Synthesis of Low-Crest Waveforms for Multicarrier CDMA System

T. F. Hc)* and Victor E(. Weit Department of Information Engineering, The Chinese University of Hong Kong

Shatin, N.T. , Hong Kong.

Abstract - The crest factor of a waveform is the ratio of

its peak power to its average power. A low crest factor

increases the efficiency of the transmitter power ampli-

fier. We study the problem of synthesizing signals with

low crest factor for multi-carrier code division multiple ac-

cess (MC-CDMA) systems. By using Golay complemen-

tary sequences as spreading codes, the crest factor is upper

bounded by 6 dB regardless of the length of the spread-

ing code. In comparison, the crest factor of DS-SS-CDIMA

pulses with certain chip shapes can grow unbounded. We

also present methods for constructing a large number of

pulses to accommodate multiple-access users.

I. INTRODUCTION

The multi-carrier code division multiple access (hlC-

CDMA) is an alternative spread-spectrum method for mul-

tiple access applications. A duality exists between MC-

CDMA and (single carrier) direct-sequence (DS) CDlMA

[l, 21. In DS-CDMA, the spectrum spreading is accom-

plished by multiplying user's data with a signature se-

quence in the time domain; while in MC-CDMA, the

spreading is done in the frequency domain.

This paper is aimed at providing methods to ease the re-

alization of low-crest MC-CDMA waveforms. In sectioin 2 ,

the MC-CDMA system is introduced in an asynchronous

multiple access environment. The crest factor (ratio of

peak power to average power) of MC-CDMA signals is dis-

cussed in section 3. Using m-sequences as spreading codes

results in large crest factors, which can grow unbounded

as the spreading gain N grows to infinity. Using Golay

complementary sequences as spreading codes, on the other

* tfho4@ie.cuhk.hk t kwweiQie.cuhk.hk

0-7803-2509-5195 US$4.00 0 1995 IEEE 131

t - RC(Cl(2*.'+%))

Figure 1: System model of MC-CDMA.

Figure 2: Receiver model of MC-CDMA.

hand, limits the crest factor to below 6 dB for all values of

N . Methods of synthesizing multitone signals for multiple

access are proposed in section 4.

11. SYSTEM MODEL

The block diagram of the MC-CDMA system with M

users is shown in Fig. 1. For each user, the input data sig-

nal is modified by his spreading code { cn} in every symbol

period, T,, prior to transmission. The transmitted signal

of user i, whose data sequence is { a t ' } , can be expressed

as M

k = - w

A common choice of the pulse is b

4

-3

-4 - The peak-to-average ratio or crest factor (CF) of the

3

Ts Ts (2) 1 2

t N 2ant w(z) ( t> = Cck ) cos(-)rect(-).

2 ’ .%

n = l

The receiver for the i-th user is shown in Fig. 2, where the Z 0 p -’ g -2 - despreader is a correlator with coefficients { E k ’ } ~ . “7

Signel denved from subseq 01 S-R sequence (-)

Signal dsnved fr-m subseq of m-sequence ( )

-

power amplifier.

dissipation class C amplifiers can be used.

If the crest factor is low, then low-

If the crest C F ( P ) I - - (5)

factor is high, then other amplifiers have to be used result- Q.E.D. ing in higher power dissipation and harmonic distortions.

If we use random sequences or m-sequences as the spread-

ing code { c n } F , then its crest factor can grow unbounded

(- m) as N grows to infinity [4]. However, if we use

other sequences, such as Sharipo-Rudin sequences or Go-

lay complementary, then the crest factor of the signal is

upper bounded by 2 (i.e. 6 dB) for all values of N.

A pair of sequences ( { p n } , { y n } ) , each of length N is

called a (Golay) complementary pair if

(4) 2N , i = O

0 , otherwise

N- i N - i

PnPn+i + ynyn+i = n=l n=l

Either member of a (Golay) complementary pair is a (Go-

lay) complementary sequence.

A large subclass of Golay complementary sequences [3]

with length 2n called Shapiro-Rudin sequences were an-

alyzed in [4]. A comparison of the MC-CDMA signals

generated from the the subsequences of long m-sequence

(with 18 stages) and that of S-R sequences is shown in

Fig. 3, where the designated bandwidth is 48MHz and the

transmission bit rate equals 32kb/s. The crest factor of

the transmitted signal constructed from the S-R sequences

was found to be 3.5dB lowered than the signals constituted

from the subsequences of long m-sequence in average.

In this paper, we restrict ourselves to cn = +1 or

- 1. There are polyphase sequences and “chirp” sequences

which can also be used to construct multicarrier wave-

Theorem 1

{pn}? is a complementary sequence, then C F ( w ) 5 2. Let w(t) = E:=’=, pn cos( y ) r e c t ( $-). If shapers with low crest factors [4, 5, 61

111. CREST FACTORS

Proof: Let { p n } and { y n } form a complementary pair,

each with length N [3]. Let u(t) = E:=,’=, yn cos(2ant/Ts).

Then,

We study the crest factors of MC-CDMA and DS-

CDMA signals. For simplicity we consider a single pulse

w( t ) and

n = l n=l A . Time-Limited Pulses by the Autocorrelation Theorem in Fourier Transform.

In the DS-CDMA system, the time-limited base-

band transmitted signal consist of rectangular waveforms

132

Then 9 , I

5 5 . : moving wmdow average of the expenmental results (-)

I

I

B. Ideally Band-Limited Pulses

In MC-CDMA, the pulse

N w(t) = cn cos(2.irnt/T,)sinc(t/T,)

n = l

is ideally band-limited. If {cn} is a complementary se-

quence, then its crest factor is bounded by 6 dB as a result

of the following theorem (proof omitted):

Theorem 2 Let w(t) = p( t )h ( t ) . Then C F ( w ) 5 CF(p)CF(h) if

The ideally band-limited pulse for a DS-CDMA system

is

N N t - nT, w( t ) = c,sinc( 1. Ts n = l

Assume {cn}F is a PN-sequence, N = 2" - 1, e is an wen

integer. Then there exist i such that

The first expression grows in the order of loglogN. All

logaritms in this paper have base 2. It is unbounded as

N grows to infinity. The second expression corresponds to

other terms of smaller magnitude. If we assume the coef-

ficients are random, then the expected value of the second

expression is zero. Therefore, the crest factor of ideally

band-limited DS-CDMA pulse is likely to grow unbounded

(- log log N ) if the spreading code {en} is randomly cho-

sen or is a PN-sequence. The theoretical estimate and the

experimental data of the crest factor against the number

of chips used are shown in Fig. 4. The solid line is the

moving window average of the experimental data obtained

on every 21 sample-frame.

C. Shaped Pulses

1) DS-CDMA

With raised-cosine chip shaping, the pulse is

where

rcos(z) = sinc(x) [ ;"";;:))2] Assume c, is a PN-sequence, N = 2" - 1, e is an even

integer. Then there exists i and special bit pattern (ci-"p,

c ~ - ~ J ~ + ~ , ..., c ~ + ~ J ~ - ~ ) such that

1 1 2 2

"12

= 2 Ircos(n - -11 + c,rcos(n - -). -

n = l other n

The first expression is bounded as N grows to infinity; but

the bound is larger for smaller r. Note that the raised co-

sine function equals the sinc function when r = 0. The

133

DS-CDMA signals with Raised-Cosine (r-0.i) chipshaper 1

5 1 : expenmental re~ulls (:)

moving window averags of the sxpenmentai result5 (-) -;

6 4 ,‘ 1 8 1

200 300 400 SO0 600 700 800 900 1M)O number of chips per symbol

Figure 5 : Signal performance in DS-CDMA system (Raised-Cosine chips).

expected value of the second expression is zero, provided

the coefficients e,, for “other” n , are random. The charac-

teristics of the pulse envelope using the raised-cosine filter

are shown in Fig. 5. I t remains an open problem to deter-

mine whether the asymptotic crest factor is bounded.

2) M C - C D M A

In MC-CDMA, the shaped pulse is w ( t ) = p ( t ) h ( t )

where N

Theorem 3

p ( t ) be as above.

arbitrary waveform. Then

Let { e n } be a complementary sequence,

Let w(t) = p( t )h ( t ) where h(t) is an

sup lw(t)12 5 2Nsup Ih(tjl2 t t

Proof: Let ({en}, {d,}) be a complementary pair, v ( t ) =

q(t)h(t) and N

n = l

The autocorrelation of W(f) is

1 W(f)W* (f + z)df

n m J

r r

= 2N / H(f)H*(f + z)df.

Taking the Fourier transform of both sides (via the Auto-

correlation Theorem) and setting 2 = 0, we obtain

Q.E.D.

Remark A common shaping is the raised cosine

I t C O S ( T T t / T , ) h ( t ) = sinc(-)[ T, 1 - (2~t/T,)’

Therefore, at least one of C F ( w ) and C F ( V ) is bounded

by 6 dB.

Remark: If we use complex modulations, i.e. p ( t ) =

C e n e j 2 n n t / T 3 , then the bound on its crest factor can be

reduced to C F ( p ) 5 4 i.e. 3 dB.

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IV. CONSTRUCTING MANY PULSES V. CONCLUSION

In a multiple-access environment, we need to supply dis-

tinct pulses to a large number of users. It is desirable that

these pulses have noise-like appearances, low cross corre-

lations, thumbtack-like autocorrelations, and small crest

factors. One practical method of constructing these pulses

is by cropping a long “mother” pulse as follows:

Let T, be the symbol period of each user, and let N/T,

be the available bandwidth. Let m >> 1 and let {c,}TN

be a complementary sequence. The mother pulse is mN 2 ~ n t

wmother = C c n cos(-). n = l mT,

The signal pulse for user i is

where ri is a random delay. Time-varying pulses can also

be used, so that

The resulting pulses are likely to possess all the clesir-

able properties mentioned above. Preprocessing before ac-

tual deployment can further eliminate “bad apples” among

them. These pulses can be synthesized by digital direct

synthesis. Let W ( i ) ( f ) be the Fourier transform of d i ) ( t ) .

We can store N or more samples of the pulse di)(t)1 and

synthesize in the time domain. Or we can store N or more

samples of its Fourier transform Wci)( f ) and synthesize

the pulse via the architecture in Figure 1.

There are other methods to accommodate more

An interesting method is to con- multiple-access users.

struct more pulses using “interleaved” carriers, i.e.

By our theorems, its crest factor is bounded by 6 dEl. We

also have a “trellis”-based search algorithm for finding new

complementary sequences. (Omitted here.)

This paper proposed methods to construct MC-CDMA

pulses with low crest factors. It is shown that Golay com-

plementary sequences are good candidates as spreading

codes. The resulting crest factors are compared to DS-

CDMA under several conditions. One important open

problem is whether DS-CDMA pulses with raised-cosine

chip shaping has bounded crest factors asymptotically.

Methods of synthesizing distinct MC-CDMA pulses for a

large number of users are discussed.

REFERENCES

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Carrier Code Division Multiple Access (MC-CDMA)

Modem Design,” 45th IEEE Vehicular Technology

Conference, pp. 1670-1674, 1994.

[3] J . E. Golay, “Complementary Series,” IRE Transac-

tions on Information Theory, pp. 82-87, April 1961.

[4] S. Boyd, “Multitone Signals with Low Crest Fac-

tor,” IEEE Transactions on Circuits and Systems, Vol.

CAS-33, No. 10, pp. 1018-1022, October 1986.

[5] M. R. Schroeder, “Synthesis of Low-Peak-Factor Sig-

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tion” IEEE Transactions on Information Theory, Vol.

IT-16, pp. 85-89, January 1970.

[6] B. M. PopoviC, “Synthesis of Power Efficient Multi-

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