3332 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

Irregular Convolutional Codes in MultiantennaBit-Interleaved Coded Modulation Under

Iterative Detection and DecodingAkrum Elkhazin, Konstantinos Plataniotis, Senior Member, IEEE, and Subbarayan Pasupathy, Life Fellow, IEEE

Abstract—This paper develops a family of irregular convolu-tional codes for bit-interleaved coded modulation (BICM) systemsunder iterative detection and decoding. Irregular convolutionalcodes are constructed through irregular puncturing over multiplemother codes of different memory. Strategies based on fixed- andvariable-size trellises are proposed to connect different memorymother codes. The use of irregular puncturing and code mem-ory yields improved coding efficiency with the aid of extrinsicinformation transfer charts. Under ergodic and quasi-static fad-ing conditions, multiantenna BICM systems using the proposedcodes outperform comparable turbo-coded systems and multilevelcoding strategies.

Index Terms—Bit-interleaved coded modulation (BICM), ex-trinsic information transfer (EXIT), irregular convolution codes,iterative detection and decoding (IDD), multiple-antenna systems,tail biting.

I. INTRODUCTION

B IT-INTERLEAVED coded modulation (BICM) is abandwidth-efficient coded modulation scheme that can

outperform symbol-interleaved trellis-coded modulation overfading channels [1], [2]. At the receiver, improved perfor-mance is possible through an iterative detection and decoding(IDD) strategy [3], which iteratively exchanges soft-coded bitinformation between the detector and decoder. The decodingconvergence of BICM-IDD systems can be analyzed usingEXtrinsic Information Transfer (EXIT) charts [4], [5].

The match between the outer code, channel, and mappingfunction is critical to good decoding performance. Within anEXIT chart paradigm, successful decoding is predicted by adecoding tunnel between the channel and code EXIT curves,whereas the area between these two curves represents thecapacity loss [5], [6]. Efficient coding requires a close matchbetween the channel and code EXIT curves, which is achievedthrough the application of two strategies jointly or separately.

Manuscript received May 28, 2009; revised September 28, 2009 andJanuary 20, 2010; accepted March 1, 2010. Date of publication April 8, 2010;date of current version September 17, 2010. The review of this paper wascoordinated by Dr. H. Lin.

A. Elkhazin is with Magnum Semiconductor, Milpitas, CA 95035 USA.K. N. Plataniotis is with the University of Toronto, Toronto, ON M5S 3J9,

Canada, and also with the School of Computer Science, Ryerson University,Toronto, ON M5B 1E9, Canada.

S. Pasupathy is with the Department of Electrical and Computer Engineering,University of Toronto, Toronto, ON M5S 3J9, Canada.

Digital Object Identifier 10.1109/TVT.2010.2047739

The first strategy is to introduce irregularity in the mappingfunction to control the channel EXIT curve shape. This ap-proach was used in [7]–[14], through a multidimensional map-ping function, which was determined through an optimizationprocedure, such as the binary switching algorithm. The secondstrategy is to introduce irregularity in the outer code to controlthe shape of its EXIT curve. In [5] and [13]–[18], an irregularconvolutional code was formed using a low-rate mother codeand a rate-compatible set of puncture patterns that determinedthe code EXIT curve shape. For turbo codes, the shape of theEXIT curve shape can be manipulated to some extent usingregular bit puncturing or repetition [19].

This paper extends the irregular convolutional coding frame-work in [5] and [13]–[17] by introducing multiple mother codesof different memory. In [5], a single low-rate mother codewas used with a rate-compatible set of puncture patterns toproduce a set of subcodes. These subcodes generate a familyof nonintersecting S-shaped EXIT curves, which are approx-imately a horizontal shift of each other. Irregular puncturingalone allows limited flexibility in determining the code EXITcurve profile, leading to a capacity loss for some channelsand mapping functions due to EXIT curve mismatch. Themismatch between the channel and code EXIT curves canbe minimized using subcodes that differ in both transitionregion slope and horizontal alignment. The steepness of thetransition region is determined by the memory or constraintlength of the underlying convolutional subcode. We proposea systematic procedure to recursively find mother codes andpuncture patterns for a family of subcodes, up to an arbitrarymaximum code memory. Subcodes of different memory needto be connected in a way to ensure proper state transitions. Twotrellis concatenation strategies are proposed; the first strategyconverts all trellises to the same size, and the second strategyuses puncture and termination bits to control rate loss andstate transitions. Both strategies are applied in BICM-IDDsystems over the additive white Gaussian noise (AWGN) chan-nel and a flat-fading multiple-input–multiple-output (MIMO)channel, under ergodic and quasi-static Rayleigh fadingconditions.

The remainder of this paper is organized as follows: The sys-tem model is described in Section II, followed by an overviewof trellis termination and concatenation strategies in Section III.The design and analysis of proposed codes using EXIT chartsare discussed in Section IV, followed by simulated results inSection V and a summary in Section VI.

0018-9545/$26.00 © 2010 IEEE

ELKHAZIN et al.: IRREGULAR CONVOLUTIONAL CODES IN MULTIANTENNA BICM UNDER IDD 3333

Fig. 1. System model of a multiantenna BICM-IDD system.

II. SYSTEM MODEL

Consider the BICM-IDD transceiver structure shown inFig. 1 for a multiantenna system with Nt transmitter antennasand Nr receiver antennas. An outer code is interleaved andrandomly mapped to bits in the symbol vector. The modulatedsymbol s ∈ C

Nt is produced from a length L binary vectorb according to s = M(b). Assuming a frequency flat-fadingchannel, the channel observation is given by

y = Hs + n (1)

where n ∈ CNr is an AWGN noise source of covariance

INrσ2

N , and H is an Nr × Nt channel matrix. The channelmatrix H is assumed to have a spatially independent Rayleighfading distribution, which remains constant over a packet du-ration under quasi-static fading conditions and independentlychanges between symbols under ergodic fading conditions. Thechannel matrix H is assumed to be known at the receiver but notthe transmitter, and we define the signal-to-noise ratio (SNR) asρ = E[s2]/σ2

N .At the receiver, the detector and decoder iteratively exchange

coded bit information until the decoder produces hard bit deci-sions for the uncoded bits, after a prescribed number of turboiterations. The optimal detector produces an extrinsic outputfor the lth bit in the binary symbol vector b = [b1, . . . , bL]T

according to

λE [bl] = −λI [bl]

+ log

∑b∈BL:bl=1 exp

(−‖y−HM(b)‖2

σ2N

)∏Lm=1

bλI [bm]me +b̄m

1+eλI [bm]∑b∈BL:bl=0 exp

(−‖y−HM(b)‖2

σ2N

)∏Lm=1

bλI [bm]me +b̄m

1+eλI [bm]

(2)

where λI [bl] is the intrinsic input for bit bl, and b̄ is the binaryNOT operator on a bit b. The first part of the main expression in(2) corresponding to the a posteriori probability b was trans-mitted, given the channel observations y, and the second partis the a posteriori probability b that was transmitted, given theset of intrinsic inputs {λI [b1], . . . , λI [bL]} from the decoder.The decoder produces soft bit decisions using the Bahl–Cocke–Jelinek–Raviv (BCJR) [20], [21] algorithm.

III. TRELLIS TERMINATION AND CONCATENATION

Trellis termination and concatenation are central to thedesign of irregular convolutional codes with good decodingperformance. Efficient trellis termination strategies eliminatefractional rate loss by ensuring that the initial and final statesare the same. Circular trellises can be created through a code ex-

tension and puncturing procedure [22], or through a tail-biting[23], [24] solution with a two-pass encoding procedure. Usingthe same feedforward and feedback polynomials to encode alluncoded bits leads to a regular convolutional code with a uni-form state trellis. A time-varying set of code parameters leadsto an irregular convolution code, whose state transitions canbe represented using nonuniform state trellis. State transitionsfor convolutional codes of different memory can be representedon a trellis of maximum size. Ensuring proper state transitionswhen concatenating or juxtaposing different trellises is criticalto good irregular code performance.

Consider an binary convolutional encoder with memory Mthat produces a N coded output bits for each input bit. The out-put bits may further be punctured, and the effects of puncturingwill be considered in Section IV. For an irregular convolutioncode, the encoder feedback and feedforward polynomials area function of time index k. The next state xk+1 ∈ B

M andencoder output yk ∈ B

N are specified by the following time-varying state-space equations:

xk+1 =Akxk + Bkuk

yk =Ckxk + Dkuk (3)

where uk ∈ B is the binary input, and the matrix parametersAk, Bk, Ck, and Dk represent time-varying code constraints.For a memory M = 2 recursive systematic convolutional(RSC) code with generator g = [7 5]oct that is shown in Fig. 2,the state-space parameters take on constant values

Ak =[

1 11 0

]Bk =

[10

]Ck =

[0 01 0

]Dk =

[10

]

and generate a trellis that is also shown in the figure. The state-space equations in (3) can be used to represent RSC codesthat have different numbers of delay elements. Consider rep-resenting the sample g = [7 5]oct code in an M = 3 dimensionspace, by adding an addition delay element that is not includedin the feedback loop. This can be achieved by either feedingforward the value x1 to the new delay element input or forcingthe delay input to zero. The M = 3 trellises generated fromboth strategies are also shown in Fig. 2. Feeding forward themost significant state bit leads to a fixed size since all 23 statesare accessible from previous states. Forcing the delay input tozero yields a variable-size trellis since only the first 22 statesare accessible from previous states.

The difference between the fixed- and variable-size trellisstrategies becomes significant when concatenating differentmemory subcodes. Consider a block of uncoded bits encodedwith the g = [7 5]oct code from Fig. 2, followed by a blockencoded with a g = [15 17]oct memory M = 3 RSC code. Inthe variable-size trellis approach, the first bit encoded with theg = [15 17]oct code can be punctured since there is only onevalid transition per state if the most significant state bit is forcedto zero. The drawback is that a circular trellis must start and endon the same state, so an additional termination bit is required tolimit the terminal state range to the first 22 states. In contrast,the fixed-size trellis approach does not limit state transitions,

3334 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

Fig. 2. RSC code with generator g = [7 5]oct and corresponding trellis. Also shown is the projected trellis in M = 3 space using fixed- and variable-size trellisprojection strategies.

so there is no need for a termination and puncture bit whenconcatenating different memory subcodes.

We now generalize the fixed- and variable-size trellis con-structions to larger values of M . Consider a tap delay filter withM delay elements that can represent RSC codes of memorym ∈ [1 . . . M ]. Let αm be the number of uncoded bits encodedwith a memory m subcode, and let mmin ≤ M be the smallestmemory subcode being used. A total of K =

∑αm informa-

tion bits are encoded. We assume that all bits encoded witha memory m subcode are encoded with a single set of codeparameters in a contiguous block. At the receiver, the BCJRalgorithm from [21] is applied with the time-varying constraintsspecified in (3).

For the variable-size trellis construction, the subcodes areordered in ascending order of memory. The output bits for thefirst mmin information bits can be punctured, because they canbe inferred from the state at time mmin. In addition, for twoadjacent subcodes with memories {m1,m2},m2 > m1, thecoded bits corresponding to the first m2 − m1 trellis sectionsin the second trellis can be punctured since the states in thesesections only have one valid transition per state. Thus, codedbits from M − mmin trellis sections at memory transitions arealso punctured. After K uncoded bits have been encoded, thecode is extended by encoding M termination bits that are usedto create trellis circularity. The termination bits are chosen suchthat xK+M = xmmin and may be found either directly fromthe linear feedback register block diagram or by constructing aspanning tree in the state trellis starting at xK . Since the numberof termination bits is equal to the number of punctured bits, acircular trellis is produced without a fractional rate loss.

For the fixed-size trellis construction, a tail-biting strategysimilar to [23] and [24] is used to create a circular trellis.A circular trellis over [0 K − 1] is defined by the conditionxK = x0. The initial/terminal state can be found by expressingthe encoder state at time k as

xk = x[zi]k + x[zs]

k (4)

which is the superposition of a zero-input solution x[zi]k defined

by ul = 0∀ l and a zero-state solution x[zs]k that is defined

by x0 = 0. The zero-state solution x[zs]k can be found by

encoding the uncoded bit sequence from the zero encoder state.Let the state-space parameters Ak, Bk, Ck, and Dk take onvalues Am, Bm, Cm, and Dm if the kth trellis section isgenerated from a mother code of memory m. Substitution ofx[zi]

K = (AαM

M × · · · × Aα11 )x0 into (4) and imposing the cir-

cularity condition xK = x0 yield

(AαM

M × · · · × Aα11 + IM )x0 = x[zs]

k (5)

where matrix Am is defined as

Am =[

0M×1 IM−1

01×M−m fbm

]

with fbm being the binary feedback vector for the memory mmother code. We are interested in finding the initial state x0

in (5), given the zero-state solution x[xs] that has been foundon a preencoding pass. If the matrix on the left-hand side of(5) is invertible, the initial state x0 leading to a circular trelliscan be found by binary matrix inversion. If, on the other hand,the matrix in (5) is singular, then, for some values of x[xs], acircular trellis cannot be constructed since (5) is not satisfied.

If a trellis is found to be possibly noncircular based on (5),there are a couple of adjustments that can be made to bringabout circularity. Since matrix multiplication is not commu-tative, the order of the trellis blocks affects trellis circularity.If a natural trellis order does not produce a circular trellis,other orderings can randomly be generated, thereby changingthe order of the Am matrices in (5). If a circular trellis cannotbe found by reordering, the number of trellis sections of onemother code can be incremented, and another one can bedecremented by randomly choosing mother codes. Reorderingcan then be applied on the modified trellis.

Comparing the variable- and fixed-size trellis constructionapproaches, each has its relative merit. The variable-size trellisstrategy punctures some systematic bits, thereby reducing theaverage channel message mutual information per bit for shorterpacket transmissions. The fixed-size trellis solution has bits thatare prone to decoding errors near high-to-low memory transi-tions, affecting the error floor behavior of longer packet trans-missions. This is because the last few bits of the longer memorycode are not part of the shorter memory code feedback loop.

ELKHAZIN et al.: IRREGULAR CONVOLUTIONAL CODES IN MULTIANTENNA BICM UNDER IDD 3335

Fig. 3. EXIT curves for rate r = {1/2, 1/4} convolutional outer codes forcode memory m = {1, . . . , 7}.

IV. SUBCODE DESIGN USING EXIT CHARTS

As the outer code is composed of a set of convolutional sub-codes, understanding the relationship between the EXIT curveshape and the choice of generator polynomials and puncturepatterns is of key importance. Following the work in [5], weconsider RSC mother codes. All one-to-one RSC codes have anEXIT curve that starts IE(0) = 0 and terminates at IE(1) = 1.The EXIT characteristics of unpunctured RSC codes are largelydetermined by the code memory and the number of generatorpolynomials. Consider a family of rate r = 1/4 RSC con-volutional codes having generators g1 = [3 2 3 3]oct, g2 =[5 7 6 6]oct, g3 = [15 17 13 11]oct, g4 = [31 23 25 35]oct,g5 = [63 45 73 65]oct, g6 = [121 157 173 113]oct, and g7 =[301 257 365 356]oct, where the first element is also thefeedback polynomial. A set of rate r = 1/2 codes is formedby omitting the last two polynomials in each case. The EXITcurves for these 14 codes are shown in Fig. 3. All the curveshave an “S” shape with an initial region gradually increasingaway from zero, followed by a steep transition region and afinal region of gradual increase toward unity. The area under theEXIT curve is approximately equal to the reciprocal of the coderate [6]. Adding more generators naturally increases this area.The effect of changing the code rate appears as a horizontal shiftthat can be seen between the r = 1/4 and r = 1/2 curves. For agiven code rate, the corresponding set of subcode EXIT curvesdiffers in transition region slope and is observed to intersect ata crossover point near IE(II) = 0.5. Fixing the code memorylimits the maximum transition region slope and flatness in theextrema. Of course, the codes whose EXIT curves are shown inFig. 3 are well chosen, and an arbitrarily chosen long memorycode does not necessarily have a steep transition region. For agiven memory, the goal is to find generators that maximize thetransition slope or flatness in the extrema. A set of convolutionalmother codes can recursively be found as follows: Let g{m}

n

be a length-n vector of generator polynomials that represents arate-1/n RSC convolutional code of memory m. Rate r = 1/2codes can be found by examining the set of 22(m+1) possiblememory m codes and choosing generators that maximize theregion of the EXIT curve to the right of the transition region.This is equivalent to minimizing error rate performance in thelow-BER region. For a test intrinsic mutual information of I∗I =max(r + 0.15, 0.95), the extrinsic mutual information IE(I∗I )can be evaluated for different generators through simulated

trials with a Gaussian intrinsic message density. There aretypically several candidate codes that have maximal extrinsicmutual information, and any one of them can be chosen.Finding lower rate codes proceeds in a recursive manner. Therate-[1/(n + 1)] code can be found from the rate-1/n code byconsidering possible generators of the form g{m}

n+1 = [g,g{m}n+1]

and by selecting the g that maximize IE(I∗I ). A set of convolu-tional mother codes found in this manner is shown in Table I forcode memory M = {1 . . . 7} and rates of r = {1/2, . . . , 1/10}in octal notation. The feedback polynomial and first generatorpolynomial are the same.

The use of multiple generator polynomials of different mem-ory only affords limited degrees of freedom and cannot beused for higher rate codes. Puncture patterns can further beapplied to fix the code rate, thereby determining the horizontalalignment of the crossover point. The set of puncture patternsfor a particular code should generate rates that uniformlyspan the EXIT space to ensure good curve-matching potential.A particular puncture pattern and code pair may generate asubcode that is not one to one, having a terminal EXIT pointbelow unity, or a low weight subcode having a less steeplysloped transition region. As such situations are undesirable, themaximum extrinsic mutual information IE(I∗I ) can be used tochoose between different puncture patterns of the same rate.

Consider a set of 17 uniformly spaced subcode rates r ={0.1, 0.15, . . . , 0.85, 0.9} that can be generated using puncturepatterns with corresponding periods Np = {1, 3, 1, 1, 3, 7, 2, 9,1, 11, 3, 13, 7, 3, 4, 17, 9}. The puncture patterns are of mini-mal length to achieve the desired rate. A set of 200 randompuncture patterns for each subcode rate and puncture periodwas generated, and the pattern maximizing the IE(I∗I ) functionwas chosen. Initially, only systematic puncture patterns wereconsidered, which produced satisfactory patterns for all ratesand generators, except r = 0.8. For the r = 0.8 subcodes, anonsystematic puncture pattern was required for the M ={2, 5, 6, 7} generators. A set of puncture patterns producedin this manner is shown in Tables II–VIII for trellis memoryM = {1 . . . 7} using the mother codes shown in Table I. Theentry {r,Np, (p1, p2, . . . , pnr

))} has rate r and puncture periodNp with a puncture pattern (p1, p2, . . . , pnr

) specified in octalnotation. For example, the rate-0.45 entry for the M = 1 codegenerators in Table II has a puncture period of 9 and a puncturepattern of (777, 367, 245) in octal or (111111111, 011110111,010100101) in binary. This is a systematic puncture patternsince none of the first generator bits are punctured. The setof EXIT curves produced from the generators in Table I andpuncture patterns from Tables II–VIII are shown in Fig. 4. For agiven memory, the set of rate-compatible EXIT curves does notintersect, and for a particular rate, codes of different memoryonly intersect at the terminal points and a crossover point nearIE(II) = 0.5. The crossover point slightly decreases for highrate codes.

The overall irregular convolutional code is constructed byencoding different fractions of bits with different mother codesand by puncturing with the appropriate patterns. Let λm,p bethe fraction of coded bits produced by the memory m motherencoder and punctured with the corresponding pth puncturepattern in the mth subset. It is convenient to index over m

3336 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

TABLE IGENERATORS FOR RSC MOTHER CODES

TABLE IIPUNCTURE PATTERNS FOR m = 1 RSC MOTHER CODES (OCTAL)

TABLE IIIPUNCTURE PATTERNS FOR m = 2 RSC MOTHER CODES (OCTAL)

TABLE IVPUNCTURE PATTERNS FOR m = 3 RSC MOTHER CODES (OCTAL)

TABLE VPUNCTURE PATTERNS FOR m = 4 RSC MOTHER CODES (OCTAL)

TABLE VIPUNCTURE PATTERNS FOR m = 5 RSC MOTHER CODES (OCTAL)

TABLE VIIPUNCTURE PATTERNS FOR m = 6 RSC MOTHER CODES (OCTAL)

ELKHAZIN et al.: IRREGULAR CONVOLUTIONAL CODES IN MULTIANTENNA BICM UNDER IDD 3337

TABLE VIIIPUNCTURE PATTERNS FOR m = 7 RSC MOTHER CODES (OCTAL)

Fig. 4. EXIT curves for punctured subcodes of memory m = {1, . . . , 7} andrates r = [0.1, 0.15, . . . , 0.9] shown in four subplots. Each cluster of sevencurves corresponds to a particular rate, with the leftmost cluster correspondingto the lowest rate code in the subplot.

and p using a single variable i = mP + p, where P = 17 isthe number of puncture patterns per code. The optimal degreedistribution {λi}∗ can be found as the solution to the followingconstrained rate maximization problem:

arg max{λi}

PM∑i=1

λiri

s.t.∑

λi = 1∑λif

{i}(II) > f−1ch (II) ∀II (6)

where ri is the rate of the ith subcode, f{i}c (II) is the ith

subcode EXIT curve, and f−1ch (II) is the inverse channel EXIT

function. The problem in (6) can be solved using a numericalgradient search algorithm. The solution in (6) can be appliedto the AWGN and independent Rayleigh fading channels,where the channel EXIT curve is constant between channelrealizations.

Under quasi-static fading conditions, the channel EXIT curveis not constant and varies with individual channel realizations.From a code design perspective, (6) can be used to design anirregular code profile for any given channel realization andmapping function. The channel EXIT curve is, however, a ran-dom entity whose capacity varies between channel realizations,and is not known at the transmitter. Provided that the channel-

fading statistics are known at the transmitter, a code profile canbe constructed based on a sample set of channel realization.

Consider the problem of designing a rate-r code to operate ata prescribed frame error rate (FER) of P̂ER, or better, based ona set of Nch channel realization. The channel EXIT curve foreach channel realization can be evaluated through simulationat a set of predefined SNR values. A design channel curveIch∗E (II) can be constructed at each SNR value, as a curve

of maximal area that intersects the channel EXIT curve for�NchP̂ER� channel realization. Through interpolation, a designchannel EXIT curve can be constructed at an arbitrary SNR,and an irregular code profile can be produced using (6) bysubstituting fch(II) = Ich∗

E (II). Under ideal decoding, a codeprofile constructed using (6) and the design channel curve willlead to a decoding tunnel for at least �Nch(1 − P̂ER)� channelrealizations and a predicted FER of P̂ER. Different SNR valuesmust be considered until the resulting code rate is equal to thedesign rate of r.

A design channel curve that minimizes the FER canbe found by evaluating the channel realization curves overa set of uniformly distributed points {xi} ∈ [0 1]. LetIchE ({xi},M(·),H, ρ) be the channel EXIT curve for a map-

ping function M(·), channel realization H, and SNR ρ, and let{yi} = Ich∗

E ({xi}) be the design channel curve evaluated over{xi}. The FER minimizing design channel curve can be foundas the solution to

arg max{yi}

∑i

yi

s.t. fER ({yi}) ≤ P̂ER

yi > yj ∀xi > xj (7)

where fER({yi}) = EH[maxi I(yi < IE(xi,H, ρ))] is the er-ror function, and I(x) is an indicator function that is 1 if x istrue and 0 otherwise. The monotonic constraint yi > yj ∀ xi >xj in (7) can be omitted since all the IE({xi},H, ρ) func-tions are monotonic, leading to a piecewise linear monotonicconstraint function fER({yi}). The monotonic nature of theobjective and constraint functions allows for a simple solutionto (7) using the iterative procedure given here.

1: {yi} ← 0;2:while δ > tol3: n = arg mini f i

ER(yi + δ);4: yn ← yn + δ;5: if fER({yn}) ≥ P̂ER

6: {yi} ← max({yi} − δ, 0); δ ← δ/2;7: end if8:end while

3338 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

The preceding procedure iteratively increases one point in theset {yi} using a fixed increment size, provided that the resultingerror rate does not exceed the target FER. The incrementedpoint is chosen based on the point error function f i

ER(y, ρ) =EH[I(IE(xi,H, ρ) < y)], which is the FER rate consideringonly the ith EXIT point. If the error rate of the incremented setis above the target FER, the increment value is reduced, and allthe EXIT points are backed off by an appropriate amount beforecontinuing.

V. SIMULATED RESULTS

This section uses simulated examples to evaluate the per-formance of the proposed code over an AWGN channel and aMIMO channel with two transmitter and two receiver antennas.The MIMO channel is Rayleigh fading, and limiting cases of er-godic and quasi-static fading conditions are considered. For thequasi-static fading simulations, 2000 test channel realizationswere generated to determine the design channel EXIT curve.For all systems, Monte Carlo simulations were performed until300 frame errors occurred or until 10 000 trials were performed.On each Monte Carlo trial, 40 turbo iterations were performedwith the proposed codes. For the reference designs, five globaliterations with 20 decoder subiterations were used with theturbo-coded BICM-IDD system, and 200 LDPC decoder it-erations were used at each decoding stage of the referencemultilevel code (MLC) design with multistage decoding (MSD)[25]. Minimal performance improvement was achieved usingmore turbo iterations.

Consider the EXIT-chart-aided design of a BICM-IDDsystem over an AWGN channel using the proposed codes anddifferent 16-state quadrature amplitude modulation (16-QAM)mapping functions. Gray and natural labeling are considered,along with three kinds of random mapping functions: 1) randomamplitude shift keying (ASK); 2) random QAM; and 3) randommultidimensional QAM (MQAM). The random ASK andQAM schemes generate random assignments over a subset ofconstellation points and then merge these subsets to producethe overall mapping function. The random MQAM labelingfunctions applies a random mapping across the two transmitterantennas in the MIMO channel and across two timeslots inthe AWGN channel, leading to an overall constellation of256 points for a 16-QAM modulation format. Using the codeoptimization in (6), irregular code profiles were generated forthe different mapping functions. The channel SNR was chosenin each case such that the resulting code rate was r = 1/2.The EXIT curves of the memory M = 7 designs are shown inFig. 5 for the different mapping functions. The random QAMand MQAM mapping functions have a steeper slope, whichreduces the area mismatch between the detector and decoderEXIT curves. The code profiles for the random QAM andrandom MQAM formats were produced at a channel SNR ofρ∗c = {5.64 dB, 5.23 dB}, respectively, which is quite closeto the constrained modulation channel capacity threshold ofρ∗ch = 5.11 dB for a rate r = 2 transmission. The nonzerocoefficient of the random QAM code profile are specified byλ3,3...5 = {0.0016, 0, 0, 0.0311}, λ4,5...9 = {0.0413, 0.0379,0, 0.0139}, λ5,2...8 = {0.0021, 0, 0, 0, 0.0297, 0.0137, 0.0353},

Fig. 5. Optimized outer code EXIT curve for different 16-QAM mappingfunctions over the AWGN channel at a binary code rate of 1/2. Code profileswere produced at channel SNR values of 8.48 dB (Gray), 7.08 dB (Natural),6.98 dB (Random ASK), 5.64 dB (Random QAM), and 5.35 dB (RandomMQAM).

λ6,6 ...10 = {0.0295, 0.0319, 0.0291, 0.0749, 0.1090}, andλ7,5 ...11 = {0.0067, 0.0031, 0.0040, 0.0287, 0.0756, 0.2005,0.2004}, and the coefficients for random MQAM labeling caseare specified in Table IX.

Practical detection and decoding performance for finite-length transmissions falls short of the predicted code thresholdvalues. This loss is influenced by the packet length and mappingfunction, as well as the trellis concatenation strategy. We firstconsider the performance of the fixed- and variable-size trellisconcatenation strategies for different packets lengths in Fig. 6for transmissions of NSym = {250, 2000} symbols. The fixed-trellis-size scheme performed better in the short packet system,whereas the variable-trellis-size scheme performed better in thelonger packet system. The short-packet fixed-size trellis systemand long-packet variable-size trellis system led to high errorfloors. Random MQAM mapping was used for the NSym =2000 system, and random QAM mapping was used for theNSym = 250 system. Although the random MQAM mappedsystem has a lower code SNR threshold, the random QAMlabeling leads to less detector message correlation and betterperformance in the shorter packet system.

The performance of the proposed codes over the AWGNchannel is compared to reference turbo-coded and MLC-MSDsystems in Fig. 7 for transmissions of NSym = {250, 2000}symbols. Fixed-size trellis construction was used for theNSym = 250 symbol system, and a variable-size trellis con-struction was used for the NSym = 2000 symbol system. Thereference BICM-IDD turbo-coded system used Gray labeling,an S-random interleaver [26] with spreading factor Sparm = 17,and a binary turbo code with generator g = [37 23]oct. Thereference MLC-MSD system [27] used set partitioning labelingand generalized low density parity check (GLDPC) subcodes[25]. Although, in the NSym = 250 system, the proposed codesslightly outperform the reference turbo codes near the errorfloor 10−5 > BER > 10−6, the reference designs generally out-perform the proposed codes over the AWGN channel.

We next consider the performance of the proposed codesover the 2 × 2 Rayleigh MIMO channel under ergodic fadingconditions. For code memories of M = {4, 7}, the performanceof the proposed codes is shown in Fig. 8 for NSym = 1000symbol blocks using random MQAM labeling and in Fig. 9

ELKHAZIN et al.: IRREGULAR CONVOLUTIONAL CODES IN MULTIANTENNA BICM UNDER IDD 3339

TABLE IXAWGN CODE COEFFICIENTS FOR MEMORY M = 7 DESIGN WITH RANDOM MQAM LABELING

Fig. 6. BER performance of a 16-QAM BICM-IDD system over an AWGNchannel using M = 7 memory fixed- and variable-sized trellises. RandomQAM was used for NSym = 250 symbol transmissions, and random MQAMwas used for NSym = 2000 symbol transmissions.

Fig. 7. AWGN channel performance of the proposed codes and referencedesigns for NSym = {250, 2000} symbol transmissions. With the proposedcodes, random QAM and random MQAM labeling were used for the NSym ={250, 2000} symbol transmissions, respectively.

for NSym = 125 symbol blocks using random QAM labeling.Turbo-coded and GLDPC-coded MLC-MSD designs are alsoshown for reference. The various design were generated attransmission rates of r = {8/3, 4.8} bits per channel use. These

Fig. 8. 2 × 2 ergodic Rayleigh fading performance of the proposed codes withrandom MQAM labeling and reference designs for 1000 symbol transmissions.

Fig. 9. 2 × 2 ergodic Rayleigh fading performance of the proposed codes withrandom QAM labeling and reference designs for 125 symbol transmissions.

rates correspond to binary code rates of rb = {1/3, 0.6}, whichallow for a simple puncture pattern to be used with the referenceturbo code designs. For NSym = 1000 and r = 4.8 transmis-sions, the proposed M = 7 code outperformed the referenceturbo and GLDPC codes by 0.35 dB at BER = 10−4. Forshorter NSym = 125 symbol packets, the M = 7 design outper-formed the reference turbo code and GLDPC code designs by

3340 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

Fig. 10. 2 × 2 quasi-static Rayleigh fading performance of the proposedcodes with random MQAM labeling and reference designs for 1000 symboltransmissions.

Fig. 11. 2 × 2 quasi-static Rayleigh fading performance of the proposedcodes with random MQAM labeling and reference designs for 250 symboltransmissions.

approximately 1 and 2 dB, respectively, at the r = 4.8 designrate and by a slightly smaller margin at the r = 8/3 design rate.In addition, the M = 4 design performed better than the M = 7design in the NSym = 125 system at error rates above BER =10−3. This is because the short memory design produced lessdecoder message correlation, which significantly affects short-packet performance. The relatively poor performance of theNSym = 125 symbol GLDPC-coded system stems from errorpropagation in the MSD decoder for short subcode lengths.

The performance of the proposed codes over the quasi-staticfading MIMO channel is shown for NSym = 1000 symboltransmissions in Fig. 10 and for NSym = 250 symbol trans-missions in Fig. 11. In addition, shown is the turbo-codedBICM-IDD system performance, as well as the performanceof two MLC-MSD systems. The first is the stratified BLAST(S-BLAST) [28]–[30] architecture that uses a nonuniform setof layer code rates, and the second is from [31] and [32], whichreplaces the Gray labeling used in the S-BLAST system withrandom MQAM labeling. Random MQAM labeling is a keyingredient to good quasi-static fading system performance, asdiscussed in [31] and [32]. In both the BICM-IDD and MLC-MSD configurations, the random MQAM-labeled systems out-performed the corresponding Gray-labeled systems by about3.5 dB at FER = 10−3. The random MQAM BICM-IDD sys-

tem using the proposed codes led to an additional 0.6–0.8-dBgain over the random MQAM MLC-MSD strategy from [31]and [32]. In the shorter packet system, the less complex M = 4design using the proposed codes had the best performance ofthe designs considered.

VI. SUMMARY

A framework for constructing and decoding irregular convo-lutional codes has been developed in this paper. An irregularconvolutional code is formed using irregular puncture patternsand code memory. Strategies based on fixed- and variable-size trellises have been developed to connect different memorymother codes. The proposed codes have been designed forBICM-IDD systems with the aid of EXIT charts. MultiantennaBICM-IDD systems designed with the proposed irregular con-volutional codes outperform binary turbo codes, as well asMLC-MSD structures in many scenarios. The performancegains are larger for shorter packet transmissions, higher trans-mission rates, and under nonergodic fading conditions.

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Akrum Elkhazin received the B.A.Sc. degree in systems design engineeringfrom the University of Waterloo, Waterloo, ON, Canada, in 2000 and theM.A.Sc. degree in electrical engineering and the Ph.D. degree from the Uni-versity of Toronto, Toronto, ON, in 2002 and 2009, respectively.

He is currently with Magnum Semiconductor, Milpitas, CA, as a Video Algo-rithms Designer. His research interests include video transcoding, multiantennawireless communication, channel coding, and iterative receivers.

Dr. Elkhazin was the recipient of an OGSST scholarship in 2002, a NationalSciences and Engineering Research Council of Canada scholarship in 2000,and first place in a Canadian Engineering competition in the corporate designcategory in 2000.

Konstantinos Plataniotis (SM’03) is currently a Professor with TheEdward S. Rogers Sr. Department of Electrical and Computer Engineering,University of Toronto, Toronto, ON, Canada, and an Adjunct Professor withthe School of Computer Science, Ryerson University, Toronto. He is also theChair of the Communications Group, Department of Electrical and ComputerEngineering, Director of The University of Toronto’s Knowledge Media DesignInstitute (www.kmdi.utoronto.ca), and the Director of Research for the Identity,Privacy and Security Institute, University of Toronto (www.ipsi.utoronto.ca).His research interests include biometrics, communications systems, multimediasystems, and signal and image processing.

Dr. Plataniotis is a registered professional engineer in the province ofOntario. He is the Editor in Chief (2009–2011) of the IEEE SIGNAL

PROCESSING LETTERS. He served on the IEEE Educational Activities Board(EAB) and was the Chair of the IEEE EAB Continuing Professional Ed-ucation Committee (2008–2009). He served as Chair of the IEEE TorontoSignal Processing Chapter (2000–2002), Chair of the IEEE Toronto Section(2004–2005), and member of the 2006 and 2007 IEEE Admissions andAdvancement Committees. He was the 2005 recipient of the IEEE Canada’sOutstanding Engineering Educator Award “for contributions to engineeringeducation and inspirational guidance of graduate students” and a co-recipientof the 2006 IEEE TRANSACTIONS ON NEURAL NETWORKS OutstandingPaper Award for the paper entitled “Face Recognition Using Kernel DirectDiscriminant Analysis Algorithms,” which was published in 2003. He is amember of the Technical Chamber of Greece.

Subbarayan Pasupathy (LF’07) was born in Chennai, India. He received theB.E. degree in telecommunications from the University of Madras, Chennai,the M.Tech. degree in electrical engineering from the Indian Institute ofTechnology, Chennai, and the M.Phil. and Ph.D. degrees in engineering andapplied science from Yale University, New Haven, CT.

He is currently a Professor Emeritus with the Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON, Canada, where hehas been a Faculty member since 1972. During 1980–1983, he was an AssociateEditor for the Canadian Electrical Engineering Journal. His research interestsover the last three decades have been statistical communication theory andsignal processing and their applications to digital communications.

Dr. Pasupathy was named Fellow of the IEEE “for contributions to bandwidthefficient coding and modulation schemes in digital communication,” a Fellowof the Engineering Institute of Canada in 2004, and a Fellow of the CanadianAcademy of Engineering in 2007. He is a registered Professional Engineer inthe province of Ontario. He has served as the Chairman of the Communica-tions Group and as the Associate Chairman of the Department of ElectricalEngineering, University of Toronto. During 1982–1989, he was an Editor forData Communications and Modulation for the IEEE TRANSACTIONS ON

COMMUNICATIONS. During 1979–1982, he served as Technical AssociateEditor for IEEE Communications Magazine for which, during 1984–1998, hewrote a regular humor column entitled “Light Traffic.” He was the recipient ofthe Canadian Award in Telecommunications in 2003 from the Canadian Societyof Information Theory. He has been identified as a “highly cited researcher” bythe ISI Web of Knowledge, and his name is listed in ISIHighlyCited.com.