Frequency-domain orthogonal multiuser communications over frequency-selective slowly fading channels

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Frequency -Domain Orthogonal Multiuser

Communications over Frequency-Selective

Slowly Fading Channels

Cha'o-Ming Chang Kwang-Cheng Chen

Institute of Communications Engineering, National Taiwan University, Taiwan,

R.O.C.

lepidus@santos.ee.ntu.edu.tw

chenkc@cc.ee.ntu.edu.tw

Abstract-Inheriting the properties of the frequeucy-domaiu approach 111 to avoid MA1 in the asynchronous multiuser communications and sharing the rationale behind OFDM systems to mitigate ISI, we propose the asynchronous multiuser eommuuica- tions systems over frequency-selective slowly fading channels such that the optimal demodulation with similar complexity to point- to-point communications is possible. In addition, by spreading the symbol energies to the whole transmission frequency band, we introduce another asynchronous multiuser communications systems to utilize the frequency diversity gain. These proposed systems not only mitigate MA1 and IS1 efficiently with simple complexity hut also are flexible for variable rates transmission to suggest their applications to practical multimedia services.

1. INTRODUCTION

One of the most challenging issues in the wireless communications systems is the mitigation of interference to improve the communications performance. In the direct sequence code division multiple access (DS-CDMA) multiuser communications systems, all users transmit information spread hy their assigned different signature waveforms in the same frequency band simultaneously, and users are interfered by the other users known as the multiple access interference (MAI). Starting from Verdu's pioneer work [ 2 ] , a great amount of researchers have been working on the multiuser detection to minimize the impact of MA1 to DS-CDMA multiuser communications. By taking the feasible implementation into consideration, however, these approaches are of suboptimal performance and are prevailed hy conventional detection employing time- domain orthogonal signature sequences (such as the Walsh- Hadamard sequences).

It is required to maintain synchronous channels perfectly such that the conventional and simple demodulation approach is possible by employing the time-domain orthogonal signature sequences. The communications channels are said to be synchronous when the received timings from all users are the same and the demodulation performance degrades severely whenever the received timings are not perfectly aligned [31. In the reverse link (also known as the up-link) transmission wherein the stations simultaneously transmit their signals to the base station, the maintenance of synchronous channel is not an easy task.

To solve this problem, we proposed the frequency-domain approach to DS-CDMA multiuser communications systems over additive white Gaussian noise (AWGN) channels [I].

By employing the novel hut equivalent spreading approach and by utilizing the frequency-domain orthogonal signature sequences, the signature sequences are orthogonal even in the asynchronous channels such that the simple decoding approach is realized with optimal performance. The rationale behind this approach is to maintain signature sequences oahogonal over frequency domain instead of time domain, because the timing shift comesponds to the scaler multiplication over frequency domain. And therefore the signature sequences remain orthogonal in the asynchronous channels.

By extending this frequency-domain approach to frequency- selective slowly fading channels [4], this frequency-domain approach is interfered by inter-symbol interference (1st) introduced by frequency-selective channels. The frequency- selective slowly fading channels can be modelled as a tapped delay line [ 5 ] and therefore the received signal is interfered by its delayed copies, known as ISI. On the other hand, unlike the

Fig. I. Novel signal spreading in frequency-domain approaches.

conventional direct sequence spreading approach, each symbol in the frequency-domain orthogonal spreading is first spread by the signature sequence and then uniformly interleaved into the whole frame as shown in Figure 1. This feature makes the frequency-domain approach more vulnerable to IS1 as

the effective symbol length is reduced from T to

T,,

where T N,T,. Fortunately, this frequency-domain approach is

still not interfered by MAL

In this paper, sharing the similar rationale behind the orthogonal frequency division multiplexing (OFDM) technologies to mitigate ISI, we shall introduce a family of frequency-domain orthogonal multiuser communications systems to combat the

IS1 with simple complexity.

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11. FREQUENCY-DOMAIN ORTHOGONAL MULTIUSER

COMMUNICATIONS WITH INVERSE TRANSFORMATION

(FDOMC-IT) OVER FREQUENCY-SELECTIVE SLOWLY

FADING CHANNELS

A. Signal Models and Assumptions

We consider the asynchronous multiuser communications systems with I< users over frequency-selective slowly fading channels such that the channel impulse responses of all users are time-invariant within a couple of frames. In addition, assuming that the null-to-null bandwidth of the transmitted signal is W , the frequency-selective slowly fading channel for

user k can he modelled as the tapped delay line having Lk

laps with tap spacing 1/W [ 5 ] .

Suppose that each user transmits Ati,,, information symbols in a frame. For example,

{Bi"")(m)lm

= 0,1,. . .

,

AIa,f,- 1) for k = 0,1, . .

,

K - 1

are

the A{,,,, information symbols of user k in the m'th frame, where m' E 2 and

2 denotes the field of integers. We assume that the kth user emp!oys the binary phase shift keying (BPSK) modulation (i.e.

Elm

' ( m ) E il) though the results in the sequel are equally applicable to other linear modulated signals. It is remarkable to note that different users could use different modulation schemes without affecting the receiving performance of other users such that the data rates could he variable for users.

Before transmission, we first take Ali,fo-point inverse dis- crete Fourier,transform (IDFT) on the infoyation symbols and have

iim

) ( m )

&

~ ~ ~ f o - l i ? ~ m ) ( n ) e ' e n m , for m = O? 1,. . .

,

AIi,,, - 1 and k = 0,1,.

. . ,

Ii

- 1. In addition, for each user, we insert

At,

guard symbols before spreading, where these A I g symbols are the cyclic prefix of the information symbols after performing IDFT and

&Ig

is

a

uon-negative integer such that MgTc 2 maxk=o,l, ... ,*-I

2 ,

wherein T, is the chip time of the signature sequences. In order

to simplify the illustrations, we assume A.I,,,, 2 A4, for the rest of this paper. For the special case that

hl,

>

n/1~,f0, we only need to pretend that the number of information symbols is

A&

by padding "0"s after

&Ii+,

information symbols. By employing the frequency-domain oxthogonal spreading in [ l ] , the m th transmitted frame from user

k

with

AI

=

n/r,,,,+M,

symbols is SL")(t) =

En:;

p k ( n )

Cmz0

bk (m)II(t -

mT,-nTns), wherepk(n) e j z " ' forn = 0 , l ; . . , N c - l is the signature sequence of user I;; Arc is the number of chips within a signature sequence such that

N,

2 K ; TM MT,;

n(t)

is the indicator function with support [O,T,); and

~ i " ) ( m

+

M,,,,~ - A{,)

N I nr-1 .+')

if

o

5

m

<

AI,

if AIg

5

m

<

M .

for m = 0,1,

'.

. ,AI - 1 is the transmitted sequence of user k after guard symbols insertion.

Let Tdiff be the maximal difference between the received

timings of users. That is, Tdiff

b ma^^,^,=^,^

,..., K - l Irk -

rk,l, where r k is the received tinnng of user k . And define

r '

p

' ( m - Illg)

si")(m)

E?

L

TG G Tdiff

+

maxk=o.l, ... +r,-1

+,

which is the maxi-

mal possible distance in time between users considering the frequency-selective channels. In order,to avoid the inter-frame interference, before transmitting

iim

) ( t ) ,

we further insert another guard time before each frame, which is the cyclic prefix of Sim ' ( t ) with duration TG. To ease the development, we temporarily assume that TG is smaller than MT. Therefore, the m th frame of user k with cyclic prefix between frames can be written as S i " " ) ( t

+

AIT - T G ) 0 5 t < T ~ TC

5 t

<

A4T +Tc

{ o ,

otherwise. ..(") sk

( t )

= Sim

' ( t

- TG)

The frame structure and transmitter of this approach are plotted in Figure 2 and Figure 3(a).

NCTc =

T ;

AIg

+

A&,, = Al. Fig. 2.

nicntions systems.

Frame sfmcfllre of frequency-domain orthogonal multiuser cammu-

As the frames can be independently demodulated with guard intervals between frames, we demodulate frame 0 in the sequel as an example. The received low-pass equivalent signal r L p ( t ) with K users in the frequency-selective slowly fading channels is r L p ( t ) =

xf2r1

p k , l i P ) ( t - Tk -

W )

+

t L p ( t ) , for

t

E [Tc

+

mink,o,l ,..., K - - l r k , A t T

+

TG

+

mink,o,l ,... , K - I r k ) , where pk.l is the complex-valued random variable whose magnitude (resp. phase) stands for the received amplitude (resp. phase) of user

k

at the Ith tap; and z L p ( t ) is the low-pass equivalent additive white Gaussian noise whose two-sided power spectral density is N 0 / 2 .

Assume that the multiuser timing recovery (such as the one in

[ 6 ] )

and channel estimation schemes

are

available to the

re-

ceiver such that r k and p k , ~ are known for all k and 1. Without loss of generality, by letting 0 ,... , K - ~ i-k

5

maxk=o,l, ..., K - 1 T k

=

T K - ~ and defining t'

=

t

- TG,

we have, for

i

E O,&IT), r L p ( t ) = r L p ( t '

+,

TG) =

+

Z L p ( t

+

TG).

where ( . ) M T

=

(.) mod AIT. This novel approach, like the OFDM system, transmits the inverse Fourier transformed information symbols to the channels such that the IS1 could be perfectly avoided with simple complexity.

B. Maximum A Posteriori Probability (MAP) Demodulation

rate Mi = 2/Tc I

TO

C k - 0 K - I

Ci=o

L i - I P k , l g p

c

((i

- T k - w ) M T ) 1

Sampling the received low-pass equivalent signal r L p ( t ) at

1/T, according to the Nyquist criterion,

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tion as an example to verify the performance of our proposed system.

T,

is set to be 0.1 @sec and

N,

= 16. Therefore, this system supports up to 16 users with symbol duration

T

= N,T, = 1.6 psec and null-to-null bandwidth W = 20

MHz. By setting AT, = 10, the receiver is not interfered by IS1 for channels with RMS delay spread smaller than 100 nsec. As can be seen in Figure 4, wherein E6 stands for the bit energy, the receiver's performance is not degraded by IS1 introduced by frequency-selective channels.

Nevertheless, the possible frequency diversity inherited from wide-band systems is lost. By concentrating the energy of a symbol into some small portion of the frequency band, each symbol experiences the independent and identical flat fading channel when the coherent bandwidth of channel is not very small. Therefore, this kind of wide-hand systems does not enjoy the frequency diversity inherited by wide-band systems.

A therapy to earn the frequency diversity back is to es- tablish the correlations between symbols by employing chan- nel coding and interleaving as IEEE 802.11a [SI. With the establishment of correlations between symbols, those deeply faded symbols may be possibly recovered by the other symbols experiencing independent and identical fading effects. Another approach to utilize the frequency diversity is to spread the symbol energy to the whole frequency band as the direct sequence spread spectrum (DSSS) communications systems. With this approach, each symbol could be recovered by the portion of the frequency band which does not deeply fade and therefore the frequency diversity is earned. In the sequel, we shall propose another frequency-domain orthogonal multiuser communications system with direct spreading to e m frequency diversity gain.

111. FREQUENCY-DOMAIN ORTHOGONAL MULTt USER

COMMUNICATIONS WITH DIRECT SPREADING

(FDOMC-DS) OVER FREQUENCY-SELECTIVE SLOWLY FADING CHANNELS

A . Signal Models and Assumptions

The signal models and assumptions in FDOMC-DS are exactly the same as FDOMC-IT in Section 11-A except that the ipformation symbols for user I; in the m' frame are

{&ip

)(m)lm = 0 , 1 , . . .

,

Mi,,fo - 1) instead of

{Blm

'(m)lm

= 0 , 1 ,

".

,

M,,,fo

-

1). That is, before the signal transmission, as shown in Figure 5(a), we do not perform IDIT on information symbols.

B. Linear-Complexity Demodularion

With the identical procedures in Section 11-B, we obtain R(I;,m,q) = B ' $ ( m ) H k ( m , q j

+

Z(I;,m,q), for k = 0 , 1 , . . . , K - 1, m = O , l , . . . , M i n f o - 1, and q = 0 , l as in (2). Defining three ATinf, x 1 column vectors

&,.

bk,

and

2,.

over C"'-fax' with the nth entries be- iFg R ( k , n , 9). ir'cn), F d Z ( k , n , q ) respectively, we have &k,q = H k , , G D F T k k

fZk,,,

where

c

iS the field Of Complex numbers: H k . q is the diagonal matrix with Hk(n,q) being its entry in the nth row and the nth column; and GDFT is the

.

-(a) Tnnsmirter of user k

(b) Optimal receiver of user k

Fig. 5.

tions systems with direct spreading for user k.

Transceiver of frequency-domain onhogonal multiuser communica-

Alcnf, x Minf, square matrix whose enuy in the mth row and nth column is

Because Z(I;, m , q ) are i.i.d. complex-valued zero-

mean Gaussian random variables, the maxi.mum

likelihood (ML) estimation of

bk

is

bk,A,L

- -

6 )

= arg m=bEC&lcRfoX1 P r ( B k , O , B k , l l & - -

G-l H - l Z

f

E : = O G E b T H L , : & , q = 4 k

+

4 E:=,

D F T k.q-k,q, which is the transmitted information

4,

contaminated by Gaussian noise. Therefore, this ML estimator is the "zero-forcing" equalizer to have zero ISI.

Due to the frequency selectivity of fading channels, the zero-forcing equalizer suffers the noise enhancement problem to degrade its performance. To solve this problem, we propose the linear minimum mean square error

(LMMSE) estimation by designing the Mi,,,

x

ATi,f, square matrix

C,

E C"""f*x"'-J~ whose entry in mth row and nth column is C ( m - l , n - 1) E C such bf'(m)12 is minimized. According to the olthogonality principle [9], we have

A<;nfo-I h , " j e - 1

that

Et=, EmCO

EI

En=,

c q ( m , n ) W , m, 4 ) -

These proposed ML and LMMSE estimators are simpler than the linear-complexity estimators proposed in [4] due to the introduction of M g guard symbols. In addition, as G;lFTb

is the Al,,f,-point IDFT of

h

for

4

E C"'.-J*X1, the FDOMC- DS system with ML estimation has the same complexity as

the FDOMC-IT one by shifting the Alznf,-point IDFT from the receiver to the transmitter shown in Figure 5 .

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C. Numerical Analysis

(a) Probability of symbol error.

(b) Probability of h eerror.

Rg. 6.

selective slowly fading channels.

Performance evaluatton for FDOMC-DS systems over frequency-

We employ the identical simulation parameters as in Sec-

tion 11-C. As shown in Figure 6, both receiving structures e m s frequency diversity to combat channel fading impairment and therefore are superior to the FDOMC-IT system. In addition, the diversity gain introduced by frequency-selective fading channels is much more efficiently utilized for LMMSE equalization than the zero-forcing estimation. This is the consequence of avoiding the noise enhancement suffered by the ML estimation.

Iv. CONCLUSIONS

We proposed two frequency-domain orthogonal multiuser communications (FDOMC) systems: FDOMC-IT and FDOMC-DS over frequency-selective slowly fading channels which are not interfered by MA1 even in the asynchronous channels. In addition, the FDOMC-IT system is further not interfered by IS1 introduced by frequency-selective channels and can be optimally demodulated with low complexity. This is the consequence of concentrating the energies of independent information symbols into different sub-bands such that each sub-band experiences frequency-nonselective fading.

This approach does not e m the frequency diversity gain inherited from frequency-selective channels without efficiently establishing the correlation between information symbols.

By spreading the energies of independent information symbols into the whole frequency hand, the FDOMC-DS system utilizes the frequency diversity to combat channel fading impairment such that it outperforms the FDOMC-IT one with similar complexity. Moreover, it was verified by the numerical analysis that the LMMSE estimation is much more efficient than the ML estimation to utilize the frequency diversity by taking the noise enhancement effects into consideration.

These two systems represent different transmission policies over frequency-selective slowly fading channels. When the channel impulse response (CIR) is available to the transmitter, the optimal power allocation follows the water-pouring policy

[IO]

to maximize the channel capacity. In this case, the FDOMC-IT system which concentrates the symbol energies to some frequency sub-bands is easier to adopt its power allocation, signal constellations, and modulation types over different sub-bands according to the CIR to approach capacity. When the CIR is unknown to the transmitter, the FDOMC-DS system e m s the frequency diversity in its hest possib

Both systems efficiently mitigate the IS1 and MA1 with complexity similar to the conventional approaches and are flexible to accommodate variable rates services. Each user can either change the modulation schemes, signal constellations, and so

on or employ multiple signature sequences to fulfill high rate services without affecting the performance of the other users'. These valuable properties of the frequency-domain orthogonal multiuser communications systems suggest the applications to practical, multimedia, and advanced wireless communications systems.

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