BER analysis of coherent optical CDMA communication systems with transfer function matrix

Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642 www.elsevier.de/aeue

LETTER

BER analysis of coherent optical CDMA communication systems with transfer function matrix Yasutaka Igarashia,∗ , Hiroyuki Yashimab a Department of Electrical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan b Department of Management Science, Tokyo University of Science, 1-14-6 Kudan-kita, Chiyoda-ku, Tokyo 102-0073, Japan

Received 2 April 2007; accepted 17 August 2007

Abstract We analyze the bit error rate (BER) of coherent optical code-division multiple-access (CDMA) communication systems with a transfer function matrix (TFM). The coherent optical CDMA is a promising system for an access network due to its advantages of asynchronous transmission, information security, multiple-access capability. TFM is defined as the Fourier transform of the impulse response of an optical fiber, which interconnects each CDMA system. TFM has parameters such as the state of polarization (SOP), phase difference, power distribution between two orthogonal modes over an optical frequency range. We first analyze the statistic of a received signal via an optical fiber with TFM, and then apply the statistic to the BER analysis. The analytical results show that BER increases with the bandwidth of a signal, and that selecting a proper center frequency leads to the BER reduction. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: Coherent optical CDMA; Fiber-optic communications; Transfer function matrix; BER analysis

1. Introduction As a prospective scheme for the multiplexing of fiberoptic communications, Salehi et al. have proposed a coherent optical code-division multiple-access (CDMA) communication system using coherent pulses [1]. This system is based on the spectral phase encoding of the coherent pulse. The advantages of the CDMA scheme originate from its vast bandwidth, no code synchronization required, the simplicity of the system implementation, the large number of random access users to a common optical channel, and the high spreading gain of coherent pulse. In the coherent optical CDMA network employing the star topology shown in Fig. 1, unique and minimally interfering code sequences are assigned to each CDMA encoder and CDMA decoder ∗ Corresponding author. Tel.: +81 4 7122 9527; fax: +81 4 7125 8651.

E-mail address: [email protected] (Y. Igarashi). 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2007.08.004

pair to enable a common channel to be shared among all users. Users communicate by imprinting information bits onto their own unique code, which are transmitted asynchronously with regard to the other users over a common channel. A matched filter at the decoder ensures that data are detected correctly only when data is imprinted onto the proper code sequence. This approach to multiplexing enables transmission without synchronization and treats multiple access interference (MAI) as the integral part of multiplexing scheme. Optical fiber has become an important medium for telecommunication. Especially, single-mode fiber (SMF) has been widely used in transoceanic communication systems as well as continental and metropolitan networks because of its low propagation loss and high information capacity. Inherently, SMF is not single mode because it can support two degenerate modes polarized in two orthogonal directions. In ideal fiber with stress free and perfect

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Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

Optical or Electrical Signal

Optical or Electrical Signal

Optical Signal

Data Generator #1

Encoder #1

Data Generator #2

Encoder #2

Data Generator #M

Encoder #M

M×MOptical Star Coupler

Transmitter

Decoder #1

Data Decision #1

Decoder #2

Data Decision #2

Decoder #M

Data Decision #M

B

A

C

D

W t

t c

Coherent Pulse A Generator

to Receiver D Optical Fiber

Fourier B C Transform

A

Inverse Fourier Transform Data ‘0’ Spectral Phase CDMA Transmitter Generator ‘1’ Code Generator Encoder E F Receiver

from Transmitter

Fourier E Transform

Optical Fiber

Receiver

Inverse F Fourier Transform

G

G

Data ‘0’ Decision ‘1’

Conjugate Spectral CDMA Phase Code Generator Decoder

MAI

Fig. 1. Coherent optical CDMA network.

2. System configuration Coherent optical CDMA communication systems employ a star network topology as shown in Fig. 1. The same bit rate and on–off keying (OOK) signaling are assumed for all users. The mth (1m M) transmitter and the mth receiver are paired where M is the total number of users. Since data signals are transmitted asynchronously to a common channel, each user suffers from MAI. Fig. 2 shows a transmitter and receiver pair for coherent optical CDMA. The transmitter consists of a coherent pulse generator, a data generator and a spectral phase encoder. A coherent pulse (A in Fig. 2) with a flat spectrum (B) is multiplied by the data generator. When data is “0”, the coherent pulse is not fed into

P0 c

≅

P0 W

2 W -W 2

- T2

T t Coherent 2 Pulse

Chip

- 2

Chip … … -2 -1 No. -N

A( )

W 2

2 0

1 2 … …

N

(a) Intensity (a.u.)

cylindrical symmetry, both modes excited with these polarizations in x and y directions would propagate at the same speed. In practical fiber, bending, twisting, internal stress, and manufacture defects of the fiber induce birefringence and noncylindrical symmetry of core shape resulting in different group velocities between the two polarization modes. Therefore, these modes propagate along the fiber at different speed. If an input pulse excites both polarization modes, the pulse becomes broader at an output end because group velocities change randomly in response to random changes in fiber birefringence and core shape [2]. In this paper, we theoretically analyze the bit error rate (BER) of coherent optical CDMA communication systems interconnected via such a practical fiber. The performance of the fiber is characterized by a transfer function matrix (TFM), which is defined as the Fourier transform of the impulse response of the fiber. TFM has parameters such as the state of polarization (SOP), phase difference, power distribution between the two orthogonal modes over an optical frequency range. This is the first work to analyze BER with TFM to our knowledge. We first study the statistic of a received signal via the fiber, and then derive BER as the function of system parameters.

Intensity (a.u.)

Fig. 2. Transmitter and receiver pair for coherent optical CDMA.

C( ) P0 W T

Encoded Signal

-W 2

W 2

t P - W0

(b) Fig. 3. Intensity profiles and baseband spectra of coherent pulse (a) and encoded signal (b).

the encoder and no optical signal is transmitted; however, when data is “1” it is fed into the encoder. A spectral phase code generator multiplies each spectral component of the coherent pulse by a specific phase shift unique to each user (C). This operation converts a high intensity coherent pulse into a low intensity pseudo-noise burst (D). A phase-shifted spectrum (E) arrives at the decoder via an optical fiber. When the spectral phase codes for the encoder and decoder are a complex conjugate pair, the decoder removes the spectral phase shift (F) and reconstructs the original coherent pulse (G). On the other hand, when the codes do not match, the decoder reassembles but does not remove the phase shift, and the decoded signal remains the pseudo-noise burst. Finally, data is decided as “1” when the intensity of the decoded signal exceeds a threshold, and as “0” otherwise. Fig. 3 shows the intensity profiles and baseband spectra of a coherent pulse (a) and the encoded signal (b), respectively.

Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

The coherent pulse is characterized by baseband Fourier spectrum at an angular frequency , given by   ⎧√ W ⎨ P0 || , W 2 A() = (1) ⎩ 0 (elsewhere) where P0 denotes the peak power of the coherent pulse and W denotes the total bandwidth of the Fourier spectrum. The inverse Fourier transform of A() represents the temporal waveform of the coherent pulse with the full-width at halfmaximum (FWHM) duration of c  2/W . The encoder multiplies A() by the phase code consisting of some randomly bipolarized chips. N0 (=2N + 1), referred to as a code length, represents the total number of the chips. The bandwidth of each chip is  = W/N0 . The example of the phase code (N0 =11) is illustrated in Fig. 3(b). The baseband spectrum of the encoded signal Cm () at the mth encoder is given by N 

Cm () =

A(N0 ( − n)) exp(−i(m) n ),

(2)

n=−N (m)

where n denotes the spectral phase code element of the nth chip for the mth encoder, which is randomly set to either (m) 0 or  with equal probability, i.e. exp(−in ) is set to either (m) 1 or −1. Note that if n = 0 for all n, (2) reduces to (1). From (2), the electric field representation of the encoded signal at the mth encoder, cm (t), is given by



 t b(t) = sinc 2

Vm () = F[vm (t)] √ N P0  = ( − n) exp(−i(m) n ), N0

where F denotes Fourier transform and (·) denotes Dirac’s delta function.

3. Transmitted signal through optical fiber SMF supports two fundamental modes, i.e. HEx11 and y HE11 modes, those are polarized in two orthogonal directions as shown in Fig. 4(a) where  is an input polarization angle describing the input direction of a linear-polarized signal. If  deviates from 0◦ or 90◦ , an input signal is split into two components those are polarized along the two princiy pal axes of HEx11 and HE11 modes as shown in Fig. 4(b). The birefringence and noncylindrical symmetry of a practical SMF induce a relative delay between the two polarization components, which is referred to as differential group delay (DGD). DGD results in the distortion of a waveform at a fiber output [2]. When the encoded signal at the mth encoder with a spectrum Vm ( − c ) and a center angular frequency c is launched into SMF, its spectral representation at the fiber input is given by



m ≡ (3)

 (4)

(6)

n=−N

where

n=−N

where

baseband spectrum of the encoded signal Vm () for the mth encoder:

Vin m ( − c ) = Vm ( − c )m ,

cm (t) = F−1 [Cm ()]  √ N  P0  = sinc exp{−i(nt + (m) t n )} 2 N0 = b(t)vm (t),

637

ˆ m ˇ m



 =

cos m

and

exp(−i m ).

sin m

Inp Lin ut Dir e ear Pol ction ariz of atio n

(7)

(8)

y

HE11

x

HE11

√

N P0  exp{−i(nt + (m) vm (t) = n )} N0 n=−N

T for |t|  . 2

Output Optical Fiber

Input

(a) y HE11

(5) Here, sinc(x) = sin x/x. F−1 denotes inverse Fourier transform. cm (t) consists of the product of two signals b(t) and vm (t). b(t) denotes a real envelope function independent of the code element, and dominates the duration of the encoded pulse. Since the coherent pulse has the FWHM duration of c , the duration of b(t) is spread over T = 2/  N0 c by a factor N0 . vm (t) denotes the encoded signal at the mth encoder. The temporal waveform of vm (t) de(m) pends on n . The Fourier transform of vm (t) gives the

y

HE11

Input Signal

t

x

HE11

x

HE11

t Output State

Input State

(b) Fig. 4. Schematic illustration of optical fiber (a) and its input and output SOP of optical signal (b).

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Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

m is a 2 × 1 Jones vector representing an input SOP for the mth encoded signal, which is normalized to be |m | = 1. y ˆ m and ˇ m denote input states of HEx11 and HE11 modes for the mth encoded signal, respectively. m and m denote an input polarization angle and an initial phase for the mth encoded signal, respectively. We assume m is uniformly distributed on an interval (−/2, /2) for BER analysis. The characteristic of SMF is given by TFM as follows [3]:  T11 () T12 () T() ≡ , (9) T21 () T22 () where T11 () = cos () exp{−i( () + ())},

(10)

T12 () = − sin () exp{−i( () − ())},

(11)

T21 () = sin () exp{i( () − ())},

(12)

T22 () = cos () exp{i( () + ())}.

(13)

Here, () denotes a polarization angle describing power distribution between the two orthogonal modes. () is a vertical phase difference representing the phase difference between T11 () and T21 (). () is a horizontal phase differences representing the phase difference between T11 () and T12 () [4] (see (A.1) for the derivations of (), (), and ()). From (7)–(13), the spectrum of the mth encoded signal at a fiber output is expressed as in Vout m ( − c ) = T()Vm ( − c ).

(14)

The Fourier transform of a baseband spectrum Vout m () gives the electric field of the mth encoded signal, vout m (t), as follows:  vˆm (t) vout (t) ≡ = F−1 [Vout m m ()] vˇm (t)  vˆR (t) − i vˆI (t) = , (15) vˇR (t) − i vˇI (t) where √ vˆR (t) =

P0 N0

√

N P0  Rˇ n2 + Iˇn2 N0

vˇI (t) =

n=−N

ˇ × sin(nt + (m) n + ϑn + m ), Rˆ n = Re(ˆ m T11 (n ) + ˇ m T12 (n )),

(20)

Iˆn = Im(ˆ m T11 (n ) + ˇ m T12 (n )),

(21)

Rˇ n = Re(ˆ m T21 (n ) + ˇ m T22 (n )),

(22)

Iˇn = Im(ˆ m T21 (n ) + ˇ m T22 (n )),

(23)

n = c + n,

(24)

cos ϑˆ n =

cos ϑˇ n =

Rˆ n2 + Iˆn2 Rˇ n Rˇ n2 + Iˇn2

−Iˆn

,

sin ϑˆ n =

,

(25)

,

−Iˇn sin ϑˇ n =

. Rˇ n2 + Iˇn2

(26)

Rˆ n2 + Iˆn2

vˆm (t) and vˇm (t) denote the output electric field of HEx11 and y HE11 modes for the mth encoded signal, respectively. vˆR (t) and vˆI (t) are the real and imaginary parts of vˆm (t). vˇR (t) and vˇI (t) are those of vˇm (t). Re(x) and Im(x) denote the real and imaginary parts of x. Note that vout m (t) reduces to (m) out vp (t) in the case of m = 0 and n = 0 for all n:  vˆp (t) (t) ≡ vout p vˇp (t) ⎡ N

⎤  Rˆ n2 + Iˆn2 exp(−i(nt + ϑˆ n )) ⎥ √ ⎢ P0 ⎢ n=−N ⎥ = ⎢ ⎥.

N ⎦ N0 ⎣  2 2 ˇ ˇ ˇ Rn + In exp(−i(nt + ϑn )) n=−N

(27) vout p (t) represents the electric field of the coherent pulse at a fiber output.

Rˆ n2 + Iˆn2

n=−N

(16)

We assume the following for the analysis. The principal factor degrading BER is only MAI. The other degrading factors are small enough to be neglected for simplicity. Chromatic dispersion is substantially compensated. The electric field of the mth encoded signal sequences Em (t) through SMF is expressed as

n=−N

+ (m) n

× sin(nt √

N P0  N0

Rˆ n

4. BER analysis N 

ˆ × cos(nt + (m) n + ϑn + m ), √

N P0  vˆI (t) = Rˆ n2 + Iˆn2 N0

vˇR (t) =

(19)

+ ϑˆ n + m ),

(17)

Em (t) =

∞  j =−∞

for |t − j T b | 

T , 2 (28)

Rˇ n2 + Iˇn2 (m)

n=−N

ˇ × cos(nt + (m) n + ϑn + m ),

(m)

dj vout m (t − j T b )

(18)

where dj is the data produced by the mth data generator taking on either “0” or “1” for every j. vout m (t −j T b ) is a unit

Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

encoded signal with a duration T and a period Tb = KT for every j where K is a positive integer and each user’s bit rate is given by 1/Tb . Since each user transmits the signal asynchronously in coherent optical CDMA systems, the electric field of the pth decoder’s output gp (t) is given by  Em (t − tm ). (29) gp (t) = Ep (t) + m=p

The second term of the right side of (29) represents the total is a random delay from the MAI arriving at the decoder. tm mth encoder to the pth decoder, which is assumed to be uniformly distributed on an interval (−T /2, T /2) for the analysis. Ep (t) represents the electric field of the successfully decoded signal sequences traveling from the pth encoder to the pth decoder. Ep (t) at desired sampling times t = j T b is (p) (p) equal to either vout p (0), when dj = 1, or 0, when dj = 0.

gp (t) 2 is defined as the intensity of the pth decoded signal given by Ip (t) ≡ gp (t) 2 = gtp (t) g∗p (t),

(p)

PgˆR gˆI gˇR gˇI (gˆ R , gˆ I , gˇ R , gˇ I /dj , ) ⎧ ⎡ (p) 2 1 1 ⎨ (gˆ R − dj vˆpR (0)) = 2 2 2 2 exp ⎣− 2 ⎩ 4  ˆ ˇ ˆ 2 (p)

+

 is a random variable with a binomial distribution;  B(l) =

(gˇ R − dj vˇpR (0))2

ˆ 2

ˇ 2

+ ⎫⎤ (p) (gˇ I − dj vˇpI (0))2 ⎬ ⎦, + ⎭ ˇ 2

N P0  ˆ 2 ˆ2 Rn + In , 2N02 n=−N

l



l 

1 1− 2K

M−1−l .

(33)

⎧ ⎡ ⎨ (I cos2  + d (p) |vˆp (0)|2 ) I sin 2 1 j ⎣− exp 2 ⎩ 42 ˆ 2 ˇ 2 ˆ 2

 2

= 0

⎫⎤ (p) (I sin2  + dj |vˇp (0)|2 ) ⎬ ⎦ + ⎭ ˇ 2 ⎛

√ ⎞ (p) dj |vˆp (0)| cos  I ⎠ + I0 ⎝ ˆ 2 ⎛

√ ⎞ (p) dj |vˇp (0)| sin  I ⎠ d, × I0 ⎝ ˇ 2

(34)

where I0 (·) is the modified Bessel function of the first kind (p) and the zeroth order. When dj = 0, (34) reduces to (p)

PI (I /dj =

= 0, )

     1 −I −I exp − exp . 2(ˆ 2 − ˇ 2 ) 2ˆ 2 2ˇ 2

(35)

The receiver decides data as “1” when I (j T b ) exceeds the threshold Ith and as “0” otherwise. Assuming that OOK is generated with equal probability 1/2, the average probability of BER is given by 1 BER = e0 () + e1 ()  2 =

M−1 1  B()(e0 () + e1 ()), 2

(36)

=1

where (31)



∞

e0 () =

Ith

= ˇ 2 =

1 2K

(p)

where ˆ 2 =



PI (I /dj , )

(p)

(gˆ I − dj vˆpI (0))2

M −1

From (31), a conditional probability density function (p) P (I /dj , ) for Ip (j T b ) is given by (see (A.2))

(30)

where superscripts t and ∗ denote transposition and complex conjugate, respectively. With a comparison between Ip (t) and a threshold Ith , data decision is carried out. Subject to the assumption that the number of interfering users who have transmitted their signals at t = j T b is , we first derive a conditional joint probability density func(p) tion (PDF) PgˆR gˆI gˇR gˇI (gˆ R , gˆ I , gˇ R , gˇ I /dj , ) for gp (j T b ) to derive PDF for Ip (j T b ). gˆ R and gˆ I are the real and imaginary parts of HEx11 mode for gp (j T b ). gˇ R and gˇ I are y are those of HE11 mode. When N0  1 and both m and tm random variables with uniform distributions, we can model gˆ R ,gˆ I , gˇ R , and gˇ I as the joint Gaussian random process by invoking the central limit theorem. Thus their PDF is given by

639

N P0  ˇ 2 ˇ2 Rn + In . 2N02 n=−N

(32) Here, vˆpR (0) and vˆpI (0) are the real and imaginary parts of vˆp (0), respectively. vˇpR (0) and vˇpI (0) are those of vˇp (0).

(p)

PI (I /dj

= 0, ) dI

     1 −Ith −Ith 2 2  ˆ −  ˇ exp exp ˆ 2 − ˇ 2 2ˆ 2 2ˇ 2 (37)

and 

Ith

e1 () = 0

(p)

PI (I /dj

= 1, ) dI .

(38)

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Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

·  is the ensemble average with respect to . e0 () represents the probability that the intensity of MAI exceeds the threshold and e1 () represents the probability that the intensity of the sum of the intended coherent pulse and MAI does not exceed the threshold.

10-5 10-6

BER

10-7

5. Numerical result

10-8 10-9

10-5

10-10

ωc=1.204×1015-rad/s ωc=1.206×1015-rad/s ωc=1.207×1015-rad/s ωc=1.212×1015-rad/s ωc=1.213×1015-rad/s

10-11 10-12 5

6

7

8

9 10

20

30

40

Number of Users

Fig. 6. BER versus the number of users for different center angular frequencies.

10-3

10-4

10-5 BER

We assume an input signal to an optical fiber is linear polarization. SMF used here is the single-mode dispersioncompensating fiber of 6-km long, for which Jones matrix is experimentally measured [5]. Fig. 5 shows BER versus the number of users for different input polarization angles, which is calculated with (36)–(38) for N0 = 511, c = 1.208 × 1015 rad/s, W = 15 × 1012 rad/s, Ith = 0.3, P0 = 1, and K = 5. From these parameters, c ≈ 419-fs and each user’s bit rate of ≈ 934-Mbit/s are derived. Performance differences among different input polarization angles are not extremely large, namely the input polarization angle is not a critical factor in the system performance. Fig. 6 shows BER versus the number of users for different center angular frequencies where N0 = 255, W = 2.0 × 1012 rad/s,  = /6, Ith = 0.3, and K = 1. It is found that c is a critical factor in the system performance. For example, BER for c =1.204×1015 rad/s is three orders of magnitude less than BER for c = 1.213 × 1015 rad/s when the number of users is 20. Fig. 7 shows BER versus a total bandwidth for the different number of users where N0 = 255, c = 1.208 × 1015 rad/s, =0, Ith =0.3, and K =1. BER is almost flat for W 3.0 × 1011 rad/s. In this region, the performance degradation caused by DGD in the fiber is negligible small. For W 3.0 × 1011 rad/s, BER increases with the total bandwidth since DGD also increases with the bandwidth.

10-6

10-7

10-8 1011

M = 16 M = 14 M = 12 M = 10

1012 Total Bandwidth: W [rad/s]

Fig. 7. BER versus total bandwidth for the different number of users.

10-6 BER

6. Conclusions

10-7

θ=0 θ=π/12 θ=π/6 θ=π/4

10-8 10

12

14

16

18 20 22 Number of Users

24

26

28

30

Fig. 5. BER versus the number of users for different input polarization angles.

We have analyzed the BER of coherent optical CDMA communication systems interconnected via an optical fiber, which performance is characterized by TFM. This has been the first work to analyze BER with TFM to our knowledge. We have studied the statistic of the received signal including MAI, and then derived BER as the function of system parameters. The numerical results have shown that BER increases with the total bandwidth of an optical signal. An input polarization angle is not a critical factor in the system performance. Selecting a proper center frequency leads to the BER reduction.

Y. Igarashi, H. Yashima / Int. J. Electron. Commun. (AEÜ) 62 (2008) 635 – 642

gˇ R =

Appendix A

√ √

641

I sin  cos ,

(A.10)

I sin  sin .

(A.11)

A.1. Derivations of (), (), and ()

gˇ I =

The components of TFM, (), (), and (), are derived from the Jones matrix J() of an optical fiber as follows [6]:

By assigning (A.8)–(A.11), to (31), a conditional probability (p) density function PI  (I, , , /dj , ) is given by



 J12 ()J21 () () = − tan , J11 ()J22 ()    1 J21 ()J22 () () = arg − 4 J11 ()J12 ()    J21 ()J22 ()    , −i ln  J11 ()J12 ()     1 J12 ()J22 ()

() = arg − 4 J11 ()J21 ()    J12 ()J22 ()   , −i ln  J11 ()J21 ()  −1

(p)

PI  (I, , , /dj , ) (p)

(A.1)

= |J |PgˆR gˆI gˇR gˇI (gˆ R , gˆ I , gˇ R , gˇ I /dj , ) =

(A.2)

I sin 2 162 2 ˆ 2 ˇ 2 ⎧ √ ⎡ ⎨ ( I cos  cos  − d (p) vˆpR (0))2 1 j × exp ⎣− 2 ⎩ ˆ 2 √ (p) ( I cos  sin  − dj vˆpI (0))2

+

ˆ 2

√ (p) ( I sin  cos  − dj vˇpR (0))2

(A.3) +

⎫⎤ √ (p) ( I sin  sin  − dj vˇpI (0))2 ⎬ ⎦, + ⎭ ˇ 2

where Jij () is an element in the ith row and the jth column of J().

A.2. Mathematical development to derive (p) P (I /dj , )

ˇ 2

(A.12)

where J denotes Jacobian. (p) By integrating PI  (I, , , /dj , ) in terms of , , and (p)

To derive (34) from (31), we operate the transformation of stochastic variables gˆ R , gˆ I , gˇ R , and gˇ I as follows: 2 2 I = gˆ R + gˆ I2 + gˇ R + gˇ I2 ,

(A.4)

, PI (I /dj , ) is given as follows: (p)

PI (I /dj , )  = 0

 2

 2  2 0

0

(p)

PI  (I, , , /dj , ) d d d. (A.13)

gˆ I  = tan−1 gˆ R

for (0 2),

(A.5)

From (A.12) and (A.13), (34) is derived.

gˇ I gˇ R

for (0 2),

(A.6)

References

 = tan−1

 = tan

−1



2 + gˇ 2 gˇ R I 2 + gˆ 2 gˆ R I

for

0  

! . 2

(A.7)

From (A4)–(A7), gˆ R , gˆ I , gˇ R , and gˇ I are given by gˆ R = gˆ I =

√ √

I cos  cos ,

(A.8)

I cos  sin ,

(A.9)

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Yasutaka Igarashi received the B.E., M.E., and Ph.D. degrees in information and computer sciences from Saitama University, Japan, in 2000, 2002, and 2005. He was a research fellow of the Japan Society for the Promotion of Science from 2004 to 2006. Since 2006, he has been a research associate of the Tokyo University of Science. His research is involved with optical CDMA and the cryptanalysis of symmetric-key cryptography. Hiroyuki Yashima was born in Mie, Japan, in 1958. He received the B.E., M.E., and Ph.D. degrees from Keio University, Japan, in 1981, 1987, 1990, respectively, all in electrical engineering. From 1990 to 2003, he was an associate professor at the Department of Information and Computer Sciences, Saitama University, Japan. From 1994

to 1995, he was a visiting scholar at the University of Victoria, Canada. In 2003, he joined Tokyo University of Science, Japan, as a professor at the Department of Management Science. His research interests include modulation and coding, optical communication systems, satellite communication systems, and spread spectrum communication systems. Dr. Yashima received the 1989 Society of Satellite Professionals International Scholarship Award.