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Performances of Chaos Coded Modulation concatenated with Space-Time Coding extended version
Jean-Pierre Cances
To cite this version:
Jean-Pierre Cances. Performances of Chaos Coded Modulation concatenated with Space-Time Coding
extended version. [Research Report] Xlim UMR 7252. 2013. �hal-01417819�
Performances of Chaos Coded Modulation concatenated with Space-Time Coding extended version
J.P. Cances, Xlim UMR CNRS 7252 I. Introduction:
Since the pioneering work of Frey in 1993 [1], chaotic communications has been an important topic in digital communications. The seemingly unpredictable behaviour of chaotic systems renders their use in secure communication systems highly attractive. They have been studied for example in multi-user communications [2-6] for asynchronous Code Division Multiple Access (CDMA) systems to synthesize spreading waveforms which exhibit lower cross- correlation levels than classical spreading codes such as Gold, Walsh-Hadamard and Golay ones. Due to their extreme sensitivity to initial conditions which, for example, facilitates theoretically the separation of merging paths in a trellis based code, these systems have also been considered as good potential candidates for channel encoding [7-11]. This explains why chaotic modulations and channel encoders derived from chaotic systems have been extensively studied in the open literature. According to us, there are mainly two types of chaos based channel encoders depending on the size of the transmitted alphabet. The first kind of chaos based channel encoders includes non-linear generators which transmit binary messages and benefit from the correlation between successive transmitted bits to obtain some coding gain. Due to the poor spectral efficiency, it is rather easy to optimize this kind of codes to obtain a non-null free distance and to obtain reasonable good performances i.e. codes that outperform un-coded systems [12-16]. Some authors have even used these binary non-linear constituent encoders to build parallel concatenated schemes just like turbo-codes which perform quite closely to the theoretical bounds provided that the interleaver size is big enough [17-18]. The second kind of chaos based channel encoders includes those which transmit a complex quasi-continuous alphabet i.e. those which are inherently chaotic in all their characteristics. These channel encoders can be compared to Trellis Coded Modulation (TCM) schemes. Many works deal with the optimization of such coders and, among them, perhaps the most famous ones were those named Chaos Coded Modulation (CCM) schemes.
However, the weakness of such transceiver was their poor BER performance since they did
not have even better performances than un-coded systems such as Binary Phase Shift Keying
(BPSK) [19-21]. This was particularly the case for the systems which use CSK (Chaos Shift
Keying) Modulation [22-24]. Nevertheless, some recent studies have stressed the fact that
Chaos Coded Modulation (CCM) systems, working at a joint waveform and coding level, can
be efficient in additive white Gaussian noise channels [25-27]. These promising works on the
AWGN channel have been recently further extended by Escribano & al in the case of
Rayleigh flat fading channels [28]. In this work, we use Chaos Coded Modulation designs of
S. Kozic [29-30] and we optimize them using the distance spectrum. We find that the distance
spectrum distribution can be good approximated by Rayleigh probability distribution function
(pdf) or, for some particular cases, by mixture of Gaussian laws. Using this optimization step,
we study the performances of the proposed Chaos Coded Modulation designs when we
concatenate them with a Space Time Block Code (STBC) such as the famous Alamouti’s
scheme [31]. Concatenation of a Trellis Coded Modulation (TCM) with a STBC code is
recognized as a performing alternative to the use of Space Time Trellis Codes (STTC) [32-
33]. The strength of our study consists in the fact that we do not only study the case of block
fading channels but we propose new insights for the case of time selective channels. Our
closed form expressions are based on the recent work of Jun He and Pooi Yuen Kam [34] who have studied the behaviour of the Alamouti’s scheme in highly time selective channels. In all cases, we derive accurate BER bounds and we show that the use of classical Maximum Ratio Combining (MRC) equalizer makes the receiver lose all diversity gain at high SNR’s. To save some diversity gain, we propose a Zero Forcing (ZF) scheme [34] and we quantify its performances for various Doppler effects. The contributions of this technical note are thus the following ones:
- Detailed study of the distance spectra of the chaos based encoders and characterization of their distribution.
- Derivation of accurate BER bounds for quasi-static block fading channels and time selective fading channels.
The rest of the note is organized as follows. In Section II, we give new insights for the chaos coded modulation schemes proposed by S. Kozic. We propose to approximate the distance distribution with some usual laws such as the Rayleigh one. In section III, we study the performances of the concatenation of the Chaos Coded Modulation (CCM) together with the Alamouti’s STBC code for quasi static block fading channels. We do the same in Section IV for the case of time-selective channels considering MRC and ZF equalization. The concluding remarks are eventually given in Section V.
II. Chaos Coded Modulation Scheme, Distance Spectrum Study:
2.1 Chaotic coder structure:
We consider the Chaos-Coded modulation scheme of Fig. 1. This scheme was originally given by S. Kozic in his PhD works [30]. The scheme of Fig. 1 can be represented by means of a convolutional coder of rate 1/( .( n Q 1)) , where at each time step k, one bit b k enters the coder and a vector of ( Q 1 ) bits v = [v , v Q Q 1 ,..., v ] 1 T is produced. The signal constellation is realized by a weighted sum of vectors 2 . i A ( Q i 1) mod(1) where A is some matrix which optimizes the distance spectrum of the code. This mapping, due to the modulus operation, is a highly non-linear operation and serves as a chaos generator. Henceforth, we have a system which combines a convolutional coder with a high dimensional mapping in the same way as Multi-level Trellis Coded Modulation (M-TCM). The corresponding convolutional coder is classically described by:
, , 1 ,0
v ( )
( ) . ... .
( )
i Q
i i Q i Q i
h D D t t D t D
b D
(1)
S. Kozic defines several possible matrices T t i j , in his work which give good
performances:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0
shift
T ,
1 0 0 1
0 1 0 0
0 0 1 0
0 0 0
e shift
T and
1 0 0 0
1 1 0 0
1 1 1 0
1 1 1
tent
T
Concerning, the choice of the matrix A, we can write the transmitted vector at the output of the modulator:
( 1) ( 1)
0 1
2 . . ( ) 2 . ( ) mod(1)
a
a
a
Q Q
Q i
i i
k i i
i i Q
D D
x A v v (2)
Before transmitting x k on the channel propagation medium, we modulate each of its components in NRZ-BPSK in order to obtain a average zero mean value to better interface with a zero mean additive noise such as for AWGN channel i.e: x k 2 . x k 1 .
One can see that the optimization of such multi-dimensional Chaos Coded Modulation is very complicated due to the number of parameters: matrices T i j , , matrix A, parameters Q, Q a … Rather than a global optimization algorithm which should optimize the convolutional coder together with the mapping process, we choose to fix a convolutional coder structure and then we optimize the mapping process by using a particular form of matrix A. We found that the choice T i j , T shift for i = j and T i j , T tent for i ≠ j enables to obtain a large set of performing non-linear mapping with A. For example, in the case n = 2, using this choice for matrices T, we are looking for matrices A with the following structure:
21
1 1
1 a
A and we optimize
the choice of a 21 using the distance spectrum. In the case, n = 3, we use matrix structure
21
31 32
1 1 1
1 1
1 a
a a
A . The choice of the remaining parameters a i j , is done using the distance spectrum of the code.
Fig 1: Trellis chaos coded modulation encoder
0 1
. 2
QA
1 2
. 2
A
QT 0Q T 01 T 00
T 1Q T 11 T 10
T QQ T Q1 T Q0
+ x k A
Q. 2
1b k b k-n+1 b k-Q.n b k-(Q+1).n+1
v 0
v 1
v Q
The state of the coder is defined by vector S k : S k [ b k ,..., b k n 1 ,..., b k Q . n ,..., b k ( Q 1 ). n 1 ] T (3) Concerning the choice of Q, it’s clear that the Viterbi decoding algorithm is rapidly limited by the complexity in the number of states which is equal to 2 n Q .( 1) . Practically, the number
.( 1)
n Q should not exceed 12 which correspond to 4096 states. For n = 2, this gives a maximum value of Q equal to 5, and for n = 3, this gives a maximum value of Q equal to 3.
The choice of Q a , is related to the chaotic behaviour of the coder. Although it is beyond the scope of this paper to characterize the chaotic behaviour of the coder, we can illustrate it on simple examples with the phase trajectories of the encoder outputs as functions of the encoder inputs. In fact, generally speaking and depending on the choice of matrix A, varying parameter Q a reduces or increases the chaotic behaviour of the encoder. For example, we illustrate the case n = 2, with matrix
8 1 1
A 1 . The output of the encoder
) 2 (
) 1 (
k k
k x
x x is a vector of two samples. To illustrate the chaotic behaviour of the decoder, we plot on Fig. 2,
) 1
1 (
x k as function of x k ( 1 ) , x k ( 2 ) and x k 1 ( 2 ) as function of x k ( 1 ) , x k ( 2 ) for two values of 3
and 1
: a a
a Q Q
Q and for 10000 input samples b k . For each of these two examples, we compute: E [ x k 1 ( 1 ). x k ( 1 )] , E [ x k 1 ( 1 ). x k ( 2 )] , E [ x k 1 ( 2 ). x k ( 1 )] and E [ x k 1 ( 2 ). x k ( 2 )] and the chaotic behaviour of the coder is characterized by the parameter:
] ) )]
2 ( ).
2 ( [ ( ) )]
1 ( ).
2 ( [ ( ) )]
2 ( ).
1 ( [ ( ) )]
1 ( ).
1 ( [ 4 .[(
1 2
1 2
1 2
1 2
1 k k k k k k k
k x E x x E x x E x x
x E
C (4)
The smaller is C and the more chaotic the system behaves since the perfect chaotic system is a white noise with C = 0.
On Fig 2, we plot x k 1 ( 1 ) as function of x k ( 1 ) , x k ( 2 ) and x k 1 ( 2 ) as function of x k ( 1 ) , x k ( 2 ) for two values of Q a : Q a 1 and Q a 3 .
We obtain C 1 = 4 . 97 . 10 5 for the case Q a 1 and C 1 = 1.70.10 for the case -4 Q a 3 . That means that system with Q a 1 is more chaotic than those with Q a 3 .
2.2 Spectrum distance analysis and performances over AWGN channels:
In order to optimize the coders, we study their distance spectrum. To do this, we have to determine the trajectories in the trellis which start with a common state S i S i * and evolve in disjoint paths for ( L 1 ) time steps and then merge again into the same state S k S * k not necessarily equal to S i . This kind of trajectory in the trellis defines a loop and the loop is characterized by its initial state S i , its final state S k and its length L. The distance of corresponding codewords belonging to the two competing paths in the loop is:
1 2 1
* 2
,
,
L
m m m
S S L i k
d x x (5)
Fig. 2: Phase trajectory of the chaotic encoder for different values of Q a .
The problem of the computation of (5) is that, unlike linear codes when we can choose a
reference path equal to a all zero sequence, due to the non-linear mapping, we have to test all
the possible transmitted sequence for a given loop length together with all the possible
starting states. Hence, the distance spectrum computation problem is of non polynomial
complexity and in straightforward manner requires the inspection of all possible initial
conditions and all possible controlled trajectories. For example, there are 2 n ( Q 1 ) . 2 nL different
controlled trajectories of length L. In order to compute the distance spectrum with a
reasonable complexity while keeping a sufficient accuracy, we form all the possible pair of
sequences starting from a given state and both converging towards an other state after L steps
with L belonging to the interval [Qn+1, n Q m .( ) ], i.e. the length of the loop varies from
Qn+1 (the constraint length of the code plus one) to n Q m .( ) (we limit practically the search
to m = 2 or 3 in our case due to the computation burden). We have partitioned the distance
spectrum into subsets by distinguishing error events which entail one error bit, error events
which entail two error bits, error events which entail three error bits and so on... In practice,
we limit our search to error events which entail five maximum error bits since simulation
results evidenced that it was sufficient to obtain accurate upper bounds for the BER on non-
selective Rayleigh fading channels.
We obtain for example with matrices:
1 0
0 1
, j
T i for i = j and
1 1
0 1
T i,j (i.e. n = 2) for j
i and matrix
8 1 1
A 1 , Q = Q a = 3, the distance spectrum illustrated on Fig. 3.
Fig. 3: Distance spectrum of the chaos coded modulation
In fact, we found that, in a majority of cases, the shape of the distance spectrum is close to a Rayleigh distribution with the following probability density function:
j C
j m
x j
j C
m x x f
m x m e
x x
f
j j
, 0 ) (
, ) .
) ( (
2 2
/ 2 . ) ( 2
(6)
For example, with the distance spectrum plotted on Fig 3, we calculate parameters m j and
2
j to obtain the best fitting between the pdf of the distance spectrum and f C ( x , m j , 2 j ) , we obtain with classical MMSE technique: m j 2 j 6 . 7 . This corresponds to a minimum free distance of the coder equal to d free 6 . 7 . The comparison between the Rayleigh distribution and the obtained normalized distance spectrum is illustrated on Fig. 4, it is clear that the two curves feet well. However, for some cases with small free distances, we obtain a much more complicated shape for the distance distribution. For example, in the case
1 0
0 1
, j
T i for i =
j and
1 1
0 1
T i,j for i j and matrix
5 1 1
A 1 , Q = Q a = 3, the distance spectrum is
illustrated on Fig. 5.
.
Fig. 4: Distance spectrum v.s. Rayleigh distribution
Fig. 5: Distance spectrum profile for a = 5
In these cases, we model the probability density function by a mixture of Gaussian or
Rayleigh laws [35-36].
- Gaussian mixture:
) , ( . .
. . 2 . 1 )
( 2
1 .
2 / ) ( 1
2 2
n n J
n n
m x n J
n n
C x e m
f n n
(7)
In the following, unlike [35], we will not assume that n 2 2. m n , i.e. that the variance is twice the mean. Finding J, n , m n and n is a complicated multidimensional optimization problem.
Fortunately, the EM (Expectation Maximization) algorithm enables to reduce considerably the complexity. For a fixed number of mixtures J, based on the observations i , i 1 ,..., n , the parameters j , m j , j , j 1,..., J can be estimated using the EM (Expectation Maximization) algorithm.
The problem can be formulated as follows. Denote ( x ; , 2 ) as the pdf of an ( , 2 ) random variable. Then, the maximum likelihood (ML) estimate of the parameters is given by:
) 8 ( ) ,
; ( . log max arg
) ( log max ˆ arg
2 1
1 1 :
1 :
1 1
j j J
j j i
n i
m p
J
j j
J
j j
The EM algorithm is an iterative procedure for solving this ML estimation problem. In the EM algorithm, the observation is termed as incomplete data. Starting from some initial estimate ( 0 ) , the EM algorithm solves the ML estimation problem (8) by the following iterative procedure:
- E-step: Compute
log ( )
)
(
())
( X
E p
Q
ii (9)
- M-step: Solve
) (
max
arg ( )
) 1
( i i
Q
(10)
Define the following hidden data Z z i , i 1 ,..., n where z i is a J-dimensional indicator vector such that:
0 ot herwise
) , ( if
,
1 2
, i j j
j i
z m
(11) The complete data is then X ( , Z ) , we have :
n i
J j
z j j i j
j
m
ip
1 1
2
,,
; ( . )
,
( Z
(12)
The log-likelihood function of the complete data is then given by:
m C z
z
p n
i J
j j
j i j j
i n
i J
j i j j
1 1 2
2 2
1 1 , , ]
. 2
) ) (
log(
2 . .[ 1 log
. )
, (
log
Z
(13)
Where C is some constant. The E-step can then be calculated as follows:
log ( , )
) '
( E ' p Z
Q
2
, 2
1 1
( )
( ') ˆ .[log log( ) ]
2.
n J
i j
i j j j
i j j
Q z m C
(14)
Where:
J
l i l l l
j j j i J
l i i l i l
j i j
i i
i j i j
i j
i
m m
z P z
p
z P z
p
z P z
E z
1
' 2 ' '
' 2 ' '
1 ,
' ,
'
, ' ,
'
, ' ' , ' ,
).
,
; (
).
,
; (
) 1 ( ).
1 (
) 1 (
).
1 (
1 ˆ ,
(15)
In addition, the M-step is calculated as follows. To obtain j , we have:
J j
n z
Q n
i i j j
j
,..., 1 ˆ ,
1 . ) 0
, (
1 ,
'
(16)
To obtain m j , we have:
'
, 2
1
2.( )
( , )
ˆ
0 .[ ] 0, 1,...,
2.
n
i j
i j
j i j
Q m
z j J
m
,
2 , 2
1 1
ˆ
ˆ . , 1,...,
n n
j i i j
i j
i i
j j
m z
z j J
, 1
, 1
ˆ .
, 1,..., ˆ
n i i j i
j n
i j i
z
m j J
z
(17)
Once, m j have been obtained, we can obtain j as:
' 2
, 3
1
2 2
, ,
1 1
( )
( , ) 1
0 ˆ .[ ] 0, 1,...,
ˆ ˆ
. .( ) , 1,...,
n
i j
i j
j i j j
n n
j i j i j i j
i i
Q m
z j J
z z m j J
2 ,
2 1
, 1
ˆ .( )
, 1,..., ˆ
n
i j i j
i
j n
i j i
z m
j J
z
(18)
Finally, the EM algorithm for calculating the Gaussian mixture parameters for the pdf of the distance spectrum is summarized as follows.
- Given the initial set of values m j - Given the initial set of values j
- Given the distance values i , the number of mixture components J, and the total number of EM iterations I, starting from the initial parameters ( 0 ) .
For i = 1, …, I, do the following:
- Let ' ( i 1 ) , and calculate z ˆ i , j , i 1 ,..., n ; j 1 ,... J according to (15)
- Calculate j , j 1 ,..., J according to (16), and calculate m j , j 1,..., J ,
j , j 1 ,..., J according to (17-18). Set: (i ) .
In the above EM algorithm, the number of mixture components J is fixed. Note that when J increases, log p ( ) increases, or log p ( ) decreases. The minimum description length (MDL) principle can be used to determine J.
. log( )
) 2 ( log min
ˆ arg
, J n
p
J MDL J J (19)
Where, in the MDL criterion, a penalty term J / 2 log( n ) is introduced. Hence, we can first set an upper bound of the number of mixture components, J max . In addition, for each J J max , we run the above EM algorithm and calculate the corresponding MDL value. Finally, we choose the optimal J with the minimum MDL. For example, for the distribution of Fig. 5, we find a mixture of two Gaussian laws with parameters, 1 0.6, m 1 4 . 2 and 1 2 . 24 ,
2
0.4, m 2 9 . 5 and 2 2 . 44 . The result is illustrated on Fig. 6.
- Rayleigh mixture:
In this case we have:
2 2
( ) / 2. 2
2
1 1
( )
( ) . .
n n. ( , )
J J
n x m
C n n n n
n n n
f x x m e m
R (20)
Where R ( m n , n 2 ) represents a Rayleigh law of parameters: m n and n 2 . Following the same
approach as for the Gaussian mixture and using the same notations, we obtain:
2 2
, , 2
1 1 1 1
( )
log ( , ) .log .[log ( ) log( ) ]
2.
n J n J
i j
i j j i j i j j
i j i j j
p z z m m C
Z (21)
We denote here ( ; , x 2 ) as the pdf of a R ( , 2 ) random variable. We obtain:
log ( , )
) '
( E ' p Z
Q
2
, 2
1 1
( )
( ') ˆ .[log log( ) 2.log( ) ]
2.
n J
i j
i j j i j j
i j j
Q z m m C
(22)
With:
' ' '
, , ,
' '2 '
' '2 '
1
ˆ , 1
( ; , ).
( ; , ).
i j i j i j i
i j j j
J
i l l l
l
z E z P z
m m
(23)
The M-step is calculated as follows. To obtain j , we have:
J j
n z
Q n
i i j j
j
,..., 1 ˆ ,
1 . ) 0
, (
1 ,
'
(24)
To obtain m j , we have:
'
, 2
1
( )
( , ) 1
ˆ
0 .[ ] 0, 1,...,
n
i j
i j
j i i j j
Q m
z j J
m m
2 2
, 2 2
1
( )
ˆ .[ ] 0, 1,...,
.( )
n
j i j
i j
i j i j
z m j J
m
2 2 2
, ,
1 1
2 2 2
, , , ,
1 1 1 1
ˆ . ˆ .( 2. . ), 1,...,
ˆ . 2. . ˆ . . ˆ ˆ . , 1,...,
n n
i j j i j i j i j
i i
n n n n
i j i j i j i j i j i j j
i i i i
z z m m j J
z m z m z z j J
(25)
To obtain j we have the set of equations:
' 2
, 3
1
2 2
, ,
1 1
( )
( , ) 2
ˆ
0 .[ ] 0, 1,...,
ˆ ˆ
2. . .( ) , 1,...,
n
i j
i j
j i j j
n n
j i j i j i j
i i
Q m
z j J
z z m j J
(26)
2 ,
2 1
, 1
ˆ .( )
, 1,..., 2. ˆ
n
i j i j
i
j n
i j i
z m
j J
z
(27)
The set of equations (26-27) is a set of coupled non-linear equations and we use the optimization toolbox with the function fsolve to solve (26-27) at each maximization step.
Fig. 6: Normalized Probability Density Distribution (pdf) approximation with Gaussian mixture for Fig. 5.
In the case n = 3, we have optimized the non-linear mapping with matrix A in the same way, using T i j , T shift for i = j and T i j , T tent for i ≠ j. We obtain the following results.
With
1 8 2
1 1 2
1 1 1
A , we have the following distance spectrum :
Fig. 7: Distance spectrum for
1 8 2
1 1 2
1 1 1
A , n = 3
The best results we found in our optimization is given on Fig. 8 with matrix
1 2 6
1 1 4
1 1 1
A , we obtain the following spectrum :
Fig. 8: Distance spectrum for
1 2 6
1 1 4
1 1 1
A , n = 3
The minimum distance in this case is nearly equal to: d free 13 .
To end this part, we give some BER results on AWGN channels, using the optimization obtained by the distance spectrum computation. Due to a lack of place we only give simulation results. The use of accurate upper bounds with the approximate pdf of the distance spectrum will be illustrated in the next part when the chaotic coder will be serially concatenated with a Space Time Block code (STBC).
For n = 2, we obtain the following result:
Fig. 9: Performances of Trellis Chaos-Coded Modulation over AWGN channels
for n = 2, Q = 3
The chaotic coder outperforms uncoded BPSK at high SNR’s due to good asymptotic properties with a moderate high free distance. The weakness of this kind of code is their poor coding rate. There are several solutions to improve this. The first is to make input bits enter the coder by groups of k bits. In this case, the coding rate becomes equal to:
) 1 .( Q n
k . However, this considerably reduces the correlation degree between consecutive states and renders the trellis non-binary. We found that the penalty encountered by this method too much important (using k = 2 results in 4 dB losses compared to k = 1) so we prefer using puncturing to increase the coding rate of our proposed coders. We added the case of punctured codes on Fig. 9 with the best puncturing patterns we found for rate 8/7 and 4/3. The performances of punctured codes are improved at low or medium SNR’s and they outperform the optimized 1/8 chaotic coder whilst at high SNR’s they exhibit a smaller free distance and their diversity gain (i.e. the slope of the BER curb) is worse than those of the optimized 1/8 chaotic coder.
As predicted by the spectrum distance, the performances of the code with mapping matrix
5 1 1
A 1 are degraded when compared to the case
8 1 1 A 1 .
The conclusions are nearly the same, considering the case n = 3 as it is illustrated on Fig. 10.
Fig. 10: Performances of Trellis Chaos-Coded Modulation over AWGN channels for n = 3, Q = 2
The number of states for n = 3 and Q = 2 is 2 9 = 512. Due to a better free distance than for the case n = 2, the BER performances are improved and the chaotic coder outperforms uncoded BPSK at a threshold SNR level equal to 8.5 dB. The best punctured codes with puncturing rates 9/7 and 3/2 are unable to outperform the uncoded BPSK even at high SNR’s due to a rapid decrease of the free distance value.
The best performances of chaotic coders for n = 3, Q = 2 are illustrated on Fig. 11. This time,
with a free distance equal to 13 for the optimized code with rate 1/9, the punctured codes (9/7)
are able to outperform uncoded BPSK at high SNR’s. To complete this overview of BER
performances over AWGN channels, it is important to say that using the approximate pdf’s of
the distance spectrum, we are able to accurately predict the BER at high’s SNR’s. To
complete the results, we give on Fig. 12 the best performances we found with n = 3, Q = 3
(i.e. the number of states is 4096).
Fig. 11: Performances of optimized Trellis Chaos-Coded Modulation over AWGN channels for n = 3, Q = 2
In fact, as it is expected, increasing the quantization level for a given dimensionality n, entails some losses. Compared to Fig. 11, the loss in terms of SNR for a BER of 10 -4 , 10 -5 is approximately 1 dB and, once again, punctured codes are nable to outperform uncoded BPSK.
.
Fig. 12: Performances of optimized Trellis Chaos-Coded Modulation over AWGN channels
for n = 3, Q = 3 (4096 states)
III. Chaos Coded Modulation Scheme concatenated with STBC, performances over quasi-static block fading channels:
The purpose of this report is to study the behaviour of the chaotic coder on a more general and complex context, when it is concatenated with a STBC (Space-Time Block Code) for a MIMO system. This coder is concatenated with a (Space Time Block Code) STBC as the well known Alamouti’s scheme as illustrated on Fig. 13.
Fig. 13: Concatenation of the chaos coded modulation encoder and a STBC code We consider to simplify the case where n = 2; i.e x k is a vector of two transmitted analog
symbols:
) (
) (
2 1
k x
k x
x k . These two symbols are transmitted using the well known Alamouti’s
scheme
( ) ( ) ) ( )
(
* 1 2
* 2 1
k x k x
k x k
S x or, in the case of real transmitted symbols,
( ) ( ) ) ( )
(
1 2
2 1
k x k x
k x k
S x .
3.1- Computation of the Pairwise Error Probability (PEP):
In the case of quasi-static block fading channels, channel remains constant over the duration of a transmitted packet but changes from packet to packet and there is no need to use an interleaver between the chaos-based channel encoder and the STBC. The received signal within two consecutive time slots can be written in the following way:
) ( ) ( )
( )
( ) 1 (
) ( ) ( )
( )
(
2
* 1 21
* 2 11 2
1
1 2
21 1
11 1
n n n x h n x h n y n
y
n n n x h n x h n y
(29)
h 11 and h 21 are two complex random variables with mean zero and variance 2. h 2 . n 1 and n 2 are two additive complex white Gaussian noise samples with mean zero and variance 2. 2 . At the output of the Maximum Ratio Combiner (MRC) we form the two decision variables
)
1 ( n
Z and Z 2 ( n ) :
) ( . ) ( . ) ( ).
( ) (
) ( . ) ( . ) ( ).
( ) (
* 11 2 1
* 21 2
2 21 2 11 2
* 2 21 1
* 11 1
2 21 2 11 1
n n h n n h n x h h
n Z
n n h n n h n x h h
n Z
(30)
Remark: Z 1 (n) and Z 2 (n) are obtained classically by Maximum Ratio Combining (MRC) technique:
Chaos
Encoder STBC
code
Channel MRC
Combiner Viterbi
Decoder
) 1 ( . ) ( . ) (
) 1 ( . ) ( . ) (
* 11 1 1
* 21 2
* 1 21 1
* 11 1
n y h n y h n Z
n y h n y h n Z
Setting B ( h 11 2 h 21 2 ) , there will be an error event when having sent the sequence x
= ( x 1 ( m ), x 2 ( m ),..., x 1 ( m L 1 ), x 2 ( m L 1 )) , each time the decoder chooses x ' x , both sequences starting in the same state and merging again in possibly other state after L steps, when the decision metric implying x’ will be inferior to those with x. Assuming ML decoding, this is equivalent to:
-1 2 2
2 2 2 2
11 21 1 11 21 1 2 11 21 2
-1 2 2 ' 2 2 2 ' 2
1 11 21 1 2 11 21 2
( ' , , ) Pr oba( ( ) ( ). ( ) ( ) ( ). ( )
( ) ( ). ( ) ( ) ( ). ( ) ) (31)
L m e
n m L m
n m
P h h Z n h h x n Z n h h x n
Z n h h x n Z n h h x n
