The idea of using a multiple (more than two) symbol observation interval to improve error probability performance is applied to differential detection of trellis coded MPSK over a mobile satellite (fading) channel. Results are obtained via computer simulation. It is shown that only a slight increase (e.g., one symbol) in the length of the observation interval will provide a significant improvement in bit error probability performance both in AWGN and fading environments

Divsalar, D.

Jedrey, T.

Lay, N.

Rafferty, W.

Simon, M. K.

English

NASA Technical Reports Server

N92-2,  C0
Combined Trellis Coding and
Feedforward Processing for
MSS Applications
D. Divsalar, M. K. Simon, T.Jedrey, N. Lay, W.Rafferty
Jet Propulsion Laboratory
4800 Oak Grove Drive
Pasadena, CA 91109 USA
ABSTRACT
The idea of using a multiple (more than two)
symbol observation interval to improve error
probability performance is applied to differential
detection of trellis coded MPSK over a mobile
satellite (fading) channel. Results are obtained via
computer simulation. It is shown that only a slight
increase (e.g., one symbol) in the length of the
observation interval will provide a significant
improvement in bit error probability performance
both in AWGN and fading environments.
1.0 INTRODUCTION
In a previous paper [1], the notion of using a
multiple symbol observation interval for
differentially detecting uncoded multiple phase-
shift-keying (MPSK) was introduced. In
particular, the technique made use of maximum-
likelihood sequence estimation of N-1 (N > 2)
phases rather than symbol-by-symbol detection as
in conventional (N = 2) differential detection. The
amount of improvement gained over conventional
differential detection was shown to be a function of
the number of phases, M, and the number of
1The research described in this paper was carried
out by the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the
National Aeronautics and Space Administration.
additional symbols (N-2) added to the observation.
Furthermore, as the number of symbols, N, in the
observation interval theoretically approached
infinity, the performance was shown to be identical
to that corresponding to ideal coherent detection.
In [2], this idea was extended to trellis coded
modulations (TCM), in particular, MPSK. There,
it was shown that a combination of a multiple
trellis coded modulation (MTCM) [3] with
multiplicity (number of trellis code output symbols
per input symbol) equal to N-1 combined with a
multiple symbol differential detection scheme
analogous to that in [1] can potentially yield a
significant improvement in performance, even for
small N, over that corresponding to conventional
trellis coded multilevel DPSK (MDPSK).
In this paper, we give further evidence of the
gain demonstrated in [2] as well as extending these
notions to fading channels.
2.0 SYSTEM MODEL
Figure 1 is a simplified block diagram of the
system under investigation. Input bits occurring at a
rate R b are passed through a rate nk/(n+l)k
multiple trellis encoder (k is the multiplicity of
the code) producing an encoded bit stream at a rate
R s = [(n+l)k/nk]R b. Next, the encoded bits are
divided into k groups of n+l bits each and each
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group is mapped into a symbol selected from an M =
2 n+l - level PSK signal set according to a set
partitioning method for multiple trellis codes [3]
analogous to that proposed by Ungerboeck [4] for
conventional (unit multiplicity) codes. Since the
MDPSK symbol rate is Rb/n, it is reasonable, from a
conservation of bandwidth standpoint, to compare
the performance of this system to an uncoded M = 2n
level DPSK system with the identical input bit
rate.
At the receiver, the noise-corrupted signal is
differentially detected and the resulting symbols
are then fed to the trellis decoder which is
implemented with the Viterbi algorithm. In
selecting a decoding metric, a tradeoff exists
between simplicity of implementation and the
optimality associated with the degree to which
the metric matches the differential detector output
statistics.
For the case of uncoded MDPSK, a metric based
on minimizing the distance between the received
and transmitted signal vectors is optimum in the
sense of a minimum probability of error test. The
specific forms of this metric for conventional and
multiple differential detection were described in
[1]. For conventional trellis-coded MDPSK, the
metric takes on the form of a minimum squared-
Euclidean distance metric. For multiple symbol
detection of MTCM, the form of the distance metric
is quite different. Nevertheless, as was shown in
[5], by a suitable modification of the multiple
trellis code design, the appropriate distance metric
can be converted once again into a squared-
Euclidean distance metric. The so-called
"equivalent" multiple trellis code that results from
this modification then becomes the key tool used
for analyzing the performance of the system.
3.0 CHANNEL MODEL
The mobile satellite fading model is shown in
Figure 2. A complex representation of the
mathematical model for the fading channel, used
to assess the performance of MTCM systems, is also
illustrated in Figure 2. In this figure, F(t)
represents the fading process, m(t) represents the
lognormal shadowing process, N(t) is a complex
Gaussian noise process, and to d = 2rcfd is the
Doppler spread in units of rad/sec.
In this paper, the effect of shadowing is not
accounted for. Furthermore, it is assumed that the
fading is slowly varying with a normalized
amplitude p = IF(t) l which is Rician distributed,
i.e.,
p(p) = 2p(1 + K)exp[-K -p2 (1 + K)]
Xlo(2p_/K(l+K)); p>-O (1)
where the parameter K is the ratio of the power in
the coherent (line-of-sight and specular) component
to that in the noncoherent (diffuse) component. A
special case of the Rician fading model is the
Rayleigh fading channel (corresponding to K = 0)
which characterizes terrestrial mobile radio
systems.
4.0 ANALYSIS MODEL
We denote a coded symbol sequence of length N s
by
X___ = (Xl, X2," • ",XN ) (2)
where the kth element 2 of x_, namely, Xk,
represents the transmitted MPSK symbol in the
kth transmission interval and, in general, is a
nonlinear function of the state of the encoder and
the nk information bits at its input. Before
transmission over the channel, the sequence x is
differentially encoded producing the sequence s. In
2Hero k isused as an index.As alreadymentioned,we
shallalsouse k todenotethemultiplicityofthe code.The
particularcase inpointshouldbe clearfromthecontextof
the usage.
176
International Mobile Satellite Conference, Ottawa, 1990
phasor notation, sk and Sk+ 1 can be written as
S, = ._'P-e jc_k
_/-_e j (¢' +aO'*l) V_ .i¢. ._Sk+l: SkXk + 1 : -_- e
(3)
where E s = rE b is the energy per MDPSK symbol
and
Xk = e ja¢* (4)
is the phasor representation of the MPSK symbol
A(_ k assigned by the mapper in the k/h
transmission interval.
The corresponding received signal in the kth
transmission interval is
rk = F,s,e j°. + nk (5)
where n k is a sample of zero mean complex
Gaussian noise with variance
2 2N0
G. - (6)
T
and Ok is an arbitrary phase introduced by the
channel which, in the absence of any side
information, is assumed to be uniformly distributed
in the interval (-_d¢). F k is a sample of a complex
Gaussian fading process and thus IFkl is Rician
distributed.
In [1] it was shown that for uncoded MPSK the
maximum-likelihood decision statistic based on an
observation of N successive MPSK symbols (the
present one, i.e., the kth, and N-1 in the past) is
_'R2_¢,=0 It
N-2 -J Y aCk ,, ,,,
77 : rk_N+ 1 + _.._q_.e (7)
n=O
with l = 1 or 2. If (7) is used as a branch metric for
the coded case then, for low SNR, ¢ = 2 whereas, for
high SNR, ¢ = 1. Since the first phase in this
sequence acts as the reference phase, this statistic
allows a simultaneous decision to be made on N-1
phases in accordance with the particular data
phase sequence A0k_N_2, A0k_N. 1 ..... AOk that
maximizes 7l.
To apply the notion of multiple symbol
differential detection to trellis coded MPSK, the
decision statistic of (7) must be associated with a
branch in the trellis diagram. To do this, we
construct a multiple trellis code of multiplicity k =
N-1. Thus, we can envision the transmitted
sequence, x_,of (2) as being partitioned into B = Ns/k
= Ns/(N-1 ) subsequences 3, i.e.,
cro) _.c2) .,x<S)) (8)
with each subsequence x (i) = (xil , xi2 .... ,Xik)
representing an assignment to a trellis branch.
Similarly, a received sequence, r_, of length N s is
associated with a path of length B branches in the
trellis diagram. Once this association is made,
computation of bit error probability for the system
follows along the lines of the approach taken in [3].
The details of the analysis are presented in [2].
The total metric proposed for the multiple
trellis decoder with channel state information
(CSI) is 4
I ir/:%_ F (') r_'__,, + _4[): "=°n=0 (9)
5.0 EXAMPLE - 16-STATE TRELLIS CODE
Consider a 16 state, rate 2/3 trellis coded 8PSK
using conventional (N = 2) and multiple (N = 3)
symbol differential detection. This code, which is
optimum on the AWGN, has the transition matrix
[5; Fig. 7]
3Since N s is arbitrary, we can choose it such that Ns/(N-1 )
is integer.
4Note that in (9), the subscript "k" on Fk has been omitted
since we assume that the fading amplitude is constant
along any given branch consistingf of N - 1 symbols.
International Mobile Satellite Conference, Ottawa, 1990
177
042
260
T=
4062
6240
6
15
4
37
51
37
4062
5
15
6240
7
73
0426
15
17
(10)
3 5 1
3 7
7 3 5 1
2 6 0 4
3 7 1 5
The output of the encoder enters the block
interleaver. We use 128 symbol interleaving with
depth of 8 and span of 8 symbol pairs (i.e., symbol
pair by symbol pair interleaving). The output
symbols of the interleaver are differentially
encoded after which they enter the 8PSK
modulator with 100% excess bandwidth root raised
cosine pulse shaping. The received signal enters
the demodulator with Doppler correction which is
also shown in Figure 1. Every 2T-sec., three
samples r k, rk_ 1, rk_ 2 enter the triple symbol by
triple symbol deinterleaver. The output of the
deinterleaver enters the multiple (double) trellis
decoder whose metric is computed based on (9). The
decoder structure is based on the Cartesian product
TxT [21.
The system in Figure 1 for the 16 state code
with multiplicity two has been simulated over an
additive white Gaussian noise channel with and
without the inclusion of Rician fading. We assume
a Rician parameter K = 10 and Doppler spreads of
20 Hz and 104 Hz. The metric used in the
simulations was that given in (9) with ! = 2 and no
CSI, i.e., IF(i) l = 1 for all i. The results of the
simulations are shown in Figures 3 and 4 for unfaded
and faded cases, respectively. We observe from
these figures that in going from N = 2 to N = 3, an
improvement in Eb/N 0 performance of at least 0.75
dB is obtained for the range of bit error
probabilities illustrated.
6.0 CONCLUSIONS
As was true for the uncoded case, the use of
multiple symbol differential detection of trellis
coded MPSK can offer an improvement in error
probability performance over conventional (two-
symbol observation) differential detection of the
same coded modulation. Again only a slight
increase in the length of the observation interval is
necessary to demonstrate a significant improvement
both for faded and unfaded channels.
7.0 REFERENCES
[1] D. Divsalar, M.K. Simon, "Multiple Symbol
Differential Detection of MPSK", March issue of
IEEE Transactions on Communications, 1990, pp. 300
-308.
[2] D. Divsalar, and M. K.
Performance of Trellis Coded
Multiple Symbol Detection,"
publication in the IEEE
Communications.
Simon, "The
MDPSK with
accepted for
Transactions on
[3] D. Divsalar, M.K. Simon, "Multiple Trellis
Coded Modulation (MTCM)," JPL Publication 86-44
(MSAT-X Report No. 141), November 15, 1986. Also
in IEEE Transactions on Communications, Vol. 36,
No. 4, April 1988, pp. 410-419.
[4] G. Ungerboeck, "Channel Coding with
Multilevel/Phase Signals," IEEE Transactions on
Information Theory, Vol, IT-28, No. 1, January 1982,
pp. 55-67.
[5] M.K. Simon, D. Divsalar, "The Performance of
Trellis Coded Multilevel DPSK on a Fading Mobile
Satellite Channel," JPL Publication 87-8 (MSAT-X
Report No. 144), June 1, 1987. Also in IEEE
Transactions on Vehicular Technology, Vol. 37, No.
2, May 1988, pp. 78-91.
178
International Mobile Satellite Conference, Ottawa, 1990
BINARY|
DATA
IN [ TRELLIS H
ENCODER H DIFFERENTIAL L_J DIGITAL PULSE SHAPER IINTERLEAVER ENCODER I - I AND 8PSK MODULATOR
DEMODULATOR
INPUT
! PS. UED_ _I_TC. HED FILTER DIFr_..FERENTIAI. DETECT_R
L ,
r_--T_:,_)__ _ ,_.xP2o_
DOPPLER ESTIMATOR
MULTIPLE TRELLIS DECODER
OUTPUT
r k
v fk-1
_lP"rk. 2
EXP (j 2o) d T )
1
1
I ,_°,_°1CHANNEL
BINARY
DATA
OUT
°_, I _N_
LEAVER DETECTOR i _
Figure 1. System Block Diagram.
Intemational Mobile Satellite Conference, Ottawa, 1990
179
A(t)e jq(t)
TRANSMITTED
SIGNAL
F(t) =
F(t) N(t)
R(t)
RECEIVED
SIGNAL
m(t)e jet + nd(t)eW(t)
LONG TERM DIFFUSE
SIGNAL PROCESS
FADING (RAYLEIGH)
SHADOWING
/ /
/
/
,DIFFUSE
, COMPONENT
/
," (RAYLEIGH
,/ DISTRIBUTED)
/
...._.',-,
COMPLEX RECEIVED FADED SIGNAL:
R(t) = M(t) • (Rdir(t) + Rspec(t)) + Rdif(t)
M(t) LONG TERM SIGNAL FADING
(LOGNORMAL DISTRIBUTED)
Figure 2. Mobile Satellite Channel Propagation Model.
180
International Mobile Satellite Conference, Ottawa, t990
101
10 .2
10 "4
10-5
6.0
N=3
16 State TCM
Interleaving Size = 128 symbols
Interleaving Depth = 8 branches
N=2
6.5 7.0 7.5 8.0
Eb/N o, dB
10 -1
10 -2
rr
10 3
LLJ
10 -4
10 -5
8.5
N=3, DF=104Hz
N=2, DF=104
7.0 7.5 80
Intedeaver= 128 sym.
Decoder Buffer = 32 sym
N=2, DF=20 Hz
•N = 3, DF = 20 Hz
85 90 95 100 105 110 11512.0
Bit SNR dB
Figure 3. Simulation Results for Bit Error
Probability of 16 State, Rate 2/3
Trellis Coded 8PSK With
Conventional (N = 2) and Multiple
(N = 3) Symbol Differential Detection
(No Fading).
Figure 4. Simulation Results for Bit Error
Probability of 16 State, Rate 2/3
Trellis Coded 8PSK With
Conventional (N = 2) and Multiple
(N = 3) Symbol Differential Detection
(Rician Fading With K = 10).
International Mobile Satellite Conference, Ottawa, 1990
181


Combined trellis coding and feedforward processing for MSS applications

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