Research Article Analytical Computation of Information Rate for MIMO Channels

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1 Hindai Computer etorks and Communications Volume 7, Article ID 6958, 6 pages Research Article Analytical Computation of Information Rate for MIMO Channels Jinbao Zhang, Zhenhui Tan, and Song Chen Beijing Jiaotong University, Beijing, China Correspondence should be addressed to Jinbao Zhang; jbzhang@bjtu.edu.cn Received 5 September 6; Revised December 6; Accepted January 7; Published 8 February 7 Academic Editor: Peng Cheng Copyright 7 Jinbao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited. Information rate for discrete signaling constellations is significant. Hoever, the computational complexity makes information rate rather difficult to analyze for arbitrary fading multiple-input multiple-output (MIMO) channels. An analytical method is proposed to compute information rate, hich is characterized by considerable accuracy, reasonable complexity, and concise representation. These features ill improve accuracy for performance analysis ith criterion of information rate.. Introduction Information rate plays an important role in performance analysis for discrete signaling constellations (m-psk, m- QAM, etc.) [ ]. Currently, there have been three metrics to evaluate information rate. They are accurate values, loer or upper bounds, and intermediate variables. According to definition of information rate, direct computation is rather hard for arbitrary fading MIMO channels [5, 6]. Therefore, Monte Carlo (MC) trials turn out to be a direct and accurate computation [5]. To reduce computational complexity, an improved particle method is proposed in [6]. Hoever, it is iterative and implicit, hich makes it ambiguous to analyze. Recently a bitise computation ith concise analytical expression is proposed []. Unfortunately, furtherstudieshaveshonthatitislimitedtosomescenarios, single-input single-output (SISO) and MIMO channels ith constellation of BPSK, QPSK, 6QAM, and 6QAM. On the other hand, to the issue of complexity, loer or upper bounds are used to profile information rate instead [7 9]. Hoever, differences beteen bounds and accurate information rate are still notable. Meanhile there are also researches hich suggest intermediate variables to implement qualitative analysis [ ]. Hoever, they are handling some special MIMO channels, such as diagonal MIMO channel. In this ork, e are focusing on analytical computation of information rate for arbitrary fading MIMO channels and proposing a symbol-ise algorithm. It is characterized by considerable accuracy, reasonable complexity, and analytical expression,hichenableirtobeapplicableforanalysis.this ork is organized as follos. In Section, a basic revie on analytical computation is presented. Then demonstration of the proposed symbol-ise computation is detailed in Section. In Section, comparison on accuracy and computational complexity beteen MC simulation and symbol-ise analytical computation is presented. Finally, conclusions are dran in Section 5.. Basic Revie of Analytical Computation Consider the problem of computing information rate I (s; y) =I([s,s,...,s T ] T ; [y,y,...,y R ] T ) () beteen input vector s =[s,s,...,s T ] T and output vector y = [y,y,...,y R ] T over MIMO channels ith additive hitegaussiannoise(awg).use R T dimensional matrix H to denote coefficient for arbitrary fading MIMO channels. T and R are numbers of transmitting and receiving antennas, respectively. Then e have [, ] y = Hs +, ()

2 Computer etorks and Communications here is AWG vector. Using Complex-C(, σ ) denotes complex Gaussian ith zero mean and variance of σ and submits to ={,,..., R }; k iid. C (, σ ), k=,,..., R. Every element s k (k =,,..., T )ins is selected from the kth finite subconstellation Ω k uniformly and independently. Therefore, s is uniformly distributed over finite discrete signaling constellations Ω, andω is the Cartesian product of all subconstellations. Assuming that size of Ω k is k, probability density function (PDF) for s is p (s) = k= δ(s q k ); Ω ={q, q,...,q }, = T k= k. Provided channel states H, definition of information rate gives in [] as I(s; y) = k= W π e /σ R σ R e /σ log (/) d, m= e H(q k q m )+ /σ here W is domain of AWG vector and a represents -norm for vector a. Generally speaking, (5) requires at least R dimensional integral. Therefore, it is difficult to implement directly. And then MC method is used to compute accurate information rate in [8] as I(s; y) = n= k= lim { e n /σ log (/) }. m= e H(q k q m )+ n /σ either MC computation is simple, nor it can reveal explicit relation beteen information rate and channel states. Consequently, bitise computation is developed, using sum of several adjusted Gaussians to approximate PDF of logise likelihood ratio (LLR), and then information rate is computed by I(s; y) L log l= L l= k= p l (l q k )I l (l q k )dl J( β l σ l ), () () (5) (6) (7) here L means the number of adjusted Gaussians hich is determined by preliminary simulations. Adjustment of β l is also determined by simulations. σ l denotes variance of the lth Gaussian, defined in []. And J(x) is J (x) = πx e (l x /) /x log ( + e l )dl = { a x +b x +c x : x.66 { { e a x +b x +c x+d : x >.66. This bitise computation achieves acceptable accuracy for single-input single-output (SISO) and MIMO channels ith BPSK, QPSK, 6QAM, and 6QAM [].. Proposed Analytical Computation Since that bitise computation of information rate is limited to some scenarios, e propose a symbol-ise algorithm. In this section, e present strict demonstration and extend this computation to general MIMO scenarios ith the help of mutual distance vector as Information rate (5) is reritten as here I(s; y) = log ( (8) d k,m = q k q m. (9) k= p () k,m e Hd k,m + k,m /σ )d, m= () k,m = tr (Hd k,m H + d k,m H H H ). () Because is AWG vector, the PDF of k,m is Gaussian, k,m (, Hd k,m σ ). () And (, d k,m σ ) denotes Gaussian ith zero mean and variance of d k,m σ. ormalize Gaussian variance as We have I(s; y) = log ( k= (, ) ; k,m = Hd k,m σ. π e / () () e Hd k,m /σ Hd k,m /σ )d. m=

3 Computer etorks and Communications Consequently, it is pivotal to compute numerical integral. Use Taylor expansion as π e / log ( = π e / e a m a m )d m= ( ) p+n p/ log [ χ p!n! p+n p ]d. p= n= The denotation χ p+n is defined as χ p+n = (5) p+n a m m=. (6) Equation (6) points out that χ p+n is arithmetic mean of progression [a,a,...,a ] to the poer of (p + n).forthe sake that it is difficult to compute (5) ith χ p+n directly, a suboptimal computation is proposed. Regarding {χ p+n } as a progression of elements, e are trying to find another geometric progression {χ p+n }. This geometric progression {χ p+n } is characterized by minimum mean square error to the original progression {χ p+n }. And then, this characteristic guarantees the minimum mean square error beteen computations of (5) ith {χ p+n } and {χ p+n }.Consequently, e accomplish this geometric progression ith least-squares fitting, χ=min χ { lim [ R R umerical approximation gives χ log e ( Thus, integral is approximated as π e / log ( π e / log p= n= R r= (χ r χ r ) ]}. (7) e χ m /( e χm / ) ). (8) m= e a m a m )d m= ( ( )p+n p/ p χ p+n )d p!n! = π e / log (e χ χ )d = log ( e χ m /( e χm / ) ). m= (9) BPSK SR (db) 56QAM 8PSK Figure : umerical results for SIMO. 6QAM Recall (), e get analytical computation of information rate as I(s; y) m= k=. umerical Results log ( e ( Hd k,m /σ )/( e Hd k,m /σ /) ). () This section presents numerical results for validation. Accurate information rate is computed ith MC method (6) as basic reference for comparison. It is clear that information rate is determined by digital signaling constellation Ω,channel states H, andawgvarianceσ. To assure that the numerical results are self-contained, e ill classify [Ω, H and σ ] into several orthogonal spaces... umerical Results for Arbitrary Fading SIMO Channels. We analyze single-input multiple-output (SIMO) channels first. The simplest scenario, single-input single-output (SISO) channels, can be seen as a subset of SIMO. Consider y = Hs+, () ith maximum ration combination (MRC); this transmission is effective to y =s+ ; (, σ () H ). For generality, constellations BPSK, 8PSK, 6QAM, and 56QAM are assigned to s, respectively. umerical results on computation of information rate are illustrated in Figure. It is clear that the proposed method achieves considerable accuracyandtellstheaccurateinformationratefordifferent digital signaling constellations.

4 Computer etorks and Communications H ith [BPSK, QPSK, 8PSK, 6QAM] 5 SR (db) H ith [BPSK, QPSK, 8PSK] H ith [BPSK, QPSK] Figure : umerical results for MISO H ith [BPSK, QPSK, 8PSK, 6QAM] SR (db) H ith [BPSK, QPSK, 8PSK] H ith [BPSK, QPSK] Figure : umerical results for MIMO... umerical Results for Arbitrary Fading MISO Channels. Then e consider multiple-input single-output (MISO) channels. Consider example as follos: y=[h,h,...,h T ][s,s,...,s T ] T +. () Since h k (k=,,..., T ) is complex, MISO channel is of at least ( T )degreesfreedom,soitisimpossibletoprofile full classification. Therefore, e have to make the folloing yields. () T is selected as,, and, for example. () For each value of T, h k (k =,,..., T ) is randomly chosen ith complex Gaussian. () Assuring generality, constellations BPSK, QPSK, 8PSK, and 6QAM are assigned to each symbol in vector s, respectively. umerical results are illustrated in Figure. It is also clear that symbol-ise computation achieves considerable accuracy for SIMO channels... umerical Results for Arbitrary Fading MIMO Channels. As to MIMO channel, H is consisted of R T complex coefficients, so e make similar yields. () R is, and T is selected as,, and, for example. () All elements of H are randomly chosen ith complex Gaussian. () Assuring generality, constellations BPSK, QPSK, 8PSK, and 6QAM are assigned to each symbol in vector s, respectively. umerical results illustrated in Figure sho that symbolise computation achieves considerable accuracy also... umerical Results for Resolution of Information Rate. Consider another problem as computing information rate of any component ithin input vector accurately. Firstly, it is proven that information rate can be resolved as follos []: I(s s ; y) =I(s; y) I(s r ; y), () Transceiver ith 8PSK Transceiver ith 6QAM.5 Transceiver ith BPSK SR (db) Transceiver ith QPSK Figure : umerical results for resolution of information rate. here s r is residual subvector by excluding s s from s. This tells that e can compute arbitrary information rate provided computation of (5). Consequently, this part of results validates () ith symbol-ise computation of information rate. For example, that H is and randomly chosen similarly as previous sections. Used constellations are BPSK, QPSK, 8PSK, and 6QAM for each symbol in vector s, respectively. Interesting components are individual symbol in vector s. Information rate of every transceiver is illustrated in Figure. It is the same as before that symbol-ise computation achieves considerable accuracy..5. umerical Results for Erasure Channel. Besides MIMO channels mentioned before, there is a very special kind of channel as follos: y=[, ] [s,s ] T +, (5) here s and s are both BPSK modulated. This kind of transmission is a typical erasure channel, here R is and T is. umerical results are illustrated in Figure 5. It also validates the proposed symbol-ise computation.

5 Computer etorks and Communications SR (db) for s for s for st transceiver for st transceiver for nd transceiver for nd transceiver methods is dependent to scale of MIMO and constellation, because it tells in [6] that particle methods need a sequence length of to obtain convergent calculation, hile the suggested method in this ork is of complexity varying ith. To sum up, the proposed computation makes sense that it is interpreted in a general and concise analytical expression, so that it facilitates further studies on performance and optimization of ireless MIMO transmissions ith information rate criterion. Disclosure This ork as presented in part at International Conference on Communications. Competing Interests 5. Discussion Figure 5: umerical results for erasure channel. We have presented an analytical computation of information rate for arbitrary fading MIMO channels. Based on simulation, e have further discussion as follos. For the Generality of the proposal, similarly as presented in [5, 6, 9, ], e demonstrate computation of information rate for MIMO channel, ithout supplemental conditions except the knoledge of constellations Ω, poer of AWG, and channel status H to the receiver. And then, e carry out validation ith numerical results on selected MIMO cases. To ensure that the selected MIMO cases are general, numbers of transmitting and receiving antenna ( T, R ) vary from to, as presented in Section ; the adopted constellations vary from QPSK to 56QAM; and simulated channel status H are randomly generated. In addition, () (insection.)canbeusedtocomputeinformationratefor each MIMO stream, hich also improves generality, hereas bitise computation is quite limited by tuning factors related to selected MIMO scenarios []. As to Accuracy of the proposal, validations in Section sho that the maximum gap beteen information rate computed by proposed and MC methods is loer than.6 bits/symbol. Reference information rate is computed by MC method [5], and particle method achieves exactly accurate numerical results [6]. With SR based intermediate variables, estimation [] and upper/loer bounds [9, ] are proposed. The gap beteen reference information rate by MC and upper/loer bounds is about.5 bits/symbol [9, ], hich is a little orse than computation proposed in this ork.theaccuracyofestimationin[]isnotcomparedfor thesakethattherearequitealotmimoandconstellations cases hen information rate is unavailable. Then e turn to Complexity. Computation shon as () ill need exponential processes and logarithms, hich is approximately equivalent to those presented in [9, ].ThisismuchsimplerthanMCmethod[5].Thecomparison on complexity beteen the proposed and particle The authors declare that there is no conflict of interests regarding the publication of this paper. Acknoledgments ThisorkisfinanciallysupportedbyationalaturalScience Foundation of China (SFC) 67 and 66 and ResearchProjectofRailayCorporation(6J-H). References [] J.Hu,T.M.Duman,M.F.Erden,andA.Kavcic, Achievable information rates for channels ith insertions, deletions, and intersymbol interference ith i.i.d. inputs, IEEE Transactions on Communications, vol. 58, no., pp.,. [] R.-R. Chen and R. Peng, Performance of channel coded noncoherent systems: modulation choice, information rate, and Markov chain Monte Carlo detection, IEEE Transactions on Communications,vol.57,no.,pp.8 85,9. []J.Zhang,H.Zheng,Z.Tan,Y.Chen,andL.Xiong, Link evaluation for MIMO-OFDM system ith ML detection, in Proceedings of the (ICC ) IEEE International Conference on Communications, pp. 6,CapeTon,SouthAfrica, May. [] K. Sayana, J. Zhuang, and K. Steart, Short term link performance modeling for ML receivers ith mutual information per bit metrics, in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM 8),pp. 8,e Orleans, La, USA, December 8. [5] A. B. Oen, Monte Carlo extension of quasi-monte Carlo, in Proceedings of the th Conference on Winter Simulation (WSC 98), vol. 6, pp , Washington, DC, USA, December 998. [6] J. Dauels and H.-A. Loeliger, Computation of information rates by particle methods, IEEE Transactions on Information Theory,vol.5,no.,pp.6 9,8. [7]. Guney, H. Deliç, and F. Alagöz, Achievable information rates of PPM impulse radio for UWB channels and rake reception, IEEE Transactions on Communications,vol.58,no.5,pp.5 55,. [8] A. Steiner and S. Shamai, Achievable rates ith imperfect transmitter side information using a broadcast transmission

6 6 Computer etorks and Communications strategy, IEEE Transactions on Wireless Communications, vol. 7, no., pp. 5, 8. [9] P. Sadeghi, P. O. Vontobel, and R. Shams, Optimization of information rate upper and loer bounds for channels ith memory, IEEE Transactions on Information Theory,vol.55,no., pp , 9. [] O. Shental,. Shental, S. Shamai, I. Kanter, A. J. Weiss, and Y. Weiss, Discrete-input to-dimensional Gaussian channels ith memory: estimation and information rates via graphical models and statistical mechanics, Institute of Electrical and Electronics Engineers. Transactions on Information Theory, vol. 5, no., pp. 5 5, 8. [] P. Sadeghi and P. Rapajic, On information rates of time-varying fading channels modeled as finite-state Markov channels, IEEE Transactions on Communications, vol.56,no.8,pp.68 78, 8. [] X. Jin, J.-D. Yang, K.-Y. Song, J.-S. o, and D.-J. Shin, On the relationship beteen mutual information and bit error probability for some linear dispersion codes, IEEE Transactions on Wireless Communications,vol.8,no.,pp.9 9,9. [] W.Dai,Y.Liu,B.Rider,andV.K.Lau, Ontheinformationrate of MIMO systems ith finite rate channel state feedback using beamforming and poer on/off strategy, IEEE Transactions on Information Theory,vol.55,no.,pp.5 57,9.

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