In his seminal 1948 paper 'The Mathematical Theory of Communication,' Claude E. Shannon derived the 'channel coding theorem' which has an explicit upper bound, called the channel capacity, on the rate at which 'information' could be transmitted reliably on a given communication channel. Shannon's result was an existence theorem and did not give specific codes to achieve the bound. Some skeptics have claimed that the dramatic performance improvements predicted by Shannon are not achievable in practice. The advances made in the area of coded modulation in the past decade have made communications engineers optimistic about the possibility of achieving or at least coming close to channel capacity. Here we consider the possibility in the light of current research results

Costello, Daniel J., Jr.

Courturier, Servanne

Levy, Yannick

Mills, Diane G.

Perez, Lance C.

Wang, Fu-Quan

English

NASA Technical Reports Server

N94-16507
- Bandwidth Efficient Coding:
Theoretical Limits and Real Achmvements
Interim Report
Error Control Techniques
for Satellite and Space Communications
NASA Grant NAGS-557
September 1993
Faculty
Dr. Daniel J. Costello, Jr.
Graduate Students
Servanne Couturier
Yannick Levy
Diane G. Mills
Lance C. Perez
Post-Doctoral Associate
Fu-Quan Wang
n
w
University of Notre Dame
Department of Electrical Engineering
Notre Dame, IN 46556
https://ntrs.nasa.gov/search.jsp?R=19940012034 2020-06-16T20:13:57+00:00Z
Introduction
_o In his seminal 1948 paper "The Mathematical Theory of Communica-
tion", Claude E. Shannon derived the "channel coding theorem"
._ which gave an explicit upper bound, called the channel capac-
ity, on the rate at which "information" could be transmitted
- reliably on a given communication channel.
• Shannon's result was an existence theorem and did not give spe-
cific codes to achieve the bound. Some skeptics have claimed
- that the dramatic performance improvements predicted by Shan-
non are not achievable in practice.
w
• The advances made in the area of coded modulation in the past
m
capacity. Here we consider this possibility in the light of current
research results.
w
decade have made communications engineers optimistic about
the possibility of achieving or at least coming close to channel
Channel Capacity
_, With respect to coding and coded modulation, the most relevant
of Shannon's results is the "noisy channel coding theorem for
- continuous channels with average power limitations."
:i This theorem states that for any transmission rate R less than
or equal to the channel capacity, C, there exists a coding scheme
that achieves an arbitrarily small probability of error!
, Conversely, if R is greater than C, no coding scheme can achieve
- reliable communication, regardless of complexity.
Shannon then shows that the capacity, C, of a continuous addi-
w
tive white Gaussian noise
and assuming Nyquist signaling is given by
(AWGN) channel with bandwidth B
C= B log2 (l + _o) bits/sec, (1)
where E_ is the average signal energy in each signaling interval
T and N0/2 is the two sided noise power spectral density.
This bound represents the absolute best performance possible
- for a communication system on the AWGN channel.
Restatement of the Capacity Bound
i Shannon's capacity bound can be put in a form more useful for
the present discussion by introducing the parameter rl, called
"- spectral efficiency, to represent the average number of informa-
tion bits transmitted per signaling interval.
J
-, From Shannon's bound, it follows that
0 <__R <_ C bits/sec,
- and hence
0 < _7<_ C/B bits signal.
, Substituting the relation
E_/No = _TEb/No,
where Eb is the average energy per information bit, into equation
(1) and performing some minor manipulations yields
27 - i
Eb/No >_ , (2)
which relates the spectral efficiency, rl, to the
: ratio (SNR), Eb/No.
signal-to-noise
-_ The bound of equation (2) manifests the fundamental tradeoff
between spectral efficiency and SNR. That is, increased spectral
efficiency can be reliably achieved only with a corresponding
increase in SNR.
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Interpretation of the Capacity Curve
V
w
i
3.5
3.0
2.5
2.0
o_
_" 1.5
l.O
0.5
_- 0.0
-2.0
w
......... I ......... I ......... I ......... I ...... _ ''1 .........
-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
Eb/No (dB)
-. Shannon's bound gives the minimum signal-to-noise ratio (SNR)
_ required to achieve a specific bandwidth efficiency with an at-
- bitrarily small probability of error.
-. Example: With r/= 2 information bits per channel signal, there
exists a coding scheme that operates reliably with an SNR of
- 1.76dB.
= :
-• Conversely, any coding scheme sending W = 2 information bits
per signal with an SNR less than 1.76dB will be unreliable, regard-
- less of complez'ity.
=m
w
w
mm_
Interpretation of the Capacity Curve
3.5
3.0
2.5
2.0
,.Q
"-" 1.5
1.0
0.5
0.0
-2.0 -,.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
_No (dS)
• Alternatively, Shannon's bound gives the maximum achievable
spectral efficiency for a specific signal-to-noise ratio (SNR).
• Example: With an SNR of Eb/No = 1.76db, there exists a coding
scheme cabable of transmitting reliably with a spectral efficiency
of _ = 2 bits per signal.
• Conversely, any coding scheme operating with an SNR of Eb/No =
1-.76dB and attempting to transmit more than _ = 2 bits per signal
will be unrealiable regardless of complexity.
INASA (2,1,6) Convolutional Code
• Historically,
below capacity.
• Example: The NASA standard rate (2,1,6) convolutional code
with QPSK modulation achieves a spectral efficiency of r/= 1
_ bit/signal and requires a signal-to-noise ratio (SNR) of Eb/No =
4.15 dB to achieve error free (10 -5 bit error rate) communication.
J
• An ideal system operating with the same Eb/No - 4.15413 can
achieve error free communication with a spectral efficiency as
high as rl = C = 3.235 bits per signal.
OR
An ideal system operating with the same spectral efficiency of
= 1 bit per signal would require an SNR of only Eb/No = 0.0
dB.
A New Optimal (2,1,14) Convolutional Code
c'J
O
..d
.!--i
10_
10 "1
10":
10 .3
10 .4
10"5
106
(2,1,14) Union Bound
(2,1,14) Simulation
(2,1,6) Simulation
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Eb/No (dB)
• A computer search has found the optimum distance spectrum
(ODS) (2,1,14) convolutional code.
• This code has optimum minimum free Hamming distance, d/,.e_ =
18, and the smallest number of nearest neighbors, Ny,._e = 26, of
any constraint length 15 code.
• The performance of this code is 1.65 dB better than the (2,1,6)
code, but is still 2.5dB away from capacity.
ii
m
n
Optimal (2,1,14) Convolutional Code
Eb/No (dB)
_ • The new optimal (2,1,14) convolutional code requires a signal-
to-noise ratio (SNR) of Eb/No = 2.5 dB to achieve error free (10 -5
_ bit error rate) communication.
• An ideal system operating with the same Eb/No = 2.5dB can
achieve error free communication with a spectral efficiency as
high as _ = C = 2.4 bits per signal.
OR
An ideal system operating with the same spectral efficiency of
r/= 1 bit per signal would require an SNR of only Eb/No = 0.0
dB.
UPractical Bounds
m
• In real communication systems, there are many practical consid-
erations that take precedence over Shannon's bound in design
decisions.
• For example, satellite communication systems that use nonlin-
ear travelling wave tube amplifiers (TWTA's) require constant
envelope signaling such as M-ary phase shift keying (MPSK).
• Thus, even if Shannon's results firmly stated that capacity at
_ a spectral efficiency of r/ = 3 bits per signal can be achieved
with a (4,3,8) convolutional code using 16 QAM, it would not
_ be feasible to do so on the TWTA satellite link.
• It therefore seems reasonable to ask what is the minimum SNR
required to achieve reliable communication, given a particular
modulation scheme and a spectral efficiency, r/.
m
=m-m
A Signal Specific Bound
- • For the discrete input, continuous output, memoryless AWGN
channel with M-ary one dimensional amplitude modulation (AM)
L
- or two dimensional (PSK, QAM) modulation and equiprobable
signaling, the capacity bound becomes
= E log 2 exp
rf log2 (M) _I i=o j=o -No ' (3)
• Here
{aJ,j=O, 1,...M-1} (4)
is an M-ary modulations set, aJ is a channel signal, n is a Gaus-
sian distributed noise random variable with mean 0 and variance
No 2, and E is the expectation operator.
For a specified signaling method and spectral efficiency, this
bound can be used to compute the minimum SNR required to
achieve reliable communication.
Interpretation of the Signal Specific Bound
3.0
8PSK
QPSK
BPSK
°°2.0 -_.0 0.0 _.0 2.0 3.0 4.0 5.0
- Eb/No(clB)
- • To send q = 1.5 information bits per signaling interval, an ideal
system using QPSK modulation requires a minimum SNR of Eb/No
1.64dB. This is 0.76 dB more than an ideal system without any
modulation constraints.
• To send q = 1.5 information bits per signaling interval, an ideal
system using 8PSK modulation requires a minimum SNR of Eb/No
1.22dB. This is 0.34 dB more than an ideal system without any
modulation constraints.
mw
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uU
Notes About the Graph
All results are with soft decision decoders.
1 NASA Codes: denoted by ×
Galileo: This is the concatenated code used on the Galileo probe.
(4,1,14) BPSK The is the (4,1,14) convolutional code developed by JPL.
Voyager: This is the concatenated code used on the Voyager probe.
(2,1,6) BPSK: This is the NASA standard (2,1,6) convolutional code with BPSK modu-
lation.
(2,1,6) QPSK: This is the NASA standard (2,1,6) convolutional code with QPSK modu-
lation.
i
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2 Pietrobon-Costello Codes: denoted by 0
(6,5,4) 2x8PSK: This is a rate 5/6, 16 state, trellis code using 2x8PSK modulation with
Viterbi decoding. It has a spectral efficiency of _7= 2.5 bits/signal and is 45* rotation-
ally invariant. A Viterbi decoder for this code has been built by Steven Pietrobon and
tested at New Mexico State.
(9,8,6) 4x8PSK: This is a rate 8/9, 64 state, trellis code using 4xSPSK modulation with
Viterbi decoding. It has a spectral efficiency of rI = 2.0 bits/signal and is 45 ° rotation-
ally invariant.
(11,10,6) 4x8PSK: This is a rate 10/11, 64 state, trellis code using 4xSPSK modulation
with Viterbi decoding. It has a spectral efficiency of rI = 2.5 bits/signal and is 45*
rotationally invariant.
(7,6,6) 2xl6PSK: This is a rate 6/7, 64 state, trellis code using 2xl6PSK modulation
with Viterbi decoding. It has a spectral efficiency of rI = 3.0 bits/signal and is 45*
rotationally invariant.
(4,3,6) 16QAM, nonlinear: This is a nonlinear rate 3/4, 16 state, trellis code using
16QAM modulation with Viterbi decoding. It has a spectral efficiency of 7/ = 3.0
bits/signal and is 90 ° rotationally invariant.
Mn
=
3 Wang-Costello Codes: denoted by []
(3,2,17)SPSK: This is a rate
sequential decoding with
rI = 2.0 bits/signal and is
2/3, memory 17, trellis code using 8PSK modulation and
a modified Fano algorithm. It has a spectral efficiency of
180 ° rotationally invariant.
(4,3,16) 16PSK: This a rate
sequential decoding with
r1 = 3.0 bits/signal and is
3/4, memory 16, trellis code using 16PSK modulation and
a modified Fano algorithm. It has a spectral efficiency of
180 ° rotationally invariant.
(4,3,16) 16QAM: This is a rate 3/4, memory 16, trellis code using 16QAM modulation
and sequential decoding with a modified Fano algorithm. It has a spectral efficiency
of 77= 3.0 bits/signal and is 180 ° rotationally invariant.
4 Lin Codes: denoted by
(32,16,8) RM: This is a rate 16/32=0.5, 64 state Reed-Muller code using BPSK modula-
tion and Viterbi decoding.
(64,42,8) RM: This is a rate 42/64=0.656, 1024 state Reed-Muller code using BPSK mod-
ulation and Viterbi decoding.
(17,16,2) 7xSPSK, TBCM: This is rate 16/17, 4 state, block coded modulation scheme
using 7xSPSK modulation with Viterbi decoding. It has a spectral efficiency of 7/=
2.286 bits/signal and is 180 ° rotationally invariant.
(18,16,6) 8x8PSK, TBCM: This is a 2-level trellis code using 8xSPSK modulation. The
first level has 64 states and is decoded with a Vit,'.rbi decoder. The second level has
8 states and is deocded with a Viterbi decoder. It has a spectral efficiency of rt = 2.0
bits/symbol and is 1800 rotationally invariant.
(3069,2799) 16x8PSK, PBCM: This is a 3x3 product block coded modulation scheme.
The horizontal codes are BCH codes and the vertical code is a 3-level block code. It is
decoded using suboptimal multi-stage decoding. It has a spectral efficiency of r/= 2.1
bits/signal and is 45 ° rotationally invariant.
5 Viterbi Pragmatic Code: denoted by •
(3,2,6) 8PSK, Pragmatic: This is a rate 2/3, 64 state, trellis code using 8PSK modulation
and Viterbi decoding. It uses the NASA standard (2,1,6) convolutional code as its basis
and is suboptimal. It can be decoded using essentially the same Viterbi decoding chip
that is used to decode the NASA standard convolutional code.
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42
m
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r._
¢)
-.o
_S
0
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0
0
0
0
d
0
Notes About the Graph
All results are with soft decision decoders.
m
1 Uncoded Systems: denoted by x
16QAM: Performance of uncoded 16QAM with a spectral efficiency of rt = 4 bits/symbol
from simulation results.
32QAM: Performance of uncoded 32QAM with a spectral efficiency of rI = 5 bits/symbol
from simulation results.
64QAM: Performance of uncoded 64QAM with a spectral efficiency of rt = 6 bits/symbol
from simulation results.
128QAM: Performance of uncoded 128QAM with a spectral efficiency of 7/= 7 bits/symbol
from simulation results.
2
Wei
Wei
Wei Codes: denoted by 0
4D, 8 state: This is Wei's 8 state code using a 4-dimensional constellation (of any
size). Performance taken from "Coset Codes - Part 1: Introduction and Geometrical
Classification," by G. David Forney. The performance was estimated taking into ac-
count the minimum squared Euclidean distance and the number of nearest neighbors.
Thus, this point shows effective coding gain at a bit-error-rate (BER) of 10 -s.
4D, 16 state: This is Wei's 16 state code using a 4-dimensional constellation (of any
size). Performance taken from "Coset Codes - Part 1: Introduction and Geometrical
Classification," by G. David Forney. The performance was estimated taking into ac-
count the minimum squared Euclidean distance and the number of nearest neighbors.
Thus, this point shows effective coding gain at a bit-error-rate (BER) of 10 -s.
This code with a 2x192QAM constellation is being considered by CCITT for the
V.FAST ultimate modem standard.


Bandwidth efficient coding: Theoretical limits and real achievements. Error control techniques for satellite and space communications

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