analysis and design of generalized bicm-t system

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 9, SEPTEMBER 2014 3223

Analysis and Design of Generalized BICM-T SystemMuhammad Talha Malik, Student Member, IEEE, Md. Jahangir Hossain, Member, IEEE, and

Mohamed-Slim Alouini, Fellow, IEEE

Abstract—The performance of bit-interleaved coded modula-tion (BICM) using convolutional codes in nonfading channels canbe significantly improved if the coded bits are not interleaved atall. This particular BICM system is referred to as BICM trivial(BICM-T). In this paper, we analyze a generalized BICM-T systemthat uses a nonequally spaced signal constellation in conjunctionwith a bit-level multiplexer in an additive white Gaussian noise(AWGN) channel. As such, one can exploit the full benefit ofBICM-T by jointly optimizing different system modules to furtherimprove its performance. We also investigate the performance ofthe considered BICM-T system in the Gaussian mixture noise(GMN) channel because of its practical importance. The presentednumerical results show that an optimized BICM-T system canoffer gains up to 1.5 dB over a non-optimized BICM-T systemin the AWGN channel for a target bit error rate of 10−6. Thepresented results for the GMN channel interestingly reveal thatif the strength of the impulsive noise component, i.e., the noisecomponent due to some ambient phenomenon in the GMN, isbelow a certain threshold level, then the BICM-T system performssignificantly better as compared to traditional BICM system.

Index Terms—Bit-interleaved coded modulation, bit error rate,bit-level multiplexer, hierarchical modulation, Gaussian mixturenoise, L-values.

I. INTRODUCTION

IN traditional bit interleaved coded modulation (BICM) [1],[2], the channel encoder is connected to the modulator via

a bit-level interleaver and at the receiver’s side, the reliabilitymetrics for the coded bits are calculated by the demapper whichare then de-interleaved and fed to the binary decoder. Due toits advantages in fading channel over other coded modulation(CM) schemes, e.g., trellis coded modulation (TCM) [3] andmultilevel coding [4], BICM is a de-facto standard for manycontemporary wireless systems. In non-fading channel, com-pared with TCM, BICM gives a smaller minimum Euclideandistance, and also a smaller constrained capacity [2]. In par-ticular, depending on the employed binary labeling rule, the

Manuscript received October 1, 2013; revised March 5, 2014 and May 27,2014; accepted June 18, 2014. Date of publication July 2, 2014; date of currentversion September 19, 2014. This work was supported in part by the NaturalSciences and Engineering Research Council of Canada under Discovery GrantProgram and in part by an NPRP grant from Qatar National Research Fund(a member of Qatar Foundation). This work was presented in part at the IEEEInternational Symposium on Personal, Indoor and Mobile Radio Communica-tions (PIMRC’13), London, U.K., Sep. 2013, and in part at the IEEE WirelessNetworking and Communication (WCNC’14), Istanbul, Turkey, Apr. 2014.The associate editor coordinating the review of this paper and approving it forpublication was M. Valenti.

M. T. Malik and M. J. Hossain are with the School of Engineering, Universityof British Columbia, Kelowna, BC V1V 1V7, Canada.

M.-S. Alouini is with the Computer, Electrical and Mathematical Sci-ences and Engineering, King Abdullah University of Science and Technology,Thuwal 23955-6900, Saudi Arabia.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2014.2335199

so-called BICM capacity exhibits a gap to the CM capacityof the used signal alphabet. Stierstorfer and Fischer studiedand compared BICM capacity for binary labelings in [5] and[6]. Suboptimality of the binary reflected Gray code (BRGC)labeling in terms of maximizing the BICM capacity for a givenequally spaced (ES) input alphabet and uniform input distribu-tions was also shown in [7, Ch. 3]. Since the capacity loss issmall when a constellation with the Gray labeling is used, BICMis still considered a valid option for CM in non-fading channel.

A global random interleaver advocated by Caire et al. [2]is often considered in the literature due to the simplicity ofanalysis of BICM system. In fact, in the initial work on BICM[1], Zehavi used three independent random bit interleaversfor transmission with 8-ary phase shift keying. It was alsoshown that the BICM transmission with single interleaver suf-fers from performance degradation and multiple interleavers(M-interleavers) should be used [8]–[10]. Similarly, M-interleavers were used for the turbo coded BICM [11], [12],and for serially concatenated systems [13]. Based on the bitlevel capacity notion, an innovative adaptive interleaver designapproach was proposed in [14].

Recent works in [15] and [16] showed that the performanceof BICM using convolutional codes in non-fading channelcan be improved significantly when the interleaver is removedfrom BICM system. This system was referred to as BICMtrivial (BICM-T) which was formally analyzed in [16] andis shown to be asymptotically as good as Ungerboeck’s onedimensional (1D)-TCM. The authors in [16] analyzed a simpleBICM-T system where the code rate 1/2 with 16-ary quadratureamplitude modulation (QAM) constellation is used. For thiselementary example, the number of encoder’s output matchesthe modulation order and it was shown that the bit error rate(BER) performance of BICM-T system is significantly bettercompared to the traditional BICM system in non-fading chan-nel. However, due to the use of an ES signal constellation in thesystem, where the distance between neighboring signal pointsis equal, it leads to a suboptimal BICM-T system.1 In particular,the BER performance can be further improved by providingproper unequal error protection (UEP) to the coded bits [9],[10]. Moreover, the design and analysis of a more general setup,where the code rate, i.e., the number of encoder’s outputs doesnot necessarily match the modulation order, are not considered.For such a generalized setup, the important question is how toconnect the channel encoder’s outputs to the modulator in orderto minimize the BER performance.

In [10], the performance of BICM system is optimizedby using hierarchical constellations, a bit-level multiplexer

1For BICM-T system, we aim at the minimization of BER. Therefore, theoptimality of BICM-T system is defined in terms of minimizing the BER.

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3224 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 9, SEPTEMBER 2014

and M-interleavers. However, this system in [10] withM-interleavers is shown to be suboptimal in non-fading chan-nel. Motivated by the results presented in [10], in this paperwe analyze a generalized BICM-T system2 by using a bit-levelmultiplexer3 and hierarchical constellations. Using our gener-alized system, the designer can exploit different degrees offreedom to notably improve the BER performance of BICM-Tsystem. Moreover, it offers a flexibility in adapting transmissionrate by changing the modulation order without changing thecode rate due to the presence of multiplexer between the chan-nel encoder and the modulator. This flexibility makes general-ized BICM-T system appealing compared to other CM schemesfor easy rate adaptation in the frequency selective multicarriernon-fading channel where different carriers can have differentchannel qualities [17], and in the quasi-static fading channelwhere the channel remains constant over the length of a dataframe, and varies independently from frame to frame [18].

In many physical channels, such as the power line com-munication channel [19], the ambient noise is known throughexperimental measurements to be non-Gaussian due to theimpulsive nature of man-made electromagnetic interference.While traditional BICM and BICM-T systems have been thor-oughly investigated and optimized in the AWGN channel (seefor examples, [9], [10], [15], [16], [20], and references therein),the analysis of BICM transmission impaired by non-Gaussiannoise has received relatively little attention [21]–[23]. Thecontributions of this paper are summarized as follows.

• We consider and analyze a general BICM-T system, wherethe code rate, i.e., the number of encoder’s outputs doesnot necessarily match the modulation order and the mod-ulation parameters of hierarchical constellations can bevaried. This particular BICM-T system is not analyzed in[10] or [16].

• Using the so-called zero-crossing model [24], we haveapproximated, in closed-forms, the probability densityfunction (PDF) of metrics that are relevant to characterizethe decoder’s error performance of generalized BICM-Tsystem in terms of constellation parameters of the BRGChierarchical QAM (HQAM) in the AWGN channel. Thenthese PDF expressions are used to develop union bound(UB) on the BER of generalized BICM-T system. Thisdeveloped bound provides a design tool to analyze andoptimize code, modulation parameters and multiplexer.

• We also analyze the performance of BICM-T system inthe GMN channel because of its practical importance. Ouranalysis using the developed UB on the BER of BICM-Tsystem in the GMN channel reveals the answer to aninquisitive question of using or removing the interleaver inBICM system in the GMN channel. Interestingly our anal-ysis shows that if the strength of impulsive component, i.e.,the noise component due to some ambient phenomenonsuch as man-made electromagnetic interference, radarclutter, and sudden electric switching in the GMN is below

2As in [15], [16], we consider non-iterative decoding based receiver for thesake of low delay and low complexity.

3The role of this multiplexer is to rearrange the coded bits and to assign themto particular bit positions in the modulator.

a certain threshold level, an optimized BICM-T systemshould be used as it offers gain over traditional BICMsystem. On the other hand, if the impulsive noise strengthis above a threshold, traditional BICM should be used as itperforms better than BICM-T. The threshold depends onthe code and the value of the signal-to-noise ratio (SNR).

The presented numerical results for two practically relevantspectral efficiencies 1 bit/dimension and 1.5 bits/dimensionshow that an optimized BICM-T system can offer gains up to0.9 dB and 1.7 dB, respectively over a non-optimized BICM-Tsystem in the AWGN channel for a target BER of 10−6.Presented results for the GMN channel show that an optimizedBICM-T system can offer gains up to 1.5 dB over traditionalBICM system for a target BER value of 10−6 if the strengthof impulsive noise component in the GMN is below a certainthreshold level. However, if this impulsive noise strength isabove a threshold, traditional BICM performs better thanBICM-T.

The rest of this paper is organized as follows. In Section II,we present the system model of generalized BICM-T systemthat uses a bit level multiplexer and hierarchical constellations.Section III presents the performance evaluation of the systemmodel presented in Section II. In Section IV, we derive thePDFs of metrics that are relevant to characterize the decoder’serror performance. These PDFs are then used to derive the UBexpressions for BICM-T transmission in the AWGN channel fortwo particular cases: 16-HQAM and 64-HQAM. In Section V,we extend our analysis to derive the UB expression for BICM-Ttransmission in the GMN channel. Selected numerical resultsare presented in Section VI. Finally, Section VII concludes thepaper.

Notations: All through this paper, boldface letters bl =[bl,1, . . . , bl,N ] are used to denote row vectors of length N

and also to denote matrices b = [bT1 , . . . , bTM ]

Tof M rows,

where (·)T represents transposition. The total Hamming weightof a binary matrix b is denoted by dH(b). Probability isdenoted by Pr(·) and the PDF of a random variable B isdenoted by pB(b). The convolution operation between twoPDFs is represented by pB1

(b) ∗ pB2(b) and pB(b)

∗m rep-resents the m-fold self convolution of the PDF pB(b). TheGaussian function with mean μ and variance σ2 is defined

as ψ(z;μ, σ)Δ= (1/(

√2πσ)) exp(−((z − μ)2/2(σ2))), the Q-

function as Q(x)Δ= (1/

√2π)∫∞x exp(−u2/2) du, and the set

Wi(l) is defined as all the combinations of i non-negative

integers such that the sum of the elements is l, i.e., Wi(l)Δ=

{wg = (wg,1, . . . , wg,i) ∈ (Z∗)i : wg,1 + · · ·+ wg,i = l}.

II. GENERALIZED BICM-T SYSTEM

The model of our generalized BICM-T system is shown inFig. 1. Next we describe the operation of different blocks ofthis transmission scheme.

A. Encoder and Multiplexer

The kc vectors of data bits uj = [uj,1, . . . , uj,Nc] with j =

1, . . . , kc are encoded by a convolutional encoder (ENC) of

MALIK et al.: ANALYSIS AND DESIGN OF GENERALIZED BICM-T SYSTEM 3225

Fig. 1. System model of BICM-T system. A channel encoder is followed by the multiplexer (MUX), the hierarchical M -PAM modulator, the AWGN/GMNchannel, the hierarchical M -PAM de-modulator, the demultiplexer (DEMUX), and the decoder at the receiver side.

Fig. 2. HPAM (M = 8) constellation with BRGC labeling.

rate R = kc/n to yield the vectors of coded bits cl = [cl,1,. . . , cl,Nc

] with l = 1, . . . , n. We denote the convolutional ENCused for transmission by C. A bit level multiplexing unit(MUX) bijectively maps c = [cT1 , . . . , c

Tn]

Tfrom the encoder

output onto c = [cT1 , . . . , cTq ]

Twith ck = [ck,1, . . . , ck,Ns

] andk = 1, . . . , q. Without losing generality, we suppose thatnNc = qNs. The k-th MUX output is linked to the k-th bitposition of a modulator which maps the multiplexed coded bitsck onto symbols using M -ary HQAM constellations labeled bythe BRGC, where q = log2 M .

The detailed description of the MUX can be found in [10],however for the completeness of this paper, we present a briefdefinition here. In general, the MUX can be defined as a one-to-one mapping between the blocks of nNc and qNs bits,i.e., {0, 1}n×Nc ↔ {0, 1}q×Ns . It is defined by using a n×Nc

matrix K, as in [10] with k ∈ {1, . . . , q} and t ∈ {1, . . . , Ns},indicating that the bit cl,t′ is assigned to the k-th MUX’s output(i.e., k-th bit position in the modulator) at time instant t, i.e.,ck,t = cl,t′ . Although this definition of the MUX is completelygeneral, for a practical reason the MUX configurations thatfunction periodically over blocks of nJ bits are considered.With such periodic MUX configurations, K can be expressedas a concatenation of Nc/J matrices Kτ , each having dimen-sions n× J , i.e., K = [K0, . . . ,KNc/J−1], where J is theperiod of the MUX. The elements of Kτ are (k, t+ τnJ/q)where t ∈ {1, . . . , nJ/q} and k ∈ {1, . . . , q}. Without losinggenerality, it can be considered that (Nc mod J) = 0 and that(nJ mod q) = 0.

Example 1: Suppose kc = 1 and J = 3. Consider an 8-aryconstellation (q = 3) and rate R = 1/2 (n = 2) encoder. Forthis case, one possible MUX configuration can be

Kτ =

[(1, 1 + 2τ) (2, 1 + 2τ) (3, 1 + 2τ)(3, 2 + 2τ) (2, 2 + 2τ) (1, 2 + 2τ)

], (1)

which results in the following MUX

K =

[(1, 1) (2, 1) (3, 1) (1, 3) (2, 3) (3, 3) . . .(3, 2) (2, 2) (1, 2) (3, 4) (2, 4) (1, 4) . . .

].

The one-to-one mapping between c and c is then

c =

[c1,1 c1,2 c1,3 c1,4 c1,5 c1,6 . . .c2,1 c2,2 c2,3 c2,4 c2,5 c2,6 . . .

]⇐⇒

c =

⎡⎣ c1,1 c2,3 c1,4 c2,6 . . .c1,2 c2,2 c1,5 c2,5 . . .c1,3 c2,1 c1,6 c2,4 . . .

⎤⎦ . (2)

Since Kτ matrix is merely a permutation of the set {1, . . . ,nJ/q} × {1, . . . , q}, we can create (nJ)! different matricesKτ . However, we can reduce the possible number of MUXconfigurations that we need to investigate since trivial oper-ations can be applied to Kτ without affecting the systemperformance. To ease the notation, for the rest of this paper wewill refer to the matrix Kτ as K.

B. H-PAM Constellations

UEP to the transmitted coded bits can be intentionally intro-duced, e.g., by changing the binary labeling of constellation,using non-equally spaced (NES) constellations or non-equallylikely symbols. Unequal protection in different bit positions ofa transmitted symbol caused by the binary labeling is studiedearlier in order to properly design component encoders for dif-ferent levels in multilevel coding [4]. In this paper, we considera single encoder and use the BRGC hierarchical constellation[25] to offer an UEP to the transmitted coded bits that aregenerated from this encoder.

Due to the BRGC, each symbol in such constellations can berepresented by a superposition of independent real/imaginaryparts, so we consider an equivalent M -ary hierarchical pulseamplitude modulation (HPAM) constellation. The multiplexedcoded bits ck from the MUX output at any time instant t aremapped to a HPAM symbol xt ∈ X = {xt,0, . . . , xt,M−1}using a memoryless mapping M : {0, 1}q → X . The HPAMconstellations (for example see Fig. 2 for 8-HPAM) aredefined by the distances dk with k = 1, . . . , q. In this figure,black circles represent the M signal constellation points andthe triangles/squares are “virtual” symbols that assist the


understanding of the HPAM signal constellation. The bitposition of the binary labeling is denoted by k = 1, . . . , q,where k = 1 represents the most significant bit. At first, thebit value for the position (k = 1) chooses one of the squaresin Fig. 2. Then the bit value for the position (k = 2) choosesone of the two triangles surrounding the previously selectedsquare. Finally, given the bit values for first two bit positions,the bit value for the least significant bit (k = 3) selects oneof the two black symbols surrounding the triangle selectedpreviously which is then transmitted to the receiver. The

constellation parameters are defined as αkΔ= dk+1/d1, with

k = 1, . . . , q − 1. The constellation is normalized for unitenergy, i.e., Es = 1 and other constraints on αk must be addedto restrict the constellation to the BRGC, as in [10].

To simplify the analysis, we use a similar approach to“symmetrize” the asymmetric binary-input soft-output (BISO)channel as used in [2]. The coded bits ck from the MUX outputare randomly inverted before mapping to the HPAM sym-bol, i.e., c = c⊕ s, where the entries of the matrix s = [sT1 ,. . . , sTNs

] ∈ {0, 1}q×Ns , with st = [s1,t, . . . , sq,t], are randomvectors of bits and ⊕ is modulo-2 element-wise addition. Sucha scrambling symmetrizes the BISO channel but it does noteliminate the UEP.

The real part of the signal received from the transmitter isyt = xt + zt where zt is the AWGN or GMN noise. The SNRis defined as γ � Es/N0 = 1/N0. The receiver computes theL-values using the received signal which for the bit position k,is given by [16],

Lk,t = logpYt

(yt|ck,t = 1)

pYt(yt|ck,t = 0)

. (3)

Since ck,t = ck,t ⊕ sk,t, we can write

Lk,t = (−1)sk,tLk,t, (4)

i.e., to reverse the scrambling, the sign of the L-values isaltered by (−1)sk,t . These L-values are reorganized by thedemultiplexer unit (DEMUX) which performs the reverseoperation done by the MUX. The resulting vectors of L-valuesare sent to a soft input Viterbi decoder [26] which estimates thetransmitted information bits from the transmitter.

C. Decoder and the Decoding Errors

The MUX determines the correspondence between the en-coder’s output and the modulator’s input. A binary codewordfrom the encoder’s output is converted to an equivalent code-word at the MUX’s output, i.e., the modulator’s input. There-fore we define the combination of the ENC and the MUX asan equivalent encoder (cf. Fig. 1). Similarly on the receiverside, the DEMUX and the DEC make an equivalent decoder(cf. Fig. 1). The decoder makes a false decision if instead ofthe transmitted equivalent codeword c, it detects an equivalentcodeword c′. The probability of this event is so called pairwiseerror probability (PEP) and is given by [16]

PEP(c → c′) = Pr

{Ns∑t=1

(e1,tL1,t + · · ·+ eq,tLq,t) ≥ 0

},

(5)

where we represent elements of the “error” codeword, e =c′ − c (where “−” is a modulo-two operation) as ek,t. We notethat q consecutive multiplexed coded bits at each time instant tare transmitted using the same symbol xt over the same channeland noise realization. Therefore, in the PEP expression (5),each sequence L1,L2, . . . ,LNs

contains q dependent L-values.As such, a different approach as compared to BICM-S, whereL-values are independent because of the random interleaver,must be adopted to calculate the PEP [16]. In the next section,we describe how to evaluate the performance of our BICM-Tsystem.

III. PERFORMANCE EVALUATION

As the channel is symmetric due to scrambling, we assumethat the transmitter transmitted a codeword with all zeros. LetE be the set of equivalent codewords corresponding to the pathsof the code diverging at time instant t from the zero-state,and remerging after T stages. The equivalent codeword that

belongs to the set E is denoted as eΔ= [eT1 , . . . , e

TT ] where et =

[e1,t, . . . , eq,t]. Let Λe be the accumulated metric associated tothe equivalent codeword e. This metric is a sum of independentrandom variables, i.e.,

ΛeΔ= Λ(t) + Λ(t+1) + Λ(t+2) + · · · , (6)

where Λ(t) =∑q

k=1 ek,tLk,t. Therefore, we can write,

Λ(t) ≡ Λ(et, st) =

q∑k=1

(−1)sk,tek,tLk,t. (7)

Since Lk,t are random variables that depend on k and xt, forthe relevant cases in (7), we need (M − 1) PDFs pΛπi

(λ|cπi,t)where πi ∈ Π, i = 1, . . . ,M − 1 and Π is the power set ofthe set {1, . . . , q} excluding the empty set.4 The vector ofmultiplexed coded bits whose indices belong to πi is given bycπi,t. We note that pΛπi

(λ|cπi,t) is not always conditioned on asingle coded bit as for the BICM-S system. In fact for differentπi, pΛπi

(λ|cπi,t) is conditioned on 1, 2, . . . or q bits. To clarifyour notations, consider the following example.

Example 2: Assuming 8-ary constellation (q = 3) and using(7), we can express the metrics as

Λt =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if et = [0, 0, 0](−1)s1,tL1,t, if et = [1, 0, 0]

(−1)s2,tL2,t, if et = [0, 1, 0]

(−1)s3,tL3,t, if et = [0, 0, 1]

(−1)s1,tL1,t + (−1)s2,tL2,t, if et = [1, 1, 0]

(−1)s2,tL2,t + (−1)s3,tL3,t, if et = [0, 1, 1]

(−1)s1,tL1,t + (−1)s3,tL3,t, if et = [1, 0, 1]∑3k=1(−1)sk,tLk,t if et = [1, 1, 1].

(8)

For this example, Π = {{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1,2, 3}} and for the relevant cases in (8), we need seven PDFs,i.e., pΛπi

(λ|cπi,t) where πi ∈ Π, i = 1, . . . , 7. We note that theconditional PDF pΛπ7

(λ|cπ7,t) is conditioned on three codedbits cπ7,t = [cπ1,t, cπ2,t, cπ3,t] rather than a single coded bit.

4We do not need to include {} ∈ Π as it gives Λt = 0 in (7).


From (6), and due to the independence of each value ofΛ(et, st) in (7), the PEP can be expressed as5

PEP(we,Π) =

∞∫0

pΛπ1(λ|cπ1,t = 0)∗we,π1 ∗ · · ·

∗ pΛπM−1

(λ|cπM−1,t = 0

)∗we,πM−1 dλ, (9)

where, we,Π = {we,π1, . . . , we,πM−1

}, we,πirepresents the

number of columns in e where the bit positions with indicesbelonging to πi have a bit value equal to 1. Then, the UB canbe expressed as [16],

UB =1

kc

∑wΠ

PEP(wΠ)∑

e∈CwΠ

dH(ue)

=1

kc

∑wΠ

PEP(wΠ)βCwΠ,K , (10)

where CwΠ

Δ={e ∈ E : wπi

=we,πi, i=1, . . . ,M−1}, dH(ue)

is the Hamming weight of the input sequence ue associatedwith the equivalent codeword e and βC

wΠ,K =∑

e∈CwΠdH(ue)

is an equivalent weight distribution spectrum (EWDS) of thecode C.

The EWDS of the code C considers the generalized weight[wπ1

, . . . , wπM−1] of codewords and the MUX configuration

matrix K. The correspondence between the encoder’s output cand the MUX output, i.e., the modulator’s input c is determinedby the matrix K, and by the instantaneous time at which thesequence diverge from an all-zero state. However, only J timeinstants must be considered because of the periodic structure ofK. Therefore, we can express the EWDS as

βCwΠ,K =

1

J

J∑n=1

βCwΠ,K(n), (11)

where the EWDS is given by βCwΠ,K(n) when the decoder

diverges at time t+ n with arbitrary t.Example 3 (EWDS of the Code (5, 7)8): Consider the con-

straint length K = 3 convolutional code with polynomial gen-erators (5, 7)8. The free distance of the code is dfreeH =5,and βC

5 = 1, i.e., there is one divergent path at Hammingdistance five from the all-zero codeword, and the inputHamming weight of the path is one. Consider the MUX inExample 1 with period J = 3 and 8-ary (q = 3) constel-lation. The J = 3 possible input sequences with Hammingweight one [. . . , 0, 1, 0, 0, 0, 0, . . .], [. . . , 0, 0, 1, 0, 0, 0, . . .] and[. . . , 0, 0, 0, 1, 0, 0, . . .] which result in the following c matrices,

c(1) =

[. . . 1 0 1 0 0 . . .. . . 1 1 1 0 0 . . .

],

c(2) =

[. . . 0 1 0 1 0 . . .. . . 0 1 1 1 0 . . .

],

c(3) =

[. . . 0 0 1 0 1 . . .. . . 0 0 1 1 1 . . .

], (12)

5For notational convenience, we used cπi = 0, where 0 is a vector of samesize as cπi .

which by using (1) yield

c(1) =

⎡⎣ . . . 1 1 0 0 . . .. . . 0 1 0 0 . . .. . . 1 1 0 0 . . .

⎤⎦ ,

c(2) =

⎡⎣ . . . 0 1 1 1 . . .. . . 1 1 0 0 . . .. . . 0 0 0 0 . . .

⎤⎦ ,

c(3) =

⎡⎣ . . . 0 1 0 1 . . .. . . 0 0 1 1 . . .. . . 1 0 0 0 . . .

⎤⎦ . (13)

If we consider only the error event at the minimum Hammingdistance, the final EWDS given by (11) is obtained by comput-ing the number of columns of c(n) with n = 1, 2, 3, where thebit positions with indices belonging to πi, with i = 1, . . . , 7,have bit value equal to 1, i.e.,

βCwΠ,K =

⎧⎨⎩

13 , if wΠ = {0, 0, 0, 0, 0, 1, 1}13 , if wΠ = {0, 1, 2, 1, 0, 0, 0}13 , if wΠ = {1, 1, 1, 1, 0, 0, 0}.

The spectrum βCwΠ,K can be numerically calculated using

a breadth first search algorithm [27]. Clearly, the spectrummust be truncated so that only diverging sequences with totalHamming weight w1 + · · ·+ wq ≤ w are considered.

As we mentioned earlier that since the MUX configurationmatrix K is merely a permutation of the set {1, . . . , nJ/q} ×{1, . . . , q}, we can create (nJ)! different matrices K. However,in terms of the performance of the system, this number can bereduced as some of these matrices result in an identical EWDSin (11), and therefore, do not modify the UBs developed inSections IV and V.

IV. PERFORMANCE IN AWGN CHANNEL

In this section we analyze the performance of our BICM-Tsystem in the AWGN channel. In order to calculate the PEP in(9), we need to compute the conditional PDFs pΛπi

(λ|cπi,t) forπi ∈ Π. We develop the approximations for these PDFs usingthe so called zero-crossing model [24].

This L-values in (3) can be approximated using max-logsimplification [26] by

Lk,t(yt|st)≈γ

[min

x∈Xk,0

(yt − x)2− minx∈Xk,1

(yt − x)2], (14)

where Xk,b represents the symbols with a bit value b at bitposition k. The L-values depend on the transmitted symbolxt, however, since a codeword with all zeros is transmitted([c1,t, . . . , cq,t] = [0, . . . , 0]), xt is completely determined byst. As such, we use Lk,t(yt|st) to emphasize that the L-valuesdepend on the received signal yt and the scrambling sequence st.

For a given transmitted symbol xt, the received signal ytis a Gaussian random variable with mean xt and varianceN0/2. Therefore each metric Λ(et, st) in (7) is a sum of piece-wise Gaussian functions. This piecewise linear relationship wasshown in [16, Fig. 3] for uniform 4-PAM constellation. Tofacilitate the analysis, we use the zero-crossing approximation


TABLE IVALUES OF a(et, st) AND b(et, st) FOR HPAM (M = 8)

TABLE IIVALUES OF μ(et, st) AND σ2(et, st) FOR HPAM (M = 8)

of the metrics which replaces all the Gaussian pieces by oneGaussian function. According to this approximation,

Λ(yt|et, st) ≈ a(et, st)yt + b(et, st), (15)

where a and b are determined by the linear piece crossingx-axis and closest to the symbol xt. Using (15), we can modelthe conditional metrics as Gaussian random variables whosemean and variance depend on γ, et and st, i.e.,

pΛ(λ|st) = ψ(λ;μ(et, st), σ

2(et, st)), (16)

where the mean value and variance are given by

μ(st) =xta(et, st) + b(et, st), (17)

σ2(et, st) = [a(et, st)]2 N0

2. (18)

The generic closed-form expressions for the PDFs of themetrics Λ(et, st) in (7) for a general M -ary signal modulationis difficult to develop. However, these PDFs for different orderHQAM can be obtained on case by case basis. In what follows,

we develop the UB on the BER of BICM-T system in terms ofconstellation distance parameters and the MUX configurationfor 64-HQAM and then the UB for 16-HQAM is developedas a special case of 64-HQAM. The UB for other HQAMconstellations can be developed using the same approach.

A. 64-HQAM

1) PDF of Λ(et, st): In this section, we consider an8-ary constellation (q = 3) which is defined by two constel-lation parameters, i.e., α1 and α2. Because there are threebit positions, we need to compute seven conditional PDFs, asexplained in Example 2, i.e., pΛπi

(λ|cπi,t) where πi ∈ Π, i =1, . . . , 7. Since the metrics Λ(et, st) are piecewise linear, wepresent the values of slope a(et, st) and intercept b(et, st) interms of constellation parameters required for the zero-crossingapproximation (15) in Table I for all possible values of et andst. In Table II we present corresponding means and variancesexpressed in (17), (18) where for convenience we have defined

=1√

1 + α21 + α2

2

. (19)


By averaging (16) over the scrambling sequence st, we obtainthe unconditional PDFs of the metrics Λ(et, st) in (7) given bythe following equation

pΛ(λ)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

14

∑4p=1 ψ

(λ;μp,γ,α1,α2

, σ2p,γ,α1,α2

), et = [1, 0, 0]

12

∑6p=5 ψ

(λ;μp,γ,α1,α2

, σ2p,γ,α1,α2

), et = [0, 1, 0]

ψ(λ;μ7,γ,α1,α2

, σ27,γ,α1,α2

), et = [0, 0, 1]

12

∑9p=8 ψ

(λ;μp,γ,α1,α2

, σ2p,γ,α1,α2

), et = [1, 1, 0]

ψ(λ;μ10,γ,α1,α2

, σ210,γ,α1,α2

), et = [0, 1, 1]

14

(∑12p=11 ψ

(λ;μp,γ,α1,α2

, σ2p,γ,α1,α2

)+2ψ

(λ;μ13,γ,α1,α2

, σ213,γ,α1,α2

)), et = [1, 0, 1]

14

∑17p=14 ψ

(λ;μp,γ,α1,α2

, σ2p,γ,α1,α2

), et = [1, 1, 1],

(20)

where the values ofμp,γ,α1,α2and σ2

p,γ,α1,α2inψ(λ;μp,γ,α1,α2

,σ2p,γ,α1,α2

) for different values of p are summarized in Table III.2) UB on the BER: Using (20), we can derive the expres-

sions for elements in the integration defining the PEP as (21),shown at the bottom of the page. Using (21) in (9), we get theexpressions for the PEP as follows

PEP(wΠ) =

∞∫0

(1

2

)wT∑jΠ

∏πi∈Π

(wπi

jπi

)ψ(λ;μwΣ

, σ2wΣ

)dλ.

(22)

TABLE IIIVALUES OF μp,γ,α1,α2 AND σ2

p,γ,α1,α2IN (20)

FOR DIFFERENT VALUES OF p

Using (22) in (10), we get the following expression for the UB

UB =1

kc

∑wΠ

βCwΠ

(1

2

)wT∑jΠ

∏πi∈Π

(wπi

jπi

)Q

(√μ2wΣ

σ2wΣ

).

(23)

where jπ1∈ W4(wπ1

), jπ2∈ W2(wπ2

), jπ3∈ W1(wπ3

),jπ4

∈ W2(wπ4), jπ5

∈ W1(wπ5), jπ6

∈ W3(wπ6), jπ7

∈W4(wπ7

), and wT, μwΣ, and σ2

wΣ, respectively, are given in

(24)–(26), shown at the bottom of the next page.

pΛπ1(λ|cπ1,t = 0)∗wπ1 =

(1

4

)wπ1∑jπ1

(wπ1

jπ1

)ψ

(λ;

4∑i=1

μi,γ,α1,α2jπ1,i, σ

21,γ,α1,α2

wπ1

)

pΛπ2(λ|cπ2,t = 0)∗wπ2 =

(1

2

)wπ2∑jπ2

(wπ2

jπ2

)ψ

(λ;

6∑i=5

μi,γ,α1,α2jπ2,i−4, σ

25,γ,α1,α2

wπ2

)

pΛπ3(λ|cπ3,t = 0)∗wπ3 =ψ

(λ;μ7,γ,α1,α2,α2

wπ3, σ2

7,γ,α1,α2wπ3

)pΛπ4

(λ|cπ4,t = 0)∗wπ4 =

(1

2

)wπ4∑jπ4

(wπ4

jπ4

)ψ

(λ;

9∑i=8

μi,γ,α1,α2jπ4,i−7, σ

28,γ,α1,α2

wπ4

)

pΛπ5(λ|cπ5,t = 0)∗wπ5 =ψ

(λ;μ10,γ,α1,α2,α2

wπ5, σ2

10,γ,α1,α2wπ5

)pΛπ6

(λ|cπ6,t = 0)∗wπ6 =

(1

4

)wπ6∑jπ6

(wπ6

jπ6

)

· ψ(λ;

12∑i=11

(μi,γ,α1,α2jπ6,i−10 + 2μ13,γ,α1,α2

jπ6,3) , σ211,γ,α1,α2

wπ6

)

pΛπ7(λ|cπ7,t = 0)∗wπ7 =

(1

4

)wπ7∑jπ7

(wπ7

jπ7

)

· ψ(λ;

17∑i=14

μi,γ,α1,α2jπ7,i−13,

15∑m=14

σ2m,γ,α1,α2

(jπ7,m−13 + jπ7,m−12)

)(21)


TABLE IVVALUES OF μp,γ,α1 AND σp,γ,α1 in (27) FOR DIFFERENT VALUES OF p

B. 16-HQAM

For 16-HQAM, there is a single constellation parameter, i.e.,α1. In this case Π = {{1}, {2}, {1, 2}}. Therefore, we need tocompute three PDFs, i.e., pΛπi

(λ|cπi,t), πi ∈ Π, i = 1, . . . , 3.to calculate the PEP in (9). Given the fact that a 16-HQAMconstellation is a special case of 64-HQAM constellation withα2 = 0, the means (μp,γ,α1

) and variances (σ2p,γ,α1

) of therelevant three PDFs can be obtained from Table III by settingα2 = 0 that are shown in Table IV where = 1/

√1 + α2

1.The unconditional PDF of the metrics Λ(et, st) in (7) for16-HQAM constellations can be expressed as

pΛ(λ) =

⎧⎪⎨⎪⎩

12

∑2p=1 ψ

(λ;μp,γ,α1

, σ2p,γ,α1

), if et = [1, 0]

ψ(λ;μ3,γ,α1

, σ23,γ,α1

), if et = [0, 1]

ψ(λ;μ4,γ,α1

, σ24,γ,α1

), if et = [1, 1].

(27)

Now using the similar procedure in Section IV-A2, the PEPfor 16-HQAM can be written as

PEP(wΠ)=

∞∫0

(1

2

)w1wπ1∑j=0

(wπ1

j

)ψ(λ;μΠ,j , σ

2Π)dλ, (28)

where

μΠ,j=−4γ(1−α1)

2

1+α21

(wπ1

+α21wπ2

(1−α1)2+

wπ3

(1−α1)2+

2α1j

1−α1

),

(29)

σ2Π=8γ

(1−α1)2

1+α21

(wπ1

+α21

(1−α1)2wπ2

+1

(1−α1)2wπ3

)(30)

Using (29), (30) in (28), and (10), we get the following expres-sion for the UB

UB=1

kc

∑wΠ

βCwΠ,K

(1

2

)wπ1wπ1∑j=0

(wπ1

j

)

·Q

⎛⎜⎜⎝√√√√√(wπ1

+α2

1

(1−α1)2wπ2

+wπ3

(1−α1)2+ 2jα1

1−α1

)2wπ1

+α2

1

(1−α1)2wπ2

+ 1(1−α1)2

wπ3

2γ(1−α1)2

1+α21

⎞⎟⎟⎠.

(31)

Note that for α1 = 1/2, (31) is identical to the UB expressiondeveloped in [16] for the ES 4-PAM constellation.

V. PERFORMANCE IN GMN CHANNEL

In this section we analyze the performance of BICM-Tsystem in the GMN channel. In particular, we develop an UB onthe BER of BICM-T system in terms of constellation distanceparameters and the MUX configuration for the case presentedin Section IV-B. For simplicity of notation and succinct ex-planation of the procedure to develop the PDF of metrics thatare relevant to characterize decoding error rate performance ofBICM-T system in the GMN channel6, we only present theanalysis for the particular case presented in Section IV-B. TheUB for other cases can be developed using the same approach.The real signal received from the transmitter is yt = xt + ztwhere zt is the GMN sample at time instant t. We considerthat the GMN samples are uncorrelated over time t. The noisesamples are distributed according to the zero-mean Gaussianmixture distribution [21],

pZ(z) =

N∑n=1

εn2πσ2

zn

exp

(− |z|22σ2

zn

), (32)

6We assumed that the system employs the max-log approximation basedEuclidean distance bit metric at the demapper which is then supplied to thedecoder for the binary code in order to estimate the binary transmitted data. Thisis an instance of mismatched decoding [28] in the presence of GMN, however,it is widely used in the literature (e.g., see [21]) with negligible performanceloss.

wT =2 (wπ1+ wπ6

+ wπ7) + wπ2

+ wπ4(24)

μwΣ=

(4∑

i=1

μi,γ,α1,α2jπ1,i +

6∑i=5

μi,γ,α1,α2jπ2,i−4 + μ7,γ,α1,α2

wπ3+

9∑i=8

μi,γ,α1,α2jπ4,i−7

+

12∑i=11

μi,γ,α1,α2jπ6,i−10 + 2μ13,γ,α1,α2

jπ6,3

17∑i=14

μi,γ,α1,α2jπ7,i−13

)(25)

σ2wΣ

=σ21,γ,α1,α2

wπ1+ σ2

5,γ,α1,α2wπ2

+ σ27,γ,α1,α2

wπ3+ σ2

8,γ,α1,α2wπ4

+ σ210,γ,α1,α2

wπ5

+ σ211,γ,α1,α2

wπ6+ σ2

14,γ,α1,α2(jπ7,1 + jπ7,3) + σ2

15,γ,α1,α2(jπ7,2 + jπ7,4) (26)


where,

N∑n=1

εn =1, (33)

N∑n=1

εnσ2zn =

1

2γ, (34)

where γ is the SNR at the receiver and σ2zn is the variance of

n-th noise component. Instead of directly computing the PDFspΛπi

(λ|cπi,t) in (9), we first compute the PDFs conditioned ona given noise component. We introduce an auxiliary randomvariable ξt which identifies the n-th noise component of thePDF (32) to which zt belongs. The distribution of this noisestate variable ξt is Pr{ξt = n} = εn. The PDF of a componentnoise random variable Zξt is [21]

pZn(z) =1

2πσ2zn

exp

(− |z|22σ2

zn

). (35)

The PDFs pΛπi(λ|cπi,t) in (9) can be considered as a weighted

sum of the PDFs pΛπi|n(λ|cπi,t) conditioned on the state of the

GMN ξt = n,

pΛπi(λ|cπi,t) =

N∑n=1

εnpΛπi|n (λ|cπi,t) . (36)

For a given transmitted symbol xt and the n-th noise compo-nent, the received signal yt is a Gaussian random variable withmean xt and variance σ2

zn .For the n-th noise component, the PDFs pΛ|n(λ), is same as

pΛ(λ) given in (27) for the AWGN channel. Therefore, using(36) and (27), elements in the integration in (9) can be derivedas (37), shown at the bottom of the page. Using (37) and (10),the expression for the UB on the BER can be derived as

UB =1

kc

∞∑l=dfree

∑wπ1

+wπ2+2wπ3

=l

βCwΠ,K

∑rΠ

g(rΠ)Q (h(rΠ)) ,

(38)

where g(rΠ) and h(rΠ) are given in (39), shown at the bottomof the page.

VI. NUMERICAL RESULTS

Numerical results are presented in this section to illustratethe performance of generalized BICM-T system with optimizedsystem modules. Blocks of 10,000 information bits are encodedby the convolutional codes with rate R, constraint length K atthe transmitter. A soft-input Viterbi algorithm without mem-ory truncation is used for decoding at the receiver. The UBin general depends on the SNR (γ), the signal constellationparameters (α1, . . . , αq), the MUX configuration (K) and

[pΛπ1

(λ|cπ1,t = 0)]∗wπ1 =

[N∑

n=1

εn2

[ψ

(λ;−1 + α1

1− α1ρn, 2ρn

)+ ψ(λ;−ρn, 2ρn)

]]∗wπ1

=∑

rπ1∈W2N (wπ1

)

(wπ1

rπ1

)

· ψ(λ,−

N∑n=1

(rπ1,2n−1

1 + α1

1− α1+ rπ1,2n

)ρn,

N∑n=1

2 (rπ1,2n−1 + rπ1,2n) ρn

)N∏

n=1

(εn2

)(rπ1,2n−1+rπ1,2n)

[pΛπ2

(λ|cπ2,t = 0)]∗wπ2 =

[N∑

n=1

εnψ

(λ;− α2

1

(1− α1)2ρn, 2

α21

(1− α1)2ρn

)]∗wπ2

=∑

rπ2∈WN (w2)

(wπ2

rπ2

)ψ

(λ,−

N∑n=1

rπ2,nα21

(1− α1)2ρn,

N∑n=1

2rπ2,nα21

(1− α1)2ρn

)N∏

n=1

εrπ2,nn

[pΛπ3

(λ|cπ3,t = 0)]∗wπ3 =

[N∑

n=1

εnψ

(λ;− 1

(1− α1)2ρn, 2

1

(1− α1)2ρn

)]∗wπ3

=∑

rπ3∈WN (wπ3

)

(wπ3

rπ3

)ψ

(λ,−

N∑n=1

rπ3,n1

(1− α1)2ρn,

N∑n=1

2rπ3,n1

(1− α1)2ρn

)N∏

n=1

εnrπ3,n (37)

g(rΠ) =∏πi∈Π

(wπi

rπi

) N∏n=1

(εn2

)(rπ1,2n−1+rπ1,2n) N∏n=1

ε(rπ2,n+rπ3,n)n

h(rΠ) =

∑Nn=1 −

(rπ1,2n−1

(1− α2

1

)+ rπ1,2n(1− α1)

2 + rπ2,nα21 + rπ3,n

)ρn[∑N

n=1 2 ((rπ1,2n−1 + rπ1,2n) (1− α1)2 + rπ2,nα21 + rπ3,n) ρn

] 12

(39)


Fig. 3. BER performance for spectral efficiency 1 bit/dimension using 16-HQAM. The simulations are shown by markers and the UB with solid lines.

the code (C). Therefore, for an optimal system, C, K and(α1, . . . , αq) should be jointly optimized for each value ofSNR. The optimum values for a given SNR are defined asC∗(γ), α∗

1(γ), . . . , α∗q(γ),K

∗(γ), i.e.,[C∗(γ), α∗

1(γ), . . . , α∗q(γ),K

∗(γ)]

= argminC,α1,...αq,K

{UB(C, α1, . . . , αq,K)} (40)

where for a given γ, the UB is a function defined by the code C,constellation parameters α1, . . . , αq and the MUX configura-tion matrix K. Since the UB function is potentially non-convex,in order to find the optimal code, the MUX configuration andconstellation parameters for a given SNR, modulation orderand constraint length, an exhaustive search can be carriedover the allowable range of constellation parameters, possibleMUX configurations and codes. This exhaustive search basedoptimization can be performed offline and the results can betabulated to be used in practice. All UB computations arecarried out considering a truncated spectrum of the code, i.e.,wπi

≤ 40 ∀πi ∈ Π. Using more number of terms does notaffect the UB in the SNR range that provides a BER of about10−6 however the UB becomes less tight at the low SNR region.

As the value of the MUX period, J increases, the numberof possible MUX configurations increases. So, for the sakeof simplicity of exhaustive search, we have used the shortestpossible value of J for Example 5 (J = 1) and Example 6(J = 3) whereas for Example 4 we have used J = 2.

A. Results for AWGN Channels

1) Spectral Efficiency 1 Bit/Dimension: We present the re-sults for this case using two examples: 16-HQAM with rate1/2 code and 64-HQAM with rate 1/3 code.

Example 4: In this example, we consider rate R = 1/2 codestwo constraint lengths, i.e., constraint lengths K = 3 and K =4. Rate 1/2 codes together with a 4-ary constellation give aspectral efficiency of 1 bit/dimension. In this case, α1 is theonly constellation parameter. A MUX with period J = 2 isconsidered.

A joint optimization of the UB was performed over C,K and α1 for the SNR range which gives the UB in (31)below 10−3. For example, for K = 3, the optimal constellationparameter for γ[dB] ∈ {6, 6.5, . . . , 9} is

α∗1(γ) = [.49, .46, .44, 0.42, 0.40, 0.38, 0.37] (41)

and the generator polynomials for C∗ are found to be (5, 7)8which are same as the ODS code [29]. For K = 4, the generatorpolynomials for C∗ are found to be (13, 17)8 which are sameas the AOCC code found in [16]. The optimum MUX for thetwo constraint lengths considered in this example are givenin Table V.

The BER performance of the BICM-T system with optimizedcode, constellation parameters and the MUX configuration ispresented in Fig. 3. In this figure, we also present the BERperformance of the optimal HQAM-BICM system [10], theBICM-T system with an ES signal constellation [16] and thetraditional BICM-S system [2] that use ODS codes, i.e., (5, 7)8for K = 3 and (15, 17)8 for K = 4 [29]. From Fig. 3, we canobserve that for K = 3 when compared to the non-optimizedBICM-T system [16] that uses an ES 16-QAM and K∗, theoptimized BICM-T system offers gains up to 0.2 dB for a targetBER of 10−6. From this figure, we can observe that for K = 4,this gain is about 0.3 dB. For K = 4, it is also obvious that thejointly optimized BICM-T system with the optimal code, theMUX and constellation parameter provides an additional gainover the BICM-T system that uses the ODS code, the optimalMUX and the optimal constellation parameter.

Example 5: In this example, we consider the 8-ary constella-tion with a rate R = 1/3 code that gives a spectral efficiency of1 bit/dimension. In this case there are two constellation parame-ters, i.e., α1 and α2. We consider the MUX configurations withthe shortest possible period, i.e., J = 1, for which there will bea total of six different MUX configurations given below:

K(1) =

⎡⎣ (1, 1)(2, 1)(3, 1)

⎤⎦ , K(2) =

⎡⎣ (1, 1)(3, 1)(2, 1)

⎤⎦ , K(3) =

⎡⎣ (2, 1)(1, 1)(3, 1)

⎤⎦

K(4) =

⎡⎣ (2, 1)(3, 1)(1, 1)

⎤⎦ , K(4) =

⎡⎣ (3, 1)(1, 1)(2, 1)

⎤⎦ , K(6) =

⎡⎣ (3, 1)(2, 1)(1, 1)

⎤⎦ .(42)

For sake of simplicity of exhaustive search, in this exampleonly the MUX configuration and the constellation parametersare optimized for a particular convolutional code, i.e., the ODScode with generator polynomial (5, 7, 7)8 and constraint lengthK = 3. Joint optimization of the code, the MUX and theconstellation parameters can provide additional gain as shownby the results in Example 4.

For the SNR γ[dB] ∈ {6, 7, . . . , 9} (which give a UB in(23) below 10−3), we obtained the optimal MUX configura-tion K∗(γ) and constellation parameters (α∗

1(γ), α∗2(γ)) using

the UB (23) and an exhaustive search for the given code.The optimal MUX for all γ[dB] ∈ {6, 7, . . . , 9} was foundto be K∗(γ) = K(1). Moreover, the optimal constellation


Fig. 4. BER performance for spectral efficiency 1 bit/dimension using64-HQAM. The simulations are shown by markers and the UB with solid lines.

parameters do not change significantly for all γ[dB] ∈{6, 7, . . . , 9}. The obtained values of optimal constellationparameters are

(α∗1(γ), α

∗2(γ)) ≈ (0.47, 0.12). (43)

The BER performance of the considered system with theoptimized constellation parameters and the MUX configura-tion is presented in Fig. 4. Here we also present the BERperformance of the optimal HQAM-BICM system of [10],non-optimized BICM-T system that uses an ES QAM con-stellation and the traditional single interleaver BICM system[2]. From this figure, we can observe that when comparedto the non-optimized BICM-T system, the optimized systemoffers gains up to 0.9 dB for a target BER of 10−6. FromFig. 4, it is also obvious that the optimized BICM-T can gainsup to 2.6 dB over the optimized BICM system with multipleinterleavers [10].

2) Spectral Efficiency 1.5 Bits/Dimension:Example 6: In this example, we consider rate R = 1/2 codes

with two constraint lengths, i.e., constraint lengths K = 3 andK = 8. Rate 1/2 codes together with an 8-ary constellation givea spectral efficiency of 1.5 bits/dimension. In this case thereare two constellation parameters, i.e., α1 and α2. We considerthe MUX configurations with the shortest possible period, i.e.,J = 3, for which there will be a total of 156 different MUXconfigurations.

In this example we have used the so-called asymptoti-cally optimal convolutional code (AOCC) [16] for both con-straint lengths. For the SNR γ[dB] ∈ {8, 9, . . . , 14} (whichgive the UB in (23) below 10−3), we obtained the opti-mal MUX configuration K∗(γ) and constellation parameters(α∗

1(γ), α∗2(γ)) using the UB (23) and an exhaustive search.

The optimal MUX for all γ[dB] ∈ {8, 9, . . . , 14} and the twoconstraint lengths considered in this example are given inTable V. Moreover, the optimal constellation parameters do notchange significantly for all γ[dB] ∈ {8, 9, . . . , 14}. For exam-ple, for K = 3, the obtained values of optimal constellationparameters are

(α∗1(γ), α

∗2(γ)) ≈ (0.46, 0.01). (44)

The BER performance of the considered system with theoptimized constellation parameters and the MUX configurationis presented in Fig. 5. Here we also present the BER perfor-mance of the non-optimized BICM-T that uses an ES QAMconstellation, the optimal HQAM-BICM system of [10], andthe traditional single interleaver BICM system [2]. For thiscase, for K = 3 when compared to the non-optimized BICM-Tsystem, the considered system offers gains up to 1.5 dBfor a target BER of 10−6. For K = 8, this gain is about 1 dB.From this figure forK = 8we can also observe that the BICM-Tsystem that uses the AOCC, the optimal MUX and constella-tion parameters provides an additional gain over the BICM-Tsystem that uses ODS code, optimal MUX and constellationparameters.

3) Optimal MUX Configurations for Different Rate R = 1/2AOCC Codes: In this section, we consider rate R = 1/2 AOCCcodes [16] with different constraint lengths and present theMUX configurations K∗ in BICM-T that results in the opti-mal BER performance. We also present the amount of gainsachieved over traditional BICM-S and BICM-T (ES-QAM,K∗) systems, that use ODS codes for a target BER of ≈ 10−6

in Table V. We present the results for two particular cases of16-QAM and 64-QAM which together with various rate R =1/2 codes that result in two practically relevant spectral effi-ciencies of 1 bit/dimension and 1.5 bits/dimension respectively.

B. Results for GMN Channels

We consider ε-mixture noise which is an important instanceof the general GMN with two noise components. This is awell used GMN model in the literature (see [30] and referencestherein for details). The first component represents an impulsivenoise due to some ambient phenomenon, such as man-madeelectromagnetic interference, radar clutter and sudden switch-ing while the second component accounts for the backgroundthermal noise. The ε-mixture noise parameters can be expressedas [21],

ε1 = ε, ε2 = 1− ε,

σz1 =

√κ√

2γ(1 + κε− ε),

σz2 =1√

2γ(1 + κε− ε), (45)

where κ = σ2z1/σ2

z2 is a measure for the strength of the impul-sive component compared to the thermal noise. In the follow-ing, we specify the parameters of ε-mixture noise by (ε, κ).

We consider a rate R = 1/2 convolutional codes with16-HQAM which has only one constellation parameter, i.e.,α1. A MUX with period J = 2 is considered and a jointoptimization was performed over K and α1 for a given code.

In order to answer the question of using or removing the in-terleaver in BICM system in the GMN channel, we compare theperformance of a BICM-T system that uses an ES constellation

and K∗ =

[(2, 1) (2, 2)(1, 1) (1, 2)

]with traditional BICM-S system

for different values of κ and ε = 0.1 for two different codes;one is (5, 7)8 code with K = 3 and the other is (247, 371)8 with


TABLE VOPTIMAL MULTIPLEXER CONFIGURATION AND THE GAINS COMPARED TO BICM-S (G1) AND THE BICM-T (ES-QAM, K∗) (G2) SYSTEMS USING

ODS CODES, FOR DIFFERENT RATE R = 1/2 AOCC CODES [16] WITH 16-QAM AND 64-QAM FOR A TARGET BER OF 10−6.ALL GENERATOR POLYNOMIALS ARE IN OCTAL NOTATION

Fig. 5. BER performance for spectral efficiency 1.5 bits/dimension using64-HQAM. The simulations are shown by markers and the UB with lines.

Fig. 6. Comparison of UB for BICM-T with BICM-S for different values κ,γ and ε = 0.1. The simulations are shown by markers and the UB with solidlines.

K = 8. In particular in Fig. 6, we present the UB versus κ forgiven values of the SNR and using an ES signal constellation. Itis interesting to note that depending on the value of κ, BICM-Tor BICM-S should be used. It is obvious from this figure that

Fig. 7. BER performance of BICM-T and BICM-S for κ = 10 and 100 andε = 0.1. The simulations are shown by markers and the UB with solid lines.

the threshold value of κ depends on the code and the value ofthe SNR. The results show that for (5, 7)8 code with K = 3 ifκ < 34, then the performance of BICM-T is better as comparedto BICM-S system. Otherwise, BICM-S outperforms BICM-Tsystem. For the (247, 371)8 code with K = 8, the result showsthat if κ < 48, then the performance of BICM-T is better ascompared to BICM-S system.

In Fig. 7 we compare the BER performance of BICM-Tthat uses an optimal MUX and constellation parameters withBICM-S system for (5, 7)8 code. In this figure we use κ =10 and κ = 100. Fig. 7 illustrates that for κ = 10, the per-formance of BICM-T using optimal constellation α∗

1(γ) =[.49, .48, .47, 0.46, 0.46, 0.45] for γ[dB] ∈ {7, 8, . . . , 12} and

multiplexer K∗ =

[(2, 1) (2, 2)(1, 1) (1, 2)

]is much better as com-

pared to BICM-S. On the other hand, the figure shows that forκ = 100, the performance of traditional BICM-S is better ascompared to BICM-T.

VII. CONCLUSION

In this paper, we analyzed a generalized BICM-T systemthat uses HQAM signal constellations in conjunction with a


bit-level multiplexer in the AWGN and GMN channels. Wedeveloped the UB on the BER performance of our generalizedBICM-T system that provides a design tool to analyze andoptimize code, modulation parameters and multiplexer. As suchone can efficiently choose different modules of such systemto minimize the BER. Presented numerical results showed thatthe gains achieved by an optimized BICM-T system depend onthe code and modulation order. The presented numerical resultsfor two practically relevant spectral efficiencies 1 bit/dimensionand 1.5 bits/dimension showed that the optimized system canoffer gains up to 0.9 dB and 1.7 dB, respectively over the non-optimized BICM-T system in AWGN channel for a BER of10−6. The results for the GMN channel interestingly revealedthat if the strength of impulsive noise component in the GMNis below a certain threshold level, then BICM-T system per-forms significantly better as compared to the traditional BICMsystem. The threshold depends on the code and the value of theSNR. The results presented for the GMN channel showed thatan optimized BICM-T system can result in gains up to 1.5 dBfor a target BER of 10−6.

In this work, we considered a non-iterative decoding basedreceiver. However, the MUX and constellation design for BICMwith iterative decoding (BICM-ID) system in the AWGN chan-nel and performance comparison of an optimized BICM-IDsystem with traditional BICM-ID are interesting problems thatcan be pursued in a future work.

REFERENCES

[1] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans.Commun., vol. 40, no. 5, pp. 873–884, May 1992.

[2] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,”IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946, May 1998.

[3] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEETrans. Inf. Theory, vol. 28, no. 1, pp. 55–67, Jan. 1982.

[4] H. Imai and S. Hirakawa, “A new multilevel coding method using error-correcting codes,” IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377,May 1997.

[5] C. Stierstorfer and R. F. H. Fischer, “(Gray) Mappings for bit-interleavedcoded modulation,” in Proc. IEEE VTC-Spring, Dublin, Ireland,Apr. 2007, pp. 1703–1707.

[6] C. Stierstorfer and R. Fischer, “Asymptotically optimal mappings forBICM with M-PAM and M2-QAM,” IET Electron. Lett., vol. 45, no. 3,pp. 173–174, Jan. 2009.

[7] A. Alvarado, “Towards fully optimized BICM transmissions,” Ph.D. dis-sertation, Chalmers University of Technology, Göteborg, Sweden, 2010.

[8] C. Stierstorfer and R. Fischer, “Intralevel interleaving for BICM in OFDMscenarios,” in Proc. 12th Int. OFDM Workshop, Hamburg, Germany,Aug. 2007, pp. 1–5.

[9] A. Alvarado, E. Agrell, L. Szczecinski, and A. Svensson, “ExploitingUEP in QAM-based BICM: Interleaver and code design,” IEEE Trans.Commun., vol. 58, no. 2, pp. 500–510, Feb. 2010.

[10] M. J. Hossain, A. Alvarado, and L. Szczecinski, “Towards fully optimizedBICM transceivers,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3027–3039, Nov. 2011.

[11] I. Abramovici and S. Shamai, “On turbo encoded BICM,” Annales desTelecommun., vol. 54, no. 3, pp. 225–234, Mar./Apr. 1999.

[12] U. Hansson and T. Aulin, “Channel symbol expansion diversity improvedcoded modulation for the Rayleigh fading channel,” in Proc. IEEE ICC,Dallas, TX, USA, Jun. 1996, vol. 2, pp. 891–895.

[13] A. Nilsson and T. M. Aulin, “On in-line bit interleaving for serially con-catenated systems,” in Proc. IEEE ICC, Seoul, Korea, May 2005, vol. 1,pp. 547–552.

[14] C. Stierstorfer and R. Fischer, “Adaptive interleaving for bit-interleavedcoded modulation,” in Proc. 7th Int. ITG Conf. SCC, Ulm, Germany,Jan. 2008, pp. 1–6.

[15] C. Stierstorfer, R. F. H. Fischer, and J. B. Huber, “Optimizing BICM withconvolutional codes for transmission over the AWGN channel,” in Proc.Int. Zurich Seminar Commun., Mar. 2010, pp. 106–109.

[16] A. Alvarado, L. Szczecinski, and E. Agrell, “On BICM receivers for TCMtransmission,” IEEE Trans. Commun., vol. 59, no. 10, pp. 2692–2702,Oct. 2011.

[17] I. Kalet, “The multitone channel,” IEEE Trans. Commun., vol. 37, no. 2,pp. 119–124, Feb. 1989.

[18] C. Shen, T. Liu, and M. P. Fitz, “On the average rate performance of hybridARQ in quasi-static fading channels,” IEEE Trans. Commun., vol. 57,no. 11, pp. 3339–3352, Nov. 2009.

[19] M. Zimmermann and K. Dostert, “Analysis and modeling of im-pulsive noise in broadband powerline communications,” IEEE Trans.Electromagn. Compat., vol. 44, no. 1, pp. 249–258, Feb. 2002.

[20] M. T. Malik, J. Hossain, and M.-S. Alouini, “Generalized BICM-Ttransceivers: Constellation and multiplexer design,” in Proc. IEEE Int.Symp. Pers., Indoor Mobile Radio Netw., Jun. 2013, pp. 466–471.

[21] A. K. Anhari and L. Lampe, “Performance analysis for BICM trans-mission over Gaussian mixture noise fading channels,” IEEE Trans.Commun., vol. 58, no. 7, pp. 1962–1972, Jul. 2010.

[22] T. Li, W. Mow, and M. Siu, “Bit-interleaved coded modulation in thepresence of unknown impulsive noise,” in Proc. IEEE ICC, Jun. 2006,vol. 7, pp. 3014–3019.

[23] A. Nasri and R. Schober, “Performance of BICM-SC and BICM-OFDM systems with diversity reception in non-Gaussian noise and in-terference,” IEEE Trans. Commun., vol. 57, no. 11, pp. 3316–3327,Nov. 2009.

[24] A. Alvarado, L. Szczecinski, R. Feick, and L. Ahumada, “Distributionof L-values in Gray-mapped M2-QAM: Closed-form approximations andapplications,” IEEE Trans. Commun., vol. 57, no. 7, pp. 2071–2079,Jul. 2009.

[25] P. K. Vitthaladevuni and M.-S. Alouini, “A recursive algorithm for theexact BER computation of generalized hierarchical QAM constellations,”IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 297–307, Jan. 2003.

[26] A. J. Viterbi, “An intuitive justification and a simplified implementation ofthe MAP decoder for convolutional codes,” IEEE J. Sel. Areas Commun.,vol. 16, no. 2, pp. 260–264, Feb. 1998.

[27] J. Belzile and D. Haccoun, “Bidirectional breadth-first algorithms for thedecoding of convolutional codes,” IEEE Trans. Commun., vol. 41, no. 2,pp. 370–380, Feb. 1993.

[28] A. Martinez, A. Guillén i Fàbregas, and G. Caire, “Bit-interleaved codedmodulation revisited: A mismatched decoding perspective,” IEEE Trans.Inf. Theory, vol. 55, no. 6, pp. 2756–2765, Jun. 2009.

[29] J.-J. Chang, D.-J. Hwang, and M.-C. Lin, “Some extended results on thesearch for good convolutional codes,” IEEE Trans. Inf. Theory, vol. 43,no. 5, pp. 1682–1697, Sep. 1997.

[30] X. Wang and H. V. Poor, “Robust multiuser detection in non-Gaussianchannels,” IEEE Trans. Signal Process., vol. 47, no. 2, pp. 3316–3327,Feb. 1999.

Muhammad Talha Malik (S’09) received the Bach-elor’s degree in electronic engineering from GhulamIshaq Khan Institute of Engineering Sciences andTechnology, Topi, Pakistan, in 2011 and the Master’sdegree in electrical engineering from the Universityof British Columbia, Kelowna, BC, Canada, in 2013.He is currently with the School of Engineering,University of British Columbia. His current researchinterests include the design, optimization, and per-formance analysis of wireless communication sys-tems. Mr. Malik was a recipient of the Graduate

Entrance Scholarship (2012) and the University Graduate Fellowship (2013)Award from the University of British Columbia.


Md. Jahangir Hossain (S’04–M’08) receivedthe B.Sc. degree in electrical and electronics engi-neering from Bangladesh University of Engineeringand Technology (BUET), Dhaka, Bangladesh, theM.A.Sc. degree from the University of Victoria,Victoria, BC, Canada, and the Ph.D. degreefrom the University of British Columbia (UBC),Vancouver, BC.

He served as a Lecturer at BUET. He was aResearch Fellow with McGill University, Montreal,QC, Canada; the National Institute of Scientific Re-

search, Quebec, QC; and the Institute for Telecommunications Research, Uni-versity of South Australia, Mawson Lakes, Australia. His industrial experiencesinclude a Senior Systems Engineer position with Redline Communications,Markham, ON, Canada, and a Research Intern position with CommunicationTechnology Lab, Intel, Inc., Hillsboro, OR, USA. He is currently an AssistantProfessor with the School of Engineering, UBC Okanagan campus, Kelowna,BC. His research interests include designing spectrally and power-efficientmodulation schemes, quality of service issues, and resource allocation inwireless networks. Dr. Hossain regularly serves as a member of the TechnicalProgram Committee of the IEEE International Conference on Communications.He was an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICA-TIONS. He was a recipient of the Natural Sciences and Engineering ResearchCouncil of Canada Postdoctoral Fellowship.

Mohamed-Slim Alouini (S’94–M’98–SM’03–F’09)was born in Tunis, Tunisia. He received the Ph.D.degree in electrical engineering from the CaliforniaInstitute of Technology, Pasadena, CA, USA, in1998. He served as a Faculty Member at the Univer-sity of Minnesota, Minneapolis, MN, USA, then inthe Texas A&M University at Qatar, Education City,Doha, Qatar, before joining King Abdullah Uni-versity of Science and Technology, Thuwal, SaudiArabia, as a Professor of electrical engineering in2009. His current research interests include the mod-

eling, design, and performance analysis of wireless communication systems.

analysis and design of generalized bicm-t system

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