数理解析研究所講究録
1011 巻 1997 年 89-106

89

Chaotic Binary Sequences with
Their.Applications to Communications
Tohru KOHDA and Akio TSUNEDA
of Computer Science andKyushu University$\mathrm{c}_{0111\mathrm{m}}11\mathrm{n}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{D}\mathrm{e}..\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}_{1}11\mathrm{e}\mathrm{n}\mathrm{t}$Engineering,

Abstract: In spread specfrllm. a lot of
nulnbers with good properties are required as spreading sequences, most of which are generated by linear feedback
shift register (LFSR). In this paper, we propose chaotic binary sequences which are basedon chaos generated by one-dilnensional nonlinear ergodic, and evaluate their perfornlance.S in spread spectrum systems. Furthermore, we propose an illlage
based on SS techniques which utilize characteristics of chaotic binary sequences.
$\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{l}\iota \mathrm{l}\mathrm{s}$$\mathrm{p}_{\mathrm{S}\mathrm{e}\mathrm{u}\mathrm{d}-}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\ln$$111\mathrm{a}\mathrm{p}_{\mathrm{S}}$$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{l}\mathrm{l}\iota \mathrm{l}\mathrm{n}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{s}_{v}\mathrm{v}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}1$1Introduction

Spread spectrum techniques are due primarily to properties of spreading sequences (orsequences)[1]. Various classes of PN sequences have been proposed mostpseudonoiseof which are generated by LFSR (linear feedback shift registers) stlch as the familiesof the Gold sequences and of the Kasami sequences with low even-correlation values [1].
On the other hand. we proposed
methods to obtain binary sequences from chaotic
trajectories generated by nonlinear ergodic maps whose even and odd correlation functions
can be theoretically given. The elnpirical distributions of correlation values of chaotic bit
sequences are shown to tend to the Gaussian distribution.
In this paper. we investigate theof chaotic binary sequences as spreading
sequences empirically and theoretically. Furthermore, we propose ancommunication
telll based on SS techniques which utilize characteristics of chaotic binary sequences.
$(\mathrm{p}\uparrow \mathrm{q}\mathrm{A})$$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}$$\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\Gamma 11\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$$\mathrm{i}_{1}\mathrm{n}\mathrm{a}\mathrm{g}\mathrm{e}$$\mathrm{s}\mathrm{y}‘ \mathrm{s}$2CDMA System Based on Direct Sequence Spread
Spectrum

In direct seqllence spread spectrum
systems, data sylnbols are directly multipliedbv a pseudonoisecode or a spreading code, which is independent of the data. Suchsystelns are due primarily to properties of spreading sequences(or PN sequences) [1]. For
the most fundamental technique, both data symbols and code symbols are bipolar. Figure
techniques. For simplicity, we
1 shows a model of a CDMAon such, spread spectrulll signals of users,
consider baseband comnlunications. In such a, are translllitted through a colnmon channel simultaneously.
$j=1,2,$$(\mathrm{D}\mathrm{S}/\mathrm{S}\mathrm{S})$$(\mathrm{P}\wedge \mathrm{N})$$\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}\mathrm{e}}\ln \mathrm{b}\mathrm{a}\mathrm{S}\mathrm{e}\mathrm{d}n$$\mathrm{D}\mathrm{S}/\mathrm{S}\mathrm{S}$$\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{n}$$s^{(j)}(f),$$\cdots,$

$J$

$J$

90

Figure 1: A model of a baseband CDMA system
The data signal
written asof the j-th user and the assigned PN code signal$d^{(j)}(t)$$d^{(j)}(t)$$= \sum_{p=-\infty}^{\infty}d_{p}(j)u\tau(t-pT)$$c^{(j)}(t)$where$?l_{T}(t)$.

$= \sum_{q=-\infty}^{\infty}Cu_{T_{c}}((qtj)-qT_{c}),$$c^{(j)}(t)$can be. $-1\}$(1)$c_{q}^{(j)}\in\{1, -1\}$(2)$d_{p}^{(j)}\in\{1$

is defined by$?l_{T}(\dagger)=\{$

1 for $0\leq t\leq T$
otherwise.

(3)

$0$

that the j-thWesequence
has period $N=T/T_{c}$ so that
, $c_{1}.\cdots.c_{N-1}$ per data
there is a PN sequence. For simplicity. assume that$T_{r}=1$ through this paper. For ais referred to as asystem, the PN code bit
chip. Thus the baseband spread spectrunl signal
is given by$\{C^{(j)}\}pp=0^{1}\mathrm{A}--$$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\ln e$$\mathrm{u}.\mathrm{s}\mathrm{e}\mathrm{r}’ \mathrm{S}\mathrm{p}_{-}(j)(j)(j)\mathrm{v}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}$$\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{l}\mathrm{b}_{\mathrm{o}1}$$c_{0}$$c_{q}^{(j)}$$\mathrm{D}\mathrm{S}/\mathrm{S}\mathrm{S}$$S^{(j)}(t)$$s^{(j)}(t\mathrm{I}=c^{(}j)(t)d(j)(t)$For asynchronous)$\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{l}$$\mathrm{l}\mathrm{s}$.

(4). the received signal $r(t)$ can be represented as(5)$r( \mathrm{t})=\sum_{j=1}^{J}c^{(}(t-t(j))j)d^{(j)}(t-\dagger)(j)+n(t)$whereis thedelay in the j-th channel and $n(t)$ is a
If the received signal $r(t)$ is the input to the correlation
output during the p-th tinue-interval is$t^{(j)}$$\mathrm{t}\mathrm{i}_{1}\mathrm{n}\mathrm{e}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{n}$channel noise process.
to
. the

$\mathrm{r}\mathrm{e}\mathrm{C}\mathrm{e}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{r}.\mathrm{m}\mathrm{a}\mathrm{t}_{\mathrm{C}\mathrm{h}\mathrm{e}\mathrm{d}}$$Z_{p}^{(i)}= \int_{pT}^{()}p+1Ttr()_{C}(i)(t)dt$.

$S^{(i)}(t)$

(6)

91

are constrained to be integral
that the system is quasi-synchronous (all of). Then we can writeof ) and $7?(t)=0$ and define$t^{(j)}$$\mathrm{A}\mathrm{s}\mathrm{s}\mathrm{U}\mathrm{n}\mathrm{l}\mathrm{e}$(7)$N+I_{J,p}^{()}i$$=$$Z_{\mathit{1})}^{()}i$$\ell_{ij}=\mathrm{f}^{(i)}-t^{(j}$$T_{c}$$111\mathrm{U}\mathrm{l}\mathrm{t}\mathrm{i}_{1})1\mathrm{e}\mathrm{s}$$= \sum_{j=1}^{J}\{\frac{d_{p}^{(j)}+dp+1(j)}{2}.R^{E}(\ell ij;\{c^{()}\}i.

$I_{J,p}^{(i)}$\{c^{(}\}j))+\frac{d_{P}^{(j)}-dp+1(j)}{2}R^{o()}(\ell_{ij}; \{ci\}, \{_{C}(j)\})\}$(8)$j\neq i$

where
and

$I_{J,p}^{(i)}$-channel interference from other $J-1$ channels and $R^{E}(P;\{c^{(i)}\}, \{C\}(j))$
denote the even and the odd cross-correlation filnctions, respectively. Thev are defined by [1]denotes$R^{O}(l_{\backslash }.

$\mathrm{c}\mathrm{o}$\{c^{(\uparrow)}\}, \{c^{(j)}\})$$(0\leq[\leq\lrcorner\backslash ^{\tau}-1)$$R^{E}(l;\{c^{(}i)\}$$\cdot$. $\{C\}(j))=R^{A}(\ell,\cdot\{c\}(i). \{c)(j\})+R^{A}(N-\ell.;\{C(j)\}. \{c^{(i})\}\mathrm{I}$.

$R^{o}(\zeta_{\backslash }.\{c\}(i), \{C\}(j))=RA(t_{\backslash }’\cdot\{C(i)\}.!\{C\}(j))-R^{4}(\mathit{1}\mathrm{V}-\ell’;\{c^{(j})\}$$\{c^{(}.i)\})$

(9)
(10)

is called an aperiodic cross-correlation function or a partial corwhererelation filnction. defined by [1]$R^{A}(C_{\backslash }.\{c^{(i)}\}.

\{c^{(j)}\})$$R^{A}(C’; \{c\}(i), \{c)(j\})=\sum_{0\eta=}^{N}-1-lc_{n}^{(}c_{n+\ell}i)(j)$

(11)

.

of such even and odd crossIn order to reduce -channel interference. absoluteof PN sequences, are desired to be small.
correlation fiunctions. which depend on the$\mathrm{v}\mathrm{a}\mathrm{l}\iota \mathrm{l}\mathrm{e}\mathrm{S}$$\mathrm{c}\mathrm{o}$$\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{l}\mathrm{y}$3Chaotic Bit Sequences

3.1

Generation

To giveference$\mathrm{n}1\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}_{\mathrm{S}}$for generating chaotic bit sequences., we$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\iota \mathrm{l}\mathrm{s}\mathrm{s}$the solutions of the dif-$\mathrm{e}\mathrm{q}_{11\mathrm{a}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}[2],[3]$$\omega_{n+1}=’,$ $(x_{n}),$$\omega_{7?}\in I=[d, e]$. $n=0.1,2$ .

(12)$\cdots$

where is a piecewise continuous function which lllaps some interval into itself. It is well
may be chaotic. For exalnple, the Chebyshev map ofknown that the sollltions ofdegree with $I=[-1.1]$ defined by [4]$I$$\tau$$\mathrm{e}\mathrm{q}.(12)$$k$$\tau(\omega)=\cos(k\cos-1\omega),$$k=2.3.4$ .

$\cdots$(13)

have chaotic trajectories. In our previous study [5]., we proposed two simple methods to. both of which canobtain binary sequences fronl chaotic real-valued sequences
give efficient methods to generate simultaneously different sequences of i.i.d. binary randonl
variables for sollle ergodic maps. The first is used by the second nlethod.
$\{\tau^{n}(\omega)\}_{n=^{0}}\infty$

92

Method-l: Using the threshold function defined by$0$$\ominus_{t}(\omega)=\{$

we can obtain a binary sequenceseqnence (called a Chebyshev threshold1$\{\ominus_{t}(\omega_{n})\}_{n=0}^{\infty}$

for $\omega<t$
,
for

(14)

$\omega\geq t$, which is referred to as a chaotic threshold$seq_{\mathrm{I}le}nce$when$\tau(\cdot)$is a Chebyshev polynomial).

The second method was based on a binary expansion of the absolute value ofas follows.

$\omega$when$|_{A}‘)|\leq 1$

Method-2: We write the value of$\omega(|\omega|\leq 1)$$|\omega|=0.A_{1(\omega})A_{2}(\omega)\cdots A_{i(\omega})\cdots$

The i-th bit$A_{i}(\omega)$

in a binary representation:,$A_{i}(\omega)\in\{0,1\}$,$|\omega|\leq 1$

.

(15)

can be expressed as$A_{i}( \omega)=2i-1\sum_{r=1}(-1)r-1\{1-\ominus-\frac{\gamma}{\circ l,\sim}(\omega)+\frac{\mathrm{r}}{9^{l},\sim}(\omega)\}$

.

(16)

which we call a chaotic bit sequence (orThus we can obtain a binary sequenceis a Chebyshev polynomial).
a Chebyshev bit sequence when$\{A_{i}(\omega_{n})\}^{\infty}n=0$$\tau(\cdot)$

3.2

Correlation Properties

For anyfunction$L_{1}$$F(\cdot)$, consider the sunu defined by$(1^{-}7)$$F_{l}.( \omega)=\frac{1}{N}\sum F(\mathcal{T}^{n}(\omega \mathit{1}n=0\mathrm{v}-1\mathrm{I}).$

According to the Birchoff individual ergodic theorem [3]., we have
$Narrow\infty 1\mathrm{i}_{111}F_{N(\omega})=\langle F\rangle$

is mixing onthat
under theinvariant (or briefly ACI) nleas\iota lre, denoted byof over , defined by$\mathrm{a}\mathrm{S}\mathrm{S}\mathrm{U}\mathrm{n}\mathrm{l}\mathrm{P}^{\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}}$$F$$\tau(\cdot)$$I$$\mathrm{a}.\mathrm{e}$

(18)

.

with respect to an absolutely continuousis the enselllble-average, where$f^{*}(\omega)d\omega$$\langle F\rangle$$I$$\langle F\rangle=\int_{I}F(\omega)f^{*}(\omega)d\omega$

.

(19)

It is known that the enselllble average technique is useful in theoretically evaluating statistics of chaotic sequences, such as lneans and correlation functions. Applying this technique, we can get the fact that the chaotic binary sequences have good
to several ergodic$\mathrm{n}\mathrm{u}\mathrm{a}_{\mathrm{P}^{\mathrm{S}}}$

statistical properties [5].

93

Now we consider two $\{0,1\}$ -valued sequencesand$\{g(\tau^{n}(\omega))\}_{n=}\infty 0$$\{h(\tau^{n}(\omega))\}_{n=}\infty 0$. Define(20)$R_{N}( \omega.\ell;g.h)--\frac{1}{N}Nn=0\sum^{-1}(2g(\tau(n\omega))-1)(2h(\tau^{n}(+l)\omega)-1)$

which is the cross-correlation function between these two sequences
a seed , whereis thedelay.
Let $X$ be a set of seeds and let $M$ be its cardinality. We introduce the average values
ofoverseeds given by$\mathrm{f}\mathrm{r}\mathrm{o}\ln$$\omega$$\ell$$\mathrm{t}\mathrm{i}_{1}\mathrm{n}\mathrm{e}$$\wedge Wf$$R_{\mathrm{V}}.(_{\iota I}.’\cdot g\ell_{\backslash }.h)$$\hat{R}_{\mathrm{A}}\mathrm{v},M(C;g, h)=\frac{1}{\mathit{1}\iota,f}\omega\in X\sum_{0m}RN(\omega 0m\cdot\ell;g.h)$Then this function
and $M$ areseeds.
. Next. let
quences(21)

whenan empirical one, approaches tocal..l.ed
is regarded as the average over all possible
large because$\hat{R}_{\mathrm{A}}\mathrm{V}_{\angle,\vee}.’$\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\overline{\mathrm{n}}\mathrm{t}\overline{\mathrm{l}}\mathrm{y}$.
$\langle R(P;g, h)\rangle$\mathrm{V}l.(\ell\backslash g, h).$$N$$\langle R(t;g.h)\rangle$

be an aperiodic correlation fllnction between two binary seanddefined as [1]$R_{\mathrm{V}\mathrm{A}}^{A}(\omega.\ell\backslash g.h)$$\{g(\tau^{n}(\omega))\}_{\eta=0^{1}}^{N-}$$\{h(\tau^{n}(d))\}_{n=0^{1}}^{\backslash }\gamma-$$R_{\perp \mathrm{V}} \wedge 4(\omega.

C\text{ノ};g, h)=\frac{1}{\mathit{1}\backslash ^{\tau}}\sum^{N}(2g(\mathcal{T}^{n}(\omega \mathrm{I})-1)(2h(\mathcal{T}^{n+}\ell n=0(-1-l\omega))-1)$(22)

which gives the even and odd correlation function to be considered in evaluating the perforlllance of SS systems. Its
correlation function is given by$\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{i}\mathrm{r}\mathrm{i}_{\mathrm{C}\mathrm{a}}1$(23)$\hat{R}_{\mathrm{V}\mathrm{A}M}^{A},(\ell\backslash g_{;}h)=(1-,\frac{\ell}{\wedge \mathrm{V}}\mathrm{I}\hat{R}N-l,M(\ell;g.\prime h).$) alsoThefunction
sufficiently large. Thus the enselnble average$\hat{R}_{N-l,M(;}\ell g.h$$\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{i}\mathrm{r}\mathrm{i}_{\mathrm{C}}\mathrm{a}1$$\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{c}_{\lrcorner}\mathrm{h}\mathrm{e}\mathrm{S}$to$\langle R_{\mathit{1}\mathrm{v}(;}^{A}\ell g, h)\rangle$

when $N$ and $M$ arecan be written as$\langle R(C;g, h)\rangle$

.

(24)

$\langle R_{N}^{E}(\ell;g.\prime h)\rangle$

and odd correlation func-$\langle R_{N}^{4}(c’\cdot g\backslash ’ h)\rangle=(1-\frac{p}{N})\langle R(\ell;g, h)\rangle$

Therefore. the theoretical even correlation functiontion
are respectively given by$\langle R_{N}^{O}(\ell’;g, h)\rangle$

3.3$\langle R_{N}^{E}(\ell;g.h)\rangle$$=$$\langle R_{N}^{A}(\ell;g, h)\rangle+\langle R_{N}^{A}(N-\ell;h, g)\rangle$

,

(25)

$\langle R_{\mathrm{V}}^{O}.(p;g, h)\rangle$$=$$\langle R_{N}^{A}(\ell;g, h)\rangle-\langle R_{N}^{A}(N-\ell;h.g)\rangle$

.

(26)

Numerical Examples

Some numerical examples of even and odd correlation functions of Chebyshev bit sequences
are shown. Figures 2 and 3 show the theoretical even (respectively odd) correlation func) . indicated by solid lines, and the em(respectively
tions$Aj)$ ), indicated by dotted lines, where(respectively
pirical ones$\langle R_{N}^{E}(p;A_{i}.Aj)\rangle$$\langle R_{\mathrm{V}}^{O}.(\ell;Ai,$$Aj)\rangle$$’$$\hat{R}_{N,M}^{E}(\ell;A_{i}, Aj)$$\hat{R}_{N,M}^{O}(\ell;Ai,$

94$N=127$ and $M=100$ . In each figure, the degree of Chebyshev maps and the bit num-

ones.
bers are indicated. The theoretical values are in good agreement with the
Besides. we can find that their cross-correlation values become low when the degree andthe bit numbers are large. Note that their auto-correlation values except $\ell=0$ are similar
to their cross-correlation values.
$\mathrm{e}\ln_{\mathrm{P}^{\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}1}}$$k$(a) even(b) odd

andFigllre 2: (a) (where $\wedge \mathrm{V}=127.M’=100$ , and $k=2$ .
$R_{N}^{E}(c;A_{2,}.A_{3})\rangle$$\hat{R}_{N,M}^{E}(\ell;A_{-},.A3),$(a) evenand$\hat{R}_{N}^{o_{\mathrm{w}(p}},.;A_{2},$$A_{3}$)(b) odd

andFigure 3: (a), where -V $=127.M=1\prime \mathit{1}00$. and $k=16$ .
$\langle R_{N}^{E}(\ell;A8, A9)\rangle$$(\mathrm{b})\langle R_{N}^{O}(\ell:A_{2}.,$ $A3)\rangle$$\hat{R}_{\mathit{1}}^{E}\mathrm{V},M(\ell;A8\cdot A9).!(\mathrm{b})\langle R_{N}^{O}(\ell;A8, A9)\rangle$

and$\hat{R}_{\mathrm{V}}^{o_{M(\cdot A_{9}}}.,,\ell_{!}A_{8},$)954Distributions of Statistics of Chaotic Sequences$c^{};A_{i,j}A$ ) and) have large scattered values
It should be noted that
because they are randonl variables. Thus we lnust investigate distributions of such randonl
because bit error probabilities in asynchronous
variables for a certain set of seeds
depend on distributions of correlation values. called empirical distributions.
and the ones of
Figure 4 shows such empirical distributions oftend. where $M=$ 8001. We can observe that theseto Gaussian distributions [6] defined$\ell;A_{i},$ $A_{j}$$R_{l\backslash (}^{o_{\tau}}\omega,$$R_{\mathit{1}\backslash ^{}(\omega}^{E},$$\{\omega 0_{m}\}_{m=1}^{M}$$\mathrm{C}\mathrm{D}_{\sim}\backslash \mathrm{I}\mathrm{A}\mathrm{s}\mathrm{y}_{\mathrm{S}}\mathrm{t}\mathrm{e}\mathrm{n}1.\mathrm{s}$$\{R_{N}^{E}(\omega_{0m}.p_{:^{A_{i,j}}}.A)\}^{M}m=1$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}_{1}1\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$$\{R_{\mathrm{v}}^{o_{(;A_{i}},\iota}.\omega 0_{m}\cdot\ell Aj)\}^{\angle}m=1l$$\mathrm{b}_{\vee}\mathrm{v}$$O( \omega)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp[-\frac{(\omega-\nu)^{2}}{2\sigma^{2}}]$

,

(27)

$(-\infty<\omega<\infty)$

where is lllean andis variance. Of course, the theoretical estinlates of lnean, denoted
are respectively given byrespectively by
and$\sigma^{2}$$\nu$$\nu_{\tau;}^{O}$$\nu_{\tau}^{E}$$\nu_{\tau}^{E}$$\nu_{\tau}^{O}$$=$$\langle R_{\mathrm{A}}^{E}4’(^{\ell};Ai, Aj)\rangle$$=$$\langle R_{\mathrm{V}}^{O}.(\ell;Ai, Aj)\rangle$

and those of variance. respectively denoted by$(\sigma_{\mathcal{T}}^{E})^{2}-\sim$

$=$

and

$(\sigma_{\mathcal{T}}^{E})^{2}$

(28)

,

(29)
$(\sigma_{\tau}^{O})^{2}$, are also given by$\langle(R_{\mathit{1}\mathrm{V}(\omega.\ell^{\mathfrak{d}};}^{E}Ai.\prime Aj))^{2}\rangle-\langle R_{N}^{E}(\omega, \ell;A_{i}, A_{j})\rangle.2$.

$(\sigma_{\tau}^{O})^{2}---\langle(R_{f\backslash ^{r}(}^{O}.\omega.\ell;Ai\cdot Aj))2\rangle-\langle R_{N}o_{(i,j}A)\rangle 2\omega.\ell;A$

(30)

(31)

) and variance(respectively
Gaussian distribution with nlean(respectively
as the estimated distribution of. Figure 4 leads us to find that the theoretical estilluates of lllean
and variance are given by

NVe refer to$\mathrm{t}\mathrm{h}\mathrm{e}\lambda$$\nu_{\tau}^{O}$$\nu_{\tau}^{E}$$(\sigma_{\tau}^{E})^{2}$$\{R_{\mathrm{A}}^{E}\mathrm{V}(\omega 0m’\zeta;A_{i}, A_{j})\}_{m1}^{\mathrm{t}t}\mathit{1}=$$(\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{I}^{)}}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}}1_{\}^{\vee}}(\sigma_{\mathcal{T}}^{E})^{2})$$\{R_{N}^{o_{(;A_{i}}.\mathrm{A}}.\omega_{0}.c7\eta Aj)\}_{m1}^{\psi I}=)$$\nu_{\tau}^{E}$$\simeq$

$0$

(32)

$\nu_{\tau}^{O}$$\simeq$

$0$

(33)

$(\sigma_{\mathcal{T}}^{E})^{2}$$\simeq$$(\sigma_{\mathcal{T}}^{O2})$$\simeq$(34)$\frac{1}{N}$(35)$\frac{1}{N}$and the delay when the degree of
each of which is independent of the bit nunlbers
the Chebyshev map and the bit nunlbers are large. It is easily checked that the estimatedwith the empirical ones.
are in good
For colnparison, the distributions of correlation values of Gold sequences are shoxvn inFigure .
$\dot{i},$$\mathrm{d}\dot{\iota}\mathrm{s}\mathrm{f}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{a}\mathrm{g}_{\Gamma \mathrm{e}\mathrm{e}\mathrm{n}1}\mathrm{e}\mathrm{n}\mathrm{t}$$\check{\mathrm{o}}$$j$$\ell$

96(a) even$\mathrm{F}\mathrm{i}\mathrm{g}_{11\mathrm{r}\mathrm{e}}4$(b) odd: Distributions of correlation values of Chebyshev bit$(i,j)=(8,9)$ , and $k=16$ .

$\mathrm{s}\mathrm{e}\mathrm{q}_{11}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}$, where $N=127.!$$\wedge 1\prime I=8\mathrm{o}\mathrm{o}1,$(b) odd(a) evellof correlation vallles of 8001 pairs of Gold se($\mathrm{F}\mathrm{i}\mathrm{g}_{11}\mathrm{r}\mathrm{e}5:\mathrm{D}\mathrm{i}_{\mathrm{S}\mathrm{t}\mathrm{r}}\mathrm{i}\mathrm{b}_{1\iota}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$5$1^{11\mathrm{e}}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{S}$, where $N=127$ .

Evaluation of Bit Error Probabilities

As is well known, bit error ) )in asynchronolls CDMA systems depelld on distributions of even and odd correlation vallles between each pair of.
).Nassume that users communicate
a CDMA system
andThtls the distriblltion of the -channel interference to the i-th userfrom other $.J-1$ channels areby the Gatlssianwith mean andvariance $(J-1)N$ because of the additive property of the Gaussian distribution. Fig)
ureshows numericalof
distributions of the -channel interference inCDMA systems using Chebyshev bit. For reference. Figure
shows the ones of the systems tlsing Gold seqllences. We can find that the distributions for$\mathrm{r}\mathrm{o}l$$\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{S}$$1$$\mathrm{s}_{\mathrm{I}^{)\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{i}}}\mathrm{n}\mathrm{g}\mathrm{S}\mathrm{e}(11\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}‘ \mathrm{S}$$o\mathrm{w}$$J$$\mathrm{t}\mathrm{h}_{\Gamma \mathrm{O}11}\mathrm{g}\mathrm{h}$$\mathrm{c}_{1\mathrm{y}1}11\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{s}\mathrm{n}\mathrm{c}\mathrm{h}_{1}\cdot \mathrm{o}\mathrm{n}\mathrm{o}\iota 1\mathrm{S}\mathrm{y}.$$\mathrm{e}11\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}1.\mathrm{v}$$\mathrm{c}\mathrm{o}$$\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}_{111}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$$6(\mathrm{a})$$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}_{1}$$\mathrm{d}\mathrm{i}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}_{1\iota}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$0$$\mathrm{s}\iota \mathrm{l}\mathrm{c}\mathrm{h}$$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{m}_{\mathrm{I}^{1\mathrm{e}\mathrm{S}}}$$\mathrm{c}o$$\mathrm{a}\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{c}1_{1}\mathrm{r}\mathrm{o}\mathfrak{U}0\iota 1\mathrm{S}$$\mathrm{s}\mathrm{e}\mathrm{q}\iota \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{S}$$6(\mathrm{b})$97

Chebyshev bit sequences tend to the Gaussian distribution irrespective of the number ofchannels. On the other hand, the ones for Gold sequences are not the Gaussian distribution
when the number of channels, , is small. However, it is shown that the distributions forGold sequences tend to the Gaussian distribution as the number of channels increases.
$J$

0.1$0\uparrow 0$$N^{-}B\mathit{2}$$\mathrm{o}\underline’$

64.48795.880127.728$[]$$J=\mathit{3}\circ$$009$$\circ$,$\theta$$J_{--}\mathit{4}*$

$008$
$0$

.

0.07

’
$J_{--}\mathit{5}\circ$$\mathrm{r}^{*}‘ \mathrm{r}$1$\mathfrak{o}*_{\theta}$$\mathrm{o}^{*}\mathrm{o}$$*\circ$’$0^{*}$$\mathrm{f}\mathrm{i}\mathrm{o}\mathrm{q})$$0_{0}$$0\circ$$*$$*$$\approx\xi 0)0.\infty$’$\mathrm{o}0.05$$\mathrm{c}$$\circ\circ$$\circ l$$\mathfrak{g}\circ$$004$$\circ*$$\circ*$$0_{9^{\mathrm{i}}}*$.

$0^{\circ}$$\mathrm{t}$0.03$\square$$\mathfrak{o}$0.02$,.\circ$$\mathrm{O}^{*^{9}}$$0$$\mathrm{O}^{*}$001$\circ\star$,‘$.\mathrm{e}_{\mathrm{O}}$.

$0^{\mathrm{g}\circ}$$c$$0_{*}^{\mathrm{o}_{*\mathrm{I}\circ}^{\mathrm{O}}}$$\circ;^{\mathrm{O}}\^{\mathrm{o}_{\mathrm{O}}}$

$c$

$c$

$0$

$- 40$$- 20$

40

20

$0$

interference(b) Gold sequences(a) Chebvshev bit sequences$arrow$: Distributions of -channel interference frolll $J-1$ channels in an asynchronousCDMA system, where $N=32$ .

$\mathrm{F}\mathrm{i}\mathrm{g}_{11\mathrm{r}}\mathrm{e}6$$\mathrm{c}\mathrm{o}$The bit errors occllr when the -channel interference is greater than $N$ if the i-th
if the i-th
infomation bit during the p-th tinle-interval is $d_{p}^{(i)}=-1$ (or is snlaller
for all and). Hence, if
bit is$n(t)=0$ , the bit err\^Or
ab..ility can be estimated by$\mathrm{c}\mathrm{o}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}-\mathrm{j}^{\tau}$$\inf_{\mathrm{o}\mathrm{r}111}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{P}\mathrm{r}\{d_{p}^{(j})=1\}=\mathrm{P}\mathrm{r}\{d_{p}(j)=-1\}=\underline{‘\frac{1}{)}}$$d_{p}^{(i)}\neg-1$$j$$\mathrm{p}\mathrm{r}\mathrm{o}..\mathrm{b}$$\simeq$

where$\overline,.$. ,:.::$P_{e}$$l–\backslash \backslash \sim\backslash \cdot\backslash \triangleleft’..$

–

.

$=$

$\int_{N}^{\infty}\frac{1}{\sqrt{2\pi\sigma-}},\exp[-\frac{\omega^{2}}{2\sigma^{2}}$$=$$Q( \frac{N}{\sigma})$

. .
:.

.

,

’

(36)

$\mathrm{P}\mathrm{r}\{I_{J,p}^{(}i)>N|d_{p}(i)=-1\}$$]d\omega$

(37)

(38)

$\cdot$$..\cdot\cdot.-..$.

$\vee^{-}\cdot$$\sigma^{-}$$=$$(J-1)N$$Q(x)$$=$$\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}\exp[-\frac{\omega^{2}}{2}]d\omega$

(39)

.

(40)

98

and
correspond respectively toindicates that $N$ and
The above quantity$[10],[11]$and. wherein the well-known bit error probability of a coherent BPSK systemdenote the bit energy and the noise power spectral density, respectively.
Finally, we show the bit error probabilities in asynchronous CDMA systems by computercan find that theoretical estimates are in good agreement within Figure 7.
the results by sinlulatioll for Chebyshev bit sequences. It is interesting that the results forGold sequences are also in good agreement with the theoretical estimates for Chebyshev
bit sequences when the nulnber of c,hannels is large.
$\sigma^{2}$$\sqrt{E_{b}}$$-\mathrm{V}_{J}/2$$P_{e}$$E_{b}$$\wedge l\backslash _{J}^{\gamma}/2$$\mathrm{s}\mathrm{i}_{\mathrm{l}\mathrm{U}\mathrm{U}}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{W}^{r}\mathrm{e}$$\sim \mathrm{h}$

.–

$.–$
$s_{\alpha}$$9^{\circ}\mathrm{h}\mathrm{f}\mathrm{i}\mathrm{O}$$\mathrm{h}\mathrm{O}$$\mathrm{h}\mathrm{h}\Phi$$. \frac{.}{\dot{S}}$number$\mathrm{F}\mathrm{i}\mathrm{g}11\Gamma e7$$\mathrm{o}\mathrm{r}_{\mathrm{C}}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\mathrm{s}d$: Bit error probabilities in an asynchronous CDMA system. where $N=63$ .

99

6Applications to Image Communications, the main probIn image transmission systems using spread spectrum. In an image coding,to transmit the image with enorlnous data
lenu isthat inlages have different
the discrete cosine transform (DCT) is extensively used.
. term.)significant DCT coefficients.(The most significant coefficient is known to be theIn general. a wide bandwidth is required for the transmission of images. In a CDMA
] $-[11]$,system. spreading sequences of the sallle period are usually assigned to theand hence each of the erroneously transmitted bits occurs nearly equiprobably. Obviously, alllllch wider bandwidth is required for the transmission of images using such CDMA systems.
To redllc such a bandwidth. we consider a CDMA system in which spreading sequencesof longer period are assigned to lllore significant bits than to less. Such a system,
which is asynchronous. permits
tothe error probabilities of more significant bits.
This technique is analogolls to the Shannon-Fano encoding. The quality of reconstructed
images is shown to be drastically improved within such a
bandwidth even if thesystem is asynchronous.
$(\mathrm{S}\mathrm{s})\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{S}[10],[11]$$\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{y}^{[\prime]}\neg-[9]$$\mathrm{h}\mathrm{o}\mathrm{w}\sim$$\wedge\tilde{\mathrm{N}}\mathrm{o}\mathrm{t}\mathrm{e}$$\mathrm{D}.\mathrm{C}$$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}1_{\mathrm{S}}11$$\mathrm{e}$$\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{s}^{[3]}1$$\mathrm{r}\mathrm{e}\mathrm{d}_{11\mathrm{C}}\mathrm{e}$$11\mathrm{S}$$1\mathrm{i}_{1}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{d}$Image Communications Based on A..synchronous
Systems $[13],[14]$6.1$\mathrm{D}\mathrm{S}/\mathrm{C}\mathrm{D}\mathrm{M}\mathrm{A}$A basic inlage
llsing spread spectrum technique is illustrated inpixels), and the discrete. First, images are )into small blocks ((2-D) signal
cosine transfornl (DCT) ofin each block is computed.
the 2-D DCT coefficients $q(?l, v)$ are quantized and appropriate nunlbers of bits are
ck by block.
assigned to them. Note that encoding and decoding are implemented
The DCT coefficients are nlllnbered as shown in Figure 9. We assign more bits to low
coefficients than to high ones. The bit allocation map we use is shown in Figure.
10.
transmit only the first 15 DCT coefficients. nanlely
Fllrtherll)ore. the, foris denoted by-th ssignificant bit of the m-th coefficientis theandexalnple.
is the most significant bit (MSB) of the DCT coefficient
least significant bit (LSB). In this paper. a gray-scale
per pixel. $720\cross 576$ Pixels,
called‘Barbara” is used, as shown in Figure 11.
$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}11\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}}\mathrm{e}\mathrm{n}1$$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{d}$$\mathrm{F}\mathrm{i}\mathrm{g}_{11}\mathrm{r}\mathrm{e}8$$1$$\mathrm{t}\mathrm{w}\mathrm{o}^{-}\mathrm{d}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}1$$8\cross 8$$p(\dot{\uparrow},j)$$\mathrm{N}e\mathrm{x}\mathrm{t}$$\mathrm{b}1\mathit{0}$$\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}_{1}1e\mathrm{n}\mathrm{c}\mathrm{y}$$\mathrm{U}^{i^{\vee}}\mathrm{e}$$54\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k},$$7?$$q_{m-n}$$q_{m}$$q_{0-1}$$0.84\mathrm{b}\mathrm{i}\mathrm{t}/\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$$q_{0}$$q_{0-_{8}}$$\mathrm{i}_{111}\mathrm{a}\mathrm{g}\mathrm{e}(8\mathrm{b}\mathrm{i}\mathrm{f}\mathrm{S}$$\cdot$$6480\mathrm{b}1_{0}\mathrm{c}\mathrm{k}\mathrm{S})$100

Figure 8: Illlage$\mathrm{C}\mathrm{o}\mathrm{m}111\mathrm{u}\mathrm{n}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{f}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}$Figure 9: Coefficient nunlbers.

Figure 11: The original image

using SS techniques.

$\mathrm{F}\mathrm{i}\mathrm{g}_{11}\mathrm{r}\mathrm{e}10$(($\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{a}$: Bit allocation lnap.

’

101

For an image, we assign each bit
appropriately. Now we define$t_{m- n}=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{A}_{\mathrm{S}\mathrm{S}1}1111\mathrm{e}$that the total

to CDMA channels (6 channels in this paper)$q_{m^{-}n}$period of the spreading sequence to be assigned to$\mathrm{n}\iota \mathrm{l}\mathrm{n}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}$$q_{m^{-}n}$

.

(41)

of chips in each channel defined by$T_{i}=$$\sum$$q_{m^{-}n}\in \mathrm{t}\mathrm{h}\mathrm{e}$$t_{m-n}$.

$i=1,2,$$\cdots,$$6$.

i-th channel

is equal to others, that is,$T_{i}=T$.

for all .
$i$

We call the “block-period”
Various kinds of models can be considered according toof each bitto
6 channels. First. we proposed the sinlplest model as shown in Figure 12, called]‘Model-l”,
where spreading sequences of fixed-period are used.
$T$$\mathrm{a}_{\mathrm{A}}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{S}$chann#lelchannel$q_{m-n}\sim$$d^{(l)}.\ovalbox{\tt\small REJECT}_{q\mathit{0}\mathit{1}:}:::q\mathit{0}z:q\mathrm{p}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\circ \mathrm{f}\mathrm{s}\mathrm{p}m\mathit{0}St\mathrm{d}\mathrm{m}\mathrm{g}\mathrm{s}_{4:}:q\mathit{0}\mathrm{b}1_{0}\mathrm{c}\mathrm{k}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{c}\mathrm{q}:q\alpha \mathit{5}:\mathit{0}:^{q}::T\delta:_{q}\mathit{0}7:q\mathit{0}\mathit{8}q\mathit{1}\mathit{1}|$$d^{(2)}$$n$$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\# 3d^{(\mathit{3}J}$:$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\# 4d^{(\mathit{4})}$$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\# 5d^{(\mathit{5}J}$.
$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\# 6d^{(\mathit{6}J}$

Figure 12: Model-l with spreading sequences of fixed-period.
Now define the mean absolute error (MAE) by

MAE$= \frac{1}{6480\cross 8\cross 8}\sum_{k=1}^{8}\sum_{=i0j}\sum_{0}^{\overline{/}}6407=|p_{k}(i,j)-\hat{p}_{k}(i, j)|$(42)

where $p_{k}(i,j)$ anddenote intensity values of the $(i, j)$ element of the k-th blockin the original
and in the reconstructed image, respectively. In order to investigate the significance of each bit, we compute the values of MAE when each bit
is transmitted, as shown in Figure 13. Furthermore, the insignificance ofis calculated
MAE-lower bound ofwhich means the number$\hat{p}_{k}(i,j)$$\mathrm{i}_{1}\mathrm{n}\mathrm{a}\mathrm{g}\mathrm{e}$$q_{m-n}$$\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{S}\mathrm{l}\mathrm{y}^{[1]}\underline’$$q_{m-7?}$$q_{m- n}$$\mathrm{f}\mathrm{r}\mathrm{o}\ln$$($$\mathrm{M}\mathrm{A}\mathrm{E})^{-1}$102of the erroneously transmitted bit

Figure 13: MAE when each bit
nlitted erroneously.

$q_{m- n}$

to cause a constant MAE, as shown in Figure 14.

is trans-$q_{m^{-}n}$,Figure 14: Insignificance of the bit
namely. the nunlber of erroneously translnitto cause a constant MAE.
ted bits$q_{m-n}$$q_{m^{-}n}$period of spreading sequences

Figure 15: Bit error rates in asynchronous CDMA systems using Chebyshev bit sequences.
is equiprobably transmitted in error. As
bitModel-l is not efficient becauseof channels, spreading sequences of longer periods canis well known. for a constant
reduce bit error rates than ones of shorter periods. This motivates us to assign spreading
according to its significance. To do this,of an appropriate period to the bit
we should investigate the bit error rates for various periods of spreading sequences and forvarious nulnbers of channels. In this paper, as spreading sequences of variable-period, we use$\mathrm{e}\mathrm{a}\mathrm{C}_{\text{ノ}}\mathrm{h}$$q_{m- n}$$\mathrm{n}\mathrm{l}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}$$\mathrm{s}\mathrm{e}\mathrm{q}_{11\mathrm{e}}\mathrm{n}\mathrm{c}e\mathrm{S}$$q_{m- n}$

103

]”
which are generated by the Chebyshev lllap, Such sequences“Chebyshev bit
sequences, and Goldare quite different from LFSR sequences such as sequences,. We have already calculated the bit error rates for the Chebyshev bit sequences$\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}^{15}$$\mathrm{h}\mathrm{I}$$\mathrm{I}\backslash ^{r}\mathrm{a}\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{i}$$\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}^{[]}1$

as shown in Figure 15.
Using the results of Figures 14 and 15, we can construct “Mode1-2” where spreading
sequences of variable-period are used as shown in Figure 16. Figure 17 shows the desired
is equal to. indicated by the solid line, when the nuinimum
for each bit
periods
given by the solid line in Figure 17 are llsed, then $T_{i}(i=1,2, \cdots, 6)$
15. If the periods
in Mode1-2. indicated by the dotted line
are not equal to each other. Thus the periods$tm-_{n}$$q_{m-n,}$$t_{\mathfrak{m}-n}$$t_{m- n}$$t_{m- n}$

in Figure 17. are used to satisfy $T_{i}=240$ for all .
$i$Figure 16: Mode1-2 with spreading sequences of variable-period.

Figure 17: Fig.10: Periods$t_{m^{-}n}$for each bit$q_{m- n}$

.

104

6.2

Computer Simulation

to evaluate the quality of reconstructed images, we use the mean square error(MSE) defined by

As a$\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}_{l}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{n}$

MSE

$=$

(43)

$\frac{1}{N\cross 8\cross 8}\sum_{k=1i}^{N}\sum_{0=j}^{l}’\sum^{7}=0(pk(i,j)-\hat{p}k(i.j’))^{2}$indicated by the dotted lineIn. we use the periods obtained by multiplying
in Figure 17 by a constant value according to given block-periods. Figures 18 and 19 show
bit error rates versus block-periods and MSEs versus block-periods, respectively. We canfind that the bit error rates in the two models are silnilar to each other. On the otherhand. the MSE perforlnance of Mode1-2 has shown to be much better than that of Model-l.
This implies that the quality of reconstructed images in Mode1-2 is drastically improved asshown inand 21.
$t_{m^{-}n}$$\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}1-2_{\mathrm{t}}$$\mathrm{F}\mathrm{i}\mathrm{g}_{11\mathrm{r}}\mathrm{e}\mathrm{s}20$$\mathrm{O}\mathrm{l}\mathrm{O}\mathrm{C}\mathrm{K}^{-}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{o}\mathfrak{a}$Figure 18:
periods.

Bit error rates versus block-

block-penod

Figure 19: MSEs versus block-periods.

105

Figure 20: The reconstructed image inModel-l for block-period 99.

7

Figure 21: The reconstructed image in
Mode1-2 for block-period 96.

Concluding Remarks

Bit error probabilities in asynchronous CDMAusing chaotic binary sequences havebeen discussed. It is observed that the central limit theorem holds for correlation values ofchaotic bit sequences. This enable us to theoretically estinlate such bit error probabilities.
Since the central limit theorenl
hold for LFSR sequences such as Gold sequences,as shown in Fig.5, it is difficult to theoretically estimate the bit error probabilities in thesystenl using LFSR sequences. However, it is noteworthy that the bit error probabilities
for chaotic bit sequences are capable of estimating the ones for Gold sequences when thentlnlber of channels is large.
Furtheremore, inlage conlmunication systems using CDMA channels with spreading seof variable-period are proposed. We have given an efficient method for assigningperiods of spreading sequences based on the insignificance of each bit of DCT coefficients.
The CDMA systems using spreading sequences of variable-period have the following advantages: 1) The quality of reconstructed images are drastically improved within a linlited
bandwidth; and hence 2) The bandwidth is not so much for the transmission of images.
Note that such techniques can be also applied to color inlage communications.
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[3] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Springer-Verlag, 1994.

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