On the Ergodic Rate for Compute-and-Forward

Transcription

1 O the Ergodic Rate for Compute-ad-Forward Ami Sakzad, Emauele Viterbo, Yi Hog Dept. Elec. & Comp. Sys. Moash Uiversity, Australia Joseph Boutros Dept. Electrical Egieerig, Texas A&M Uiversity at Qatar Abstract A key issue i compute-ad-forward for physical layer etwork codig scheme is to determie a good fuctio of the received messages to be reliably estimated at the relay odes. We show that this optimizatio problem ca be viewed as the problem of fidig the closest poit of Z[i to a lie i the -dimesioal complex Euclidea space, withi a bouded regio aroud the origi. We the use the complex versio of the LLL lattice basis reductio CLLL algorithm to provide a reduced complexity suboptimal solutio as well as a upper boud to the miimum distace of the lattice poit from the lie. Usig this boud we are able to fid a lower boud to the ergodic rate ad a uio boud estimate o the error performace of a lattice costellatio used for lattice etwork codig. We compare performace of the CLLL with a more complex iterative optimizatio method as well as with a simple quatized search. Simulatios show how CLLL ca trade some performace for a lower complexity. Idex Terms Ergodic rate, compute-ad-forward, CLLL algorithm, quatized error, successive refiemet. I. INTRODUCTION AND MOTIVATION Suppose that a complex iteger lattice Λ = Z[i ad a vector h C are give. We cosider the followig optimizatio problem mi Qa, α a,α Λ C where Qa, α σ α + αh a. The optimizatio problem i has bee ivestigated i the recet physical layer etwork codig schemes [, 4, 6 0,, 5 ad MIMO liear receivers [6. I [4, 8, the optimal o-zero Gaussia iteger vector a is chose based o a exhaustive search withi the ball of radius + h σ ad the the optimum α was give to be the MMSE coefficiet i order to maximize the computatio rate of a real-valued AWGN etwork. However, it has bee show i [ that the degree of freedom for compute ad forward is less tha. I [, the search for the optimum vector a is modeled as a shortest vector problem for a lattice Λ. However, a efficiet approach to joitly fid the optimum value of a, α is ot preseted i these papers. It is clear that for small σ the searchig regio is large ad fidig the optimal a is expesive i terms of computatioal complexity. For a set X, we let X = X {0}. I our paper, we cosider the lattice reductio strategy to miimize Qa, α, a critical problem appeared i compute-ad-forward protocol [8 ad lattice etwork codig scheme [4. Differet from previous approaches i [4, 8,, we restrict our result to o-zero vector a Z[i ad ozero scalar α C. We summarize our cotributios as follows. We use the complex versio of the LLL lattice reductio algorithm CLLL algorithm [5 to joitly fid a, α that miimize Qa, α. The algorithm also provides a upper boud o Qa, α. Furthermore, the CLLL algorithm eables us to fid simultaeously a, α Z[i Z[i with much lower complexity, whe compared to the simple quatized approach. We defie the ergodic rate for compute-ad-forward protocol i physical layer etwork codig [8. We derive a lower boud o the ergodic rate i terms of a, α derived from the CLLL algorithm [5. We compare this boud with the ergodic capacity [3 of a multiple access chael MAC-MISO upper boud. We also study the average error probability of a lattice etwork protocol [4 ad derive a uio boud estimate o the average error probability. We propose ad compare the followig three search methods to fid a ad α that miimize Qa, α. The aive algorithm uses a simple quatized search over all possible complex values of α. The quatizig αh yields the iteger vector a. The CLLL algorithm for a + -dimesioal complex lattice is cosidered. I order to relax α from Z[i to C, we propose a successive refiemet search aroud the iitial iteger value of α. 3 A iterative MMSE based [8 quatizatio approach is used to refie the α provided by. The rest of the paper is orgaized as follows. I Sectio II, we review the CLLL-reduced basis of [5. I Sectio III, we study the miimizatio problem with applicatios to computead-forward protocol [8 ad lattice etwork code desig [4. I Sectio IV, we preset solutios to the optimizatio problem, which are based o the successive refiemet of CLLL ad a iterative MMSE-based quatizatio. I Sectio V, we show some experimetal results. We coclude our results i Sectio VI. Notatios. Boldface letters are used for colum vectors, ad capital boldface letters for matrices. We let Z, C, R, ad

2 Z[i deote the rig of ratioal itegers, the field of complex umbers, the field of real umbers, ad the rig of Gaussia itegers, respectively, where i =. We let I deote a idetity matrix. We let the operatios T ad H deote traspose ad Hermitia traspositio. The operatios R ad I deote the real ad imagiary parts of a complex umber. We let deote the absolute value of a real umber, or the modulus of a complex umber. The operatio deotes the Euclidea orm of a vector. The Hermitia product of two vectors a ad b is deoted by a, b b H a. The set of orthogoal vectors geerated by the Gram-Schmidt orthogoalizatio procedure are deoted by {b GS, b GS,..., b GS } spaig the same space of {b, b,..., b }. The operatio E deotes mea of a radom variable. We let x deote the closest iteger to x. II. COMPLEX LATTICE BASIS REDUCTION A complex lattice Λ with basis {g, g,..., g }, where g k C, icludes poits represeted as a liear combiatio of basis vectors with Gaussia iteger coefficiets. Let us defie the geerator matrix for Λ as the complex matrix G [ g g g. Oe ca express Λ as {p = Gz z Z[i }. I the lattice reductio, we let B = GU, where U is a uimodular matrix. Let us defie µ l,j = < b l, b GS j > b GS j where l, j. A geerator matrix B = [ b b b is said to be CLLL-reduced if the followig two coditios are satisfied [5 for j < l ; Rµ l,j / Iµ l,j / b GS k δ µ k,k b GS k where < k, δ /, is a factor selected to achieve a good quality-complexity tradeoff. I [5, a algorithm was itroduced to compute a CLLLreduced basis matrix B for a lattice Λ with a give geerator matrix G. Specifically, for a iput matrix G ad a factor δ, the algorithm outputs the CLLL-reduced basis matrix B ad the uimodular matrix U such that B = GU. It was show i [5 that the first colum of B, deoted by b, satisfies b β 4 volλ where β = /δ / ad volλ is the volume of the lattice Λ. This provides a upper boud to the legth of the shortest vector of a complex lattice Λ. The tightest upper boud is foud for δ =, which leads to β =. III. MAIN RESULTS I this sectio, we first solve the optimizatio problem usig CLLL algorithm. We the use this solutio to derive a lower boud o the ergodic rate of compute-ad-forward protocol for physical layer etwork codig. Similarly, we derive a uio boud estimate o the average probability of decodig error for physical layer etwork codig. A. Methodology Let h = h,..., h T C ad e l be the l-th uit vector i C +. Let us defie a + dimesioal complex vector as h = h,..., h, σ T for some oise variace σ. Let Λ be the lattice geerated by It follows that G = [ e e e h. volλ = detg = σ. The + dimesioal CLLL-reduced basis geerator matrix B = [ b b b b + of Λ ca be computed usig the aforemetioed algorithm i [5. For example, for the first colum of B, we ca compute where b volλ + = σ + 3 b = a e + + a e + a + h for some Gaussia itegers a l s, l +. Defiig a = a,..., a ad takig a + = α, we obtai B. Applicatio b = αh a + σ α = Qa, α σ +. 4 The above optimizatio ca be straightforwardly applied to the compute-ad-forward protocol over fiite rigs [8, 4. I [8, the optimizatio problem is i fact to maximize the computatio rate [8 Rh, a = max α C log+ σ α + αh a where log + x = max{logx, 0}. Maximizig 5 is equivalet to miimizig Qa, α. Usig, we defie R h max a,α Λ C log+ 6 Qa, α The the defiitio of the ergodic rate is give below. Defiitio : The ergodic rate R e of a compute-adforward protocol is defied as R e E R h, where the mathematical expectatio is take over the chael coefficiet vector h. We have the followig theorem. 5

3 Theorem : The ergodic rate of the compute-ad-forward protocol is lower bouded by R e log + σ + whe the CLLL algorithm discussed i Sectio III-A is used to fid a, α, give h. Proof: The CLLL algorithm determies a, α accordigly for a give h. Thus the ergodic rate is averaged over various chael coefficiets h. Moreover, sice the fuctio gx log/x is a decreasig covex fuctio for a positive umber x, we have R e = E R h = E max a,α Λ C log+ Qa, α E log + σ α + αh a log + Eσ α + αh a 7 log +. 8 σ + where the first iequality follows from the fact that a, α is suboptimal. The iequality i 7 holds because of Jese s iequality ad the iequality i 8 holds because of 4 ad the fact that gx is a mootoe fuctio. I the compute-ad-forward protocol, we assume that each relayig ode receives data trasmitted simutaeously from users each with a sigle atea. This sceario is very similar to a user MAC case with oe trasmit atea for each user ad oe receiver atea at the destiatio termial. I this protocol, assumig that x l is trasmitted by the l th user, l, over a fadig chael with coefficiet h l, the received sigal at the relayig ode is give by y = h l x l + CN 0, σ, l= ad a upper boud o the istataeous capacity is MISO boud C h = log + h l /σ l= Cosiderig h l CN 0,, we have the upper boud o the ergodic capacity MISO boud EC h 0 log + t/σ e t t /! dt. 9 I the ext sectio we will compare the ergodic rate R e ad the ergodic capacity EC h umerically. Let Λ/Λ be a -dimesioal ested complex lattice costellatio ad u be the desired liear combiatio of messages with coefficiet vector a Z[i, ad α be the scalar. The the uio boud estimate of the coditioal probability of decodig error is give by [4 d P e h, a K exp 4σ α + 4 αh a 0 where d is the miimum ier coset distace ad K is the umber of shortest vectors i the set Λ Λ. Accordig to [, ote that the sig ca be replaced by the otatio. This meas a approximate upper boud, oe that becomes closer ad closer to a true boud as σ goes to zero. Miimizig P e h, a i 0 is equivalet to miimizig Qa, α by choosig the proper vector a ad scalar α. By substitutig the upper boud 4 to the uio boud estimate 0, we have the followig result. Theorem 3: Let Λ/Λ be a ested lattice code ad u be the desired liear combiatio of messages with coefficiet vector a 0 ad α 0 be the scalar both comig from CLLL algorithm. The the uio boud estimate for the average probability of decodig error is E P e h, a K exp d σ + + IV. SEARCHING APPROACHES. I [8, for a give a Z[i, the optimum value of α is give by SNR < h, a > α MMSE = + SNR h where SNR = σ. However the problem of fidig o-zero a Z[i is ot well-addressed. I fact, the authors of [8 suggest to fid this vector by doig a brute force searchig i a bouded sphere of square radius + SNR h. Here, we propose ad aalyze the followig search techiques to fid the best a ad α. I the followig proposed methods except successive refiemet of CLLL, we discretize α i its orm ad phase ad let the absolute value of α vary from to a maximum value ad phase of α vary from 0 to 89 degrees. A. Simple Quatized Search To obtai a = {a l }, l, we quatize a l = αh l. We the compute Qa, α ad select the lowest oe ad its correspodig a ad α. Note that i this case a is a fuctio of α. We call this method as simple quatized search. Specifically, this approach ivolves two passes: We let α vary from to a maximum value i iteger steps ad the phase of α vary from 0 to 89 degrees. We let α be the value that miimizes Qa, α. We refie the search aroud this α i steps of 0. aroud the real ad imagiary parts i order to fid smaller values of Qa, α. Note that i this case a is a fuctio of α.

4 B. Successive Refiemet of CLLL The CLLL algorithm fids the best α Z[i ad a Z[i simultaeously, while previous techique performs a compoetwise quatizatio of the vector αh for each α. Sice the derived α is a Gaussia iteger, we perform successive refiemet to search for the best α i C. I particular, we search aroud the best value of α delivered by the CLLL algorithm to fid smaller values of Qa, α. Let α = α r + iα i be such value. We the compute Qa, α, where α = α r + iα i, α r < α r < α r + ad α i < α i < α i +. The α i ad α r are chose i steps of η = 0.0 i our simulatios. If oe results i smaller Qa, α < Qa, α, we replace α by α. We refer this method as successive refiemet of CLLL. C. Iterative MMSE-based Quatizatio As i the first method we search through may values of α to fid the best vector a by quatizig αh compoetwise. For each α we substitute the vector a ito to compute α MMSE. We let α = α MMSE ad iterate the above procedure util we reach a maximum umber of iteratios. We select the best α ad a that miimize Qa, α. This method is amed iterative MMSE-based quatizatio. D. Complexity Compariso Let us assume that the ergodic rate ad the parameter Qa, α for M samples of -dimesioal chael vector h at a fixed SNR are of iterest. We fid the complexity of the above three proposed methods ad the exhaustive search provided i [4, 8. A Sice we search for all possible o-zero iteger vectors where their square orms are less tha + SNR h, the complexity of the brute force search i [4, 8 is of order O M SNR. B At high SNR s, the optimal α MMSE is roughly equivalet to a h. Therefore, the search space for α ca be upper bouded by SNR. The best complexity for a sortig algorithm for each h is O M SNR log SNR. Hece the complexity order of simple quatized search ad iterative MMSE-based quatizatio is at most O M SNR log SNR. C The complexity of successive refiemet of CLLL equals to the complexity of fidig a + -dimesioal complex reduced basis which is O M + log +. The complexity of successive refiemet of CLLL is lower tha all the other schemes, sice it oly varies by. However the complexity of the other methods is based o SNR. We ote that the highest complexity is for exhaustive optimal search preseted i [4, 8. V. EXPERIMENTAL RESULTS We evaluate ad compare the performace of the differet methods for the -dimesioal complex space i.e., the two user case. The compoets of h are assumed to be circularlysymmetric complex Gaussia, with mea zero ad uit variace, as i the Rayleigh fadig chael model. For the simple quatized search, we let α vary from to 300 i iteger steps ad the phase of α vary from 0 to 89 by steps of degrees. I a secod pass we refie the search aroud α i steps of 0.05 to fid smaller values of Qa, α. Similar refiemet was performed for the CLLL method. For the iterative MMSE method we selected a maximum of 0 iteratios. CCDF Origial exhaustive search Quatized Error Plus Sacaled Noise Fig.. Complemetary cumulative distributio fuctio for Qa, α at SNR = 0 db. The complex chael gai is circularly-symmetric Gaussia distributed CN 0,. Figs. ad compare the complemetary cumulative distributio CCDF of the miimum Qa, α at SNR = 0 ad0 db respectively. Fig. 3 shows the CCDF of the miimum Qa, α at SNR = 40 db for the three methods. We observe that iterative MMSEbased quatizatio method exhibits the best performace at the highest complexity. The CLLL method has the lowest complexity but exhibits a slight degradatio. I high SNR s the curve of CLLL algorithm via successive refiemet breaks the simple quatized search oe. I Fig. 4, for differet values of SNR, we plot the upper boud σ + / agaist the average values of the miimum Qa, α s for simple quatized search, successive refiemet of CLLL ad iterative MMSE-based quatizatio. The bars o curves show the stadard deviatio of the miimum Qa, α s for that specific SNR. Fially, a compariso betwee the ergodic rates for the differet strategies is give i Fig. 5. For compariso we show the capacity of a two-user MAC ad the lower boud i 8.

5 Upper boud CCDF Quatized Error Plus Scaled Noise Quatized Error Plus Sacaled Noise Fig.. Complemetary cumulative distributio fuctio for Qa, α at SNR = 0 db. The complex chael gai is circularly-symmetric Gaussia distributed CN 0,. Exhaustive search is itractable at this SNR value. CCDF Quatized Error Plus Sacaled Noise Fig. 3. Complemetary cumulative distributio fuctio for Qa, α at SNR = 40 db. The complex chael gai is circularly-symmetric Gaussia distributed CN 0,. Exhaustive search is itractable at this SNR value. VI. CONCLUSIONS A ovel method is itroduced to fid the closest poit of a Z[i lattice to a lie, withi a bouded regio aroud origi. This is also used to maximize the computatio rate of a compute-ad-forward protocol for a physical layer etwork SNRdB Fig. 4. Compariso betwee average miimum Qa, α for the differet approaches. The complex chael gai is circularly-symmetric Gaussia distributed CN 0,. Exhaustive search is itractable at this SNR value. codig. A lower boud o the ergodic rate ad a estimatio of the error performace of a lattice costellatio for lattice etwork schemes is obtaied. We propose three methods for solvig the optimizatio problem for Qa, α: simple quatized search, successive refiemet of CLLL ad iterative MMSE-based quatizatio. The successive refiemet of CLLL has two specific properties differet from the simple quatized search ad the iterative MMSE-based quatizatio. First, CLLL algorithm determies the vector a, α i oe step. Secod, the complexity of this method is lower tha the others. Simulatios are carried out to reveal the effectiveess of CLLL algorithm alog with successive refiemet. The result of usig successive refiemet of CLLL is approximately equivalet to origial exhaustive search for low SNR. I particular, successive refiemet of CLLL trades a little bit of performace for a reduced complexity. However the simple quatized search ad iterative MMSE-based quatizatio method outperforms the CLLL setups. Give the vector a, the best α is the oe delivered by, which is equivalet to a h at low σ s. Sice a has to large [, if h be very small, the α MMSE will be large as well. This caot happe because α MMSE has to be upper bouded by σ. This issue arises whe we fix a for each h ad differetiate Qa, α to get the best α. This is also impractical i ergodic situatio because fidig the best a is costly at low σ s. The experimetal results suggest us to fid α first ad the put a = αh. This alog with successive refiemet give better performace tha the above approach

6 Bits Capacity Lower boud SNRdB Fig. 5. Compariso betwee ergodic rates for the differet approaches. Twouser MAC capacity ad lower boud derived based o CLLL are also plotted. at much lower complexity. Overall, we thik the solutio of miimizig Qa, α has to be foud joitly, because a ad α are depedet i geeral. I future work, we will geeralize these methods to wireless etwork codig over fiite rigs [4. We will further study good lattice costellatios derived from strog ifiite lattices icludig the oes itroduced i [3. VII. ACKNOWLEDGMENTS This work was performed at the Moash Software Defied Telecommuicatios Lab supported by the Taleted Ehacemet Scheme through the Moash Professorial Fellowship MPF program. REFERENCES [ S. Avestimehr, S. Diggavi, ad D. Tse, Wireless etwork iformatio flow: A determiistic approach, IEEE Tras. If. Theory, vol. 57, pp , 0. [ S. Beedetto ad E. Biglieri, Priciples of Digital Trasmissio, with Wireless Applicatios Kluwer Academics/ Pleum Publishers, 999. [3 E. Biglieri, J. Proakis, ad S. Shamai, Fadig Chaels: Iformatio-Theoretic ad Commuicatios Aspects, IEEE Tras. If. Theory, vol. 44, No. 6, pp , 998. [4 C. Feg, D. Silva, ad F.R. Kschichag, A Algebraic Approach to Physical-Layer Network Codig, submitted to IEEE Tras. If. Theory, arxiv: v. [5 Y. H. Ga, C. Lig, ad W. H. Mow, Complex Lattice Reductio Algorithm for Low-Complexity Full-Diversity MIMO Detectio, IEEE Tras. o Sigal Processig, vol. 57, o. 7, pp , 009. [6 S.-C. Liew, S. Zhag, ad L. Lu, Physical-layer etwork codig: Tutorial, survey, ad beyod, i Phys. Commu., 0 [Olie. Available: [7 S.H. Lim, Y.-H. Kim, A. El Gamal, ad S.-Y. Chug, Noisy etwork codig, IEEE Tras. If. Theory, vol. 57, pp , 0. [8 B. Nazer ad M. Gastpar, Compute-ad-Forward: Haressig Iterferece through Structured Codes, IEEE Tras. o If. Theory, vol. 57, o. 0, pp , 0. [9 K. Narayaa, M.P. Wilso, ad A. Spritso, Joit physical layer codig ad etwork codig for bi-directioal relayig, i Proc. 45th Aual Allerto Coferece o Commuicatios, Cotrol ad Computig, Moticello, IL, Sep [0 U. Niese, B. Nazer, ad P. Whitig, Computatio Aligmet: Capacity Approximatio without Noise Accumulatio, submitted to IEEE Tras. If. Theory, arxiv: 08.63v. [ U. Niese ad P. Whitig, The Degrees of Freedom of Compute-ad-Forward, to appear i IEEE Tras. If. Theory, arxiv:0.8v. [ A. Osmae ad J.C. Belfiore, The Compute-ad- Forward Protocol: Implemetatio ad Practical Aspects, submitted to IEEE Commuicatios Letters, arxiv: v. [3 A. Sakzad, M.-R. Sadeghi, ad D. Paario, Costructio of turbo lattices, i Proc. 48th Aual Allerto Coferece o Commuicatio, Cotrol, ad Computig, Allerto, Chicago, USA, pp. 4, 00. [4 E. Viterbo, Y. Hog, ad J. Boutros, Wireless Network Codig over Fiite Rigs, Workshop o Algebraic Structure i Network Iformatio Theory, Baff Iteratioal Research Statio, Aug. 0. [5 M.P. Wilso, K. Narayaa, H. Pfister, ad A. Spritso, Joit physical layer codig ad etwork codig for bidirectioal relayig, IEEE Tras. If. Theory, vol., pp , Nov. 00. [6 J. Zha, B. Nazer, U. Erez, ad M. Gastpar, Itegerforcig Liear Receivers, Iformatio Theory Proceedigs ISIT, 00 IEEE Iteratioal Symposium o, pp. 0 06, 00.