bit error probability and bit outage rate in chaos ... - Springer Link

C IRCUITS S YSTEMS S IGNAL P ROCESSING VOL . 24, N O . 5, 2005, P P. 519–534

c Birkh¨auser Boston (2005)
DOI: 10.1007/s00034-005-2404-9

B IT E RROR P ROBABILITY AND B IT O UTAGE R ATE IN C HAOS C OMMUNICATION * Anthony J. Lawrance1 and Gan Ohama2

Abstract. This paper investigates the notion of the probability of bit error (PBE) and its distribution in chaos-based communication systems; these are seen as being the fundamental quantities to both the well-known bit error rate (BER) and the new concept in chaos communications of bit outage rate (BOR). The form of the distribution illustrates the degree to which bit error rate is a stable representation of performance. Bit outage rate is another measure of performance which gives practically helpful information about bit error. For a simple coherent chaos-shift-keying system the distribution of bit error probability is derived exactly, and theoretically exact formulas for the bit outage rate and bit error rate are presented. Two specific cases are developed to obtain useful qualitative and quantitative information. The cases concern independent Gaussian spreading, as a lower benchmark and logistic map spreading, as typical of effective chaotic spreading. Comparisons are obtained between these spreading distributions and between different extents of their spreading, calibrated against per bit signal to noise ratio. A general conclusion is that bit outage and bit error rates are complementary measures of performance. Key words: Bit error rate, bit outage rate, chaos-shift keying communication systems, distribution of bit error probability.

1. Introduction The performance assessment of chaos-based communication systems has in the past mainly been limited to bit error rate (BER); by being a rate, it is an average over a distribution of many bits, and the relevant probability distribution has not previously been investigated. Another general idea of performance in ∗ Received February 15, 2005; revised May 30, 2005; This work was supported in part by the UK

Engineering and Physical Sciences Research Council (EPSRC) under Grant GR/M74795 and by the Shiga University Sabbatical Fund. 1 Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. E-mail: [email protected] 2 Faculty of Economics, Shiga University, Hikone, Shiga 552-8522, Japan. E-mail: [email protected]

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communication systems is bit outage rate (BOR), the main focus of this paper, and not previously discussed for chaos-based communication systems, it would seem. The concept tries to capture the rate of transmission for which a bit error is likely, according to a probability quantification of “likely,” rather than the average rate at which errors actually occur. Thus it refers to the probability of bit error (PBE) directly and the rate at which this has a specified probability. The aim of this paper is to set out an approach to the exact calculation of the probability of bit error and its accurate approximation. This is then linked to both bit error rate and bit outage rate, developing the authors’ earlier exact approach to bit error rate [12]. Exemplification will be with coherent chaos-shift-keying systems employing their correlation decoders and taking both independent Gaussian and logistic map spreading. A key theoretical concern has to be with the distribution of bit energy, typically not constant with chaos-based systems, as first noted in [6], and which features in PBE. With admittedly unrealistic independent Gaussian spreading, the exact distribution of bit energy is chi-squared, but it is not exactly tractable in other cases. However, for the more practically relevant chaotic map spreading, the bit energy distribution can be approximated by a scaled chi-squared distribution [15], and this allows much easier analytical calculation of bit outage rates. The aims of this paper are motivated by the earlier work on chaos communication systems, those employing chaotic waves in place of conventional sinusoidal waves. This work mostly gave noise performance assessment based on simulation and approximate analytics, rather than exact theory. Among the earlier work, a trio of papers, [7], [8], and [9], introduced and developed the basic ideas of chaos communication, while in [1], [2], and [14] there were the first developments of their noise performance approximate analytics. The monograph [11] coordinated much existing research in the area and also introduced further topics. There is another area of research, seen from [3], concerned with optimal maximum likelihood decoders. The general mathematical approach here is to use discrete-time base-band models which are designed to be equivalent to continuous-time random process models as far as performance studies are concerned. They simplify much of the traditional continuous-time mathematical basis of the communications modeling.

2. Probability of bit error and CSK systems Although the basic ideas underlying this paper are not specific to any particular chaos-based spread spectrum communication system, the well-known chaosshift-keying (CSK) system in its binary (antipodal) bit and coherent form will be used as the basis of the development. Suppose the CSK system uses chaotic spreading sequences {X i } which are generated by the chaotic map τ (z), −c < z < c, with a symmetric invariant distribution, thus of mean 0, and with variance σ X2 ; there is no lack of generality

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with these convenient assumptions. Assume an AWGN channel with spectral density and equivalent discrete-time variance N0 /2 ≡ σ 2 . A spreading extent N and spreading segment (X 1 , X 2 , . . . , X N ) are used in transmitting each binary bit b. A +1 bit value is signified by transmitting the spreading segment unaltered, while for a −1 the spreading values are reflected about their mean of zero. The received bit sequence R = (R1 , R2 , . . . , R N ) is thus of the form Ri = bX i + εi ,

i = 1, 2, . . . , N ,

b = ±1.

(2.1)

Because of the system coherence, at the receiver a synchronized exact copy of the spreading segment (X 1 , X 2 , . . . , X N ) is assumed known. Thus the correlation decoder C(X, R) ≡

N 

X i Ri

(2.2)

i=1

can be used to decide bit type, positive values indicating a +1 and negative values a −1. In fact, this decoder is optimal in a likelihood sense for the coherent CSK system. The customary communications performance measure is the bit error rate (BER) which for a +1 bit error is given by BER+1 = P {C(X, R) < 0|b = 1}   N N   2 =P Xi + εi X i < 0 . i=1

(2.3)

i=1

This is usefully the same as that for a −1 bit error, BER−1 , in this CSK case, and thus is the overall bit error rate. Indeed, for systems when BER+1  = BER−1 , it is likely that the decoder is not optimal. Following [12], the BER of (2.3) can be evaluated by noting that its inner term, as a linear function of the random variables (ε1 , ε2 , . . . , ε N ), is itself a Gaussian random variable conditional on (X 1 , X 2 , . . . , X N ). Then, with (·) denoting the distribution function of a standardized Gaussian variable, the probability of bit error (PBE), which is conditional on (X 1 , X 2 , . . . , X N ), can be introduced as     N    PBE(X 1 , X 2 , . . . , X N ) =  −

σ , (2.4) X i2   i=1 where X i = τ i (X ) is the ith iteration of the map τ acting on X , a random variable with the natural invariant distribution of the map τ . More succinctly, (2.4) can be written as  N  σ   X PBE(X 1 , X 2 , . . . , X N ) =  − |S N ,τ | , where S N2 ,τ = X i2 σ X2 . σ i=1 (2.5)

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PBE provides the basis of all subsequent results. It will be regarded as a random function of (X 1 , X 2 , . . . , X N ). Additionally, it may be noted that for extensive spreading, that is, large N , by the central limit theorem, (2.5) will be approximately true for most noise distributions, not just Gaussian. The probability of bit error is seen to be a function of the chaotic random variables X 1 , X 2 , . . . , X N through |S N ,τ |, the nonnegative square-root of their corrected and scaled sum-of-squares S N2 ,τ . This is a standardized form of bit energy, typically a randomly variable quantity in chaos-based systems, as indicated in [7]; later in [9] the authors suggested keeping it constant, and in [10] calling the difficulties of doing this “the estimation problem.” The average of the bit error probability is the customary bit error rate BER, and so is given by    σ X BER ≡ E X {PBE(X 1 , X 2 , . . . , X N )} = E S 2  − |S N ,τ | (2.6) N ,τ σ or, in terms √ of the complementary error function erfc, where (−x) = 1 erfc(x/ 2), 2    σX 1 BER = 2 E S 2 erfc (2.7) √ |S N ,τ | . N ,τ σ 2 The crucial role of bit energy for BER is openly demonstrated by this exact result and is mainly a consequence of the correlation decoding. On the assumption that X i = ±c equally, there is var (X i ) = c2 and S N2 ,τ = N , and if defining E b /N0 = N σ X2 /2σ 2 as the per bit signal-noise-ratio, (2.7) reduces to the √ well-known classical binary phase shift-keying (BPSK) result for BER, 12 erfc( E b /N0 ), e.g., [4], [13]. Here the definition SNR = N σ X2 /2σ 2 will be mathematically employed, while in standard dB notation, E b /N0 will denote 10 log10 (N σ X2 /2σ 2 ) and will be used in plotting. In [12] the BPSK result was shown to be the lower bound to the exact result (2.6) for coherent CSK systems and which can thus be inaccurate when used analytically. This result was first asserted in [5] which included the subsequently disputed but justified claim that coherent CSK with correlator demodulation could not improve on BPSK as far as BER was concerned. Interest is usually in BER as a function of the signal-to-noise ratio (SNR). Then (2.6) becomes     BER = E S 2  − (2SNR/N )|S N ,τ | (2.8) N ,τ

√ with the lower bound (− 2SNR). The distribution of PBE in (2.5), rather than the BER (2.8), is of central interest in this paper, with bit outage rate being derived from it in Section 3. The exact distribution of S N2 ,τ is fundamental to both PBE and bit outage rate, as well as to bit error rate. But it can be complicated, when in the chaotic case S N2 ,τ from (2.5) is a function of the initial random variable X and map iterations

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τ i (X ), and so taking the form S N2 ,τ (X ) =

N 

 {τ i−1 (X )}2 σ X2 .

(2.9)

i=1

The calculation requires a set of disjoint x-intervals such that x: S N2 ,τ (x) ≤ z which can be quite onerous to obtain for N > 2. The range of S N2 ,τ is upper bounded by N c2 /σ X2 due to the range (−c, c) of the spreading distribution, and in chaotic cases is nonnegatively lower bounded, this being expressed as Nlm2 /σ X2 ; there is an illustration with N = 2 for logistic map spreading in Section 6. Approximating the distribution of S N2 ,τ can avoid these problems, as exemplified in Section 7, although the exact approach is needed initially. An implication of S N2 ,τ being bounded is that BER is bounded according to     c √ lm √  − 2SNR ≤ BER ≤  − 2SNR . (2.10) σx σX with the interval seen to be within (0, 12 ). 3. Bit outage rate Bit outage rate is apparently a new concept in chaos communications, although known in related areas. A web search revealed that it is used with many meanings but broadly to convey “probability of nonavailability.” Here the meaning is the quantification of the likelihood of error of an individual bit. Bit outage rate is defined as the probability that PBE from (2.4) is greater than α, where α is a severity parameter with range(0, 12 ); values such as 0.001, 0.01, 0.1 can be regarded as practically realistic. The smaller the value of α the more severe or demanding the definition of outage. A poor communication system will have a high proportion of bits whose PBE’s are greater than a small α. The bit outage rate (BOR) is therefore the probability that PBE > α over all possible spreading sequences X 1 , X 2 , . . . , X N , that is, BORα = P{PBE(X 1 , X 2 , . . . , X N ) > α}.

(3.1)

Thus, BOR and the traditional BER are not equivalent but both are aspects of the distribution of PBE; one operational difference is their permissible ranges. As was done in [12] for the bit error rate (2.6), (3.1) can be examined exactly. Expressing (3.1) in terms of PBE from (2.5),     σ   −1 (1 − α) X BORα = P  − |S N ,τ | < α = P |S N ,τ | < σ σ X /σ  2  −1  (1 − α) , (3.2) = FS 2 N ,τ σ X /σ

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where FS 2 (·) is the distribution function of S N2 ,τ . The key expression (3.2) will N ,τ allow further specific results to be obtained in Sections 5 to 7. Particularly, there will be concern with the effect of the system’s SNR on its bit outage rate. Thus (3.2) in terms of SNR, and taking into account the range of S N2 ,τ , can be expressed as BORα (SNR)   1, 0 ≤ SNR < 12 [−1 (1 − α)]2 (σ X /c)2 ,    F 2 (N [−1 (1 − α)]2 (2SNR)−1 ), 1 [−1 (1 − α)]2 (σ X /c)2 ≤ SNR S N ,τ 2 =  < 12 [−1 (1 − α)]2 (σ X /lm )2 ,    0, 1 −1 2 2 2 [ (1 − α)] (σ X /lm ) ≤ SNR < ∞. (3.3) This formula indicates the plausible general form of BORα as a function of SNR; namely, there is an initial interval of low SNR values which give BORα = 1, followed by a period of decrease in BORα to zero, followed by constancy at zero as SNR indefinitely increases.

4. The exact distribution of bit error probability The distribution of bit error probability (2.4) or (2.5) over all spreading sequences is not usually considered but it extends the understanding of bit error and provides a further way of assessing a communications system; clearly, there is an advantage in having a fairly constant bit error probability so that its expectation, which is a bit error rate, provides a good description of bit-by-bit behavior. This distributional aspect might also be useful when comparing systems. By (2.5) the distribution function of PBE is    −1  2  −1 (1 − z)  (1 − z) 2 P(PBE < z) = P |S N ,τ | > , = P S N ,τ > (σ X /σ ) (σ X /σ ) (4.1) or, in terms of the distribution function of S N2 ,τ , FP B E (z)   0,     2  −1 (1−z) = 1 − FS 2 , (σ X /σ ) N ,τ    1,

√ 0 ≤ z < (−c N /σ ), √ √ (−c N /σ ) ≤ z < (−lm N /σ ), (4.2) √ (−lm N /σ ) ≤ z < ∞.

The breaks in form arise from the minimum and maximum values of S N2 ,τ . The distribution of S N2 ,τ is again the key aspect here. The probability density function

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(pdf) of PBE follows by differentiation of (4.2) as    −1   −1 2  −1 (1 − z)  (1 − z)  (1 − z) f P B E (z) = 2 , f S2 N ,τ σ X /σ σ X /σ σ X /σ √ √ (−c N /σ ) ≤ z < (−lm N /σ ), (4.3) 

and zero otherwise, where −1 (·) is the differential of −1 (·); it can be expressed as 1/φ{−1 (·)} where φ(·) is the standardized Gaussian pdf. The lower bound in (4.3) reflects the fact that the PBE cannot be eliminated in communication using this CSK system. The formulas (4.2) and (4.3) will be the basis of specific results in Sections 5, 6, and 7. A version of (4.3) in which the signal-to-noise ratio SNR is made explicit is given by 

f P B E (z) = N (SNR)−1 −1 (1−z)−1 (1−z) f S 2 (N [−1 (1−z)]2 (2SNR)−1 ), 1

N ,τ

1

(−(c/σ X )(2SNR) 2 ) ≤ z < (−(lm /σ X )(2SNR) 2 ).

(4.4)

This will be used in Sections 5 and 7 to plot the pdf of PBE. A further use of (4.2) could be to determine an interval for the bit error probability, instead of its average, the bit error rate. This acknowledges that there is variability associated with BER. A simple approach would to determine an interval of PBE values with coverage of, say, 80%, allowing 10% for each of the low and high extremes. The general form of this interval giving a coverage of 100(1 − 2β)% will be       −1 −1 1/2 1/2  − 2SNR/N [F 2 (1 − β)] ,  − 2SNR/N [F 2 (β)] , S N ,τ

S N ,τ

(4.5) 2 where F −1 2 (·) is the inverse cumulative distribution function of S N ,τ . S N ,τ

5. Distribution of bit error probability and bit outage rate under Gaussian spreading While interest is primarily in chaotic spreading sequences, and independent Gaussian ones usually give a poor performance, they do provide a lower bench mark for the use of chaotic spreading, and are very tractable. Furthermore, as will be seen in Section 7, a chi-squared approximation for S N2 ,τ , which is exact in the case of Gaussian spreading, can be very useful, needing very similar results to those of the Gaussian spreading. Thus, (2.6), (3.2), and (4.3) apply in the Gaussian case (where c = ∞, lm = 0) under the assumption that the random variable S N2 ,τ has a chi-square distribution with N degrees of freedom. An illustration is given in Figure 1 for the pdf of PBE, and BER results in this case were given in [12]. For N = 1, 2, 3, 4, 20 and SNR = 1, Figure 1 illustrates the distribution (4.4) of bit error probability and shows that extensive spreading through large N is

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PBE

Figure 1. Distribution of probability of bit error (PBE) for independent Gaussian spreading and SNR = 1 for N = 1 (lower decreasing solid line), N = 2 (decreasing dashed line), N = 3 (modal lower solid line), N = 4 (modal upper dotted line), and N = 20 (strongly modal solid line).

required for its average BER to represent a stable probability of bit error. Higher values of SNR would also make the distribution of PBE less variable and lead to its stronger modality and better representation of BER. Next, the bit outage rate (3.3) as a function of SNR can be expressed in terms of the distribution function Fχ 2 (·) of a χ N2 random variable as N

BORα (SNR) = Fχ 2 (N [ N

−1

(1 − α)]2 (2SNR)−1 ),

0 ≤ SNR < ∞.

(5.1)

The result (5.1) shows that as a function of α, bit outage rate decreases from 1 at α = 0 to 0 at α = 12 . The plots in Figure 2 illustrate the decrease in bit outage rate for increasing N as SNR increases, and the corresponding increase in bit outage rate to one as the SNR decreases. 6. Exact results for S N2 ,τ and BORα with logistic map spreading when N = 2 Although chaotic spreading with the factor N = 2 will be quite unrealistic for practical use, this simplest case does illustrate the route to exact calculation of PBE and BORα ; as well as offering theoretical insight, it justifies the practical 2 (x), from (2.9), in necessity of the approximate results of Section 7. First for S2,τ the case of the logistic map over (−1, 1) which has τ (x) = 2x 2 − 1, σ X2 = 12 , there is 2 S2,τ (x) = (x 2 + (2x 2 − 1)2 )/σ X2 = 2(4x 4 − 3x 2 + 1).

(6.1)

B IT E RROR AND O UTAGE

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Outage

Eb/N0 Figure 2. Plot of bit outage rate BORα against E b /N0 for α = 0.01 and independent Gaussian spreading when N = 1 (upper solid line), N = 2 (dashed line), N = 3 (long dashed line), N = 4 (dotted line), and N = 20 (lower solid line).

Ssqdx

–

–

x

2 (x) as a function of x illustrating calculation of the distribution function of Figure 3. Plot of S N ,τ 2 (X ) for N = 2 with logistic map spreading. SN ,τ

A plot of this function is given in Figure 3 which shows that there is a local maximum of 2 at 0, global maxima of 4 at x = ±1, and global minima of 78 at  7 x = ± 38 , and giving lm2 = 32 . These are used in calculating the distribution 2 (X ) as needed by (3.2) for BOR . function of S2,τ α 2 (X ) is seen to be Guided by Figure 3, the distribution function of S2,τ  0, 0 ≤ z < 78 ,     P(l < X < l ) + P(r < X < r ), 7 1 2 2 1 8 ≤ z < 2, FS 2 (z) = N ,τ  2 ≤ z < 4, P(l1 < X < r1 ),    1, 4 ≤ z < ∞,

(6.2)

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2 Figure 4. The distribution function of S N ,τ in the case N = 2 for logistic spreading.

where X has the beta( 12 , 12 ) invariant distribution of the logistic map, and l1 , l2 , r2 , r1 are the roots in ascending magnitude of the equation 2{4x 4 − 3x 2 + 1} = z. These roots are explicitly   l1 = − 18 {3 + (8z − 7)},   r2 = 18 {3 − (8z − 7)},

  l2 = − 18 {3 − (8z − 7)},   r1 = 18 {3 + (8z − 7)}.

(6.3)

(6.4)

Figure 4 gives a plot of the distribution function (6.2) and illustrates the break in 2 (x). form of the function at z = 2 due to the local maximum of S2,τ Starting from equation (3.3) for the logistic map, and hence using σ X2 = 12 , BORα is seen to be 1 for low SNR in the range 0 ≤ SNR < 14 [−1 (1 − α)]2 . With outage severity parameter values α = 0.0001, 0.001, 0.01, the minimum SNR values giving BORα less than 1 are, respectively, 5.39, 3.78, 1.31 dB. For the main interval 14 [−1 (1 − α)]2 ≤ SNR < (1/4lm2 )[−1 (1 − α)]2 , the results (6.2) and (6.4) allow (3.3) to be written in the form BORα (SNR)  1,     p(SNR), =  q1 (SNR) + q2 (SNR),    0,

0 ≤ SNR < 14 [−1 (1 − α)]2 , 1 1 −1 2 −1 2 4 [ (1 − α)] ≤ SNR < 2 [ (1 − α)] , 1 8 −1 2 −1 2 2 [ (1 − α)] ≤ SNR < 7 [ (1 − α)] , 8 −1 2 7 [ (1 − α)] ≤ SNR < ∞, (6.5)

where the functions p(SNR), q1 (SNR), q2 (SNR) derive from the probabilities in (6.2). Figure 5 uses (6.5) to give BORα (SNR) for three typical values of the

B IT E RROR AND O UTAGE

529

Outage

Eb/N0 Figure 5. Plot of bit outage rate BORα against E b /N0 for logistic map spreading (N = 2) and α = 0.01, 0.001, 0.0001 as second, third, and fourth curves from the left.

severity parameter α. It demonstrates the effect on the bit outage rate on the severity parameter α. The calculations here illustrate the complexity in extending the exact analysis to more extensive spreading. The polynomials corresponding to (6.1) will have many more turning points which will significantly add to the complexity of (6.2) and the computational burden. Hence there is an advantage in seeking effective approximation in calculating bit outage rates, as undertaken in Section 7.

7. Use of a chi-square approximation to accurately calculate bit outage rates The previous key exact formulas (2.6), (3.2), and (4.1) for bit error rate, bit outage rate, and bit error probability all require aspects of the distribution of S N2 ,τ which will be intractable when the spreading extent N becomes much larger than 2. However, in this situation, an approximation from [15] can be restated. By matching the mean and variance of S N2 ,τ to that of a scaled chi-squared random 2 variable κ1 χκ N  , there is the following expression, κ=

2 , (µ X,4 /σ X4 ) − 1

(7.1)

where µ X,4 is the fourth central moment of X and µ X,4 /σ X4 is the kurtosis of X . The N -modification factor κ applies only when the map produces quadratically uncorrelated sequences, as does a logistic map, otherwise there is the factor  −1  N −1   2 k 1+2 1− . (7.2) ρ X 2 (k) N (µ X,4 /σ X4 )−1 k=1

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Outage

Eb/N0 Figure 6. Illustration of the chi-square approximation in the plot of bit outage rate BORα=0.01 against E b /N0 for logistic spreading for N = 1, 2, 3, 4, 20, which is the eventual order of the curves, and comparable to the exact Gaussian spreading of Figure 2.

The approximation (7.1) has been found highly accurate, at least for the logistic map with its zero quadratic autocorrelations when N > 5, and is so even in this somewhat extreme case of a beta( 12 , 12 ) invariant distribution. Note that when µ X,4 = 3σ X4 , as it does for an independent Gaussian spreading sequence, (7.1) reduces to the exact χ N2 distribution. Since, for the logistic map with its beta( 12 , 12 ) invariant distribution, σ X2 = 12 and µ X,4 = 38 , the N -modification factor κ is 4. The approximating distribution 2 and can now replace that of S 2 is thus 14 χ4N N ,τ in (2.6) and (2.8) for BER, in (3.2) and (3.3) for BORα , and in (4.3) and (4.4) for the pdf of PBE. 2 approximation to the distribution of S 2 With the 14 χ4N N ,τ in (3.3), the bit outage rate for logistic spreading becomes BORα (SNR)  1, 0 ≤ SNR < 12 [−1 (1 − α)]2      ×(σ X /c)2 ,     F 2 (4N [−1 (1 − α)]2 (2SNR)−1 ), 1 [−1 (1 − α)]2 (σ /)2 ≤ SNR X χ4N 2 ∼ = 1 −1  < 2 [ (1 − α)]2 (σ X /lm )2 ,     1 −1 2 2  0,  2 [ (1 − α)] (σ X /lm )   ≤ SNR < ∞. (7.3) This approximation has been used to produce Figure 6 which is comparable to Figure 2 for Gaussian spreading. The generally lower outage rates of logistic spreading are clearly visible; this is emphasized by noting that the range in Figure 2 is more than 35 dB, whereas in Figure 6 the range is less than 8 dB. Note

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Table 1. Bit outage rates with severity parameter values α = 0.01, 0.001 for logistic spreading with N = 10 by the chi-square approximation for selected E b /N0 values.

E b /N0 =

3 dB

4 dB

5 dB

6 dB

7 dB

8 dB

α = 0.01 α = 0.001

0.9342 1

0.6595 0.9994

0.2730 0.9798

0.0611 0.8191

0.0077 0.4443

0 0.1324

Outage

Eb/N0 Figure 7. Comparison of bit outage rate BORα=0.01 values for logistic spreading with N = 2 obtained by exact result (dotted line) and chi-squared approximate result (solid line).

that as in the case of Gaussian spreading, extensive spreading is an advantage in bit outage terms for the higher SNR but that the reverse is true for the lower SNR. The low and high cut-offs in Figure 6 are due to the piecewise form of (7.3). Some representative numerical values of bit outage rate when N = 10, using the chi-squared approximation of (7.3), are given in Table 1. The two choices of severity parameter illustrate that the lower its value the higher is the bit outage probability. The choice of severity parameter will depend on the risk assessment of the system relative to its purpose. Figure 7 compares the exact and approximate bit outage rates for logistic spreading in the case N = 2, taken individually from Figures 5 and 6. The quality of the approximation is seen to be fairly good even at N = 2. 2 approximation for the distribution This section is concluded by giving the 14 χ4N of PBE using (4.4). The approximate pdf of PBE with logistic map spreading follows as 

f P B E (z) = 4N (SNR)−1 −1 (1 − z)−1 (1 − z) f χ 2 (2N [−1 (1 − z)]2 (SNR)−1 ), 4N √ √ 1 1 (−(c 2)(2SNR) 2 ) ≤ z < (−(c 2)(2SNR) 2 ). (7.4) In comparison with Gaussian spreading and Figure 1, the lower variability of

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PBE 2 Figure 8. Distribution of the probability of bit error (PBE) for logistic map spreading by the 14 χ4N approximation with SNR = 1 and N = 1, 2, 3, 4, 20, with the curves identified in this order by the increasing height of their modal values.

Figure 8 can be seen, most clearly for the large N = 20 case. This to be expected because the logistic spreading is usually best among the common maps.

8. Discussion This paper has introduced the ideas of bit outage rate and probability of bit error in chaos-based communications and emphasized that both bit outage rate and bit error rate are aspects of the probability of bit error, and are complementary measures of performance. Bit error rate is seen as the average of the distribution of bit error probability. Bit outage rate gives the rate at which a bit error is likely according to some probability and, therefore, is a much more conservative measure than bit error rate. It has been noted that the distribution of bit error probability allows a fixed-inclusion probability interval to be given, corresponding to and including the bit error rate The theory in the paper has been developed on an exact basis for coherent chaos-shift-keying systems with correlation decoding, and openly shows the key role of the randomly varying bit energy of such systems and that it is due to the correlation decoding. Results for extensive spreading when exact theory becomes computationally infeasible have been obtained for bit outage rate by use of an effective chi-square approximation for bit energy. This could equally well be used to obtain accurate but approximate analytical results for bit error rate.

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Acknowledgments The authors are grateful to Professor G. Mazzini of the Universities of Bologna and Ferrara Chaos Engineering Group for bringing the idea of bit outage in chaos communications to our attention. We are also grateful for his comments on a much earlier version of this paper.

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