San Jose State University
From the SelectedWorks of Robert Henry Morelos-Zaragoza
April, 2015
On Performance Improvements with Odd-Power
(Cross) QAM Mappings in Wireless Networks
Quyhn Quach
Robert H Morelos-Zaragoza
Available at: http://works.bepress.com/robert_morelos-zaragoza/46/
On Performance Improvements with Odd-Power
(Cross) QAM Mappings in Wireless Networks
Quyhn Quach and Robert H. Morelos-Zaragoza
Abstract—Modern wireless networks use QAM modulation
with mappings that are derived from square signal constellations
in which the number of signal points is an even power of two. In
the particular case of IEEE 802.11 networks, these mappings are
4-QAM, 16-QAM and 64-QAM and are hereto referred as evenpower QAM mappings. This paper considers the performance
improvements that are obtained by adding odd-power QAM
mappings (obtained from cross QAM constellations) to the set of
available mappings. These odd-power mappings are associated
with 22m-1-QAM cross constellations that are carved out from
larger 22m-QAM square constellations. For the specific cases of 8QAM, 32-QAM and 128-QAM mappings, the improvements in
throughput are quantified assuming a Rayleigh fading condition.
Furthermore, in order to illustrate the use of bit metrics
generated by a demapper, also examined is the performance of
bit-interleaved 8-QAM and 32-QAM mappings with binary codes
over an AWGN channel.
Index Terms—Wireless networks; Quadrature-amplitude
modulation; Throughput; Demapping; Channel coding.
Second, as channel coding forms an integral part of modern
wireless networks, illustrative examples of demappers and their
application in maximum likelihood decoding of a short ReedMuller code are presented for 8-QAM and 32-QAM mappings.
The paper is organized as follows. Section II presents a good
approximation on the average bit error probability (ABEP) of
QAM mappings under additive white Gaussian noise
(AWGN). This expression of the ABEP is used in Section III
to determine the increase in throughput that is achieved by the
introduction of odd-power QAM mappings. In section IV, for
the specific cases of 8-QAM and 32-QAM, the performance is
examined of demappers for bit-interleaved odd-power QAM
mappings combined with a first-order RM code and a binary
LDPC code over an AWGN channel. Finally, section V gives
final remarks and directions for future research.
II. ODD-POWER 𝑀-QAM MAPPINGS
The odd-power 2!ℓ𝓁!! -QAM mappings that are considered in
this paper are those associated with cross QAM constellations
I. INTRODUCTION
carved out from larger square 2!ℓ𝓁 -QAM constellations [4].
HE physical layer of current wireless networks uses bits-to- Figures 1 to 3 show the 8-QAM, 32-QAM and 128-QAM
symbol mappings based on square 𝑀-QAM constellations signal constellations with ℓ𝓁 = 2, 3 and 4 respectively studied
[1-3]. This is to say that the number of signal points is an even in this paper.
power of two: 𝑀 = 2!ℓ𝓁 , 1 ≤ ℓ𝓁 ≤ 5, with the integer 𝑚 = 2ℓ𝓁
equals to the number of bits per symbol. These mappings are
referred to as even-power QAM mappings. While it is
advantageous to have a square QAM constellation in which
each signal point coordinate (in-phase and quadrature matchedfilter outputs) is in turn a signal point in a PAM constellation,
recent advances in signal processing platforms and algorithms
allow for the extension of the set of available mappings to
include odd-power 𝑀-QAM mappings with 𝑀 = 2!ℓ𝓁!! .
T
The purpose of this paper is to explore the advantages of
including odd-power QAM mappings in the set of even-power
Fig. 1. 8-QAM signal constellation.
QAM mappings that are already implemented. Two aspects are
considered: The first one is the improvement in throughput that
In this section an overview is given of a good approximation
results from adding a new set of odd-power QAM mappings. It
is shown in particular that, with data transmission in IEEE on the average bit error probability (ABEP). The expression is
802.11-2012 wireless networks under Rayleigh fading used it to determine ranges of values of the average symbol
conditions, up to a 9% increase in throughput is possible. energy-to-noise ratio 𝐸! /𝑁! for which a target value of ABEP
is not exceeded. This helps to determine the throughput of a
wireless network with a given set of available QAM mappings
Paper submitted to GLOBECOM 2015 on April 1, 2015.
The authors are with the Department of Electrical Engineering, San José and a probability density function of the received signal
State University, San Jose, CA 95192-0084 USA. (Contact e-mail: energy. This is discussed in section III.
[email protected]).
TABLE 1
QAM SIZE, ERROR COEFFICIENT AND NORMALIZED MSED
M
𝑁!
𝐷 ! /𝐸!
4
8
16
32
64
128
256
2
2
9/4
7/2
7/2
15/4
15/4
2
2/3
2/5
2/10
2/21
2/41
2/85
Fig. 2. 32-QAM signal constellation.
QAM mappings in Figure 4. In Section III these theoretical
expressions are used to estimate the increase in throughput that
is achievable by the introduction of odd-power QAM
mappings.
Fig. 3. 128-QAM signal constellation.
A. Bit error probability of M-QAM mappings
𝑚
Assuming Gray bits-to-signal mappings of 2 −QAM signal
constellations, the ABEP 𝑃! that is well approximated by the
expression
𝑁!
𝑃! =
𝑄
𝑚
𝐷!
, (1)
2𝑁!
Fig. 4. Average bit error probability of BPSK mapping and 2! -QAM
mappings, for 2 ≤ 𝑚 ≤ 8.
III. THROUGHPUT IMPROVEMENT UNDER RAYLEIGH FADING
To illustrate the improvements in throughput that can be
obtained with the inclusion of a set of odd-power QAM
where 𝐷 ! is the minimum squared Euclidean distance (MSED) mappings, a simple Rayleigh fading condition is assumed such
between signal points expressed in terms of the average symbol that the signal energy-to-noise ratio Γ has an exponential
energy 𝐸! , 𝑁! /2 is the AWGN power spectral density and 𝑁! probability density function:
is the error coefficient (i.e., the average number of nearest
1
𝛾
𝑝! 𝛾 = exp −
𝑢(𝛾),
neighbors). Figure 4 shows the ABEP curves based on Eq. (1)
𝛾!
𝛾!
𝑚
for BPSK mapping and 2 −QAM mappings with 2 ≤ 𝑚 ≤ 8.
In the figures, the curves with dashed lines correspond to the where 𝛾! is the average symbol energy-to-noise ratio.
BEP of the even-power QAM mappings that are currently used
A value of signal energy-to-noise ratio Γ inside an interval
in 802.11 networks [1], while curves with solid lines to
[𝛾! , 𝛾! ], corresponds to a transmission rate 𝑅(𝛾! , 𝛾! ) in bits per
indicate the BEP of odd-power QAM mappings.
symbol, or bps/Hz, using the error performance curves of
Table 1 shows the values of constellation size 𝑀, error QAM mappings in Figure 4, such that the ABEP is lower than
coefficient 𝑁! , and normalized MSED 𝐷 ! /𝐸! , for each of the a target maximum ABEP value.
is implemented with a binary first-order Reed-Muller code of
length 𝑁 = 8, denoted RM(3.1) code. The idea is simply to
illustrate the use of bit metrics with maximum-likelihood softdecision decoding for odd-power mappings.
TABLE 2
INTERVALS AND TRANSMISSION RATES FOR A TRAGET 𝑃! = 0.01
𝑚
𝛾!!
𝛾!!
𝑅(𝛾!! , 𝛾!! )
1
2
3
4
5
6
7
8
4
7.4
11.5
13
17
19.5
22.5
25.5
7.4
11.5
13
17
19.5
22.5
25.5
∞
1
2
3
4
5
6
7
8
For a given set of available QAM mappings, the average
throughput is readily obtained as
𝑅=𝐸 𝑅 =
𝑅 γ!! , 𝛾!!
!∈!
=
𝑚
!∈!
=
!!!
!!!
!!!
!!!
𝑝! 𝛾 𝑑𝛾
𝑝! 𝛾 𝑑𝛾
𝑚 exp −
!∈!
𝛾!!
𝛾!!
− exp −
𝛾!
𝛾!
,
where 𝐼 denotes the set of available QAM constellations and
for simplicity it is assumed that no channel coding is used. In
the case of even-power mappings, 𝐼! = 1,2,4,6,8 , while with
the inclusion of odd-power mappings the set is larger and given
by 𝐼! = 1,2,3,4,5,6,7,8 .
Table 2 shows the set of values of interval limits 𝛾! , 𝛾! and
transmission rate 𝑅(𝛾! , 𝛾! ) for all the QAM mappings in
Figure 1 and a target ABEP 𝑃! = 0.01. Let 𝑅! and 𝑅! denote
the average throughput of a wireless network with even-power
QAM mappings only and with all QAM mappings. Since the
set of available QAM constellations is larger when including
cross constellations, 𝐼! ⊃ 𝐼! , it is expected that the throughput
will be higher. To quantify this increase in information rate,
define the rate increase as
Fig. 5. Rate increase Δ𝑅 (%) versus average symbol energy-to-noise
ratio 𝛾! (dB), under Rayleigh fading, for a target ABEP of 0.01
A. Bit-interleaving and Demapping
For a given 𝑀-QAM mapping with 𝑀 = 2! , a rectangular
interleaver is used heretofore. A total of 𝑚 binary codewords
out of a channel encoder are arranged in an array of 𝑚 rows
and 𝑁 columns, where 𝑁 is the code length. The QAM mapper
then takes as input every column of the array and maps it to a
symbol to be modulated. Therefore, with a binary RM(3,1)
code and 8-QAM and 32-QAM mappings respectively, the
interleavers are of size 3-by-8 and 5-by-8. The transmitted
QAM symbol sequences are of length eight. As pointed out in
[5], it is essential that Gray mapping of bits to symbols be
applied so that an equivalent wireless link with 𝑚 parallel and
independent binary channels is created.
At the receiver end of the link, the eight in-phase and
quadrature
matched-filter outputs (𝑦! , 𝑦! ) are input to a
𝑅!
Δ𝑅 =
− 1.
demapper that produces 𝑚 bit metrics, one metric for every bit
𝑅!
used to label the QAM constellation points. A bit metric is
Figure 5 shows the value of Δ𝑅 as a percentage plotted as a given by the logarithm of the ratio of two a-posteriori
function of 𝛾! for a target ABEP of 0.01 with the QAM probabilities, conditioned to bit values 0 and 1 [5],
mappings in Figure 4. The results show that the maximum rate
𝒚!! !
increase is 9% and corresponds to an average signal energy-to! exp ! !
!
𝜆 𝐵! = log
,
noise ratio of approximately 17 dB.
𝒚!! !
! exp ! !
!
IV. DEMAPPING AND CHANNEL CODING
where 𝛼 represents a signal point with 𝐵! = 0, 𝛽 represents a
In this section, the error performance with bit interleaving
signal point with 𝐵! = 1, and 𝑦 = (𝑦! , 𝑦! ) is the pair of
and channel coding, in conjunction with odd-power QAM
matched-filter outputs from the quadrature demodulator.
mappings, is illustrated. For simplicity of exposition, attention
is focused on 8-QAM and 32-QAM mappings. Channel coding
Figure 6 shows the metrics for the two most-significant AWGN channel and one with either Rayleigh or Rician fading.
label bits 𝐵! and 𝐵! for the three odd-power mapings examined The results are shown in Figure 8 with BER representing the
in this paper. These metrics are the same as QPSK mapping, bit error rate and 𝐸! /𝑁! the average bit energy-to-noise ratio.
since these two bits identify the quadrant in which the signal
point lies. On the other hand, as opposed to even-power
mappings based on square 𝑀-QAM signal constellations that
be expressed as the Cartesian product of two 𝑀-PAM signal
constellations, the bit metrics of the 𝑚 − 2 least-significant
bits in odd-power QAM mappings depend on both matched
filter outputs (𝑦! , 𝑦! ). The computation of these metrics is
therefore slightly more complex compared to even-power
mappings. Figure 7 shows the bit metric for the least
significant bit 𝐵! for 8-QAM mapping.
After the demapping process, the bit metrics are written
column-by-column into an 𝑚-by-𝑁 rectangular array. This
process is known as deinterleaving. Each element in a row of
this metric array (𝑁 metrics) constitutes a BPSK-mapped noisy
binary codeword bit that is fed into a binary channel decoder in
order to estimate its value.
Fig. 7. Bit metric of the least-significant bit of 8-QAM mapping.
The simulated BER performance curves in Figure 8 show
that the bit-interleaved 32-QAM, with a block interleaver of
length eight, degrades and eventually reaches an error floor.
The reason for this behavior is that, as mentioned in [5], the
interleaver length 𝑁 is not sufficiently long relative to the
number of bits per symbol 𝑚. Since 𝑁 = 8 and 𝑚 = 5 for 32QAM mapping and the binary RM(3,1) code, the bits metrics
become highly correlated and error performance does not
improve at higher values of signal energy. The performance of
a longer binary projective geometry (273,191) low-density
parity-check (LDPC) code [6] with a block interleaver and 32QAM, shown in Figure 9, does not show a floor because the
length of the code is now much larger than the number of bits
per symbol. In the simulations, the number of iterations was set
to 8 and decoding performed with belief propagation using
Fig. 6. Bit metrics of the two most-significant bits of all the 𝑀-QAM LLR metrics from the demmaper.
mappings considered in this paper.
V. CONCLUSIONS
B. Performance with a binary RM(3,1) code
In this section, simulation results are presented on the
performance of bit-interleaved 8-QAM and 32-QAM mappings
using the rectangular interleaver and bit metrics generated by a
rectangular demapper described in the pervious section.
In the computer simulations, maximum-likelihood decoding
was implemented with a Green machine (equivalent to a trellis
decoder [6]) for the first-order binary RM(3,1) code of length
𝑁 = 8. The channel model was assumed is AWGN (i.e., no
fading), as the relative difference in error performance with
channel coding does not change significantly between an
It has been shown that augmenting the set of available
mappings in a wireless network, with the inclusion of oddpower QAM mappings, results in an increased throughput. As
an illustration, the improvement in throughput was quantified
for data transmission under Rayleigh fading conditions. The
results show that augmenting the set of modulations in the
IEEE 802.11 standard [1] by 8-QAM, 32-QAM and 128-QAM
mappings can possibly yield an increase of up to 9% in
throughput compared to the use of even-power QAM mappings
alone, with a relatively low complexity cost incurred in the
computation of bit metrics at the demapper. Moreover, the
error performance of the combination of bit interlaving and
channel coding was examined for the cases of 8-QAM and 32QAM mappings with a binary RM(3,1) code and 32-QAM
with a binary finite-geometry (273,191) LDPC code.
Future work includes a theoretical analysis of the rate
increase obtained by augmenting the set of available mappings
under different signal energy probability density functions due
to different fading scenarios. Also contemplated is a more
comprehensive study of the combination of odd-power QAM
mappings, their associated demappers and binary low-density
parity-check (LDPC) codes from the IEEE 802.11 standard [1].
Fig. 9. Error performance of a binary PG (273,191) LDPC code with a
rectangular bit-interleaver and 32-QAM mapping.
REFERENCES
[1]
[2]
[3]
Fig. 8. Error performance of a binary RM(3,1) code with a rectangular
bit-interleaver and 8-QAM and 32-QAM mappings.
[4]
[5]
[6]
Part 11: Wireless LAN Medium Access Control (MAC) and Physical
Layer (PHY) Specifications, IEEE Standard 802.11-2012.
Part 11: Wireless LAN Medium Access Control (MAC) and Physical
Layer (PHY) Specifications, Amendment 3: Enhancements for Very
High Throughput in the 60 GHz Band, IEEE Standard 802.11ad-2012.
Part 11: Wireless LAN Medium Access Control (MAC) and Physical
Layer (PHY) Specifications, Amendment 4: Enhancements for Very
High Throughput for Operation in Bands below 6 GHz, IEEE Standard
802.11ac-2013.
G. J. Foschini, R. D. Gitlin and S. B. Weinstein, “Optimization of TwoDimensional Signal Constellations in the Presence of Gaussian Noise,”
IEEE Trans. Commun., vol. COM-22, no. 1, pp. 28–38, Jan. 1974.
G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded
modulation,” IEEE Trans. Info. Theory, vol. 44, pp. 927–946, May 1998.
S. Lin and D.J. Costello, Jr., Error Control Coding, 2nd ed., Wiley 2004.