Orthogonal space–time block codes for free-space IM=DD optical links
In either of the above cases it is easy to show that the following relationship holds for symbols s0 and s1 [4]: si ðtÞ ¼ sj ðtÞ þ I ; i 6¼ j; i; j 2 f0; 1g
M. Premaratne and F.-C. Zheng Use of orthogonal space-time block codes (STBCs) with multiple transmitters and receivers can improve signal quality. However, in optical intensity modulated signals, output of the transmitter is non-negative and hence standard orthogonal STBC schemes need to be modified. A generalised framework for applying orthogonal STBCs for free-space IM=DD optical links is presented.
Introduction: For applications where line-of-sight (LOS) is available between transmitter and receiver, free-space optical communications systems offer potential advantages in bandwidth over conventional microwave technology [1]. The main benefit of free-space optical communications systems is that bandwidth is dedicated between points so that it can make a high-speed connection with data rates comparable to those of a fibre-optic link at a fraction of the deployment cost. Moreover the extremely narrow laser beams with small beam divergence in a physically small setup enable us to install a large number of free-space optical links between two given LOS points. Therefore, by combining a large number of beams, it is possible to construct a large capacity communication link between two LOS points with a bit-rate of several tens of gigabits per second. Interestingly, unlike RF or microwave systems, free-space optical communications systems require no spectrum licensing and are immune to electromagnetic interference from neighbouring systems [2]. Free-space optical communications systems are built by transmitting a modulated laser beam through the atmosphere between two LOS points. Although relatively unaffected by rain and snow, if the distance between LOS points is greater than a few hundred metres or more, inhomogeneities in the temperature and pressure of the atmosphere can deteriorate the quality of the signal received at the receiver [2]. In intensity modulation and direct detection (IM=DD) free-space optical communication systems, intensity of the laser beam is recovered and used for subsequent processing. Therefore, any change in the received optical beam due to atmospheric distortion along the transmission path causes intensity fluctuations at the receiver, often referred to as scintillation. Scintillation fluctuations can produce errors and thus degrade the performance of IM=DD free-space optical communication links [2, 3]. Even though in principle it is possible to use coherent techniques for transmitting and receiving optical signals, owing to technological complexity and cost, present-day free-space optical communication systems resort to intensity modulation (IM) techniques for symbol transmission [3]. Optical intensity refers to optical power emitted= transmitted per solid angle. In optical IM systems, the instantaneous power output of the transmitter is proportional to some function of the modulating signal. Therefore, every optical IM signal transmitted must be non-negative. Recently, Simon et al. [4] showed the application of Alamouti coding scheme [5] for free-space transmission utilising IM=DD schemes. In this Letter, we show the application of general space–time block codes (STBCs) [6] to free-space optical communication links with direct detection. Our formulation is different from the approach reported by Simon et al. [4]. We also show that IM=DD results for Alamouti scheme considered in [4] can be obtained as a special case of the method proposed in this Letter.
To continue the analysis, we define the complement of a signal xi by x¯ i to represent the opposite binary state of the signal xi (i.e. if xi ¼ s0 then x¯ i ¼ s1 and if xi ¼ s1 then x¯ i ¼ s0). Application of this definition to (3) results in the following relationship: x ðtÞ ¼ xðtÞ þ I ; xðtÞ 2 fs0 ; s1 g
s0 ðtÞ ¼ 0;
0 t T and s1 ðtÞ ¼ I ;
0tT
ð1Þ
where T is the symbol duration and I is the constant intensity of the laser at the transmitter for symbol ‘1’. For a binary pulse-position modulation (PPM) scheme we have [4]: 0; 0 t T =2 s0 ðtÞ ¼ I ; T =2 t T and I ; 0 t T =2 ð2Þ s1 ðtÞ ¼ 0; T =2 t T
ð4Þ
Noting that x(t) ¼ 0, I, the relationship (4) ensures that x¯ (t) remains non-negative. Consider a hypothetical communication system where bipolar (i.e. positive and negative valued symbols) transmission is possible in the real number domain. Without loss of generality, we consider a system with K transmitters and a single receiver. The case for multiple receivers can be derived as a straightforward extension of the presentation given here. Suppose at times ti:i ¼ 1, 2, . . . , M, each transmitter j ¼ 1, 2, . . . , K, transmits a real-valued symbol xj(tj) over the channel. It is convenient to collect such transmitted symbols into a matrix, X(x1, . . . , xK), of dimension M K called transmission-code-matrix where columns correspond to different transmitters and rows correspond to different transmission time slots. The primary property of a space time code X(x1, . . . , xK) X(x) with x ¼ (x1, . . . , xK)T is [6]: X T ðxÞX ðxÞ ¼ I K kxk2
ð5Þ
where (*)Tdenotes the transpose operation, IK is the K K unit matrix and kxk2 ¼ (x21 þ x22 þ þ x2K). However, a coding scheme satisfying (5) cannot be implemented in IM=DD systems because transmitted IM signals must be non-negative at all times. Above coding schemes show that certain transmitter outputs must be negated to get the orthogonality (5). Therefore the above coding schemes cannot be used for free-space IM=DD systems. To overcome the above problem, we introduce the following STBC matrix jXj(x, x¯ ) ¼ [(jXj(x, x¯ ))i, j]M K for the IM=DD system: ( if ðX ðx1 ¼ 1; . . . ; xK ¼ 1ÞÞi; j 0 ðX ðxÞÞi; j ð6Þ ðjX jðx; x ÞÞi; j ¼ ðX ðxÞÞi; j otherwise This definition ensures that the transmitted symbols are always nonnegative for an IM=DD system. Assuming that the free-space optical channel can be modelled as a quasi-static linear channel in M symbol periods and non-frequency selective in the band of transmission with additive, signal independent, white Gaussian noise, the received signal y ¼ (y1, y2, . . . , yM)T for M transmissions can be written as: y ¼ jX jðx; x Þh þ z T
ð7Þ
T
where x¯ ¼ (x¯ 1, . . . , x¯ K) , z ¼ (z1, z2, . . . , zM) are Gaussian noise, which may include contributions from thermal and=or shot noise [3] and h ¼ (h1, h2, . . . , hK)T is the quasi-static channel response for transmitters i ¼ 1, 2, . . . , K, respectively. Application of relationship (4) to definition (7) gives: jX jðx; x Þ ¼ X ðxÞ þ IU
Theory: Consider a binary intensity modulation (IM) scheme where symbols ‘0’ and ‘1’ are represented by s0 and s1. For an on–off keying system (OOK) we have [4]:
ð3Þ
ð8Þ
where U is a matrix with dimensions equal to matrix X (x) with its (i, j), element given by: 1 if ðX ðx1 ¼ 1; . . . ; xK ¼ 1ÞÞi;j < 0 U i;j ¼ ð9Þ 0 otherwise Assuming perfect knowledge of channel gains and the transmitted signal level, we define the following measure as a metric for estimating the signal at the receiver: x~ ¼ X T ðhÞJ M ðy I UhÞ
ð10Þ
where JM is a M M diagonal matrix with 1 in the diagonal except 1 for J1,1(i.e. J ¼ diag (1, 1, 1, . . . , 1)). Substitution of (7) and (8) into (10) gives
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x~ ¼ X T ðhÞJ M ðX ðxÞh þ zÞ
ð11Þ
Noting that space time orthogonal codes considered here satisfy X (x)h ¼ JMX(h)x, we get x~ ¼ X T ðhÞJM ðJ M X ðhÞx þ zÞ
ð12Þ
where h ¼ (h1, h2)T is the channel response vector for two transmitters. Equation (21) agrees with the results given in [4]. The decision rule for selecting xi ¼ xˆ i follows from (19): ð~xi x^ i Þ2 þ ðh21 þ h22 1Þ^x2i ð~xi xi Þ2
Using the relation JM JM ¼ IM, we finally obtain: x~ ¼ khk2 x þ X T ðhÞJ M z
This shows that x˜ is a reliable estimate for x. Note that equivalence of (10)–(13) can be easily verified for all STBC matrices of Gi type (up to eight transmitters) reported in [6]. To calculate the maximum likelihood decision metric, we calculate the distance between observed vector y and noise-free received vector mðy; xÞ ky jX jðx; x Þhk2 ¼ ky X ðxÞh I Uhk2
ð14Þ
Equation (14) can be simplified by noting that (12) allows us to write y¼
1 J M X ðhÞ~x þ I Uh khk2
ð15Þ
and hT X T ðxÞy ¼ xT x~ þ I hT X T ðxÞUh
þ ð~x xÞT ð~x xÞ þ kyk2 þ k~xk2 þ I 2 hT U T Uh T
I h U y I y Uh
ð17Þ
Ignoring the terms that are hypothesis independent, we get the following decision metric for IM=DD signals: 2
2
ð22Þ Conclusions: We have considered a generalised framework for applying orthogonal space–time block codes (STBCs) to free-space optical links (FSOLs) using intensity modulation and direct detection (IM=DD) where the transmitted signals are non-negative. Given that orthogonal STBC schemes are mainly designed for wireless radio frequency channels, complex bipolar operation of transmitters is assumed in their construction. However, in optical intensity modulated signals, the output of transmitters is non-negative and hence standard orthogonal STBC schemes need to be modified. We have shown in this Letter that, by using a complement of signal that represents the opposite binary state of the original signal, we can easily modify the standard STBCs for their application in multiple transmitter, multiple receiver free-space optical communications systems. # The Institution of Engineering and Technology 2007 1 December 2006 Electronics Letters online no: 20073712 doi: 10.1049/el:20073712
mðy; xÞ ¼ ðkhk2 1Þkxk2
T
xi 6¼ x^ i
ð16Þ
Substitution of (16) into (14) gives
T
þ ðh21 þ h22 1Þx2i ;
ð13Þ
T
mð~x; xÞ ¼ ðkhk 1Þkxk þ ð~x xÞ ð~x xÞ
ð18Þ
This reduces to the well known decision metric of the Alamouti code when two transmitters are used [4, 5]. Using (18), the decision rule for selecting xi ¼ xˆ i can be written as: ð~xi x^ i Þ2 þ ðkhk2 1Þ^x2i minfð~xi xi Þ2 þ ðkhk2 1Þx2i : xi 2 fx1 ; x2 ; . . . ; xK gg ð19Þ Example: The Alamouti scheme [5] is the simplest case of orthogonal STBCs [6] where two transmitters are used for transmission. The matrices, jXj(x, x¯ ) (see (6)) and U (see (11)) are as follows: x1 x2 0 0 X ðx; x Þ ¼ and U ¼ ð20Þ x 2 x1 1 0 Substituting these into (12) gives x h1 z1 þ h2 z2 x~ 1 ¼ ðh21 þ h22 Þ 1 þ x2 h2 z1 h1 z2 x~ 2
M. Premaratne (Department of Electrical and Computer Systems Engineering, Advanced Computing and Simulation Laboratory (AXL), PO Box 35, Clatyon, Victoria 3800, Australia) E-mail:
[email protected] F.-C. Zheng (School of Electrical Engineering, Victoria University of Technology, Footscray Park Campus, Ballarat Road, Footscray, Victoria, Australia) References 1 Ewart, R.A., and Enoch, M.: ‘Guest editorial free-space laser communications’, IEEE Commun. Mag., 2000, 38, pp. 124–125 2 Lu, X., and Kahn, J.M.: ‘Free-space optical communication through atmospheric turbulence channels’, IEEE Trans. Commun., 2002, 50, pp. 1293–1300 3 Zhu, X., and Kahn, J.M.: ‘Performance bounds for coded free-space optical communications through atmospheric turbulence channels’, IEEE Trans. Commun., 2003, 51, pp. 1233–1239 4 Simon, M.K., and Vilnrotter, V.: ‘Alamouti-type space-time coding for free-space optical communication with direct detection’, IEEE Trans. Wirel. Commun., 2005, 4, pp. 35–39 5 Alamouti, S.M.: ‘A simple transmit diversity scheme for wireless communications’, IEEE J. Sel. Areas Commun., 1998, 16, pp. 1451–1458 6 Tarokh, V., Jafarkhani, H., and Calderbank, A.R.: ‘Space-time block codes from orthogonal designs’, IEEE Trans. Inf. Theory, 2000, 45, pp. 1456–1467
ð21Þ
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