space time codes for optical comm

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EURASIP Journal on Applied Signal Processing 2002:3, 211220

c 2002 Hindawi Publishing Corporation

Space-Time Codes for Wireless Optical Communications

Shane M. Haas

Laboratory for Information and Decisions Systems, and Research Laboratory of Electronics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, USAEmail: [email protected]

Jeffrey H. Shapiro

Laboratory for Information and Decisions Systems, and Research Laboratory of Electronics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, USAEmail: [email protected]

Vahid Tarokh

Laboratory for Information and Decisions Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139, USAEmail: [email protected]

Received 1 June 2001

A space-time channel coding technique is presented for overcoming turbulence-induced fading in an atmospheric optical hetero-dyne communication system that uses multiple transmit and receive apertures. In particular, a design criterion for minimizing thepairwise probability of codeword error in a space-time code (STC) is developed from a central limit theorem approximation. Thisdesign criterion maximizes the mean-to-standard-deviation ratio of the received energy difference between codewords. It leadsto STCs that are a subset of the previously reported STCs for Rayleigh channels, namely those created from orthogonal designs.This approach also extends to other fading channels with independent, zero-mean path gains. Consequently, for large numbers oftransmit and receive antennas, STCs created from orthogonal designs minimize the pairwise codeword error probability for this

larger class of fading channels.

Keywords and phrases: space-time codes, optical communication, orthogonal designs.

1. INTRODUCTION

In atmospheric optical communication, lognormal fadingarising from refractive-index turbulence can make the recov-ery of a transmitted signal extremely difficult at the receiver.As a result, the receiver must have a redundant replica of thetransmitted signal for reliable communication. Space-timecodes (STCs) provide both spatial and temporal redundancy,or diversity, by using multiple apertures (antennas) over sev-

eral time-slots.Tarokh et al., in [1], established space-time code design

criteria for Rayleigh and Ricean fading channels. These de-sign criteria specify the pairwise properties of codewordsfrom the STC. In this paper, we derive a similar design cri-terion for the lognormal fading channel based on a centrallimit theorem approximation. Our criterion leads to STCscreated from orthogonal designs, a subset of the previouslyreported STCs for Rayleigh channels. Tarokh et al., in [2],showed that such codes have a decoding algorithm requir-ing only linear processing at the receiver. We show that theseSTCs also maximize the mean-to-standard-deviation ratio of

the received energy difference between codewords, a resultanalogous to maximal ratio combining.

Our derivation extends to other fading channels with in-dependent, zero-mean path gains. In other words, we showthat for large numbers of transmit and receive antennas,STCs created from orthogonal designs minimize the pairwisecodeword error probability regardless of the individual path-gain fading distributions.

This paper is structured as follows. Section 2 describesour channel model and the objective of space-time coding.Section 3 derives the STC design criterion for lognormal fad-ing channels based on a central limit theorem approxima-tion. Section 4 discusses the performance and presents an ex-ample of these STCs.

2. PROBLEM FORMULATION

Consider a line-of-sight atmospheric optical heterodynecommunication system that uses multiple transmit andreceive apertures, as shown in Figure 1. The space-timeencoder maps a segment of bits from the information
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212 EURASIP Journal on Applied Signal Processing

.

.

.

.

.

.

11

12

1M

NM

ReceiverSpace-time

encoderInformation

source

Figure 1: Block diagram of the transmitter, channel, and receiver.

source to a codeword of the STC. The codeword c, where c =(c1(1), c1(2) , . . . , c1(T), c2(1) , . . . , c2(T) , . . . , cN(1) , . . . , cN(T))is sent over T nonoverlapping adjacent discrete-time slotsusing N transmit apertures. During time slot t, transmitaperture n sends cn(t), a symbol from a quadrature-amplitude-modulation (QAM) signal constellation.

A laser beam propagating over a clear-weather, line-of-sight atmospheric path from transmitter exit optics to re-ceiver entrance optics experiences amplitude and phase fluc-tuations due to refractive-index turbulence [3]. A propa-gation model, established in [4], based on the extendedHuygens-Fresnel principle [5] characterizes this fading as acomplex lognormal process with correlation times on the or-der of 103 to 102 seconds. At high data rates, a single fadecan obliterate several message packets.

Because the duration of a fade is usually much longerthan the length of a message packet, we will assume thatthe fades are constant during a codeword transmission. Wewill model the path gain from transmit aperture n to re-ceive aperture m as nm = exp(nm + jnm). Here nm,

nm are independent Gaussian random variables with mo-ments var(nm) = 2, E(nm) = 2, var(nm) = 2 1,and E(nm) = 0. The log-amplitude variance, 2, typicallylies within the range 0.01 (mild fading) to 0.35 (severe fad-ing). We also assume that the spacing between elements ofthe receiver aperture array is large enough to ensure thatthe path gains for different (n, m) values are approximatelyindependent.

We will assume optical heterodyne reception, for whichthe detector output is known to consist of a frequency down-shifted version of the incident optical field plus an additivewhite Gaussian noise [6]. We will use wm(t) to denote this ad-ditive Gaussian noise for receive aperture m during time slott; it is a complex-valued, zero-mean, white Gaussian randomprocess with variance N0/2 per real dimension.

Combining the fading and additive noise fluctuations,the signal at receive aperture m {1 , . . . , M } during timeslot t {1 , . . . , T } is

rm(t) =N

n=1

nmcn(t) + wm(t). (1)

Given a received sequence {rm(t) : 1 m M, 1 t T}and knowledge of the path gains = {nm : 1 n N, 1 m M}, the minimum probability of error receiver chooses

the codeword c that minimizes

Mm=1

Tt=1

rm(t) N

n=1

nmcn(t)

2

. (2)

The exact probability of error is difficult to calculate foran STC with more than two codewords. An upper bound on

this probability of error comes from the union bound

Pe c,e

Pr(c e), (3)

where Pr(c e) is the probability of decoding codeword cas codeword e in the absence of all other codewords. Thissum is usually dominated by the terms of the closest, or min-imum distance, codeword pairs. The union bound estimateof the codeword error probabilityis the sum of pairwise errorprobabilities of the minimum distance codeword pairs

Pe Kmin Pr(c e)min, (4)

where Kmin is the average number of minimum-distancecodeword neighbors and Pr(c e)min is the pairwise prob-ability of erroneously decoding a pair of minimum distancecodewords.

Given knowledge of the path gains and assuming equally-likely codewords, the pairwise probability of incorrectly de-coding transmitted codeword c as codeword e is

Pr(c e | ) = Q

d2(c, e)

2N0

, (5)

where

d2(c, e) =M

m=1

Tt=1

N

n=1

nm

en(t) cn(t)

2

(6)

is the distance between codewords at the receiver, and Q(x) isthe area under the tail of the standard normal density func-tion. Averaging over , the unconditional probability of in-correctly decoding c as e is therefore,

Pr(c e) =

Pr(c e | )p()d, (7)

where p() is the joint probability density function of thelognormal path gains.


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Space-Time Codes for Wireless Optical Communications 213

Our ultimate objective is to construct a space-time codethat minimizes the exact probability of error, Pe. In this pa-per, however, we will focus on minimizing Pr(c e)min inthe union bound estimate of this probability.

3. DESIGN CRITERION

The integral in (7) is very difficult to evaluate analytically be-cause of the lognormal density function. We will attempt tosimplify its evaluation using a central limit theorem (CLT)approximation.

Rewriting (6) as

d2(c, e) =M

m=1

Nn=1

Nk=1

nmkmAnk, (8)

where

Ank =T

t=1en(t)

cn(t)ek(t) ck(t)

(9)

shows that d2(c, e) is the sum of MN2 complex lognormalrandom variables.1 Because the coefficients {Ank : 1 n, k N} and the central moments are bounded, no single termdominates the sum. Thus, we will use the central limit theo-rem to approximate its distribution as a truncated Gaussianwith mean and variance 2 on the interval d2(c, e) 0. Us-ing this approximation, we can rewrite (7) as

Pr(c e)

0

Q

x

2N0

pX|X0(x | X 0)dx, (10)

where

pX(x) =1

22e(1/2

2)(x)2,

pX|X0(x | X 0) =pX(x)

Pr(X 0)

=

1/

22

e(1/2

2)(x)2

1 Q(/) , for x 0.

(11)

Define A as the matrix with Ank as its nkth element.This matrix characterizes the relationship between codewordpairs of the space-time code. Our goal is to derive propertiesof

Athat minimize the CLT approximation to Pr(

c e).

We will do so by expressing (10) as a function of two nor-malized parameters that measure the fading strength and thesignal-to-noise ratio. We then find bounds on the normal-ized fading strength based on the design matrix A. We con-jecture that (10) is unimodal as a function of this fluctua-tion strength. We then show that for large numbers of trans-mit and receive apertures, minimizing the normalized fadingstrength, or equivalently choosing A to be a scaled identity

1The scaled multiplication of lognormal random variables is also a log-normal random variable.

matrix, minimizes the CLT approximation to the pairwiseprobability of error.

3.1. Normalized parameters

Our first step in minimizing (10) is to rewrite Pr(c e)in terms of normalized parameters. The first normalized pa-

rameter measures the strength of the fading. Define the nor-malized fading strength, , to be the standard-deviation-to-mean ratio of the energy difference between the codewordsat the receiver, that is, = / . The second normalized pa-rameter measures the total received signal-to-noise ratio andis defined as = /N0.

With the change of variables z= x/, (10) becomes

Pr

c e; 2, 0

Q

1

2z

pZ|Z0(z)dz, (12)

where

pZ|Z0(z) =

1/

22

e(1/22)(z1)2

1 Q(1/) , for z 0, (13)

and Z is a Gaussian random variable with unit mean andvariance 2.

3.2. Mean and variance calculations

To approximate d2(c, e) as Gaussian, we must first determineits mean and variance 2. Notice that because 2 1, wehave that E(nm) E(2nm) 0. Also, because E(nm) = 2,we find that E(

|nm

|2) = 1 and E(

|nm

|4) = e4

2 . The mean of

d2(c, e) is then

= E

M

m=1

Nn=1

Nk=1

nmkmAnk

= Mtr(A), (14)

where tr(A) =N

n=1 Ann. We define the energy difference be-tween transmitted codewords as

Ed = tr(A) =N

n=1

Tt=1

cn(t) en(t)2. (15)We can then express the total signal-to-noise ratio, , as thesum of signal-to-noise ratios at each receive aperture, that is,

= MEd/N0 = MSNR, where SNR = Ed/N0 is the signal-to-noise ratio at each receive aperture.

The second moment ofd2(c, e) is

E

d4(c, e)=

Mm1=1

Nn1=1

Nk1=1

Mm2=1

Nn2=1

Nk2=1

An1k1An2k2

En1m1 k1m1 n2m2 k2m2.(16)

To evaluate this summation, we can split it into two cases.


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For m1 = m2, we have

E

n1m1 k1m1

n2m2 k2m2

= E

n1m1 k1m1

E

n2m2 k2m2

=

1 ifn1=

k1 and n2=

k2,0 otherwise.

(17)

When m1 = m2 = m, we find that

E

n1m1 k1m1

n2m2 k2m2

= E

n1mk1m

n2mk2m

=

e42

ifn1=k1 = n2 = k2,

1 ifn1 = k1 = n2 = k2,1 ifn2 = k1 = n1 = k2,0 other wise.

(18)

From these results it follows that the second moment ofd2(c, e) is

E

d4(c, e)

=

Mm1=1

e42 N

n=1

A2nn +N

n=1

Nk=1k=n

AnnAkk

+N

n=1

Nk=1k=n

AknAnk +M

m2=1m

2=m

1

Nn=1

Nk=1

AnnAkk

= M

e42 2 N

n=1

A2nn +N

n=1

Nk=1

Ank2 + M tr(A)2,(19)

whence

2 = var

d2(c, e)

= M

e42 1 N

n=1

A2nn + 2N

n=1

n1k=1

Ank2. (20)

Notice that we have assumed that the path gains are log-normally distributed, but we have only used the fact thatthey are independent and identically distributed with zeromean, unit variance, and finite fourth moment. Therefore,our method and results extend to all fading distributions thatsatisfy these weaker conditions.

3.3. Bounds on the normalized fading fluctuation

The and that we have found are tied to the design matrixA by (14) and (20), respectively. We will now derive boundson their ratio = / expressed in terms ofA.

A lower bound, obtained via the Cauchy-Schwarz in-

equality, is

2 =M

e42

1Nn=1 A2nn + 2Nn=1n1k=1 Ank2M2

Nn=1 Ann

2

e42

1N

n=

1 A

2

nn + 2N

n=

1n1

k=

1Ank

2MN

Nn=1 A

2nn

.

(21)

Equality holds in (21) when Ann = , n = 1 , . . . , N , for somepositive real number . Furthermore, setting Ank = 0 for n =k minimizes the numerator in (21). Thus we get the bound

2 e42 1MN

, (22)

with equality when A = I, where I is the N N identitymatrix. Also, Ann =

Tt=1 |cn(t) en(t)|2 = , n = 1 , . . . , N

implies that = Ed/N. Orthogonal designs [2] provide amethod to construct STCs that satisfy the design criterion

A = (Ed/N)I and provide easy decoding at the receiver.Therefore, STCs created from orthogonal designs maximizethe mean-to-standard-deviation ratio of the received energydifference between codewords.

We start the upper bound derivation by noticing thatA ispositive semi-definite [7] because it has an NTsquare-rootmatrix B with ntth element cn(t) en(t) such that A = BB[1]. Let 1 , . . . , N denote the nonnegative eigenvalues of A.

For e42

2 0, an upper bound on is found as follows:

2 =

e4

2 2Nn=1 A2nn +Nn=1Nk=1 Ank2

MNn=1 Ann2

e42

2Nn=1Nk=1 Ank2 +Nn=1Nk=1 Ank2MN

n=1 Ann2 ,

(23)

with equality when A is a diagonal matrix. Using tr(A2) =Nn=1

Nk=1 |Ank|2 =

Nn=1

2n, this upper bound becomes

2 e42 1M

Nn=1

nN

k=1 k

2 e

42 1M

, (24)

with equality when A is a diagonal matrix of rank one. Thelast inequality follows from

Nn=1

nN

k=1 k

2

Nn=1

nNk=1 k

= 1, (25)

which is met with equality when exactly one of the eigenval-ues is nonzero.

For e42

2 < 0, an upper bound on is found by sup-pressing the first term in 2:

2 N

n=1

Nk=1

Ank2M

Nn=1 Ann

2 = 1MN

n=1

nN

k=1 k

2 1

M. (26)


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Equality in (26) requiresA to be rank one with all its diagonalterms equal to zero, an impossibility if A is positive semi-definite.

The bounds on are then

e42

1MN

2

max1, e

42 1M

. (27)

The lower bound is achieved when A = (Ed/N)I. If e42

1 1, the upper bound is achieved when A has only onenonzero diagonal element. The upper bound is unachieved

when e42

1 < 1.

3.4. Minimizing the probability of codeword error

To our knowledge, the codeword probability of error in (12)does not have a closed-form solution. In this section, we willanalyze its asymptotic behavior, and conjecture that it is uni-modal as a function of2, that is, it has only one extremum,a maximum, for a fixed .

First, we will fix a value for and examine the behaviorof Pr(c e; 2, ) as we vary. We saw in Section 3.3 that is closely tied to the STC design matrix; therefore, fixing avalue of is in essence fixing a design matrix.

For small values of , the probability of codeword errorapproaches one-half, that is,

lim0

Pr

c e; 2, = 12

. (28)

As increases without bound, Pr(c e; 2, ) decays as 1/,namely,

Pr c e; 2, large

0

Q

1

2z

pZ|Z0(0)dz

=

e1/22

22

1 Q(1/)

0

Q

1

2z

dz

=

e1/22

22

1 Q(1/)1

,

(29)

where

0 Q(

(1/2)z)dz= 1/ using integration by parts.

We will now fix the total receiver signal-to-noise ratio,

, and determine the probability of codeword error for dif-ferent values of normalized fading fluctuation, , or equiv-alently, for different design matrices. As approaches zero,the Gaussian probability density function in (12) becomessharply peaked around the value z = 1. This sampling-likebehavior results in

lim0

Pr

c e; 2, = Q

2

. (30)

Furthermore, for any fixed value of,

lim

Pr

c e; 2,

= 0, (31)

104 103 102 101 100 101 102109

108

107

106

105

104

103

102

101

Pr(c

e)

var(d2)/mean(d2)2

MSNR = 8 dBMSNR = 13 dBMSNR = 15 dBMSNR = 18 dBlimit as var(d2)/mean(d2)2 0

Figure 2: The probability of codeword error, Pr(c e; 2, ), as afunction of2 for = 8, 13, 15, 18 dB.

because the Gaussian density approaches zero for large valuesof.

The behavior of (12) for intermediate values of is moredifficult to evaluate analytically. We will, therefore, make thefollowing conjecture as supported by numerical evaluationsof Pr(c e; 2, ).

Conjecture 1. For 0 < < , Pr(c e; 2

, ) has only oneextremum, a maximum, for a given value of. Plots of (12)for different values of are shown in Figure 2 to support thisconjecture.

Assuming that Pr(c e; 2, ) is unimodal in 2, its min-imum must occur on the boundary of the allowable range for in (27). In other words, if

Pr

c e; e

42 1MN

, MSNR

< Pr

c e; max

1, e4

2 1

M, MSNR

,

(32)

then the optimal design criterion, in terms of minimizing thepairwise probability of codeword error, is A = (Ed/N)I, be-cause this design matrix meets the lower bound of with

equality. When (32) does not hold, and e42

1 1, then theoptimal design criterion is to choose A to be all zero exceptfor a single nonzero diagonal element. This design matrix,however, violates the CLT assumption that no single termdominates the summation in (8). Figure 3 shows the boundson and the probability of codeword error curve.

In the central limit theorem regime, the values of Mand N must be large in order for d2(c, e) to be approxi-mately Gaussian. We have also observed through numerical


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104 103 102 101 100 101 102109

108

107

106

105

104

103

102

Pr(ce)

var(d2)/mean(d2)2

Pr(c e)for MSNR = 18dBLimit as var(d2)/mean(d2)2 0Lower bound = (exp(42) 1)/MNUpper bound = max(exp(42

)

1, 1)/M

Figure 3: The probability of codeword error, Pr(c e; 2, ), as afunction of 2 for = 18 dB, with M = N = 4 and 2 = 0.1. The

smallest achievable error probability occurs when 2 = (e42

1)/MN, or equivalently, when A = (Ed/N)I.

1 2 3 4 5 6 7 82

3

4

5

6

7

8

9

10

11

12

Numberoftransmitan

tennas(N)

Number of receive antennas (M)

SNR = 3.15dB

SNR = 3.2 dB

SNR = 3.5 dB

Figure 4: The smallest values ofM and N such that (32) holds in

mild fading (2

=

0.01).

evaluation that the value of that maximizes Pr(c e; 2, )increases with increasing . As a result, increasing N and Mcause the bounds on given in (27) to become tighter, andthe mode of Pr(c e; 2, ) to increase until (32) eventu-ally holds. These threshold values of M and N are plottedin Figures 4, 5, and 6 for different values of SNR and fadingenvironments. For a given SNR, these plots show the small-est number of transmit and receive apertures required for

A = (Ed/N)I to be the optimal design matrix. From these

1 2 3 4 5 6 7 82

3

4

5

6

7

8

9

10

Numberoftransm

itantennas(N)


SNR = 4 dB

SNR = 5 dB

SNR = 7.5 dB

Figure 5: The smallest values of M and N such that (32) holds inmoderate fading (2 = 0.1).

1 2 3 4 5 6 7 82

3

4

5

6

7

8

9

10

11

12

13

14

N

umberoftransmitantennas(N)


SNR = 10dB

SNR = 15dB

SNR = 40dB

Figure 6: The smallest values of M and N such that (32) holds insevere fading (2 = 0.35).

plots, we conclude that in the central limit theorem regime(large values ofMand N), A = (Ed/N)I is the optimal designmatrix.

4. PERFORMANCE

In this section, we address the validity of the central limit the-orem approximation and the performance of STCs on log-normal channels.

4.1. Performance bounds for orthogonal design STCs

We will now derive the pairwise probability of decodingcodeword c as codeword e assuming that the space-time codesatisfies the design criterion A = (Ed/N)I, but without using


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the central limit theorem approximation. Under this designcriterion, d2(c, e) becomes

d2(c, e) =M

m=1

Nn=1

EdN

nm2 = EdN

MNk=1

e2k, (33)

where k, k=

1 , . . . , M N , are independent, identically dis-tributed Gaussian random variables with var(k) = 2

and

E(k) = 2. Define = (1 , . . . , MN). The probability ofdecoding c as e is then

Pr(c e) =

Pr(c e | )p()d, (34)

where p() is the multivariate Gaussian probability den-sity function for . Using the bound Q(x) exp(x2/2)/2gives

Pr(c e)

12

122

MN/2 exp

1

N

Ed4N0

MNk=1

e2k 122

MNk=1

k +

2

2d

=

1

2

122

exp

1

N

Ed4N0

e2x

exp

122

x + 2

2dx

MN

=12

Fr

1N

Ed4N0

, 0; MN

=

1

2

Fr

SNR

4N, 0;

MN,

(35)

where Fr(a, 0; b) is the lognormal density frustration func-tion given by

Fr(a,0;b)=

12b 2

exp ae2x exp

1

2b2

x + b22

dx.

(36)

Using the bound Q(x) exp(x2)/4, gives a similar lowerbound

Pr(c e) 14

Fr

SNR

2N, 0;

MN. (37)

A closed form evaluation of the frustration functiondoes not exist; therefore, we use a saddle-point integrationmethod developed by Halme in [8] to numerically evalu-ate it. For the design criterion A = (Ed/N)I, Figure 7 com-pares the probability of codeword error in (34), the central

10 8 6 4 2 0 2 4 6 8 101010

109

108107

106

105

104

103

102

101

100

Pr(c

e)=

(Pairwiseerrorprobability)

SNR/2N[dB]Log norm (exact)

Log norm (Fr LB)

Log norm (Fr UB)

CLT (exact)

CLT (large SNR)

Figure 7: Comparison of pairwise codeword probability of errorfor A = (Ed/N)I STCs with and without the central limit theoremapproximation; MN = 16, 2 = 0.1.

limit theorem approximation probability of codeword errorin (12), its asymptotic behavior in (29), and the frustrationfunction bounds in (35) and (37). This figure shows that forsmall values of SNR, or typical values of error probability,the CLT approximation seems valid for MN = 16 in mod-erate fading. Asymptotically, however, the CLT probability ofcodeword error decays slower than the actual error proba-

bility. From (29), we know that the CLT probability of code-word error decays as 1/SNR, whereas the frustration functionbounds suggest the actual curve decays faster. This discrep-ancy arises from the dissimilarities in the tails of the Gaus-sian distribution and the actual distribution as emphasizedby large values of SNR.

To measure the validity of the central limit theorem ap-proximation, we examined the difference in SNR between theerror probability expression in (34) and its approximation in(12) at a given error probability. For example, in Figure 7 foran error probability of Pr(c e) = 106, the CLT approx-imation requires 0.5 dB more SNR than the actual lognor-mal curve. Figure 8 shows this spurious SNR for different

aperture products (MN) in different fading environments(2 = 0.01, 0.1, 0.35). From Figure 8, we see that the CLTapproximation is accurate to fractions of a dB in mild fad-ing environments (2 = 0.01) for all values of MN 2. Alarger number of apertures is required for more severe fading(roughly, MN > 16 for 2 = 0.1 and MN > 64 for

2 = 0.35).

4.2. A lower bound on the probabilityof codeword error

In Section 4.1, we derived lower and upper bounds on theprobability of incorrectly decoding codeword c as codeworde under the design criterion A = (Ed/N)I without using the


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100 101 102103

102

101

100

101

102

SpuriousSN

R[dB]

Aperture product

2 = 0.01

2 = 0.1

2 = 0.35

Figure 8: Difference in SNR required to achieve a 106 pairwisecodeword error probability between the actual lognormal error ex-

pression in (34) and its central limit theorem approximation in(12).

central limit theorem approximation for d2(c, e). In this sec-tion, we derive a lower bound on this probability of errorwithout using the central limit theorem approximation thatis valid for an arbitrary design matrix A. Using the Cauchy-Schwarz inequality on (6) gives

d2(c, e) M

m=1

Tt=1

Nn=1

nm

2N

k=1

ek(t) ck(t)

2

=

Mm=1

Nn=1

Ednm2

= EdMNk=1

e2k,

(38)

where we have renumbered the sum of the MN indepen-dent lognormal random variables as in Section 4.1. Follow-ing a similar derivation to that in Section 4.1, a lower boundon the probability of error for any design matrix is

Pr(c

e)

1

4FrSNR

2

, 0; MN

. (39)

For a large number of transmit apertures, N, this bound canbe quite loose, confer the orthogonal design bound in (37).

4.3. Infinite transmit diversity performance limit

If we fix the energy difference between codewords, Ed, andhave enough receive apertures, M, such that (32) holds, then

A = (Ed/N)I minimizes the pairwise error probability, andthis design matrix gives 2 = (exp(42) 1)/MN. As we in-crease the number of transmit apertures, N, we see that ap-proaches zero, and hence (30) provides a performance limit

for infinite transmit diversity, that is,

limN

Pr

c e; e

42 1MN

, MSNR

= Q

MSNR

2

. (40)

These limits appear as circles in Figure 2 for MSNR = 8,

13, 15, 18 dB.One can view this limit as the error probability of a one

transmit, Mreceive aperture system with no fading. In otherwords, the large number of transmit apertures mitigates thefading, and the only uncertainty in the decision process arisesfrom the additive white Gaussian noise.

4.4. An orthogonal design example:the Alamouti scheme

Alamouti in [9] proposed a simple transmit diversity tech-nique using two transmit apertures (N = 2), two time-slots(T = 2), M receive apertures, and a complex QAM signal constellation of size 2b. During the first time-slot, 2b bits

arrive, determining two signal constellation points, s1 and s2that are transmitted simultaneously on the first and secondapertures, respectively. During the second time-slot, the firstaperture transmits s2, while the second sends s1. In otherwords, this STC consists of all the codewords of the formc = (c1(1), c1(2), c2(1), c2(2)) = (s1, s2, s2, s1) where s1 ands2 range over all possible signal constellation points. Tarokhin [2] showed that the Alamouti scheme is an example of aSTC created from a complex orthogonal design.

The design matrix of this STC for two codewords c =(c1, c2, c2, c1) and e = (e1, e2, e2, e1) satisfies our design cri-teria A = (Ed/2)I, where Ed = 2|c1 e1|2 + 2|c2 e2|2 is the en-ergy difference between the codewords. The performance of

this code for pairsof codewords is shown in Figure 7 for eightreceive apertures (M= 8) in moderate fading (2 = 0.1).

Orthogonal designs have the property that the sym-bol sequences on each aperture are orthogonal, that is,T

t=1 cn(t)ck

(t) = 0 for n = k. As a result, space-timecodes created from orthogonal designs, such as the Alamoutischeme, have a simple decoding algorithm. Rewriting the de-cision metric in (2) as

c = argmincNT

Mm=1

Tt=1

rm(t)2 + N

n=1

Nk=1

nmkmcn(t)c

k(t)

2 Rerm(t)N

n=1 nmcn(t)

= argmincNT

Nn=1

Tt=1

Mm=1

nm2cn(t)2

2 Re

Mm=1

rm(t)nm

cn(t)

(41)

shows that joint detection of (c1(1) , . . . , cN(T)) is equivalentto decoding each individual symbol, cn(t), separately. The


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structure of the Alamouti STC allows for further simplifica-tion, and the decision rules become

s1 = argmins

1 +2

n=1

Mm=1

nm2

|s|2

+s

Mm=1

rm(1)1m + rm(2)2m

2

,

s2 = argmins

1 +2

n=1

Mm=1

nm2

|s|2

+

s M

m=1

rm(1)

2m + r

m(2)1m

2.

(42)

5. CONCLUSIONS

In this paper, we presented a framework for developingspace-time codes for an atmospheric optical heterodynecommunication system. Through a central limit theorem ap-proximation, we found that the design criterion A = (Ed/N)Iminimized the pairwise probability of codeword error forlarge numbers of apertures. Although developed for lognor-mal fading, this method generalizes to other fading distribu-tions in which the fades are zero-mean and independent.

Our design criterion also satisfies the rank and determi-nant criteria presented in [1] for Rayleigh channels. Further-more, orthogonal designs provide a method of constructingSTCs that satisfy our criterion, and require only linear pro-cessing at the receiver [2].

ACKNOWLEDGMENTThis research was supported in part by Defense AdvancedResearch Projects Agency Grant MDA972-00-1-0012.

REFERENCES

[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-timecodes for high data rate wireless communication: performancecriterion and code construction, IEEE Transactions on Infor-mation Theory, vol. 44, no. 2, pp. 744765, 1998.

[2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-timeblock codes from orthogonal designs, IEEE Transactions onInformation Theory, vol. 45, no. 5, pp. 14561467, 1999.

[3] J. H. Shapiro andR. C. Harney, Simple algorithms forcalculat-

ing optical communication performance through turbulence,in Proc. SPIE Contr. and Commun. Tech. in Laser Sys. , vol. 295,pp. 4154, 1981.

[4] J. H. Shapiro and R. C. Harney, Burst-mode atmospheric op-tical communication, in Proc. Nat. Telecommun. Conf. Record,pp. 27.5.127.5.7, New York, NY, USA, 1980.

[5] J. H. Shapiro, Imaging and optical communication throughatmospheric turbulence, in Laser Beam Propagation in the At-mosphere, J. W. Strohbehn, Ed., chapter 6, Springer, Berlin, Ger-many, 1978.

[6] R. M. Gagliardi and S. Karp, Optical Communications, JohnWiley & Sons, New York, NY, USA, 1976.

[7] R. A. Horn and C. R. Johnson, Matrix Analysis, CambridgeUniversity Press, New York, NY, USA, 1988.

[8] S. J. Halme, B. K. Levitt, and R. S. Orr, Bounds and ap-proximations for some integral expression involving lognormalstatistics, Technical report, MIT Res. Lab. Electron. Quart.Prog. Rept., 1969.

[9] S. M. Alamouti, A simple transmit diversity technique forwireless communications, IEEE Journal on Selected Areas inCommunications, vol. 16, no. 8, pp. 14511458, 1998.

Shane M. Haas is a Ph.D. candidate inElectrical Engineering at the MassachusettsInstitute of Technology. He received B.S.and M.S. degrees in Electrical Engineer-ing in 1998, 1999 and B.S. and M.A. de-grees in Mathematics from the University ofKansas with honors and highest distinctionin 1998, 1999, respectively. His doctoral re-search concerns the theoretical and experi-mental study of coding for and the capacityof optical communication through the turbulent atmosphere. Mr.Haas is a National Defense Science and Engineering Graduate Fel-low, National Science Foundation Fellow, and a Goldwater Scholar.

In 1998, he was named the Outstanding Senior in both the Univer-sity of Kansas School of Engineering and the Department of Elec-trical Engineering and Computer Science. As an undergraduate, hewas supported by the National Science Foundation Research Ex-perience for Undergraduates program for four consecutive years.From 1997 to 2000, he worked at Flint Hills Scientific, L.L.C., de-veloping an adaptive digital signal processing algorithm to predictand detect the onset of epileptic seizures. He is currently a graduateresearch and teaching assistant at the Laboratory of Informationand Decision Systems and the Research Laboratory of Electronicsat MIT.

Jeffrey H. Shapiro received the S.B., S.M.,E.E., and Ph.D. degrees in Electrical Engi-

neering from the Massachusetts Institute ofTechnology in 1967, 1968, 1969, and 1970,respectively. As a graduate student he wasa National Science Foundation Fellow, aTeaching Assistant, and a Fannie and JohnHertz Foundation Fellow. His doctoral re-search was a theoretical study of adaptivetechniques for improved optical communi-cation through atmospheric turbulence. From 1970 to 1973, Dr.Shapiro was an Assistant Professor of Electrical Sciences and Ap-plied Physics at Case Western Reserve University. From 1973 to1985, he was an Associate Professor of Electrical Engineering atMIT, and in 1985, he was promoted to Professor of Electrical En-gineering. From 1989 until 1999 Dr. Shapiro served as AssociateDepartment Head. In 1999, he became the Julius A. Stratton Pro-

fessor of Electrical Engineering. In 2001, Dr. Shapiro was appointeddirector of MITs Research Laboratory of Electronics. Dr. Shapirosresearch interests have centered on the application of communica-tion theory to optical systems. He is best known for his work onthe generation, detection, and application of squeezed-state lightbeams, but he has also published extensively in the areas of at-mospheric optical communication and coherent laser radar. Dr.Shapiro is a fellow of the Institute of Electrical and Electronics En-gineers, of the Optical Society of America, and of the Institute ofPhysics, and he is a member of SPIE (The International Society forOptical Engineering). He has been an Associate Editor of the IEEETransactions on Information Theory and the Journal of the OpticalSociety of America.


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Vahid Tarokh received his Ph.D. degree inElectrical Engineering from the Universityof Waterloo, Ontario, Canada in 1995.FromAugust 1995 to May 1996, he was em-ployed by the Coordinated Science Labora-tory, of the University of Illinois Urbana-Champaign, as a visiting Professor. He then

joined the AT&T Labs-Research, where hewas employed as a Senior Member of Tech-nical Staff, Principal Member of TechnicalStaff, and the Head of the Department of Wireless Communica-tions and Signal Processing until August 2000. In the fall of 2000,Dr. Tarokh joined Department of Electrical Engineering and Com-puter Sciences of MIT as an Associate Professor, where he is cur-rently employed. Dr. Tarokh received numerous awards includingthe 1987 Gold Tablet of The Iranian Math Society, the 1995 Gov-ernor General of Canadas Academic Gold Medal, the 1999 IEEEInformation Theory Society Prize Paper Award, (jointly with A. R.Calderbank and N. Seshadri), and more recently the 2001 Alan T.Waterman Award.

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