Growth in satellite mobile communications leads to increasing requirements for high data rate transmission that can be met by more efficient modulation schemes (M greater than 8). The 16-PSK trellis coded modulation technique is a very promising solution. A class of new 16-PSK trellis codes with improved error rate are designed based on the criteria on fading channels

Du, Jun

Vucetic, Branka

Zhang, Lin

English

NASA Technical Reports Server

N9 2- 2 49
New 16-PSK Trellis Codes for
Fading Channels
Jun Du, Branka Vucetic and Lin Zhang
The School of Electrical Engineering
The University of Sydney, NSW 2006, Australia
Phone: 62-2-692-2154 Fax: 62-2-692-3847
Abstract
Growth in satellite mobile communications leads to
increasing requirements for high data rate transmis-
sion that can be met by more efficient modulation
schemes (M > 8). The 16-PSK trellis coded mod-
ulation technique is a very promising solution. In
this paper, a class of new 16-PSK trellis codes with
improved error rate are designed based on the cri-
teria on fading channels.
1 Introduction
The expansion of satellite communications implies
the use of higher radio frequencies (SHF and EHF
regions where a major disadvantage associated with
such systems is the signal attenuation due to fading.
Most of the TCM codes constructed for Gaussian
channels suffer significantly degradation on fading
channels. It has been pointed out[2][3114] that the
codes for fading channels should be designed by us-
ing the criteria different from those for Gaussian
channels.
Some efforts have been made to constructing new 8-
PSK TCM codes for fading channels[3]. The growth
in satellite communications, however, leads to in-
creasing requirements for high data rate transmis-
sion that can be met by more efficient modulation
schemes (M > 8). So far, the efforts on construction
of 16-PSK TCM codes were concentrated on Gaus-
sian channels [1][.5][6][71. The problem of designing
16-PSK TCM codes for fading channels is studied in
this paper. A class of new 16-PSK trellis codes with
improved performance on fading channels are pro-
posed based on the code design criteria for fading
channels. The performances of the 16-PSK TCM
systems on Rayleigh fading channels are evaluated
both theoretically and by Monte Carlo simulations.
2 System Description
In the system under consideration, the digital
source generates data digit stream at a rate of 1/Tb.
Fro the 3-input information sequence, the convolu-
tional encoder generate 4-output coded sequence.
The coded sequences are then interleaved and used
to modulate 16-PSK waveform at a rate 1/3Tb sym-
bols/s according to symbol mapping rules. The
channel is described by a slowly varying frequency
non-selective Rayleigh fading. The receiver consists
of a coherent demodulator followed by the deinter-
leaver and a Viterbi algorithm.
The channel is modelled by non-selective t/ayleigh
fading. We shall restrict attention to channels with
slow amplitude variations relative to an elementary
signaling interval and assume perfect phase tracking
of the phase perturbation process. The interleaving
depth is chosen to achieve ideal interleaving for a
given fade rate in the simulations and assumed infi-
nite in the analysis. In this model, the amplitude of
the envelope of the received signal is multiplied by
a random variable a, normalized so that E[a 2] = 1
to ensure that the received signal energy is equal to
the transmitted signal energy Es. The probability
distribution function of a is
p(a) = 2de -a2 (1)
The mean of the random variable a is
ma - (2)2
while the variance is given by
2 1 m_ (3)O"a -- --
International Mobile Satellite Conference, Ottawa. t990
pt::_ECEDtNG PAGE BLANK NOT F!LMED
481
https://ntrs.nasa.gov/search.jsp?R=19920014906 2020-03-17T12:28:11+00:00Z
3 Error Probability and Code
Design Criteria
Figure 1 shows a 16-PSK signal constellation and
the notation where A; denotes the Euclidean dis-
tance between channel symbols. Let C denote the
set of all possible coded signal sequences {ZN},
where ZN is a coded symbol sequence of length N:
ZN = (ZI,Z2,...,Zi,...,ZN)
Let us suppose that VN:
V N ---- (Vl,V2,...,Vi,...,VN)
an element of C, is the transmitted sequence. Then
vi represents the MPSK symbol transmitted at time
i. The corresponding received sequence at the input
of the decoder is:
rN = (rl,ra,...,rh...,rN)
At time i, the received symbol ri is related to the
transmitted symbol v_ by:
ri= aivi+ni
where ai is a complex random variable equal to the
envelope of the fading channel normalized to the
transmitted signal and ni is a sample of a zero mean
complex additive white Gaussian noise (AWGN)
with variance a 2 in each signal space coordinate,
The corresponding received sequence rN can be rep-
resented as
rN = aN X V N + nN
where x denotes vector multiplication.
In the Viterbi algorithm, the path metric m(rN, ZN)
is the sum of the branch metrics:
N
m(rN,ZN) = E m(ri,zi)
i=l
The branch metric used in the algorithm is equal to
the squared Euclidean distance:
mi = Iri- zil 2 (4)
Definitions
In the context of fading communications, besides
the minimum Euclidean distance, d,, and the num-
ber of paths at distance de from the correct path,
N(de), the following parameters play important
roles:
1) Effective Code Length (ECL): the minimum ef-
fective length taken over all error paths in C where
the Effective Length, denoted by $, of an error path
is the number of erroneous symbols on the error
path, and can be expressed as
M]2-1
5":. w; (5)
i=O
where M is the number of signal points in the sig-
nal constellation and wi are the weights associated
with the corresponding nonzero symbol distances
Ai along the error path. The ECL of a TCM code
is denoted by _H in this paper.
2) Minimum Product Distance: the minimum prod-
uct distance taken over all paths with the ECL
where Product Distance , denoted by Ap, is the
product of the corresponding nonzero squared sym-
bol distances along the error path and is given by
M/2-1
_P = H m2wi (6)
i=0
The minimum PD is denoted by dp in this paper.
Similar to N(de), we have N(dp) associated with
dp, which is the number of the paths at product dis-
tance dp from the correct path, In this paper both
N(de) and N(dp) are treated as average numbers in
the same way as in [7] and [10] where the average
is taken over all paths in the trellis.
Pairwise Error Probability
The pairwise error probability of choosing the path
ZN instead of VN (first error event probability) is
equal to
P(VN _ ZN)= Pr{m(rN,ZN)< m(rN, VN)}
or equivalently
P(vN  N) =
Pr{2Re[(zN - VN)*" (rN -- VN)] > m(ZN,VN)}
(7)
where * denotes the complex conjugate and • de-
notes tile scalar product of two vectors. The
482
International Mobile Satellite Conference, Ottawa, 1990
squared Euclidean distance between the transmit-
ted and error path is given by
N N
d2(-N,vN)= E Izi- =
i=l i=l
Since di is equal to one of the Aj, 0 < j < M/2,
(Fig. 1) d2(rN,ZN) can can be expressed as
M/2-1
d2( rN' ZN) -- Z wiAi2 (8)
i=0
With infinite interleaving, the sequence aN of fading
amplitudes constitutes an independent and iden-
tically distributed sequence. Equation (7) can be
rewritten as
P(VN --, ZN) = Pr{2Re [(ZN -- VN)'" nN] >
d2(ZN, VN) -- 2Re [(ZN -- VN)*" (aN X v N -- VN)])
(9)
Let us suppose for simplicity that aN is real and the
modulation scheme is M-PSK. Then the right hand
side of Eq.(9) can be simplified by noting that:
-2Re [(Z N -- VN)*" VN] -- d2(ZN, VN)
and
N
-2Re[(zN- VN)* '(aN X VN)] = _--_ aid_
i=l
Then Eq.(9) can be rewritten as
P(VN "* ZN) =
N
Pr{2Re [(ZN -- VN)*. nN]_ _ aid_} (10)
i=l
For a given realization of the fading variable aN, the
term on the left hand side of (10) is a zero-mean
Gaussian random variable with variance 4a2d 2 and
the term on the right hand side is a constant which
depends on aN. The conditional pairwise error
probability for a given realization of the fading ran-
dom variable aN is given by
[v"N ad2\
[L-,i=l i i / (11)P(VN- ZNfaN)=Q\ ]
where Q(x) is the complementary error function de-
fined by
1
q(x) - [ e-t2/2dt (12)
Jx
In this analysis, we shall distinguish two cases, de-
pending on the value of the ECL _H. For large SH,
using the approximation Q(x) < 1/2exp(-x2/2) for
x >> 1, P(VN _ ZN) can be bounded by or equiva-
lently
P(VN "_ ZN) =
1 2ad 2 (Pd,D < 2 x/4a2d 2 + aaD 4exp
with
(13)
N
D = _ d_ (14)
i=l
The formula (13) can be used for codes with large
ECL for any SNR or for codes with medium ECL
for small SNR where the influence of error paths
with high effective lengths is significant. The bit
error probability is upperbounded by
1
Pb __ V Z bd,DPd, D (15)
l'L
d,D
where bd, D is the total number of erroneous infor-
mation bits on M1 paths at the Euclidean distance
d from the transmitted path characterized by the
distance D and k is the number of information bits
in a symbol.
In the second case, when 6H is small, the Gaussian
assumption is no longer valid and the probability
has to be expressed as or equivalently
LN
For large signal-to-noise ratios, we can write
P(VN --* ZN) ----
P(VN --' ZNlaN)p(al)... p(aN)dal ... daN
(16)
1 fi 8a2d 2
P(VN -'_ ZN) _< _ dti=1
or equivalently
1 (8crd) 2_
2 r-i M/2-1 4wi
Xli=0 Ai
P(VN --_ ZN) = Pd,w -<
(17)
(18)
It should be pointed out that the formula (18) is
valid for short ECL codes and large SNR only.
The bit error probability is obtained by using the
union bound technique as
1
Pb _< _ Z bd,wPd,w (19)
d,W
International Mobile SoteBite Conference, Ottawa, _990
483
where bd,w is the total number of erroneous infor-
mation bits on all paths characterized by the Eu-
clidean distance d and the set w.
Code Design Criteria
The analytical results indicate that the design cri-
teria for fading channels will depend on ECL val-
ues. For the codes with large ECL, the error prob-
ability (Eqs. (13)) is dominated by the minimum
Euclidean distance, while for the codes with small
ECL, Eq.(18) and Eq.(19 show that the error per-
formance is mostly dependent on the ECL, the min-
imum PD and the number of erroneous bits on error
paths, bd.
To construct good codes for fading channel, it is
very important to avoid having parallel transitions
in the trellis. All the previously known 16-PSK and
16-QAM codes[6][5][7], however, have parallel tran-
sitions. An upper bound on the maximum ECL has
been obtained[8]. Table 1 shows the upper bounds
on ECL, Lm, for 16-ary trellis codes with constraint
length u.
According to the possible values of ECL, to min-
imize error probability on fading channels one
should attempt to
• when the possible ECL is small
(I-a) design the codes with maximum ECL and
(I-b) among the codes obtained form (I-a) max-
imize dp first, and then maximize de
(I-c) minimize N(dp) and N(de) among the
found codes
• when the possible ECL is large
(II-a) maximize de
(II-b) minimize N(de) among the found codes
The criterion set I can be used to construct the
codes with small possible ECL. It is possible to
construct new 16-PSK TCM codes with no paral-
lel transitions, though the minimum Euclidean dis-
tances are inferior to the known codes.
4 Code Construction
A computer search based on the defined criteria has
been carried out in order to find out the best codes
for fading channels. To minimize bit error rate, the
Gray mapping (i.e., the neighboring symbols in a
constellation differ only by one bit) is adopted in
code construction.
A stack algorithm for calculating the minimum free
Hamming distance of convolutional codes has been
used for calculating the minimum free Euclidean
distance of trellis codes by replacing the ttam-
ming weight of channel symbols by the Euclidean
weight[l]. This algorithm is referred to as Algo-
rithm 1 in this paper.
A modified version of this algorithm, which enables
to compute not only de but also N(de), has been
developed[9]. In the modified algorithm, additional
information on the multiplicity of the merging paths
to a state with the lowest Euclidean weight is added
into the stack memory. The search does not termi-
nate de is reached, but continues till all the paths at
de are found. We refer to this modified algorithm
as Algorithm 2 here.
In order to compute alp, another modification of
the previous algorithms has been made. The new
version can compute the minimum PD among the
paths of any specified effective length. We refer to
the new algorithm which computes only dp as Algo-
rithm 3. The other new algorithm which computes
dp as well as N(dp) is referred to as Algorithm 4.
From Table 1, one can see that the maximum pos-
sible ECLs for 16-ary trellis codes are not large.
According to the code design criteria in Section 3,
the computer search should be carried out following
the criteria set I.
To make the search efficient, Algorithms 3, 1, 4 and
2 are used successively for searching the best codes
in terms of maximum ECL, dp, de, minimum N(dp)
and N(de) among the codes obtained at each pre-
vious step.
Table 2 lists the parameters of some new 16-PSK
codes, where Code Fi is T-state code proposed for
fading channel. As references, the parameters of
the corresponding Ungerboeck 16-PSK TCM codes
given in [6] are also listed as Code Ui. All of the
other known 16-PSK codes[5][7] have parallel tran-
sitions and therefore they behave similarly on fad-
ing channels. In the table, the parity-check poly-
nomials are given in the octal form, and N and G
denote the natural mapping and the Gary mapping
respectively.
It can be seen from the table that the proposed
codes for fading channels have the maximum ECL
and maximum dp, but smaller de than the corre-
sponding Ui type codes. It should be noted that
484
International Mobile Satellite Conference, Ottawa, 1990
the search for codes was only based on the first
term of dp and de. It is possible to achieve a slight
improvement in error rate performance by optimiz-
ing the distance spectrum though it requires more
complicated search algorithms.
5 Performance Evaluation
The performance evaluation in terms of bit er-
ror rate (BER) has been conducted. To compute
the analytical bounds for the BER performance, a
modified Viterbi algorithm has been implemented
to compute distance spectrum. Concurrently, ex-
tensive Monte Carlo computer simulations of the
schemes above were carried out.
Fig. 2 shows the analytical and simulation BER re-
sults for the 16-state code F4 compared with code
U4, on Rayleigh channel. Fig. 3 illustrates the re-
sults for the 128-state code F7 compared with code
U7. A summary of the performance of all proposed
Fi type codes is presented in Fig. 4.
The performance results show that all the Unger-
boeck codes have error floors above BER=10 -4 on
fading channel due to parallel transitions. The pro-
posed codes achieve significant improvement. As
the simulation results show, at the BER = 10 -3 (
which is an acceptable rate for digital voice trans-
mission), the difference between the two classes of
codes varies from 1.5 dB for the 256-state codes
to 6 dB for the 128-state codes on Rayleigh fad-
ing channels. This difference is even larger at lower
error rates.
6 Conclusions
The design criteria based on ECL and PD are ap-
plied to construction of new 16-PSK trellis codes
with improved error rate performance on fading
channels. The code performance is evaluated both
analytically and by simulation. The comparison
with the previously known codes of the same com-
plexity shows that the proposed codes can achieve
significant improvement in bit error rate. The new
16-PSK trellis codes will meet the requirements for
high data rate transmission in future satellite com-
munications.
Acknowledgements
This work was carried out under the IR&D Com-
monwealth Department Grant in Satellite Mobile
Communications.
References
[1] Ungerboeck, G. 1982. "Channel Coding
with Multilevel/Phase signals", IEEE Trans.
Inform. Theory, pp. 55-67.
[2] Divsalar, D. D. and Simon, M. D. 1988.
"The Design of Trellis Coded MPSK for Fad-
ing Channels: Performance Criteria", IEEE
Trans. Commun., pp. 1004-1011.
[3] Schlegel, C. and Costello, D. J. 1989.
"Bandwidth Efficient Coding .for Fading Chan-
nels: Code Construction and Performance
Analysis", IEEE J.S.A.C., pp. 1356-1368.
[4] Vucetic, B. and Nicolas, J. 1990. "Con-
struction of M-PSK Trellis Codes and Per-
formance Analysis over Fading Channels"
ICC'90, Atlanta, USA
[5] Wilson, S. G. et al, 1984. "Rate 3/4 Convo-
lutional Coding of 16-PSK: Code Design and
Performance Study", IEEE Trans. Commu.,
pp. 1308-1315.
[6] Ungerboeck, G. 1987. "Trellis Coded Mod-
ulation with Redundant Signal Sets", IEEE
Commun. Mag., pp. 6-21.
[7] Benedetto, S., et al. 1988 "Combined Cod-
ing and Modulation: Theory and Applica-
tions" IEEE Trans. Inform. Theory, pp. 223-
236.
[8] Schlegel, C. 1988. "Bandwidth Efficient Cod-
ing for Non-Gaussian Channels", Ph.D. Disser-
tation, University of Notre Dame.
[9] Du, J. and Kasahara, M. 1989. "Improve-
ments in Information-Bit Error Rate of Trellis
Coded Modulation Systems", Trans. of IEICE
of Japan, pp. 609-614.
[10] Rouanne, M. and Costello, D. J. 1989.
"An Algorithm for Computing the Distance
Spectrum of TreUis Codes" IEEE J.S.A.C., pp.
929-940.
International Mobile Satellite Conference, Ottawa, 1990
485
5s 7A. e_
.-?
o
9 _ I d:
h_
lO0 _6
11 • _15 _-
• q,
12 _
13
Figure 1: Notation for 16-PSK Modulation
>'S
.= 6
o-
,e
t t
112 It6 20 24
I:[b/1_ (dB)
28
Figure 2: BER performance of the 16-state 16PSK
TCM codes on Rayleigh channel
Dash: Analysis; Solid: Monte Carlo simulation
Table 1 Upper Bound on Maximum ECL
u 2 3 4 5 6 7 8 9 10 11 ...
Lm 1 2 2 2 3 3 3 4 4 4 ...
Table 2 16-PSK TCM codes .
i.d. h 3
U3
F3 15
U4
F4 37
U5
F5 75
U6
F6 165
U7
F7 337
U8
FS* 771
-,.%. ..... ....
12 20 24
Eb/]'4o (dB)
Figure 3: BER performance of the 128-state 16PSK
TCM codes on Rayleigh channel
Dash: Analysis; Solid: Monte Carlo simulation
-¢i
7,
oo
,¢
"4
........... --. :.-.%.
6 8 10 12 14 16 18 20 22 24
Eb/No (dB)
Figure 4: Summary of BER performance of 16PSK
TCM codes on Rayleigh channel
Dash 1: Uncoded 8-PSK; Dash 2 to 4: Codes U3,U7
and U8; Solid 1 to 6: Codes F3,F4,F5,F6,F7 and
F8; All results are obtained by Monte Carlo simu-
lations
h _ h ] h ° _H d_ N(d_) dp N(dp) Map.
04 13 1 1.47 4.00 2.00 2.00 N
17 05 13 2 0.89 0.19 0.61 0.25 G
04 23 1 1.62 8.00 2.00 2.00 N
21 15 33 2 0.89 0.34 2.47 0.50 G
10 45 1 1.91 8.00 2.00 2.00 N
73 47 57 2 1.48 0.77 13.7 1.00 G
- 024 103 1 2.00 2.00 2.00 2.00 N
175 115 143 3 1.48 1.90 0.68 1.00 G
- 024 203 1 2.00 2.00 2.00 2.00 N
343 307 211 3 1.48 0.03 4.69 2.00 G
374 176 427 1 2.08 7.88 2.00 1.00 N
513 463 405 3 1.78 1.39 27.3 1.00 G
486
International Mobile Satellite Conference, Ottawa. 1990


New 16-PSK trellis codes for fading channels

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