Space Codes for MIMO Optical Wireless Communications: Error Performance Criterion and Code Construction
Yan-Yu Zhang, Hong-Yi Yu, Jian-Kang Zhang, Yi-Jun Zhu, Jin-Long Wang, Tao Wang
ZZHANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 1
Space Codes for MIMO Optical WirelessCommunications: Error Performance Criterionand Code Construction
Yan-Yu Zhang, Hong-Yi Yu, Jian-Kang Zhang, Senior Member, IEEE, Yi-JunZhu, Jin-Long Wang and Tao Wang
Abstract
In this paper, we consider a multiple-input-multiple-output optical wireless communication (MIMO-OWC) system in the presence of log-normal fading. In this scenario, a general criterion for the designof full-diversity space code (FDSC) with the maximum likelihood (ML) detector is developed. Thiscriterion reveals that in a high signal-to-noise ratio (SNR) regime, MIMO-OWC offers both large-scalediversity gain, governing the exponential decaying of the error curve, and small-scale diversity gain,producing traditional power-law decaying. Particularly for a two by two MIMO-OWC system withunipolar pulse amplitude modulation (PAM), a closed-form solution to the design problem of a linearFDSC optimizing both diversity gains is attained by taking advantage of the available properties on thesuccessive terms of Farey sequences in number theory as well as by developing new properties on thedisjoint intervals formed by the Farey sequence terms to attack the continuous and discrete variables
Partial results of this paper has been presented in ISIT 2015, Hong Kong. This work was supported in part by Key Laboratoryof Universal Wireless Communications (BUPT), Ministry of Education of P. R. China under Grant No. KFKT-2012103, in partby NNSF of China (No. 61271253), and in part by NHTRDP (863 Program) of China (Grant No.2013AA013603).Yan-Yu Zhang, Hong-Yi Yu, Yi-Jun Zhu, Jin-Long Wang and Tao Wang are with Department of Communication En-gineering, Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan Province (450000), China. Emails:[email protected], [email protected], [email protected], [email protected] and [email protected] Zhang is with the Department of Electrical and Computer Engineering, McMaster University, 1280 Main StreetWest, L8S 4K1, Hamilton, Ontario, Canada. Email: [email protected].
September 25, 2015 a r X i v : . [ c s . I T ] S e p HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 2 mixed max-min design problem. In fact, this specific design not only proves that a repetition code (RC)is the optimal linear FDSC optimizing both the diversity gains, but also uncovers a significant differencebetween MIMO radio frequency (RF) communications and MIMO-OWC that space dimension aloneis sufficient for a full large-scale diversity achievement. Computer simulations demonstrate that FDSCsubstantially outperforms uncoded spatial multiplexing with the same total optical power and spectralefficiency, and the latter provides only the small-scale diversity gain.
Index Terms
Multiple-input-multiple-output (MIMO), optical wireless communications (OWC), log-normal fad-ing channels, linear space code, full diversity, repetition coding and maximum likelihood detector.
I. I
NTRODUCTION
In the past decade, the demand for capacity in cellular and wireless local area networks hasgrown in an explosive manner. This demand has triggered off an enormous expansion in radiofrequency (RF) wireless communications. As an adjunct or alternative to RF communication,optical wireless communications (OWC), due to its potential for bandwidth-hungry applications,has become a very important area of research [1]–[12]. The importance of OWC lies in theadvantages of low cost, high security, freedom from spectral licensing issues etc. Furthermore,OWC links of practical interest involve satellites, deep-space probes, ground stations, unmannedaerial vehicles, high altitude platforms, aircraft, and other nomadic communication partners.Moreover, all these links can be used in both military and civilian contexts, or both indoor andoutdoor scenarios in demand of high data rate. Therefore, OWC is considered to be the nextfrontier for net-centric connectivity for bandwidth, spectrum and security issues.However, some challenges remain, especially in the mobile or atmospheric environments.For high data rate OWC systems over mobile or atmospheric channels, robustness is a keyconsideration. In mobile links, there will be inevitable impairments such as terminal-sway, aerosolscattering and non-zero pointing errors [13]–[16], etc. In addition, for atmospheric environments,atmospheric effects, such as rain, snow, fog and temperature variation, will affect the linkperformance. Therefore, in the design of OWC links, we need to consider these impairments-induced fading [17]. This fading of the received intensity signal can be described by the log-normal (LN) statistical model [18]–[23], which is considered in this paper. To combat fading,multi-input-multi-output (MIMO) OWC (MIMO-OWC) systems provide diverse replicas of
September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 3 transmitted symbols to the receiver by using multiple receiver apertures with sufficient separationbetween each so that the fading for each receiver is independent of others. Such diversity canalso be achieved by introducing a design for the transmitted symbols distributed over transmittingapertures (space) and (or) symbol periods (time). Full diversity is achieved when the decayingspeed of the error curve for the coded MIMO-OWC system is maximized.Unfortunately, unlike MIMO techniques for radio frequency (MIMO-RF) communicationswith Rayleigh fading, there are two significant challenges in MIMO-OWC communications.The first is that there does not exist any available mathematical tool that could be directlyapplied to the analysis of the average pair-wise error probability (PEP) when LN is involved.In this scenario, it is indeed a challenge to extract a dominant term of the average PEP. Letalone say how to achieve a full diversity gain. Here, it should be mentioned that there arereally mathematical formulae in literature for numerically and accurately computing the integralinvolving LN [18]–[22]. However, it can not be used for the theoretic analysis on diversity. The second is a nonnegative constraint on the design of transmission for MIMO-OWC, which is amajor difference between MIMO RF communications and MIMO-OWC. It is because of thisconstraint that the currently available well-developed MIMO techniques for RF communicationscan not be directly utilized for MIMO-OWC. Despite the fact that the nonnegative constraint canbe satisfied by properly adding some direct-current components (DC) into transmitter designs sothat the existing advanced MIMO techniques [24] for RF communications such as orthogonalspace-time block code (OSTBC) [25]–[33] could be used in MIMO-OWC, the power loss arisingfrom DC incurs the fact that these modified OSTBCs [34]–[36] in an LN fading optical channelhave worse error performance than the RC [21], [37]–[39].All the aforementioned factors greatly motivate us to develop a general criterion on thedesign of full-diversity transmission for MIMO-OWC. As an initial exploration, we considerthe utilization of a space dimension alone, and intend to uncover some unique characteristics ofMIMO-OWC. With this goal in mind, our main tasks in this paper are as follows.1)
To establish a general criterion for the design of full-diversity space code (FDSC) . Tothis end, our main idea here is that by fragmenting the integral region of the averagePEP involving LN into two sub-domains adaptively with SNR, the dominant term will beextracted. With this, we will give a necessary and sufficient condition for a space code toassure full diversity.
September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 4 To attain an optimal analytical solution to a specific two by two linear FDSC designproblem . To do that, we formulate and simplify our optimization problem with continuousand discrete mixed design variables by taking advantage of available properties as well asby developing new properties on Farey sequences in number theory for our purpose. Infact, we will prove that the RC is the optimal linear FDSC for this specific scenario in thesense of the design criterion proposed in this paper.II. C
HANNEL M ODEL A ND S PACE C ODE
In this section, we first briefly review the channel model which is considered in this paper.Then, we propose the space coding structure and formulate the design problems to be solved.
A. Channel Model with Space Code
Let us consider an M × N MIMO-OWC system having M receiver apertures and N transmitter apertures transmitting the symbol vector s , { s (cid:96) } , (cid:96) = 1 , · · · , L , which arerandomly, independently and equally likely, selected from a given constellation. To facilitatethe transmission of these L symbols through the N transmitters in the one time slot (channeluse), each symbol s (cid:96) is mapped by a space encoder F (cid:96) to an N × space code vector F (cid:96) ( s (cid:96) ) andthen summed together, resulting in an N × space codeword given by x = (cid:80) L(cid:96) =1 F (cid:96) ( s (cid:96) ) , wherethe n -th element of x represents the coded symbol to be transmitted from the n -th transmitteraperture. These coded symbols are then transmitted to the receivers through flat-fading pathcoefficients, which form the elements of the M × N channel matrix H . The received space-onlysymbol, denoted by the M × vector y , can be written as y = 1 P op Hx + n , (1)where P op is the average optical power of x and, the entries of channel matrix H are independentand LN distributed, i.e., h ij = e z ij , where z ij ∼ N (cid:0) µ ij , σ ij (cid:1) , i = 1 , · · · , M, j = 1 , · · · , N .The probability density function (PDF) of h ij is f H ( h ij ) = 1 √ πh ij σ ij exp (cid:32) − (ln h ij − µ ij ) σ ij (cid:33) (2)The PDF of H is f H ( H ) = (cid:81) Mi =1 (cid:81) Nj =1 f H ( h ij ) . September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 5
The signalling scheme of s is unipolar pulse amplitude modulation (PAM) to meet theunipolarity requirement of intensity modulation (IM), i.e., s ∈ R L × . As an example, theconstellation of unipolar p -ary PAM is B p = { , , · · · , p − } , where p is a positiveinteger. Then, the equivalent constellation of s is S = { s : s i ∈ B p , i = 1 , · · · , L } , i.e., S = B L p .Furthermore, for noise vector n , the two primary sources at the receiver front end are due tonoise from the receiver electronics and shot noise from the received DC photocurrent induced bybackground radiation [40], [41]. Although the signal intensity also results in shot noise, which issignal-dependent, the shot noise from hight-intensity background radiation dominates. Therefore,by the central limit theorem, this high-intensity shot noise for the lightwave-based OWC isclosely approximated as additive, signal-independent, white, Gaussian noise (AWGN) [41] withzero mean and variance σ n .By rewriting the channel matrix as a vector and aligning the code-channel product to form anew channel vector, we have Hx = (cid:0) I M ⊗ x T (cid:1) vec ( H ) , where ⊗ denotes the Kronecker productoperation and vec ( H ) = [ h , . . . , h N , . . . , h M , . . . , h MN ] T . For discussion convenience, wecall I M ⊗ x T a codeword matrix. Then, the correlation matrix of the corresponding error codingmatrix between I M ⊗ x T and I M ⊗ ˜ x T is given by (cid:0) I M ⊗ x T − I M ⊗ ˜ x T (cid:1) T (cid:0) I M ⊗ x T − I M ⊗ ˜ x T (cid:1) = I M ⊗ (cid:0) ee T (cid:1) (3)where e is the “distance” between distinct codewords x and ˜ x . All these non-zero e form anerror set, denoted by E . B. Problem Statement
To formally state our problem, we make the following assumptions throughout this paper.1)
Power constraint . The average optical power is constrained, i.e., E (cid:104)(cid:80) Ni =1 x i (cid:105) = P op .Although limits are placed on both the average and peak optical power transmitted, in thecase of most practical modulated optical sources, it is the average optical power constraintthat dominates [42].2) SNR definition . The optical SNR is defined by ρ op = √ Nσ n , since the noise variance perdimension is assumed to be σ n . Thus, in expressions on error performance involved in the September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 6 squared Euclidean distance, the term ρ , in fact, is equal to ρ = ρ op = 1 N σ n (4)with optical power being normalized by P op . Unless stated otherwise, ρ is referred to asthe squared optical SNR thereafter.3) Channel state information . Channel state information (CSI) at the receiver (CSIR) is knownand CSI at the transmitter (CSIT) is unavailable.Under the above assumptions, our primary task in this paper is to establish a general criterionon the design of FDSC and solve the following problem.
Problem 1:
Design the space encoder F ( · ) subject to the total optical power such that 1) ∀ s ∈ S , F ( s ) meets the unipolarity requirement of IM; 2) Full diversity is enabled for the MLreceiver.Naturally, two questions come up immediately: 1) What is full diversity referred to as forMIMO-OWC ? What is the design criterion ?
So far, both questions remain open and thus,motive us to analyse the error performance in the ensuing section.III. E
RROR P ERFORMANCE A NALYSIS A ND D ESIGN C RITERION FOR S PACE C ODE
In this section, our purpose is to analyze the error performance of the ML detector anddevelop a design criterion for FDSC. To make our presentation more clear as well as our mainidea more easily understandable, we begin with the symbol error probability (SEP) analysis onsingle-input-single-output OWC (SISO-OWC).
A. SEP of SISO-OWC with PAM
Recall that the PDF of a scalar channel h is f H ( h ) = √ πhσ exp (cid:16) − (ln h ) σ (cid:17) . In this specificcase, it is known [43] that the average SEP for p -ary PAM is given by P e ( ρ ) = c (cid:90) ∞ Q (cid:18) √ c ρhP op (cid:19) f H ( h ) dh (5)where ρ denotes the squared optical SNR defined by (4), c = 2 − − p and c = p p − . For thesake of simplicity and without loss of generality, we assume p = 1 which is the case of on-offkeying (OOK). September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 7
Lemma 1:
For OOK, P e ( ρ ) is bounded by Q (cid:0) σ (cid:1) √ π ρ − ln σ P op e − (cid:32) ln ρ − ln P op σ (cid:33) σ ≤ P e ( ρ ) ≤
12 (ln ρ ) − e − (cid:32) ln ρ ln2 ρ − ln P opσ (cid:33) σ + O (cid:18) e − (ln ρ )28 σ (cid:19) (6)The proof of Lemma 1 is postponed into Appendix A.In the proof of Lemma 1, our selection of τ is reasonable in finding the dominant termadaptively with SNR. Here, it should mentioned that the term “adaptive” is referred to the factthat the fragmentation of the integral region is done by a function of SNR. We also noticed thatthe splitting method is similar to those used in [16], [44]. However, the significant differencebetween our proposed technique and those used in [16], [44] is that our proposed integral-splittingmethod is to extract the dominant behavior by selecting τ adaptively with SNR. The SEP forSISO-OWC has been given and the adaptive fragmentation with SNR can help attack the integralinvolved in LN. In light of this, we extend this technique to the case of MIMO-OWC. B. PEP of MIMO-OWC
This subsection aims at deriving the PEP of MIMO-OWC by means of the above-mentionedadaptive fragmentation with SNR and then, establishing a general design criterion for the spacecoded MIMO-OWC system.Given a channel realization H ∈ R M × N + and a transmitted signal vector s , the probability oftransmitting s and deciding in favor of ˆ s with the ML receiver is given by [45] P ( s → ˆ s | H ) = Q (cid:18) d ( e )2 (cid:19) (7)where d ( e ) = ρNP op (cid:80) Mi =1 (cid:0) h Ti e (cid:1) with h i = [ h i , · · · , h iN ] T , i = 1 , · · · , M . Averaging (7)over H yields P ( s → ˆ s ) = (cid:90) P ( s → ˆ s | H ) f H ( H ) d H . (8)For presentation clarity, we first give the definition of the dominant term. Suppose that there aretwo terms T ( ρ ) and T ( ρ ) , each of which goes to zero when ρ tends to infinity. We say that September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 8 T ( ρ ) is a dominant term in the sum of T ( ρ ) + T ( ρ ) if lim ρ →∞ T ( ρ ) /T ( ρ ) = 0 . To extractthe dominant term of (8), we make an assumption for time being. Later on, we will prove thatthis condition is actually necessary and sufficient for a full diversity achievement. Assumption 1:
Any e ∈ E is unipolar without zero entry.We are now in a position to formally state the first main result in this paper. Theorem 1:
Under Assumption 1, P ( s → ˆ s ) is asymptotically bounded by C L (ln ρ ) − MN e − (cid:80) Mi =1 (cid:80) Nj =1 ( ln ρ +ln ( P op Ω ) − ln ( M (cid:80) Nk =1 e k )) σ ij (cid:124) (cid:123)(cid:122) (cid:125) P L ( s → ˆ s ) ≤ P ( s → ˆ s ) ≤ C U (ln ρ ) − MN e − (cid:80) Mi =1 (cid:80) Nj =1 (cid:18) ln ρ ln2 ρ +ln ( P op Ω ) − ln e j (cid:19) σ ij (cid:124) (cid:123)(cid:122) (cid:125) P U ( s → ˆ s ) + O (cid:32) e − (cid:80) Mi =1 (cid:80) Nj =1 ln2 ρ σ ij (cid:33) (9)where Ω = (cid:80) Mi =1 (cid:80) Nj =1 σ − ij , C L = (cid:81) Mi =1 (cid:81) Nj =1 σ ij (4 π ) MN exp (cid:0) − MN (cid:1) Q (cid:32) N (cid:88) k =1 e k (cid:33) − and C U = (cid:0) N P op (cid:1) MN (cid:81) Mi =1 (cid:81) Nj =1 (cid:113) σ ij exp (cid:18) − Ω8 ln (cid:18) N P op Ω M (cid:19)(cid:19) The detailed proof of Theorem 1 is provided in Appendix B.With all the aforementioned preparations, we are able to give the general design criterion forFDSC of MIMO-OWC in the following subsection.
C. General Design Criterion for FDSC
The discussions in Subsection III-B tells us that P U ( s → ˆ s ) is the dominant term of the upper-bound of P ( s → ˆ s ) in (9). With this, we will provide a guideline on the space code design inthis subsection. To define the performance parameters to be optimized, we rewrite P U ( s → ˆ s ) as follows. P U ( s → ˆ s ) = C U G c ( e ) (cid:18) ρ ln ρ (cid:19) Ω4 ln (cid:18) NP op Ω M (cid:19) − ln D s ( e ) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 9 × (ln ρ ) − MN exp (cid:18) − Ω8 ln ρ ln ρ (cid:19) (10)where D s ( e ) = (cid:81) Nj =1 | e j | (cid:80) Mi =1 σ − ij and G c ( e ) = exp (cid:16) (cid:80) Mi =1 (cid:80) Nj =1 (ln | e j | σ ij ) (cid:17) (cid:16) NP op Ω M (cid:17) ln ln D s ( e ) .Here, the following three factors dictate the minimization of P U ( s → ˆ s ) :1) Large-scale diversity gain . The exponent Ω with respect to ln ρ ln ρ governs the behaviorof P U ( s → ˆ s ) . For this reason, Ω is named as the large-scale diversity gain . The fulllarge-scale diversity achievement is equivalent to the event that all the M N terms in
Ω = (cid:80) Mi =1 (cid:80) Nj =1 σ − ij offered by the N × M MIMO-OWC are fully utilized. On the otherhand, Ω can be considered to be the reciprocal of the channel equivalent variance. Infact, if the channel is independently and identically distributed with variance being σ H ,then = σ H MN . In this case, the full large-scale diversity gain can be considered to be thereduction of the channel variance by M N , which is the maximum reduction amount. In thesense of this point, our definition of Ω being the large-scale diversity gain is parallel to thediversity order for STBC of MIMO-RF. Thus, when we design space code, full large-scalediversity must be assured in the first place to maximize the exponential decaying speedof the error curve.2) Small-scale diversity gain . D s ( e ) = (cid:81) Nj =1 | e j | (cid:80) Mi =1 σ − ij is called small-scale diversity gain ,which affects the power-law decaying in terms of ρ ln ρ . min e D s ( e ) should be maximizedto optimize the error performance of the worst error event. Since the small-scale diversitygain will affect the average PEP via the power-law decaying speed of the error curve, thesmall-scale diversity gain of the space code is what to be optimized in the second place to maximize the power-law decaying speed of the error curve.3) Coding gain. G c ( e ) is defined as coding gain . On condition that both diversity gainsare maximized, if there still exists freedom for further optimization of the coding gain, max e ∈E G c ( e ) should be minimized as the last step for the systematical design of spacecode.In what follows, we will give a sufficient and necessary condition on a full large-scale diversityachievement. From (10), we know that Assumption 1 is sufficient for FDSC. Here, from the errordetection perspective, we prove that Assumption 1 is also necessary. Let us consider the followingtwo possibilities.1) Bipolar error vector . Without loss of generality, assume there exists an error vector e September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 10 containing bipolar entries, i.e., e i < and e j > , i (cid:54) = j, i, j ∈ { , · · · , N } . Then, thereexists a positive vector h = [ e , · · · , ˆ e i , · · · , e N ] T such that (cid:0) e T h (cid:1) = 0 , where ˆ e i = − (cid:80) Nj =1 ,j (cid:54) = i e j /e i . In addition, for any ξ > , h ξ = ξ h is positive and (cid:0) e T h ξ (cid:1) = (cid:0) e T h ξ (cid:1) = ( ξ ) 0 = 0 . In other words, there exist two distinct signal vectors x and ˆx satisfying ( x − ˆ x ) T h = 0 . That is to say, this signal design is not able to provide theunique identification of the transmitted signals in the noise-free case, and, then the reliabledetection of the signal will not be guaranteed, even in a sufficiently high SNR. Therefore,the maximal decaying speed of the corresponding error curve can not be achieved. As amatter of fact, in this case, full large-scale diversity gain can not be attained.2) Error vector with zero entries . If n entries of e are zero, with index being N − n +1 , · · · , N without loss of generality, then, (cid:80) Mi =1 (cid:0) e T h i (cid:1) = (cid:80) Mi =1 (cid:16)(cid:80) N − nj =1 e j h ij (cid:17) . Therefore, theremust be M n entries of the channel matrix H that makes no contribution to PEP. In otherwords, all the degree of freedoms offered by the N × M MIMO-OWC are not utilizedand, as a consequence, full large-scale diversity can not be achieved.To sum up, if either e is bipolar or has zero-valued entries, the corresponding large-scalediversity gain will be less than M N . Hence, Assumption 1 is sufficient and necessary for FDSC,which is summarized as the following theorem:
Theorem 2:
A space code enables full large-scale diversity if and only if ∀ e ∈ E , e is unipolarwithout zero-valued entries or equivalently, ∀ e ∈ E , ee T is positive.It is known that for MIMO-RF, the maximal decaying is achieved if and only if the differencematrix of any two distinct codeword matrix is full-rank [24]. However, from Theorem 2, we cansee that a space-only transmission can assure the maximal decaying speed of the correspondingPEP, say, full large-scale diversity gain. The main reason for this significant difference is that theMIMO-OWC channels are nonnegative and the coding matrix is not necessary to be full-rank,which is verified by Theorem.Now, the questions raised at the end of Section II is answered. With these results, we canproceed to design FDSC systematically in the following section.IV. O PTIMAL D ESIGN OF S PECIFIC L INEAR
FDSCIn this section, we will exemplify our established criterion in (10) by designing a specific linear
FDSC for × MIMO-OWC with unipolar pulse amplitude modulation (PAM). For this
September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 11 particular design, a closed-form solution to the design problem of space codes optimizing bothdiversity gains will be obtained by taking advantage of the available properties on the successiveterms of Farey sequences in number theory as well as by developing new properties on thedisjoint intervals formed by Farey sequence terms.
A. Design Problem Formulation
Consider a × MIMO-OWC system with F ( s ) = Fs , where F = f f f f and ee T = e e e e e e . By Theorem 2, ee T should be positive to maximize the large-scalediversity gain. On the other hand, from the structure of ee T and (10), the small-scale diversitygain is D s ( e ) = | e e | under the assumption that CSIT is unknown. Therefore, to optimize theworst case over E , FDSC design is formulated as follows: max f ,f ,f ,f min e e e s.t. [ e , e ] T ∈ E , f ij > , i, j ∈ { , } ,e e > , f + f + f + f = 1 . (11)Our task is to analytically solve (11). To do that, we first simplify (11) by finding all thepossible minimum terms. B. Equivalent Simplification of Design Problem
For p -PAM, all the possible non-zero values of e e are e e = ( mf ± nf ) ( mf ± nf ) (cid:54) = 0 , m, n ∈ B p . (12)
1) Preliminary simplification:
After observations over (12), we have the following facts.1) ∀ m (cid:54) = 0 , m, n ∈ B p , it holds that ( mf + nf ) ( mf + nf ) ≥ f f . ∀ n (cid:54) = 0 , m, n ∈ B p , it is true that ( mf + nf ) ( mf + nf ) ≥ f f . September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 12 ∀ k (cid:54) = 0 , m + n (cid:54) = 0 , k, m, n ∈ B p , we have k ( mf − nf ) ( mf − nf )( mf − nf ) ( mf − nf ) ≥ . So, all the possible minimum of e e in (11) are f f , f f and ( mf − nf ) ( mf − nf ) ,where nm is irreducible, i.e., m ⊥ n . These terms are denoted by F = f f (cid:18) f f × f f (cid:19) , F = f f ,F mn = f f (cid:18) m f f − n (cid:19) (cid:18) m f f − n (cid:19) . After putting aside the common term, f f , we can see that F mn is the piecewise linear functionof f f and f f , respectively. So, (11) can be solved by fragmenting interval [0 , ∞ ) into disjointsubintervals. This fragmentation can be done by the breakpoints where F mn = 0 . To characterizethis sequence, there exists an elegant mathematical tool in number theory presented below.
2) Farey sequences:
First, we observe some specific examples of the breakpoint sequences.For OOK, the breakpoints , , ∞ . For 4-PAM, they are , , , , , , , , ∞ . For 8-PAM, wehave the breakpoint sequence with the former part being , , , , , , , , , , , , , , , , , , (13a)and the remaining being , , , , , , , , , , , , , , , , , ∞ (13b)Through these special examples, we find that the series of breakpoints before / (such as thesequence in (13a)) is the one which is called the Farey sequence [46]. The Farey sequence F k forany positive integer k is the set of irreducible rational numbers ab with ≤ a ≤ b ≤ k arrangedin an increasing order. The series of breakpoints after (such as the sequence in (13b)) is thereciprocal version of the Farey sequence. Thus, our focus is on the sequence before .The Farey sequence has many interesting properties [46], some of which closely relevant toour problem are given as follows. Proposition 1: If n m , n m and n m are three successive terms of F k , k > and n m < n m < n m ,then,1) m n − m n = 1 and m + m ≥ k + 1 .2) n + n m + m ∈ (cid:16) n m , n m (cid:17) and n m = n + n m + m . September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 13
However, having only Proposition 1 is not enough to solve our design problem in (11). Weneed to develop the other important new properties of Farey sequences, which will be utilizedin the FDSC design in the ensuing subsections.
Property 1:
Given k > , assume n m , n m , n m , n m ∈ F k and n m < n m < n m < n m . If n m and n m are successive, then, n + n m + m ≥ n m and n + n m + m ≤ n m .The proof of Property 1 is postponed into Appendix C. Property 2:
Assume n m , n m ∈ F k , k > and n m < n m . Then,1) n m < n + n m + m < n m holds.2) If f f , f f ∈ (cid:16) n m , n + n m + m (cid:17) , then, F m n < F m n .3) If f f , f f ∈ (cid:16) n + n m + m , n m (cid:17) , then, F m n > F m n .4) If f f = f f = n + n m + m , then, F m n = F m n .The proof of Property 2 is given in Appendix D.Using Properties 1 and 2, we attain the following property. Property 3: If n m and n m are successive in F k and f f , f f ∈ (cid:16) n m , n m (cid:17) , then, F m n and F m n are the two worst cases.The proof of Property 3 is provided in Appendix E. C. Techniques to Solve The Max-min Problem
Thanks to Farey sequences, (11) is transformed into a piecewise max-min problem with twoobjective functions. By solving this kind of problem, another main result of this paper can beformally presented as the following theorem.
Theorem 3:
The solution to (11) is determined by F F DSC = 12 + 2 p +1 p p , (14a)or F F DSC = 12 + 2 p +1 p p . (14b)The proof of Theorem 3 is postponed into Appendix F. September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 14
Theorem 3 uncovers the fact that the optimal linear space coded symbols are actually unipolar p -ary PAM symbols, since B p = { s +2 p s : s , s ∈ B p } . Therefore, in fact, we have provedthat RC [21] is optimal in the sense of the criterion established in this paper.V. C OMPUTER S IMULATIONS
In this section, we carry out computer simulations to verify our theoretical results. We firstexamine the performance bounds given in Lemma 1 and Theorem 1. As shown by Figs. 1 and2, it can be seen that in the high SNR regimes, the proposed upper-bounds have almost thesame negative slope as those of simulated results. In other words, the proposed upper-boundshave captured the behavior of the error curve with respect to the decaying speeds. As shown byFigs. 1 and 2, when SNR is sufficiently high, our proposed bounds have a horizontal shift to theright compared with the simulated results. Therefore, the tightness of the proposed asymptoticalbounds is mainly dependent on the precise estimate of some constant independent of SNR. −6 −5 −4 −3 −2 −1 B i t E rr o r R a t e simulation, σ =0.1upper−bound, σ =0.1simulation, σ =0.3upper−bound, σ =0.3 Fig. 1E
RROR PERFORMANCE OF
SISO-OWC
In the following, we simulate to verify our newly developed criterion in (10). In light ofour work being initiative, the only space-only transmission scheme available in the literatureis spatial multiplexing. Accordingly, we compare the performance of spatial multiplexing andFDSC specifically designed for × MIMO-OWC in Section IV. In addition, we suppose that
September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 15 −6 −5 −4 −3 −2 −1 Squared Optical SNR/dB A ve r a g e C od e w o r d E rr o r R a t e simulation, σ =0.1upper−bound, σ =0.1simulation, σ =0.3upper−bound, σ =0.3 Fig. 2E
RROR PERFORMANCE OF
MIMO-OWC h ij , i, j = 1 , are independently and identically distributed and let σ = σ = σ = σ = σ .These schemes are as follows:1) FDSC . The optical power is normalized in such a way that (cid:80) i,j =1 f ij = 2 yields E (cid:104)(cid:80) i,j =1 f ij s j (cid:105) = 1 . From (14), the coding matrix is F F DSC = .2) Spatial Multiplexing . We fix the modulation formats to be OOK and vary σ . So the rateis 2 bits per channel use (pcu). The transmitted symbols s , s are chosen from { , } equally likely. The average optical power is E [ s + s ] = 1 .We can see that both schemes have the same spectrum efficiency, i.e., 2 bits pcu and the sameoptical power. Through numerical results, we have following observations.1) Substantial enhancement from FDSC is achieved, as shown in Fig. 3. For σ = 0 . , theimprovement is almost 16 dB at the target bit error rate (BER) of − . For σ = 0 . ,the improvement is almost 6 dB at the target BER of − . Note that the small-scale gainalso governs the negative slope of error curve. The decaying speed of the error curve ofFDSC is exponential in terms of ln ρ ln ρ and that of spatial multiplexing is power-law withrespect to ρ . The reason for this difference is that FDSCs are full-diversity guaranteed bythe positive constraints in (11), whereas spatial multiplexing does not satisfy the positiverequirement in Theorem 2. September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 16 −4 −3 −2 −1 Squared Optical SNR/dB BE R × σ =0.1 σ =0.1 σ =0.3 σ =0.3 σ =0.01 σ =0.01SISO, σ =0.3 Spatial multiplexing
FDSC
Fig. 3BER
COMPARISONS OF
FDSC
AND SPATIAL MULTIPLEXING . −4 −3 −2 −1 Squared Optical SNR/dB BE R × σ =0.01 σ =0.1 σ =0.001benchmark σ =0.0001SISO, σ =0.3 SNR −1 Fig. 4BER
COMPARISONS OF SPATIAL MULTIPLEXING .
2) Spatial multiplexing presents only small-scale diversity gain illustrated in Fig. 4. By varyingthe variance of H , we find that in the high SNR regimes, the error curve decays as ρ − as long as the SNR is high enough. From σ = 0 . to σ = 0 . , the error curve hasa horizonal shift, which is the typical style of MIMO RF [24]. The reason is given as September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 17 follows. The equivalent space coding matrix is e T e = e e e e e e , e , e ∈ { , ± } with e + e (cid:54) = 0 . It should be noted that there exists two typical error events: e e = − and e e = 0 . From the necessity proof of Theorem 2, for e e = − , the attained large-scale diversity gain is zero, and at the same time, if e e = 0 with e + e (cid:54) = 0 , then,the attained large-scale diversity gain is only two for × MIMO-OWC. Therefore, theoverall large-scale diversity gain of spatial multiplexing is zero with small-scale diversitygain being attained.3) From Fig. 3 and 4, we notice that when σ increases, the error performances of repetitioncodes will worsen. For this phenomenon, the reason is that for repetition codes, thelarge-scale diversity is given by Ω = (cid:80) i,j =1 σ − ij , which dominates the exponentialdecaying speed. However, for spatial multiplexing, increasing σ will improves the errorperformance by providing a horizontal shift to the left. It is known that since an exacterror probability formula for OWC over log-normal fading channels is indeed hard to beobtained particularly for MIMO-OWC, it is very challenging to theoretically prove thatthe relationship between the error performance of spatial multiplexing and σ . Intuitivelyspeaking, the relationship between the error performance of spatial multiplexing and σ follows the radio frequency MIMO. That is to say, σ is the average electrical power of ln h . That’s why increasing σ will improve the error performance of spatial multiplexing,as shown by Figs. 3 and 4.We can see that the performance gain of MIMO-OWC if any will become larger againstincreasing SNR in a high enough regime. This implies that a slight improvement in thecoding structure will result in a significant enhancement of error performance in the high SNRregimes instead of only an horizonal shift to the left. Unique characteristics of MIMO-OWC areexperimentally uncovered and our established criterion are verified.VI. C ONCLUSION AND D ISCUSSIONS
In this paper, we have established a general criterion on the full-diversity space codedtransmission of MIMO-OWC for the ML receiver, which is, to our best knowledge, the firstdesign criterion for the full-diversity transmission of optical wireless communications withIM/DD over log-normal fading channels. Particularly for a × case, we have attained a September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 18 closed-form solution to the optimal linear FDSC design problem, proving that RC is the optimalamong all the linear space codes. Our results indicate that the transmission design is indeednecessary and essential for significantly improving overall error performance for MIMO-OWC.However, the design criterion and the specific code constructions for MIMO-OWC presented inthis paper are just initiative. Some significant issues are under consideration:1)
Our proposed criterion can be applicable to any non-linear designs. It remains openwhether there exists any better non-linear space code than RC .2)
It has been shown in this paper that the space dimension alone is sufficient for full large-scale diversity. Here, a natural question is: what kind of benefit can be obtained if space-time block code designs are used for MIMO-OWC ?3)
Like MIMO techniques for RF communications, what is the diversity-multiplexing tradeofffor MIMO-OWC? A PPENDIX
A. Proof of Lemma 1
Our main idea here is to split the whole integral in (5) into two parts by properly choosing τ > such that its dominant term can be extracted. In other words, the average SEP [43] canbe rewritten by P e ( ρ ) = (cid:90) τ Q (cid:18) √ ρhP op (cid:19) f H ( h ) dh + (cid:90) ∞ τ Q (cid:18) √ ρhP op (cid:19) f H ( h ) dh. (15)Now, we select τ to satisfy that when ρ → ∞ , τ → , to fragment (0 , ∞ ) into (0 , τ ) and ( τ, ∞ ) ,adaptively with SNR. This fragmentation is to find the dominant term of P e ( ρ ) in the high SNRregimes by giving the upper-bound and the lower bound of (cid:82) τ Q (cid:16) √ ρhP op (cid:17) f H ( h ) dh , and the upper-bounds of (cid:82) ∞ τ Q (cid:16) √ ρhP op (cid:17) f H ( h ) dh and then, examining their asymptotical behaviors related with τ . Temporarily, we take this fragmentation for granted and then, will explain the essential reasonlater on.
1) Upper-bound of SEP over (0 , τ ) : We integrate the first part of SEP in (15), which isdenoted by P τ ( ρ ) . Notice that when ρ → ∞ , τ → , and in this case, f H ( h ) is monotonically September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 19 increasing over (0 , τ ) . Hence, P τ ( ρ ) is upper-bounded by P τ ( ρ ) ≤ f H ( τ ) (cid:90) τ Q (cid:18) √ ρhP op (cid:19) dh ≤ f H ( τ )2 (cid:90) τ exp (cid:18) − ρh P op (cid:19) dh (16)where the last inequality is obtained by using the Chernoff bound on the Gaussian tail integral.Furthermore, we can upper-bound the last integral of (16) by (cid:90) τ exp (cid:18) − ρh P op (cid:19) dh = P op √ πρ − (cid:32) − Q (cid:32) (cid:112) ρτ P op (cid:33)(cid:33) ≤ P op √ π ρ − (17)which produces P τ ( ρ ) ≤ P op √ πf H ( τ )4 ρ − = P op exp (cid:16) σ (cid:17) √ σ ρ − exp (cid:32) − (ln τ + σ ) σ (cid:33) (18)
2) Lower-bound of SEP:
At the same time, when ρ → ∞ , we have τ ln ρ ≤ τ . This inequalityallows us to lower-bound P τ ( ρ ) by P τ ( ρ ) = (cid:90) τ Q (cid:18) √ ρhP op (cid:19) f H ( h ) dh ≥ (cid:90) τ ln ρ Q (cid:18) √ ρhP op (cid:19) f H ( h ) dh ≥ (cid:90) τ ln ρ Q (cid:18) √ ρτP op ln ρ (cid:19) f H ( h ) dh (19)where the last inequality follows from the monotonically decreasing property of Q -function.Then, by integrating f H ( h ) over (cid:16) , τ ln ρ (cid:17) , we have P τ ( ρ ) ≥ Q (cid:18) √ ρτP op ln ρ (cid:19) Q (cid:18) − ln τ + ln ln ρσ (cid:19) ≥ Q (cid:16) √ ρτP op ln ρ (cid:17) √ π σ ln ln ρτ exp − (cid:16) ln τ ln ρ (cid:17) σ (20)where the last inequality is obtained by using Q ( x ) ≥ √ πx exp (cid:18) − x (cid:19) , x ≥ √ . (21) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 20
3) Upper-bound of SEP over ( τ, ∞ ) : We now turn to the second part of P e ( ρ ) in (15), whichis denoted by ¯ P τ ( ρ ) . Letting f (cid:48) H ( h ) = 0 produces the point h satisfying ∀ h ≥ , f H ( h ) ≤ f H ( h ) = exp (cid:16) σ (cid:17) √ πσ . With this, ¯ P τ ( ρ ) can be upper-bounded by ¯ P τ ( ρ ) ≤ f H ( h ) (cid:90) ∞ τ Q (cid:18) √ ρhP op (cid:19) dh Using Chernoff bound on the Gaussian tail integral gives us ¯ P τ ( ρ ) ≤ f H ( h ) (cid:90) ∞ τ exp (cid:18) − ρh P op (cid:19) dh = 12 f H ( h ) √ πρ − / Q (cid:32) (cid:112) ρτ P op (cid:33) ≤ exp (cid:16) σ (cid:17) √ σ ρ − / exp (cid:18) − ρτ P op (cid:19) (22)
4) Determination of τ : Thus far, we have attained three bounds, respectively shown in (18),(20) and (22). Now, we examine their tightness related to τ . We can see that (18) and (20) havethe same exponential term (ln τ ) σ . To make the upper bounds in (18) to approach the upper-boundin (22) as tightly as possible, we seek to select such τ that ρτ P op = (ln τ + σ ) σ (23)An explicit solution to (23) with respect to τ is difficult to obtain. Accordingly, we propose toapproximate the solution to (23) by τ = P op ln ρ (cid:112) ρσ (24)which satisfies the following asymptotical equality. lim ρ →∞ ρτ P op (ln τ + σ ) σ = lim ρ →∞ (ln ρ ) (cid:16) ln ρ ln ρ − ln P op σ (cid:17) = 1 Finally, combining (18), (20), (22), and (24) leads to the fact that there exist three positiveconstants C L , C U and C U independent of ρ such that C L ρ − ln σ P op exp − (cid:16) ln ρ − ln P op σ (cid:17) σ ≤ P e ( ρ ) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 21 ≤
14 (ln ρ ) − exp − (cid:16) ln ρ ln ρ − ln P op σ (cid:17) σ + C U ρ − exp (cid:32) − (ln ρ ) σ (cid:33) (25)where C L = Q ( σ ) √ π , C U = and C U = exp (cid:16) σ (cid:17) √ σ . This is the complete proof of Lemma 1. (cid:3) B. Proof of Theorem 1
The condition that any e ∈ E is unipolar without zero entry implies that if ∀ τ > and ∀ e ∈ E , (cid:0) e T h i (cid:1) ≤ τ , then, we can have h ij ≤ τ | e j | , i = 1 , . . . , M, j = 1 , . . . , N . After thesepreparations, we adopt the same techniques as Subsection III-A. Temporally, assume that when ρ → ∞ , τ → . Similarly, P ( s → ˆ s ) can be adaptively fragmented with SNR as P ( s → ˆ s ) = (cid:90) ( e T h j ) ≤ τ P ( s → ˆ s | H ) f H ( H ) d H + (cid:90) ( e T h j ) >τ P ( s → ˆ s | H ) f H ( H ) d H . (26)The target in the ensuing subsections is to give the asymptotical bounds on P ( s → ˆ s ) followingthe similar procedures to the case of SISO-OWC.
1) Upper-bound of PEP over (0 , τ ) : To begin with, let us process the first part of P ( s → ˆ s ) in (26) denoted by P τ ( s → ˆ s ) . We know when ρ → ∞ , τ → . In this instance, f H ( h ij ) is monotonically increasing over (cid:16) , τ | e i | (cid:17) and then, f H ( h ij ) ≤ f H (cid:16) τ | e j | (cid:17) . Together with theChernoff bound of Q -function, P τ ( s → ˆ s ) can be upper-bounded by P τ ( s → ˆ s ) ≤ N (cid:89) j =1 M (cid:89) i =1 f H ij (cid:18) τ | e j | (cid:19) (cid:90) ( e T h i ) ≤ τ exp (cid:32) − ρ (cid:80) Mi =1 (cid:0) e T h i (cid:1) N P op (cid:33) d H (27a)In addition, by Assumption 1, e , · · · , e N have the same signs. This result allows us to furtherupper-bound P τ ( s → ˆ s ) by P τ ( s → ˆ s ) ≤ N (cid:89) j =1 M (cid:89) i =1 f H ij (cid:18) τ | e j | (cid:19) × (cid:90) h ij ≤ τ | ej | exp (cid:32) − ρ (cid:80) Nj =1 (cid:80) Mi =1 e j h ij N P op (cid:33) d H (27b)Further, integrating the last term in (27b) produces P τ ( s → ˆ s ) ≤ N (cid:89) j =1 M (cid:89) i =1 f H ij (cid:18) τ | e j | (cid:19) N (cid:89) j =1 M (cid:89) i =1 √ π (cid:18) ρ | e j | N P op (cid:19) − (cid:18) − Q (cid:18) ρτ N P op (cid:19)(cid:19) ≤ N (cid:89) j =1 M (cid:89) i =1 √ π f H ij (cid:18) τ | e j | (cid:19) (cid:18) ρ | e j | N P op (cid:19) − September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 22 = τ − MN (cid:81) Nj =1 (cid:81) Mi =1 (cid:113) σ ij (cid:18) ρN P op (cid:19) − MN exp (cid:32) − N (cid:88) j =1 M (cid:88) i =1 (ln τ − ln | e j | ) σ ij (cid:33) (27c)
2) Lower-bound of PEP over (0 , τ ) : To attain the lower bound of P τ ( s → ˆ s ) , we need thefollowing preparations. We observe that ee T is rank-one with the only non-zero eigenvalue being λ max = (cid:80) Nj =1 e j . Then, h Ti ee T h i ≤ λ max (cid:80) Nj =1 h ij , i = 1 , . . . , M . In other words, for all τ > ,it holds that { H : h i ∈ R N + , λ max N (cid:88) j =1 h ij ≤ τ , i = 1 , . . . , M }⊆ { H : h i ∈ R N + , h Ti ee T h i ≤ τ , i = 1 , . . . , M } (28a)Further, it is true that { H : 0 ≤ h ij ≤ τ √ N λ max , j = 1 , . . . , N, i = 1 , . . . , M }⊆ { H : λ max N (cid:88) j =1 h ij ≤ τ , i = 1 , . . . , M }⊆ { H : h i ∈ R N + , h Ti ee T h i ≤ τ , i = 1 , . . . , M } (28b)Now, by (28b), we can lower-bound P τ ( s → ˆ s ) by P τ ( s → ˆ s ) ≥ (cid:90) h ij ≤ τ √ N (cid:80) Nk =1 e k P ( s → ˆ s | H ) f H ( H ) d H ≥ Q √ M ρτ N P op (cid:113)(cid:80) Nk =1 e k (cid:90) h ij ≤ τ √ N (cid:80) Nk =1 e k f H ( H ) d H ≥ Q √ M ρτ N P op ln ρ (cid:113)(cid:80) Nk =1 e k N (cid:89) j =1 M (cid:89) i =1 Q − ln τ ln ρ √ N (cid:80) Nk =1 e k σ ij (29a)where the last inequality holds for high SNR such that ln ρ > . Again, by (21), we arrive atthe following lower-bound by P τ ( s → ˆ s ) ≥ exp (cid:0) MN (cid:1) (4 π ) MN Q √ M ρτ N P op ln ρ (cid:113)(cid:80) Nk =1 e k × (cid:81) Nj =1 (cid:81) Mi =1 σ ij ln MN ln ρ √ N (cid:80) Nk =1 e k τ exp − N (cid:88) j =1 M (cid:88) i =1 σ ij ln τ ln ρ (cid:113) N (cid:80) Nk =1 e k (29b) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 23
3) Upper-bound of PEP over ( τ, ∞ ) : Now, we are in a position to analyze P ( s → ˆ s ) − P τ ( s → ˆ s ) , i.e., the second term of (26), which is denoted by ¯ P τ ( s → ˆ s ) .Similarly, for h ij , f (cid:48) H ( h ij ) = 0 gives the extreme point h ij, of f H ( h ij ) , i.e., f H ( h ij ) ≤ f H ( h ij, ) , i = 1 , . . . , M, j = 1 , . . . , N . Then, we have f H ( H ) ≤ f H ( H ) and thus, ¯ P τ ( s → ˆ s ) can be upper-bounded by ¯ P τ ( s → ˆ s ) ≤ f H ( H )2 (cid:90) ( e T h i ) >τ exp (cid:32) − ρ (cid:80) Mi =1 h Ti ee T h i N P op (cid:33) d H (30a)In fact, (28b) also implies { H : h Ti ee T h i ≥ τ , i = 1 , . . . , M } ⊆ { H : h ij ≥ τ √ N (cid:80) Nk =1 e k , i =1 , . . . , M, j = 1 , . . . , N } . With this, ¯ P τ ( s → ˆ s ) can be further upper-bounded by ¯ P τ ( s → ˆ s ) ≤ f H ( H )2 (cid:90) h ij ≥ τ √ N (cid:80) Nk =1 e k exp (cid:32) − ρλ max (cid:80) Mi =1 (cid:80) Nj =1 h ij N P op (cid:33) d H = f H ( H )2 M (cid:89) i =1 M (cid:89) j =1 √ π (cid:32) ρ (cid:80) Nk =1 e k N P op (cid:33) − Q (cid:32)(cid:115) ρτ N P op (cid:33) ≤ exp (cid:18) (cid:80) Mi =1 (cid:80) Nj =1 σ ij (cid:19) (cid:81) Mi =1 (cid:81) Nj =1 σ ij (cid:32) ρ (cid:80) Nk =1 e k N P op (cid:33) − MN exp (cid:18) − M ρτ N P op (cid:19) (30b)So far, we have attained three bounds and now, examine their tightness to select τ in thefollowing subsection.
4) Selection of τ : It is noticed that the exponential terms of (27c) and (29b) are the same,i.e., (ln τ ) (cid:80) Mi =1 (cid:80) Nj =1 σ − ij . To mathch the bounds (27c) and (30b), τ is selected such that M ρτ N P op = (ln τ ) M (cid:88) i =1 N (cid:88) j =1 σ − ij (31)Since a closed-form solution to (31) is hard to attain. the solution to (31) is approximated below τ = (cid:118)(cid:117)(cid:117)(cid:116) N P op M M (cid:88) i =1 N (cid:88) j =1 σ − ij ln ρ √ ρ (32)For the selection in (32), the following asymptotical equality holds. lim ρ →∞ Mρτ NP op (ln τ ) (cid:80) Mi =1 (cid:80) Nj =1 σ − ij = lim ρ →∞ (ln ρ ) (cid:16) ln ρ ln ρ − ln (cid:16) NP op M (cid:80) Mi =1 (cid:80) Nj =1 σ − ij (cid:17)(cid:17) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 24 C L (ln ρ ) − MN e − (cid:80) Mi =1 (cid:80) Nj =1 ( ln ρ +ln ( P op Ω ) − ln ( M (cid:80) Nk =1 e k )) σ ij (cid:124) (cid:123)(cid:122) (cid:125) P L ( s → ˆ s ) ≤ P ( s → ˆ s ) ≤ C U (ln ρ ) − MN e − (cid:80) Mi =1 (cid:80) Nj =1 (cid:18) ln ρ ln2 ρ +ln ( P op Ω ) − ln e j (cid:19) σ ij (cid:124) (cid:123)(cid:122) (cid:125) P U ( s → ˆ s ) + C U ρ − MN e − (cid:80) Mi =1 (cid:80) Nj =1 ln2 ρ σ ij (cid:124) (cid:123)(cid:122) (cid:125) P U ( s → ˆ s ) (33) = 1 Then, putting (27c), (29b), (30b) and (32) together, we are allowed to arrive at the fact that thereexists three positive constants C L , C U and C U , independent of ρ shown by (33), postponedto the top of the next page, where Ω = M (cid:88) i =1 N (cid:88) j =1 σ − ij (34a) C L = (cid:81) Mi =1 (cid:81) Nj =1 σ ij (4 π ) MN exp (cid:0) − MN (cid:1) Q (cid:32) N (cid:88) k =1 e k (cid:33) − (34b) C U = (cid:0) N P op (cid:1) MN (cid:81) Mi =1 (cid:81) Nj =1 (cid:113) σ ij exp (cid:18) − Ω8 ln (cid:18) N P op Ω M (cid:19)(cid:19) (34c) C U = exp (cid:18) (cid:80) Mi =1 (cid:80) Nj =1 σ ij (cid:19) (cid:81) Ni =1 (cid:81) Mj =1 σ ij (cid:32) (cid:80) Nk =1 e k N P op (cid:33) − MN (34d)Now, we can see that in (33), P L ( s → ˆ s ) and P U ( s → ˆ s ) have the same exponential term, exp (cid:0) − Ω8 ln ρ (cid:1) , whereas the exponential term of P U ( s → ˆ s ) is exp (cid:16) − Ω8 ln ρ ln ρ (cid:17) , which decaysslower than exp (cid:0) − Ω8 ln ρ (cid:1) against high SNRs. That being said, we have attained the dominantterm, P U ( s → ˆ s ) , of the upper-bound of P ( s → ˆ s ) . This completes the proof of Theorem 1. (cid:3) C. Proof of Property 1
For m ⊥ n , m ⊥ n and m ⊥ n , we have that n + n m + m ≥ n m ⇔ m n − m n ≥ m n − m n September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 25
This is indeed true, since we obtain m n − m n = 1 by Lemma 1 and m n − m n ≥ fromthe assumption that n m < n m .In the same token, given m ⊥ n , m ⊥ n and m ⊥ n , we have that n + n m + m ≤ n m ⇔ m n − m n ≥ m n − m n This indeed holds, since we can attain m n − m n = 1 by Lemma 1 and m n − m n ≥ from the assumption that n m < n m . This completes the proof of Property 1. (cid:3) D. Proof of Property 2
First, Statement 1: n m < n + n m + m < n m can be verified by n + n m + m − n m = m ( m + m ) (cid:18) n m − n m (cid:19) < ,n + n m + m − n m = m ( m + m ) (cid:18) n m − n m (cid:19) > . When f f , f f ∈ (cid:16) n m , n m (cid:17) , we can have ( m f − n f ) > and ( m f − n f ) < . Hence, | m f − n f | − | m f − n f | = ( m + m ) f (cid:18) f f − n + n m + m (cid:19) (35a)and | m f − n f | − | m f − n f | = ( m + m ) f (cid:18) f f − n + n m + m (cid:19) . (35b)Combining (35a) and (35b) leads us to the following facts:1) If f f , f f ∈ (cid:16) n m , n + n m + m (cid:17) , then, we have − m f + n f > m f − n f > , − m f + n f > m f − n f > Thus, F m n < F m n holds.2) If f f , f f ∈ (cid:16) n + n m + m , n m (cid:17) , then, we arrive at m f − n f > − m f + n f > m f − n f > − m f + n f > . Hence, we have F m n > F m n .3) If f f = f f = n + n m + m , then F m n = F m n .This completes the proof of Property 2. (cid:3) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 26
E. proof of Property 3 ∀ n m , n m ∈ F k satisfying n m < n m < n m < n m , when f f , f f ∈ (cid:16) n m , n m (cid:17) , we have F m n , F m n , F m n , F m n > . Property 1 tells us that n + n m + m ≥ n m and n + n m + m ≤ n m . Asa result, we can have (cid:16) n m , n m (cid:17) ⊆ (cid:16) n + n m + m , n m (cid:17) ⊆ (cid:16) n m , n m (cid:17)(cid:16) n m , n m (cid:17) ⊆ (cid:16) n m , n + n m + m (cid:17) ⊆ (cid:16) n m , n m (cid:17) Using Property 2, we can attain F m n > F m n and F m n < F m n . The successivity of n m and n m ensures F m n and F m n are the last two minimum over (cid:16) n m , n m (cid:17) . This completes theproof of Property 3. (cid:3) F. proof of Theorem 3
Assume n m and n m are varying two successive items of F p − and denote x = f f and y = f f .These notations produce f (1 + x ) + f (1 + y ) = 1 . We firstly prove that if x, y ∈ (cid:16) n m , n m (cid:17) ,then the local solution to (11) is determined by x = y = n + n m + m ,f = f = m + m m + m + n + n ) (36)Letting ( xm − n ) ( ym − n ) ≤ ( xm − n ) ( ym − n ) yields x ≤ n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) .In the same token, ( xm − n ) ( ym − n ) ≥ ( xm − n ) ( ym − n ) implies y ≤ n ( xm − n ) − n ( xm − n ) m ( xm − n ) − m ( xm − n ) . Then, from Property 2 and Property 3, we have that if x, y ∈ (cid:16) n m , n m (cid:17) ,then (11) can be equivalent to the following two sub-problems:1) If x ≤ n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) , then (11) is equivalent to the following problem max f ,f ,f ,f F m n s.t. x ≤ n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) ,x, y ∈ (cid:16) n m , n m (cid:17) ,f + f + f + f = 1 . (37)Over the feasible region of x and with the increasing property of F m n , we can have F m n ≤ (cid:18) m n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) − n (cid:19) f f ( ym − n ) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 27 = m y ( m n − m n ) + n ( m n − m n ) m ( ym − n ) − m ( ym − n ) f f ( ym − n )= − f f ( ym − n ) ( ym − n ) m ( ym − n ) − m ( ym − n ) where the last equality is attained by using m n − m n = 1 and the equality holdswhen x = n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) . This upper bound of F m n , F f f is denoted by f ( y ) andits derivative is f (cid:48) ( y ) = − (( m + m ) y − ( n + n ))( m ( ym − n ) − m ( ym − n )) (( m − m ) y − ( n − n )) Letting f (cid:48) ( y ) = 0 yields that y = n − n m − m or n + n m + m . Now, we prove y = n + n m + m is themaximum point. Firstly, we verify that n − n m − m is outside (cid:16) n m , n m (cid:17) . We can have n − n m − m − n m = − m ( m − m ) n − n m − m − n m = − m ( m − m ) From [46], we know m (cid:54) = m . Therefore, for the sign of m − m , we have the followingtwo possibilitiesa) If m > m , then n − n m − m < n m < n m . We can have that over (cid:16) n m , n + n m + m (cid:17) , f (cid:48) ( y ) ispositive and over (cid:16) n + n m + m , n m (cid:17) , f (cid:48) ( y ) is negative.b) If m < m , then n − n m − m > n m > n m . Also, we can have that over (cid:16) n m , n + n m + m (cid:17) , f (cid:48) ( y ) is positive and over (cid:16) n + n m + m , n m (cid:17) , f (cid:48) ( y ) is negative.These observations indicate that y = n + n m + m is the maximum point. Combining y = n + n m + m and x = n ( ym − n ) − n ( ym − n ) m ( ym − n ) − m ( ym − n ) gives x = y = n + n m + m .2) If y ≤ n ( xm − n ) − n ( xm − n ) m ( xm − n ) − m ( xm − n ) , then (11) is transformed into the following problem max f ,f ,f ,f F m n s.t. y ≤ n ( xm − n ) − n ( xm − n ) m ( xm − n ) − m ( xm − n ) ,x, y ∈ (cid:16) n m , n m (cid:17) ,f + f + f + f = 1 . (38)In the same token, we can have the solution to (38) is x = y = n + n m + m .Putting things together tells us that with x, y ∈ (cid:16) n m , n m (cid:17) the local solution to (11) is f f = f f = n + n m + m (39) September 25, 2015HANG ET.AL, SPACE CODES FOR MIMO OWC: ERROR PERFORMANCE CRITERION AND CODE CONSTRUCTION 28
It follows that F m n = F m n from Property 2.Now, it is time to give the analytical FDSC. Combining (39) with f + f + f + f = 1 produces f + f = m + m m + m + n + n . In addition, using the geometrical and arithmetical inequality: √ ab ≤ a + b for a, b ≥ , we obtain F m n = f f ( m x − n ) ( m y − n ) ≤ ( f + f ) (cid:18) m n − m n m + m (cid:19) = ( m n − m n ) m + m + n + n ) where the equality holds if and only if f = f = n + n m + m + n + n ) and f = f = m + m m + m + n + n ) . Then, the local maximum value of the objective function is given by max x,y ∈ (cid:16) n m , n m (cid:17) min e e e = 14 ( m + m + n + n ) . (40)where we use m n − m n = 1 from Proposition 1. On the other hand, Proposition 1 reveals m + m + n + n ≥ p . This inequality implies max F min e e e ≤
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