Optimal chaos shift keying communications with ... - Semantic Scholar
Proceedings of the 2004 International Symposium on Circuits and Systems (ISCAS04), Vol.4, pp 593-596, 23-26 May 2004, Vancouver, Canada
OPTIMAL CHAOS SHIFT KEYING COMMUNICATIONS WITH CORRELATION DECODING Ji Yao School of Mathematics and Statistics, The University of Birmingham, Birmingham, B15 2TT, UK Email: [email protected] ABSTRACT The effect of the chaotic map employed in chaos shift keying (CSK) communication systems is discussed. Based on the exact bit-error rate of a CSK system, the conditions for an optimal map are deduced. One group of asymptotically optimal maps is proposed which has asymptotic performance at the level of the theoretical BPSK lower bound. Simulation results using the optimal maps are provided and compared with commonly used chaotic maps. The proposals show how to design both coherent and non-coherent CSK systems, with spreading factors of two, which have the advantage of being chaotic and have the high performance of BPSK.
the context of chaotic shift keying (CSK) communication systems. As a main result, one group of maps is proposed for which the CSK system has the asymptotically optimal performance in agreement with the theoretical lower bound. 2. ANTIPODAL CSK SYSTEM MODEL 2.1 Modulator model Here a single user discrete-time antipodal CSK communication system [1] is treated. The bit to be transmitted is b ∈ {−1, +1} , which is modulated to a
sequence s = bx , where x = {xi }, i = 0,1, 2,..., N − 1 is generated by a chaotic map τ ( x ) . 1. INTRODUCTION
spreading factor. The chaotic sequence
In most analytical performance evaluations of chaotic communication systems, the Central Limit Theorem (CLT) is applied to a sum of dependent variables, and in so doing ignores properties of the underlying chaotic map apart from the mean and variance of its natural invariant distribution. As indicated in [1], the results obtained by using the CLT are just the same as those for the BPSK system performance in the coherent case, and are only a lower bound for such communication systems. So the exact performance is always poorer than the CLT results suggest. Further, the performance of two systems with chaotic maps having the same mean and variance may unknowingly differ to an important degree from each other. This performance inaccuracy is known as the estimation problem [2] in chaos communication systems and is associated with the non-constant energy of a sample from the chaos sequence; only constant energy per bit leads to optimal performance [2], [3]. It is possible to introduce other modulation schemes to overcome this problem, such as the FM-DCSK system [4], but in this paper the effect of the map will be analyzed. Conditions will be derived that minimize the estimation problem in
;,(((
N is termed the
{X }
is assumed
to have been started with a random initial value x0 , which is chosen from the natural invariant distribution ρ ( x ) of the map τ ( x ) . The common means of {X } are denoted by µ ≡ E ( X ) , the common variances by σ X2 ≡ var [ X ]
and υ X2 ≡ var ⎡⎣ X 2 ⎤⎦ . Without loss of generality, we take
µ = 0 in the following development.
2.2 Channel model The channel is modeled as an additive white Gaussian noise (AWGN) channel. So corrupted by the channel noise, the received version of the chaotic signal sequence is ri = si + ε i , where {ε i } are independent Gaussian random variables with zero mean and variance σ N2 . In the context of a coherent system, the uncorrupted reference sequence is generated at the receiver, that is the received reference sequence ti = xi . The per bit signal noise ratio Eb N 0 is defined as ( N σ X2 ) ( 2σ N2 ) [5].
In the context of non-coherent systems, the reference sequence is also corrupted by the channel noise, so the
,9
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,6&$6
received reference sequence is ti = xi + ηi , where {ηi } are independent Gaussian random variables with zero mean and variance σ N2 . As the noise exists in both the received signal sequence and the reference sequence, Eb N 0 is
( Nσ ) σ 2 X
2 N
in this case.
2.3 Demodulator model In both coherent and non-coherent cases, correlation decoders are considered in this paper, that is, the decision about the bit value b′ is made by ⎧+ ⎪ 1 if C (t , r ) ≥ 0 , b′ = ⎨ ⎪⎩ −1 if C (t , r ) < 0 N −1
where C (t , r ) = ∑ ti ri .
The correlation decoder is
i=0
optimal in the single-user coherent system [1], but it is not so for non-coherent systems. 3. EXACT BER OF ANTIPODAL CSK SYSTEMS As shown in [1], the exact bit-error rate (BER) of coherent system is N −1 ⎡ ⎛ ⎞⎤ 2 i BERCO = E ⎢Φ ⎜ − ∑ ⎡⎣τ ( ) ( X )⎤⎦ σ N ⎟ ⎥ (3.1) ⎜ ⎟ i =0 ⎠ ⎦⎥ ⎣⎢ ⎝
(
2 ⎡ N −1 i ⎤ var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ( N σ X2 )⎥ can be considered as a ⎣ i =0 ⎦ measure of how far the exact BER is from the lower bound BER, although there is no direct relation between them. Because F ( x; r1 , r2 , λ ) is a convex function of the non-
central parameter [1], the same approach can be applied to (3.2), that is to 2 ⎞ ⎡ ⎛ 2 E ⎞⎤ ⎛ N −1 i BERNC = E ⎢ F ⎜1; N , N , ⎜ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎟ ( N σ X2 ) ⋅ b ⎟ ⎥ N 0 ⎠⎦ ⎝ i =0 ⎠ ⎣ ⎝ 2 ⎡ N −1 i ⎤ from which var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ( N σ X2 )⎥ can be taken as ⎣ i =0 ⎦ the corresponding measure. So both lower bounds can only be reached when N −1 2⎤ ⎡ i var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎥ = 0 , (4.1) ⎣ i =0 ⎦ that is when N −1 2⎤ ⎡ N −1 i ⎡ ⎤ var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎥ ≡ ⎢ N + 2∑ ( N − k ) β k ⎥ υ X2 = 0 , k =1 ⎣ i =0 ⎦ ⎣ ⎦
(
where β k = cov X 2 , ⎡⎣τ ( k ) ( X )⎤⎦
2
) var ⎡⎣ X ⎤⎦ . Thus when 2
N −1
N + 2∑ ( N − k ) β k = 0 ,
(4.2)
k =1
the estimation problem is minimised.
)
with a lower bound of BERCOL = Φ − 2 Eb N 0 , which
5. LIMITINGLY OPTIMAL MAPS
is just same as the BPSK system. In the non-coherent system [1], [6], the exact BER is 2 ⎞ ⎡ ⎛ ⎞⎤ ⎛ N −1 i BERNC = E ⎢ F ⎜1; N , N , ⎜ 2∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎟ σ N2 ⎟ ⎥ , (3.2) ⎝ i =0 ⎠ ⎠⎦ ⎣ ⎝ where F ( x; r1 , r2 , λ ) is the cdf of a non-central F-
In this section, only the simplest case N = 2 is considered. By (4.2), the condition to reach the lower bound BER is 2 + 2 β1 = 0 ⇒ β1 = −1 .
distribution with degrees r1 , r2 and non-central parameter λ . The lower bound in the non-coherent case is BERNCL = F (1; N , N , 2 Eb N 0 ) . Due to the estimation problem, this lower bound is never reached with the Bernoulli shift map, tent map or Chebyshev map.
)
( Nσ X2 ) ⋅
2 Eb N0
2
)
var ⎡⎣ X 2 ⎤⎦ = −1
(5.1)
which indicates that ⎡⎣τ ( x )⎦⎤ = ax 2 + b with a < 0 , b unrestricted, for all possible x . If τ (⋅) is a symmetric odd or even map such that 2
x → τ ( x ) : [−1,1] → [−1,1] , then we can define another
( )
Because Φ − x is a convex function [1], with (3.1) ⎡ ⎛ ⎛ N −1 i 2 ⎞ BERCO = E ⎢Φ ⎜ − ⎜ ∑ ⎡τ ( ) ( X )⎤ ⎟ ⎣ ⎦ ⎠ ⎜ ⎣⎢ ⎝ ⎝ i = 0
(
β1 = cov X 2 , ⎡⎣τ ( X )⎤⎦
map ψ (⋅) such that
4. CONDITIONS FOR MINIMISING THE ESTIMATION PROBLEM
(
By the definition of β k ,
⎞⎤ ⎟⎥ , ⎟ ⎠ ⎦⎥
2
ψ ( x ) = ⎡τ x ⎤ for 0 ≤ x ≤ 1 . ⎣ ⎦ Then ψ ( x ) has the following relation with τ ( x ) () x ⎯⎯⎯ →y τ ⋅
7
7 . ψ (⋅ )
x ⎯⎯⎯ → y2 2
,9 Authorized licensed use limited to: WARWICK UNIVERSITY. Downloaded on November 5, 2009 at 06:35 from IEEE Xplore. Restrictions apply.
(5.2)
By (5.1) and (5.2), ψ ( x 2 ) = ⎡⎣τ ( x )⎤⎦ = ax 2 + b which
be converted to negative values, i.e. take negative square roots. With the assumption that τ (⋅) is even or odd
be fully mapped from [−1,1] to [−1,1] , then ψ (⋅) will
symmetrical, the map is developed to the negative side of x by folding. Here, considering the even symmetrical case, map τ (⋅) takes the form
2
implies ψ ( x ) = ax + b for 0 ≤ x ≤ 1 . If the map τ (⋅) is to
also be fully mapped from [0,1] to [0,1] . To satisfy this condition, a = −1 and b = 1 , i.e. ψ ( x ) = − x + 1 .
However, not all maps ψ (⋅) can be converted back to
τ (⋅) because of the restriction on τ (⋅) that it should be mapped from
[−1,1]
to
[−1,1] .
Only those maps
y = ψ ( x ) which have at least two pre-images for any given
y,
i.e.
there
exist
two
different
⎧ −2 x 2 + 1 + i M ⎪ τ (x) = ⎨ ⎪⎩ − −2 x 2 + 1 + (i + 1) M
i M < x
8.5dB). Further, the performance of the proposed map with M = 2,3 is quite near the BPSK performance, which is the lower bound of coherent CSK systems.
The CSK communication system with the maps proposed in this paper has an asymptotic bit-error performance in agreement with the theoretical lower bound of CSK, both in the coherent and non-coherent cases. For each, using the asymptotically optimal maps with M = 2,3 , it is possible to get both the advantages of a chaotic communication system and the high performance of BPSK. Although these maps are designed only for spreading factors of 2, in principle it is possible to design further more complicated maps for higher spreading factors. However, the asymptotically constant energy of sequences with spreading factor 2 usually guarantees nearly constant energy for longer sequences, certainly for those with even spreading factors. Further, because in the non-coherent case the BER lower bound will increase if spreading factors increase, there seems no reason to have more complicated maps and higher spreading factors. ACKNOWLEDGEMENT I thank my research supervisor, Professor A. J. Lawrance, for useful discussions and advice. REFERENCES
Figure 3. Simulation results for the coherent and non-coherent CSK communication system with different underlying chaotic maps. The asymptotically optimal maps proposed in the paper are seen to have a strong and obvious advantage over commonly used maps.
For the non-coherent case, Figure 3 shows that to achieve a BER of 10−3 , Eb N 0 needs to be about 12dB using the proposed asymptotically optimal map with M = 1 and about 11dB with M = 2 and M = 3 . Again, these are much lower than those for the Bernoulli shift map (>14.7dB) and the Chebyshev map (>13dB). It is
[1] A. J. Lawrance and G. Ohama, “Exact calculation of bit error rates in communication systems with chaotic modulation,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 1391-1400, Nov. 2003. [2] G. Kolumban, M. P. Kennedy, Z. Jako and G. Kis, “Chaotic communications with correlator receivers: theory and performance limits,” Proceeding of the IEEE, vol. 90, May 2002, pp. 711-732. [3] A. Abel and W. Schwarz, “Chaos communication-principles, schemes, and system analysis,” Proceeding of the IEEE, vol. 90, No. 5, pp. 691-710, May. 2002. [4] G. Kolumban, G. Kis, Z. Jako, and M. P. Kennedy, “FMDCSK: A robust modulation scheme for chaotic communication,” IEICE Trans. Fund., vol. E81-A, pp. 17981802, Oct. 1998. [5] W. M. Tam, F. C. M. Lau, C. K. Tse, and M. M. Yip, “An approach to calculating the bit-error rate of a coherent chaosshift-keying digital communication system under a noisy multiuser environment,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 210-223, Feb. 2002. [6] A. J. Lawrance and G. Ohama, “Exact analysis of bit error and optimal spreading in non-coherent chaos shift keying communication”, Research Report, University of Birmingham, 2003. [7] J. G. Proakis, “Digital communication,” McGraw-Hill.
,9 Authorized licensed use limited to: WARWICK UNIVERSITY. Downloaded on November 5, 2009 at 06:35 from IEEE Xplore. Restrictions apply.
OPTIMAL CHAOS SHIFT KEYING COMMUNICATIONS WITH CORRELATION DECODING Ji Yao School of Mathematics and Statistics, The University of Birmingham, Birmingham, B15 2TT, UK Email: [email protected] ABSTRACT The effect of the chaotic map employed in chaos shift keying (CSK) communication systems is discussed. Based on the exact bit-error rate of a CSK system, the conditions for an optimal map are deduced. One group of asymptotically optimal maps is proposed which has asymptotic performance at the level of the theoretical BPSK lower bound. Simulation results using the optimal maps are provided and compared with commonly used chaotic maps. The proposals show how to design both coherent and non-coherent CSK systems, with spreading factors of two, which have the advantage of being chaotic and have the high performance of BPSK.
the context of chaotic shift keying (CSK) communication systems. As a main result, one group of maps is proposed for which the CSK system has the asymptotically optimal performance in agreement with the theoretical lower bound. 2. ANTIPODAL CSK SYSTEM MODEL 2.1 Modulator model Here a single user discrete-time antipodal CSK communication system [1] is treated. The bit to be transmitted is b ∈ {−1, +1} , which is modulated to a
sequence s = bx , where x = {xi }, i = 0,1, 2,..., N − 1 is generated by a chaotic map τ ( x ) . 1. INTRODUCTION
spreading factor. The chaotic sequence
In most analytical performance evaluations of chaotic communication systems, the Central Limit Theorem (CLT) is applied to a sum of dependent variables, and in so doing ignores properties of the underlying chaotic map apart from the mean and variance of its natural invariant distribution. As indicated in [1], the results obtained by using the CLT are just the same as those for the BPSK system performance in the coherent case, and are only a lower bound for such communication systems. So the exact performance is always poorer than the CLT results suggest. Further, the performance of two systems with chaotic maps having the same mean and variance may unknowingly differ to an important degree from each other. This performance inaccuracy is known as the estimation problem [2] in chaos communication systems and is associated with the non-constant energy of a sample from the chaos sequence; only constant energy per bit leads to optimal performance [2], [3]. It is possible to introduce other modulation schemes to overcome this problem, such as the FM-DCSK system [4], but in this paper the effect of the map will be analyzed. Conditions will be derived that minimize the estimation problem in
;,(((
N is termed the
{X }
is assumed
to have been started with a random initial value x0 , which is chosen from the natural invariant distribution ρ ( x ) of the map τ ( x ) . The common means of {X } are denoted by µ ≡ E ( X ) , the common variances by σ X2 ≡ var [ X ]
and υ X2 ≡ var ⎡⎣ X 2 ⎤⎦ . Without loss of generality, we take
µ = 0 in the following development.
2.2 Channel model The channel is modeled as an additive white Gaussian noise (AWGN) channel. So corrupted by the channel noise, the received version of the chaotic signal sequence is ri = si + ε i , where {ε i } are independent Gaussian random variables with zero mean and variance σ N2 . In the context of a coherent system, the uncorrupted reference sequence is generated at the receiver, that is the received reference sequence ti = xi . The per bit signal noise ratio Eb N 0 is defined as ( N σ X2 ) ( 2σ N2 ) [5].
In the context of non-coherent systems, the reference sequence is also corrupted by the channel noise, so the
,9
Authorized licensed use limited to: WARWICK UNIVERSITY. Downloaded on November 5, 2009 at 06:35 from IEEE Xplore. Restrictions apply.
,6&$6
received reference sequence is ti = xi + ηi , where {ηi } are independent Gaussian random variables with zero mean and variance σ N2 . As the noise exists in both the received signal sequence and the reference sequence, Eb N 0 is
( Nσ ) σ 2 X
2 N
in this case.
2.3 Demodulator model In both coherent and non-coherent cases, correlation decoders are considered in this paper, that is, the decision about the bit value b′ is made by ⎧+ ⎪ 1 if C (t , r ) ≥ 0 , b′ = ⎨ ⎪⎩ −1 if C (t , r ) < 0 N −1
where C (t , r ) = ∑ ti ri .
The correlation decoder is
i=0
optimal in the single-user coherent system [1], but it is not so for non-coherent systems. 3. EXACT BER OF ANTIPODAL CSK SYSTEMS As shown in [1], the exact bit-error rate (BER) of coherent system is N −1 ⎡ ⎛ ⎞⎤ 2 i BERCO = E ⎢Φ ⎜ − ∑ ⎡⎣τ ( ) ( X )⎤⎦ σ N ⎟ ⎥ (3.1) ⎜ ⎟ i =0 ⎠ ⎦⎥ ⎣⎢ ⎝
(
2 ⎡ N −1 i ⎤ var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ( N σ X2 )⎥ can be considered as a ⎣ i =0 ⎦ measure of how far the exact BER is from the lower bound BER, although there is no direct relation between them. Because F ( x; r1 , r2 , λ ) is a convex function of the non-
central parameter [1], the same approach can be applied to (3.2), that is to 2 ⎞ ⎡ ⎛ 2 E ⎞⎤ ⎛ N −1 i BERNC = E ⎢ F ⎜1; N , N , ⎜ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎟ ( N σ X2 ) ⋅ b ⎟ ⎥ N 0 ⎠⎦ ⎝ i =0 ⎠ ⎣ ⎝ 2 ⎡ N −1 i ⎤ from which var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ( N σ X2 )⎥ can be taken as ⎣ i =0 ⎦ the corresponding measure. So both lower bounds can only be reached when N −1 2⎤ ⎡ i var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎥ = 0 , (4.1) ⎣ i =0 ⎦ that is when N −1 2⎤ ⎡ N −1 i ⎡ ⎤ var ⎢ ∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎥ ≡ ⎢ N + 2∑ ( N − k ) β k ⎥ υ X2 = 0 , k =1 ⎣ i =0 ⎦ ⎣ ⎦
(
where β k = cov X 2 , ⎡⎣τ ( k ) ( X )⎤⎦
2
) var ⎡⎣ X ⎤⎦ . Thus when 2
N −1
N + 2∑ ( N − k ) β k = 0 ,
(4.2)
k =1
the estimation problem is minimised.
)
with a lower bound of BERCOL = Φ − 2 Eb N 0 , which
5. LIMITINGLY OPTIMAL MAPS
is just same as the BPSK system. In the non-coherent system [1], [6], the exact BER is 2 ⎞ ⎡ ⎛ ⎞⎤ ⎛ N −1 i BERNC = E ⎢ F ⎜1; N , N , ⎜ 2∑ ⎡⎣τ ( ) ( X )⎤⎦ ⎟ σ N2 ⎟ ⎥ , (3.2) ⎝ i =0 ⎠ ⎠⎦ ⎣ ⎝ where F ( x; r1 , r2 , λ ) is the cdf of a non-central F-
In this section, only the simplest case N = 2 is considered. By (4.2), the condition to reach the lower bound BER is 2 + 2 β1 = 0 ⇒ β1 = −1 .
distribution with degrees r1 , r2 and non-central parameter λ . The lower bound in the non-coherent case is BERNCL = F (1; N , N , 2 Eb N 0 ) . Due to the estimation problem, this lower bound is never reached with the Bernoulli shift map, tent map or Chebyshev map.
)
( Nσ X2 ) ⋅
2 Eb N0
2
)
var ⎡⎣ X 2 ⎤⎦ = −1
(5.1)
which indicates that ⎡⎣τ ( x )⎦⎤ = ax 2 + b with a < 0 , b unrestricted, for all possible x . If τ (⋅) is a symmetric odd or even map such that 2
x → τ ( x ) : [−1,1] → [−1,1] , then we can define another
( )
Because Φ − x is a convex function [1], with (3.1) ⎡ ⎛ ⎛ N −1 i 2 ⎞ BERCO = E ⎢Φ ⎜ − ⎜ ∑ ⎡τ ( ) ( X )⎤ ⎟ ⎣ ⎦ ⎠ ⎜ ⎣⎢ ⎝ ⎝ i = 0
(
β1 = cov X 2 , ⎡⎣τ ( X )⎤⎦
map ψ (⋅) such that
4. CONDITIONS FOR MINIMISING THE ESTIMATION PROBLEM
(
By the definition of β k ,
⎞⎤ ⎟⎥ , ⎟ ⎠ ⎦⎥
2
ψ ( x ) = ⎡τ x ⎤ for 0 ≤ x ≤ 1 . ⎣ ⎦ Then ψ ( x ) has the following relation with τ ( x ) () x ⎯⎯⎯ →y τ ⋅
7
7 . ψ (⋅ )
x ⎯⎯⎯ → y2 2
,9 Authorized licensed use limited to: WARWICK UNIVERSITY. Downloaded on November 5, 2009 at 06:35 from IEEE Xplore. Restrictions apply.
(5.2)
By (5.1) and (5.2), ψ ( x 2 ) = ⎡⎣τ ( x )⎤⎦ = ax 2 + b which
be converted to negative values, i.e. take negative square roots. With the assumption that τ (⋅) is even or odd
be fully mapped from [−1,1] to [−1,1] , then ψ (⋅) will
symmetrical, the map is developed to the negative side of x by folding. Here, considering the even symmetrical case, map τ (⋅) takes the form
2
implies ψ ( x ) = ax + b for 0 ≤ x ≤ 1 . If the map τ (⋅) is to
also be fully mapped from [0,1] to [0,1] . To satisfy this condition, a = −1 and b = 1 , i.e. ψ ( x ) = − x + 1 .
However, not all maps ψ (⋅) can be converted back to
τ (⋅) because of the restriction on τ (⋅) that it should be mapped from
[−1,1]
to
[−1,1] .
Only those maps
y = ψ ( x ) which have at least two pre-images for any given
y,
i.e.
there
exist
two
different
⎧ −2 x 2 + 1 + i M ⎪ τ (x) = ⎨ ⎪⎩ − −2 x 2 + 1 + (i + 1) M
i M < x
8.5dB). Further, the performance of the proposed map with M = 2,3 is quite near the BPSK performance, which is the lower bound of coherent CSK systems.
The CSK communication system with the maps proposed in this paper has an asymptotic bit-error performance in agreement with the theoretical lower bound of CSK, both in the coherent and non-coherent cases. For each, using the asymptotically optimal maps with M = 2,3 , it is possible to get both the advantages of a chaotic communication system and the high performance of BPSK. Although these maps are designed only for spreading factors of 2, in principle it is possible to design further more complicated maps for higher spreading factors. However, the asymptotically constant energy of sequences with spreading factor 2 usually guarantees nearly constant energy for longer sequences, certainly for those with even spreading factors. Further, because in the non-coherent case the BER lower bound will increase if spreading factors increase, there seems no reason to have more complicated maps and higher spreading factors. ACKNOWLEDGEMENT I thank my research supervisor, Professor A. J. Lawrance, for useful discussions and advice. REFERENCES
Figure 3. Simulation results for the coherent and non-coherent CSK communication system with different underlying chaotic maps. The asymptotically optimal maps proposed in the paper are seen to have a strong and obvious advantage over commonly used maps.
For the non-coherent case, Figure 3 shows that to achieve a BER of 10−3 , Eb N 0 needs to be about 12dB using the proposed asymptotically optimal map with M = 1 and about 11dB with M = 2 and M = 3 . Again, these are much lower than those for the Bernoulli shift map (>14.7dB) and the Chebyshev map (>13dB). It is
[1] A. J. Lawrance and G. Ohama, “Exact calculation of bit error rates in communication systems with chaotic modulation,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 1391-1400, Nov. 2003. [2] G. Kolumban, M. P. Kennedy, Z. Jako and G. Kis, “Chaotic communications with correlator receivers: theory and performance limits,” Proceeding of the IEEE, vol. 90, May 2002, pp. 711-732. [3] A. Abel and W. Schwarz, “Chaos communication-principles, schemes, and system analysis,” Proceeding of the IEEE, vol. 90, No. 5, pp. 691-710, May. 2002. [4] G. Kolumban, G. Kis, Z. Jako, and M. P. Kennedy, “FMDCSK: A robust modulation scheme for chaotic communication,” IEICE Trans. Fund., vol. E81-A, pp. 17981802, Oct. 1998. [5] W. M. Tam, F. C. M. Lau, C. K. Tse, and M. M. Yip, “An approach to calculating the bit-error rate of a coherent chaosshift-keying digital communication system under a noisy multiuser environment,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 210-223, Feb. 2002. [6] A. J. Lawrance and G. Ohama, “Exact analysis of bit error and optimal spreading in non-coherent chaos shift keying communication”, Research Report, University of Birmingham, 2003. [7] J. G. Proakis, “Digital communication,” McGraw-Hill.
,9 Authorized licensed use limited to: WARWICK UNIVERSITY. Downloaded on November 5, 2009 at 06:35 from IEEE Xplore. Restrictions apply.
