rate allocation for 2-user mac with mmse turbo
TRANSCRIPT
Japan Advanced Institute of Science and Technology
JAIST Repositoryhttps://dspace.jaist.ac.jp/
TitleRate allocation for 2-user MAC with MMSE turbo
equalization
Author(s)Grossmann, Marcus; Ortlepp, Thomas; Matsumoto,
Tad
CitationIEEE Transactions on Wireless Communications,
9(3): 1033-1043
Issue Date 2010-03
Type Journal Article
Text version publisher
URL http://hdl.handle.net/10119/9100
Rights
Copyright (C) 2010 IEEE. Reprinted from IEEE
Transactions on Wireless Communications, 9(3),
2010, 1033-1043. This material is posted here
with permission of the IEEE. Such permission of
the IEEE does not in any way imply IEEE
endorsement of any of JAIST's products or
services. Internal or personal use of this
material is permitted. However, permission to
reprint/republish this material for advertising
or promotional purposes or for creating new
collective works for resale or redistribution
must be obtained from the IEEE by writing to
[email protected]. By choosing to view
this document, you agree to all provisions of the
copyright laws protecting it.
Description
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010 1033
Rate Allocation for 2-User MAC withMMSE Turbo Equalization
Marcus Grossmann, Thomas Ortlepp, and Tad Matsumoto, Senior Member, IEEE
AbstractâWe consider the problem of rate allocation infrequency-selective 2-user Gaussian multiple access fading chan-nels employing turbo equalization. The turbo equalization frame-work used in this paper contains a soft cancellation frequencydomain minimum mean squared error equalizer and two aposteriori probability channel decoders. Using the relationshipbetween the rate of any code and the area under its correspondingdecoder extrinsic information transfer (EXIT) function, we derivean upper bound on the rate region of the 2-user turbo system,given the EXIT characteristic of the equalizer for a particularchannel realization. With the rate region upper bound, wethen study the problem of maximizing the sum rate of bothusers, and provide an approximate solution to this optimizationproblem. Based on the obtained result, a practical code selectionalgorithm for rate allocation at both transmitters is proposed. Inaddition, we discuss the extension of the proposed algorithm toan outage-based rate allocation approach. Numerical results ofcapacity calculations and throughput simulations are presentedto demonstrate the performance enhancement achieved by theproposed rate allocation technique over automatic repeat requestwith fixed coding rate.
Index TermsâTwo-user multiple access channel, rate allo-cation, extrinsic information transfer chart analysis, minimummean squared error equalization.
I. INTRODUCTION
RECENTLY, iterative (turbo) techniques have been rec-ognized as practical solutions to multi-user detec-
tion/equalization problems in coded communication systems.In [1], utilization of the optimal a posteriori probability (APP)equalizer (EQ) in combination with the APP-based decoder(DEC) is considered for turbo equalization in frequency-selective fading channels. The turbo EQ of [1] achievesexcellent performance, however, its complexity is prohibitivelyhigh in channels having a medium-to-large number of multi-path components. In [2], the APP detector (DET) is replacedby a less complex DET that performs soft canceling andminimum mean squared error (SC MMSE) filtering. This SCMMSE filtering approach, originally proposed for detection ofrandom coded code-division multiple-access (CDMA) signals,
Manuscript received July 8, 2008; revised March 31, 2009 and October7, 2009; accepted October 15, 2009. The associate editor coordinating thereview of this paper and approving it for publication was D. Gesbert.
M. Grossmann and T. Ortlepp are with Institute for Information Technology,Ilmenau University of Technology, Germany (e-mail: {marcus.grossmann,thomas.ortlepp}@tu-ilmenau.de).
T. Matsumoto is with Japan Advanced Institute of Science and Technology,Japan, and Center for Wireless Communication at University of Oulu, Finland(e-mail: [email protected]).
This work was supported in part by the German Research Foundation(DFG) under grant SPP1163 and in part by the Japanese government fundingprogram, Grant-in-Aid for Scientific Research (B), No. 20360168.
Digital Object Identifier 10.1109/TWC.2010.03.080894
is applied to channel equalization in [3], and to multiple-input multiple-output (MIMO) channel equalization in [4]. In[5], a turbo equalization technique for single carrier block-transmission over multiple-access channels (MACs) in thepresence of frequency-selective fading is proposed that per-forms the SC MMSE filtering in frequency domain, furtherreducing the computational complexity.
The convergence of turbo systems can be analyzed byextrinsic information transfer (EXIT) charts [6]. The EXITchart may be used to predict the convergence threshold andto visualize the mutual information (MI) exchange betweenthe soft-input soft-output components, representing the conver-gence property of turbo systems. Ashikhmin et al. [7] showedthat for any code with rate ð , the area under its correspondingDEC EXIT function is 1âð . This property has been provedso far only for the case when the DECâs a-priori informationis assumed to be from an erasure channel. However, it alsoappears to work well for Gaussian distributed a-priori log-likelihoods (LLRs) [7]. Based on the area property of theEXIT chart, it has been shown in [8], [9] that the problemof rate allocation in the single-user case, in general, reducesto a simple curve-fitting problem of the two-dimensional (2D)EXIT curves of the DET and DEC. In [8], ten Brink et al. usedthis technique to determine the optimal degree distribution oflow density parity check (LDPC) codes for single-user MIMObit interleaved coded modulation (BICM) employing iterativedetection and decoding. Based on a similar idea, the authorsof [9] investigate the design of repeat-accumulate codes withiterative detection.
The EXIT chart can also be used for rate allocation inMAC scenarios. In [10], the EXIT chart analysis is usedto optimize LDPC codes for the equal-rate 2-user GaussianMAC without fading employing iterative decoding. It is shownthat for allocating the same rate to both users very simpleLDPC code optimization approaches can be derived, similarlyto the single-user case. Similar results with LDPC codes areobtained in [11], where the authors showed that by properlychoosing the multi-user code and applying iterative decoding,any point on the boundary of the capacity region can beclosely approached without requiring time-sharing [12] orrate-splitting [13].
The EXIT chart analysis is used in [14] to optimize LDPCcodes for the equal-rate flat-fading MIMO MAC employingiterative detection and decoding. The problem of rate allo-cation is studied from a large-system perspective, i.e., thenumber of users and antennas are taken to infinity, whiletheir ratio remains fixed. For such a system, it is shownthat results from the asymptotic analysis of CDMA withrandom spreading remains valid, which allows the behavior
1536-1276/10$25.00 câ 2010 IEEE
1034 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
of the iterative receiver to be described by a single 2D EXITchart. Thus, rate allocation is performed by 2D EXIT curvematching, as in the single-user case. The results reveal thatwith the system proposed in [14] near capacity performancecan be asymptotically achieved. Asymptotic techniques forrate allocation have also been applied in [15] to designrepetition codes in the context of equal-rate CDMA systemswith iterative detection and decoding.
In this paper, we consider the problem of rate allocationfor the 2-user frequency-selective fading MAC employingsoft cancellation frequency domain MMSE (SC FD-MMSE)equalization. In contrast to previous work [10], [14], [15], wedo not restrict the rates of both users to be equal. For such aturbo system, the EQ EXIT characteristic is given by two 3Dsurfaces, and thus, rate allocation is no longer a simple 2Dmatching of the EXIT chart.
Based on the area property of EXIT functions, we derivean upper bound on the rate region of the 2-user turbo sys-tem, given the EQ EXIT functions for a particular channelrealization and receiver noise variance. The rate region upperbound is then used to study the problem of maximizing thesum rate of both users. Specifically, we show that the sum ratemaximization can be formulated as a variational problem. Weprovide an efficient solution to this optimization problem byusing an approximation on the EQ EXIT functions. As a result,the optimal DEC EXIT curves of both users with respect tothe maximum sum rate of the turbo system, are obtained.
For practical reasons, we restrict each transmitter to haveonly a finite number of codes with fixed rates. For this scenarioassumption, we propose a simple code selection algorithm forthe rate allocation that maximizes the coding rate of eachuser, given the optimal DEC EXIT curve, while satisfyingthe constraints for successful decoding. Moreover, we discussas an extension of the proposed algorithm an outage-basedrate allocation approach, and determine the achievable codingrates of both users using a specific outage constraint.
The remainder of this paper is organized as follows. Thenotation is introduced in Section II. In Section III, the systemdescription and the channel model are presented. In SectionIV-A, we define the EXIT functions for the EQ and bothDECs, and derive an upper bound on the rate region of the2-user turbo system. We then consider the optimization of thesum rate of both users in Section IV-B and propose a simplerate allocation algorithm in Section IV-C. In Section IV-D,we extend the proposed algorithm to an outage-based rateallocation approach. In Section V, we present some numericalresults to verify the performance of the proposed approaches.We summarize our results in Section VI.
II. NOTATION
The transpose and conjugate transpose operators are de-noted by (â )ð and (â )ð» , respectively. The circð
{a}
operatorgenerates an ð à ð circulant matrix having the elements ofvector a on its first column. The symbol â indicates theKronecker product. The notation a ⥠b for length-ð vectorsa and b means ðð ⥠ðð for ð = 1, .., ð . We use thesymbol â¡ to indicate that the expression on the left handside is defined by the expression on the right hand side. We
use ðŒð to represent a closed set ðŒ
ð â¡ {x â [0, 1]ð}, andâðŒ2 is used for the boundary of ðŒ
2. We denote the fourcorner points of the region ðŒ
2 by ð0 â¡ (0, 0), ð1 â¡ (0, 1),ð2 â¡ (1, 1) and ð3 â¡ (1, 0). Finally, â±1[ð, ð] denotes thespace of monotonically increasing, continuous and piecewisedifferentiable functions on the interval [ð, ð].
III. SYSTEM MODEL
Consider a single carrier cyclic prefix (CP) assisted 2-useruplink system, where a base station having ð receive antennasreceives signals from two active users, each equipped withðŸ transmit antennas. For the ease of analysis, we assume inthe following ð = 2 and ðŸ = 1. However, the extensionto more generic cases (ð > 2, ðŸ > 1) is straightforward.The transmission scheme of each user is based on BICM,where the information bit sequence is independently encodedby a rate-ðð,ð (with ð = 1, 2 being the user index) binaryencoder, randomly bit-interleaved, binary phase-shift keying(BPSK) modulated, and grouped into ð (ð = 1, .., ð ) blocksbð(ð) â¡
[ð0,ð(ð), ..., ðð,ð(ð), ..., ððâ1,ð(ð)
]ðof length ð
that are transmitted over the frequency-selective fading MAC.The binary encoder can be a single convolutional code (SCC)or a serially concatenated convolutional code (SCCC) [22].Note that throughout the paper, we use index ð to denoteuser ð.
The frequency-selective fading MAC is assumed to beconstant during the transmission of one frame (comprised ofð blocks), but varying randomly and independently frame-by-frame. We restrict ourselves to Rayleigh block-fadingchannels, where each of the 2ðŸð links is comprised ofð¿ independent and identically distributed (i.i.d.) complexcircularly-symmetric Gaussian path components hð,ð â¡[âð,ð(0), ..., âð,ð(ð¿ â 1)
]ðfor ð = 1, ...,ð . We always
assume that all channel gains are perfectly known at thereceiver.
Employing a CP of length ð = ð¿ â 1 to each transmitblock, the received signals can be expressed as
r(ð) =
2âð=1
Hðbð(ð) + v(ð), ð = 1, .., ð, (1)
where Hð =[Hðð,1 Hð
ð,2
]ðwith Hð,ð = circð
{hð,ð
}being the block-circulant channel matrix associated to the ð-th user, and v(ð) ⌠ðð© (0, ð2I) is the additive white Gaussiannoise (AWGN). Note that each Hð may be decomposed intoa diagonal-block matrix Îð by the Fourier matrix,
Hð = (I2 â Fð»)ÎðF, (2)
where F denotes the Fourier matrix of size ð, whose (ð, ð)-thelement is given by (1/
âð) exp(âð2ððð/ð), ð = ââ1, for
0 †ð, ð †ð â 1.At the receiver side, iterative processing for joint equaliza-
tion and decoding is performed. The receiver consists of an SCFD-MMSE EQ and two single-user DECs that perform APPdecoding. It should be noted here that the major outcomes ofthis paper are also applicable to other types of equalizationtechniques.
Within the iterative processing, extrinsic LLRs of the codedbits are exchanged between the EQ and two DECs, each
GROSSMANN et al.: RATE ALLOCATION FOR 2-USER MAC WITH MMSE TURBO EQUALIZATION 1035
separated by the interleaver and deinterleaver in their iterationloop, following the turbo principle [2]. Inputs to the EQ arethe received signals r(ð) and the a priori LLR sequences
ð»ð(ð) â¡[ð0,ð(ð), ..., ðð,ð(ð), ..., ððâ1,ð(ð)
]ðfor all ð, ð,
(3)
where
ðð,ð(ð) â¡ log
(Prob
(ðð,ð(ð) = +1
)Prob
(ðð,ð(ð) = â1
)). (4)
The EQ computes the extrinsic LLR for each transmitted bitðð,ð(ð),
ðð,ð(ð) â¡ log
(Prob
(ð§ð,ð(ð)â£ðð,ð(ð) = +1
)
Prob(ð§ð,ð(ð)â£ðð,ð(ð) = â1
))
(5)
where ð§ð,ð(ð) is the SC FD-MMSE filter output as defined inequation (18) in [5]. Similarly to (3), we arrange the ðð,ð(ð)âsinto vectors ðð(ð) â¡
[ð0,ð(ð), ..., ððâ1,ð(ð)
]ðfor all ð, ð.
Note that during the first iteration of turbo equalization,ðð,ð(ð) is zero for all ð, ð, ð, and later on ðð,ð(ð) is providedvia the interleaver in the form of extrinsic LLRs of the ð-thDEC.
The receiver also selects the code to be used for each userfrom an available code set according to the criterion derivedin Section IV-C, where both users are notified of the codesselected through separated feedback links. We assume zero-delay and error-free feedback links.
IV. RATE ALLOCATION
A. Definition of EXIT Functions and Area Property
Let the MI between the transmitted bits ðð,ð(ð) and thecorresponding LLRs ðð,ð(ð) be denoted as [5]
ðŒð,ð â¡ limðââ
1
ððI(bð(1), ...,bð(ð);ðð(1), ...,ðð(ð)
),
and let the MI between ðð,ð(ð) and ðð,ð(ð) be denoted as
ðŒð,ð â¡ limðââ
1
ððI(bð(1), ...,bð(ð); ð»ð(1), ..., ð»ð(ð)
).
In the 2-user case the convergence characteristic of the EQ isdefined by two EXIT functions,
fð : Ið â fð â¡(ðð,1(Ið), ðð,2(Ið)
) â ðŒ2,
which depend on the MI Ið â¡ (ðŒð,1, ðŒð,2) â ðŒ2 of both
DECs. In [5] it is shown that these two EXIT functions canbe efficiently computed by modeling the LLRs ðð,ð(ð) âŒ
ð© ((ð2ð/2)ðð,ð(ð), ð
2ð
)as independent Gaussian random vari-
ables, where ðð = ðœâ1(ðŒð,ð) with ðœâ1 being the inverse ofðœ defined as [6]
ðœ(ð) = 1â 1â2ðð
â« +â
ââðâ
(ð¥âð2/2)2
2ð2 log2(1 + ðâð¥)dð¥. (6)
As stated in [3], the filter output ð§ð,ð(ð) of the EQ can beclosely approximated by an equivalent AWGN channel havingðð,ð(ð) as its input,
ð§ð,ð(ð) = ðððð,ð(ð) + ðð,ð(ð), (7)
0 0.2 0.4 0.6 0.8 1
00.20.40.60.810
0.2
0.4
0.6
0.8
a3
fd,1
Id,1
a2
D
fe,2
a0Id,2
fe,1
I e,k
,k=
1,2
a1
decodingtrajectory
Fig. 1. EQ EXIT functions ðð,1 and ðð,2 for a single random channelrealization and DEC EXIT function ðð,1 for a constraint length 5 rate-1/2SCC. A possible decoding trajectory visualizing the MI exchange over theiterations is plotted as projection onto the plane region ð° . ð = 128, ð¿ = 10and ðžð /ð0 = 0 dB.
where ðð is the equivalent channel gain, which is conditionedon the FD channel response (Î1, Î2) and the receiver noisevariance ð2, and ðð,ð(ð) ⌠ð© (0, ðð(1 â ðð)/2). The signal-to-noise ratio (SNR) in (7) is given by SNRð = 2ðð/(1â ðð).Based on (5), the value of ðŒð,ð is given by the average MI ofthe equivalent memoryless binary-input Gaussian channel, as[6]
ðŒð,ð = ðð,ð(Ið) = ðœ(2â
SNRð). (8)
Similarly, the convergence characteristics of both DECs aredefined by the two EXIT functions1 ðð,ð : ðŒð,ð â ðð,ð(ðŒð,ð) âðŒ. In addition, these two functions have the extreme valuesðð,ð(0) = 0 and ðð,ð(1) = 1 for ð = 1, 2. We obtain the ðð,ðâsby a Monte Carlo method [6]. In the following, we assumethat fð, ðð,1 and ðð,2 are monotonically increasing, continuousand differentiable.
An example of the two EQ EXIT functions fð and the DECEXIT function ðð,1 is shown in Fig. 1. Also shown is a possi-ble decoding trajectory of the MI exchange, which is plotted asa projection onto the plane region ð° â¡ {Ið : Ið â ðŒ
2}. Notethat the DEC EXIT function ðð,2 (not shown) is drawn inthe ðŒð,2-coordinate. For the computation of the trajectory, thecodes of both users were in this case assumed to be identical,and hence the shapes of their EXIT functions are exactly thesame.
Let ð be a region defined by
ð â¡ {Ið : ðð,ð(Ið) ⥠ðð,ð(ðŒð,ð), ð = 1, 2}. (9)
The region in (9) is shown in Fig. 1, and is referred to as thefeasible region of fð and ðð,ð, ð = 1, 2. Note that ð0 â ð,since ðð,ð(ð0) ⥠ðð,ð(0) = 0 for ð = 1, 2.
Let {v(ð)}, v(ð) â ð, v(0) = ð0, ð = 0, ..., ð be a sequenceof Ið-tuples that models the decoding trajectory (projectedonto the plane region ð°) according to a specific activation
1Note that ðð,ð , denoted in this paper as the DEC EXIT function,corresponds to the inverse DEC EXIT characteristic defined in [6].
1036 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
ordering of the EQ and the two DECs over ð iterations.The monotonicity of all EXIT functions together with thedefinition in (9) imply that v(ð+1) ⥠v(ð) for all ð, andthus, the sequence {v(ð)} converges monotonically to a limitpoint Iâð = limðââ v(ð). As shown in Theorem 1 of [16], thislimit point is unique and independent of the actual activationordering. Convergence of turbo equalization is achieved, whenthe decoding trajectory attains the maximum point Iâð = ð2.This is possible for ð being sufficiently large, if the followingtwo constraints hold:
ð is pathwise connected2 and ð2 â ð. (10)
Let ðŽð â¡â«â«
ð dIð be the area of ð. Assume now that eachDEC EXIT function ðð,ð is matched to the corresponding EQEXIT function ðð,ð so that only an infinitesimally small opentube between the four surfaces remains, where the trajectorycan go from v(0) = ð0 to Iâð = ð2. Note that such DECEXIT functions imply 1) an ideally designed code for eachuser of infinite block length to achieve a nearly zero BERand 2) an infinite number of iterations between the EQ andthe two DECs3. Under this assumption, the size of the areaðŽð is close to zero and the region ð can be characterizedby a curve ð®, which is referred to as convergence curve inwhat follows. The convergence curve ð® is parameterized by avector-function
u(ð¡) â¡ (ð¢1(ð¡), ð¢2(ð¡)): ðŒâ ð®, ð¢ð â â±1[0, 1], ð = 1, 2,
(11)
where each ð¢ð(ð¡), ð¡ â ðŒ is monotonically increasing in theparameter ð¡ and has the prescribed boundary values ð¢ð(0) = 0and ð¢ð(1) = 1.
Let wð(u) be a three-dimensional space curve, obtained byprojection of u on ðð,ð,
wð(u) â¡(ð¢1(ð¡), ð¢2(ð¡), ðð,ð(u(ð¡))
) â ðŒ3, (12)
and let ð (u)ð : ðŒð,ð â ð
(u)ð (ðŒð,ð) â ðŒ be a function obtained
by projection of wð onto the ðŒð,ð-ðŒð,ð-plane. The convergencecurve u â ð satisfies the constraints in (10), which impliesthat
ðð,ð(ðŒð,ð) < ð(u)ð (ðŒð,ð), âðŒð,ð â [0, 1), for ð = 1, 2. (13)
From (13), we easily obtain the following bound:
ðŽð < ðŽ(u)ð , (14)
where ðŽð and ðŽ(u)ð denote the areas under ðð,ð(ðŒð,ð) and
ð(u)ð (ðŒð,ð), respectively,
ðŽð â¡â« 1
0
ðð,ð(ðŒð,ð)dðŒð,ð, (15)
ðŽ(u)ð â¡
â« 1
0
ð(u)ð (ðŒð,ð)dðŒð,ð. (16)
2A set ð is said to be pathwise-connected if for every ð, ð â ð there aretwo real numbers ð, ð with ð †ð and a continuous mapping ð such thatð(ð) = ð, ð(ð) = ð, and ð([ð, ð]) â ð [18].
3Note that this does not lead to a conclusion that the whole transmissionchain can achieve capacity, since the use of sub-optimal (SC FD-MMSE)equalization already incurs a loss in rate [7].
Id,1
Id,2
u(t)
fe,2(u(t))
fe,1(u(t))
R1
f(u)1 (Id,1)
f(u)2 (Id,2)
R2
a0
a1
a2
a3
Fig. 2. Example of the parametric convergence curve u(ð¡) with the twocorresponding EXIT space curves and the two areas defining the achievablerates.
In [7], it is shown that the area under the EXIT function ðð,ð ofan APP-based DEC for a rate-ð ð code satisfies the propertyð ð = ðŽð . Combining this result with (14) yields an upperbound for the rate ð ð of user ð with respect to the convergencecurve u,
ð ð < ðŽ(u)ð . (17)
Equivalently, the area ðŽ(u)ð in (16) can be expressed as
line integral of ðð,ð along ð® in ðŒð,ð-direction, ðŽ(u)ð =â«
ð® ðð,ð(Ið)dðŒð,ð. Therefore, we can also express (17) as
ð ð <
â«ð®ðð,ð(Ið)dðŒð,ð (18a)
=
â« 1
0
ðð,ð(u(ð¡)
)ð¢â²ð(ð¡)dð¡, (18b)
where ð¢â²ð(ð¡) denotes the first derivative of ð¢ð(ð¡). The equality
in (18b) follows directly from the curve parameterizationin (11). An example of the convergence curve with thetwo corresponding EQ EXIT space curves and related areasdefining the achievable rates of both users is shown in Fig. 2.
Let ð« be the set of admissible parametric curves in theplane region ð° ,
ð« â¡ {p : ðð â â±1[0, 1], ðâ²ð(ð¡) ⥠0, âð¡, ð = 1, 2,
p(0) = ð0,p(1) = ð2}. (19)
From the inequality in (18) and with the definition in (19),we finally obtain an upper bound for the rate region of bothusers as
â â¡âªpâð«
{(ð 1, ð 2) : ð ð <
â«p
ðð,ð(Ið)dðŒð,ð, ð = 1, 2}.
(20)Fig. 3 illustrates an example of the rate region in (20), whereðð,1 and ðð,2 have been computed using (8) for a randomchannel realization. The rates at the corner point ð1 can beachieved by successive equalization and decoding techniques,where the signal from user 1 is detected first through iterationsonly between the EQ and user 1âs DEC, such that only theMI ðŒð,1 increases with the iterations, while the MI ðŒð,2 stayszero. After decoding user 1âs signal, the signal from user 2 isiteratively detected, while ðŒð,1 = 1. Thus, for achieving the
GROSSMANN et al.: RATE ALLOCATION FOR 2-USER MAC WITH MMSE TURBO EQUALIZATION 1037
0.6 0.65 0.7 0.75 0.8 0.850.75
0.8
0.85
0.9
0.95
R1 [bpc]
R2
[bpc
]
V1
V2
Rmax = 1.62 bpc
Fig. 3. Rate region of 2-user MAC with SC FD-MMSE turbo equalizationfor a random channel realization at ðžð /ð0 = 5 dB, numerically computedby generating a large number of different admissible convergence curves (agray dot corresponds to one curve). ð = 32, ð¿ = 10.
corner point ð1, the convergence curve ð® must be given by apath on the boundary âð° , which connects the points ð0, ð3and ð2, ð® = â(ð0, ð3, ð2), where â(ð0, ð1, ..., ðð£) is definedas the union of ð£ straight line segments, connecting the pointsðð â ðŒ
2, ð = 0, ..., ð£:
â(ð0, ð1, ..., ðð£) â¡ð£â1âªð=0
{ðð + ð(ðð+1 â ðð) : ð â [0, 1]
}.
(21)
The rate tuple at ð2 can similarly be achieved by firstiteratively detecting the signal from user 2, followed by user1âs signal detection. In this case the convergence curve mustbe given by ð® = â(ð0, ð1, ð2). Note that the rate region in(20) is non-convex, in general, where the dominant face ofthis region strongly depends on the particular realization ofthe EQ EXIT functions fð.
B. Maximization of Sum Rate
To identify the rates of both users and the correspondingconvergence curve that maximize the sum rate, we are inter-ested to solve the following variational optimization problem:
ð ððð¥ â¡ maxpâð«
2âð=1
â«p
ðð,ð(Ið)dðŒð,ð. (22)
Using (18), the functional in (22) can also be expressed as
ð [p(ð¡)] â¡â« 1
0
ð(p(ð¡)
)dð¡, (23)
where
ð (p) =
2âð=1
ðð,ð(p(ð¡)
)ðâ²ð(ð¡). (24)
A first-order necessary condition of optimality to the vari-ational problem is given by the Euler-Lagrange differentialequations [17]:
âð (p)
âððâ d
dð¡
âð (p)
âðâ²ð
!= 0, ð = 1, 2. (25)
Using (24), one can easily check, that the differential equationsin (25) reduce to one algebraic equation,
âðð,2(p)
âð1â âðð,1(p)
âð2
!= 0. (26)
The solutions of (26), if they exist, are the candidates satisfy-ing the optimality requirement. However, a direct computationof (26) is not possible, since ðð,1 and ðð,2 are not given inclosed form. Moreover, the candidate curves of (26) are sta-tionary paths, which generally do not satisfy the monotonicityand boundary conditions of (19). Thus, the extremal cannotbe obtained directly from (26).
An approximate solution to problem (22) may be derivedwhen each EQ EXIT function ðð,ð is approximated by a 2Dquadratic form4
ðð,ð(Ið) â ðŒð,0 + ðŒð,1ðŒð,1 + ðŒð,2ðŒð,2 + ðŒð,3ðŒð,1ðŒð,2
+ ðŒð,4ðŒ2ð,1 + ðŒð,5ðŒ
2ð,2 (27)
with the coefficients ðŒð,ð obtained from a standard regressionmethod [18]. Under this assumption, the Euler-Lagrange equa-tions in (25) reduce to one linear algebraic equation,
ð (Ið) â¡ âðð,2(Ið)
âðŒð,1â âðð,1(Ið)
âðŒð,2
!= 0, (28)
which leads to three possible outcomes: (i) no solution, (ii)infinite number of solutions, or (iii) a unique solution.
(i) If (28) has no solution, the partial derivativesâðð,2(Ið)/âðŒð,1 and âðð,1(Ið)/âðŒð,2 are non-equal constantsin â. In Appendix A it is shown that in such a case either theboundary curve â(ð0, ð3, ð2) or â(ð0, ð1, ð2) solves problem(22). Thus, the sum rate is maximized at the rate region cornerpoint of either ð1 or ð2.
(ii) If (28) has an infinite number of solutions, i.e., curl fð =0 for all Ið â ðŒ
2, the functional in (23) is path-independent[18]. Thus, ð [p] is identical for all p â ð« and the two cornerpoints ð1 and ð2 are connected by a straight line segmenthaving a decay of â1.
(iii) If (28) has a unique solution, the solution curveâ° â¡ {Ið : ð (Ið) = 0
}can be parameterized by a continuous
vector-function g : ðŒ â â° , where the first derivative of gsatisfies ðâ²ð(ð¡) ⥠0 or ðâ²ð(ð¡) †0 for all ð¡ â ðŒ. This propertyallows us to formulate the following theorem.
Theorem 1: Let â° â ð° be parameterized by a vector-function g : ðŒ â â° . Let the following two conditions besatisfied:
ðâ²ð(ð¡) ⥠0, ð = 1, 2 for all ð¡ â ðŒ, (29a)
ð (Ið)
{⥠0 for Ið â ð¢1,†0 for Ið â ð¢2,
(29b)
where the regions ð¢ð, ð = 1, 2 are defined as ð¢1 â¡{Ið :
ðŒð,1 †ð1(ð¡), ðŒð,2 ⥠ð2(ð¡), âð¡ â ðŒ}
and ð¢2 â¡{Ið : ðŒð,1 â¥
ð1(ð¡), ðŒð,2 †ð2(ð¡), âð¡ â ðŒ}
. Then, the convergence curve
ð® = Ⱐ⪠â(ð0,g(0)) ⪠â(g(1), ð2) (30)
4Note that extensive simulations show that using a 2D quadratic form givesa close approximation to ðð,ð . Note also that a polynomial approximation onthe EXIT functions has been used in [8] to design LDPC codes for iterativeMIMO systems.
1038 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
Id,1
Id,2pÌ : S
M1
M2
p(t2) = pÌ(t2)
p(t1) = pÌ(t1)
p(t3) = pÌ(t3)
G1
p
G2
g : E
g(0)
g(1)
Fig. 4. Curve g : â° , corresponding convergence curve pÌ : ð® , and partitionof pÌ for an arbitrary parametric curve p â ð« in the plane region ð° .
comprised of the curve â° and the two straight line segments(all shown in Fig. 4) is optimal with respect to (22).
The following proof of Theorem 1 is based on Greenâstheorem [18] and is similar to that presented in [19].
Proof: For any admissible parametric curve p â ð« , wecan find a parameterization pÌ â ð« of ð® and a partitioningâªð[ð¡ð, ð¡ð+1] of the interval ðŒ such that for each interval
[ð¡ð, ð¡ð+1], one has p(ð¡ð) = pÌ(ð¡ð), p(ð¡ð+1) = pÌ(ð¡ð+1), andp(ð¡) â ð¢1 or p(ð¡) â ð¢2, for all ð¡ â [ð¡ð, ð¡ð+1]. An exampleof such a partitioning is shown in Fig. 4. Based on the parti-tioning, we can express the sum rate difference ð [pÌ] â ð [p]between pÌ and p as
ð [pÌ]âð [p] =âð
(Îð ð â¡
â« ð¡ð+1
ð¡ð
ð (pÌ)dð¡ââ« ð¡ð+1
ð¡ð
ð (p)dð¡).
(31)Each increment Îð ð can be written as line integral along theclosed curve ðð â¡
{p(ð¡), pÌ(ð¡) : ð¡ â [ð¡ð, ð¡ð+1]
}yielding
Îð ð = ðŸð
â®ðð
ðð,1(Ið)dðŒð,1 + ðŸð
â®ðð
ðð,2(Ið)dðŒð,2, (32)
where ðŸð = 1 when p(ð¡) â ð¢1, for all ð¡ â [ð¡ð, ð¡ð+1], and ðŸð =â1 otherwise. Let each simple closed curve ðð be orientedin the positive direction with the bounded regionâ³ð â ð° onthe left. Then, applying Greenâs theorem to (32) allows us torewrite Îð ð as
Îð ð = ðŸð
â«â«â³ð
ð (Ið)dðŒð,1dðŒð,2. (33)
With the conditions in (29), we can state that Îð ð ⥠0 sinceeither â³ð â ð¢1 or â³ð â ð¢2, as shown in Fig. 4. Thus, weconclude that ð [pÌ] ⥠ð [p] for any p â ð« , which yields theoptimality of ð® in (30).
The optimality of ð® in (30) is given only when bothconditions (29) are satisfied. In the cases when at least one ofthese conditions is violated, we can use Greenâs theorem asin (31) and (32) to show that candidate solutions to problem(22) are the convergence curves â0(ð¢) â¡ â(ð0, ð0,1, ð0,2, ð2)and â1(ð¢) â¡ â(ð0, ð1,1, ð1,2, ð2), which are comprised of theunion of three straight-line segments connecting the pointsð0,1 = (ð¢, 0), ð0,2 = (ð¢, 1) and ð1,1 = (0, ð¢), ð1,2 = (1, ð¢)
with ð¢ â ðŒ. Using the candidate solutions, we show inAppendix B that problem (22) can be rewritten as a quadraticextremal problem, which may be solved with standard meth-ods of classical calculus [18].
The three possible outcomes of (28) discussed above, cannow be used to calculate the convergence curve that solvesproblem (22), which is summarized in Algorithm 1.
Algorithm 1 Algorithm for computing the convergence curvethat maximizes the sum rate
1: Calculate the coefficients in (27) for each EQ EXITfunction ðð,ð using a standard regression method [18].
2: if (45) is satisfied then3: Output the boundary curve that solves (46).4: else if (45a) is satisfied and (45b) is violated then5: Output an arbitrary convergence curve p â ð« .6: else if (29) is satisfied then7: Output the convergence curve in (30).8: else9: Output the convergence curve âð(ð¢), ð = 1, 2, ð¢ â ðŒ
with (ð, ð¢) being the solution to (51).10: end if
C. Rate Allocation using EXIT Functions
Let pÌ â ð« be the convergence curve that solves problem(22). An upper bound of the rate of user ð with respect to pÌ
is then given by the area under the EXIT function ð(pÌ)ð , ð ð =â« 1
0ð(pÌ)ð (ðŒð,ð)dðŒð,ð, as illustrated in Fig. 2. To closely approach
the rate ð ð, the code of user ð should have its DEC EXITfunction ðð,ð as close to ð
(pÌ)ð as possible, while satisfying the
constraints in (10). In practice, however, optimizing ðð,ð byadjusting the available code parameters such that optimalityin a strict sense is guaranteed may not be possible, since thecode parameters are presumably limited. Thus, for practicalreasons, we restrict each transmitter to have only a finite setof codes ð = {ð¶1, ..., ð¶ð}, with ð being the number of thecodes in the set, with fixed rates. Then, a simple approach forrate allocation is to select, for each ð, the code ð¶ð â ð withthe highest possible rate at which convergence is achieved,while its DEC EXIT function ð
(ð¶ð)ð,ð best fits to ð
(pÌ)ð . Based
on this concept, we now propose a rate allocation algorithm,which is summarized as follows.
1) For each realization of the FD channel matrices (Î1,Î2), calculate the EQ EXIT functions ðð,1 and ðð,2 using(8).
2) Calculate the convergence curve pÌ â ð« that solves (22)using Algorithm 1.
3) Calculate ð (pÌ)ð by projecting the space curve wð(pÌ) onto
the ðŒð,ð-ðŒð,ð-plane.4) To obtain high information rate, select the channel code
for each user that satisfies:
ðð,ð = maxð¶ðâð
{ð(ð¶ð) : ð
(pÌ)ð (ðŒð,ð) ⥠ð
(ð¶ð)ð,ð (ðŒð,ð)
+ ð§(ðŒð,ð), âðŒð,ð â [0, 1)
}, (34)
GROSSMANN et al.: RATE ALLOCATION FOR 2-USER MAC WITH MMSE TURBO EQUALIZATION 1039
where ð(ð¶ð) denotes the rate of code ð¶ð, and ð§(ðŒð,ð)is a function to control the speed of convergence of thedecoding trajectory to the maximum point Iâð = ð2.
5) If ðð,ð is NULL, select the code with the lowest possiblerate in ð,
ðð,ð = minð¶ðâð
ð(ð¶ð). (35)
6) Output the selected coding rates ðð,1 and ðð,2.
D. Outage-based Rate Allocation
In this section, we extend the above code selection algo-rithm to an outage-based rate allocation approach for non-ergodic fading channels. In the context of turbo equalization,the coding rate ðð,ð of the ð-th user should be no greater thanðâð,ð(pÌ) â¡ maxð¶ðâð
{ð(ð¶ð) : ð
(pÌ)ð (ðŒð,ð) ⥠ð
(ð¶ð)ð,ð (ðŒð,ð) +
ð§(ðŒð,ð), âðŒð,ð â [0, 1)}
with pÌ â ð« being the solution to (22),such that the constraints in (10) are satisfied. Assume that foreach code ð¶ð â ð we have ð
(ð¶ð)ð (ðŒð) ⥠ð
(ð¶ð)ð (ðŒð), for all
ðŒð â [0, 1) if ð(ð¶ð) ⥠ð(ð¶ð) with ð¶ð â ð. Then, the setof code rate pairs, at which convergence of turbo equalizationis achieved for the specific FD channel matrices (Î1, Î2), isgiven by
ð¯ (pÌ) â¡ {(ðð,1, ðð,2) : ðð,1 = ð(ð¶ð) †ðâð,1(pÌ),
ðð,2 = ð(ð¶ð) †ðâð,2(pÌ), ð¶ð, ð¶ð â ð}. (36)
For an outage-based rate allocation, we view the rate pair(ðâð,1(pÌ), ð
âð,2(pÌ)
)as random variables since pÌ depends on
(Î1, Î2) which are random matrices whose particular re-alizations change independently, frame-by-frame. If the twousers are transmitting at rates (ðð,1, ðð,2), an outage eventoccurs if either ðð,1 > ðâð,1(pÌ) or ðð,2 > ðâð,2(pÌ). Thus, wedefine the outage probability for the code rate pair (ðð,1, ðð,2)as ððð¢ð¡(ðð,1, ðð,2) â¡ Prob
((ðð,1, ðð,2) /â ð¯ (pÌ)). The 2-user
outage rate region is then given by
âðð¢ð¡ð â¡{(ðð,1, ðð,2) : ððð¢ð¡(ðð,1, ðð,2) †ð
}, (37)
where ð denotes the outage probability. The maximum sumrate at which reliable transmission for the two users is possiblefor (1 â ð) â 100% of the channel realizations, can then beexpressed as solution to the problem:
ð ðð¢ð¡ð = max(ðð,1,ðð,2)ââðð¢ð¡
ð
ðð,1 + ðð,2. (38)
For the prescribed constraint ððð¢ð¡(ðð,1, ðð,2) †ð, the codingrates (ðð,1, ðð,2) satisfying (38) are the largest rates over the2-user MAC for the given codes in ð. In this paper, we obtainthe region in (37) by a Monte Carlo method, as described inSection V-C.
V. NUMERICAL RESULTS
In this section, results of capacity calculations and sim-ulations conducted to evaluate the throughput enhancementachieved by the proposed two rate allocation approaches arepresented. We consider a single-carrier block-cyclic transmis-sion over frequency-selective Rayleigh block-fading channels,
â20 â16 â12 â8 â4 0 4 8 120.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Es/N0 [dB]
Spec
tral
eff
icie
ncy
[bpc
]
GaussianBPSK capacityBPSK turbo SC FD-MMSE
Fig. 5. Spectral efficiency of a 2-user turbo system with SC FD-MMSEequalization when optimally designed user codes are assumed for transmis-sion, and ergodic sum capacity with Gaussian and with BPSK inputs. ð = 32,ð¿ = 3, ð = 2.
where typical values of ð¿ (ð¿ = 3, ð¿ = 5 and ð¿ = 32) areassumed. We define the average SNR at the receiver as
ðžð ð0â¡âðð=1
âð¿â1ð=0 E
[â£âð,ð(ð)â£2]ðž0
ðð2, (39)
where E[â ] denotes expectation, and ðž0 is the total energy persymbol at the transmitters.
A. Maximum Spectral Efficiency
Fig. 5 shows the maximum achievable spectral efficiencyð â¡ ð
ð+ð ï¿œÌï¿œ of the 2-user turbo system, when assuming(optimally designed) user codes whose DEC EXIT functionssatisfy:
ðð,ð = ð(pÌ)ð , for ð = 1, 2. (40)
In (40), pÌ is the solution to problem (22) for a specific channelrealization given. The rate ï¿œÌï¿œ has been calculated by averagingð =
â2ð=1
â« 1
0ð(pÌ)ð (ðŒð,ð)dðŒð,ð over a large number of channel
realizations. For comparison, the ergodic i.i.d. Gaussian-inputsum capacity [20] and the ergodic BPSK-input sum capacity ofthe frequency-selective Rayleigh fading channel (ð¿ = 3) withevenly allocated transmit power are also shown. The BPSK-input capacity result has been obtained by a Monte Carlomethod, as described in [21]. We assume that the channelstate information is known only at the receiver. Comparingthe numerical results in Fig. 5, we find that the loss incurredby the use of SC FD-MMSE equalization with respect tothe BPSK-input capacity slightly increases for large valuesof ðžð /ð0. We also observe that there is almost no differencebetween the three curves up to ðžð /ð0 = â8 dB, indicatingthat the 2-user turbo system with BPSK-inputs, using SCFD-MMSE equalization with user codes whose DEC EXITfunctions satisfy (40), is nearly optimal in the low ðžð /ð0
region.
B. Rate Allocation with Practical Codes
The rate-compatible punctured SCCCs proposed in [22],consisting of a rate-ðð outer encoder and a recursive rate-
1040 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
TABLE ISELECTED CODE RATE PAIRS AND INFORMATION RATE PAIRS OF BOTH
USERS FOR TWO DIFFERENT CONVERGENCE CURVES.
ðð,1 ðð,2 ðð ð 1 ð 2 ð Max-Sum-Rate 0.75 0.85 1.60 0.84 0.90 1.74Min-Sum-Rate 0.90 0.45 1.35 0.98 0.58 1.56
1 inner encoder with polynomials (ðð, ð0) = (3, 2) (ðð de-notes the feedback polynomial) in octal notation, are assumedfor rate allocation at both users. The outer encoder of theSCCC is selected from a set of 17 subcodes with ratesðð = 0.05 â (1 + ð), ð = 1, .., 17. The length of a frameis fixed to ðð = 16384 BPSK symbols, and ð§(ð¥) in (34) isdefined as ð§(ð¥) = 0.025 â 0.025ð¥. The turbo EQ performs15 iterations between the EQ and both SCCC DECs, and 20iterations between the inner and outer DEC. Note that ð§(ð¥)is preliminarily optimized by computer simulations to find anacceptable trade-off between performance of the turbo EQ andnumber of iterations needed to ensure convergence.
Table I shows the selected code rate pairs at ðžð /ð0 =4.75 dB with respect to (34) and the information ratepairs satisfying (22) for a single random channel realization.The channel coefficients are given by â1,1 = [0.045 +0.027ð,â0.054+0.045ð,â0.134â0.022ð]ð , â1,2 = [â0.124+0.125ð,â0.038â0.010ð,â0.193+0.590ð]ð , â2,1 = [â0.042+0.020ð,â0.140 â 0.036ð,â0.145 â 0.121ð]ð , and â2,2 =[â0.157 + 0.186ð,â0.153 + 0.078ð, 0.198 + 0.312ð]ð withð =
ââ1. For comparison, the selected code rate pairsðð,ð and information rate pairs ð ð =
â«pÌðð,ð(Ið)dðŒð,ð,
computed for the convergence curve satisfying pÌ â¡argminpâð«
â2ð=1
â«p ðð,ð(Ið)dðŒð,ð , are also shown and re-
ferred to as âMin-Rate-Sumâ. The results in Table I indicatethat the selected code rate pairs of both users, and thus thetotal achievable rate of the turbo system, strongly dependon the convergence curve chosen for the particular channelrealization.
Fig. 6 shows the BER performance of the turbo EQ fora single channel realization having the same coefficientsas given above. The usersâ codes are selected with respectto (34) at SNR ðžð /ð0 = 4.75 dB. Also shown is theSNR ðžð /ð0â£min at which the instantaneous BPSK-input sumcapacity (for the given channel realization) is equal to thetotal rate
â2ð=1 ðð,ð = 1.60 bit per channel use (bpc) of the
turbo system. It is observed that the selected codes showconvergence starting at 3.6 dB and 4.1 dB ðžð /ð0 for user1 and user 2, respectively. Thus, the SC FD-MMSE turboEQ indeed satisfies the convergence constraints (10) for thedesired SNR ðžð /ð0â£des. Further, we observe that the turbo EQoperates within about 2 dB ðžð /ð0 to its respective capacitylimit at a BER of 10â6.
For evaluating the throughput efficiency, a selective-repeatautomatic repeat-request (ARQ) system with infinite buffering[23] is assumed. Fig. 7 shows the average total throughput ofthe turbo system versus ðžð /ð0 for ð¿ = 32 channels. Thespectral efficiency ð (see Section V-A) of the turbo systemwith user codes whose DEC EXIT functions satisfy (40),is shown as a reference. Also shown is the average totalthroughput performance for an ARQ scheme with fixed codingrates ðð,1/2 = 0.1 â ð, ð = 1, .., 9 of both users. As observed
0 1 2 3 4 5 610
â6
10â5
10â4
10â3
10â2
10â1
100
Es/N0[dB]
BE
R
Es/N0|min Es/N0|des
user 1user 2average
Fig. 6. BER performance of the turbo EQ for a fixed ð¿ = 3 channelrealization with user codes having the coding rates ðð,1 = 0.75 and ðð,2 =0.85. ð = 32, ð¿ = 3.
â6 â4 â2 0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Es/N0 [dB]
Ave
rage
thro
ughp
ut [
bpc]
proposed methodBPSK turbo SC FD-MMSE
Fig. 7. Average total throughput of both users versus ðžð /ð0 for the proposedrate allocation scheme and for ARQ with fixed coding rates ðð,1/2 = 0.1 â ðwith ð = 1, .., 9 (dashed curves, from bottom to top). ð = 128, ð¿ = 32.
in Fig. 7, substantial throughput gain is obtained with therate allocation approach proposed in Section IV-C over fixedrate ARQ. Further, we find that the throughput performanceis only 1.5 dB away (in the high ðžð /ð0 region) from thethroughput achieved when optimally designed user codes areassumed for transmission. Notably, however, there is still agap of almost 2.5 dB between both curves in the low ðžð /ð0
region. It should be noted though, that by using more flexiblecoding techniques, such as irregular SCCCs (e.g., see [22])or irregular LDPC codes (e.g., see [8]) which allow a codedesign, given the channel realizations, the performance losscan be reduced.
C. Outage-based Rate Allocation and Outage Capacity Re-gion
In this section, the outage-based rate allocation techniqueproposed in Section IV-D is applied to ð¿ = 5 channels. Thesame code set as in Section V-B is assumed for rate allocation
GROSSMANN et al.: RATE ALLOCATION FOR 2-USER MAC WITH MMSE TURBO EQUALIZATION 1041
TABLE IICODE RATE PAIRS AND ACHIEVABLE INFORMATION RATES OF A 2-USER
SYSTEM WITH BPSK INPUTS FOR OUTAGE PROBABILITY OF ð = 0.1.
ðžð /ð0 [dB] ðð,1 ðð,2 ðð ð1 ð2 ð0 0.35 0.35 0.70 0.47 0.53 1.001 0.40 0.40 0.80 0.55 0.61 1.162 0.50 0.50 1.00 0.64 0.67 1.313 0.55 0.60 1.15 0.70 0.76 1.46
0.60 0.554 0.65 0.65 1.30 0.79 0.82 1.61
at both users. In addition, we include the numerical results forthe boundary calculation for the outage binary-input capacityregion [24], [25] with various values of outage probabilitiesas references. The outage binary-input capacity region of the2-user MAC is defined as the set of all rate pairs (ð1, ð2)satisfying [24]
ððð¢ð¡ð â¡{(ð1, ð2) : ðð(ð1, ð2) †ð
}, (41)
where ðð(ð1, ð2) â¡ Prob((ð1, ð2) /â ðð(Î1,Î2)
)is the outage
probability constraint for the rate pair (ð1, ð2) with ðð(Î1,Î2)being the binary-input capacity region conditioned on thespecific FD channel realization (Î1,Î2). The binary-inputcapacity region ðð(Î1,Î2) can be expressed as follows [12]:
ðð(Î1,Î2) â¡{(ð1, ð2) :
â
ðâðµðð †I(r; ðð, ð â ðµ â£â£ðð, ð â ðµ )
},
(42)where ðµ denotes any subset of {1, 2}, and ðµ its complement5.Fig. 8 shows the code rate pairs (ðð,1, ðð,2) that maximize sumrate among those combinations of the codes in the code setð for outage probabilities ð = 0.01, 0.1, and 0.5. In all thecases, the ðžð /ð0 is set at 4 dB. To obtain the code rate pairsfor the outage constraint ððð¢ð¡(ðð,1, ðð,2) †ð, we computedthe 2-user outage rate region âðð¢ð¡ð by transmitting over ðindependent channel realizations. Specifically, we determinedthe set of code rate pairs ð¯ (pÌ) for each of the ð channelrealizations according to (36), and recorded the total numberð of outage events ((ðð,1, ðð,2) /â ð¯ (pÌ)) for a given code ratepair (ðð,1, ðð,2). According to the definition of âðð¢ð¡ð , all thecode rate pairs (ðð,1, ðð,2) â âðð¢ð¡ð satisfy the condition ð
ð †ð,and the code rate pairs that maximize sum rate can easily becomputed with (38). Also shown in Fig. 8 are the outagebinary-input capacity regions ððð¢ð¡ð for the three differentvalues of ð, obtained by a Monte Carlo method [21]. Theachievable rate pairs on the region boundaries that maximizeoutage capacity, i.e., (ð1, ð2) = max(ð1,ð2)âððð¢ð¡
ðð1 + ð2 are
marked as well. As it is evident from this figure, the largerthe desired outage probability, the larger is the resultingoutage binary-input capacity region. Furthermore, as expected,the code rate pairs that satisfy (38) are contained by theirrespective outage binary-input capacity region. We observethat these rate pairs are almost symmetric with respect tothe line ð1 = ð2 = ð ðð¢ð¡ð /2. This is apparently obvioussince we expect that the two usersâ rates obtained by (22)are equal in the sense of average. Note that, due to a finitenumber of Monte Carlo trials, the rate pairs are not exactlysymmetric. The code rate pairs for a given outage probabilityof ð = 0.1 at different ðžð /ð0-values are listed in Table II.
5Note that the indices ð and ð in the variable ðð,ð(ð) have been omittedin (42) for notational simplicity.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r1 [bits/use]
r 2[b
its/
use]
ε = 0.01ε = 0.1ε = 0.5
ε = 0.01
ε = 0.5
ε = 0.1
Fig. 8. Achievable code rate pairs (markers: â+â, âââ, and âââ) for the twousers that maximize sum rate for the given code set ð and respective 2-user outage binary-input capacity regions when ð = 0.01, 0.1, 0.5, andðžð /ð0 = 4 dB. The âoâ markers indicate the achievable rates of the twousers that maximize outage capacity with the desired outage probability.
For comparison, the maximal achievable rates (ð1, ð2) are alsolisted. As observed in Table II, there is still a performanceloss of about 0.3 bpc in sum rate over the considered ðžð /ð0
range of the proposed outage-based rate allocation approach,compared to the outage capacity results. Reducing this lossby using appropriate but still practical coding techniques isan interesting further research topic that will be considered inour future work.
VI. CONCLUSION
In this paper, we consider the problem of rate allocationin frequency-selective 2-user Gaussian multiple access fadingchannels employing SC FD-MMSE turbo equalization. Thearea property theorem for EXIT charts is used to derivean upper bound on the rate region of the turbo system.Using this bound, we study the problem of maximizing thesum rate of both users, given the EXIT function of theEQ, and present an approximate solution to this optimizationproblem. In addition, a simple code selection algorithm forrate allocation, using the EXIT curves obtained as a resultof the sum rate optimization, is proposed. Numerical resultsshow that the achievable throughput of the proposed algorithmusing rate-compatible punctured SCCCs for each user, is only1.5 dB away for high ðžð /ð0 from the maximal supportablethroughput achieved with optimally designed user codes foreach channel realization. Furthermore, the extension of thisalgorithm to an outage-based rate allocation approach isdiscussed. It should be emphasized here that the proposed twoalgorithms are only applicable for the 2-user case. In [26],the authors discuss a heuristic approach based on a dynamicprogramming principle that extends the proposed algorithmsto more generic cases having more than two users.
APPENDIX A
Using (27), the functional in (23) can be written asâ« 1
0
ð (p)dð¡ â ð+
â« 1
0
â1(ð1)ðâ²2dð¡+
â« 1
0
â2(ð2)ðâ²1dð¡, (43)
1042 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
where â1(ð1) â¡ ðŒ2,1ð1 + (ðŒ2,4 â ðŒ1,3/2)ð21 and â2(ð2) â¡
ðŒ1,2ð2 + (ðŒ1,5 â ðŒ2,3/2)ð22 are quadratic polynomials in ð1
and ð2, respectively, and ð is a constant. The Euler-Lagrangeequation in (28) can be written with (27) as
ðŒ2,1 â ðŒ1,2 + (2ðŒ2,4 â ðŒ1,3)ðŒð,1 â (2ðŒ1,5 â ðŒ2,3)ðŒð,2!= 0.
(44)
In the case when (28) has no solution, the coefficients of thepolynomials â1(ð1) and â2(ð2) satisfy:
ðŒ1,3 = 2ðŒ2,4, ðŒ2,3 = 2ðŒ1,5, (45a)
ðŒ2,1 â ðŒ1,2 â= 0. (45b)
By (43) and (45), the variational problem in (22) becomes
ð ððð¥ â ð+maxpâð«
{(ðŒ1,2 â ðŒ2,1)
â«p
ðŒð,2dðŒð,1
}. (46)
Now, it is easy to confirm that the sum rate expression givenby (46) is maximized, either for p(ð¡) â â(ð0, ð1, ð2), for allð¡ â ðŒ if ðŒ1,2 > ðŒ2,1 or p(ð¡) â â(ð0, ð3, ð2), for all ð¡ â ðŒ ifðŒ1,2 < ðŒ2,1.
APPENDIX B
Using the integral expression in (43), we can express thesum rates ð 0(ð¢) and ð 1(ð¢) for â0(ð¢) and â1(ð¢), respec-tively, as functions of ð¢ â ðŒ, as
ð 0(ð¢) â¡ ð+
â« ð¢
0
â2(0)dðŒð,1 +
â« 1
0
â1(ð¢)dðŒð,2
+
â« 1
ð¢
â2(1)dðŒð,1, (47)
ð 1(ð¢) â¡ ð+
â« ð¢
0
â1(0)dðŒð,2 +
â« 1
0
â2(ð¢)dðŒð,1
+
â« 1
ð¢
â1(1)dðŒð,2, (48)
Integrating each term in (47) and (48) allows us to rewriteð 0(ð¢) and ð 1(ð¢) as
ð 0(ð¢) = ð+ â1(ð¢) + (1â ð¢)â2(1), (49)
ð 1(ð¢) = ð+ â2(ð¢) + (1â ð¢)â1(1). (50)
Thus, the variational problem in (22) reduces to:
ð ððð¥ â maxðâ{0,1}
maxð¢â[0,1]
{ð ð(ð¢)
}, (51)
of which solution can easily be obtained with the standardframework of classical calculus, since the objective functionis quadratic in ð¢ [18].
REFERENCES
[1] C. Douillard, et al., âIterative correction of intersymbol interference: turboequalization," European Trans. Telecommun., vol. 6, no. 5, pp. 507-511,Sept. 1995.
[2] X. Wang and H. V. Poor, âIterative (turbo) soft interference cancellationand decoding for coded CDMA," IEEE Trans. Commun., vol. 47, no. 7,pp. 1046-1061, July 1999.
[3] M. TÃŒchler and J. Hagenauer, âTurbo equalization: principles and newresults," IEEE Trans. Commun., vol. 50, no. 5, pp. 754-767, May 2002.
[4] T. Abe and T. Matsumoto, âSpace-time turbo equalization in frequency-selective MIMO channels," IEEE Trans. Veh. Technol., vol. 52, no. 3, pp.469-475, May 2003.
[5] K. Kansanen and T. Matsumoto, âAn analytical method for MMSEMIMO turbo equalizer EXIT chart computation," IEEE Trans. WirelessCommun., vol. 6, no. 1, pp. 59-63, Jan. 2007.
[6] S. ten Brink, âConvergence behavior of iteratively decoded parallelconcatenated codes," IEEE Trans. Commun., vol. 49, no. 10, pp. 1727-1737, Oct. 2001.
[7] A. Ashikhmin, G. Kramer, and S. ten Brink, âExtrinsic informationtransfer functions: model and erasure channel properties," IEEE Trans.Inf. Theory, vol. 50, no. 11, pp. 2657-2673, Nov. 2004.
[8] S. ten Brink, G. Kramer, and A. Ashikhmin, âDesign of low-densityparity-check codes for modulation and detection," IEEE Trans. Commun.,vol. 52, no. 4, pp. 670-678, Apr. 2004.
[9] S. ten Brink and G. Kramer, âDesign of repeat-accumulate codes foriterative detection and decoding," IEEE Trans. Signal Process., vol. 51,no. 11, pp. 2764-2772, Nov. 2003.
[10] A. Roumy and D. Declercq, âCharacterization and optimization ofLDPC codes for the 2-user Gaussian multiple access channel," EURASIPJ. Wireless Commun. and Netw., vol. 2007, Article ID 74890, 10 pages,2007.
[11] A. Amraoui, S. Dusad, and R. Urbanke, âAchieving general points inthe 2-user Gaussian MAC without time-sharing or rate-splitting by meansof iterative coding," in Proc. IEEE Int. Symp. Inf. Theory, Lausanne,Switzerland, July 2002.
[12] T. M. Cover and J. A. Thomas, Elements of Information Theory. NewYork: John Wiley &. Sons, Inc., 1991.
[13] B. Rimoldi and R. Urbanke, âA rate-splitting approach to the Gaussianmultiple-access channel," IEEE Trans. Inf. Theory, vol. 42, no. 2, pp.364-375, Mar. 1996.
[14] A. Sanderovich, M. Peleg, and S. Shamai (Shitz), âLDPC coded MIMOmultiple access with iterative joint decoding," IEEE Trans. Inf. Theory,vol. 51, no. 4, pp. 1437-1450, Apr. 2005.
[15] C. Schlegel, Z. Shi, and M. Burnashev, âOptimal power/rate allocationand code selection for iterative joint detection of coded random CDMA,"IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4286-4294, Sept. 2006.
[16] F. BrÀnnström, L. K. Rasmussen, and A. J. Grant, âConvergence analysisand optimal scheduling for multiple concatenated codes," IEEE Trans. Inf.Theory, vol. 51, no. 9, pp. 3354-3364, Sept. 2005.
[17] J. Jost and X. Li-Jost, âCalculus of variations," Cambridge Studies inAdvanced Mathematics, vol. 64. Cambridge University Press, 1998.
[18] E. Kreyszig, Advanced Engineering Mathematics, 9th ed. New York:John Wiley & Sons, 2005.
[19] A. Miele, âExtremization of linear integrals by Greenâs theorem,"Optimization Technics, G. Leitmann Ed. New York: Academic Press, pp.69-98, 1962.
[20] W. Rhee and J. M. Cioffi, âErgodic capacity of multiantenna Gaussianmultiple-access channels," in Proc. 35th Asilomar Conf. Signals, Systems,Computers, vol. 1, pp. 507-512, Nov. 2001.
[21] Z. Zhang et al., âAchievable information rates and coding for MIMOsystems over ISI channels and frequency-selective fading channels," IEEETrans. Commun., vol. 52, no. 10, pp. 1698-1710, Oct. 2004.
[22] M. TÃŒchler, âDesign of serially concatenated systems depending on theblock length," IEEE Trans. Commun., vol. 52, no. 2, pp. 209-218, Feb.2004.
[23] K. Kansanen et al., âMultilevel-coded QAM with MIMO turbo-equalization in broadband single-carrier signaling," IEEE Trans. Veh.Technol., vol. 54, no. 3, pp. 954-966, May 2005.
[24] J. Haipeng and A. Acampora, âBounds on the outage-constrained ca-pacity region of space-division multiple-access radio systems," EURASIPJ. Appl. Signal Proc., vol. 2004, no. 9, pp. 1288-1298, 2004.
[25] T. Guess, H. Zhang, and T.V. Kotchiev, âThe outage capacity of BLASTfor MIMO channels," in Proc. IEEE Conf. Commun., vol. 4, pp. 2628-2632, May 2003.
[26] M. Grossmann and T. Matsumoto, âRate allocation for ðŸ-user MACwith turbo equalization," in Proc. International ITG Workshop SmartAntennas 2009, vol. 1, pp. 281-288, Feb. 2009.
Marcus Grossmann received the Diplom-Ingenieur(M.S.) degree in electrical engineering and computerscience, from Ilmenau University of Technology,Germany, in 2004. He is currently working towardsthe Ph.D. degree at the same university. His researchinterests are in the area of communication systems,and signal processing for wireless communications.
GROSSMANN et al.: RATE ALLOCATION FOR 2-USER MAC WITH MMSE TURBO EQUALIZATION 1043
Thomas Ortlepp received the Dipl.-Math. degree inmathematics from Ilmenau University of Technol-ogy, Germany, in 1999. He moved to the Instituteof Information Technology and received the Dr.-Ing.degree in electrical engineering in 2004. T. Ortleppworks as a lecturer for electromagnetics, relativis-tic electrodynamics and superconductive electronics.His main research interest is mixed signal circuitanalysis with main focus on circuit simulation, de-sign, analysis of noise induced errors, and jitter aswell as new circuit topologies.
Tad Matsumoto (Sâ84, SMâ95) received his B.S.,M.S., and Ph.D. degrees from Keio University,Yokohama, Japan, in 1978, 1980, and 1991, respec-tively, all in electrical engineering. He joined Nip-pon Telegraph and Telephone Corporation (NTT) inApril 1980. Since he engaged in NTT, he was in-volved in a lot of research and development projects,all for mobile wireless communications systems. InJuly 1992, he transferred to NTT DoCoMo, wherehe researched Code-Division Multiple-Access tech-niques for Mobile Communication Systems. In April
1994, he transferred to NTT America, where he served as a Senior TechnicalAdvisor of a joint project between NTT and NEXTEL Communications. InMarch 1996, he returned to NTT DoCoMo, where he served as a Head ofthe Radio Signal Processing Laboratory until August of 2001; He workedon adaptive signal processing, multiple-input multiple-output turbo signaldetection, interference cancellation, and space-time coding techniques forbroadband mobile communications.
In March 2002, he moved to University of Oulu, Finland, where he servedas a Professor at Centre for Wireless Communications. In 2006, he served asa Visiting Professor at Ilmenau University of Technology, Ilmenau, Germany,funded by the German MERCATOR Visiting Professorship Program. SinceApril 2007, he has been serving as a Professor at Japan Advanced Instituteof Science and Technology (JAIST), Japan, while also keeping the positionat University of Oulu.
T. Matsumoto has been appointed as a Finland Distinguished Professorfor a period from January 2008 to December 2012, funded by the FinnishNational Technology Agency (Tekes) and Finnish Academy, under whichhe preserves the rights to participate in and apply to European and Finnishnational projects. He is a recipient of IEEE VTS Outstanding Service Award(2001), Nokia Foundation Visiting Fellow Scholarship Award (2002), IEEEJapan Council Award for Distinguished Service to the Society (2006), IEEEVehicular Technology Society James R. Evans Avant Garde Award (2006), andThuringia State Research Award for Advanced Applied Science (2006), 2007Best Paper Award of Institute of Electrical, Communication, and InformationEngineers of Japan (2008), and Telecom System Technology Award by theTelecommunications Advancement Foundation (2009).