Exact Probability of Error of ST-Coded OFDMSys tem s w ith Freq u en cy Offs et in Flat

R ayleig h Fadin g Ch an n elsK . Sath an an th an an d C. R . N . A th au dag e

Abstract— In this paper, we analyze the error perfor-m anc e of sing le-inpu t-sing le-ou tpu t ( S IS O ) and spac e-tim ec od ed ( S T -c od ed ) orthog onal freq u enc y d iv ision m u ltiplex -ing ( O F D M ) system s with c arrier freq u enc y offset (C F O )in freq u enc y-fl at R ayleig h fad ing c hannels. E x ac t analytic alex pressions are d eriv ed for the sym b ol error prob ab ility(S E P ) for B P S K and Q P S K m od u lation sc hem es. T he anal-ysis presented has the following k ey featu res - it ex plic itlyac c ou nts for (i) the non-G au ssian natu re of the inter-c arrier-interferenc e (IC I) noise when the total nu m b er ofsu b c arriers is sm all, and (ii) the d epend enc y of the post-F F T sig nal and the IC I noise term s d u e to su b c arrierc orrelations. T hu s, the proposed m ethod prov id es ac c u rateS E P form u lae, ov erc om ing the inac c u rac ies inv olv ed inG au ssian approx im ation b ased m ethod s reported in liter-atu re. T he nu m eric al resu lts d em onstrate the sensitiv ity ofthe rec eiv er error perform anc e to C F O in S IS O -O F D M andS T -c od ed O F D M system s in fl at R ayleig h fad ing c hannels.A lso, the S E P resu lts presented c an b e interpreted as thelower-b ou nd of error perform anc e in a g eneral freq u enc y-selec tiv e R ayleig h fad ing c hannel.

I n d e x T e rm s— O F D M , spac e-tim e c od ing , inter-c arrier-interferenc e (IC I) , c arrier freq u enc y offset (C F O ) ,freq u enc y-fl at R ayleig h fad ing , sym b ol error prob ab ility(S E P ) .

I . I N TR ODU CTI ON

Orth og on al freq u en cy div is ion m u ltip lexin g (OFDM)h as recen tly becom e a k ey m odu lation tech n iq u e for h ig hdata rate w ireles s an d w irelin e ap p lication s . OFDM, als ok n ow n as m u lticarrier m odu lation (MCM), in corp oratesa larg e n u m ber of orth og on ally s elected s u bcarriers totran s m it a h ig h -data-rate s tream in p arallel in th e fre-q u en cy dom ain . Sp ectral efficien cy an d m u ltip ath im m u -n ity are tw o m ajor adv an tag es of th e OFDM tech n iq u e.It h as been adop ted for dig ital au dio broadcas tin g (DA B )an d dig ital terres trial telev is ion broadcas tin g (DV B ) [ 1 ] .OFDM h as als o been u s ed for dig ital s u bs criber loop(xDSL ) an d w ireles s local area n etw ork ( W L A N ) ap -p lication s [ 1 ]- [ 2 ] . On th e oth er h an d, s p ace-tim e (ST)com m u n ication s ys tem s in corp oratin g m u ltip le tran s m itan d/or receiv e an ten n as (MIMO s ys tem s ) p rov ide larg eg ain s in ach iev able data rates (ch an n el cap acity) [ 3 ] .

K . Sath an an th an is w ith th e Sch ool of Com p u ter Scien ce an d Soft-w are En g in eerin g , Facu lty of In form ation Tech n olog y, Mon as h U n i-v ers ity, Clayton , A u s tralia. Em ail: satha@csse.monash.edu.au

C. R . N . A th au dag e is w ith th e A R C Sp ecial R es earch Cen terfor U ltra-B roadban d In form ation N etw ork s ( A ffiliated Prog ram ofN ation al ICT A u s tralia), Dep artm en t of Electrical an d Electron icEn g in eerin g , Th e U n iv ers ity of Melbou rn e, V ictoria 3 0 1 0 , A u s tralia.E-m ail: cath@ee.mu.oz.au

Th u s , com bin in g MIMO tech n iq u es w ith OFDM, i.e.MIMO-OFDM, is a p rom is in g m arriag e for fu tu re fou rthg en eration ( 4 G ) w ireles s com m u n ication s ys tem s , e.g .G ig abit w ireles s [ 4 ] .

A m ajor draw back of OFDM is its relativ ely h ig hs en s itiv ity to carrier freq u en cy offs et (CFO) errors , com -p ared to a s in g le carrier s ys tem [ 5 ]- [ 7 ] . Th e freq u en cyoffs et error is cau s ed by th e m is alig n m en t in carrierfreq u en cies at th e receiv er du e to flu ctu ation s in receiv erR F os cillators or a ch an n el’s Dop p ler freq u en cy. Th is fre-q u en cy offs et can des troy th e s u bcarrier orth og on ality ofth e OFDM s ig n al, in trodu cin g in ter-carrier-in terferen ce(ICI). Th e ICI res u lts in s ev ere deg radation of th ebit-error-rate (B ER ) p erform an ce of OFDM s ys tem s .A lth ou g h freq u en cy s yn ch ron iz ation tech n iq u es u s in gbas eban d s ig n al p roces s in g can larg ely com p en s ate forCFO, an y res idu al error in freq u en cy s yn ch ron iz ation(CFO es tim ation error) con tribu tes to th e deg radation ofreceiv er p erform an ce.

I n th is p ap er w e in v es tig ate th e p robability of er-ror of s in g le-in p u t-s in g le-ou tp u t OFDM (SISO-OFDM)s ys tem s an d s p ace-tim e coded OFDM s ys tem s (ST-coded OFDM) w ith CFO in frequency-flat Rayleigh

fad ing channels . I n fin ite s eries bas ed exact exp res s ion sfor s ym bol error p robability are deriv ed for B PSK an dQ PSK con s tellation s w ith n o as s u m p tion s m ade on th es tatis tical ch aracteris tic of th e ICI n ois e du e to CFO. Its h ou ld be n oted th at th e m ajority of th e w ork av ailablein th e literatu re on th is s u bject [ 8 ]- [ 1 0 ] u s es a G au s s ianas s u m p tion for ICI n ois e an d th erefore is n ot exact. Th eexact exp res s ion s deriv ed in th is p ap er are exten s ion s ofth e bas ic res u lts p res en ted in [ 6 ] for th e A W G N cas e tofadin g ch an n els an d ST-coded s ys tem s .

Th e au th ors p rev iou s w ork [ 1 0 ] on p robability of erroran alys is w as p erform ed for SISO-OFDM an d ST-codedOFDM s ys tem s for frequency-s electiv e Rayleigh fad ing

channels . Th e ch an n el w as as s u m ed to be com p letelyfreq u en cy-s electiv e (i.e. s u bcarrier ch an n el res p on s es arei.i.d.) , an d als o th e ICI n ois e w as as s u m ed to be G au s -s ian dis tribu ted. Th e in dep en den t (from s ig n al) G au s s ianas s u m p tion for ICI n ois e leads to v alid p robability oferror exp res s ion s for a totally freq u en cy-s electiv e ch an -n el [ 1 0 ] . H ow ev er, w h en th e ch an n el becom es les s an dles s freq u en cy s electiv e (freq u en cy-flat cas e is th e lim itof th is ) th e dep en den cy betw een th e p os t-FFT s ig n alan d th e ICI n ois e in creas es , m ak in g it n eces s ary to tak e

0 -7 8 0 3 -9 0 0 7 -5 /0 5 /$ 2 0 .0 0 c©2 0 0 5 IEEE 2 1 A u s CTW 2 0 0 5

this dependency into account for an accurate probabilityof error analysis. The method described in this paperexplicitly accounts for the frequency-fl at nature of thechannel and the non-Gaussian nature of the ICI noise.Exact expressions for probability of symbol error areprovided for BPSK and QPSK modulation schemes.

This paper is organized as follows. Section II givesthe error analysis for the SISO-OFDM case. The erroranalysis for the ST-coded OFDM case is given in SectionIII. As a reference technique, error analysis with aGaussian approximation for ICI noise is given in SectionIV. Numerical results are given in Section V. Finally,some conclusions are given in Section VI.

II. ERROR ANALY SIS FOR SISO-OFDM SY STEMS

In this section we provide the probability of erroranalysis for a conventional SISO-OFDM system withCFO in a fl at Rayleigh fading channel. The results forthe AWGN channel case are given fi rst as a prelude tothe Rayleigh fading case.

A. AWGN Channel

The signal for the k-th subchannel after the receiverdiscrete Fourier transform (DFT) processing can bewritten as [6]

Rk = S0Xk +N−1∑

l=0,l 6=k

Sl−kXl + Wk (1)

where Xk is a modulated data symbol on k-th subcarrierand Wk is a complex Gaussian noise sample. N is thenumber of subcarriers in OFDM. The sequence Sk isknown as the ICI coeffi cients and is given by

Sk =sin π(k + ε)

N sin πN

(k + ε)e x p [j π(k + ε)] (2)

where ε is the normalized frequency offset defi ned asa ratio between the frequency offset and the subcarrierspacing. We assume that the common phase rotation,ej πε , is removed before data detection at the receiver.Therefore, the decision variable can be expressed as

Yk = |S0|Xk + e−j πε

N−1∑

l=0,l 6=k

Sl−kXl + Wk

. (3)

1 ) B P S K M o dulatio n: We use the method in [6] tocalculate the exact probability of symbol error. For BPSKmodulation, Xk ∈ {1,−1} and it will be suffi cient toconsider the real part of (3). The characteristic function(CHF) of < (Yk) can be expressed as [6]

φ(ω) = ej ω Xk|S0|−ω2σ2

2

N−1∏

l=0, l 6=k

c o s(ω<{e−j πε Sl−k})

(4)where <{z} denotes the real part of z and σ2 is the noisevariance. Without loss of generality, we consider the fi rstsubcarrier and the transmitted symbol X0 = 1. Now, adecision error occurs if the real part of Y0 (3) is less

than zero. Thus, the probability of symbol error is givenby (5) (shown on top of the next page) [6] where ω0 =2π/ T , T is a parameter governing the sampling rate inthe frequency domain and γ is the average signal-to-noise ratio (SNR) given by γ = E{|Xk|

2}/ E{|Wk|2} =

σ2X / σ2

W . The index set No = {1, 3, 5, . . . , ∞ } is the setof all positive odd integers. We rewrite (5) as

Pb p sk(γ) =1

2−

∑

n∈No

αne−ω20

n2

2γ (6)

where

αn =2 sin [n ω0|S0|]

n π

N−1∏

l=1

c o s(n ω0<{e−j πε Sl}). (7)

2 ) Q P S K M o dulatio n: We consider the QPSK signalas two orthogonal BPSK signals. Therefore, the symbolerror probability can be expressed as in (8) (shown onthe next page).

B . F requency-F lat Rayleigh F ading Channel

In a frequency-fl at Rayleigh fading channel the post-FFT received signal for the kth subcarrier can be writtenas

Rk = S0HXk +N−1∑

l=0,l 6=k

Sl−kHXl + Wk (9)

where H is the frequency response (common for all sub-carriers) of the fl at Rayleigh fading channel. The post-equalized (phase-rotation removed) decision variable canbe expressed as

Yk = |S0||H|2Xk+e−j πε

|H|2N−1∑

l=0,l 6=k

Sl−kXl + HWk

(10)where (.) is complex-conjugate operator. The averagesymbol error probability can be expressed as

Pa v g =

∫ ∞

0

Ps(γ)pγ(γ)d γ (11)

where Ps(γ) and pγ(γ) are the symbol error prob-ability and the probability density function (pdf), re-spectively, for the instantaneous SNR given by γ =|S0|

2|H|2σ2X / σ2

W . The pdf of γ is a Chi-squared dis-tribution of degree 2 (equivalent to an exponential dis-tribution), and is given by [10], [12]

pγ(γ) =1

γe x p

(

−γ

γ

)

(12)

where γ is the average SNR given by γ = E{|γ|2} =|S0|

2σ2X / σ2

W . The average channel power is assumed tobe unity, i.e. E{|H|2} = 1.

22

Pbpsk(γ) =1

2−∑

n∈No

2 sin [nω0|S0|]e−ω20n2

2γ

nπ

N−1∏

l=1

cos(nω0<{e−jπεSl}) (5)

Pq psk(γ) =1 − (1 − Pbpsk(γ))2

=3

4−∑

n∈No

αne−ω20n2

2γ −∑

n∈No

∑

m∈No

αnαme−ω20(n2+m2)

2γ (8)

1) BPSK Modulation: We can express the average biterror probability of BPSK using (6) and (12) as

Pavg =

∫ ∞

0

(

1

2−∑

n∈No

αne−ω20n2

2γ

)

1

γexp

(−γ

γ

)

dγ

=1

2− 1

γ

∑

n∈No

αnIn (13)

where

In =

∫ ∞

0

exp

(

−ω20n2

2γ− γ

γ

)

dγ. (14)

Using the results in ( [13]-3.371.9), (14) simplifi es to

In =√

2ω20n2γK1

√

2ω20n2

γ

(15)

where, Kν (x) is the modifi ed Bessel function of the sec-ond kind [13]. The parameter ν is an integer. Therefore,the average symbol error probability becomes

Pavg =1

2− β

∑

n∈No

nαnK1 (βn) (16)

where β =√

2ω20

γ.

2) QPSK Modulation: Similar to BPSK, we canexpress the average symbol error probability of QPSKusing (8) and (12) as in (17).

III. ERROR ANALYSIS FOR ST-CODED OFDMSYSTEMS

In this section we analyze the effect of the frequencyoffset in a space-time coded OFDM system. In particular,we consider an Alamouti coded OFDM system [10]- [11]incorporating two transmit antennas and a single receiveantenna as shown in Fig. III. For the 2×1 MISO-OFDMsystem (Fig. III), the post-FFT signal Rk at the receiveantenna for the kth OFDM subcarrier becomes

Rk =1√2

1∑

m=0

S0HmXmk

+1√2

1∑

m=0

N−1∑

l=0,l 6=k

Sl−kHmXml

+ Wk (18)

where, Xmk is the transmit data symbol from the mthtransmit antenna on the kth subcarrier. The channel

response from the mth transmit antenna to the receiveantenna (for all the subcarriers) is given by Hm. Thefactor 1/

√2 scales the total transmit power from the 2

antennas to σ2X . The above (18) can be written as

Rk =1√2

1∑

m=0

S0HmXmk + Zk (19)

where, Zk = Ik + Wk is the total noise term consistingof the interference term Ik and the AWGN term Wk. Theinterference term Ik that accounts for both inter-antennaand inter-carrier interference is given by

Ik =1√2

1∑

m=0

Hm

N−1∑

l=0,l 6=k

Sl−kXml

. (20)

Alamouti coding can be applied to each subcarrier in-dependently to achieve space-time diversity. Let the twosymbols transmitted simultaneously from antennas 0 and1 on the kth subcarrier at symbol time t = 0 be X0k andX1k, respectively. According to Alamouti coding [11],at symbol time t = 1, antennas 0 and 1 transmit thesymbols −X1k and X0k, respectively. Let the frequencyresponse of the channel 0 and 1 (for all the subcarriers)at the symbol time t to be H

(t)0 and H

(t)1 , respectively.

An assumption made in Alamouti coding is that H(t)0 and

H(t)1 are constant over two consecutive symbol periods,

that is

H(2t)m (k) = H(2t+ 1)

m (k), m = 0 , 1. (21)

With a single receive antenna, the post-FFT receivesignals R

(0)k and R

(1)k over the two consecutive symbol

periods t = 0 and t = 1 for the kth subcarrier can beobtained using (19) as[

R(0)k

R(1)k

]

=1√2

[

S0H0 S0H1

S0H1 −S0H0

] [

X0k

X1k

]

+

[

Z(0)k

Z(1)k

]

(22)where, H

(0)0 = H

(1)0 = H0 and H

(0)1 = H

(1)1 = H1. The

total noise terms including both ICI interference noiseand AWGN channel noise are given by Z

(0)k = I

(0)k +

W(0)k and Z

(1)k = I

(1)k + W

(1)k . Let H0 = ejπε H0 and

H1 = ejπε H1 be the effective channel resp onses thatinclude both the actual channel response and the phase-rotation due to the self-interference term S0 = |S0|ejπε .We assume that the effective channels H0 and H1 areknown, i.e. perfectly estimated, for Alamouti decoding.

23

Pavg =

∫ ∞

0

(

3

4−∑

n∈No

αne−ω20n2

2γ −∑

n∈No

∑

m∈No

αnαme−ω20(n2+m2)

2γ

)

1

γexp

(−γ

γ

)

dγ

=3

4− β

∑

n∈No

nαnK1 (βn) − β∑

n∈No

∑

m∈No

√

n2 + m2αnαmK1

(

β√

n2 + m2)

(17)

OFDMMod 1

TransmitSymbols Symbols

Receive

Tx 0

Mod 2OFDM

OFDMDemod

Rx

Tx 1ST

EncoderST

DecoderChannel 1

Channel 0

1: Block schematic of an Alamouti coded OFDM system incorporating two transmit antennas and one receive antenna.

Thus, the decision variables Y0k (for X0k) and Y1k (forX1k) derivable by combing the receive signals R

(0)k and

R(1)k (Alamouti decoding) becomes

Y0k = e−jπεH0R(0)k + ejπεH1R

(1)k (23)

Y1k = e−jπεH1R(0)k − ejπεH0R

(1)k . (24)

We choose the symbol X0k, i.e. decision variable Y0k,to analyze the effect of ICI noise on the performance ofthe ST-coded OFDM system. Substitution of (22) in (23)gives

Y0k =1√2|S0|

(

|H0|2 + |H1|2)

X0k

+ e−jπεH0

(

I(0)k + W

(0)k

)

+ ejπεH1

(

I(1)k + W

(1)k

)

. (25)

By substituting the ICI terms I(0)k and I

(1)k from (20),

the post-equalized decision variable Y0k at the output ofthe ST-decoder (Fig. III) becomes

Y0k =1√2|S0|

(

|H0|2 + |H1|2)

X0k

+1√2e−jπε

(

|H0|2 + |H1|2)

N−1∑

l=0,l 6=k

Sl−kX0l

+ e−jπεH0W(0)k + ejπεH1Wk

(1)(26)

where both H0 and H1 are Rayleigh random variables.In deriving (26) Alamouti coding was assumed on eachsubcarrier thus X

(1)0l = −X

(0)1l = −X1l and X

(1)1l =

X(0)0l = X0l. Also, ICI coeffi cients have the property

Sk = |Sk|ejπε, i.e. all Sk have the same phase-angleaccording to (2). The instantaneous post-equalized SNR(for a given channel realization) of space-time coded

OFDM becomes γ =|S0|

2(|H0|2+|H1|

2)σ2X

2σ2W

. Therefore,γ is a Chi-squared random variable of degree 4 and thecorresponding pdf is given by [10], [12]

pγ(γ) =4γ

γ2 exp

(−2γ

γ

)

(27)

where γ is the average SNR given by γ =|S0|

2σ2X

σ2W

.

A. BPSK Modulation

We can express the average bit error probability ofBPSK using (6) and (27) as

Pavg =

∫ ∞

0

(

1

2−∑

n∈No

αne−ω20n2

2γ

)

4γ

γ2 exp

(−2γ

γ

)

dγ

=1

2− 4

γ2

∑

n∈No

αnI′

n (28)

where

I′

n =

∫ ∞

0

γ exp

(

−ω20n2

2γ− γ

2γ

)

dγ. (29)

Using the results in ( [13]-3.371.9), (29) simplifi es to

I′

n =ω2

0n2γ

2K2

2

√

ω20n2

γ

. (30)

Therefore, the average symbol error probability becomes

Pavg =1

2− β2

∑

n∈No

n2αnK2

(

βn√

2)

. (31)

B. QPSK Modulation

Similar to BPSK, we can express the average symbolerror probability of QPSK using (8) and (27) as in (32)(shown on top of the next page).

IV. ERROR ANALYSIS WITH GAUSSIANAPPROX IMATION

In this section we provide closed-form expressionsfor the probability of error based on a Gaussianapproximation for the ICI noise as a reference technique.Both SISO-OFDM and ST-Coded OFDM cases are an-alyzed in a frequency-fl at Rayleigh fading channel. Theaccuracy of the proposed method is highlighted againstthis reference method. The Gaussian approximationmethod have two major error sources, i.e., the resultingexpressions for SEP are not exact expressions. These are:

24

Pavg =

∫ ∞

0

(

3

4−∑

n∈No

αne−ω20n2

2γ −∑

n∈No

∑

m∈No

αnαme−ω20(n2+m2)

2γ

)

4γ

γ2 exp

(−2γ

γ

)

dγ

=3

4− β2

∑

n∈No

n2αnK2(βn√

2) − β2∑

n∈No

∑

m∈No

(n2 + m2)αnαmK2

(

β√

2(n2 + m2))

(32)

1) The Gaussian approximation for the ICI term itselfis not totally accurate. According to the CentralLimit Theorem, only the limiting case (N → ∞) isperfectly Gaussian.

2) The ICI term (total power) is considered an inde-pendent random variable with respect to the signalterm. This is not true as the channel variable |H|introduces some dependency between the ICI termand the signal term (see (10) for the SISO-OFDMcase and (26) for the ST-Coded OFDM case).

For large N , the error due to 2) can be more predominantthan that due to 1). It is important to notice that in theproposed method both of the above error sources areeliminated, as exact expressions for the SEP are used.

A. SISO-OFDM System

The post-FFT received signal for the SISO-OFDMcase is given by (9). The average power of the sig-nal, ICI noise, and AWGN noise in (9) becomes|S0|2σ2

X , (1 − |S0|2)σ2X , and σ2

W , respectively. Notethat

∑N−1k=0 |Sk|2 = 1 and thus

∑N−1l=0,l 6=k |Sl−k|2 =

1−|S0|2. Therefore, the signal-to-interference-and-noiseratio (SINR), Λ, can be expressed as

Λ(ε, ρ ) =|S0|2γ

(1 − |S0|2)γ + 1(33)

where Λ = Λ(ε, ρ ) is a function of the frequency offset(ε) and the average channel SNR γ = σ2

X/σ2W . The

average symbol-error-probability for BPSK modulationin a Rayleigh fading channel can thus be given as [10],[12]

Pavg:b p sk =1

2

(

1 −√

Λ

Λ + 1

)

. (34)

For QPSK modulation the average symbol-error-probability can be given as [10]

Pavg:q p sk =3

4−√

Λ

Λ + 2

(

1 − 1

πta n−1

{√

Λ + 2

Λ

} )

.

(35)

B. ST -Coded OFDM System

Using the results of [10] the average symbol-error-probability with a Gaussian approximation for the ICInoise for BPSK modulation can be given as

Pavg:b p sk =1

2

(

1 − Λ + 3

Λ + 2

√

Λ

Λ + 2

)

(36)

where Λ is given by (33). For QPSK modulation theaverage symbol-error-probability can be given as (37)(shown on top of the next page) [10].

V. NUMERICAL RESULTS

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 501 0

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No

O F D M (ε=0)

O F D M (ε=0.15):Exact

O F D M (ε=0.15):Simulation

O F D M (ε=0.15):Approximation

PSfrag re p lac e m e n ts

SE

P

2: Symbol error probability (SEP) of OFDM with frequency offset ina fl at R ayleig h fad ing channel for B PSK mod ulation.

Simulations were carried out to d emonstrate the ac-curacy of the ex pressions d eriv ed for the probability oferror. T he SEP was ev aluated throug h simulations for anOFDM system using B PSK and Q PSK mod ulation forN = 128 and ε = 0 .15 . T he constant phase rotationd ue to C FO was remov ed at the receiv er before symbold etection. Fig . 2 shows the SEP of the B PSK scheme ina fl at R ayleig h fad ing channel. T he simulation resultsperfectly ag ree with the proposed ex act method (1 6 )while the G aussian approx imation method (3 4 ) showsmuch d eg rad ation in SEP and a premature error fl oor.Fig . 2 also rev eals that the SN R d eg rad ation d ue to ε of0 .1 5 is 2 d B at SEP of 10

−5. However, the Gaussianap p rox im ation shows very p essim istic results with anerror fl oor of ap p rox im ately 2×1 0

−2. I t should b e notedthat the p rop osed ex ac t c alc ulation m ethod s are b ased onevaluating the sum of infi nite series. However, the seriesc an b e trunc ated b ased on the req uired ac c urac y . W ehave c alc ulated the num eric al values of the series for anac c urac y of 1 2 sig nifi c ant d ig its.

F ig . 3 shows the S E P of the Q P S K sc hem e in a fl atR ay leig h fad ing c hannel. A g ain, the p rop osed tec hniq ueis p rec ise, while the Gaussian ap p rox im ation is g reatlyinac c urate. However, the Gaussian ap p rox im ation ag reeswith the sim ulation results at low S N R . T his is d ue tothe d om inanc e of A W GN noise over IC I , and thus theassum p tions on IC I noise is not sig nifi c ant. T herefore,in the very low S N R reg ion the Gaussian ap p rox im ationis ac c ep tab le.

F ig . 4 and F ig . 5 show the error p erform anc e of S T -

2 5

Pavg:qpsk =3

4−

1

π(Λ + 4)

[

Λ

(Λ + 2)+ (Λ + 6 )

√

Λ

Λ + 4

(

π − ta n −1

√

Λ + 4

Λ

) ]

(37)

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 501 0

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No

O F D M (ε=0)

O F D M (ε=0.15):Exact

O F D M (ε=0.15):Simulation

O F D M (ε=0.15):Approximation

PSfrag re p lac e m e n ts

SE

P

3: Symbol error probability (SEP) of OFDM with frequency offset ina fl at R ayleig h fad ing channel for Q PSK mod ulation.

0 5 1 0 1 5 2 0 2 5 3 01 0

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No

ST−OFDM (ε=0)

S T − O F D M (ε=0.15):Exact

S T − O F D M (ε=0.15):Simulation

ST − O F D M (ε=0.15):Approximation

PSfrag re p lac e m e n ts

SE

P

4: Symbol error probability (SEP) of ST-coded OFDM with frequencyoffs et in a fl at R ayleig h fading channel for B PSK modulation.

coded B PSK and Q PSK s chemes in a fl at R ayleig hfading channel. The s imulation res ults perfectly ag reewith our ex act method while the G aus s ian approx imationmethod dev iates from s imulation res ults s ig nifi cantly,ex hibiting a premature error fl oor. Fig . 4 als o s hows thatthe SN R deg radation for B PSK due to ε of 0 .1 5 is 4dB at a SEP of 10

−5. In the SISO-OFDM cas e, thisdeg radation was only 2 dB (Fig . 2 ). This implies thatST-OFDM is more s ens itiv e to C FO than OFDM.

V I . C ON C L U SI ON S

In this paper, we pres ented an ex act method of analyt-ically ev aluating the error performance of SISO-OFDMand ST-coded OFDM s ys tems with carrier frequencyoffs et in frequency-fl at R ayleig h fading channels . Themethod is ex act due to (i) the non-G aus s ian nature ofthe IC I nois e, and (ii) the dependency of the s ig nal and

0 5 1 0 1 5 2 0 2 5 3 0 3 51 0

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No

S T − O F D M (ε=0)

S T − O F D M (ε=0.15):Exact

S T − O F D M (ε=0.15):Simulation

ST − O F D M (ε=0.15):Approximation

PSfrag re p lac e m e n ts

SE

P

5: Symbol error probability (SEP) of ST-Coded OFDM with frequencyoffs et in a fl at Rayleig h fading channel for Q PSK modulation.

ICI nois e term due to s ubcarrier correlations are bothaccounted for in the propos ed method. Comparis on withs imulation res ults confi rms the accuracy of the deriv edformulae for s ymbol error probabilities . W e as s umed thatcons tant phas e rotation was remov ed at the receiv er byproper channel es timation. H owev er, the effect of phas e-rotation can s till be tak en into account in this method byappropriately s electing the characteris tic function. This isnot pos s ible in the conv entional G aus s ian approx imationmethod s ince it s imply us es the av erag e s ig nal power andICI nois e power.

V I I . A CK N OW L EDG EMEN T

The s econd author, C. R. N . A thaudag e, would lik eto ack nowledg e the s upport by the A us tralian Res earchCouncil (A RC).

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