Principle of Link Evaluation

Link Evaluation (LE) is proposed in system evaluation to reduce complexity. It is important to practical systems also for link adaptation. Current algorithms for link evaluation are developed by simulation method, lacking of theoretical description. Although they provide some good accuracy for some scenarios, all of them are not universal. With the help of information theory, a universal principle of link evaluation is proposed in this paper, which explains current algorithms and leads to a universal algorithm to implement link evaluation for common wireless transmissions. This paper proposes an Extended Received Block Information Rate (ERBIR) algorithm for universal link evaluation, which is extended from current RBIR algorithm by the help of the principle presented in this paper. Mainly the universality and accuracy are highlighted. Simulation results verify all the algorithms mentioned in this paper. Both the principle and ERBIR are validated by simulation with various wireless scenarios.


Introduction
LINK evaluation aims to estimate the instant performance of transmissions for given channel status information (CSI), by a computational model with reasonable complexity.
As to wireless transmissions, due to macro and micro fading, the CSI is varying within both time and frequency domains, so fading will influence wireless transmission a lot.Consequently, link evaluation is significant to analysis and design for real wireless system.
For that the instant performance for wireless transmissions under given CSI can be computed by link evaluation quite simply and accurately, it is possible for System Level Simulation (SLS) to hold down real coding and decoding procedures, reducing a lot of complexity [1].Meanwhile, wireless system can dynamically choose the proper transmitting mode with the help of link evaluation to enhance system performance, which is referred as link adaptation [1][2][3].
Accuracy is very important to link evaluation.For SLS, obviously it directly determines if the simulation results are reasonable.For link adaptation, accurate link evaluation ensures that the transmitting mode is selected properly.If the link performance is overestimated, the transmitter will always choose a mode which can not be supported by instantaneous CSI, introducing too much transmission error; while the link performance is underestimated, potential gain exists.Both of the above cases will lead to loss of system performance.
Currently, there are several algorithms to implement link evaluation, like Effective Exponential Signal-tonoise-ratio Mapping (EESM) [4], Mean Instantaneous Capacity (MIC) [5], Received Block Information Rate (RBIR) [6] and Mean Mutual Information per Bit (MMIB) [7].Here RBIR and MMIB are Mean mutual Information (MI) based algorithms, with different RBIR calculation.Unfortunately, all of them are just simulation methods, lacking of theoretical analysis.Moreover, when it comes to nonlinear detection, there are still problems with all these algorithms mentioned above.This paper proposes a universal principle for link evaluation, and extends RBIR to common wireless scenarios.Firstly, background knowledge is introduced, including models of common transmission and link evaluation; a universal principle for link evaluation is proposed; and then RBIR is extended to ERBIR with the help of this principle.Simulation results show that the proposed algorithm provides more accuracy for different scenarios.Finally, conclusions are drawn.

Background
To analyze link evaluation, common models of wireless transmission and link evaluation are presented in this section.

Common Model of Wireless Transmission
Following assumptions are made for analysis in this paper.
1) Multi-Input Multi-Output Orthogonal Frequency Division Multiplexing (MIMO OFDM) is adopted in wireless transmission.NT and NR indicate the number of transmitting and receiving antennas respectively.NOFDM indicates the number of subcarriers in OFDM symbol.As to SISO or single subcarrier case, there is NT= NR=1 or NOFDM=1; 2) Perfect channel estimation and the channel response is flat fading on each OFDM subcarrier; 3) Detection with interference cancellation is not taken into consideration; 4) Source bits are random and iterative coding and decoding is used, for example Turbo; 5) Link evaluation interests in statistical BLock Error Rate (BLER) [1] for given CSI.Let Nu indicate the number of subcarriers mapped to the interested wireless resource block.
Disregarding subcarrier index, MIMO OFDM transmission can be written as [2] where y is N R ×1 dimensional receiving signal vector; H c is N R ×N T channel response matrix; F is N T ×N S transmitting precoding matrix; x is N S ×1 independent transmitting signal vector, with unit transmitting power; H I is N R ×N S interference channel response matrix; x I is N S ×1 independent interference signal vector, with unit transmitting power; n is N R ×1 AWGN vector, which is consisted of N R independent AWGN elements with power of σ2 (given SNR, σ2=10  ; See Appendix A for a proof.Here He is N R ×N S effective channel response matrix.||A|| F refers to the Frobenius norm of matrix A. I(N) is N×N identity matrix.σ e 2 is effective AWGN power.

Detection Algorithms
Consider detection at receiver.There are mainly three types of detection algorithms [8]: Minimum Mean Square Error (MMSE), Zero Forcing (ZF) and Maximum Likelihood (ML).Since MMSE and ZF are homologous, MMSE and ML are emphasized, and ZF is similar to MMSE.
For MMSE detection, the output signal is where M is N S ×N R dimensional equalizing matrix.Then this MIMO transmission can be divided into NS SISO transmissions with NS different Output Signal to Inter-ference and Noise Ratio (OSINR), written as γ i , i = 1, 2, … , NS.
where n(i) is independent AWGN with power of 1/γ i .For MMSE, M and OSINR for each output signal are detailed in Appendix B. As to ML detection, let Ω(x) mean the vector aggregate of every possible value of x, then output signal is Note that there is an exception of Alamouti MIMO.Only one symbol can be transmitted by each transmission for Alamouti MIMO.This Alamouti MIMO is effective to SISO transmission [8], where n is AWGN with power of σ 2 .
For both linear and nonlinear detection, the iterative coding and decoding is adopted.The implementation of such system is described in reference [9].

Common Model of Link Evaluation
There have been already several algorithms to carry out link evaluation, such like EESM, MIC, RBIR and MMIB.Common model of link evaluation is shown as the following figure.
Link evaluation follows these procedures: Step 1: Channel estimation outputs CSI of this block; Step 2: According link evaluation algorithm, indicator S k for the k th subcarrier is computed from CSI; Step 3: Compute average S with all these indicators; Step 4: Once the relation between S and BLER of this block is definite, BLER is computed from S, without Monte Carlo simulation.
If necessary, Packet Error Rate (PER), Frame Error Rate (FER) and so on can be computed also, using bellowing equation Here N B is the number of blocks in the packet or frame.

Universal Principle of Link Evaluation
EESM, MIC, RBIR and MMIB algorithms are developed by simulation method, without strict theoretical deduction.A universal principle of link evaluation is proposed in this section, making them clear.

Mathematical Model of Link Evaluation
Since link evaluation mainly interests in BLER for given CSI, it should be deduced from block transmission error rate.As the transmitting block is consisted of N u subcarriers, the uncoded BLER is computed as Here BLER u means the statistical uncoded BLER.With the help of information theory, there is lemma 1: When MCS of the transmitted block is given, BLER is one-one to the BLER u .Written as See Appendix C for a proof.Then the universal principle of link evaluation is described as: find a unified and accurate indicator to reflect the BLER of the current transmitting block.

Current Link Evaluations
Generally speaking, there are three indicators which reflect the Symbol Error Rate (SER) under given CSI.They are OSINR, Channel Capacity and MI.Current link evaluation falls into EESM, MIC and RBIR algorithms.

EESM Link Evaluation
As γ k is known, the Chernoff limit of SER k is approximated as [8] where β is MCS related parameter.Use the mathematical average of all SER k to approximate SER ave of this block, then Then there is data from Link Level Simulation (LLS).
d SNR is given, the channel The effective OSINR defined by ( 13) is exactly the same as in EESM [4].The mapping function between γ eff and BLER, and parameter β can be decided by training

MIC Link Evaluation
Since channel response H k an capacity for this transmission is Here |A| means the determinant of matrix A. The capa where A and B are MCS related pa city decides the lower bound of SER k [5], so Here A 1 and A 2 are optimized by training d LL ata from S, and A1 and A2 are listed in Table 2.Then, This is exactly the same as [5].Then MIC is validated by x, and the receiving sym-the same LLS data base in previous section.

RBIR Link Evaluation
Let the transmitting symbol is bol is y after distortion by fading channel and pollution by interference and noise.Then MI for this symbol is Generally speaking, for multi-subcarriers transmis each sym Reconsider the uncoded BLER u

 (21)
Compare ( 20) and ( 21 btained by LL on is used, it is not the same.So ERBIR is extensi O transmission, the received symbol is x y e to RBIR u of ock.From lemma 1, BLER is one-one to RB u also.
According to different calculations of RBIR u, there are RBIR and MMIB algorithms.
As to RBIR, RBIR u is computed by OSINR [6], so there are same problems as EESM, not to support ML scenario.But as it is strictly in accordance to the BLER model, RBIR shows better accuracy than EESM.
As to MMIB, computation of RBIR u is from bit Logwise Likelihood Ratio (LLR), which is presented in reference [7], so it can support ML scenario.Also as it is strictly in accordance to the BLER model, MMIB should be of the same accuracy as RBIR.

Principle of Link Evaluati
There are two parts for the principle o based on previous analysis.Firstly, BLER should be computed from the BLER model presented before; secondly, RBIR is the most accurate indicator of BLER computation.

Erbir Lin
Previous analysis shows that RB sion error probability accurately.Thus link evaluation should be based on mean mutual information indicator.This section proposes extension for RBIR, obtaining a unified and accurate ERBIR algorithm for common wireless transmissions.

General Procedur
ERBIR link evaluation is implemented following these steps: 1) Get instantaneous CSI from channel estimation.The interes 2) According to detection algorithms, normalized MI 'I k ' for each transmitted symbol is computed; 3) Average all the I k in this block to get RBIR; 4) Finally BLER is computed from RBIR according to RBIR to BLER mapping function which is o S.
In Step (2), computation is the same as conventional RBIR when it comes to MMSE detection.While ML detecti on for RBIR, which is homologous to RBIR and MMIB, but providing more accurate and universal RBIR computation.

Normalized MI Computation for SISO
For SIS The normalized MI 'I' is computed as (23) See Appendix D for details.And the following figure shows that it is accurate for a random selected channel 'H'.mplify analysis, take 2×2 MIMO as example, and The

Normalized MI Computation for MIMO
To si analysis is similar for MIMO with more antennas.received symbol is Here assume The normalized MI 'I 1 ' for the 1 st transmitted symbol is computed as This computation is detailed in Appendix E. And Figure 3 shows that it is accurate for a random selected channel 'H'.

Validation by Static LLS
All algorithms for link evaluation are validated by static LLS, based on WiMAX II down link.

St
CSI is given, and then the block transmission is trialed by a lot of Monte Carlo simulations to get real BLER under the given CSI.Then the CSI and real BLER are stored.The CSI is processed by link evaluation to get computed BLER.Obviously t more different between real and computed BLER, the worse the link evaluation algorithm is.Configuratio stat Configuration atic LLS means that the he n of ic LLS is shown as the following Table 3.

Simulation Results
In these results figures, the black bold curve is the computing function of indicator S to BLER obtained by training data from LLS; and the marked point is plotted with real BLER and S computed by the adopted link evaluation algorithm.The more deviation between the marked point and black bold curve, the more inaccurate is the link evaluation algorithm.Here simulation results also validate the theoretical conclusions.MIC chooses the upper bound of SER, so all the real BLER are bigger than computed BLER.And MMIB uses approximation in MI computation, so there is a little inaccuracy.

Link Evaluation for VEC
Firstly, VEC transmission with MMSE detection is validated by different link evaluation algorithms, shown as the following figures.
These figures show that although EESM, MIC an MIB algorithms can also obtain quite accurate link evaluation, RBI oreover, RBIR/ERBIR algorithm doesn't need any nel related tuning parameters.Then, VEC transmission with ML detection is validated by different link evaluation algorithms, shown as the following figures.
Figure 7(a), Figure 7(b) and Figure 7(c) show that EESM and RBIR algorithms are invalid, and MIC and MMIB algorithms show too much inaccuracy.ERBIR algorithm betters the accuracy of link evaluation for VEC ML transmissions a lot, although there is still some inaccuracy.
Here, MIC algorithm only provides the upper bound of wireless transmissions, and it is of the worst accuracy.Although MMIB seems a little better, for the sake of limed parameters presented in reference [7], the RBIR not very accurate, so MMIB shows worse results than

Link Evaluation for HEC
Firstly, HEC transmission with MMSE detection is validated by different link evaluation algorithms, shown as the following figures.These figures show that although EESM, MIC and MMIB algorithms can also obtain quite accurate link evaluation, RBIR/ERBIR algorithm is the most accurate.Moreover, RBIR/ERBIR algorithm doesn't need any channel related tuning parameters.
Then, HEC transmission with ML detection is valithms, shown as dated by different link evaluation algori the following figures.
Figure 9 shows that EESM, RBIR, MIC and MMIB algorithms are invalid at all.Only ERBIR algorithm can achieve link evaluation for HEC ML transmissions.

Further Results Comparisons and Analysis
To ensure the universality of the simulation, more MCS levels are simulated.Following configuration in Table 3, MCS levels are set to MCS 1~8 with different MIMO schemes respectively.And the average difference is listed in the following tables.The average difference is measured by Mean Square Error Root (MSER) between computed and real BLER values.

Validation by Link Adaptation and SLS
Link re v d S MAX II do ence caused by inaccuracy of link evaluation.Since previous results show that ERBIR is accurate, and MIC is not, link adaptation and SLS with ERBIR and MIC link evaluations are implemented.

Validation by Link Adaptation
Basic configuration of dynamical LLS is the same as  Then enable HARQ with maximum retransmission times of 3. Simulation results are listed in the following Table 6.
Compare the results of link adaptation with/without HARQ, it is obvious that accurate ERBIR link evaluation will ensure wireless system to choose proper MCS level, obtaining better BLER and throughput, and reducing the retransmission times.While using inaccurate MIC link evaluation, it is shown that MIC will overestimate the in simulation results in prend retransmission time will become

mical LLS results with HARQ
Extended MI MIC link performance, as shown vious section.So BLER a s increase, and throughput decreases.

Validation by SLS
Configuration of dynamical SLS is listed in Table 6.In SLS, link evaluation is used to hold down real coding and decoding procedures, reducing SLS complexity, as described in Reference [1].Because the BLER in SLS is computed by link evaluation, the SLS results

Conclusions
Link evaluation aims to provide a fading insensitive performance metric for common transmissions.It is proven from the view of information theory how that the proposed ERBIR algorithm ry well for common transmissions, solving th s existing in current link evaluatio s.OFDM System-Level Simulations.
[5] WiMAX Forum.Mobile WiMAX-Part I: A Technical Ovview and Performance Evaluation.white paper, August 2006.

Ef
The original wireless transmission is out interference symbol but the correlation, and H is consisted of correlated Gaussia Gaussians.
This iden view of capacity.Let |A| means the determinant of matrix A. Ch f the original transmission is Equation ( 34) and ( 35) indicates that the two transmissions are effective.

E{( H
Where eans the i th row and i th column element of matrix A.

A
LER is one-one to RBIR.Then consider the uncoded block f (40) x  ccording to Equation ( 20) and ( 21), B H x n Since the MCS is given, it is pointed out that Extrinsic Information Transfer (EXIT) is definite [11].So RBIR for the coded block after iterative decoding is determined by So there is This is referred to lemma 1.

Normalized MI for SISO
d symbol is * } = σ 2 = 10 −SNR/10 (44) For SISO transmission, the receive Since x is random selected from the constellation, t P(x=q i )=1/N QAM Where q i is the i th mapping point in the modulation co points in the co hen (46) nstellation, and NQAM is the number of nstellation.So nsider the probability of P(y | x), Since x 1 and x 2 are random selected from the constel-lation, Here q 1,i and q 2,j are the i th and j th mapping points i the constellation for x 1 and x 2 respectively.N QAM is the number of points in the constellation.Given transmitting vector, q l = [q 1,i, q 2,j ] T ; l = 1,2,…,N QAM 2 ; i, j = 1,2,…,N QAM (58) Here the tuning parameter is

Figure 1 .
Figure 1.Common model of link evaluation ), RBIR u is one-on the ted CSI indicators are channel response matrixes of [H 1 , H 2 , …, H Nu ], and AWGN power of SNR;

Figure 2 .
Figure 2. SISO normalized MI computation Firstly, SISO transmission with MMSE detection is validated by different link evaluation algorithms, shown as the following figures.From these figures, it is obvious that although EESM, MIC and MMIB algorithms can obtain accurate eno k evalu tain most for simulated transmission ver, RBIR/ERBIR algorithm ×2 Normalized MI co ugh lin a ation.RBIR/ERBIR algorithm can ob ccurate link evaluation Monte Carlo trials.Moreo doesn't need any channel related tuning parameters, which makes RBIR/ERBIR more universal.

Figure
Figure 4(a).EESM LE for SISO MMSE

Figure 4 (
Figure 4(d).MMIB LE for SISO MMSE Then, SISO transmission with ML detection is valiated by different link evaluation algorithms, shown as the following figures.e figures, it is obvious that EESM w d in

Figure
Figure 7(a).MIC LE for VEC ML Figure 8(a).EESM LE for HEC MMSE great support.upported in part by the Hi-tech research t program of China (2007AA01Z277), tional Natural Science Foundation of China (6077 2035), University Doctorial Foundation of China (2007 0004010), and Intel China Research Centre.

y
= H e x + n e ; E{xx H } = I(N S ); E{ n e n e H } = σ e 2 I(N R ) (36) Let x o = My = M(H e x + Because receiver knows nothing ab Then the channel capacity of the effective t I ns, H I x I + n is approximated as correlated Assume R I is known to the receiver.

Let E{n 1
n 1 H } = I(N R ), and TT H = R I + σ 2 I(N R ).So E{(Tn 1 )(Tn 1 ) } = E{(H I x I + n)(H I x I + n) } (30) This means Tn 1 is effective to H I x I + n, so the original wireless transmission is effective to y = HcFx + Tn 1 ; E{ n e 2 |πeE{( H I x I + n H I x I + n) = diag(MH e ), N = diag(σ e 2 MM H ) and I = MH e −D, then OSINR for each symbol in the transmitting signal vector is Let n e = (Δ l,mH n+n H Δ l,m ) / || Δ l,m || F , so n e

Table 2
ER is 0.1, SNR is [5 tomatic Repeat reQu F disabled, and simulation results are listed in the following Table5.

Table 8 . Tuning parameter for 2×2 MIC 6
Then compute MI 2 , and let Δ 2,j,t = q 2,j − q 2,t ,The computation of I 1 and I 2 are similar, so