228
IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
BER Computation of 4=M -QAM Hierarchical
Constellations
Pavan K. Vitthaladevuni, Student Member, IEEE and Mohamed-Slim Alouini, Member, IEEE
Abstract—Hierarchical constellations offer a different degree of
protection to the transmitted messages according to their relative
importance. As such they found interesting application in digital
video broadcasting systems as well as wireless multimedia services.
Although a great deal of attention has 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 of hierarchical QAM constellations. Indeed the only available expressions
are “leading-term” approximate BER expressions for 4/16-QAM
and 4/64-QAM. In this paper, we obtain exact and generic expresfor the BER of the 4
-QAM (square and rectansions in
gular) constellations over additive white Gaussian noise (AWGN)
and fading channels.
For the AWGN case, these expressions are in the form of a
weighted sum of complementary error functions and are solely
dependent on the constellation size
, the carrier-to-noise
ratio, and a constellation parameter which controls the relative
message importance. Because of their generic nature, these new
expressions readily allow numerical evaluation for various cases
of practical interest. In particular numerical results show that
the leading-term approximation gives significantly optimistic
BER values at low carrier-to-noise ratio (CNR) in particular over
Rayleigh fading channels but is quite accurate in the high CNR
region.
Index Terms—BER computation, digital broadcasting, embedded modulation, fading channels, gray mapping, hierarchical
modulations, multiresolution transmission, QAM constellations.
I. INTRODUCTION
I
N HIS STUDY of broadcast channels, Cover [1] showed
about three decades ago that one strategy to guarantee basic
communication in all conditions is to divide the broadcasted
messages into two or more classes and to give every class a
different degree of protection according to its importance. The
goal is that the most important information (known as basic or
coarse data) must be recovered by all receivers while the less
important information (known as refinement, detail, or enhancement data) can only be recovered by the “fortunate” receivers
which benefit either from better propagation conditions (e.g.,
closer to the transmitter and/or with a direct line-of-sight path)
or from better RF devices (e.g., lower noise amplifiers or higher
antenna gains). Motivated by this information–theoretic study,
Manuscript received August 13, 2001. This work is supported in part by the
National Science Foundation grant CCR-9983462. This is an expanded version
of work which is going to be presented at the 12th IEEE International Symposium on Personal, Indoor and Mobile Communications (PIMRC 2001), San
Diego, CA, September 2001.
The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:
{pavan; alouini}@ece.umn.edu).
Publisher Item Identifier S 0018-9316(01)10594-9.
many researchers have shown since then that one practical way
of achieving this goal relies on the idea of hierarchical modulations (known also as embedded or multiresolution modulations) which are constituted of constellations with nonuniformly
spaced signal points [2]–[4]. This concept was studied further in
the early nineties for digital video broadcasting systems [3], [5]
and has gained more recently new actuality with
i) the demand to support multimedia services by simultaneous transmission of different types of traffic, each with
its own quality requirement [6]–[8], and
ii) a possible application in the DVB-T standard [9] in which
hierarchical modulations can be used on OFDM subcarriers.
Yet the exact bit error rate (BER) evaluation performance of
these types of modulations in additive white Gaussian channel
(AWGN) or fading channels has not been investigated. The
only available expressions are approximate BER expressions
for 4/16-QAM and 4/64-QAM [5], [6] and for multicast
-PSK [7]. In this paper, we focus on the BER computation
-QA5M family of modulation wherein a 4-QAM
of the
-QAM constellation
constellation is embedded into an
(see Fig. 1). For this family of hierarchical constellation
refinement information bits are sent along two
basic information bits per channel access.
Recent work related to this topic is described in what follows. Generic (in ) BER approximate expressions for uniform -QAM has been developed in [10] and [11] based on
signal–space concepts and a recursive algorithm, respectively.
Exact expressions for the BER of 16-QAM and 64-QAM were
derived in [12]. More recently, Yoon et al. [13] obtained the
exact and generic (in ) expression for the BER of uniform
square QAM. Motivated by this elegant and general result we
set out to see if a similar generic BER expression could be
-QAM constellations. Starting with
found for hierarchical
some relatively simple examples (4/16-QAM, 4/64-QAM, and
4/256-QAM) this paper shows that it is actually possible to find
for
a pattern and to obtain an exact and generic expression in
the BER of these constellations first for square QAMs then for
nonsquare QAMs.
The remainder of this paper is organized as follows. Section II
presents the system model and parameters as well as the method
to compute the average BER of the base and refinement information bits in terms of the BER of the in-phase and quadrature-phase bits. Section III reviews the “leading-term” approximation approach developed in [5], [6] to find the BER of such
constellations. Section IV tracks the pattern in the BER of the
in-phase and quadrature-phase bits and obtains the resulting new
generic BER expressions. Section V presents some numerical
00018–9316/01$10.00 © 2001 IEEE
VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF
-QAM HIERARCHICAL CONSTELLATIONS
229
Fig. 1. 4=M -QAM square and rectangular (nonsquare) hierarchical constellations.
results as well as their interpretation. Finally, Section VI concludes the paper by summarizing the main contributions.
II. SYSTEM MODEL AND PARAMETERS
A. System Model
-QAM constellation with Karnaugh map style
For a
Gray mapping (see Fig. 2 for the 4/16-QAM example). We assume that we have two incoming streams of data. One carrying
the basic information and the other carrying the refinement
refinement
information. For every channel access
bits are sent along with 2 basic bits, as shown in Fig. 3. The two
basic bits are assigned the most significant bit (MSB) position
in the in-phase (I) and quadrature-phase (Q). In the square
-QAM case
,
refinement
bits are assigned to the in-phase and the rest
refinement bits are assigned to the quadrature-phase. On the
-QAM case
,
other hand, for the rectangular
refinement bits are assigned to the in-phase
refinement bits are assigned
and the rest
to the quadrature-phase. In this fashion the base bits
Fig. 2. Embedded 4/16-QAM constellation with constellation points Gray
coded in i q i q fashion.
230
IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
M ) 0 2 refinement bits are sent along with 2 basic bits.
Fig. 3. System Model: log (
can be viewed as being transmitted by selecting one of the four
fictitious symbols, whereas the refinement bits can be viewed as
symbols surrounding the
being transmitted via one of the
selected fictitious symbol.
For a
-QAM rectangular constellation, it can be shown
is given by
that the average energy per symbol
B. System Parameters
represents the dis1) Distances: As shown in Fig. 2,
represents
tance between the two fictitious symbols whereas
the distance between two neighboring symbols within one quadwhich is the minimum
rant. Another important distance is
distance between two symbols in adjacent quadrants. Note that
-QAM case, can be related to and by
in the square
(1)
as the constellation size parameter and
We define
as the constellation priority parameter which controls the
we have a uniform
relative message priorities. When
corresponds to a uniform 16-QAM. When
4-QAM, while
the basic bits are given a higher priority, while these
.
bits are given less protection for
-QAM square constellation, it can
2) Energies: For a
be shown that the average energy per symbol 1 is given by
(5)
In this case the first two terms represent the energy in the base
bits whereas the last term represents the energy of the refinement bits. Hence the ratio of the energy in the fictitious 4-QAM
constellation to the total average symbol energy is given in the
rectangular constellation case by
(6)
3) BER Parameterization: As we will see in Section III
-QAM over AWGN
the following section, the BER of
channels is a weighted sum of complementary error functions
whose arguments are in the following form:
(7)
(2)
or equivalently
(3)
The first term in the above expression (3) represents the average
symbol energy of a 4-QAM system with constellation points
. On the other hand, the second term
separated by a distance
-QAM
in (3) represents the average symbol energy of an
. We define
with constellation points separated by a distance
the ratio of the energy in the fictitious 4-QAM constellation
to the total average symbol energy:
(4)
s
1Throughout the paper the superscript will refer to square QAM and
refer to rectangular (nonsquare) QAM.
r will
where is nonnegative integer, is a strictly positive integer
is the two-sided power spectral density of
number, and
the AWGN. These arguments can be expressed solely in terms
as
of , , and the carrier-to-noise ratio (CNR)
(8)
where for the square
-QAM case
(9)
-QAM case (see equation (10)
whereas for the rectangular
at the bottom of the page). Note that alternatively the overall
BER can be expressed solely in terms of , , and .
(10)
VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF
-QAM HIERARCHICAL CONSTELLATIONS
231
C. BER for the Base and Refinement Bits
-QAM: The BER of the base information
1) Square
is given by
(11)
and
denote the BER of the th
where
-QAM constellain-phase and quadrature phase bits of a
tion, respectively. On the other hand, the BER of the refinement
, is obtained by averaging over all the reinformation,
maining in-phase and quadrature bits, yielding
(12)
-QAM: We lose symmetry between
2) Rectangular
in-phase and quadrature phase bits. However the in-phase bits
-QAM rectangular constellation will behave in the
of a
-QAM square
same manner as the in-phase bits of a
constellation. Hence we can write the BER for these in-phase
, as
bits,
(13)
-QAM rectOn the other hand the quadrature bits of a
angular constellation will behave in the same manner as the
-QAM square constellation. Hence
quadrature bits of a
, as
we can write the BER for these quadrature bits,
Fig. 4. The I sub-channel for 4/16-QAM case.
2 subchannels, I and Q. In the case of square constellations,
the I and Q sub-channels have the same average bit error
probability. So, we show the derivation for the I subchannel
alone. Consider the I subchannel bits shown in Fig. 4. The
in symbol 1000 is given by
probability of error for the bit
. Similarly, the bit error probability
.
for bit in symbol 1010 is given by
We consider just the leading term, as an approximation for the
for bit , i.e.,
bit error probability
(17)
Now, consider the second subchannel . The bit error probability for this subchannel in symbol 1000 is given by
(18)
For symbol 1010, it is
(19)
(14)
Therefore the overall BER of the base information
is given by
So, the average bit error probability
channel is given by
(15)
whereas the overall BER of the refinement info is given by (see
equation (16) at the bottom of the page).
Based on all these equations, we can conclude that the BER
-QAM
computation of the base and refinement bits of a
(square and rectangular) constellation reduces to finding an expression for the BER of the in-phase bits only.
III. LEADING-TERM APPROXIMATION
The BER expressions for
-QAM can be approximated to
the leading term in error function, as has been done by [5] for
the 4/16-QAM and 4/64-QAM constellations.
A. 4/16-QAM Constellation
This approach yields good results for 4/16-QAM constellations over AWGN channels. The data bits are divided into
for the sub-
(20)
Neglecting the higher order terms, we just have
(21)
B. 4/64-QAM Constellation
Similar to what has been done in the case of 4/16-QAM, we
can extend the result to 4/64-QAM. In the case of 64-QAM,
we have 6 bits per symbol. That gives 3 bits per subchannel.
The base bit has the following protection distances with equal
,
,
,
. Considering
probability:
just the leading term, the probability of error is
(22)
(16)
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
Fig. 5. The I sub-channel for 4/64-QAM case.
Similarly, for bit
and
, we have the error probability as
(23)
and,
(24)
(29)
IV. EXACT BER
As we will see later Section V, the approximate approach
yields good results for the bit error probability at high CNR end.
But, at low CNR, it appears under estimating the BER. The performance deteriorates further, if the channel fades. In this section, we derive the exact BER expressions.
A. 4/16-QAM Constellation
C. 4/256-QAM Constellation
The BER expression for this constellation helps us discover
the pattern, using which we can write the general expression
-QAM constellations. Similar to
for the BER for general
Section III, we can write
Referring to Fig. 2 we can write
(25)
(26)
B. 4/64-QAM Constellation
(30)
Referring to Fig. 5 we can write
(27)
(28)
(31)
VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF
Fig. 6.
-QAM HIERARCHICAL CONSTELLATIONS
233
Three sets of coefficients have been tracked. The sum of these three coefficients gives the coefficients for the uniform case.
(33)
D. General Expression for Square
in AWGN
-QAM Constellations
As can be seen in the equations of the Section III, there are
three types of terms to be tracked. First, coefficients of the
functions whose arguments are of the form
alone. Second, coefficients of the
functions whose argusuch that
.
ments are of the form
Finally, we track the rest of the coefficients. The behavior
of these coefficients is illustrated in Fig. 6. Based on these
observations, a pattern can be found for these sets of terms.
is different from that for the other
This pattern for the bit
we can write
bits. In particular, for
(32)
(34)
and for
, we have the following general expression
234
Fig. 7.
IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
Comparison between the exact and the average BER of 4/256-QAM with R
= 0 8 over AWGN channels.
:
(38)
and for
where
by
is the floor function,
(35)
(36)
and
(37)
Recall that knowledge of the BER of the in-phase bits of
square QAMs as given in a generic fashion in (34) and (35), is
sufficient for the computation of the overall BER of any square
-QAM as discussed in Section II-C. As a
or rectangular
check for the validity of the proposed expressions (34) and (35),
it can be easily verified that they reduce to the corresponding ex(uniform QAM case).
pression of [13], [14] when
Using the notation of Section II-B, the new generic BER expressions (34) and (35) over AWGN channels can be expressed
in terms of the CNR , as
(39)
where,
(40)
VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF
Fig. 8.
-QAM HIERARCHICAL CONSTELLATIONS
Effect of Rayleigh fading on the BER performance of 4/256-QAM with R
= 0 8.
:
Fig. 9. Effect of the variation of R on the BER of the base and refinement bits over AWGN channels.
235
236
IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
Fig. 10.
Effect of the variation of
R on the BER of the base and refinement bits over Rayleigh fading channels.
Fig. 11.
Effect of the variation of
M on the BER of the base and refinement bits over AWGN channels.
VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF
Fig. 12.
Effect of the variation of
-QAM HIERARCHICAL CONSTELLATIONS
237
M on the base and refinement bits over Rayleigh fading channels.
E. Extension to Flat Fading Channels
Since the BER expressions over AWGN are in the form of
weighted sum of complementary error functions, these results
are readily extendable to flat fading channels with and without
diversity reception. Therefore, the AWGN expressions (38) and
(39) are valid for flat fading channels provided the definition
function is changed accordingly. For example, for
of
Rayleigh channels
where
is the Marcum -function,
is the 0th-order
modified Bessel function of the first kind, and
(44)
where
is the Rician factor.
(41)
V. NUMERICAL RESULTS
where is the average CNR. For Nakagami-
channel channels
(42)
is the Gauss hypergeometric function and
where
is the Nakagami fading parameter. For the Rician fading channels
(43)
We note that the analytical expressions derived in this paper,
and the corresponding numerical results presented in this section have been verified extensively by Monte Carlo simulations
for the various constellations under consideration. Thus, the
reader can be totally confident in the correctness of the newly
derived closed form expressions and the accuracy of the numerical results illustrated below.
Figs. 7 and 8 compare the exact and the approximate BER
expressions for 4/256-QAM over AWGN and Rayleigh fading
channels, respectively. These numerical results show that the
leading-term approximation gives significantly optimistic BER
values at low CNR in particular over Rayleigh fading channels.
In the case of Rayleigh fading channels, even at a high average
received CNR, the leading term approximation is significantly
optimistic.
Figs. 9 and 10 illustrate the effect of the variation of on the
BER of the base and refinement bits over AWGN and Rayleigh
238
Fig. 13.
IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001
Generalized hierarchical QAM constellations.
fading channels, respectively. When
(uniform 256QAM), we find that the base bits are less protected than the
, we find that base bits are
refinement bits. When
relatively well protected, at the cost of the BER of the refinement
bits.
on
Figs. 11 and 12 illustrate the effect of the variation of
the base and refinement bits over AWGN and Rayleigh fading
channels, respectively. As increases, the BER of base bits increases by a negligible amount, but the BER of refinement bits
increases significantly. This is because, the fictitious 4-QAM
symbols move by a negligible amount, but the refinement symbols around these fictitious 4-QAM symbols get crowded.
VI. CONCLUSION
We derived a generic (in ) closed form expression for the
-QAM constellations. For the AWGN case, these
BER of
expressions are in the form of a weighted sum of complementary
error functions and are solely dependent on the constellation
size , the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance. Because
of their generic nature, these new expressions readily allow numerical evaluation for various cases of practical interest. Numerical results show that the leading-term approximation underestimates the BER at the low CNR end, in particular over
Rayleigh fading channels but is quite accurate at the high CNR
end. More recently, we have derived a recursive way of computing the BER of generalized square -QAM constellations
as shown in Fig. 13 (see [15]). This recursive algorithm gives
an alternative way of computing the exact BER of generalized
-QAM conhierarchical -QAM constellations, of which
stellations form a special case.
ACKNOWLEDGMENT
The authors wish to thank Dr. D. Yoon of the Department of
Information and Communications Engineering, Taejon University, Taejon, Korea for the pre-print of [14].
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-QAM HIERARCHICAL CONSTELLATIONS
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[14] K. Cho and D. Yoon, “On the general BER expression of -ary square
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239
Mohamed-Slim Alouini (S’94-M’99) was born
in Tunis, Tunisia. He received the “Diplôme d’Ingénieur” degree from the Ecole Nationale Supérieure
des Télécommunications (TELECOM Paris), Paris,
France, and the “Diplôme d’Etudes Approfondies
(D.E.A.)” degree in Electronics from the University
of Pierre & Marie Curie (Paris VI), Paris, France,
both in 1993. He received the M.S.E.E. degree from
the Georgia Institute of Technology (Georgia Tech),
Atlanta, GA, USA, in 1995, and the Ph.D. degree in
Electrical Engineering from the California Institute
of Technology (Caltech), Pasadena, CA, USA, in 1998.
While completing his D.E.A. thesis, he worked with the optical submarine
systems research group of the French national center of telecommunications
(CNET-Paris B), on the development of future transatlantic optical links. While
-band satellite channel
at Georgia Tech, he conducted research in the area of
characterization and modeling. From June 1998 to August 1998, he was a PostDoctoral Fellow with the Communications group at Caltech carrying out research on adaptive modulation techniques and on CDMA mobile communications. He joined the department of Electrical and Computer Engineering of the
University of Minnesota, Minneapolis, in September 1998, where his current research interests include statistical modeling of multipath fading channels, adaptive modulation techniques, diversity systems, and digital communication over
fading channels.
Dr. Alouini has published several papers on the above subjects and he is
co-author of the recent Wiley Interscience textbook Digital Communication
over Fading Channels. He is a recipient of a National Semiconductor Graduate
Fellowship Award, the Charles Wilts Prize for outstanding independent research
leading to a Ph.D. degree in Electrical Engineering at Caltech, and co-recipient
of the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference
(VTC’99-Fall), Amsterdam, The Netherlands, for his work on the performance
evaluation of diversity systems. He was awarded a 1999 CAREER Award from
the National Science Foundation, a McKnight Land-Grant Professorship by the
Board of Regents of the University of Minnesota in 2001, and a Top Instructor
Award (ECE Department) by the Institute of Technology Student Board at the
University of Minnesota in June 2001. He is an editor for the IEEE Transactions on Communications (Modulation & Diversity Systems) and for the Wiley
Journal on Wireless Systems and Mobile Computing.
K
Pavan K. Vitthaladevuni (S’2000) received
the Bachelor of Technology (B.Tech) degree in
Electrical Engineering from Indian Institute of Technology, Madras, India, in 1999 and the M.S.E.E.
degree from University of Minnesota, Twin Cities
in 2001. He is currently a graduate research assistant
in the Department of Electrical and Computer
Engineering at the University of Minnesota, Twin
Cities, and is working toward his Ph.D. His research
interests include Digital Communications over
fading channels and Information Theory.