Research ArticleAnalytical Computation of Information Ratefor MIMO Channels
Jinbao Zhang Zhenhui Tan and Song Chen
Beijing Jiaotong University Beijing China
Correspondence should be addressed to Jinbao Zhang jbzhangbjtueducn
Received 5 September 2016 Revised 10 December 2016 Accepted 4 January 2017 Published 8 February 2017
Academic Editor Peng Cheng
Copyright copy 2017 Jinbao Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Information rate for discrete signaling constellations is significant However the computational complexity makes information raterather difficult to analyze for arbitrary fading multiple-input multiple-output (MIMO) channels An analytical method is proposedto compute information rate which is characterized by considerable accuracy reasonable complexity and concise representationThese features will improve accuracy for performance analysis with criterion of information rate
1 Introduction
Information rate plays an important role in performanceanalysis for discrete signaling constellations (m-PSK m-QAM etc) [1ndash4] Currently there have been three metrics toevaluate information rate They are accurate values lower orupper bounds and intermediate variables
According to definition of information rate direct com-putation is rather hard for arbitrary fading MIMO channels[5 6] Therefore Monte Carlo (MC) trials turn out to bea direct and accurate computation [5] To reduce computa-tional complexity an improved particle method is proposedin [6] However it is iterative and implicit which makes itambiguous to analyze Recently a bitwise computation withconcise analytical expression is proposed [4] Unfortunatelyfurther studies have shown that it is limited to some scenariossingle-input single-output (SISO) and 2 times 2 MIMO channelswith constellation of BPSK QPSK 16QAM and 64QAM Onthe other hand to the issue of complexity lower or upperbounds are used to profile information rate instead [7ndash9]However differences between bounds and accurate informa-tion rate are still notable Meanwhile there are also researcheswhich suggest intermediate variables to implement qualita-tive analysis [10ndash13] However they are handling some specialMIMO channels such as diagonal MIMO channel
In this work we are focusing on analytical computationof information rate for arbitrary fading MIMO channels and
proposing a symbol-wise algorithm It is characterized byconsiderable accuracy reasonable complexity and analyticalexpression which enable IR to be applicable for analysisThiswork is organized as follows In Section 2 a basic reviewon analytical computation is presented Then demonstrationof the proposed symbol-wise computation is detailed inSection 3 In Section 4 comparison on accuracy and compu-tational complexity betweenMC simulation and symbol-wiseanalytical computation is presented Finally conclusions aredrawn in Section 5
2 Basic Review of Analytical Computation
Consider the problem of computing information rate
between input vector s = [1199041 1199042 119904119873119879]119879 and output vectory = [1199101 1199102 119910119873119877]119879 over MIMO channels with additivewhite Gaussian noise (AWGN) Use 119873119877 times 119873119879 dimensionalmatrix ndash H to denote coefficient for arbitrary fading MIMOchannels 119873119879 and 119873119877 are numbers of transmitting andreceiving antennas respectively Then we have [10 13]
y = Hs + w (2)
HindawiJournal of Computer Networks and CommunicationsVolume 2017 Article ID 6495028 6 pageshttpsdoiorg10115520176495028
2 Journal of Computer Networks and Communications
where w is AWGN vector Using Complex-119862N(0 1205901199082)denotes complex Gaussian with zero mean and variance of1205901199082 and w submits to
Every element 119904119896 (119896 = 1 2 119873119879) in s is selected from thekth finite subconstellation Ω119896 uniformly and independentlyTherefore s is uniformly distributed over finite discretesignaling constellations ndash Ω and Ω is the Cartesian productof all subconstellations Assuming that size of Ω119896 is 119873119896probability density function (PDF) for s is
119901 (s) = 1119873119873sum119896=1
120575 (s minus q119896) Ω = q1 q2 q119873
119873 = 119873119879prod119896=1
119873119896(4)
Provided channel states ndash H definition of information rategives in [10] as
whereW is domain of AWGN vector ndashw and a2 represents2-norm for vector a Generally speaking (5) requires atleast 2119873119877 dimensional integral Therefore it is difficult toimplement directly And thenMCmethod is used to computeaccurate information rate in [8] as
Neither MC computation is simple nor it can reveal explicitrelation between information rate and channel states Con-sequently bitwise computation is developed using sum ofseveral adjusted Gaussians to approximate PDF of logwiselikelihood ratio (LLR) and then information rate is computedby
where 119871 means the number of adjusted Gaussians which isdetermined by preliminary simulations Adjustment of 120573119897 isalso determined by simulations 1205902119897 denotes variance of thelth Gaussian defined in [4] And 119869(119909) is
= 11988611199093 + 11988711199092 + 1198881119909 119909 le 163631 minus 11989011988621199093+11988721199092+1198882119909+1198892 119909 gt 16363
(8)
This bitwise computation achieves acceptable accuracy forsingle-input single-output (SISO) and 2 times 2 MIMO channelswith BPSK QPSK 16QAM and 64QAM [4]
3 Proposed Analytical Computation
Since that bitwise computation of information rate is limitedto some scenarios we propose a symbol-wise algorithm Inthis section we present strict demonstration and extend thiscomputation to general MIMO scenarios with the help ofmutual distance vector as
Because w is AWGN vector the PDF of 1199081015840119896119898 is Gaussian1199081015840119896119898 sim N (0 21003817100381710038171003817Hd119896119898
1003817100381710038171003817221205901199082) (12)
And N(0 2d119896119898221205901199082) denotes Gaussian with zero meanand variance of 2d119896119898221205901199082 Normalize Gaussian varianceas
119908 sim N (0 1) 119908 = 1199081015840119896119898radic2 1003817100381710038171003817Hd119896119898
Equation (16) points out that 120594119901+2119899 is arithmetic mean ofprogression [1198861 1198862 119886119873] to the power of (119901 + 2119899) For thesake that it is difficult to compute (15) with 120594119901+2119899 directly asuboptimal computation is proposed Regarding 120594119901+2119899 asa progression of 119873 elements we are trying to find anothergeometric progression 120594119901+2119899 This geometric progression120594119901+2119899 is characterized by minimum mean square error tothe original progression 120594119901+2119899 And then this character-istic guarantees the minimum mean square error betweencomputations of (15) with 120594119901+2119899 and 120594119901+2119899 Consequentlywe accomplish this geometric progression with least-squaresfitting
This section presents numerical results for validation Accu-rate information rate is computed with MC method (6) asbasic reference for comparison It is clear that informationrate is determined by digital signaling constellationΩ chan-nel states H and AWGN variance 1205901199082 To assure that thenumerical results are self-contained we will classify [Ω Hand 1205901199082] into several orthogonal spaces41 Numerical Results for Arbitrary Fading SIMO ChannelsWe analyze single-input multiple-output (SIMO) channelsfirstThe simplest scenario single-input single-output (SISO)channels can be seen as a subset of SIMO Consider
y = H119904 + w (21)
withmaximum ration combination (MRC) this transmissionis effective to
For generality constellations BPSK 8PSK 64QAM and256QAM are assigned to 119904 respectively Numerical results oncomputation of information rate are illustrated in Figure 1It is clear that the proposed method achieves considerableaccuracy and tells the accurate information rate for differentdigital signaling constellations
4 Journal of Computer Networks and Communications
minus20 minus10 0 10 20 30 40 500123456789
10
SNR (dB)
1 times 4 H with[BPSK QPSK 8PSK 16QAM]
1 times 2 H with[BPSK QPSK]
1 times 3 H with[BPSK QPSK 8PSK]
Accurate information rateSymbol-wise computation
I(s y
) (bi
tssy
mbo
l)
Figure 2 Numerical results for MISO
42 Numerical Results for Arbitrary Fading MISO ChannelsThenwe considermultiple-input single-output (MISO) chan-nels Consider example as follows
Since ℎ119896 (119896 = 1 2 119873119879) is complex MISO channel is of atleast 2(119873119879 minus 1) degrees freedom so it is impossible to profilefull classification Therefore we have to make the followingyields
(1) 119873119879 is selected as 2 3 and 4 for example(2) For each value of 119873119879 ℎ119896 (119896 = 1 2 119873119879) is ran-
domly chosen with complex Gaussian(3) Assuring generality constellations BPSK QPSK
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results are illustrated in Figure 2 It is alsoclear that symbol-wise computation achieves considerableaccuracy for SIMO channels
43 Numerical Results for Arbitrary Fading MIMO ChannelsAs to MIMO channel H is consisted of 119873119877 times 119873119879 complexcoefficients so we make similar yields
(1) 119873119877 is 3 and119873119879 is selected as 2 3 and 4 for example(2) All elements of H are randomly chosen with complex
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also
44 Numerical Results for Resolution of Information RateConsider another problem as computing information rate ofany component within input vector accurately Firstly it isproven that information rate can be resolved as follows [11]
119868 (s119904 y) = 119868 (s y) minus 119868 (s119903 y) (24)
Figure 4 Numerical results for resolution of information rate
where s119903 is residual subvector by excluding s119904 from s Thistells that we can compute arbitrary information rate providedcomputation of (5) Consequently this part of results vali-dates (24) with symbol-wise computation of information rateFor example that H is 3 times 4 and randomly chosen similarlyas previous sections Used constellations are BPSK QPSK8PSK and 16QAM for each symbol in vector s respectivelyInteresting components are individual symbol in vector sInformation rate of every transceiver is illustrated in Figure 4It is the same as before that symbol-wise computationachieves considerable accuracy
45 Numerical Results for Erasure Channel Besides MIMOchannels mentioned before there is a very special kind ofchannel as follows
where s1 and s2 are both BPSK modulated This kind oftransmission is a typical erasure channel where 119873119877 is 1 and119873119879 is 2 Numerical results are illustrated in Figure 5 It alsovalidates the proposed symbol-wise computation
Journal of Computer Networks and Communications 5I(s y
) (bi
tssy
mbo
l)
minus20 minus15 minus10 minus5 0 5 100
05
1
15
SNR (dB)
Accurate information rate for s
Symbol-wise computation for s
Accurate information rate for 1st transceiverSymbol-wise computation for 1st transceiverAccurate information rate for 2nd transceiverSymbol-wise computation for 2nd transceiver
Figure 5 Numerical results for erasure channel
5 Discussion
We have presented an analytical computation of informationrate for arbitrary fading MIMO channels Based on simula-tion we have further discussion as follows
For the ldquoGeneralityrdquo of the proposal similarly as pre-sented in [5 6 9 10] we demonstrate computation ofinformation rate for MIMO channel without supplementalconditions except the knowledge of constellations ndashΩ powerof AWGN and channel status ndash H to the receiver And thenwe carry out validation with numerical results on selectedMIMO cases To ensure that the selected MIMO cases aregeneral numbers of transmitting and receiving antenna (119873119879119873119877) vary from 1 to 4 as presented in Section 4 the adoptedconstellations vary from QPSK to 256QAM and simulatedchannel status ndashH are randomly generated In addition (24)(in Section 44) can be used to compute information rate foreach MIMO stream which also improves generality whereasbitwise computation is quite limited by tuning factors relatedto selected MIMO scenarios [4]
As to ldquoAccuracyrdquo of the proposal validations in Section 4show that the maximum gap between information ratecomputed by proposed and MC methods is lower than0063 bitssymbol Reference information rate is computedby MC method [5] and particle method achieves exactlyaccurate numerical results [6] With SNR based intermediatevariables estimation [4] and upperlower bounds [9 10] areproposedThe gap between reference information rate byMCand upperlower bounds is about 015 bitssymbol [9 10]which is a little worse than computation proposed in thiswork The accuracy of estimation in [4] is not compared forthe sake that there are quite a lot MIMO and constellationscases when information rate is unavailable
Then we turn to ldquoComplexityrdquo Computation shown as(20) will need119873times119873 exponential processes andN logarithmswhich is approximately equivalent to those presented in [910] This is much simpler than MC method [5] The com-parison on complexity between the proposed and particle
methods is dependent to scale of MIMO and constellationbecause it tells in [6] that particle methods need a sequencelength of 104 to obtain convergent calculation while thesuggested method in this work is of complexity varying withN
To sum up the proposed computationmakes sense that itis interpreted in a general and concise analytical expressionso that it facilitates further studies on performance and opti-mization of wireless MIMO transmissions with informationrate criterion
Disclosure
This work was presented in part at 2010 International Confer-ence on Communications
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is financially supported by National Natural Sci-ence Foundation of China (NSFC) 61471030 and 61631013 andResearch Project of Railway Corporation (2016J011-H)
References
[1] J Hu T M Duman M F Erden and A Kavcic ldquoAchievableinformation rates for channels with insertions deletions andintersymbol interference with iid inputsrdquo IEEE Transactionson Communications vol 58 no 4 pp 1102ndash1111 2010
[2] R-R Chen and R Peng ldquoPerformance of channel codednoncoherent systems modulation choice information rate andMarkov chain Monte Carlo detectionrdquo IEEE Transactions onCommunications vol 57 no 10 pp 2841ndash2845 2009
[3] J Zhang H Zheng Z Tan Y Chen and L Xiong ldquoLinkevaluation for MIMO-OFDM system with ML detectionrdquo inProceedings of the (ICC rsquo10)mdash2010 IEEE International Confer-ence on Communications pp 1ndash6 Cape Town South AfricaMay 2010
[4] K Sayana J Zhuang and K Stewart ldquoShort term link per-formance modeling for ML receivers with mutual informationper bit metricsrdquo in Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM rsquo08) pp 4313ndash4318 NewOrleans La USA December 2008
[5] A B Owen ldquoMonte Carlo extension of quasi-Monte Carlordquo inProceedings of the 30th Conference on Winter Simulation (WSCrsquo98) vol 16 pp 571ndash577 Washington DC USA December1998
[6] J Dauwels and H-A Loeliger ldquoComputation of informationrates by particle methodsrdquo IEEE Transactions on InformationTheory vol 54 no 1 pp 406ndash409 2008
[7] NGuneyHDelic and FAlagoz ldquoAchievable information ratesof PPM impulse radio for UWB channels and rake receptionrdquoIEEE Transactions on Communications vol 58 no 5 pp 1524ndash1535 2010
[8] A Steiner and S Shamai ldquoAchievable rates with imperfecttransmitter side information using a broadcast transmission
6 Journal of Computer Networks and Communications
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
Every element 119904119896 (119896 = 1 2 119873119879) in s is selected from thekth finite subconstellation Ω119896 uniformly and independentlyTherefore s is uniformly distributed over finite discretesignaling constellations ndash Ω and Ω is the Cartesian productof all subconstellations Assuming that size of Ω119896 is 119873119896probability density function (PDF) for s is
119901 (s) = 1119873119873sum119896=1
120575 (s minus q119896) Ω = q1 q2 q119873
119873 = 119873119879prod119896=1
119873119896(4)
Provided channel states ndash H definition of information rategives in [10] as
whereW is domain of AWGN vector ndashw and a2 represents2-norm for vector a Generally speaking (5) requires atleast 2119873119877 dimensional integral Therefore it is difficult toimplement directly And thenMCmethod is used to computeaccurate information rate in [8] as
Neither MC computation is simple nor it can reveal explicitrelation between information rate and channel states Con-sequently bitwise computation is developed using sum ofseveral adjusted Gaussians to approximate PDF of logwiselikelihood ratio (LLR) and then information rate is computedby
where 119871 means the number of adjusted Gaussians which isdetermined by preliminary simulations Adjustment of 120573119897 isalso determined by simulations 1205902119897 denotes variance of thelth Gaussian defined in [4] And 119869(119909) is
= 11988611199093 + 11988711199092 + 1198881119909 119909 le 163631 minus 11989011988621199093+11988721199092+1198882119909+1198892 119909 gt 16363
(8)
This bitwise computation achieves acceptable accuracy forsingle-input single-output (SISO) and 2 times 2 MIMO channelswith BPSK QPSK 16QAM and 64QAM [4]
3 Proposed Analytical Computation
Since that bitwise computation of information rate is limitedto some scenarios we propose a symbol-wise algorithm Inthis section we present strict demonstration and extend thiscomputation to general MIMO scenarios with the help ofmutual distance vector as
Because w is AWGN vector the PDF of 1199081015840119896119898 is Gaussian1199081015840119896119898 sim N (0 21003817100381710038171003817Hd119896119898
1003817100381710038171003817221205901199082) (12)
And N(0 2d119896119898221205901199082) denotes Gaussian with zero meanand variance of 2d119896119898221205901199082 Normalize Gaussian varianceas
119908 sim N (0 1) 119908 = 1199081015840119896119898radic2 1003817100381710038171003817Hd119896119898
Equation (16) points out that 120594119901+2119899 is arithmetic mean ofprogression [1198861 1198862 119886119873] to the power of (119901 + 2119899) For thesake that it is difficult to compute (15) with 120594119901+2119899 directly asuboptimal computation is proposed Regarding 120594119901+2119899 asa progression of 119873 elements we are trying to find anothergeometric progression 120594119901+2119899 This geometric progression120594119901+2119899 is characterized by minimum mean square error tothe original progression 120594119901+2119899 And then this character-istic guarantees the minimum mean square error betweencomputations of (15) with 120594119901+2119899 and 120594119901+2119899 Consequentlywe accomplish this geometric progression with least-squaresfitting
This section presents numerical results for validation Accu-rate information rate is computed with MC method (6) asbasic reference for comparison It is clear that informationrate is determined by digital signaling constellationΩ chan-nel states H and AWGN variance 1205901199082 To assure that thenumerical results are self-contained we will classify [Ω Hand 1205901199082] into several orthogonal spaces41 Numerical Results for Arbitrary Fading SIMO ChannelsWe analyze single-input multiple-output (SIMO) channelsfirstThe simplest scenario single-input single-output (SISO)channels can be seen as a subset of SIMO Consider
y = H119904 + w (21)
withmaximum ration combination (MRC) this transmissionis effective to
For generality constellations BPSK 8PSK 64QAM and256QAM are assigned to 119904 respectively Numerical results oncomputation of information rate are illustrated in Figure 1It is clear that the proposed method achieves considerableaccuracy and tells the accurate information rate for differentdigital signaling constellations
4 Journal of Computer Networks and Communications
minus20 minus10 0 10 20 30 40 500123456789
10
SNR (dB)
1 times 4 H with[BPSK QPSK 8PSK 16QAM]
1 times 2 H with[BPSK QPSK]
1 times 3 H with[BPSK QPSK 8PSK]
Accurate information rateSymbol-wise computation
I(s y
) (bi
tssy
mbo
l)
Figure 2 Numerical results for MISO
42 Numerical Results for Arbitrary Fading MISO ChannelsThenwe considermultiple-input single-output (MISO) chan-nels Consider example as follows
Since ℎ119896 (119896 = 1 2 119873119879) is complex MISO channel is of atleast 2(119873119879 minus 1) degrees freedom so it is impossible to profilefull classification Therefore we have to make the followingyields
(1) 119873119879 is selected as 2 3 and 4 for example(2) For each value of 119873119879 ℎ119896 (119896 = 1 2 119873119879) is ran-
domly chosen with complex Gaussian(3) Assuring generality constellations BPSK QPSK
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results are illustrated in Figure 2 It is alsoclear that symbol-wise computation achieves considerableaccuracy for SIMO channels
43 Numerical Results for Arbitrary Fading MIMO ChannelsAs to MIMO channel H is consisted of 119873119877 times 119873119879 complexcoefficients so we make similar yields
(1) 119873119877 is 3 and119873119879 is selected as 2 3 and 4 for example(2) All elements of H are randomly chosen with complex
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also
44 Numerical Results for Resolution of Information RateConsider another problem as computing information rate ofany component within input vector accurately Firstly it isproven that information rate can be resolved as follows [11]
119868 (s119904 y) = 119868 (s y) minus 119868 (s119903 y) (24)
Figure 4 Numerical results for resolution of information rate
where s119903 is residual subvector by excluding s119904 from s Thistells that we can compute arbitrary information rate providedcomputation of (5) Consequently this part of results vali-dates (24) with symbol-wise computation of information rateFor example that H is 3 times 4 and randomly chosen similarlyas previous sections Used constellations are BPSK QPSK8PSK and 16QAM for each symbol in vector s respectivelyInteresting components are individual symbol in vector sInformation rate of every transceiver is illustrated in Figure 4It is the same as before that symbol-wise computationachieves considerable accuracy
45 Numerical Results for Erasure Channel Besides MIMOchannels mentioned before there is a very special kind ofchannel as follows
where s1 and s2 are both BPSK modulated This kind oftransmission is a typical erasure channel where 119873119877 is 1 and119873119879 is 2 Numerical results are illustrated in Figure 5 It alsovalidates the proposed symbol-wise computation
Journal of Computer Networks and Communications 5I(s y
) (bi
tssy
mbo
l)
minus20 minus15 minus10 minus5 0 5 100
05
1
15
SNR (dB)
Accurate information rate for s
Symbol-wise computation for s
Accurate information rate for 1st transceiverSymbol-wise computation for 1st transceiverAccurate information rate for 2nd transceiverSymbol-wise computation for 2nd transceiver
Figure 5 Numerical results for erasure channel
5 Discussion
We have presented an analytical computation of informationrate for arbitrary fading MIMO channels Based on simula-tion we have further discussion as follows
For the ldquoGeneralityrdquo of the proposal similarly as pre-sented in [5 6 9 10] we demonstrate computation ofinformation rate for MIMO channel without supplementalconditions except the knowledge of constellations ndashΩ powerof AWGN and channel status ndash H to the receiver And thenwe carry out validation with numerical results on selectedMIMO cases To ensure that the selected MIMO cases aregeneral numbers of transmitting and receiving antenna (119873119879119873119877) vary from 1 to 4 as presented in Section 4 the adoptedconstellations vary from QPSK to 256QAM and simulatedchannel status ndashH are randomly generated In addition (24)(in Section 44) can be used to compute information rate foreach MIMO stream which also improves generality whereasbitwise computation is quite limited by tuning factors relatedto selected MIMO scenarios [4]
As to ldquoAccuracyrdquo of the proposal validations in Section 4show that the maximum gap between information ratecomputed by proposed and MC methods is lower than0063 bitssymbol Reference information rate is computedby MC method [5] and particle method achieves exactlyaccurate numerical results [6] With SNR based intermediatevariables estimation [4] and upperlower bounds [9 10] areproposedThe gap between reference information rate byMCand upperlower bounds is about 015 bitssymbol [9 10]which is a little worse than computation proposed in thiswork The accuracy of estimation in [4] is not compared forthe sake that there are quite a lot MIMO and constellationscases when information rate is unavailable
Then we turn to ldquoComplexityrdquo Computation shown as(20) will need119873times119873 exponential processes andN logarithmswhich is approximately equivalent to those presented in [910] This is much simpler than MC method [5] The com-parison on complexity between the proposed and particle
methods is dependent to scale of MIMO and constellationbecause it tells in [6] that particle methods need a sequencelength of 104 to obtain convergent calculation while thesuggested method in this work is of complexity varying withN
To sum up the proposed computationmakes sense that itis interpreted in a general and concise analytical expressionso that it facilitates further studies on performance and opti-mization of wireless MIMO transmissions with informationrate criterion
Disclosure
This work was presented in part at 2010 International Confer-ence on Communications
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is financially supported by National Natural Sci-ence Foundation of China (NSFC) 61471030 and 61631013 andResearch Project of Railway Corporation (2016J011-H)
References
[1] J Hu T M Duman M F Erden and A Kavcic ldquoAchievableinformation rates for channels with insertions deletions andintersymbol interference with iid inputsrdquo IEEE Transactionson Communications vol 58 no 4 pp 1102ndash1111 2010
[2] R-R Chen and R Peng ldquoPerformance of channel codednoncoherent systems modulation choice information rate andMarkov chain Monte Carlo detectionrdquo IEEE Transactions onCommunications vol 57 no 10 pp 2841ndash2845 2009
[3] J Zhang H Zheng Z Tan Y Chen and L Xiong ldquoLinkevaluation for MIMO-OFDM system with ML detectionrdquo inProceedings of the (ICC rsquo10)mdash2010 IEEE International Confer-ence on Communications pp 1ndash6 Cape Town South AfricaMay 2010
[4] K Sayana J Zhuang and K Stewart ldquoShort term link per-formance modeling for ML receivers with mutual informationper bit metricsrdquo in Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM rsquo08) pp 4313ndash4318 NewOrleans La USA December 2008
[5] A B Owen ldquoMonte Carlo extension of quasi-Monte Carlordquo inProceedings of the 30th Conference on Winter Simulation (WSCrsquo98) vol 16 pp 571ndash577 Washington DC USA December1998
[6] J Dauwels and H-A Loeliger ldquoComputation of informationrates by particle methodsrdquo IEEE Transactions on InformationTheory vol 54 no 1 pp 406ndash409 2008
[7] NGuneyHDelic and FAlagoz ldquoAchievable information ratesof PPM impulse radio for UWB channels and rake receptionrdquoIEEE Transactions on Communications vol 58 no 5 pp 1524ndash1535 2010
[8] A Steiner and S Shamai ldquoAchievable rates with imperfecttransmitter side information using a broadcast transmission
6 Journal of Computer Networks and Communications
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
Equation (16) points out that 120594119901+2119899 is arithmetic mean ofprogression [1198861 1198862 119886119873] to the power of (119901 + 2119899) For thesake that it is difficult to compute (15) with 120594119901+2119899 directly asuboptimal computation is proposed Regarding 120594119901+2119899 asa progression of 119873 elements we are trying to find anothergeometric progression 120594119901+2119899 This geometric progression120594119901+2119899 is characterized by minimum mean square error tothe original progression 120594119901+2119899 And then this character-istic guarantees the minimum mean square error betweencomputations of (15) with 120594119901+2119899 and 120594119901+2119899 Consequentlywe accomplish this geometric progression with least-squaresfitting
This section presents numerical results for validation Accu-rate information rate is computed with MC method (6) asbasic reference for comparison It is clear that informationrate is determined by digital signaling constellationΩ chan-nel states H and AWGN variance 1205901199082 To assure that thenumerical results are self-contained we will classify [Ω Hand 1205901199082] into several orthogonal spaces41 Numerical Results for Arbitrary Fading SIMO ChannelsWe analyze single-input multiple-output (SIMO) channelsfirstThe simplest scenario single-input single-output (SISO)channels can be seen as a subset of SIMO Consider
y = H119904 + w (21)
withmaximum ration combination (MRC) this transmissionis effective to
For generality constellations BPSK 8PSK 64QAM and256QAM are assigned to 119904 respectively Numerical results oncomputation of information rate are illustrated in Figure 1It is clear that the proposed method achieves considerableaccuracy and tells the accurate information rate for differentdigital signaling constellations
4 Journal of Computer Networks and Communications
minus20 minus10 0 10 20 30 40 500123456789
10
SNR (dB)
1 times 4 H with[BPSK QPSK 8PSK 16QAM]
1 times 2 H with[BPSK QPSK]
1 times 3 H with[BPSK QPSK 8PSK]
Accurate information rateSymbol-wise computation
I(s y
) (bi
tssy
mbo
l)
Figure 2 Numerical results for MISO
42 Numerical Results for Arbitrary Fading MISO ChannelsThenwe considermultiple-input single-output (MISO) chan-nels Consider example as follows
Since ℎ119896 (119896 = 1 2 119873119879) is complex MISO channel is of atleast 2(119873119879 minus 1) degrees freedom so it is impossible to profilefull classification Therefore we have to make the followingyields
(1) 119873119879 is selected as 2 3 and 4 for example(2) For each value of 119873119879 ℎ119896 (119896 = 1 2 119873119879) is ran-
domly chosen with complex Gaussian(3) Assuring generality constellations BPSK QPSK
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results are illustrated in Figure 2 It is alsoclear that symbol-wise computation achieves considerableaccuracy for SIMO channels
43 Numerical Results for Arbitrary Fading MIMO ChannelsAs to MIMO channel H is consisted of 119873119877 times 119873119879 complexcoefficients so we make similar yields
(1) 119873119877 is 3 and119873119879 is selected as 2 3 and 4 for example(2) All elements of H are randomly chosen with complex
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also
44 Numerical Results for Resolution of Information RateConsider another problem as computing information rate ofany component within input vector accurately Firstly it isproven that information rate can be resolved as follows [11]
119868 (s119904 y) = 119868 (s y) minus 119868 (s119903 y) (24)
Figure 4 Numerical results for resolution of information rate
where s119903 is residual subvector by excluding s119904 from s Thistells that we can compute arbitrary information rate providedcomputation of (5) Consequently this part of results vali-dates (24) with symbol-wise computation of information rateFor example that H is 3 times 4 and randomly chosen similarlyas previous sections Used constellations are BPSK QPSK8PSK and 16QAM for each symbol in vector s respectivelyInteresting components are individual symbol in vector sInformation rate of every transceiver is illustrated in Figure 4It is the same as before that symbol-wise computationachieves considerable accuracy
45 Numerical Results for Erasure Channel Besides MIMOchannels mentioned before there is a very special kind ofchannel as follows
where s1 and s2 are both BPSK modulated This kind oftransmission is a typical erasure channel where 119873119877 is 1 and119873119879 is 2 Numerical results are illustrated in Figure 5 It alsovalidates the proposed symbol-wise computation
Journal of Computer Networks and Communications 5I(s y
) (bi
tssy
mbo
l)
minus20 minus15 minus10 minus5 0 5 100
05
1
15
SNR (dB)
Accurate information rate for s
Symbol-wise computation for s
Accurate information rate for 1st transceiverSymbol-wise computation for 1st transceiverAccurate information rate for 2nd transceiverSymbol-wise computation for 2nd transceiver
Figure 5 Numerical results for erasure channel
5 Discussion
We have presented an analytical computation of informationrate for arbitrary fading MIMO channels Based on simula-tion we have further discussion as follows
For the ldquoGeneralityrdquo of the proposal similarly as pre-sented in [5 6 9 10] we demonstrate computation ofinformation rate for MIMO channel without supplementalconditions except the knowledge of constellations ndashΩ powerof AWGN and channel status ndash H to the receiver And thenwe carry out validation with numerical results on selectedMIMO cases To ensure that the selected MIMO cases aregeneral numbers of transmitting and receiving antenna (119873119879119873119877) vary from 1 to 4 as presented in Section 4 the adoptedconstellations vary from QPSK to 256QAM and simulatedchannel status ndashH are randomly generated In addition (24)(in Section 44) can be used to compute information rate foreach MIMO stream which also improves generality whereasbitwise computation is quite limited by tuning factors relatedto selected MIMO scenarios [4]
As to ldquoAccuracyrdquo of the proposal validations in Section 4show that the maximum gap between information ratecomputed by proposed and MC methods is lower than0063 bitssymbol Reference information rate is computedby MC method [5] and particle method achieves exactlyaccurate numerical results [6] With SNR based intermediatevariables estimation [4] and upperlower bounds [9 10] areproposedThe gap between reference information rate byMCand upperlower bounds is about 015 bitssymbol [9 10]which is a little worse than computation proposed in thiswork The accuracy of estimation in [4] is not compared forthe sake that there are quite a lot MIMO and constellationscases when information rate is unavailable
Then we turn to ldquoComplexityrdquo Computation shown as(20) will need119873times119873 exponential processes andN logarithmswhich is approximately equivalent to those presented in [910] This is much simpler than MC method [5] The com-parison on complexity between the proposed and particle
methods is dependent to scale of MIMO and constellationbecause it tells in [6] that particle methods need a sequencelength of 104 to obtain convergent calculation while thesuggested method in this work is of complexity varying withN
To sum up the proposed computationmakes sense that itis interpreted in a general and concise analytical expressionso that it facilitates further studies on performance and opti-mization of wireless MIMO transmissions with informationrate criterion
Disclosure
This work was presented in part at 2010 International Confer-ence on Communications
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is financially supported by National Natural Sci-ence Foundation of China (NSFC) 61471030 and 61631013 andResearch Project of Railway Corporation (2016J011-H)
References
[1] J Hu T M Duman M F Erden and A Kavcic ldquoAchievableinformation rates for channels with insertions deletions andintersymbol interference with iid inputsrdquo IEEE Transactionson Communications vol 58 no 4 pp 1102ndash1111 2010
[2] R-R Chen and R Peng ldquoPerformance of channel codednoncoherent systems modulation choice information rate andMarkov chain Monte Carlo detectionrdquo IEEE Transactions onCommunications vol 57 no 10 pp 2841ndash2845 2009
[3] J Zhang H Zheng Z Tan Y Chen and L Xiong ldquoLinkevaluation for MIMO-OFDM system with ML detectionrdquo inProceedings of the (ICC rsquo10)mdash2010 IEEE International Confer-ence on Communications pp 1ndash6 Cape Town South AfricaMay 2010
[4] K Sayana J Zhuang and K Stewart ldquoShort term link per-formance modeling for ML receivers with mutual informationper bit metricsrdquo in Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM rsquo08) pp 4313ndash4318 NewOrleans La USA December 2008
[5] A B Owen ldquoMonte Carlo extension of quasi-Monte Carlordquo inProceedings of the 30th Conference on Winter Simulation (WSCrsquo98) vol 16 pp 571ndash577 Washington DC USA December1998
[6] J Dauwels and H-A Loeliger ldquoComputation of informationrates by particle methodsrdquo IEEE Transactions on InformationTheory vol 54 no 1 pp 406ndash409 2008
[7] NGuneyHDelic and FAlagoz ldquoAchievable information ratesof PPM impulse radio for UWB channels and rake receptionrdquoIEEE Transactions on Communications vol 58 no 5 pp 1524ndash1535 2010
[8] A Steiner and S Shamai ldquoAchievable rates with imperfecttransmitter side information using a broadcast transmission
6 Journal of Computer Networks and Communications
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
Since ℎ119896 (119896 = 1 2 119873119879) is complex MISO channel is of atleast 2(119873119879 minus 1) degrees freedom so it is impossible to profilefull classification Therefore we have to make the followingyields
(1) 119873119879 is selected as 2 3 and 4 for example(2) For each value of 119873119879 ℎ119896 (119896 = 1 2 119873119879) is ran-
domly chosen with complex Gaussian(3) Assuring generality constellations BPSK QPSK
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results are illustrated in Figure 2 It is alsoclear that symbol-wise computation achieves considerableaccuracy for SIMO channels
43 Numerical Results for Arbitrary Fading MIMO ChannelsAs to MIMO channel H is consisted of 119873119877 times 119873119879 complexcoefficients so we make similar yields
(1) 119873119877 is 3 and119873119879 is selected as 2 3 and 4 for example(2) All elements of H are randomly chosen with complex
8PSK and 16QAM are assigned to each symbol invector s respectively
Numerical results illustrated in Figure 3 show that symbol-wise computation achieves considerable accuracy also
44 Numerical Results for Resolution of Information RateConsider another problem as computing information rate ofany component within input vector accurately Firstly it isproven that information rate can be resolved as follows [11]
119868 (s119904 y) = 119868 (s y) minus 119868 (s119903 y) (24)
Figure 4 Numerical results for resolution of information rate
where s119903 is residual subvector by excluding s119904 from s Thistells that we can compute arbitrary information rate providedcomputation of (5) Consequently this part of results vali-dates (24) with symbol-wise computation of information rateFor example that H is 3 times 4 and randomly chosen similarlyas previous sections Used constellations are BPSK QPSK8PSK and 16QAM for each symbol in vector s respectivelyInteresting components are individual symbol in vector sInformation rate of every transceiver is illustrated in Figure 4It is the same as before that symbol-wise computationachieves considerable accuracy
45 Numerical Results for Erasure Channel Besides MIMOchannels mentioned before there is a very special kind ofchannel as follows
where s1 and s2 are both BPSK modulated This kind oftransmission is a typical erasure channel where 119873119877 is 1 and119873119879 is 2 Numerical results are illustrated in Figure 5 It alsovalidates the proposed symbol-wise computation
Journal of Computer Networks and Communications 5I(s y
) (bi
tssy
mbo
l)
minus20 minus15 minus10 minus5 0 5 100
05
1
15
SNR (dB)
Accurate information rate for s
Symbol-wise computation for s
Accurate information rate for 1st transceiverSymbol-wise computation for 1st transceiverAccurate information rate for 2nd transceiverSymbol-wise computation for 2nd transceiver
Figure 5 Numerical results for erasure channel
5 Discussion
We have presented an analytical computation of informationrate for arbitrary fading MIMO channels Based on simula-tion we have further discussion as follows
For the ldquoGeneralityrdquo of the proposal similarly as pre-sented in [5 6 9 10] we demonstrate computation ofinformation rate for MIMO channel without supplementalconditions except the knowledge of constellations ndashΩ powerof AWGN and channel status ndash H to the receiver And thenwe carry out validation with numerical results on selectedMIMO cases To ensure that the selected MIMO cases aregeneral numbers of transmitting and receiving antenna (119873119879119873119877) vary from 1 to 4 as presented in Section 4 the adoptedconstellations vary from QPSK to 256QAM and simulatedchannel status ndashH are randomly generated In addition (24)(in Section 44) can be used to compute information rate foreach MIMO stream which also improves generality whereasbitwise computation is quite limited by tuning factors relatedto selected MIMO scenarios [4]
As to ldquoAccuracyrdquo of the proposal validations in Section 4show that the maximum gap between information ratecomputed by proposed and MC methods is lower than0063 bitssymbol Reference information rate is computedby MC method [5] and particle method achieves exactlyaccurate numerical results [6] With SNR based intermediatevariables estimation [4] and upperlower bounds [9 10] areproposedThe gap between reference information rate byMCand upperlower bounds is about 015 bitssymbol [9 10]which is a little worse than computation proposed in thiswork The accuracy of estimation in [4] is not compared forthe sake that there are quite a lot MIMO and constellationscases when information rate is unavailable
Then we turn to ldquoComplexityrdquo Computation shown as(20) will need119873times119873 exponential processes andN logarithmswhich is approximately equivalent to those presented in [910] This is much simpler than MC method [5] The com-parison on complexity between the proposed and particle
methods is dependent to scale of MIMO and constellationbecause it tells in [6] that particle methods need a sequencelength of 104 to obtain convergent calculation while thesuggested method in this work is of complexity varying withN
To sum up the proposed computationmakes sense that itis interpreted in a general and concise analytical expressionso that it facilitates further studies on performance and opti-mization of wireless MIMO transmissions with informationrate criterion
Disclosure
This work was presented in part at 2010 International Confer-ence on Communications
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is financially supported by National Natural Sci-ence Foundation of China (NSFC) 61471030 and 61631013 andResearch Project of Railway Corporation (2016J011-H)
References
[1] J Hu T M Duman M F Erden and A Kavcic ldquoAchievableinformation rates for channels with insertions deletions andintersymbol interference with iid inputsrdquo IEEE Transactionson Communications vol 58 no 4 pp 1102ndash1111 2010
[2] R-R Chen and R Peng ldquoPerformance of channel codednoncoherent systems modulation choice information rate andMarkov chain Monte Carlo detectionrdquo IEEE Transactions onCommunications vol 57 no 10 pp 2841ndash2845 2009
[3] J Zhang H Zheng Z Tan Y Chen and L Xiong ldquoLinkevaluation for MIMO-OFDM system with ML detectionrdquo inProceedings of the (ICC rsquo10)mdash2010 IEEE International Confer-ence on Communications pp 1ndash6 Cape Town South AfricaMay 2010
[4] K Sayana J Zhuang and K Stewart ldquoShort term link per-formance modeling for ML receivers with mutual informationper bit metricsrdquo in Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM rsquo08) pp 4313ndash4318 NewOrleans La USA December 2008
[5] A B Owen ldquoMonte Carlo extension of quasi-Monte Carlordquo inProceedings of the 30th Conference on Winter Simulation (WSCrsquo98) vol 16 pp 571ndash577 Washington DC USA December1998
[6] J Dauwels and H-A Loeliger ldquoComputation of informationrates by particle methodsrdquo IEEE Transactions on InformationTheory vol 54 no 1 pp 406ndash409 2008
[7] NGuneyHDelic and FAlagoz ldquoAchievable information ratesof PPM impulse radio for UWB channels and rake receptionrdquoIEEE Transactions on Communications vol 58 no 5 pp 1524ndash1535 2010
[8] A Steiner and S Shamai ldquoAchievable rates with imperfecttransmitter side information using a broadcast transmission
6 Journal of Computer Networks and Communications
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
Journal of Computer Networks and Communications 5I(s y
) (bi
tssy
mbo
l)
minus20 minus15 minus10 minus5 0 5 100
05
1
15
SNR (dB)
Accurate information rate for s
Symbol-wise computation for s
Accurate information rate for 1st transceiverSymbol-wise computation for 1st transceiverAccurate information rate for 2nd transceiverSymbol-wise computation for 2nd transceiver
Figure 5 Numerical results for erasure channel
5 Discussion
We have presented an analytical computation of informationrate for arbitrary fading MIMO channels Based on simula-tion we have further discussion as follows
For the ldquoGeneralityrdquo of the proposal similarly as pre-sented in [5 6 9 10] we demonstrate computation ofinformation rate for MIMO channel without supplementalconditions except the knowledge of constellations ndashΩ powerof AWGN and channel status ndash H to the receiver And thenwe carry out validation with numerical results on selectedMIMO cases To ensure that the selected MIMO cases aregeneral numbers of transmitting and receiving antenna (119873119879119873119877) vary from 1 to 4 as presented in Section 4 the adoptedconstellations vary from QPSK to 256QAM and simulatedchannel status ndashH are randomly generated In addition (24)(in Section 44) can be used to compute information rate foreach MIMO stream which also improves generality whereasbitwise computation is quite limited by tuning factors relatedto selected MIMO scenarios [4]
As to ldquoAccuracyrdquo of the proposal validations in Section 4show that the maximum gap between information ratecomputed by proposed and MC methods is lower than0063 bitssymbol Reference information rate is computedby MC method [5] and particle method achieves exactlyaccurate numerical results [6] With SNR based intermediatevariables estimation [4] and upperlower bounds [9 10] areproposedThe gap between reference information rate byMCand upperlower bounds is about 015 bitssymbol [9 10]which is a little worse than computation proposed in thiswork The accuracy of estimation in [4] is not compared forthe sake that there are quite a lot MIMO and constellationscases when information rate is unavailable
Then we turn to ldquoComplexityrdquo Computation shown as(20) will need119873times119873 exponential processes andN logarithmswhich is approximately equivalent to those presented in [910] This is much simpler than MC method [5] The com-parison on complexity between the proposed and particle
methods is dependent to scale of MIMO and constellationbecause it tells in [6] that particle methods need a sequencelength of 104 to obtain convergent calculation while thesuggested method in this work is of complexity varying withN
To sum up the proposed computationmakes sense that itis interpreted in a general and concise analytical expressionso that it facilitates further studies on performance and opti-mization of wireless MIMO transmissions with informationrate criterion
Disclosure
This work was presented in part at 2010 International Confer-ence on Communications
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is financially supported by National Natural Sci-ence Foundation of China (NSFC) 61471030 and 61631013 andResearch Project of Railway Corporation (2016J011-H)
References
[1] J Hu T M Duman M F Erden and A Kavcic ldquoAchievableinformation rates for channels with insertions deletions andintersymbol interference with iid inputsrdquo IEEE Transactionson Communications vol 58 no 4 pp 1102ndash1111 2010
[2] R-R Chen and R Peng ldquoPerformance of channel codednoncoherent systems modulation choice information rate andMarkov chain Monte Carlo detectionrdquo IEEE Transactions onCommunications vol 57 no 10 pp 2841ndash2845 2009
[3] J Zhang H Zheng Z Tan Y Chen and L Xiong ldquoLinkevaluation for MIMO-OFDM system with ML detectionrdquo inProceedings of the (ICC rsquo10)mdash2010 IEEE International Confer-ence on Communications pp 1ndash6 Cape Town South AfricaMay 2010
[4] K Sayana J Zhuang and K Stewart ldquoShort term link per-formance modeling for ML receivers with mutual informationper bit metricsrdquo in Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM rsquo08) pp 4313ndash4318 NewOrleans La USA December 2008
[5] A B Owen ldquoMonte Carlo extension of quasi-Monte Carlordquo inProceedings of the 30th Conference on Winter Simulation (WSCrsquo98) vol 16 pp 571ndash577 Washington DC USA December1998
[6] J Dauwels and H-A Loeliger ldquoComputation of informationrates by particle methodsrdquo IEEE Transactions on InformationTheory vol 54 no 1 pp 406ndash409 2008
[7] NGuneyHDelic and FAlagoz ldquoAchievable information ratesof PPM impulse radio for UWB channels and rake receptionrdquoIEEE Transactions on Communications vol 58 no 5 pp 1524ndash1535 2010
[8] A Steiner and S Shamai ldquoAchievable rates with imperfecttransmitter side information using a broadcast transmission
6 Journal of Computer Networks and Communications
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
strategyrdquo IEEE Transactions on Wireless Communications vol7 no 3 pp 1043ndash1051 2008
[9] P Sadeghi P O Vontobel and R Shams ldquoOptimization ofinformation rate upper and lower bounds for channels withmemoryrdquo IEEE Transactions on InformationTheory vol 55 no2 pp 663ndash688 2009
[10] O Shental N Shental S Shamai I Kanter A J Weiss andY Weiss ldquoDiscrete-input two-dimensional Gaussian channelswith memory estimation and information rates via graphicalmodels and statistical mechanicsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol54 no 4 pp 1500ndash1513 2008
[11] P Sadeghi and P Rapajic ldquoOn information rates of time-varyingfading channels modeled as finite-stateMarkov channelsrdquo IEEETransactions on Communications vol 56 no 8 pp 1268ndash12782008
[12] X Jin J-D Yang K-Y Song J-S No and D-J Shin ldquoOnthe relationship between mutual information and bit errorprobability for some linear dispersion codesrdquo IEEETransactionson Wireless Communications vol 8 no 1 pp 90ndash94 2009
[13] W Dai Y Liu B Rider and V K Lau ldquoOn the information rateof MIMO systems with finite rate channel state feedback usingbeamforming and power onoff strategyrdquo IEEE Transactions onInformation Theory vol 55 no 11 pp 5032ndash5047 2009
![1 Constant Envelope Signaling in MIMO Channels · arXiv:1605.03779v1 [cs.IT] 12 May 2016 1 Constant Envelope Signaling in MIMO Channels Borzoo Rassouli and Bruno Clerckx Abstract](https://static.documents.pub/doc/80x56/5f0322117e708231d407b393/1-constant-envelope-signaling-in-mimo-channels-arxiv160503779v1-csit-12-may.jpg)
