1 nk 1, 1   hk*1, 2 d sk 1, 1 

hk*1, 1d sk 1, 2 nk , 1   hk , 2 d sk , 2  hk , 1d sk , 1 nk* 1, 2 

h

s

k 1, 1d k 1, 1

 hk 1, 2 d sk 1, 2 nk*, 2

(11b)

Ignoring the correlated coefficient of fading gains between two consecutive block intervals, we can easily obtain hk 1, id =hk , id (i 1, 2). Substituting it into Eq. (11) and simplifying yields 2 2 2 2 ­b hk , 1d  hk , 2 d sk 1, 1  sk 1, 2 bk , 1 +III1 ° k, 1 (12) ® 2 2 2 2 °bk , 2 hk , 1d  hk , 2 d sk 1, 1  sk 1, 2 bk , 2 +III 2 ¯ where the variables III1 and III 2 are white Gaussian noise

 

 



as defined in Eqs. (11a) and (11b), respectively. Since both S1 and S2 transmit signal with unit power, Eq. (12) can be simplified to 2 2 ­b hk , 1d  hk , 2 d bk , 1 +III1 ° k, 1 (13) ® 2 2 °bk , 2 hk , 1d  hk , 2 d bk , 2 +III 2 ¯ and bˆ of b and b can So the estimations bˆ

 



k, 1

k, 2

k, 1

k, 2

be obtained as below 2 ­bˆ arg ˆmin bk , 1  bˆk , 1 °° k , 1 bk , 1 B (14) ® 2 °bˆk , 2 arg min bk , 2  bˆk , 2 bˆk , 2 B °¯ where B represents the N-ary symbol set. Now, the differential detection process is complete.

3 BER analysis of differential space - time detection From Eq. (11), the expression of bk , 1 can also be rewritten in the following form bk , 1 X 1Y1*  X 2Y2*

(15)

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ZOU Yu-long, et al.: BER analysis of MPSK space-time code with differential detection over…

where X 1 hk , 1d , X 2

Y1*

hk , 2 d

(16)

O1*  [1* , Y2* O2*  [ 2*

(17)

and § · h hk*1, 1d ( sk , 1sk* 1, 1  sk , 2 sk* 1, 2 )  ¨ hk*1, 2 d  k 1, 1d hk*, 2 d ¸ sk , 1sk* 1, 2 ¨ ¸ h k , 1d © ¹ § · h hk*1, 1d bk , 1  ¨ hk*1, 2 d  k 1, 1d hk*, 2 d ¸ sk , 1sk* 1, 2 ¨ ¸ hk , 1d © ¹ * * h h h sk , 1nk* 1, 1  k , 1d sk , 2 nk 1, 2  k 1, 1d sk* 1, 1nk , 1  k 1, 1d sk* 1, 2 nk*, 2 hk , 1d hk , 1d hk , 1d

O1*

* 1

[

O2*

21

statistical calculation. Figures 2 and 3 show the numerical solutions of the integrations in Eq. (19) for U b equivalent to 0.997 94 and 0.979 94, respectively. For notational convenience, the real line denotes the theoretical PDF curve of standardized normal variate (SNV) and the circle and square points denote the accumulated distribution function (CDF) of the real part and the imaginary part of Yi , respectively. From the figures, the CDF of Yi coincide with the theoretical PDF curve of SNV well, which can further verify that variables Y1 and Y2 meet the criteria for the complex Gaussian distribution.

§ · h hk*1, 2 d ( sk , 1sk* 1, 1  sk , 2 sk* 1, 2 )  ¨ hk*1, 1d  k 1, 2 d hk*, 1d ¸ ˜ ¨ ¸ hk , 2 d © ¹ § · h hk*1, 2 d bk , 1  ¨ hk*1, 1d  k 1, 2 d hk*, 1d ¸ sk , 2 sk* 1, 1 ¨ ¸ hk , 2 d © ¹ * * h h sk , 2 nk* 1, 1  k , 2 d sk , 1nk 1, 2  k 1, 2 d sk* 1, 2 nk , 1  hk , 2 d hk , 2 d

sk , 2 sk* 1, 1

[ 2*

hk 1, 2 d hk , 2 d

sk* 1, 2 nk*, 2

Under the assumptions given in Sect. 2.1, X 1 and X 2 (a)

(b)

are i.i.d. zero-means complex Gaussian random variables. The variables Y1 and Y2 require more careful consideration because

Fig. 2

they are dependent on {Q q }q

normal variate for U b =0.994 97

Q1 Q5

hk*, 1d hk , 1d * k , 2d

h

hk , 2 d

, Q2

hk*1, 1d

, Q6

hk , 1d

, Q3

1, 2, 3, 4, 5, 6

hk 1, 1d hk , 1d

, where

, Q4

The CDF of Y1 and Y2 vs PDF of standardized

hk 1, 2 d hk , 2 d

* k 1, 2 d

h

(18)

hk , 2 d

Referring to [10], the joint probability density function of the fading gains at two consecutive block intervals can be expressed as r1r2 ˜ p (r1 , r2 , T1 , T 2 ) (2ʌP ) 2 (1  U b 2 ) § r 2  r2 2  2r1r2 U b cos (T 2  T1  I ) · exp ¨  1 ¸ 2 P (1  U b 2 ) © ¹ where r1 and r2 denote the amplitude of fading gains at the two consecutive block interval, respectively. Thus, the probability distributed function (PDF) of the real and imaginary part of Yi (i 1, 2) are given by

FRe(Yi ) ( y )

FIm(Yi ) ( y )

Re( Oi  [i )  y

p( r1 , r2 , T1 , T 2 )dT1dT 2dr2dr1

PPDF  Im(Yi )  y

³³³³

Im( Oi  [i )  y

PPDF  Im(Oi  [i )  y

(b)

Fig. 3 The CDF of Y1 and Y2 vs PDF of standardized normal variate for U b =0.979 94

Furthermore, the mean-square error of fitting PDF of SNV and CDF of Yi (i 1, 2) are given in Table.

PPDF  Re(Yi )  y Pr  Re(Oi  [i )  y

³³³³

(a)

(19)

p( r1 , r2 , T1 , T 2 )dT1dT 2dr2dr1

With regards to the above complicated equations, by using integral formulae we cannot obtain the solutions in closed form. But, the arithmetical solutions could be obtained using

Table 2

U 0.994 97 0.997 84 0.979 94 0.955 14

Mean-square error of fitting Fitting error Real part of Yi

Imagine part of Yi

0.000 124, 0.000 122 0.000 159, 0.000 157 0.000 173, 0.000 169 0.000 241, 0.000 243

0.000 121, 0.000 124 0.000 158, 0.000 156 0.000 166, 0.000 171 0.000 248, 0.000 245

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The Journal of CHUPT

By using the result of Appendix A, it is easy to get * mY1Y2 E (YY 0 1 2 )

(20)

Hence, pairs ( X 1 , Y1 ) and ( X 2 , Y2 ) are uncorrelated complex-valued, zero-mean Gaussian. Form Appendix B, we have ­mxx E ( X 1 2 ) E ( X 2 2 ) 1 ° 2 2 ° (21) ®myy E ( Y1 ) E ( Y2 ) 1  2 1  U b  4 N 0 ° * * °¯mxy E ( X 1Y1 ) E ( X 2Y2 ) U b Thus, the normalized cross-correlation is given by mxy Ub (22) P mxx m yy 4 1  2(1  U b )  Js Here, J s

2008

simulation are molded as Jakes’ channel that is temporally correlated Rayleigh fading channel. The numerical results are illustrated in Figs. 4 and 5, with normalized Doppler frequencies BdTs equal to 0.011 3 and 0.022 6, respectively. To simplify the expression, let real line denote the analytic BER curve and let discrete points represent the simulation results.

Es / N 0 =1/ N 0 is signal-to-noise ratio (SNR) per

symbol and U b

J 0 (4ʌBdTs ) . Referring to Eq. (15), the PDF

of phase of received signal can be given by (1  P 2 )2 (2  P 2 cos 2 T ) p (T )  4ʌ(1  P 2 cos 2 T ) 2

Fig. 4 BER performance versus mean SNR from source node to destination node for U b =0.994 97

3P (1  P 2 )2 cosT arccos( P cosT ) (23) 4ʌ(1  P 2 cos 2 T )5 / 2 where P is defined as Eq. (22). From Eq. (23), the symbol error probability (SEP) of MPSK is 2ʌ M  P( M ) ³ ʌ p(T )dT ª 2 2 ʌ º M 8ʌ «1  P cos M »¼ ¬ ʌ ª ʌº ʌ P sin «3  P 2  2P 2 cos 2 » 1  P cos M¬ M¼ M arccot 3/ 2 ʌ ª 2 2 ʌ º 1  P 2 cos 2 4ʌ «1  P cos » M M ¬ ¼ (24) Hence, the BER of BPSK, quaotrature phase shift keying (QPSK), and octal phase shift keying (OPSK) space-time code with differential detection over correlated block-fading Rayleigh channel are calculated using the following equations ­ ʌ ° P2e 2³ ʌ p (T )dT 2 P (2) 2 ° 3ʌ °° ʌ §4· (25) ® P4e ³ ʌ4 p (T )dT  2³ 3ʌ p(T )dT P (4)  P ¨ ¸ 3¹ © 4 4 ° ° 2 ʌ 2 ʌ 2ª § 8 ·º ° P8e ʌ p (T )dT  3ʌ p (T )dT « P (8)  P ¨ 3 ¸ » ³ ³ 3 3 3 © ¹¼ ¬ 8 8 ¯° where P (2), P (4), and P (8) are defined as Eq. (24). ʌ

4

M 1  2M

P 2 (1  P 2 )sin

Simulation results and analysis

The system simulation is done on the condition defined in assumption 1)4) and all the fading channels employed in

Fig. 5 BER performance versus mean SNR from source node to destination node for U b =0.979 94

Figure 4 shows BER performance versus mean SNR from source to destination of the diversity system, with Ub equal to 0.994 97. The BER performance of DBPSK, DQPSK, and DOPSK are also given in Fig. 4. From the figure, the BER performance of double-antenna system improves distinctly compared with the single-antenna system for any differential modulation mode (e.g., DBPSK, DQPSK, and DOPSK). And the region between the BER performance curve of differential space-time scheme and that of single-antenna system with differential transmitting scheme can be explained as the available region of differential space-time system. As can be seen, the available region is fairly large, which could account for the effectiveness of the differential space-time scheme. In addition, there always exists the irreducible error floor in the high SNR region for the differential detection over the correlated block-fading channel in Fig. 4. From Eq. (11), it is

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ZOU Yu-long, et al.: BER analysis of MPSK space-time code with differential detection over…

obvious that the received signal with the differential detection includes the desired item, interference item, and noise item. When the SNR approaches to infinity, the noise item is approximately equal to zero, and then the interference item becomes the key factor for determining the error decision, that is to say, the system performance is dependent on the signal-to-interference ratio, not signal-to-noise ratio with respect to J s o f .Therefore, the error floor emerges when the SNR is very high for the differential modulation system. Figure 5 shows the BER performance versus the mean SNR from source to destination of the diversity system, with U b

zero-means complex Gaussian random variables, we can obtain § · §h · E (O1* ) E ( hk*1, 1d bk , 1 )  ¨ E ( hk*1, 2 d )  E ¨ k 1, 1d ¸ E ( hk*, 2 d ) ¸ ˜ ¨ h ¸ ¨ ¸ © k , 1d ¹ © ¹ sk , 1sk* 1, 2

we also see that the simulated BER performance and analytical results coincide well, which can also confirm the correctness of our analytical derivations.

5

Conclusions

In this article, we have presented the multiple antenna diversity protocol based on differential space-time block coding which can achieve diversity gain without the need for channel state information (CSI). A closed-form expression for BER of the differential space-time scheme was derived for temporally correlated block-fading Rayleigh channel. Also, the system simulation was done for different modulation mode (i.e., BPSK, QPSK and OPSK). The numerical results show that the DSTBC-based double-antenna system obtains the distinct diversity gain compared with the single-antenna system, with a fairly large region in the BER performance figures, which can explain the effectiveness of the differential space-time detection. Last but not the least, the numerical results coincide well with the analytical results, which conforms to our derivations. Acknowledgements This work is supported by the National Natural Science Foundation of China (60372107), Key Project of Nature Science Funding of Jiangsu Province (BK2007729), and Major Development

Program

of

Jiangsu

Educational

Committee

(06KJA51001).

Appendix A Cross-correlation analysis between Y1 and Y2 Noting that both hk 1, 1d , hk , 1d and hk 1, 2 d , hk , 2 d are i.i.d.

§ §h 0 ˜ bk , 1  ¨ 0  E ¨ k 1, 1d ¨ h ¨ © k , 1d ©

· · * ¸¸ ˜ 0 ¸ sk , 1sk 1, 2 ¸ ¹ ¹

0 (A.1)

Also, nk , i is the zero-mean Gaussian white noise, so E ([1* )

equal to 0.979 94. As shown in the figure, the BER performance of this case is worse than that of U b equal to 0.994 97 and the BER performance of double-antenna system-based differential space-time block code is better than that of single-antenna system that employs differential modulation (i.e., DBPSK, DQPSK and DOPSK). Meanwhile, the error floor raises considerably with the decrease of U b . From Fig. 4 and Fig. 5,

23

§ h* sk , 1 ˜ 0  sk , 2 E ¨ k , 1d ¨h © k , 1d §h · sk* 1, 2 E ¨ k 1, 1d ¸ ˜ 0 ¨ h ¸ © k , 1d ¹

Hence E (Y1 ) E (Y1* )

E (O1*  [1* )

· § hk*1, 1d * ¸¸ ˜ 0  sk 1, 1E ¨¨ ¹ © hk , 1d

· ¸¸ ˜ 0  ¹

0

(A.2)

E (O1* )  E ([1* ) 0

(A.3)

In a similar way, the expectation of Y2 can be calculated by E (Y2 ) 0

(A.4)

Thus, the cross-correlation coefficient between Y1 and Y2 can be given by R Y1 , Y2 E ª¬Y1  E (Y1 ) Y2*  E (Y2* ) º¼ E Y1 Y2* E (O1 O2* )  E (O1 [ 2* )  E ([1 O2* )  E ([1 [ 2* )

(A.5)

where

E (O1 O2* )

* ­ º h °ª E ® « hk*1, 1d bk , 1  ( hk*1, 2 d  k 1, 1d hk*, 2 d ) sk , 1sk* 1, 2 » ˜ hk , 1d °¬ ¼» ¯« ª * º ½° hk 1, 2 d * * hk , 1d ) sk , 2 sk* 1, 1 »¾  « hk 1, 2 d bk , 1  ( hk 1, 1d  hk , 2 d ¬« ¼» ¿° 2

bk , 1 E (hk 1, 1d ) E (hk*1, 2 d ) 0

(A.6)

and * ­ª º °« » § · h ° E (O1 [2* ) E ®«hk*1, 1d bk , 1  ¨ hk*1, 2d  k 1, 1d hk*, 2d ¸ sk , 1sk*1, 2 » ˜ ¨ ¸ » hk , 1d °«  © ¹  

» « °¬ ¼ C1 ¯

ª º½ « » °° hk*, 2d hk*1, 2d * hk 1, 2d * * * sk , 1nk 1, 2  sk 1, 2nk , 1  sk 1, 2nk , 2 » ¾ « sk , 2nk 1, 1  hk , 2d hk , 2d hk , 2d «   

» ° ¬« ¼» ¿° C2 (A.7) Since nk 1, 1 , nk , 1 , nk 1, 2 , nk , 2 and hk 1, 1d , hk , 1d澤hk 1, 2 d ,

hk , 2 d are independent of each other, the component C1 and C2 are mutually independent. Thus, Eq. (A.7) is simplified to

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The Journal of CHUPT

* ­ª º § § hk 1,1d · · ° * E (O1 [ ) E (C1 ) E (C ) ®«0 ˜ bk ,1  ¨ 0  E ¨ ¸¸ ˜ 0 ¸ ˜ sk ,1sk 1,2 » ˜ ¨ h ¨ ¸ »¼ °¯«¬ © k ,1d ¹ ¹ © * ­ º ª § § hk 1,1d · · °ª * ¸¸ ˜ 0 ¸ sk ,1sk 1,2 » « sk ,2 ˜ 0  ® « 0 ˜ bk ,1  ¨¨ 0  E ¨¨ ¸ »¼ «¬ °¯ «¬ © hk ,1d ¹ ¹ © * 2

* 2

§ h* sk ,1E ¨ k ,2 d ¨h © k ,2 d

· § hk*1,2 d * ¸¸ ˜ 0  sk 1,2 E ¨¨ ¹ © hk ,2 d

· ¸¸ ˜ 0  ¹

ª § h* 2 « E nk* 1, 1 sk , 1 2  E ¨ k , 1d « ¨ hk , 1d © ¬ § h* E ¨ k 1, 1d ¨ hk , 1d ©



§h E ¨ k 1,2 d ¨ © hk ,2 d

(A.8)

(A.9)

Substituting Eqs. (A.6), (A.8) and (A.9) into Eq. (A.5) and simplifying yields R (Y1 , Y2 ) E (O1 O2* )  E (O1 [ 2* )  E ([1 O2* )  E ([1 [ 2* ) 0

The autocorrelation coefficient of Y1 is defined as

E YY



E ª¬ O1  [1  O  [ º¼ * 1

 

E O1

2

 E [1

2



2

2

* 1 1

* 1 1

E O1  [1  O [  [ O



 2Re E  O [

* 1 1



· § h 2 2 ¸ E nk , 1 sk* 1, 1  E ¨ k 1, 1d ¸ ¨ hk , 1d ¹ ©





2 * k 1, 2

º » » ¼







2

· ¸˜ ¸ ¹

Since nk 1, 1 , nk , 1 , nk 1, 2 , nk , 2

and hk 1, 1d , hk , 1d hk 1, 2 d ,

hk , 2 d are independent of each other, and O1 and [1 are mutually independent. Thus ­° ª º § · h E (O1[1* ) ® E « hk*1,1d bk ,1  ¨ hk*1,2 d  k 1,1d hk*,2 d ¸ sk ,1sk* 1,2 » ˜ ¨ ¸ hk ,1d °¯ ¬« © ¹ ¼»

ª h* h* E « sk ,1nk* 1,1  sk ,2 k ,1d nk 1,2  sk* 1,1 k 1,1d nk ,1  sk* 1,2 ˜ hk ,1d hk ,1d «¬ * º § §h · · hk 1,1d * º ½° °­ª nk ,2 » ¾ ®«0 ˜ bk ,1  ¨ 0  E ¨ k 1,1d ¸ ˜ 0 ¸ sk ,1sk* 1,2 » ˜ ¨ ¸ ¨ ¸ hk ,1d »¼ ¼» ¿° °¯«¬ © hk ,1d ¹ ¹ © *½ ° ª¬ sk ,1 ˜ 0  sk ,2 ˜ 0  sk* 1,1 ˜ 0  sk* 1,2 ˜ 0 º¼ ¾ 0 °¿ Hence, the autocorrelation coefficient of Y1 is

Autocorrelation analysis of Yi (i 1, 2)

* 1

s

· 2 2 ¸ E nk 1, 2 sk , 2  ¸ ¹

(B.2)

independent.

* 1 1

2

2

2

4N0

(A.10) Therefore, the random variables Y1 and Y2 are mutually

Appendix B





2

E nk*, 2

½ · º° s ¸¸ ˜ 0 » ¾ 0 ¹ ¼» ° ¿ Similarly, it is easy to get E ([1 O2* ) 0, E ([1 [ 2* ) 0 * k 1,2

E [1

2008

* E YY 1  2 1  U b  4 N 0 1 1

Similarly, we also can obtain * E Y2Y2* E YY 1  2 1  U b  4 N 0 1 1

(B.3)

(B.4) (B.5)

where



E O1

2

­° ª º h E ® « hk*1,1d bk ,1  (hk*1,2 d  k 1,1d hk*,2 d ) sk ,1sk* 1,2 » ˜ h k ,1d ¼» ¯° ¬« *

½ ° ¾ °¿

§ h 2 bk ,1  E ¨ hk*1,2 d  k 1,1d hk*,2 d ¨ hk ,1d ©

2

ª * º hk 1,1d * * hk ,2 d ) sk ,1sk* 1,2 » « hk 1,1d bk ,1  ( hk 1,2 d  h k ,1d ¬« ¼» ­ ° * ® E hk 1,1d °¯



2



References 1. Alamouti S M. A simple transmitter diversity scheme for wireless communications. IEEE Journal on Selected Areas in Communications, 1998, 16 (10): 14511458

· ¸˜ ¸ ¹

2

sk ,1sk 1,2  E ( hk*1,1d ) E ( hk 1,2 d )bk ,1sk*,1sk 1,2  § h* · E ¨ k *1,1d hk*1,1d ¸ E ( hk ,2 d )bk ,1sk*,1sk 1,2  ¨ h ¸ © k ,1d ¹ §h · E (hk 1,1d ) E ( hk*1,2 d )bk*,1sk ,1sk* 1,2  E ¨ k 1,1d hk 1,1d ¸ ˜ ¨ h ¸ © k ,1d ¹ ½° E (hk*,2 d )bk*,1sk ,1sk* 1,2 ¾ 1  2(1  U b ) (B.1) °¿ Here, U b J 0 (4ʌBdTs ) denotes the correlation coefficient between the fading gains at two consecutive space-time block intervals. Meanwhile

2. Tarokh V, et al. Space-time block code from orthogonal designs. IEEE Transactions. on Information Theory, 1999, 45(7): 14561467 3. Xu Kai, Luo Tao, Yin Chang-chuan, et al. A new performance enhancement algorithm for space-time block coding in OFDM systems. Journal of China Universities of Posts and Telecommunications, 2004, 11(3): 6265 4. Proakis J G. Digital communications. McGraw-Hill, 3rd edition, 1995 5. Wang X, Poor H V. Wireless communication systems: advanced techniques for signal reception. Englewood Cliffs, NJ, USA: Prentice-Hall, 2004 6. Voelcker H. A note on error statistics in fading radio teletype circuits. IEEE Transactions on Information Theory, 1960, 6(5): 558558 7. Kam P Y. Bit error probabilities of MDPSK over the nonselective Rayleigh fading channel with diversity reception. IEEE Transactions. on Communications, 1991, 39(2): 220224

Issue 2

ZOU Yu-long, et al.: BER analysis of MPSK space-time code with differential detection over…

25

8. Tarokh V, Jafarkhani V. A differential detection scheme for

ZHENG Bao-yu, full professor and doctoral

transmit diversity. IEEE Journal on Selected Areas in Communications, 2000, 18(7): 11691174

advisor at Nanjing University of Posts and

9. Hughes B L. Differential space-time modulation. IEEE Transactions Information. Theory, 2000, 46(7): 25672578

doctoral advisor at Shanghai Jiao Tong University.

Telecommunications, an adjunct professor and His research interests are the intelligent signal

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processing,

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signal

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Biographies: ZOU Yu-long, Ph. D. Candidate of Nanjing University of Posts and Telecommunications. His current research interests span the broad area of scalable wireless communication and networking, with emphasis on the cooperative relay techniques and opportunistic cooperation for next generation networks (NGNs).

From p. 17

6. LU Yan-hui, LUO Tao, YIN Chang-chuan, et al. Adaptive radio resource allocation for multiple traffic OFDMA broadband

Acknowledgements

This work is supported by the National Basic

Research Program of China (2007CB310604) and the National Natural Science Foundation of China (60772108).

wireless access system. The Journal of China Universities of Posts and Telecommunications, 2006, 13(4): 16 7. Shen Zu-kang, Andrews J G, Evans B L. Adaptive resource allocation in multiuser OFDM systems with proportional rate

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of multiple access for OFDM transmission technique in Rayleigh

(in Chinese) 10. Baldick R. Optimization of engineering systems course notes. Austin, TX, USA: University of Texas [2006-12-5]. http://www. ece.utexas.edu/~baldick/ 11. IEEE 802.20-PD-08, 802.20 Channel Models Document for IEEE 802.20 MBWA System Simulations. 2005

Biography: XU Wen-jun, Ph. D. Candidate in Beijing University of Posts and Telecommunications, interested in wireless communications.