Constellation Design via Capacity Maximization Constellation Designs Inc. Maged F. Barsoum, David Bailey, Christopher R. Jones [email protected] [email protected] [email protected]
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No Slide Titleo Previous Work on Constellation Design
o Failure of Traditional Constellation Designs with Modern Codes
o Proposed Constellation Design
o Conclusions
Shaping
Gap
• This 25% improvement has been elusive
due primarily to complexity / coding system
compatibility constraints
CJOINT CPD
• Shaping gain has typically been thought of as a signal
processing / coding procedure by which points from a
given constellation are transmitted with differing
frequencies. The frequencies are chosen such that the
PDF formed by the final waveform has a Gaussian
distribution.
inherent difficulties associated with encoding and decoding
points with differing frequencies.
• Labeling
• Probability Distribution
• Maximize performance under a constellation energy constraint i.e. at any given SNR
• Equal energy constellations are sometimes preferred
2
2
• Equiprobable Signaling
– Reasonable with classic codes requiring high SNR
– Could fail with modern codes that operate at low SNR
• Geometrically shaped constellations
– Gaussian Mimicking (Duan, Rimoldi & Urbanke: 1997, Sommer & Fettweis: 2000)
– Capacity maximization approach (Barsoum & Jones)
• Non-Equiprobable Signaling (Probabilistic Shaping)
(data dependent rate, buffering, synchronization)
• Pseudo-probabilistic Shaping:
- Shell Shaping (V.34 modems)
Minimum Distance-Based Constellation Design
dmin criteria is based on an approximation of the probability of error using the
union bound of the pair-wise error probabilities for an uncoded model
• Coded model (Practical system)
distance criteria
• Traditional QAM
minimum distance criteria
distance criteria
• Minimum distance based enhanced
• Conjectured to be the best 16 points QAM
constellation
• Better minimum distance constellations in
multi-dimensions -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(Conway and Sloane 1983)
Geometric Design Example 2: Warped Constellations
• Betts, Calderbank, Laroia (1994) demonstrated a gain of 0.25 dB when using
a warped constellation with a trellis coded modulation system.
* AT&T, “Nonlinear encoding for V.fast,” Contribution 21, TIA-TR 30.1, Ft. Lauderdale, FL, Feb 1992
• A uniformly spaced constellation is
transformed to a warped
constellation using some warping
-3 -2 -1 0 1 2 3 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
code rates.
decoding capacity with 32
PAM at rate ½
SNR’s
guaranteed to be optimum
Does not work as expected with LDPC or Turbo Codes!
• 16 points Voronoi constellation in 4-D
• Minimum distance is 0.74 dB better than QPSK
• However, is more than 2 dB worse with a modern code!!!
0 1 2 3 4 10
-4
User rate = 0.5bit/dimension
• Gaussian input distribution
achieves maximum capacity
• Uniformly-spaced infinite PAM
capacity
asymptotically achieve the
• Capacity constrained to signal constellation X
and to a soft bit demapper (w/o feedback) in
the channel
• Label dependent
• Label Independent
B YX
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
g C
e r
d im
-4
• User rate = 0.5 bit/constellation dimension
• Voronoi is the ‘best distance’ constellation design, but has 2dB worse
capacity (why optimize for block size == 1 ?)
dB2
Observation1: Constellation Design Methodology
• Optimize the location of the constellation points to maximize capacity or
parallel decoding capacity:
• Design for a specific SNR:
Capacity or Parallel Decoding Capacity is computed at a given SNR using numerical
integration.
A non-linear optimizer is used to find the location of the points to give the highest
capacity.
• Design for a specific capacity:
Involves an iterative process as the SNR required to yield the required capacity is not
know apriori.
1. Optimize a constellation for an initial SNR guess.
2. Find the SNR at which the designed constellation gives the required capacity.
3. Optimize a constellation for the new SNR.
4. Repeat until there is no further significant reduction in SNR.
Optimization for PD Capacity yields labelings in addition to location
Constellation Designs 18
• Geometric shaping is a superset of probabilistic shaping given
cardinality growth is permitted
4S 8X
p(s) = 1/8 3/8 3/8 1/8 p(x) = 1/8 1/8 1/8 1/8 1/8
1/8 1/8 1/8
S X
* M. Barsoum, C. Jones and M. Fitz, “Constellation Design via Capacity Maximization”, ISIT, Nice 2007
22 SEXE
CJOINT CPD
M=2 M=4 M=8 M=16 M=32
Signaling with Regularly Spaced PAM
SNRPD(R) - SNRGauss(R)
SNRJoint(R) - SNRGauss(R)
-5 0 5 10 15 20 -4
-3
-2
-1
0
1
2
3
4
optimization specifies
(L o c a ti o n )
Evolution of point locations vs. SNR
-5 0 5 10 15 20 25 -10
-5
0
5
10
• Can be constructed using orthogonal optimized PAM constellations
• Optimization in 2 or more dimensions could give better results but is
more complex
-8 -6 -4 -2 0 2 4 6 8 -8
-6
-4
-2
0
2
4
6
8
Constellation Designs 24
Opt. for Joint / Gap to Joint Opt. for PD / Gap to PD
M=2 M=4 M=8 M=16 M=32 M=2 M=4 M=8 M=16 M=32
0.7 dB
1.4 dB
0.3 dB
1.8 dB
Constellation Designs 25
Code Rate Rule
where M is the constellation size
• Conventional Code Rate Rule:
an M point constellation
M=2 M=4 M=8 M=16 M=32
PAM
3/5 with PAM-32 vs 3/4
with PAM-16
7/10 with QAM-1024 vs
Constellation Designs 26
1. Optimized constellations can provide significant gains using the conventional code rate rule
– For example: 0.72 dB gain in joint capacity for PAM-32 at the
conventional code rate 4/5 compared to traditional PAM-32
2. Furthermore, the optimized constellations could provide further gains at code rates lower than the conventional rates
– For example: 0.88 dB gain in joint capacity for PAM-32 at rate 3/5
compared to traditional PAM-32
– 1.5 dB gain in parallel decoding capacity for PAM-32 at rate ½
compared to traditional PAM-32
• Protograph LDPC AR4JA rate-1/2
k = 4096, 16384 (k=4096 w/ and w/o decoder / detector iterations)
• PAM-32 system
-4
P DC
• Protograph LDPC
AR4JA rate-4/5
• k = 4096
• 802.11ac
8 8.5 9 9.5 10 10.5 11 11.5 12 10
-7
R
80211.AC Rate 1/2 Length 1944 Code with Standard GREY QAM 256 and Constellation Designs QAM 256
1944 Rate 1/2 256 CDQAM BER (4 bps)
1944 Rate 1/2 256 QAM BER (4 bps)
GREY Mapped 16 QAM BER Uncoded (4 bps)
Constellation Designs 30
• 802.11ac
• Grey QAM 256
• CD QAM 256
8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10 10
-5
F E
R /B
E R
80211.AC Rate 1/2 Length 1944 Code with Standard GREY QAM 256 and Constellation Designs QAM 256
1944 Rate 1/2 256 CDQAM BER (4 bps)
1944 Rate 1/2 256 CDQAM FER (4 bps)
1944 Rate 1/2 256 QAM BER (4 bps)
1944 Rate 1/2 256 QAM FER (4 bps)
Constellation Designs 31
Constellation Design Comparison
Uncoded Systems Coded Systems
Uniformly-spaced points Geometrically-shaped points
Code rate rule:
Constellation Designs 32
Conclusions
• Significant gains in capacity / SNR gap to Gaussian capacity have been achieved
using equiprobable but unequally spaced constellations.
• Constellations and labels are designed by direct optimization of the location of
the points.
• For PAM 32, a 1.5 dB gain in parallel decoding capacity has been achieved at
rate 1/2, and a gain of 0.7 dB at rate 4/5.
• Performance of systems using modern codes are shown to reflect these gains.
• Besides the conventional use of a (log2(M)-1)/log2(M) code rate with an M-point
constellation for bandwidth efficient communications, the optimized constellations
could offer further gains at lower code rates (unlike traditional constellations).
• Some margin in the target rate should be allowed when designing the
constellations.
• Significant gains in Joint and PD capacity have also been achieved for
the Raleigh fading channel assuming no Channel State Information (CSI)
at the transmitter and perfect CSI knowledge at the receiver.
Constellation Designs 33
Publications on this topic
• M. F. Barsoum, C. Jones, and M. Fitz, “Constellation design via capacity
maximization,” IEEE International Symposium on Information Theory,
Nice, France, June, 2007.
Tech Briefs, 2007, accepted.
method and apparatus for signaling with capacity optimized
constellations,” United States Patent Application 20090097582, April 2009.
• M. F. Barsoum, C. Jones, and M. Fitz, “Power and bandwidth efficient
communications using capacity maximizing constellations,” IEEE
Transactions on Information Theory, in preparation.
Related topic (optimality of gray labels on traditional constellations):
• M. Samuel, M. F. Barsoum, and M. Fitz, “On the suitability of gray bit
mappings to outer channel codes in iteratively decoded BICM,” IEEE
Asilomar Conference on Signals, Systems and Computers, November,
2009.
Citing Articles
• Optimal alphabets and binary labelings for BICM at low SNR, E Agrell, A Alvarado
• A novel BICM-ID system approaching Shannon-limit at high spectrum efficiency, Y Zhixing, XIE
Qiuliang, P Kewu
• BICM transmission using non-uniform QAM constellations: Performance analysis and design,
MJ Hossain, A Alvarado
• Constellation design for improved iterative LDPC decoding, EL Valles
• On the Capacity of BICM with QAM Constellations, A Alvarado, E Agrell, A Svensson
• Efficient utilization of channel state information in modern wireless communication systems, C
Shen
• Signal Shaping for BICM at Low SNR, E Agrell, A Alvarado
• Towards fully optimized bicm transceivers, MJ Hossain, A Alvarado
• Constrained capacities of dvb-s2 constellations in log-normal channels at ka band, S Enserink,
MP Fitz
• Achieving the Shannon limit with probabilistically shaped BICM, E Agrell, A Alvarado
• Joint signallabeling optimization under peak power constraint, F Kayhan, G Montorsi
• Constrained Capacities in Log-Normal Channels with Site Diversity at Ka Band, S Enserink, MP
Fitz
• A utility maximization approach to the design of unequal error protection with multilevel codes, C
Shen, MP Fitz
Constellation Designs 35
Citing Articles
• Constellation Shaping for SISO/MIMO Systems, B Gretarsson, SM Pérez, M Loncar, LPB
Christensen
• Channel Coding for IDM: High-Rate Convolutional Code Concatenated with Irregular Repetition
Code, M Noemm, PA Hoeher, Y Wang
o Failure of Traditional Constellation Designs with Modern Codes
o Proposed Constellation Design
o Conclusions
Shaping
Gap
• This 25% improvement has been elusive
due primarily to complexity / coding system
compatibility constraints
CJOINT CPD
• Shaping gain has typically been thought of as a signal
processing / coding procedure by which points from a
given constellation are transmitted with differing
frequencies. The frequencies are chosen such that the
PDF formed by the final waveform has a Gaussian
distribution.
inherent difficulties associated with encoding and decoding
points with differing frequencies.
• Labeling
• Probability Distribution
• Maximize performance under a constellation energy constraint i.e. at any given SNR
• Equal energy constellations are sometimes preferred
2
2
• Equiprobable Signaling
– Reasonable with classic codes requiring high SNR
– Could fail with modern codes that operate at low SNR
• Geometrically shaped constellations
– Gaussian Mimicking (Duan, Rimoldi & Urbanke: 1997, Sommer & Fettweis: 2000)
– Capacity maximization approach (Barsoum & Jones)
• Non-Equiprobable Signaling (Probabilistic Shaping)
(data dependent rate, buffering, synchronization)
• Pseudo-probabilistic Shaping:
- Shell Shaping (V.34 modems)
Minimum Distance-Based Constellation Design
dmin criteria is based on an approximation of the probability of error using the
union bound of the pair-wise error probabilities for an uncoded model
• Coded model (Practical system)
distance criteria
• Traditional QAM
minimum distance criteria
distance criteria
• Minimum distance based enhanced
• Conjectured to be the best 16 points QAM
constellation
• Better minimum distance constellations in
multi-dimensions -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(Conway and Sloane 1983)
Geometric Design Example 2: Warped Constellations
• Betts, Calderbank, Laroia (1994) demonstrated a gain of 0.25 dB when using
a warped constellation with a trellis coded modulation system.
* AT&T, “Nonlinear encoding for V.fast,” Contribution 21, TIA-TR 30.1, Ft. Lauderdale, FL, Feb 1992
• A uniformly spaced constellation is
transformed to a warped
constellation using some warping
-3 -2 -1 0 1 2 3 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
code rates.
decoding capacity with 32
PAM at rate ½
SNR’s
guaranteed to be optimum
Does not work as expected with LDPC or Turbo Codes!
• 16 points Voronoi constellation in 4-D
• Minimum distance is 0.74 dB better than QPSK
• However, is more than 2 dB worse with a modern code!!!
0 1 2 3 4 10
-4
User rate = 0.5bit/dimension
• Gaussian input distribution
achieves maximum capacity
• Uniformly-spaced infinite PAM
capacity
asymptotically achieve the
• Capacity constrained to signal constellation X
and to a soft bit demapper (w/o feedback) in
the channel
• Label dependent
• Label Independent
B YX
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
g C
e r
d im
-4
• User rate = 0.5 bit/constellation dimension
• Voronoi is the ‘best distance’ constellation design, but has 2dB worse
capacity (why optimize for block size == 1 ?)
dB2
Observation1: Constellation Design Methodology
• Optimize the location of the constellation points to maximize capacity or
parallel decoding capacity:
• Design for a specific SNR:
Capacity or Parallel Decoding Capacity is computed at a given SNR using numerical
integration.
A non-linear optimizer is used to find the location of the points to give the highest
capacity.
• Design for a specific capacity:
Involves an iterative process as the SNR required to yield the required capacity is not
know apriori.
1. Optimize a constellation for an initial SNR guess.
2. Find the SNR at which the designed constellation gives the required capacity.
3. Optimize a constellation for the new SNR.
4. Repeat until there is no further significant reduction in SNR.
Optimization for PD Capacity yields labelings in addition to location
Constellation Designs 18
• Geometric shaping is a superset of probabilistic shaping given
cardinality growth is permitted
4S 8X
p(s) = 1/8 3/8 3/8 1/8 p(x) = 1/8 1/8 1/8 1/8 1/8
1/8 1/8 1/8
S X
* M. Barsoum, C. Jones and M. Fitz, “Constellation Design via Capacity Maximization”, ISIT, Nice 2007
22 SEXE
CJOINT CPD
M=2 M=4 M=8 M=16 M=32
Signaling with Regularly Spaced PAM
SNRPD(R) - SNRGauss(R)
SNRJoint(R) - SNRGauss(R)
-5 0 5 10 15 20 -4
-3
-2
-1
0
1
2
3
4
optimization specifies
(L o c a ti o n )
Evolution of point locations vs. SNR
-5 0 5 10 15 20 25 -10
-5
0
5
10
• Can be constructed using orthogonal optimized PAM constellations
• Optimization in 2 or more dimensions could give better results but is
more complex
-8 -6 -4 -2 0 2 4 6 8 -8
-6
-4
-2
0
2
4
6
8
Constellation Designs 24
Opt. for Joint / Gap to Joint Opt. for PD / Gap to PD
M=2 M=4 M=8 M=16 M=32 M=2 M=4 M=8 M=16 M=32
0.7 dB
1.4 dB
0.3 dB
1.8 dB
Constellation Designs 25
Code Rate Rule
where M is the constellation size
• Conventional Code Rate Rule:
an M point constellation
M=2 M=4 M=8 M=16 M=32
PAM
3/5 with PAM-32 vs 3/4
with PAM-16
7/10 with QAM-1024 vs
Constellation Designs 26
1. Optimized constellations can provide significant gains using the conventional code rate rule
– For example: 0.72 dB gain in joint capacity for PAM-32 at the
conventional code rate 4/5 compared to traditional PAM-32
2. Furthermore, the optimized constellations could provide further gains at code rates lower than the conventional rates
– For example: 0.88 dB gain in joint capacity for PAM-32 at rate 3/5
compared to traditional PAM-32
– 1.5 dB gain in parallel decoding capacity for PAM-32 at rate ½
compared to traditional PAM-32
• Protograph LDPC AR4JA rate-1/2
k = 4096, 16384 (k=4096 w/ and w/o decoder / detector iterations)
• PAM-32 system
-4
P DC
• Protograph LDPC
AR4JA rate-4/5
• k = 4096
• 802.11ac
8 8.5 9 9.5 10 10.5 11 11.5 12 10
-7
R
80211.AC Rate 1/2 Length 1944 Code with Standard GREY QAM 256 and Constellation Designs QAM 256
1944 Rate 1/2 256 CDQAM BER (4 bps)
1944 Rate 1/2 256 QAM BER (4 bps)
GREY Mapped 16 QAM BER Uncoded (4 bps)
Constellation Designs 30
• 802.11ac
• Grey QAM 256
• CD QAM 256
8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10 10
-5
F E
R /B
E R
80211.AC Rate 1/2 Length 1944 Code with Standard GREY QAM 256 and Constellation Designs QAM 256
1944 Rate 1/2 256 CDQAM BER (4 bps)
1944 Rate 1/2 256 CDQAM FER (4 bps)
1944 Rate 1/2 256 QAM BER (4 bps)
1944 Rate 1/2 256 QAM FER (4 bps)
Constellation Designs 31
Constellation Design Comparison
Uncoded Systems Coded Systems
Uniformly-spaced points Geometrically-shaped points
Code rate rule:
Constellation Designs 32
Conclusions
• Significant gains in capacity / SNR gap to Gaussian capacity have been achieved
using equiprobable but unequally spaced constellations.
• Constellations and labels are designed by direct optimization of the location of
the points.
• For PAM 32, a 1.5 dB gain in parallel decoding capacity has been achieved at
rate 1/2, and a gain of 0.7 dB at rate 4/5.
• Performance of systems using modern codes are shown to reflect these gains.
• Besides the conventional use of a (log2(M)-1)/log2(M) code rate with an M-point
constellation for bandwidth efficient communications, the optimized constellations
could offer further gains at lower code rates (unlike traditional constellations).
• Some margin in the target rate should be allowed when designing the
constellations.
• Significant gains in Joint and PD capacity have also been achieved for
the Raleigh fading channel assuming no Channel State Information (CSI)
at the transmitter and perfect CSI knowledge at the receiver.
Constellation Designs 33
Publications on this topic
• M. F. Barsoum, C. Jones, and M. Fitz, “Constellation design via capacity
maximization,” IEEE International Symposium on Information Theory,
Nice, France, June, 2007.
Tech Briefs, 2007, accepted.
method and apparatus for signaling with capacity optimized
constellations,” United States Patent Application 20090097582, April 2009.
• M. F. Barsoum, C. Jones, and M. Fitz, “Power and bandwidth efficient
communications using capacity maximizing constellations,” IEEE
Transactions on Information Theory, in preparation.
Related topic (optimality of gray labels on traditional constellations):
• M. Samuel, M. F. Barsoum, and M. Fitz, “On the suitability of gray bit
mappings to outer channel codes in iteratively decoded BICM,” IEEE
Asilomar Conference on Signals, Systems and Computers, November,
2009.
Citing Articles
• Optimal alphabets and binary labelings for BICM at low SNR, E Agrell, A Alvarado
• A novel BICM-ID system approaching Shannon-limit at high spectrum efficiency, Y Zhixing, XIE
Qiuliang, P Kewu
• BICM transmission using non-uniform QAM constellations: Performance analysis and design,
MJ Hossain, A Alvarado
• Constellation design for improved iterative LDPC decoding, EL Valles
• On the Capacity of BICM with QAM Constellations, A Alvarado, E Agrell, A Svensson
• Efficient utilization of channel state information in modern wireless communication systems, C
Shen
• Signal Shaping for BICM at Low SNR, E Agrell, A Alvarado
• Towards fully optimized bicm transceivers, MJ Hossain, A Alvarado
• Constrained capacities of dvb-s2 constellations in log-normal channels at ka band, S Enserink,
MP Fitz
• Achieving the Shannon limit with probabilistically shaped BICM, E Agrell, A Alvarado
• Joint signallabeling optimization under peak power constraint, F Kayhan, G Montorsi
• Constrained Capacities in Log-Normal Channels with Site Diversity at Ka Band, S Enserink, MP
Fitz
• A utility maximization approach to the design of unequal error protection with multilevel codes, C
Shen, MP Fitz
Constellation Designs 35
Citing Articles
• Constellation Shaping for SISO/MIMO Systems, B Gretarsson, SM Pérez, M Loncar, LPB
Christensen
• Channel Coding for IDM: High-Rate Convolutional Code Concatenated with Irregular Repetition
Code, M Noemm, PA Hoeher, Y Wang
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