IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 297
A Recursive Algorithm for the Exact BER Computation ofGeneralized Hierarchical QAM Constellations
Pavan K. Vitthaladevuni, Student Member, IEEE,andMohamed-Slim Alouini, Member, IEEE
Abstract—Hierarchical constellations offer a different degree of protec-tion to the transmitted messages according to their relative importance. Assuch, they find interesting application in digital video broadcasting systemsas well as wireless multimedia services. Although a great deal of attentionhas been devoted in the recent literature to the study of the bit error rate(BER) performance of uniform quadrature amplitude modulation (QAM)constellations, very few results were published on the BER performance ofhierarchical QAM constellations. In this correspondence, we obtain exactand generic expressions in for the BER of the generalized hierarchical
-PAM (pulse amplitude modulation) constellations over additive whiteGaussian noise (AWGN) and fading channels. We also show how these ex-pressions can be extended to generalized hierarchical -QAM constella-tions (square and rectangular). For the AWGN case, these expressions arein the form of a weighted sum of complementary error functions and aresolely dependent on the constellation size , the carrier-to-noise ratio, anda constellation parameter which controls the relative message importance.Because of their generic nature, these new expressions readily allow numer-ical evaluation for various cases of practical interest.
In his study of broadcast channels, Cover [1] showed about threedecades ago that one strategy to guarantee basic communication inall conditions is to divide the broadcasted messages into two or moreclasses and to give every class a different degree of protection accordingto its importance. This strategy, known as unequal error protection,helps in the recovery of the most important information (known as basicor coarse data) by all receivers. The less important information (knownas refinement, detail, or enhancement data) however, can only be recov-ered by the “fortunate” receivers which benefit either from better prop-agation conditions (e.g., closer to the transmitter and/or with a directline-of-sight path) or from better RF devices (e.g., lower noise ampli-fiers or higher antenna gains). Motivated by this information-theoreticstudy, many researchers have shown since then that one practical way ofachieving this unequal error protection relies on the idea of hierarchicalmodulations (known also as embedded or multiresolution modulations)which consist of constellations with nonuniformly spaced signal points[2]–[4]. This concept was studied further in the early 1990s for dig-ital video broadcasting systems [3], [5] and has gained more recentlynew actuality with i) the demand to support multimedia services by si-multaneous transmission of different types of traffic, each with its ownquality requirement [6]–[8], and ii) a possible application in the DVB-T
Manuscript received August 14, 2001; revised May 13, 2002. This workwas supported in part by the National Science Foundation under GrantCCR-9983462 and in part by the Center of Transportation Studies (CTS)through the Intelligent Transportation Systems (ITS) Institute, Minneapolis,MN. The material in this correspondence was presented in part at the IEEEGlobal Communications Conference (GLOBECOM’2001), San Antonio, TX,November 2001.
The authors are with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]; [email protected]).
Communicated by G. Caire, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2002.806159
standard [9] in which hierarchical modulations can be used on orthog-onal frequency division multiplexing (OFDM) subcarriers.
Yet the exact bit error rate (BER) evaluation performance of thesetypes of modulations in additive white Gaussian channel (AWGN)or fading channels has not been investigated. The only availableexpressions are approximate BER expressions for4=16-QAM and4=64-QAM [5], [6] and for “multicast”M -PSK (phase-shift keying)[7]. In this correspondence, we focus on the exact BER computationof generalizedM -PAM constellations (see Figs. 1 and 2). We thenaddress the BER computation of generalized square QAM (see Figs. 3and 4) and rectangular QAM (see Fig. 5) constellations.
Recent work related to this topic is described in what follows.Generic (inM ) BER approximate expressions for uniformM -QAMhave been developed in [10] and [11] based on signal–space conceptsand a recursive algorithm, respectively. Exact expressions for theBER of 16-QAM and64-QAM were derived in [12]. More recently,Yoon et al. [13] obtained the explicit and generic (inM ) expressionfor the BER of uniform square QAM. These results were extendedby the authors to hierarchical square and nonsquare4=M -QAMconstellations [14]. In this correspondence, we adopt another approachfor BER computation. Starting with some relatively simple examples(2=4-PAM and 2=4=8-PAM), this correspondence shows that it isactually possible to find a repetitive pattern and to obtain, as a result,an exact recursive expression inM for the BER of these constellationsfirst for PAMs and then for various families of QAMs (square andrectangular).
The remainder of this correspondence is organized as follows. Thenext section presents the system model and parameters for generalizedhierarchical PAMs. Section III tracks the pattern in the BER of gen-eralized hierarchicalM -PAM constellations and obtains the resultingnew recursive BER expressions. Section IV shows the way in whichthese expressions can be extended to generalized hierarchicalM -QAMconstellations. Section V presents some numerical examples as well astheir interpretation. Finally, Section VI summarizes the results of thiscorrespondence.
II. SYSTEM MODEL AND PARAMETERS FORGENERALIZED
HIERARCHICAL PAMS
A. System Model
We consider a generalizedM -PAM (M = 2m) constellation withKarnaugh map style Gray mapping (see Fig. 2 for the generalized8-PAM example). The highest priority bit (biti1) is assigned the mostsignificant bit (MSB) position. We will also refer to this as subchanneli1. Bits ik (k = 2; 3; . . . ; m), with lower priorities are assigned thesubsequent positions of lower significance. For instance, the bit withthe second highest priority (biti2 or subchanneli2) is assigned thesecond most significant position, and so on, until the least prioritybit (bit per subchannelim) is assigned the least significant bit (LSB)position. This can be viewed as a2=4=8 � � � =M -PAM constellation.
In both Figs. 1 and 2, the black and grey symbols represent only fic-titious symbols. The actual transmitted symbols are the white symbolsthat are labeled with respect to their nearest fictitious black and grey
Fig. 3. Generalized4=16-QAM constellation with constellation points Gray coded ini q i q fashion.
symbols. For example, the symbol “F” in the2=4=8-PAM of Fig. 2 islabeled as011. In general, for a2=4= � � � =M -PAM, the gray codingcan be done using the following well-known procedure. Label the con-stellation points from the left to right (orvice versa) with integersstarting from0 to M � 1. In Figs. 1 and 2, the labeling has beendone from right to left. Then, convert the integer labels to their bi-nary form. For thekth symbol(k = 1; 2; . . . ; M), let them-digitbinary equivalent beb1; kb2; k � � � bm;k. Then, the corresponding graycode(g1; kg2; k � � � gm; k) is given by
g1; k = b1; k
gi; k = bi; k � bi�1; k; i = 2; 3; . . . ; m (1)
where� represents modulo-2 addition.
B. System Parameters
1) Distances: As shown in Fig. 2, the distances we use in thiscorrespondence evolve in a hierarchy.2d1 represents the distancebetween the points in the fictitious black-colored binary phase-shiftkeying (BPSK) constellation. We refer to this as the first level ofhierarchy.2d2 (second level of hierarchy) represents the distancebetween the points in the fictitious grey-colored BPSK constellationcentered around the fictitious black points. Finally,2d3 represents thedistance between points in the actual BPSK constellation centeredaround the fictitious grey points. Then for a generalized hierarchicalM -PAM constellation, 2d1; 2d2; . . . ; 2dm�1 represent distancesbetween points in the first, second,. . . ; (m� 1)th levels of hierarchy,respectively. Finally,2dm represents the distance in the final levelof hierarchy(M = 2m). To simplify the notation in our proposed
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 299
Fig. 4. Generalized hierarchical4=16=64-QAM constellations which can beviewed as4=16=64-QAM constellations.
algorithm, we define the distance vector asddd = [d1; d2; . . . ; dm](wheredi � 2di+1) and the priority vectorppp as
ppp = [p1; p2; . . . ; pm�1; pm] =d1dm
;d2dm
; . . . ;dm�1dm
; 1 : (2)
This vector controls the relative message priorities. The larger the ratiopi=pi+1 the greater is the protection for the bit in positioni than thebit in position(i + 1).
2) Energies: The average symbol energy for the generalizedM -PAM constellation can be computed as follows. The coordinate forthekth symbol,xk (k = 1; 2; . . . ; )M , in the constellation (shownin Fig. 2) is of the form
xk = �m
i=1
(2bi; k � 1)di = pppaaaTk (3)
whereb1; k, b2; k, . . ., bm;k are the bits (0 or1) in the binary code for thekth symbol (as presented in Section II-A), andaaak is a coordinate row
vector for thekth symbol. For example, the symbol “E” in Fig. 2 hasvectoraaa = [1; �1; �1]d3. The average energy of the constellation,Es, can be written as
whereIII is them �m identity matrix.3) BER Parameterization:As we will see in the following section,
the BER expressions are weighted sums of complementary error func-tionserfc(�) whose arguments are in the form
CdmpN0
(7)
whereC is the distance between a symbol and a threshold used to de-fine a decision region, andN0=2 is the two-sided power spectral den-sity of the AWGN. These arguments can be expressed solely in termsof ppp, and the carrier-to-noise ratio (CNR) = Es=N0 as
CdmpN0
= � G(C; ppp) (8)
where for the generalizedM -PAM case
G(C; ppp) =C2
ppppppT: (9)
III. EXACT BER COMPUTATION OFGENERALIZED M -PAM
We now propose the general recursive algorithm for BER evalua-tion of generalizedM -PAMs. The generalized4-PAM example shownbelow forms the root of the recursion. The generalized8-PAM exampleshows how the recursion works. Since these two constellations are verywell known, we just mention the BER results for4--PAM and8-PAM.
A. Generalized4-PAM Constellation
Consider the4-PAM constellation as shown in Fig. 1. The averageprobability of error for biti1, Pb(4; ddd; i1), is given by
300 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Fig. 6. For calculating the BER for biti in a generalized2=4=8-PAM constellation, it is convenient to visualize the upper subplot as the superposition of thetwo lower subplots. We note that the decision boundaries for biti remain the same even in the resultant constellations.
B. Generalized8-PAM Constellation
We use this example to show how the recursion works. The proba-bility of error for bit i1, Pb(8; ddd; i1), can be easily shown to be givenby
Pb(8; ddd; i1) =1
8erfc
d1 � d2 � d3pN0
+ erfcd1 � d2 + d3p
N0
+ erfcd1 + d2 � d3p
N0
+ erfcd1 + d2 + d3p
N0
: (12)
This can be expressed in terms of the BER of two4-PAM constellationsas
Pb(8; ddd; i1) =1
2[Pb(4; d+d+d+; i1) + Pb(4; d�d�d�; i1)] (13)
where
ddd� = [d1; d2 � d3]: (14)
This is equivalent to the dividing the symbols into two groups: onegroup has symbols with biti3 as0, and the other has symbols with biti3 as1 (see Fig. 6). It is easy to see that bitsi1 andi2 in the constella-tion points A′ through H′ (in the resultant4-PAMs) have the same deci-sion boundaries as those in the constellation points A through H (in the8-PAM constellation), respectively. This is the reason why we are ableto writePb(8; ddd; i1) as in (13). In general, splitting anM -PAM con-stellation in this fashion ensures that the decision boundaries for bitsi1throughim�1 remain the same in the newly createdM=2-PAMs. Thisargument proves the recursion.
Similarly, it is possible to write the BER of biti2, Pb(M; ddd; i2), as
Pb(8; ddd; i2) =1
2[Pb(4; ddd+; i2) + Pb(4; ddd�; i2)] (15)
whereddd� have the same definition as in (14). Now, for biti3, the error,Pb(8; ddd; i3), can be shown to be given by
Pb(8; ddd; i3) =1
84erfc
d3pN0
+ 2erfc2d2 � d3p
N0
+ erfc2d1 � 2d2 � d3p
N0
� 2erfc2d2 + d3p
N0
� erfc2d1 � 2d2 + d3p
N0
� 2erfc2d1 � d3p
N0
+ 2erfc2d1 + d3p
N0
+ erfc2d1 + 2d2 � d3p
N0
� erfc2d1 + 2d2 + d3p
N0
: (16)
As we shall see later, the BER for the LSB can be written in a closedform, using the fact that the LSB changes once every two symbols.Indeed, the LSB changes according to the pattern0-1-1-0-0-1-1-0 � � �from one end of the PAM to the other, giving, as we shall show later, asimple expression for the BER.
C. Recursive Expression forM -PAM Constellations in AWGN
We have already developed the expressions for exact BER of bitsi1 andi2 in the case of4-PAM (m = 2). Hence, we use these BERexpressions as the root of this algorithm. To compute the BER for bitsik, Pb(M; ddd; ik), (k = 1; 2; . . . ; m), where bitim represents theLSB, andm > 2, we use a recursive algorithm, whose pseudocode isgiven as follows.
If k < m
Pb(M; ddd; ik) =1
2Pb
M
2; ddd+; ik + Pb
M
2; ddd�; ik (17)
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 301
where
ddd� = [d1; d2; . . . ; dm�2; dm�1 � dm] (18)
are (m � 1)-dimensional row vectors (ddd is anm-dimensional rowvector). As we proceed through the recursion, the constellation sizegoes on decreasing until we either reach a stage where the vectorsddd+andddd� are of length2 (in the case of bitsi1 andi2), or bitik (for k > 2)becomes the LSB. In the former case, we use the result from4-PAM(given in Section III-A), and come out of the recursion. Ifk = m or inthe event that bitik becomes the LSB at some stage through the recur-sion, we need an LSB algorithm. It is possible to write the BER of theLSB in a closed form, given by
Pb(M; ddd; im) =1
2m(P0 + P1) (19)
where
P0 =
2
i=1
2
j=1
1
2(�1)j+1erfc
ddd0(i)�BBB(j)pN0
(20)
and
P1 =
2
i=1
1 +
2
j=1
1
2(�1)jerfc
ddd1(i)�BBB(j)pN0
(21)
where
• B is the vector of decision boundary positions for the LSB,with respect to (w.r.t.) the origin, it has the positions where LSBchanges from1 to 0 and0 to 1. It can be very easily generatedbecause of the fact that the LSB change from symbol to symbolhas the pattern0-1-1-0-0-1-1-0 � � �. It can be easily shown thatthe entries of the vectorBBB,BBB(j) are given by
• ddd0 is the vector of positions of constellation points whose LSB is0 w.r.t. the origin and is given by
ddd0(1) =x1
ddd0(2m�1) =x2 (23)
and fori = 2; 3; . . . ; 2m�1 � 1
ddd0(i) =x2i; if i is even
x2i�1; if i is odd.(24)
• ddd1 is the vector of positions of constellation points whose LSBis 1, w.r.t. the origin, and can be generated similarly using thefollowing equation.
For i = 1; 2; . . . ; 2m�1
ddd1(i) =x2i; if i is odd
x2i�1; if i is even.(25)
• 1
2P0 represents the LSB’s BER averaged over those symbols
with LSB = 0.
• 1
2P1 represents the LSB’s BER averaged over those symbols
with LSB = 1.
In (20) and (21), the inner sum gives the BER of a symbol, while theouter sum averages the BER over all symbols. For the reader’s conve-nience, our MATLAB computer programs (along with a readme file)are available at [15] to allow one to immediately compare the perfor-mance of various generalized hierarchical constellations.
D. Extension to Flat-Fading Channels
Averaging the conditional BER over the fading distribution, it can beshown that the average BER of the in-phase bits is given by the samerecursion, with the exception that at the final stage in the recursion,when we have reduced the situation to that of a4-PAM case, we usethe functionI(�) in place oferfc, which is defined as follows. For theAWGN channel
I(C; ppp; ) = erfc � G(C; ppp) (26)
depending upon the sign ofC.For Rayleigh channels, it can be shown that
I (C; ppp; ) = 1� G(C; ppp)
1 +G(C; ppp) (27)
if C is positive, and
I (C; ppp; ) =G(C; ppp)
1 +G(C; ppp) (28)
if C is negative. In (27) and (28), is the average CNR. ForNakagami-m channels, it can be shown that
I (C; ppp; )=G(C; ppp)
�
mm
(m+G(C; ppp) )m+1=2
�(m+ 1=2)
�(m+ 1)
�2F1 1; m+ 1=2; m+ 1;m
m+G(C; ppp) (29)
if C is positive, and
I (C; ppp; )=1� G(C; ppp)
�
mm
(m+G(C; ppp) )m+1=2
�(m+1=2)
�(m+1)
� 2F1 1; m+1=2; m+1;m
m+G(C; ppp) (30)
if C is negative, where, in (29) and (30),2F1(�; �; �; �) is the Gausshypergeometric function andm is the Nakagami fading parameter(m � 1
2). For the Rician fading channels
I (C; ppp; ) = Q(u; v)� 1
21 +
w
1 + we� I0(uv) (31)
if C is positive, and
I (C; ppp; ) = 1� Q(u; v)
�1
21 +
w
1 + we� I0(uv) (32)
if C is negative, where, in (31) and (32),Q(�; �) is the MarcumQ-func-tion,I0(�) is the zeroth-order modified Bessel function of the first kind,and
w =G(C; ppp)
K + 1
u =pK
1 + 2w
2(1 + w)� w
1 + w
1=2
v =pK
1 + 2w
2(1 + w)+
w
1 + w
1=2
(33)
whereK is the Rician factor.
IV. EXTENSION TO GENERALIZED HIERARCHICAL
M -QAM CONSTELLATIONS
A generalized hierarchical squareM -QAM constellation(M = 22m) can be modeled as follows. We assume that there
302 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Fig. 7. It is well known that QAM constellations are two PAMs inquadrature. They can be analyzed using the recursive procedure shown inthis correspondence. This figure shows a2 -QAM constellation withnin-phase andp quadrature bits.
arem bit streams of data. Each one of these incoming streams carriesinformation of a particular priority. For every channel access twobits are chosen from each level of priority. The two highest prioritybits are assigned the most MSB positions in the in-phase (I) andthe quadrature phase (Q), respectively. Bits with lower priorities areassigned the subsequent positions of lower significance. For instance,the two bits with the second highest priority are assigned the secondmost significant positions in the I-phase and Q-phase, and so on,until the two least priority bits are assigned the LSB position in theI- and Q-phase. This can be viewed as a4=16=64 � � � =M -QAMconstellation.
In the case where every level of priority is not restricted to twobits per channel access, many other QAM constellations can arise.The recursive algorithm developed in this correspondence for the exactBER computation can be readily adapted to treat all these cases. Fig. 7shows a2n+p-QAM implementation. As an example of the applica-bility of the proposed algorithm, we will consider in this correspon-dence a family of rectangularM -QAM (i.e.,M = 22m+1) constella-tions withm+1 incoming streams of data. For this considered familyof constellations, the highest priority level is assigned one bit in thein-phase whereas all the other levels are assigned two bits in similarfashion as the square QAM case described above. This can be viewed asa2=8=32 � � � =M -QAM constellation. As an illustration of this familyof constellations. Fig. 5 shows a generalized32-QAM constellation.
A. N=M -QAM
The generalizedM -QAM constellations under consideration reduceto theN=M -QAMs when two data streams (known as the basic and therefinement streams) with two level of priorities are transmitted. Then most significant bits(n = 1
2log2N) are referred to as base bits.
The remaining(m � n) bits (m = 1
2log2M) are referred to as the
refinement bits. In this case, the priority vector is given by
ppp = [(�2m�1); (�2m�2); . . . (�2m�n); (2m�n�1);
(2m�n�2); . . . ; 4; 2; 1] (34)
where� > 1 controls the degree of prioritization between the two datastreams. These constellations can be easily analyzed using the recursive
algorithm developed above, as illustrated in the section on numericalexamples.
B. Generalized HierarchicalM -QAM
The procedure derived in Section III can also be used for the mostgeneral QAM constellations.
1) M = 22m Case: In this case, we have two distance vectorsdddiii
anddqdqdq, and two priority vectorspipipi andpqpqpq defined as follows:
dididi = [di1; di2; . . . ; d
im] (35)
and
dqdqdq = [dq1; dq2; . . . ; d
qm]: (36)
Without loss of generality, let us assume thatdim < dqm. Then, the twopriority vectors are defined as
pipipi =di1dim
;di2dim
; . . . ; 1 (37)
and
pqpqpq =dq1dim
;dq2dim
; . . . ;dqmdim
: (38)
The symbol energy of the constellation,Ess ,1 can be written as
Ess = dddii
idddiii
+ dddqqqdddqqq (39)
which can be further written as
Ess = pppii
ipppiii
+ pppqqqpppqqq
dim: (40)
Then we apply the recursive algorithm independently to the in-phaseand quadrature phase. As we have seen in Section III, the BER ex-pressions are weighted sums of complementary error functionserfc(�)whose arguments are in the form
CdimpN0
(41)
whereC has been defined in (7) andN0=2 is the two-sided powerspectral density of the AWGN, and the superscripti=q refers to thein-phase/quadrature-phase priority vectors, respectively. These argu-ments can be expressed solely in terms ofpppiii=qqq, and the CNR =Es=N0 as
CdimpN0
= � G(C; pppiii=qqq) (42)
where
G(C; pppiii=qqq) =
C2
pppiiipppiii + pppqqqpppqqq: (43)
The BER of the in-phase bitsP sb (M; dddiii; dddqqq; ik) can be written as
P sb (M; dddii
i; dddqq
q; ik) = Pb(
pM; dddii
i; ik);
k = 1; 2; . . . ;1
2log2 M: (44)
The BER of the quadrature phase bitsP sb (M; dddiii; dddqqq; qk) can be
written as
P sb (M; dddii
i; dddqq
q; qk) = Pb
pM; dddqq
q; ik ;
k = 1; 2; . . . ;1
2log2M: (45)
1We use the superscript “s” for square QAM and “r” for rectangular QAM.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 303
Fig. 8. Comparison of different methods (Yang and Hanzo [11], Cho and Yoon [13] for the computation of the BER of uniform64-QAM obtained by settingppp = ppp = ppp = [4; 2; 1] over an AWGN channel.
2) Nonuniform Nonsquare QAM,M = 22m+1: The above equa-tions hold even for the generalized rectangularM -QAM case, the onlydifference being, the length of the distance (and priority) vectors in thein-phase channel and the quadrature phase channel. Without loss ofgenerality, let us assume that the length of the in-phase distance vectorism+1. The distance vectors in the in-phase and quadrature phase are
dddiii= [di1; d
i2; . . . ; d
im+1] (46)
and
dddqqq= [dq1; d
q2; . . . ; d
qm]: (47)
The priority vectors would then be
pppiii=
di1dim+1
;di2
dim+1
; . . . ; 1 (48)
and
pppqqq=
dq1dim+1
;dq2
dim+1
; . . . ;dqmdim+1
: (49)
The symbol energy of the constellation can be written as
Ers = dddii
idddiii
+ dddqqqdddqqq (50)
which can be further written as
Ers = pppii
ipppiii
+ pppqqqpppqqq
dim+1: (51)
As we have seen in Section III, the BER expressions are weightedsums of complementary error functionserfc(�) whose arguments arein the form
Cdim+1pN0
(52)
whereC has been defined in (7) andN0=2 is the two-sided powerspectral density of the AWGN, and the superscripti=q refers to thein-phase/quadrature-phase priority vectors, respectively. These argu-ments can be expressed solely in terms ofpppiii=qqq, and the CNR =Es=N0 as
Cdim+1pN0
= � G(C; pppiii=qqq) (53)
where
G(C; pppiii=qqq) =
C2
pppiiipppiii + pppqqqpppqqq: (54)
The BER of the in-phase bitsP rb (M; dddiii; dddqqq; ik) can be written as
P rb (M; dddii
i; dddqq
q; ik) = Pb
p2M; dddii
i; ik ;
k = 1; 2; . . . ;1
2log2 2M: (55)
304 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Fig. 9. Comparison of the various methods for the computation of the BER for the hierarchical4=256-QAM constellation obtained by settingp vectorppp = ppp =ppp = [9; 4; 2; 1].
The BER of the quadrature phase bitsP rb (M; dddiii; dddqqq; qk) can be
written as
P rb (M; dddii
i; dddqq
q; qk) = Pb
M
2; dddqq
q; ik ;
k = 1; 2; . . . ;1
2log
2
M
2: (56)
V. NUMERICAL EXAMPLES
Since the number of different cases covered by our analysis is quitelarge, we just present in this section some numerical examples thatdemonstrate the usefulness of our analytical tools for a variety of sce-narios of practical interest. We further note that the analytical expres-sions derived in this correspondence, and the corresponding numer-ical results presented in this section have been verified extensively byMonte Carlo simulations for the various constellations under consider-ation. Thus, the reader can be totally confident in the correctness of thenewly derived recursive expressions and the accuracy of the numericalresults illustrated below. We have usedppp = pppiii = pppqqq here.
Fig. 8 deals with the uniform64-QAM case and shows the perfectmatch between our exact recursive algorithm and the exact generic ex-pression obtained by Yoonet al. [13] as well as the Monte Carlo simu-lations. Note that the performance of an8-PAM constellation would besimilar. The approximate recursive expression obtained in [11] comesvery close to the exact result whereas the leading term approximation2
gives an optimistic result at low CNR. Fig. 9 verifies our algorithmwith Monte Carlo simulations, and compares it with the leading termapproximation, for the case of hierarchical4=256-QAM (N=M -QAM)
2By leading term, we mean just the dominanterfc(�) term.
constellation. The performance of a2=16-PAM constellation wouldbe similar. Fig. 10 illustrates the effect of the variation of the priorityvectorppp, for the special case of4=256-QAM constellation. We see thatas the ratiop1=p2 increases, for a given CNR, the base bits(i1; i2)get better protection at the expense of a lower protection for the refine-ment bits (i.e.,i2; i3; . . . ; i8). Also note that these numerical resultsshow that the leading-term approximation gives significantly optimisticBER values at low CNR but is quite accurate in the high CNR region.Fig. 11 illustrates anotherN=M -QAM constellation with a special caseof 16=64-QAM in which we have four base bits and two refinementbits. Finally, Fig. 12 deals with the most general case of4=16=64-QAMwith three different levels of priority. It shows that higher the priorityratio p
p, the better the BER performance.
VI. CONCLUSION
In this correspondence, we have argued that the recursive way ofGray coding a constellation ensures the existence of a recursive algo-rithm for the computation of the BER. We have obtained exact andgeneric expressions (inM ) for the BER of the generalized hierarchicalM -PAM constellations over AWGN and fading channels. We havealso showed how these expressions can be extended to generalizedhierarchicalM -QAM constellations (square and rectangular). For theAWGN case, these expressions are in the form of a weighted sum ofcomplementary error functions and are solely dependent on the constel-lation sizeM , the CNR, and a constellation parameter which controlsthe relative message importance. Because of their generic nature, thesenew expressions readily allow numerical evaluation for various cases ofpractical interest. In particular, numerical examples have shown that theleading-term approximation gives significantly optimistic BER valuesat low CNR, but is quite accurate in the high CNR region.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 305
Fig. 10. Effect of variation ofp on hierarchical4=256-QAM, with i andi representing the base bits, and the rest being the refinement bits(ppp = ppp = ppp =[7:5; 4; 2; 1], [8; 4; 2; 1], and[9; 4; 2; 1]).
Fig. 11. BER of16=64-QAM obtained by settingppp = ppp = ppp = [5; 2:5; 1].
306 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Fig. 12. Effect ofppp on the BER of generalized64-QAM with three levels of priority(ppp = ppp = ppp = [3; 1:75; 1], [4; 2; 1], and[5; 2:25; 1]).
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 307
[2] C.-E. W. Sundberg, W. C. Wong, and R. Steele, “Logarithmic PCMweighted QAM transmission over Gaussian and Rayleigh fading chan-nels,”Proc. Inst. Elec. Eng., vol. 134, no. 6, pp. 557–570, Oct. 1987.
[3] K. Ramchandran, A. Ortega, K. M. Uz, and M. Vetterli, “Multiresolutionbroadcast for digital HDTV using joint source/channel coding,”IEEE J.Select. Areas Commun., vol. 11, pp. 6–23, Jan. 1993.
[4] L.-F. Wei, “Coded modulation with unequal error protection,”IEEETrans. Commun., vol. 41, pp. 1439–1449, Oct. 1993.
[5] M. Morimoto, H. Harada, M. Okada, and S. Komaki, “A study onpower assignment of hierarchical modulation schemes for digitalbroadcasting,”IEICE Trans. Commun., vol. E77-B, pp. 1495–1500,Dec. 1994.
[6] M. Morimoto, M. Okada, and S. Komaki, “A hierarchical image trans-mission system in a fading channel,” inProc. IEEE Int. Conf. UniversalPersonal Communicationa (ICUPC’95), Oct. 1995, pp. 769–772.
[7] M. B. Pursley and J. M. Shea, “Nonuniform phase-shift-key modulationfor multimedia multicast transmission in mobile wireless networks,”IEEE J. Select. Areas Commun., vol. 17, pp. 774–783, May 1999.
[8] , “Adaptive nonuniform phase-shift-key modulation for multimediatraffic in wireless networks,”IEEE J. Select. Areas Commun., vol. 20,pp. 1394–1407, Aug. 2000.
[9] DVB-T standard: ETS 300 744, Digital Broadcasting Systems for Televi-sion, Sound and Data Services: Framing Structure, Channel Coding andModulation for Digital Terrestrial Television, ETSI Draft, vol. 1.2.1, no.EN300 744, 1999-1.
[10] J. Lu, K. B. Letaief, J. C.-I. Chuang, and M. L. Liou, “M -PSK andM -QAM ber computation using signal-space concepts,”IEEE Trans.Commun., vol. 47, pp. 181–184, Feb. 1999.
[11] L. L. Yang and L. Hanzo, “A recursive algorithm for the error probabilityevaluation ofM -QAM,” IEEE Commun. Lett., vol. 4, pp. 304–306, Oct.2000.
[12] M. O. Fitz and J. P. Seymour, “On the bit error probability of QAM mod-ulation,” Int. J. Wireless Inform. Networks, vol. 1, no. 2, pp. 131–139,1994.
[13] K. Cho and D. Yoon, “On the general BER expression of one and twodimensional amplitude modulations,”IEEE Trans. Commun., vol. 50,pp. 1074–1080, July 2002. Conference version inProc. IEEE Vehic-ular Technology Conf. (VTC’2000–Fall), Boston, MA, Sept. 2000, pp.2422–2427..
[14] P. K. Vitthaladevuni and M.-S. Alouini, “BER computation of general-ized hierarchical 4/M -QAM constellations,” inProc. IEEE Int. Symp.Personal, Indoor and Mobile Radio Commun. Conf. (PIMRC’2001), vol.1, San Diego, CA, Sept. 2001, pp. 85–89. See also:IEEE Trans. Broad-casting, vol. 47, pp. 228–239, Sept. 2001.
[15] , MATLAB programs for BER computation of generalizedhierarchicalM -QAM constellations. [Online]. Available: Availableat: http://www.ece.umn.edu/users/pavan/Generalized-Hierarchical-Qam.html.
Signal Constellations for Bit-Interleaved Coded Modulation
Stéphane Y. Le Goff
Abstract—Bit-interleaved coded modulation (BICM) is a bandwidth-ef-ficient coding technique consisting of serial concatenation of binaryerror-correcting coding, bit-by-bit interleaving, and high-order modula-tion. BICM is capable of achieving excellent error performance providedthat powerful codes, such as for example turbo codes or low-densityparity-check (LDPC) codes, are employed. In this correspondence, weaddress the problem of finding the signal sets that are the most suitableones for designing power-efficient BICM schemes over an additive whiteGaussian noise (AWGN) channel. To this end, we exploit the expressionof the BICM capacity limit, and evaluate it for several 8- and 16-aryconstellations. The bit-error rate (BER) performance of some BICMschemes made up of turbo codes and various signal sets is also investigatedby computer simulations so as to illustrate the theoretical results. We showthat, for spectral efficiencies of practical interest, the most attractive signalsets are those for which Gray mapping is possible, provided that theirsymbol error rate performance is “sufficiently close” to the optimum. Thisexplains why some constellations having a simple structure, such as8-PSKand 16-QAM, perform very well when combined with a powerful code.At the same time, the constellations displaying optimal error performancewithout coding are, generally, not of interest for BICM.
Index Terms—Bit-interleaved coded modulation (BICM), channelcapacity, Gray mapping, signal constellation, turbo code.
I. INTRODUCTION
Bit-interleaved coded modulation (BICM) is a bandwidth-efficientcoding technique based on serial concatenation of binary error-cor-recting coding, bit-by-bit interleaving, and high-orderM -ary modu-lation [1], [2]. The basic idea behind BICM is simply to map the en-coded bits, after interleaving, to a certain constellation using Gray orquasi-Gray mapping. The decoding is performed by first computing thelog-likelihood ratios of the coded bits, and then, after de-interleaving,using a binary decoder as if these log-likelihood ratios were the ob-servations at a binary phase-shift keying/quaternary phase-shift keying(BPSK/QPSK) channel output.
Despite its simplicity, BICM has proven to be a very power-effi-cient approach provided that state-of-the-art codes, such as turbo codes[3], [4] or low-density parity-check (LDPC) codes [5], [6], are em-ployed. For instance, when a rectangular quadrature amplitude modu-lation (QAM) is used in association with a turbo code, BICM is capableof achieving bit-error rate (BER) performance similar to that obtainedwith more complicated turbo-coded modulation systems [7]–[9].
It is possible to design BICM schemes by employing any two-di-mensional constellation. In this correspondence, we address theproblem of finding the most suitable signal sets for designingpower-efficient BICM schemes, on additive white Gaussian noise(AWGN) channels, by using the concept of channel capacity limit. Theexpression of the BICM capacity is first exploited so as to determinesome design rules for BICM. We then evaluate the capacity limit ofBICM for various 8- and 16-ary constellations, and thus determine,among all these signal sets, those being potentially the most attractiveones for BICM design. The idea of evaluating capacity limits to
Manuscript received February 18, 2002; revised September 2, 2002. Thematerial in this correspondence was presented in part at the CommunicationSystems, Networks, and Digital Signal Processing Symposium (CSNDSP’02),Stafford, U.K., July 2002.
The author is with the Department of Communication Engineering, EtisalatCollege of Engineering, Sharjah, United Arab Emirates (e-mail: [email protected]; [email protected]).
Communicated by G. Caire, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2002.806152
