My first academic paper, derived from my graduation work. Published in 1999 at TELSIKS international conference

1. Differential 8-PSK code with multisymbol interleaving
Sa{a or|evi}
Abstract-Conventional differential systems are known to have
poor performance when used with Rice- or Rayleigh- fading
channel. These systems are incapable of working properly due to
time changing channel parameters. This difficulty is overcome
by using multiple symbol differential detection (MSDD). This
paper deals with one simplification of these systems.
Keywords-Rayleigh-fading channel, M-ary PSK, Multilevel
encoder, Differential detection
I. Introduction
Since the first mentioning of trellis code modulation in the
beginning of 80's it has become one of most popular methods
of achieving code gain without expanding bandwidth. The
general principle is to use expanded signal set (more than
used by uncoded signal), and thus having more efficient
communication. There are two different types of TCM:
i) lettice type (PAM and QASK)
ii)type with variable phase and constant amplitude (M-PSK).
Latter types have less power efficiency in comparison with
former types, but have more tolerance on errors.
In this paper second type of TCM (8-PSK) will be
discussed. It will be investigated performance over Rayleighfading channel of this code. Due to time variant parameters of
Rayleigh-fading channel coherent detection is almost
impossible to achieve so different approach must be used. In
this case differential modulation is used, to simplify detection.
To prevent deep fades of completely destroying signal
interleaver is implemented in system. But, using interleaving
will break the relations between two consecutive symbols
(which is important in case of differential detection)
interleaving must be used over block of L symbols instead of
one symbol. Thus we have Lx8-PSK. Here will be weighed
performance of 2x8-PSK code.
In Section II system model will be introduced. Section III
shows methods for partitioning 2x8-PSK signal set. Encoder
example is given in section IV. Theoretical overwiev of code
performance is given in section V. In the section VI
numerical results are presented, and in last section a
conclusion is given.
Saša Đorđević is with the Mihajlo Pupin Institute, Volgina 15,
11060 Beograd, Yugoslavia, E-mail: sasha@kondor.imp.bg.ac.yu
II. System model
Block diagram of the system used is shown in figure 1. Bit
stream enters the multilevel encoder, which gives a stream of
LxM-PSK symbols as its output. This stream passes through
an ideal interleaver which rearranges these symbols as units.
Furthermore, these interleaved symbols pass through the
differential encoder and modulator, end then are emitted over
'flat' Rayleigh-fading channel. Fading is simulated by time
variant signal g(t). Additive white Gaussian noise (AWGN) is
also present (on the block diagram denoted by n(t)) and added
to the already faded signal.
Input
Multilevel
encoder
L-symbol
inter leaver
Differ ential
encoder
Decoder
L-symbol
deinter leaver
PSK
modulator
Demodulator
Output
n(t)
g(t)
Fig. 1. System model
On the receiver side hard decision is used to determine which
signal is being received. Although many authors ([1], [2], [3])
choose optimal filtering on the receiver's input, in this
particular case it is much faster to determine which symbol is
received when hard decision is used, and not much
degradation is made. After the demodulation, the signal
passes through an ideal deinterleaver, which reorganizes
symbols in original order.
Decoding is done afterwards, in the hard decision Viterbi
decoder (again for faster detection), but if optimal filtering is
being used , a soft decision Viterbi decoder can be utilized.
III. Constellation partitioning
Partitioning of the 8-PSK constellation is done by forming
of binary normal subgroup chain with minimum squared
subset distance as shown on Fig.2.

2. Correspondingly to the results given above one 2x8-PSK
signal can be defined with 6 bits, zi, i=0,1,2,3,4,5 as shown in
Eq. 2.
y0=0
y1=0
⎡y ⎤
⎡4 ⎤
⎡0⎤
y ( z) = ⎢ 1 ⎥ = Ω 6 ( z) = z 5 ⎢ ⎥ + z 4 ⎢ ⎥
⎣4 ⎦
⎣4 ⎦
⎣ y2 ⎦
⎡2 ⎤
⎡0⎤
⎡1⎤
⎡0⎤
+ z 3 ⎢ ⎥ + z 2 ⎢ ⎥ + z 1 ⎢ ⎥ + z 0 ⎢ ⎥ (mod 8)
2⎦
2⎦
1⎦
⎣
⎣
⎣
⎣1⎦
1
1
0
1
(2)
By means of this equation we can uniquely encode 6
different signals and add them together to get one 2x8-PSK
signal. This operation is done in encoder.
y2=0
1
0 1
0 1
0 1
IV. Multilevel encoder
The block of main importance in this system is multilevel
encoder. An example of decoder used in this simulation is
shown in Fig. 3.
Fig. 2. 8-PSK constellation partitioning
Input 1 Convolutional
encoder
#1
In case of 2x8-PSK constellation partitioning it is necessary
to define 2x3 matrix as shown on Eq. 1.
⎡ y ⎤ ⎡ y2
y = ⎢ 1 ⎥ = ⎢ 12
⎣ y2 ⎦ ⎣ y2
1
y1
1
y2
y10 ⎤
0⎥
y2 ⎦
Input 2 Convolutional
encoder
#2
(1)
...
Let us define Cmi as a block code which consists of vectors
y i i=0,1,2. Thus Cm0 has least significant bits of matrix y.
Value m shows which block is in use. For L=2 only three
block codes are used: C0, which is a (2,2) block code with
Hamming distance d0=1 (and code words [0 0]T, [0 1]T,[1 1]T,
[1 0]T); C1 is a (2,1) block code with d0=2 (and code words [0
0]T, [1 1]T) and C2 is a (2,0) block code with d0=∞ (and only
one code word [0 0]T). For more information see [4].
By means of the rule shown above a table for partitioning
2x8-PSK signal set can be made. Results are shown in Table
1.
Partition
level (p)
0
1
2
3
5
6
7
Ωp
2
d pz
Ω(C0 ,C0 ,C0)
Ω(C0 ,C0 ,C1)
Ω(C0 ,C0 ,C2)
Ω(C0 ,C1 ,C2)
Ω(C0 ,C2 ,C2)
Ω(C1 ,C2 ,C2)
Ω(C2 ,C2 ,C2)
0.586
1.172
2.0
4.0
4.0
8.0
∞
Generator ( t p )T
[0 1]
[1 1]
[0 2]
[2 2]
[0 4]
[4 4]
---
...
Input 6 Convolutional
encoder
#6
Figure 3. Encoder example
Fig 2. Multilevel encoder
This is a six level code over 2x8-PSK. Every partition
level uses identical rate 2/3 convolutional encoder with two
states (ν=1) shown on Fig. 4. Encoder input is a bit stream
split into 6 substreams.
Each substream enters its corresponding convolutional
encoder, which emits two types of symbols:
i) identity element (0,0) if the output of the encoder
is zero
ii) corresponding coset representative if output is one
x1
y1
y2
x2
Table 1. 2x8-PSK signal set partitioning
...
Output
y3
Figure 4. 2/3 rate convolutional encoder

3. Fig.5 shows the trellis structure of convolutional encoders
used in all six encoder levels.
P(Z ⇒ Z ') ≤
( 2 l − 1) !
l !( l − 1) !
K
∏
k =1
Zk ≠Zk '
1
( E s N 0 ) d z2k
( 3)
where l is the number of code blocks Zk and Zk' different
between Z and Z' , d z2 is differential product distance, and
k
00/000
01/001
11/
10/ 010
011
Es/N0 is signal-to-noise ratio (SNR). In the end we can define
the squared product distance as shown in Eq. 4:
2
d pz =
1
/11
00 110
01/
10/100
11/101
∏d
η
k∈
2
pk
( 4)
where η is the set of k that Zk≠ Zk'. Hence good code should
be designed by means of maximizing Hamming distance
dH=⎥η⎜ between the sequence of L symbols, and then
differential product distance
V. Numerical results
Fig 5. Trellis structure of binary convolutional code
As Figure 5. shows there are four transition branches
leaving and entering all four states. Which transition branch
will be used (and what would output be) is decided by pair of
input bits. Output is then mapped over G(s) by the means of
coset representatives for each encoder level.
Coset representatives for all encoder levels are presented in
Table 2. Constellation partitioning is achieved as it is shown
in Section II, and in [4].
I
1
2
3
4
5
6
Numerical simulations have been done for SNR in range 020dB, and for the fading parameter fD T taking values of 0.1,
0.01 and 0.001 (see [1] and [5]). Fig. 6. represents simulation
results.
Coset representative
(4,4)
(0,4)
(2,2)
(0,2)
(1,1)
(0,1)
Table 2.
Coset representatives are denoted by (m,n) where both
numbers represent rotation of 2π/M radians e.g. (2,5)
represents rotation of 450 in the first case, and 2050 in the
second case. Encoder output is the sum of all selected coset
representatives; it is also an element of 2x8-PSK
constellation.
V. Code performance
Generally, code performance is hard to determine (except
by simulation). But, over slower fading (fDT≈0) an accurate
relation between code performances and channel parameters
can be made.
In [1] detail equations can be found, therefore here will be
shown only pairwise probability that emitted encoded
sequence will be understood wrong P(Z ⇒ Z') (Eq 3.)
Figure 6. Simulation results
As expected, the performance of the system gradually
improves as SNR increases. System is also much more
efficient over slower fading (fD T = 0.001). Performance can
be improved by increasing the constraint level of each
component code (from ν=1 to ν=2, or even more). Using
larger values for L, e.g. 3x8-PSK, a larger tolerance to a faster
fading can be obtained. In that case an encoder has to be
designed with nine 2/3 rate convolutional encoders and new,
nine level partitions. However cost of this improvement is an
increase in the number of encoders, and in slower detection.
This improvement is effective only over faster fading, but is
almost identical over slow fading as distance properties of the
last partition is almost the same as for 2x8-PSK. For
performance of 3x8-PSK and more information on this type
of code see [1].

4. VI. Conclusion
This code is shown to be very useful in systems where are
slow fading, although it ca be used with channels with
relatively fast fading. This properties makes this code
effective in satellite communications.
Numerical results shows that curves are rather steep. But
for higher SNR it can be shown that this tendency will not
follow. This is the result of phase variations of the fading
channel, even if there is no AWGN. However, high SNR is
more theoretical than practical case, so it was not mentioned
in this paper. For moor e information on this subject see [1]
and [3].
Although many authors have solving this problem, it is
interesting that only small number of authors decides to
simplify system, but more complications are implemented.
The main contribution of this paper is simplification of the
most complicated blocks in order to achieve greater speed of
signal processing with small degradation of signal quality,
which in some applications can be tolerated. This
simplification is (of course) optional and can be combined
with other similar systems (such as all LxM-PSK signal sets).
References
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R. Van Nobelen , D. P. Taylor , “Multiple symbol
differentially detected multilevel codes for the Rayleighfading channel ,” IEEE Trans. Commun., vol 45, pp.
1529-1537, Dec 1997.
[2]
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QAM modulacije u mobilnom radio kanalu,"
Telekomunikacije broj 3/4, str. 10-15, Nov. 1998.
[3]
F. Edbauer , “Perfomance of interleaved trellis-coded
differential 8-PSK modulation over fading channels,”
IEEE J. Select. Areas Commun., vol 7, pp. 1340-1346,
Dec 1989.
[4]
S.S. Pietrobon ,et al. ,”Trellis-coded multidimensional
phase modulation,” IEEE Trans. Inform. Theory , vol 36
pp. 63-89, Jan 1990.
[5]
G.Petrović, D.Drajić, D.Bajić, Dj.Zrilić, “A Gilbert-like
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