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Proc IE E E 63rd Vehicular Technology Conference/ Spring, Melbourne, May 2006, pp. 1560-1564.

DOI: 10.1109/VETECS.2006.1683108

Channel Prediction Using Lumpable Finite-State

Markov Channels in OFDMA Systems

Tommy K , Chee, Cheng-Chew Lun

School of Electrical and Electronic Engineermg

The University of Adelaide, SA 5005, Australia

Email: {tchee»cclim}@eleceng.adelaide.edu.au.

J inho Choi

School of Electrical Engineering and Telecommunications

University of New South Wales, NSW 2052, Australia

E mail: j.choi@unsw.edu.au.

Abstract—In this paper, Baylelgh fadlug channel with an

OFDMA system is modeled as a toite-state Markov channel

(FSMC) by partitioning (lie received signal envelope into several

Intervals. With the aid of sub-band formation and property of

lumpabiUty, the size of feedback information can be reduced.

This approach Involves reducing an exponentially Increased states

of Markov channel to multiple luiupable Markov channels.

The conresponding state tronsifion probabilities and sleady-state

probabilities are used to predict channel states information in

multiple symbol durations ahead. Some simulation examples are

presented to Illustrate the capability of the liunpablc FSMC in

channel prcdicllon.

I, INTRODUCTION

The study of the finite-state Markov channel (FSMC) was

initiated by Gilbert [1] and Elliolt [2], They scarted wilh a two-

state Markov channel known as Gilbert-Blliott channel model.

In this channel model, each state conesponds to a specific

channel quality, which is either noiseless (represent? by T)

or noisy (represents by '0'}. However, two-state Oilbert-Elliott

cbanTiel is inadequate to describe the channel especially if

the channel quality varies dramatically. In [3], Wang and

Moayeri utilized the idea of finite-staie Markov mfldel to

partition the range of received signal-to-noise ratio (SNK) into

a finite number of intervals, hpnce each interval'forms the

state of the Markov chain. Wang and Chang further rejfined

the model by using analytical first-order $tatistics to obtain

model parameters [4]> The first-order Markovian assumption

for Rayleigh fading channel model is verified to be adequately

accurate compared to the higher order Markov models of much

higher complexity. In [5], Zhang and Kassam adopted the view

of [3] and develop a methodology to partition the received

SNR into a finite number of states based on the time duration

of each state.

Quantization of the received SNR of Rayleigh fading chan-

nel for FSMC model is a common practice. Some literatures

[6], [7] proposed to model die received signal envelope's

amplitude. Botli the maximum likelyhood sequence estimation

(MLSE ) and maximum a postenori (MAP) channel state

estimation over FSMC were studied in [7]. It sliows that

FSMC gives close to optimum eiror perfonnance for medium

SNR ranges. Ideally, the research on adaptive transmissions,

e.g. power allocation, modulation and coding, requires cliau-

nel stats iuformation (CSI) to be known perfectly to both

transmitter and receiver. Imperfect channel state information

may cause severe degradation in performance especially for

systems tfaat rely on adaptive techniques [8]. To realize die

potential of adaptive transmissions, channel state information

has to be reliably predicted in advance at the transmitter.

Conventionally, the feedback channel coosists of the exact

CSI from the previous transmission, which is a waste of

feedback bandwidth, If the tiae-varying fading channel is

modeled as a FSMC, the feedback is reduced to a quantized

vector which carries the instantaneous states of the curreut

channel. However, as the number of sutwiha^nels in a muld-

carrier system increases, the matrix size df FSMC grows

exponentially. In this paper, a no^iel approach is proposed to

reduce the size of the feedback iuformatioa. It is outlined as

a two-step method which involves the fonaation of sub-band

and lumpable FSMC. We extend the concept oflumpability to

reduce a generic Mw"state Markov channel to N/b parallel

(M-1)64- 1-state Markov chaanels while maintaining similar

behaviour. The corresponding state transition probabilities and

steady-state probabilities are tlien applied to predict chanoel

states information.

This paper is organized such that the general intfoducdoa

of PSMC is discussed in Section H-A and a brief introduction

on the fading model is ouffined in Section II-B. In Section

ID, the two-step approach is presented, followed by ftie

implementation of FSMC in channel prediction in Section IV.

Simulation results and discussion are given in Section V and

Section VI concludes this paper.

II. FINITB.STATE MARKOV CHANNEL

A. Markov Model

Let S = {si, S3,.,,, 8^,1} denote a finite set of states and

{Si}, t == 0,1,,.., be a constant Markov process, which

has stationary transitions [3]. Let ay tie the state transition

probability and ?r, be the steady-state probability, which can

be written as

_ J 'Pr(5n.i = s^ |5( = 5(), for |, - j\ ^ 1

^ 0, otherwise.

w< = Pr(5( = s,-),

(D

(2)

respectively, for all ij e {0,1,., .,M - 1}. With (1) and

(2), we can define a (M x M) transition matrix, A and a

(M x 1) steady-state probabilily vector, w, with the propeides

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that the sum of the elements on each row of A is equalled to

li i.e. £^iol ai,j = l> a.ndl ttle sum of all the elements in n-

is equalled to 1, i.c, E ^' 1-1 = 1.

In a typical mullipath propagation cliannel, the received

signal envelope can be modeled with Rayleigh distribution,

Let 7 denotes (he received signal-to-noise ratio (SNR) which

is proportional to the square of the signal envelope. One can

sliow that 7 is exponentially distributed [9] with its probability

density function given as

P(7)—H).

(3)

where 7 is the mean and also the standard deviation of -7. The

expected number of limes per second (also known as level

crossing rate} the received SNR '/ passes downward across a

quantized level • • /„, is given by

(4)

where fd is the maximum Doppler frequency, which is defined

as /d = vfc/c, in whicli v/c is tlie ratio of moving speed

of mobile terminal to speed of liglit and /c is the carrier

frequency. With (lie tliresholds of (lie quantized SNR levels

define as 0 = 70 < Ti < • • . < 7A/-1 = oo> the Rayleigli

fading channel is said to be in state sm if the received SNR

is witliin the interval of ['Ymi'ym+i). With the exponential

distributed SNR, llie stcady-siate probability for each state is

given as

r7m+l i..._ ( ^

-exp| --

An. 7~"''\ . 7/

:P(-^)-«P(-^

exp

'w^

The transition probability can then be approximated by

Mn+lT

ftm,m+l »

ttin,m-l w

I'm

r'

"I

m == 0,1, ...,M - 2

m = 1,2,..., M - 1

(5)

(6)

(7)

wliere T represenls the lime interval for each transmission

over the channel, i.e, symbol duration,

B. Fading Model

Consider iin OPDMA system of K users and N snb-carriers

witli a time-varying, frequency-selective fading channel, As-

sume that the sub-camer separation is smaller than the coher-

ent bandwidth, thus each sub-can'ier can be considered as a flat

fading sub-channel. Assume unity average transmission power,

the downlink received signal is modeled as Y = HX + u,

where H is the channel matrix, X and Y are the transmitted

and received signals, respectively, and n is additive white

Gaussian noise with a noise variance of <rj „ = o'^/Ar for

user k at sub-carrier n,

For an arbiirary user k, its chanuel vector is extracted

from the channel matrix H and is expressed as a vector,

h = l/hi/ia, • .., /iw) where (lie corresponding SNR for n"'

sub-chaiincl is expressed as |/tr>|2/cr2. This SNR is modeled as

an M-state Markov chain where |/in I/o- 6 {0,1,..., M— 1}

indicates the quantized SNR level.

Assume that at each time instant, no more than one sub-

channel shall vary no more than one quantized level at either

direction, By expanded process, eacli state is formed with all

N consecutive conditions in the channel vector, h. An MN-

state Markov channel is formed. Consider the Markov chain

with states sg = (00...0), si = (0,,. 01), ..,, SA/N_I =

(M- 1.. .M- 1), the state transition probability, Og,^, and

steady-state probabilily, Wgp are defined as

_ ;Pr?+i = s^\ St = s<), for |s; - s^l < 1

0, othcrwii

0,

7T,( = Pr(S( = Bi),

otherwise.

(9)

where transition from state s, to state Sj, which originally

occurs on N successive steps with the original chain, is

restricted to one step transition such that only one snb-channel

is susceptible lo state Iransilion. For example, the transition

probability from state so to state si can be approximated by

one step transition of sub-channel n"' from sub-state 0 to sub-

state 1 as all sub-states in state so and state si remain at O's

except one. Thus, the transiliou probability takes the form of

aso,si w <4"i)/Ar, Where Ar is the number of sub-channels and

a^ is the nth sub-channel's transition probability from state

0 to slate 1. The transition probability a^ belongs to the M-

state Markov chain, Other state transition probabilities can be

obtained in similar manner. On the other hand, the steady-state

probabilities, We = \ va,},fori = 0,l,...,MA'-l,ofthis

MN-slste Markov chain can be computed by

= ^.^x

= ^XTT^X

Vso

7T»

X 7T,

X 7T;

w

0 i

w

(10)

V,aM"-l

,(1) ^ ^(2) ^ ... ^ ^(N)

W'M-1 x WA/'-1 ><• • • >< ^A/^.1.

wliere 7r^"), 7r{"\ ,.., TT^LI are tlie n"' sub-cliannel's steady-

slate probabilities for state 0, slate 1,,.., and stale M -lot

the original chain, respectively,

Theorem 1: Given that ihe sum of all steady-state proba-

bilities for an arbitrary nth sub-channel, w<jn\ of an M-state

Markov cliain is 1, the sum of all steady-state probabilities,

n-s,, of the expanded MN-stnte Markov chain is given by

yA^-1^ ^^Af^i

i-ii=0 "sf — • • — ' •

For all sub-channels subject to variation, an (MN X M")

transition matrix. A,; = fag, ^}, for i,^' +0,1,... ,MN - 1,

is defined from (8). In any practical system, the SNR level

can be quaniized lo at leasl 2 levels whereas the number of

sub-channels of an OFDM system can go up to 512. Simply

relating these figures to M and N, the matrix dimension can

approach to 2512 w 1.34 x 101("i, which is too big to be

practical. We need to define a way to reduce the size of the

malnx in order to realize channel prediction at transmilter.

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III. LUMPABILITY

In this section, we reduce Uic size of Ihc transition mnirix

of the M'v-state Markov cliannel based on two steps:

A. Formation ofSnb-bands

For an OFDMA system witli N sub-channels, if one sub-

channel is qunnlized and modeled as an A//-staie Markov

channel, there are N parallel sub-dinnncls whicli are modeled

as M-stnte Markov chnnncls. If eacli of the sub-channels is

correlated to its adjacent sub-channels, we can model all of

tliem as an M^-slatc Mnrkov clianncl, wliicli generates n

transition matrix which is too large to be handled, Thus, we

propose to galhcr b sub-chnnnels as one sub-band, llie original

N sub-cliannels model with MN -state Markov chain cnn then

be transformed into Ar/& parallel sub-bands, eiicli with an M6-

state Markov chain. In other words, the (MN xMN} iransition

matrix, Ac, can be iransformed into N/b parallel (Mb x Mb)

transilion matrice, A(sx\ wliere ,r 6 {0,1, ,,, N/b-1}. Since

it becomes n parallel problem, the notation of A, will be used

tliroughout this paper for simplicity purpose.

The M''-state Markov chnin can be illustrated similar to

MA/-statc Markov chain except tlinl now eacli state only com-

prises a sequence of b components. Referring to Theorem 1,

by replacing N with b, tlie sum of all steady-stale probabilities

in w, still satisfy the property of Sjlo 7r»i =: !•

Consider a transition process from state so of & consecutive

sub-slaie O's to state si with a sub-slate 1 and (b - 1)

consecutive sub-state O's, Its state transition probability can

be written ns as,,gi w a^/fc, wlicre (4"i is the nth sub-

clinnnel's Iransition probability from stnte 0 to slale 1 of (lie

original M-state Markov chain and b is llie number of sub-

channels within a sub-band, Other slate transition probabilities

can be obtained in similar manner and hence they form the

transition matrices, A,,

The choice of b is decided by the frequency correlation

belween sub-cliannels. For a severe t'reqiiency-selective fading

environment, nil sub-channels are non-con-elated, which leads

b to take a \ a\ w as small as 1 because each sub-channel can be

treated as an independent clinonel. As the correlation between

sub-channels increases, the value of b increases too. When all

sub-channels are fully correlated, b is equalled to N because

all sub-channels are now identical and they shall be treated

as a single channel, In other words, fully correlated N sub-

channels leads the problem from a frequency-selective fading

environment back to a flat fading environment,

B. Liimpable Stales

The property of lumpabilily of a Markov chain lias been

previously addressed in [10], [11]. In essence, the properly

of lumpability means that there is a partition of aggregated

states of a Markov chain and yet the behavior of Ihe Markov

chain remains in a similar manner as far as the stale dynamics

and observation statistics are concerned. We first observe (lie

pattern of the A'/l)-state Markov cliain. 'nien, we present tlic

concept of lumpabilily to form an aggregated state L from state probabilities in ni must also be 1.

multiple atomic states of a finite-state Markov chain and obtain

ils eventual transition matrix.

Definition 1: Consider an A//('-state Markov chnin with

states s< = (s,is,2-"Sib) where each of the sub-states,

Sfk € {0,1,..., M - 1} for all i = 0, l,...,Mb - 1 and

k == 1,2,.,,,b. All Mb slates can then be divided into

Q = (M — l)b + 1 lumpable partitions, of which the <jth

partition is defined as

=<S.IE Sifc=^w=°'l'-'-'^i>-1}'

lk=l

(11)

where q =0,1,.. .,Q— 1,

Definition 2: Let q 6 {0,1,..., [M - !)(>} be the index of

lumpable partitions, it can be written in the form of

AJ -1

r/=0.no+l'ni+..,+(M-l).nA/_i = ^ ?'.nf

(=0

(12)

where n; is the number of occurrence of sub-state i m a given

stale for all i = 0,1,..., M - 1, which is subjected to a

constraint of

M-l

no -I- ni -f-... + n.A/-1 = ^ MI = &.

»=0

(13)

Hence, the number of aggregated slates in q11' partition can be

defined as

#L ,=£

b\

,(-'))«")l...«w I'

j "0"!ni"!-"nM-l

(14)

wliere j is tlic number of combinations for n.^> wliich satisfies

both (12) and (13),

Definition 3: For (lie pai'ti lions, L = {Lo, Li,..., liQ-i},

assume that (lie cliain before lumping has R = Mb states

and after lumping has Q = (M - l)fc +1 states, Let U be the

(Q x R) matrix wliosc • i11' row is the probability veclor having

equal componenls for stales in L, and 0 for tlie remaiuing

states. Let V be the (R x Q) matrix with tlie j11' column is a

vcclor with 1's in the componenls corresponding to states in

Lj and O's otlierwise. Given Uiat (lie iransition matrix of tlie

Mb-slMc Markov chain is A,, tlic lumped transition matrix

[10] is defined as

A, = UA,V,

(15)

where A( = [ai.,,ij, for x,y = 0,1,.. .,Q- i.

Dcfmilion 4: Given that the steady-slate vector of the Mb-

state Markov chain is ffg, die lumped sleady-state probability

vector is defined as

^(=7T ,V , (16)

where TT( = |TTL,]> for x = 0,1,..., Q — 1. Since the sum of

all steady-slate probabililies in IT, is 1, the sum of all steady-

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IV, CHANNEL PREDICTION

Channel prediciion can be divided into two main tasks as

estimates cliannel condition and predicls channel condition in

advance. Throughout this paper. Die receiver is assumed to

cslimaie the cliiinncl conditions perfectly and llius tlie received

SNR is used to eslimalc the average SNR for tlie timc-varying

channel and it can be modeled as an M-slateMnrkov channel,

At cnch transmission, ktl1 user's receiver lins perfect estimation

on (he corresponding SNR for all its sub-chiinnels, The clinnge

in average SNR, "j, will directly change (3), (4) and (5). Hence,

a new sel of transition inalrix and steady-staie probnbility

vector are required, There are two possible options lo obtain

the predicted state,

A, Feedback with States ofSnb-bands

In an N sub-carrier receiver, lliere are N/b sub-bands and

each sub-band has its (Mb x M1') transilion matrix, Aw,

and its (Mb x 1) stendy-state probability vector, TT^, for

x = l,2,...,Ar/b. L et S( = s, be (lie current state of

channel condilion where S( = (siis,2--'s;i,) and S(+i = sj

be one of the potendnl state at the next time frame with a

probability of

Pr(S<+i =s^) = Pr(S(+i = Sj|S< == s,). Pr(S( = s,)

= "S .,8^s,. (17)

At the next transmission time frame, llicre ore multiple

polential slnlcs for the cliannel condition, Tlnis two selection

policies are introduced to obtain the predicted state, Sy,

Policy 1; For all possible stales, n weighted average

is^ computed to oblnin llic corresponding predicted slalc,

s{WA^ = (suisu2"-sut)> of which each sub-stnie is

determined by

Su»=^^.Pr(S(+i=s,), V v=l,2,...,& (18)

3

where suu e {0,1,, .,,M - 1},

Policy 2: Among all possible slates, one parlicular state

siHf>^ with (lie highest probability, as obtained from (17),

Pr(S(+i=sJ >Pr(S(+i=s,), V; ^ u (19)

is recognized as the prcdicled state.

Will) llie prcdicled channel condition from Policies 1 and

2 given at stale su, the receiver is able to retrieve eacli sub-

channel condilion, i,e. snb-stnte suu, of one sub-biind in terms

of a finite integcr witliin the rnnge of 0 to M - 1. to otlier

words, the receiver requires to feedback N finilc integers,

wliicli is equivalent to log;; (M) bits per sub-camer,

D. Limited Feedback with Liimpable Slates

Based on the lumpability property presented in Section

III-B, an A/(>-state Markov channel can be reduced to an

\ (M - 1) &+Ij-state Markov channel, hence it is capable

of reducing the size of the feedback information.

Let S; = Sj be the current state of channel condition where

s, belongs to die partition

L,, At the next transmission time

4,6

-0- Full Fwdback

United FeeAack, bs4

—0— United Faodback. b-B

Uiriled Feedback, b-16

-a

e"

10 16

Nwriberol Finite Slal««,M

Fig. I, Number of feedback bin per sub-carrier required for each lime

frame, note (liiil coses with liimpnbitity fealnre may have less than 1 bit per

sub-cairier due to llielr feedbacks are per siib-baiid basis, wliich it of size fc.

frame, the channel condition is predicted to be S(+i == sj,

wliere Sj c Lr, witli n probability of

Pr(S<+i=s,) = Pr(§t+i=L ,)

Pr(St+i=L ,,|S(=L ,).Pr(St=L ,)

«L ,,L ^L ,. (20)

Since the reduced model of [(M - 1)6 + l]-state Markov

channel is a typical birili-death process, there are only three

siluations al (he next irausinissiou time frame: (i) transit to a

set of lumpable states of higher order, (ii) transit to a set of

lumpable states of lower order aud (iii) remain in the same

sel of lumpable states. A specific selection policy is npplied

to obtain Uw predicted partilion, Lu,

Policy 3: Among tlie three possible options, the partition

Li, will) the highest probability, as obtained from (20),

PI-(S(+I = Lu) > Pr(S(+i = Lr), Vr ^ u (21)

is expected lo be llie predicted partition.

With llic predicted partition from Policy 3 given as L||,

where this partition may consist of more than one atomic state,

tlic receiver is not able to retrieve each sub-cliannel condition

of one sub-band. However, [lie receiver can quaniize the sub-

band condition in terms of a finite integer witliin the range of

0 lo (M - l)b, where 0 indicates nil sub-channels within the

sub-bancl are cxpiiricncing worsl fading and (M-l)b indicntcs

otherwise. In other words, (lie receiver requires to feedback

less information at N/b finite intcgcrs, whicli is equivalent

to \ ogy\ (M - l)fc+l] bits per sub-band. Pig, 1 shows the

compnrison between number of feedback bits required for

the case with full feedback (without lumpnbility) and limited

feedback (wilh lumpabilily),

V. SIMULATION AND DISCUSSION

In the simulation study, the system is configured to have 512

sub-carriers and a received bandwidth of 5.12MHz, Sub-carrier

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m-30

£-35

1-40

[U

Q.

&-45

CP1 (Full feedback)

. x... CP2 (Full feedback)

-?- CP3 (Limited feedback)!

0.015

CP1 (FulUeedback)

..x.. CP2 (FulHeedba.ck)

-7- CP3 tLimited feedback)

345678

Prediction Horizon, no. of syrrbols ahead

Fig. 2. Prediction error of Ml>-state Markov channels for channel variation pig. 3. prediction error of 3(>-state Markov channel for prediclion horizon

in terms of f^T, where the size of sub-band is taken as 6 = 4. of 1 to 10 symbols ahead, where the size of sub-band is taken as b = 4.

spacing is determined as A/ = lOlcHz. Assume that carrier

frequency is 2.4GHz and cyclic prefix is 11/AS, thus symbol

period is defined as T= ^y+ 11 = 111/^s. Throughout this

section, the size of sub-band is b = 4.

By varying the speed of mobile users from lOkm/h to

lOOkm/h, the average prediction error of Mb -state Markov

channels for M = 2,3,4 is illustrated in F ig. 2. As fdT

increases, it is shown that the case with lumpable states

experiences up to 3dB lesser in error than both cases without

lumpable states. Since FSMC is an equivalent quantization of

the real channel, it is understandable that the larger the size

of M, the more accurate the real channel is represented by

the M-state Markov channel. In the contrary, the smaller the

size of M, the less room for error. Hence, it explains the

phenomena that higher M experiences more prediction error.

The prediction policies are put into a test to investigate the

performance in several symbol durations ahead. Fig.3 shows

one example of the prediction error of an 3i'-state Markov

channel for a prediction depth of 1 to 10 symbol durations

ahead for slow (fdT = 0.0025), moderate {fdT = 0.01) and

fast (fdT = 0.025) fading channels. In a microscopic point of

view, for each of the three fading channels, it can be observed

that the gap of prediction error between policy with lumpable

states and policies without lumpable states is increasing as

the depth of prediction increases. In a macroscopic point of

view, this gap increases too as the channel proceeds from slow

fading to fast fading. This result indicates that the proposed

method is still capable to provide reliable prediction with no

more than 3dB error separating every adjacent horizon within

the same policy. In general, the scheme with lumpable states

experiences less prediction error and requires less feedback

bits as compared to the schemes with only sub-band formation.

VI. CONCLUSION

This paper focuses on the reduced size of feedback informa-

don and its reliability in channel prediction. Conventionally,

channel state information is constructed with detailed current

channel conditions in Ae form of amplitude or SNR. In this

paper, the Rayleigh fading channel with an OFDMA system is

modeled as a finite-state Markov channel by parddoning the

received signal envelope into several quantized levels. With

the aid of sub-band formation and property of lumpability,

the size of feedback information is reduced from N\ ogy(M)

bits to ^ • log2[(M - 1)& + 1] bits, where N, M and & are

number of sub-carriers, number of states and size of sub-band,

respectively.

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