Recent Patents on Signal Processing, 9,, -7 Enhanced Algorith for MIESM R. Sandanalakshi *, Shahid Mutaz * and Kazi Saidul * Open Access University of Aveiro, Aveiro, Portugal Abstract: The link adaptation technique based on MIESM (Mutual Inforation based exponential SR Mapping) has been extensively used in the literature for 86 based systes. The previous work on MIESM uses equal odulation order for all the subcarriers in an OFDM block. We have proposed the concept of unequal odulation for the subcarriers in the single OFDM block, and derived a atheatical odel based on bivariate Gaussian distribution, based on this a atheatical equation has been derived for joint PDF and the average probability of perforance equation has been obtained. The difficulty in generating a probability of error for bivariate Gaussian is the otivation for this paper. Results show that for bivariate case the perforance is related to the correlation paraeter. Keywords: 86, MIESM, OFDM, link adaptation. I. ITRODUCTIO ext generation cellular systes support ultiple transission odes, which can be used to iprove the perforance of such systes by adapting to current channel conditions. This process is referred to as link adaptation. Typically, these transission odes include different odulation and coding, schees (MCS) and different ultiple antenna arrangeents odes such as bea-foring, spacetie coding and spatial ultiplexing as the transission becoes ultidiensional in space, tie, and frequency doain. Orthogonal Frequency Division Multiplexing (OFDM) is the air interface for 8, 86 (WiMAX), and 3GPP Long Ter Evaluation (LTE) systes. The resources typically referred to as subcarriers, available in an OFDM frae, can be defined on a tie frequency grid []. The perforance of a binary code depends on the channel condition obtained over the allocated subcarriers. Typically, the channel is frequency selective, and a ean SR etric is only sufficient to obtain a long ter perforance etric of the channel. On the other hand, short ter perforance etrics, which are also key to obtaining perforance enhanceents with feedback, are obtained fro the actual instantaneous channel realization. A well-known approach to link perforance odeling and link quality prediction is the Effective Exponential SIR Metric (EESM) ethod, which coputes an effective SR (also referred to as AWG equivalent SR) etric by taking as input the individual sub carrier SRs and using an exponential cobining function. Once coputed, the block error rate is obtained fro looking up an AWG perforance curve. This approach has been widely applied to OFDM link layers and is based on the perforance approxiation by asyptotic union bounds []. One of the disadvantages of the EESM approach is that a noralization paraeter (usually represented by a scalar, ) ust be coputed for each odulation and coding (MCS) schee. In particular, for broader link-syste apping *Address correspondence to these authors at the University of Aveiro, Aveiro, Portugal; E-ails: sutaz@av.it.pt, kazi.saidul@av.it.pt, sandenelakshi@pec.edu applications, it can be inconvenient to use EESM when cobining codeword apped onto different odulation types, where the EESM ethod can require the use of socalled sybol de-apping penalties. Seeking a eans to overcoe soe of the shortcoings of EESM, we focus here on the Mutual Inforation based approach to link perforance prediction [3]. Approaches based on utual inforation (MI) are proposed in literature [4, 5], which provides advantages over paraetric EESM approaches. However, ost of the link perforance prediction ethods proposed so far are based on apping fro SR to an associated MI etric, by which new algorith for MIESM Link Adaptation with unequal odulation has been proposed. It has been found by the literature survey [5, 6] that the Doppler spreading of OFDM systes follows joint probability distribution function iplies the transission will also be dependent on Bivariate Gaussian distribution and gives better PDF, CDF. By the concept of unequal odulation technique, the perforance of 86 networks can be enhanced as the difference in odulation leads to different channel gain. MMIB has been worked for fast link adaptation in 86 systes where bit interleaved coded odulation is used. The MI is dependent upon the code rate not on the odulation order. But it has been found that for higher odulation the perforance of MMIB to BLER does a not atch well and again there is a bound liit for this assuption. The encoded bits follow different odulation in the channel but at the receiver, it uses the sae decoder, hence, the decoder perforance will be different for soft decision bits but will not be a proble with hard code decision bits [6]. Hence, the MMIB to BLER apping ust be dependent on the odulation as well as the code rate for perfect perforance results. Hence, the proble of using different odulation in the sae channel for different sybols can be considered as two diensional that is bivariate Gaussian distribution. Keeping this concept as otivating point the unequal odulation is designed by bivariate distribution where two odulation orders are used, expressions has been derived for PDF and average probability of error perforance. This paper is organized as follow: Section II explains the link 877-64/9 9 Bentha Open
Recent Patents on Signal Processing, 9, Volue Sandanalakshi et al. adaptation, Section III explains the unequal odulation in 86, and conclusion is presented in section IV. II. 86 LIK ADAPTATIO For counication systes like OFDM where ultiple channel states ay be obtained on a transitted codeword, link perforance prediction, in general, is based on deterining a function I (SIR, SIR, ), which aps ultiple physical SIR observations into a single effective SIR etric SIR eff (or equivalent), which can then be input to a second apping function B (SIR eff ) to generate a block error rate (BLER) estiated for a hypothesized codeword transission. We assue the access to a set of SIR easures, denoted SIR n, n <. ote that the precise definition of these observations will depend on the SISO/MIMO transission ode and a receive type, but for the siple SIOS case, the SIR easures ay be assued to correspond to SIR observations of individual data subcarriers (and therefore, of associated QAM sybols) transporting the hypothesized codeword of interest. Fig. () shows the BICM odel used in this paper [7, 8]. where I (.) is a function that depends on the odulation type identified by and the associated bit labeling in the constellation, where {, 4, 6} corresponding to QPSK, 6- QAM, 64-QAM, respectively. I (.) aps the subcarrier SIR to the ean utual inforation between the log- likelihood ratio and the binary codeword bits coprising the QAM sybol. M = I ( SIR) = I,i SIR i = (4) We will refer to the above quantity as Mutual Inforation per coded Bit or MIB, with the understanding that it is derived by averaging over the bit channels. Furtherore, ean utual inforation per bit (MMIB) is used to refer to the ean obtained over different channel states or SR easures. Thus, the odulation order follows Gaussian distribution and is shown in Figs. (, 3), for the QPSK and 6 QAM. We can note that, as expected, for BPSK, the distribution is Gaussian with ean / n = 4 E s / o =.65 (SR = 5 db). Predictably for QPSK, the distribution is also Gaussian with a ean, which is one half of the BPSK ean. The plot between SR and MMIB is given by the CDF of Gaussian curve is shown in Figs. (4, 5). PDF of QPSK MOdulation Fig. (). BICM odel. The first apping function I, and effective SIR etric SIR eff, ay be generally defined as: SIR eff = I = SIR eff () where and are constants (and aybe constrained to be equal), which ay be MCS-specific, and ay correspond to a defined statistical easure. I (.) is a reference function usually selected to represent a perforance odel. Exponential ESM is derived by using an exponential function, which is based on using Chernoff approxiation to the union bounds on the code perforance. Siilarly, other perforance easures like capacity or utual inforation can be used. The accuracy of the odel to soe extent is dependent on how closely the reference odel represents the code perforance (with sufficient paraeterization, a given odel can yield a reasonably good accuracy as in EESM). In the ethod proposed here, is the ean utual inforation per coded bit (MMIB), or siply denoted as M, and and are discarded (i.e., set to unity). M = I( SIR eff ) = I ( SIR n ) () Pdf Fig. (). QSPK bit-wise conditional distributions. Pdf.8.6.4. - -5 5 5 5 (QPSK) 8 6 4.8.6.4 PDF of 6QAM MOdulation ( SIR eff ) = I I ( SIR n ) (3). - -5 5 5 (6 QAM) Fig. (3). 6 QAM bit-wise conditional distributions.
Enhanced Algorith for MIESM Recent Patents on Signal Processing, 9, Volue 3 MMIB.9.8.7.6.5.4.3. CDF of QPSK MOdulation shown in Fig. (). ote that for higher-order odulations, the bit positions in the signal constellation are asyetric; therefore, each bit location of one sybol will experience a different bit channel and will have different statistical properties [8, 9]. BLER.9.8.7.6.5 Fig. (4). MIB vs Es/o for QPSK. MMIB - -5 5 5 5 Eb/o.9.8.7.6.5.4.3. CDF of 6QAM MOdulation - -5 5 5 Eb/o Fig. (5). MIB vs Es/o for 6 QAM. The BS can store the AWG reference curves for different MCS levels in order to ap the MMIB to BLER [9]. Another alternative is to approxiate the reference curve with a paraetric function. For exaple, we consider a Gaussian cuulative odel with 3 paraeters, which provides a close fit to the AWG perforance curve, paraeterized as y = a erfc x b, c (5) c where a is the transition height of the error rate curve, b is the transition center, and c is related to the transition width (transition width =.349 c) of the Gaussian cuulative distribution. In the linear BLER doain, the paraeter a can be set to, and the apping requires only two paraeters as shown in Fig. (6). III. 86 UEQUAL MODULATIOS BIVARIATE DISTRIBUTIO In a bit-interleaved coded odulation (BICM) syste, the bit interleaver breaks the eory of the odulator such that the equivalent channel in Fig. () is represented by an equivalent odel of parallel independent bit channels as Fig. (6). Mapping of MMIB to BLER. Taking into account the above rearks and assuing the bits in a sybol have equal probabilities, the utual inforation per coded bit (MIB) is considered as the average of the bit-wise utual inforation easures, being the nuber of bits per sybol []: I = I,i i= ( c i, ( c i )) where i, ( i, ( i) ) I c c is the MI between the input to the -QAM apper and the output for the i th bit position in the sybol. Ii, ( ci, ( c i) ) depends on the SIR of the QAM sybol, therefore, equation becoes: I ( SIR)= I,i i= ( SIR) Fro the above equation, considering the codeword is transitted over channel sybols, the ean utual inforation per coded bit (MMIB) can be coputed as: MMIB = I (n) ( SIR n ) where (n) is the odulation order of the n th sybol and SIR n is the corresponding SIR. It is also shown in [] that I can be nuerically approxiated, and then stored for later use for link perforance prediction. The MI per coded bit is given by: I,i.4.3...3.4.5.6.7.8.9 MMIB ( c i, ( c i ))= + p c i, { } p ( r c i ) log r c i dr p ( r c i = )+ p ( r c i = ) where r is a integration variable and ( ) p r c is the s conditional PDF for the i th bit position in the QAM sybol. i
4 Recent Patents on Signal Processing, 9, Volue Sandanalakshi et al. It can be deonstrated that for BPSK and QPSK odulations the bit s are Gaussian distributed and their ean value has the property μ = : p I,i ( r c i )= ( c i, ( c i ))= + exp r μ c i exp r μ c i log ( + er )dr J ( ) where J ()is a nonlinear function, which can be approxiated by [3]: J( x) a x3 + b x + c x, for < x <.6363 exp( a x 3 + b x + c x + d ), for x.6363 where the coefficients are: a =.466 b =.95 c =.648 a =.849 b = 4675 c =.854 d =.54968 For BPSK, we have = 8E s, while for QPSK = 4E s. This leads to the following expression of the MI per coded bit for BPSK: I ( SIR)= I, ( c, ( c ))= J 8SIR While for QPSK we have equation a : I ( SIR)= I,i c i, ( c i ) = J 4SIR = J 4SIR i= i= The proble with higher-order odulations is that they cannot be expressed in a closed for like above. But in [], an approxiation of the MIBs for these odulations is given, based on the observation that the conditional PDFs can be approxiated by a ixture of Gaussian distributions that are non-overlapping at high SIR values. Thus, the MIB is expressed as a su of J () functions in equation b I K K ( SIR)= a k J b k SIR with a k = k= [] States that liiting the su in equation a to 3 ters (3 doinant Gaussians) gives a good approxiation for the MIB function. In this way, the following expressions are obtained for 6QAM and 64QAM: I 4 ( SIR)= J b 4 ( SIR )+ 4 J b 4 ( SIR )+ 4 J b 4 ( SIR 3 ) I 6 ( SIR)= 3 J b 6 ( SIR )+ 3 J b 6 ( SIR )+ 3 J b 6 ( SIR 3 ) k= The b k coefficients are obtained by nuerical siulation and curve fitting by following these steps [, ]:. Obtain through nuerical siulation, the conditional PDFs for each bit of a specific odulation at each SR in an AWG channel.. Copute the MIB by nuerical integration using above equations. Perfor curve fitting using the approxiation functions in equation b. Fig. () presents the MIB versus SR curves. ote that if the MIB approxiations for 64QAM are not accurate enough, LUTs, which store the MIB vs SIR, could be used. Having the I () functions for the odulation types of interest allows the coputation of MMIB for a specific channel realization over the coded sybols [, 3]. In the basic link adaptation for 86 based on MIESM, the utual inforation follows the Gaussian distribution, in which the OFDM block uses a single odulation. We have adapted the sae concept in which the OFDM block uses odulation schees: a low order and a high order. The purpose of such a ultilevel odulation is that the carriers in an OFDM block under deep fading can be allotted a low odulation order and the others high level odulation and as a result the SR perforance can be iproved uch better than equal odulation. Fro theory, two univariate arginal distributions following Gaussian distribution can be odeled under the sae roof as bivariate joint distribution given by f X X (x, x ) = / (x, x ) ( )( ) ( +)/ (/) ( ) exp x / + x / I x x U(x )(x ) where (.)is gaa function and I(.) is a odified Bessel function of first kind, and are scaling paraeters, x and x are rando variables for different arginal distributions, and is the correlation coefficient, When the joint pdf tends to be the product of two univariate gaa distribution [3]. Consider two OFDM odulation techniques, 6QAM and 64QAM, and the joint probability distribution for the is shown in Figs. (7, 8). akagai proposed a odel that corresponds to the scenarios of dual-diversity reception over correlated akagai- channels, which are not necessarily identically distributed. This ay therefore, apply to independent fading channels aong signal paths.
Enhanced Algorith for MIESM Recent Patents on Signal Processing, 9, Volue 5 pdf(obj,[x,y]) d = d, d =,,....8.6 = + ( ) (9).4. 5 y -5 - - Fig. (7). PDF for Bivariate noral distribution for 6QAM and 64QAM. -5 x 5 = ( ) + 4 4 ( ) where d, d =,, is the average fading power of the dth channel. By using a standard transforation of rando variables, it can be shown that the PDF of the cobined SR per sybol, pa(t), is given by p a( t ) = () ( ) t ' / () I / ( ' t )e ' t, t where the paraeters and are noralized versions of the paraeters and, and are given by ' = = ( + ) E s / ( ) ' = = ( + ) 4 / ( ) E s / ( ) () And finally the equation reduces to Fig. (8). CDF for bivariate noral distribution for 6QAM and 64QAM. In this case, the PDF of the cobined signal envelope, pa(rt), is given by p a(rt ) = r t () ( ) r t I / (r t )e r t, r t / where I( ) denotes the th-order odified Bessel function p a ( t ) = exp t exp t ( + ) ( ) cov(r =, r ), (8) var(r ) var(r ) Is the envelope correlation coefficient between the two signals and the paraeters d (d =, ),, and are defined as follows: (7) p a ( t ) = exp t exp t ( + ) ( ), t () Using the Laplace transfor it can be shown after soe anipulations that the MGF of pa(t) is given by M a ( s :, : : ) = M a (s) = ( + ) s + ( ) s (3), s It should be noted that the previous leads to an exact expression for average probability of error. For exponential correlation and average probability of error, perforance is plotted for different values of correlation coefficient, clearly the BER perforance degrades with correlation values shown in Figs. (9, ). IV. COCLUSIO The concept of unequal odulation has been odeled and studied for 86 based systes. Analysis shows that unequal odulation follows bivariate distribution. Matheatical expressions have been derived correlation paraeters and results have been plotted for average probability of error with correlation coefficient.
6 Recent Patents on Signal Processing, 9, Volue Sandanalakshi et al. ρ =.8 ρ = M.5.5.5 M ρ = -.5 M ρ = -.8 M.5 M.5.5 M.5 M Fig. (9). BER for different values of. Fig. (). Perforance of SR vs BER for different values of. Fig. (). Equivalent inforation channel. Fig. (). MIB vs SR; nuerical siulation (black) and approxiation (red).
Enhanced Algorith for MIESM Recent Patents on Signal Processing, 9, Volue 7 ACKOWLEDGEMETS The work presented in this paper was supported by the European project ICT-CODIV. REFERECES [] Sayana K, Zhuang J, Stewart K, Motorola Inc. Short ter link perforance odeling for l receivers with utual inforation per bit etrics June. [] 3GPP, TSG-C WG3.3GPP TS 5.3 v 5.5., Mediu Access Control (MAC) protocol specification. [3] IEEE 86 Broadband Wireless Access Working Group, Sayana K, Zhuang J, Stewart K, Motorola Inc. Link perforance abstraction based on ean utual inforation per bit(ib) of the llr channel ov 6. [4] Brueninghaus K, Astely D, Salzer T, Visuri S. Link perforance odels for syste level siulations of broadband radio access systes, IEEE 6th International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC) Sept 5. [5] Wan L, Tsai S, Algren M. A fading- insensitive perforance etric for a united link quality odel. Proceedings of WCC 6; -4. [6] Wang TR, Proakis JG, Masry E, Zeidler JR. Perforance Degradation of OFDM Systes Due to Doppler Spreading Sept 4. [7] Ki J, Ashkhiin A, Wijngaarden A, Soljanin E. Reverse link hybrid ARQ link error prediction ethodology based on convex etric. Lucent Technologies Jan 7. [8] IEEE 86-7/3, IEEE 86 Broadband Wireless Access Working Group: RBIR MLD PHY Abstraction for HARQ IR/CC, 3//7. [9] Sion MK, Alouin MS. Digital counication fading channels A unified approach to perforance analysis. John Wiley and Sons;. [] Sayana K, Zhuang J. Link perforance abstraction based on ean utual inforation per bit (MMIB) of the channel. IEEE Contribution, C86-7/97; May 7. [] Ericsson, Syste-level evaluation of OFDM further considerations, 3GPP TSG-RA WG, o. 35 R-333; 3. [] Bau, Kevin, L., Blankenship, Yufei, W., Classon, Brian, K., Sartori. : US58 (6). [3] Eerolainen, Lauri.: US455436 (5). Received: June 5, 9 Revised: July 9, 9 Accepted: August 3, 9 Sandanalakshi et al.; Licensee Bentha Open. This is an open access article licensed under the ters of the Creative Coons Attribution on-coercial License (http://creativecoons.org/licenses/by-nc/ 3./) which perits unrestricted, non-coercial use, distribution and reproduction in any ediu, provided the work is properly cited.