A New Method to Design CDMA Spreading Sequences

8/8/2019 A New Method to Design CDMA Spreading Sequences

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A New Method to Design CDMA Spreading Sequences

*

Bi Jianxin Wang Yingmin Tian Hongxin Yi Kechu

(National Key Lab. On ISN, Xidian University, Xian 710071, China)

*

This paper is supported by National Natural Science Funds (N0. 69872027).

Abstract: It is well-known that multi-value CDMA

spreading codes can be designed by means of a pair of

mirror multi-rate filter banks based on some

optimizing criterion. This paper indicates that there

exists a theoretical bound in the performance of its

circulate correlation property, which is given by an

explicit expression. Based on this analysis, a criterion

of maximizing entropy is proposed to design such

codes. Computer simulation result suggests that the

resulted codes outperform the conventional binary

balance Gold codes for an asynchronous CDMA

system.

Key words: Multi-rate unitary filter banks; CDMA

spreading code; Maximum entropy criteria.

. INTRODUCTION

The selection of spreading codes is essential for

asynchronous CDMA systems. Recently, it is believed

to be promising that the CDMA spreading code is

designed based on multi-rate filter bank theory. For

this purpose, Akansu[1] has converted the design of

CDMA spreading code into an optimizing problem.

Simulations suggested that the resulted multi-valued

codes outperform Gold codes. Qinghua Shi[2], et al.,

give another scheme, from frequency domain, for

spreading code generation based on multi-rate filter

bank theory and a lattice structure. This paper

indicates that there exists a theoretical bound in the

performance of spreading codes circulate correlation

property, and gives an explicit expression for it. Based

on this analysis, a criterion of maximizing entropy is

proposed to design such codes. Computer simulation

result suggests that the resulted codes behave better

than the conventional binary balance Gold codes for

an asynchronous CDMA system.

. TWO-BAND MULTIRATE UNITARY

FILTER-BANK THEORY(TBFMFB)

The coefficients of two-band unitary filter-banks

only depend on )n(h0 , which satisfies[3]

=+n

00 )k()k2n(h)n(h (1)

From the viewpoint of the frequency domain, the

parameterization representation of the two-band

unitary is given as follows:

=

111JJ

1

0

z

1)(ZRZ)(ZR)(R

)z(H

)z(HL ( 2/NJ = ) (2)

=

cossinsincos)(R

= 2z0

01Z

So, we can use 2/N parameters { }J21 ,,, L to

represent )n(h0 .

. The property of circulate correlation of the

codes based on TBFMFB

From definition of circulate correlation, it is evident

that

1

=

=+

= )1N,,5,3,1i()i(R

)2N,,4,2,0i()i(R

)i(Ra

a

b L

L

(3)

2 0)i(Rab = )2N,,4,2,0i( = L (4)

where N-length normalized vector a and vector b

are the coefficients of the analysis filter bank, and

)i(Ra denotes i-shifting circulate auto-correlation of

vector a and )i(Rab denotes i-shifting circulate cross-

correlation.

Theorem 1 If N-length b (N= u2 , Iu ) is the


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mirror of a normalized vector a , then, the following

inequality is true:

1)i(R)i(R1N

0i

2ab

1N

1i

2a +

=

=

)Zi,1Ni0( (5)

where the equality is valid if and only if)2N,,4,2k(0)k(Ra == L .

Proof: Let A and B denote two circular matrices:

1N210 aaaaA = L

1N210 bbbbB = L

where ia and ib represent two column vectors

obtained respectively by circular--shifting vector a

and b for i elements. Let

y= +

=

=

1N

0i

2ab

1N

0i

2a )i(R)i(R

=22

aBaA + = TTT a)BBAA(a + (6)

Making use of the properties of circular matrix, we

have [4]

1N22

1NN1 PaPaPaIaA

++++= L

1N22

1NN1 PbPbPbIbB

++++= L (7)

where

=

00

10

0

0

01

00

0000000

00

1

0

00

10

L

L

MMM

L

MM

L

L

P

Substituting (7) into (6) and using iNTi P)P( = , we

have

y= )PaPaPaIa)(PaPaPaIa{(a 1NN23211N221NN1 ++++++++ LL

+ T1NN23211N221NN1 a)}PbPbPbIb)(PbPbPbIb( ++++++++ LL

(8)

Using the properties of matrix P ,we can simplify 8

into

y=2+ )1N(R)1N(R)1(R)1(R)1N(R)1(R baba2a2a +++++ LL

(9)

With Lemma1 we obtain

y=2+2[ 2a2

a2

a )2N(R)4(R)2(R ++ L ] (10)

So we have

1)2N(R2)4(R2)2(R21)i(R)i(R2

a2

a2

a

1N

0i

2ab

1N

1i

2a ++ +=+

=

=L

The equation is true if and only if

)2N,,4,2k(0)k(Ra == L .

It should be noticed that )0i()i(R 2a and2

ab )i(R

represent the normalized energies of all the possible

multi-path interfere and all the possible multi-access

interfere in the output of correlation receiver,

respectively. So the theorem implies :

(1)It provides the theoretical basis for designing

CDMA spreading code based on unitary filter bank.

When we design CDMA spreading code based on

TBFMFB, if we adopt the interfere energy minimum

criterion in the output of correlator, the optimization

solutions are the coefficients of unitary filter banks

(2)It gives us the lowest bound of the sum of all

possible interfere energy in the output of correlator.

That is, if we use a pair of CDMA spreading codes

based on TBFMFB for user codes, the sum of all

possible interfere energy in the output of correlator is,

on the condition of equal power transmission, greater

than or equal to its signal energy. Moreover, theequality is valid if and only if the codes are resulted

from a unitary filer bank.

. a Proposed Method for Designing CDMA

Spreading Codes Based on Unitary Filter Bank

It is of significance that we adopt the minimum of

the maximum of side lobes of absolute value of

circulate auto-correlation and cross-correlation as an

objection function. In another viewpoint, the flatter

side lobes of absolute value of circulate auto-

correlation and cross-correlation are, the better the

performance of the corresponding asynchronous

CDMA systems will be. So,we use a criterion of

maximizing entropy. The entropy of a vector

[ ]N1 p,,pp L= =

2

ab

2

ab

2

ab

2

a

2

a

2

a )1N(R,,)3(R,)1(R,)1N(R,,)3(R,)1(R LL


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is defined as

)p(logp)p(logp)p(entropy N2N121 ++= L (11)

Furthermore, we also search the spreading code from

the viewpoint of the frequency domain (Z-domain),

and get the same merits as reference [2]. It is obvious

that p is the function of J21 ,,, L and )p(entropy

is also the function of J21 ,,, L . So, the objection

function we use for optimizing is

{ }[ ]),,,p(entropymaxarg),,,( J21,,,

J21

J21

=

LL

L

(12)

To make )n(h0 suitable for DS-CDMA systems, we

must spread )n(h0 in both time and frequency as

evenly as possible. So, by using the entropy function

defined above, we add two constrains as follows

1T c)aa(entropy

22T c))a(FFT/)a(FFT)a(FFT(entropy

where ( )aFFT denotes the Fourier transform of vector

a . In experiment, these parameters

are: 46.4cc 21 == , 32N = . The experiment results are

showed in fig.1,2, 3, and 4. Gold codes of length-31

are used for comparison. The Gold codes result fromsumming (mod 2) a pair of m-sequences. The

preferred pair of primitive polynomial is 45 and 67.

Fig.1 compares the spectra of a 32-length optimized

code for the two-user case with a 31-length balanced

Gold code. The circulate auto-correlation and

circulate cross-correlation of these codes are also

displayed in Fig.2 and 3. It is obvious from these

figures that the circulate auto-correlation, circulate

cross-correlation and frequency properties of the

multi-valued optimized code outperform the

comparable duration binary balanced Gold code.

. PERFORMANCE OF THE PROPOSED

MULTIVALUED CDMA SPREADING CODES

By using Monte Carlo computer simulation method,

the bit error rate (BER) performance of a two-user

asynchronous CDMA communication system based

on the user code type presented above are investigated

in this section. Antipodal signaling and BPSK

modulation for CDMA are used in the simulations.

The channel noise is modeled by additive white

gaussian noise (AWGN).

In the process of evaluating the BER performance,

it is assumed that for a user there are a multipath

interfere and a multiuser interfere, whose delays are

random variables evenly distributed in

[ ]cc T)1N(T0 L , where cT denotes the duration

of a chip. Because circulate correlation properties of

the optimized codes outperform those of the balanced

Gold code, it is expected that the BER performance of

the optimized codes are also better than the latter. The

result showed in Fig.4 also substantiates it.

. CONCLUDING REMARKS

Multi-rate unitary filter bank provides us powerful

mathematical tool for designing CDMA spreading

code. This paper provides the theoretical basis for

designing CDMA spreading code based on unitary

filter bank, and gives us a lower bound of the sum of

all possible interfere energy in the output of correlator.

Based on this analysis, a criterion of maximizingentropy is proposed to design such codes. It is worth

while to further investigate how to extend this method

to more generalized case where user number K is

more than 2.

REFERENCES

[1] Ali N.Akansu, M.V.Tazebay and R.A.Haddad, a newlook at digital orthogonal transmultiplexers for CDMA

communications,IEEE trans. On SP-45, No.1, January

1997.

[2] Qinghua Shi and Shixin Cheng, optimal spreadingsequence design based on PR-QMF theory,

ELECTRONICS LETTERS, 18th March 1999, Vol.35,

No.6.

[3] C.S.Burrus,R.A.Gopinath,H.Guo, introduction towavelets and wavelets transforms---a primer, prentill

hall,1998.


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[4] 1999 6

[5] Verdu Sergio. Multiuser Detection. Cambridge UniversityPress, 1998.

Fig.1 Frequency spectra of the optimized 32-length Fig.2 Circulate auto-correlation functions of the

codes and 31-length balanced Gold codes. optimized 32-length codes and 31-length balanced

Gold codes.

Fig.3 Circulate cross-correlation functions of Fig.4 BER performance of two-user asynchronous CDMA

the optimized 32-length codes and 31-length system for different user code types

balanced Gold codes.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

31-length balanced Gold code

frequency(rad)

magnitude

function

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

32-length optimized code

frequency(rad)

magnitudefunction

5 10 15 20 25 30-0.5

0

0.5

1


time shiftcirculateauto-correlationfunction

5 10 15 20 25 30-0.5

0

0.5

1


time shiftcirculate

auto-correlationfunction

5 10 15 20 25 30-0.4

-0.2

0

0.2

0.4


time shiftcirculate

cross-correlationfunction

5 10 15 20 25 30-0.4

-0.2

0

0.2

0.4


circulatecros

s-correlationfunction

time shift

-2 - 1 0 1 2 3 4 5 6 7 81 0

- 4

1 0- 3

1 0- 2

1 0- 1

1 00

SNR in dB

errorrate

2 - u s e r a s y n c h r o n o u s C D M A r e c e i v e r

s ing le us e r

3 1 - l e n g t h b a l a n c e d G o l d c o d e

3 2 - l e n g t h o p t i m i z e d c o d e

A New Method to Design CDMA Spreading Sequences

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