Ici Distribution

1488 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007

Distribution of the ICI Term inPhase Noise Impaired OFDM SystemsTim C. W. Schenk, Member, IEEE, Remco W. van der Hofstad, Erik R. Fledderus,

and Peter F. M. Smulders, Senior Member, IEEE

Abstract Orthogonality between the subcarriers of an orthog-onal frequency division multiplexing (OFDM) system is affectedby phase noise, which causes inter-carrier interference (ICI). Thedistribution of this interference term is studied in this paper. Thedistribution of the ICI for large number of carriers is derived andit is shown that the complex Gaussian approximation, generallyapplied in previous literature, is not valid and that the ICIterm exhibits thicker tails. An analysis of the tail probabilitiesconfirms these finding and shows that bit-error probabilities areseverely underestimated when the Gaussian approximation forthe ICI term is used, leading to too optimistic design criteria.Results from a simulation study confirm the analytical findingsand show the validity of the limit distribution, obtained underthe assumption of a large number of subcarriers, already for amodest number of subcarriers.

Index Terms Inter-carrier interference, limit distribution, or-thogonal frequency division multiplexing, phase noise, stochasticintegral, tail probabilities.

I. INTRODUCTION

THE multicarrier technique orthogonal frequency divisionmultiplexing (OFDM) [1] is chosen as basis for severalwireless systems, because of its high spectral efficiency andability to divide a dispersive multipath channel into parallelfrequency flat subchannels. Furthermore, most systems applya guard interval, a cyclic extension of the OFDM symbols,to protect the detected symbol against time delayed versionsof the previous symbol, which could cause inter-symbolinterference (ISI) [2]. OFDM is standardized for applicationin, amongst others, wireless local-area-networks (WLAN)[3],wireless metropolitan networks (WMAN), e.g. WiMAX [4],digital audio broadcasting (DAB) [5] and digital video broad-casting (DVB) [6].

Manuscript received August 4, 2005; revised March 14, 2006; acceptedJuly 21, 2006. The associate editor coordinating the review of this paper andapproving it for publication was M. Uysal. The work of T. C. W. Schenk, E.R. Fledderus, and P. F. M. Smulders was sponsored by the Dutch cooperativeresearch project B4 BroadBand Radio@Hand (BTS01063). The work of R.W. van de Hofstad was supported in part by the Netherlands Organisation forScientific Research (NWO).

T. Schenk was with the Telecommunications Technology and Electro-magnetics Group, Eindhoven University of Technology, Eindhoven, P. O.Box 513, 5600 MB Eindhoven, The Netherlands. He is now with PhilipsResearch Laboratories, Connectivity Systems and Networks Group, 5656 AE,Eindhoven, The Netherlands (e-mail: [email protected]).

R. van der Hofstad is with the Department of Mathematics and ComputerScience, Eindhoven University of Technology, P. O. Box 513, 5600 MBEindhoven, The Netherlands (e-mail: [email protected]).

E. Fledderus, and P. Smulders are with the Telecommunications Tech-nology and Electromagnetics Group, Eindhoven University of Technology,Eindhoven, P. O. Box 513, 5600 MB Eindhoven, The Netherlands (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TWC.2007.05601.

The high spectral efficiency of OFDM is achieved by ap-plying partly overlapping spectra for the different subcarriers.When the system is perfectly synchronized, these subcarriersare orthogonal. However, when due to imperfect local oscilla-tors (LO) at either transmitter (TX) or receiver (RX) side of thesystem carrier frequency offset or phase noise (PN) occurs, theorthogonality between the subcarriers is lost and inter-carrierinterference (ICI) occurs. These imperfections in the LOs willmore and more appear to be a factor limiting performance ofOFDM systems, when low-cost implementations or systemswith high carrier frequencies are regarded, since it is in thosecases harder to produce an oscillator with sufficient stability.Therefore, it is important to understand the influence of imper-fect oscillators, i.e. phase noise, on the system performance.

The influence of PN on the performance of an OFDMsystem has been regarded in several publications, see e.g.[7][15]. They all show that the influence can be split intoa multiplicative part, which is common to all subcarriers andtherefore often referred to as common phase error (CPE),and an additive part, which is often referred to as inter-carrier interference (ICI). Although the CPE is identified as themain performance limiting factor for coherent detection basedreceivers, many adequate correction approaches for the CPEhave been proposed previously, see, e.g. [8], [16], [17]. Someapproaches for correction of the ICI have been proposed in[18], [19], which, however, require high signal-to-noise ratios(SNRs) to achieve reasonable performance and have a highcomplexity.

Thus, since any practical coherent OFDM receiver wouldhave some kind of CPE correction, the main performancelimiting factor in PN impaired systems is the ICI term. Theproperties of the ICI term are studied by several authors, seee.g. [7][11]. Many of these papers, implicitly or explicitly,assume the ICI term is complex Gaussian distributed, due tothe central-limit-theorem. This is, however, as also noted bythe authors of [14], [15], an approximation which is only validfor some combinations of PN and subcarrier spacings. Theauthors of [14] show that the complex Gaussian approximationonly holds for fast PN, i.e., PN which changes fast comparedto the symbol time, and that error probabilities calculatedunder this Gaussian assumption are very inaccurate for othertypes of PN. The authors of [15] confirm these findings withsimulation results from Monte Carlo simulations, which showthat good agreement with the Gaussian distribution is onlyachieved when the ratio of the 3 dB bandwidth of the power-spectral-density of the LO spectrum and the subcarrier spacing

1536-1276/07$25.00 c 2007 IEEE

SCHENK et al.: DISTRIBUTION OF THE ICI TERM IN PHASE NOISE IMPAIRED OFDM SYSTEMS 1489

approaches 1. It is noted, however, that these values of PNcorrespond to such a severe system performance degradation,that for practical systems this ratio always will be chosen tobe much smaller than 1.

In this contribution we extend previous work by analyticallystudying the distribution of the ICI term due to PN in OFDMsystems. We will derive a limit distribution for the ICI term,which will be shown to exhibit thicker tails than the complexGaussian distribution with the same variance. We show thatprevious approaches, based on the Gaussian assumptions,significantly underestimate the tail probabilities of the ICIterm, and thus underestimate the bit-error probabilities (BEP)of a system experiencing PN.

The paper is organized as follows. Section II introduces thesystem model and shows the influence of PN. In Section IIIwe examine the distribution and tails of the ICI term for largenumber of carriers. Subsequently, we study the approximatedistribution for small PN values. Additionally, simulationresults are reported in this section to confirm the analyticalfindings. Finally, conclusions are drawn in Section IV.

II. OFDM SYSTEM EXPERIENCING PNIn this section we first derive the baseband system model

for a system experiencing PN. Subsequently, we show howthe induced ICI influences the BEP of such a system.

A. System ModelA transmitted OFDM signal is formed by applying the

inverse discrete Fourier transformation (IDFT) to the complexdata symbols and by adding a cyclic prefix to it by placing acopy of the last NG samples in front of the symbol. This canbe written as follows [1], [2]:

um,n =

1N

N1k=0

sm,k exp(i 2(nNG)kN

)for NG n Ntot 1

um,N+nfor 0 n NG 1,

(1)

where we introducem OFDM symbol indexn sample index within the OFDM symbol,

{0,1,...,Ntot-1}N number of subcarriersk subcarrier index, {0,1,...,N}um,n transmitted complex data signal during the

nth sample of the mth symbolNG the guard interval lengthi the imaginary unit, i2 = 1Ntot OFDM symbol length N +NGMoreover, the transmitted complex data symbol during the

mth symbol on the kth carrier is given by sm,k, which can berewritten as sm,k = Rm,k+iIm,k. We define R and I to be realzero-mean variables, which depend on the modulation schemeused for transmission. For instance, for BPSK modulationR {1, 1} with equal probability and I = 0. For QPSKor 4-QAM modulation R { 1

2, 1

2} and I { 1

2, 1

2},

again with equal probability of occurrence.

The signal u is then up converted to radio frequency (RF)fRF using the TX LO. Ideally this LO exhibits a spectrumwhich is given by a spectral line at fRF and can be expressedby exp(i2fRFt). However, any practical LO will exhibit arandom phase process, i.e., phase noise, and it is thus modeledas exp(i(2fRFt+TX(t))), where TX(t) denotes the transmitterPN process, which is a function of time t.

At the RX side, the signal is down converted to baseband,by multiplication with the receiver local oscillator processexp(i(2fRFt+RX(t))), where RX(t) denotes the receiver PNprocess. Since we regard a sampled baseband model here, wefurther regard TX,n and RX,n, which are the sampled versionsof the TX and RX phase noise, respectively.

When we define n = mNtot + n, the received basebandsignal with additive white Gaussian noise (AWGN) is givenby

rm,n = um,n exp (i(2fRFnT + TX,n)) exp (i(2fRFnT + RX,n)) + wm,n

= um,n exp (in) + wm,n, (2)where the combined TX and RX phase noise term is given byn = TX,n RX,n and wm,n denotes the complex Gaussian,zero-mean additive receiver noise with a variance of 2w.Here we assumed the system does not experience a multipathchannel.

Subsequently, the cyclic prefix is removed from rm,n bydisregarding the first NG samples of each symbol and thefrequency domain complex data symbols are then found byapplying the discrete Fourier transform (DFT) to the remainingsamples, and are given by

xm,k =

1N

Ntot1n=NG

rm,n exp(i2(nNG)k

N

). (3)

Using (1) and (2), equation (3) can be rewritten as

xm,k = gm,0sm,k +N1

l=0,l =kgm,klsm,l + nm,k, (4)

where the received signal is split into three parts: the first termrepresents the signal term multiplied with the common phaseerror (CPE), the second is the inter-carrier interference (ICI)term caused by PN and the last term, i.e., nm,k, models theAWGN in the frequency domain. The factor gm,kl modelsthe influence of the PN and is given by

gm,kl =1N

N1n=0

exp(imNtot+NG+n)ei 2(kl)nN . (5)

Here we define the increments of the phase noise processto be independent, which is the commonly used model forfree-running oscillators [20], [21], so that

n+1 = n + n+1 =n+1j=1

j , (6)

where 0 = 0, n N(0, 2

)and 2 = 4T . Here

denotes the one-sided 3 dB bandwidth of the correspondingLorentzian spectrum of the LO and T is the sample time.


Since is generally small compared to the subcarrier spacing1/(NT ), it is useful to define n = NT , which is thenormalized version of . Subsequently, we can rewrite 2 as

2 =4nN

=2

N, (7)

where 2 = 4n is independent of N .Before detection is performed to xm,k, generally correction

of the CPE gm,0 is applied. Approaches proposed in [8], [10],[16], [17] could be applied to do this. Remaining nuisancesin detection of the signal are the common AWGN and the ICIterm. The influence of the AWGN on detection is well known,but the statistical properties of the ICI term remain an openissue in previous literature. Therefore, the remainder of thispaper will focus on the properties of this term.

The ICI for the kth carrier in the mth symbol, here denotedas m,k, is given by

m,k =N1

l=0,l =kgm,klsm,l

=1N

N1l=0,l =k

sm,l

N1n=0

exp (imNtot+NG+n) ei 2(kl)nN

=1N

N1l=0,l =k

sm,l

N1n=0

ei

mNtot+NG+n

j=1j

ei2(kl)n

N

=mN

N1l=0,l =k

sm,l

N1n=0

ei

mNtot+NG+n

j=mNtot+NGj

ei2(kl)n

N

=mN

N1l=0,l =k

sm,l

N1n=0

ei(n

j=0 j)ei

2(kl)nN , (8)

where m = exp(imNtot+NG1

j=1 j

), where the elements of

j are i.i.d. according to N(0, 2

)and where {j}N1j=0 =

{mNtot+NG+j}N1j=0 is independent of {j}mNtot+NG1j=1 .In the remainder of this paper we will, without loss of

generality, regard the subcarrier k = 0 in symbol m = 0.When we then omit the subcarrier and symbol indices forbrevity, (8) reduces to

0,0 = =0N

N1l=1

sl

N1n=0

exp

i n

j=0

j

exp(i2nl

N

).

(9)The random variable 0 = exp

(iNG1

j=1 j

)is independent

of all other terms appearing in (9), and has norm 1 and, thus,gives rise to a rotation in the complex plane of the randomvariable

1N

N1l=1

sl

N1n=0

exp

i n

j=0

j

exp(i2nl

N

). (10)

For convenience, we will, therefore, in the sequel take 0 = 1and study (10), where we keep in mind that for the ICI, arandom and independent rotation should be performed in theend.

B. Bit-Error ProbabilitiesWe recall that the received signal after the DFT-processing

xm,k is given by (4). Before detection is applied to this signal,first the CPE g0,m is removed. When this is assumed to bedone perfectly, the resulting signal is given by

x = s+ + n = s+ , (11)which is input to the decision device. Here denotes theadditive error term. Note that the symbol and carrier indexare omitted in (11) for brevity.

As a measure of performance of the system the symbol-errorprobability (SEP) can be considered, which is the probabilitythat the output of the decision device for x is incorrect. TheSEP can be found by evaluating the probability of erroneousdetection conditioned on the transmitted symbol for each ofthe symbols in the regarded modulation. The SEP can then beexpressed as

Ps =1M

Mm=1

(1 P (x Rm|s = Sm)

), (12)

where Rm is the decision region corresponding to the symbolSm out of the modulation alphabet S with M elements. In asimilar way the bit-error probability (BEP) can be evaluated,which can be expressed as

Pb =1M

Mm=1

(1 1

B

Bb=1

P (x Rm,b|s = Sm))

, (13)

where Rm,b is the decision region corresponding to the bthbit of symbol Sm. Clearly, (12) and (13) largely depend onthe distribution of the error term . When consists of acomplex normal distributed error source, i.e., when = n, theexpressions for (12) and (13) are well studied for different kindof modulations, see e.g. [22], [23]. For a system experiencingICI due to PN, however, this is not the case, which justifiesthe study of the distribution of the term in (10), as presentedin Section III.

To further illustrate the dependence of the BEP on thedistribution of the error term , we further work out (13) asan example for the QPSK or 4-QAM modulation, for whichM = 4 and B = 2. It is easily verified that the BEP is givenby (14). Here I and R denote the imaginary and real part ofthe error term , respectively. When only ICI is experienced, Iand R can be replaced by I and R in (14), respectively. HereI and R denote the imaginary and real part of the ICI term ,respectively. The final expression for the BEP in (14) clearlyillustrates the dependence of the system performance on thetail probabilities of the error term, i.e., the tail probabilities ofthe ICI-term in a PN-impaired system. The foregoing justifiesa careful study of the tail probabilities of , since they fullydetermine the system performance, as illustrated by (14). Thisstudy is presented in Section III-C.

III. PROPERTIES OF THE ICI TERM

With the aim of deriving a statistical characterisation of theerror term in detection of the transmitted symbols in a PN-


PbQPSK =

14

(4 1

2

{[P

(I > 1

2

)+ P

(R 1

2

)+ P

(R > 1

2

)]

+[P

(I 1

2

)]+[P

(I

12

)), (14)

impaired system, we will, in this section, study the ICI term

=1N

N1l=1

sl

N1n=0

exp

i n

j=0

j

exp(i2ln

N

). (15)

In Section III-A, we will study the convergence of whenN . In Section III-B, we will study the distribution of thelimit of when is small, i.e., when the interference is quitesmall, and in Section III-C, we will qualitatively study the tailsof the distribution when is small. Finally, in Section III-Dwe will investigate the BEP performance of a system impairedby PN by means of simulations and show how the deriveddistribution of can be used to accurately predict the BEPperformance.

A. Convergence of the ICI Term for Large NBefore stating the main convergence result, we need some

notation and a key observation. We let {Bt}t0 be a standardBrownian motion. Then, we have that

{j}N1j=0 d={(B j+1

NB j

N

)}N1j=0

, (16)

where X d= Y when the random variables X and Y have thesame distribution. Therefore, we have that

d=

1N

N1l=1

sl

N1n=0

ei2ln

N exp

i n

j=0

(B j+1

NB j

N

).(17)

In the sequel, we will use j = (B j+1

NB j

N

), and will

identify the limit of the right-hand side of (17), which, forconvenience, we again write as .

We also define

=

lZ:l =0

sl2l

10

eiBt [1 ei2lt]dBt, (18)

which will turn out to be the limit of when N .Some words of caution are necessary here. The integral on

the right-hand side of (18) is a so-called stochastic integral,which can be defined properly. In fact, the construction of suchintegral uses limits of the form in (17), and these ideas canbe used to prove rigorously that the right-hand side of (17)converges to (18), as we will perform in more detail in theAppendix. Before continuing with the analysis, we list someproperties of stochastic integrals that we will rely on. Firstly,we will use the rules that, for functions f, g : [0, 1] R Rsuch that

10E[|f(t, Bt)|2]dt,

10E[|g(t, Bt)|2]dt < , we

have thatE

[ 10

f(t, Bt)dBt]= 0, (19)

and

E

[ 10

f(t, Bt)dBt 10

g(t, Bt)dBt]

= 10

E[f(t, Bt)g(t, Bt)

]dt. (20)

Secondly, we will use for any function f : [0, 1] R, we havethat 1

0

f(t)dBt (21)

has a normal distribution with mean zero and variance 10

f2(t)dt. References for these statements can be found in[24, Chapter 13] or [25].

We will prove the following main result:

Theorem 3.1: When N , for any > 0 fixed, in theright-hand side of (17) converges in probability to , definedin (18).

The significance of Theorem 3.1 lies in the fact that it provesthat the ICI converges when the number of subcarriers growslarge, and it identifies the limit explicitly. In the sequel ofthis paper, we make use of this explicit limit in (18) toderive properties of the ICI. We will also present simulationsthat show that the distribution of is already close to thedistribution of when N equals 64, thus showing that thelimit result in Theorem 3.1 is also of practical use.

The proof of Theorem 3.1 is deferred to the Appendix.Simulation Results: To illustrate the convergence for large

number of subcarriers in Theorem 3.1, we have carried outMonte Carlo simulations. In these simulations an OFDMsystem experiencing PN was simulated, where the PN wasmodeled as in (6). From the results of these simulation theICI-term , as expressed by (15) in the analysis above, wasevaluated. The evaluation was carried out for QPSK modula-tion and a system for which all carriers were containing datasymbols. As in the analytical derivations, no multipath channelwas simulated. For all results 105 independent experimentswere performed.

First Fig. 1 depicts the empirical cumulative distributionfunction (ECDF), denoted by F(x) in the figure, of thenormalized real part of the ICI variable, i.e., of {}/N ,as found from these simulations. The results are given for2 = 103. For a sample frequency of fs = 1/T , thismeans that the corner frequency of the PN spectrum isgiven by 8.1 102 times the subcarrier spacing fs/N . Foran IEEE 802.11a based system [3], where the subcarrierspacing fs/N equals 312.5 kHz, this corresponds to a of 25.3 kHz. Results are depicted in the figure for systems


0.04 0.02 0 0.02 0.040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N = 64N = 128N = 256N = 512

x

F(x

)

Fig. 1. The ECDF of the real part of the normalized ICI term, i.e., N

,

found from Monte Carlo simulations of an OFDM system experiencing PN.The results are depicted for systems with different number of subcarriersN . The PN process was modeled according to (6) with a variance equal to2 = 10

3.

applying different number of carriers, i.e., for N equals 64,128, 256 and 512.

From the results in Fig. 1 we can observe the convergencein distribution for high number of subcarriers N , as stated inthe main result of Theorem 3.1 and proven in the Appendix.It is concluded that already for a moderate number of 64subcarriers convergence seems to be reached, since all curveslie on top of each other. This shows that although the limitdistribution was derived for high N , it is already applicable formoderate values of N . For the imaginary part of the ICI-term(15) the same convergence occurs.

The real part of the normalized ICI term is further studiedin Fig. 2. Hereto the result of Fig. 1 for N = 512 subcarriersis depicted again, but now in a normal probability plot. Thisfigure shows the distribution of the ICI with the correspondingnormal distribution, i.e., the normal distribution with the samemean and variance. The scaling of the plot is such that anormal distribution would be depicted as a straight line.

It can be concluded from this graph that the ICI is clearlynot normally distributed and that, since the curve is aboveand below the straight line in the left and right part of thefigure, respectively, the ICI distribution has thicker tails thanthe normal distribution. Similar results were found for theimaginary part of the ICI. This result endorses the analyticalresults obtained in this section.

B. Approximation of the ICI for Small In this section, we will investigate the distribution of the

limit =

lZ:l =0

sl2l

10

eiBt [1 ei2lt]dBt. (22)

Since the expression for the limit is a stochastic integralit is difficult to apply it for analyses. Therefore we here regard

5 0 5x 103

0.0010.0030.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

0.9970.999

x

{}/N (15)Corr. normal distr.

F(x

)

Fig. 2. Normal probability plot of the real part of the normalized ICIterm, i.e.,

N, found from Monte Carlo simulations of an OFDM system

experiencing PN. The results are depicted for a system with N = 512subcarriers. The PN process was modeled according to (6) with a varianceequal to 2 = 103.

the special case where is quite small. When this is the case,it seems reasonable to assume that we can replace eiBt in(22) by 1. If we do so, then we end up with

lZ:l =0

sl2l

10

[1 ei2lt]dBt

=

lZ:l =0

sl2l

[B1

10

ei2ltdBt

]. (23)

We note that Z = B1 is standard normally distributed, so that,with

X =

lZ:l =0

sl2l

, (24)

we arrive at

XZ

lZ:l =0

sl2l

10

ei2ltdBt. (25)

Next, since t ei2lt is deterministic, we have that 10

ei2ltdBt has a complex normal distribution. Furthermore,since, for |k| = |l|,

E

[ 10

cos (2lt)dBt 10

cos (2kt)dBt

]

= 10

cos (2lt) cos (2kt)dt = 0, (26)

we have that, with Zl = 2 10 cos (2lt)dBt, the sequence{Zl}l=1 is a sequence of i.i.d. standard normal random vari-

ables, where we are using that a vector of normal randomvariables is independent when all covariances are equal tozero.

Similarly, we can see that, for l 1,Z l =

2 10 sin (2lt)dBt are i.i.d. standard normal


0.15 0.1 0.05 0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x)

{}/N{a}/

N

(a) 2 = 102

0.05 0 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x)

{}/N{a}/

N

(b) 2 = 103

0.015 0.01 0.005 0 0.005 0.01 0.0150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x)

{}/N{a}/

N

(c) 2 = 104

Fig. 3. The ECDF of the real part of the normalized ICI term, i.e., N

(in dashed lines) and normalized real part of the approximation of the limitdistribution for small , i.e., a

N(in solid lines). The results are depicted for N = 512 subcarriers and varying 2 .

variables. All these random variables are independent ofZ = B1 =

10

dBt, since

E

[ 10

cos (2lt)dBt 10

1dBt

]= 10

cos (2lt)dt = 0.

(27)Finally, by a similar argument, also {Zl}l1 and {Z l}l1 areindependent. We conclude that

a = (ZX1 +

Y+,1 Y,22

)+ i

(ZX2 +

Y+,2 + Y,12

), (28)

where we define, with sl = Rl+iIl for l > 0, and sl = Rl+iI lfor l < 0,

X1 =12

l=1

Rl +Rll

, (29)

X2 =12

l=1

Il + I ll

, (30)

Y,1 =12

l=1

Zl[Rl Rl]l

, (31)

Y,2 = 12

l=1

Z l [Il I l ]l

. (32)

We note that for the bit-error probabilities (BEPs), we haveto look at the probability that is larger than a constant, say1. This we can do when tends to zero, to investigate theBEPs when the interference decreases. We will investigatesuch probabilities in more detail in Section III-C.

Simulation Results: In the following we will numericallystudy a, as defined by (28), which is the approximation ofthe limit distribution of the ICI term , as defined in (22), forsmall values of .

Hereto Monte Carlo simulations were performed, in which,again, an OFDM system experiencing PN was simulated,where the PN was modeled as in (6). From the results of

these simulations, the ICI-term , as expressed by (15), wasfound. The evaluation was carried out for QPSK modulationand a system with N = 512 subcarriers that all contained datasymbols. Furthermore, a was simulated using (28), also forQPSK modulation.

The results from these simulations for the normalized realpart of the two resulting distributions are presented in Fig. 3,by means of their ECDF. The results are given in Fig. 3(a),Fig. 3(b) and Fig. 3(c) for 2 equal to 102, 103 and 104,respectively. This corresponds to a 3 dB oscillator bandwidth of 8.1 101, 8.1 102 and 8.1 103 times the subcarrierspacing fs/N , respectively. For all simulation results in thissection, we have performed 105 independent experiments.

It can be concluded from Fig. 3 that the resemblance ofthe two distributions increases with decreasing , which is asexpected since a was derived under the assumption of small and, thus, small . It is concluded that for the consideredsystem already for 2 = 103 reasonable agreement betweenthe CDFs seems to be achieved. This indicates that theapproximate limit distribution a in (28) correctly models thedistribution of the ICI for these values of 2 .

The ECDF of results from similar simulations, but now for2 = 104, are given in Fig. 4. Here the ECDF is depictedon a logarithmic scale, which enables us to study the tailsof the distributions. The figure, again, depicts the simulatedreal part of the normalized ICI of (15), the real part of theapproximation of the limit distribution of the ICI term of (28),but now also the corresponding normal distribution, i.e., withthe same mean and variance as the other variables. In Fig. 4,we have performed 106 experiments for each result.

It can be concluded from Fig. 4 that the limit distribution awell approximates the ICI, even in the tails of the distribution.Furthermore, it is found that the Gaussian distribution showslower tail probabilities, and has a faster fall off. For instance,the probability that the real part of the normalized ICI issmaller or equal than 0.01 is approximately 2 103, i.e.,P( 0.01) 2 103. This is correctly predicted by


0.02 0.015 0.01 0.005 0 0.005 0.01104

103

102

101

100

x

{}/NNormal distr.{a}/

N

F(x

)

Fig. 4. The ECDF of the real part of normalized ICI term N

(dottedline), as expressed in (15), and of the normalized approximation of the limitdistribution a

N(solid line), as expressed in (28), and of the corresponding

normal distribution (dashed line) with the same mean and variance. Resultsare given for N = 512 subcarriers, QPSK modulation and a PN variance of2 = 10

4.

the proposed limit distribution. The corresponding Gaussianapproximation of the ICI, however, predicts the probability tobe approximately 2 104, which underestimates it by abouta factor of 10.

C. Tail ProbabilitiesIn this section, we will analytically show that the tail

probabilities of the random variables in Section III-B aredifferent from the ones for a Gaussian approximation, whatwas already numerically illustrated in Fig. 4. Tail probabilitiesare important in the ICI case since the BEP can be rewritten interms of the tail probabilities, as was elucidated for QPSK inSection II-B. Therefore, a system with smaller tail probabili-ties performs better than a system with larger tail probabilities.We present a qualitative analysis of such tail probabilities inorder to show that the usual Gaussian assumptions lead toan underestimation for the BEPs. This will be substantiatedfurther by the results from appropriate BEP simulations inSection III-D.

We will start by computing the first two moments of therandom variables appearing in (28):

E[ZX1] = E[Y,1] = 0, (33)while, writing Var(R1) = 2R ,

E[(ZX1)2] = E[Z2]E[X21 ] = 2Var(X1)

= 22Var(R1)l=1

12l2

= 22Var(R1)

6

=22R3

, (34)

and

E[Y 2,1] = 2Var(R1)

l=1

12l2

=2Var(R1)

6

=22R6

. (35)Using further that E[ZX1Y,1] = E[ZX1Y,2] = 0, therandom variable R = ZX1+ 2 (Y+,1Y,2), which signifiesthe real part of the ICI, has mean zero and variance

Var(R) = 2E[(ZX1)2] +14(E[Y 2+,1] + E[Y

2,2]

)= 22R

(13+

112

)=

522R12

. (36)

Therefore, the usual Gaussian assumptions lead to a tailestimate of the form

P(R > y) Q( y

)= exp

( 6y

2

522R(1 + o(1))

). (37)

We will now show that the tail is in fact much larger thanthat.

In the explanation below, we will assume that Rl and Ilare 1 with equal probability, for which 2R = 1. For this, wefix a P , and we investigate the probability that Rl = Rl = 1for l P , while l>P Rl+Rll 0 and Y+,1 + Y,2 0. Bysymmetry of the random variables involved, this probabilityis at least 14P 12 12 . Also, when the above is true, then

X1 Ml=1

1l

=1logP (1 + o(1)). (38)

Therefore, in order for ZX1 y to hold, we only need thatZ y

logP, (39)

which has probability

Q

(y

logP

) exp

(

2y2

22 log2 P(1 + o(1))

). (40)

Therefore, for every P 1 and y > 0,

P(R > y) 14P 14exp

(

2y2

22 log2 P(1 + o(1))

). (41)

For example, for P = y2

2 log4 ( y ), we obtain that 14P+1 =

exp(o( y

2

2 log2 P)), so that

P(R > y) exp(

2y2

82 log2 ( y )(1 + o(1))

). (42)

The tail in (42) is much larger than the ones in (37), sothat Gaussian assumptions, as formulated in (37), lead toa systematic underestimation of the tails, and therefore ofthe BEPs. Indeed, when y = 1 and is quite small, theexponent has become a factor 48 log

2 ( 1 )

52 smaller, which issubstantial for small. This can be clearly seen in the resultsfrom simulations in Fig. 2, where we see thicker tails of theICI distribution compared to a normal random variable withthe same variance. This is even more clear from the ECDFdepicted on a logarithmic scale in Fig. 4. A similar analysiscould be carried out for the imaginary part of the ICI I.


0 5 10 15 20 25 30 35106

105

104

103

102

101

100

Average SNR (dB)

BEP

= 700 Hz = 1000 Hz = 2000 Hz = 5000 Hz

Fig. 5. BEP results from Monte Carlo simulations with an IEEE 802.11a-like system applying 16-QAM modulation. Results are included for: a systemexperiencing PN modeled as in (6) for different values of and applyingperfect CPE correction (in dashed lines), a system modeling the resulting ICIas an equivalent complex Gaussian process (in dash-dot lines) and a systemmodeling the resulting ICI according to the small approximation of thelimit distribution as in (28) (in solid lines).

D. Simulation Results

To confirm these analytical findings, we have carried outBEP simulations. Hereto an IEEE 802.11a-like system wassimulated, i.e., the number of carriers N = 64, guard intervallength NG = 16 samples, the sample frequency fs = 20 MHz,which corresponds to a sample length of 50 ns and a symbollength of (64+16)50 ns = 4 s. In the system all 64 subcarrierscontain data symbols, which for Fig. 5 are taken from the16-QAM modulation alphabet and for Fig. 6 from boththe 16-QAM and 64-QAM modulation alphabet. A systemwithout coding and not experiencing a multipath channel wassimulated. The packet length in the simulations was equal to10 symbols.

Fig. 5 shows BEP results for a system impaired by bothAWGN and PN. The BEP is depicted versus the signal-to-noise ratio (SNR) for different values of , i.e., the 3 dBbandwidth of the LO power spectral density. The PN isgenerated according to the model in (6) and the CPE isperfectly removed, leaving the ICI as the only influence ofthe PN. Results from these simulations are given by thedashed lines in the figure. The simulated s vary from 700to 5000 Hz, which corresponds to 2 values varying from1.4 104 to 103.

These results are depicted together with results from simula-tions where the influence of the ICI is modeled as an additionalzero-mean complex Gaussian noise source at the receiver, i.e.,the commonly used approach in previous literature, where thevariance of the noise equals that of the actual ICI. The resultsof these simulations are given by the dash-dot lines. Finally,results are included from simulations where the influence ofthe ICI is modeled using the small approximation of theproposed model for ICI, i.e., a, as defined by (28). These

102 103 104 105105

104

103

102

101

100

(Hz)

BEP

Actual systemGaussian approx.Small approx. a

Fig. 6. BEP results from Monte Carlo simulations with an IEEE 802.11a-like system applying 16-QAM (in solid lines) and 64-QAM (in dashed lines)modulation. Results are included for: a system experiencing PN modeledas in (6) for different values of and applying perfect CPE correction(Actual system), a system modeling the resulting ICI as an equivalent complexGaussian process (Gaussian approx.) and a system modeling the resulting ICIaccording to the small approximation of the limit distribution as in (28)(Small approx. a).

results are depicted by solid lines.It can be concluded from Fig. 5 that the BEP is severely

underestimated using the Gaussian approximation for the ICIterm. Especially for small values of , i.e., small values,the difference between the actual and predicted BEP by theGaussian model are very large, which confirms our analyticalfindings on tail probabilities in the previous section. Forinstance for = 2000 Hz and a SNR of 30 dB the BEP isunderestimated by a factor of 20 using the Gaussian approach.For lower values of this difference is even higher. Theresulting BEPs using the approximate of the proposed limitdistribution, in contrast, closely resemble those of the actualPN impaired system. It is noted that this limit distribution wasfound under the assumption of a large number of carries, but,judging the results, it already holds for a system with a modestnumber of 64 subcarriers.

Results depicted in Fig. 6 are obtained from similar simu-lations carried out for a system not experiencing additive RXnoise, i.e., the system is only impaired by PN. Results fromBEP simulations, for the same three cases as for Fig. 5, aredepicted versus for both the 16-QAM (in solid lines) and64-QAM (in dashed lines) modulation. These results can beinterpreted as the high SNR flooring of the BEP curve for acertain , i.e., the maximum achievable BEP performance fora certain level of PN.

It can be concluded from Fig. 6 that for both modulationformats the Gaussian approximation of the ICI only providesreliable BEP results for very high values of , where the BEPis so high that reliable data transfer is almost not possible. Thisconfirms findings previously presented in [14] and [15], wheresimulations were used to show the validity of the Gaussianmodel for the ICI only for very large values of . The BEPs


found from simulations with a, the approximate expressionfor our limit distribution , on the other hand, show goodagreement with the BEPs found from simulations with theactual PN impaired system. It is noted though that a smalldiscrepancy occurs at high values of , where becomes high,since the approximate results of Section III-B were derived forsmall . The results are, however, even for these high valuesreasonable. For these high values of the actual evaluationof in (18) will obtain better results.

Overall it can be concluded from the simulation results inthis section that, although derived for large N and small , theapproximate limit distribution a can be well applied to findthe BEP of a PN-impaired system. It is noted, furthermore, thatsimulations were carried out for a system with a low numberof carriers, i.e., N = 64, and that the applicability of ourresults will even be better for systems with higher number ofsubcarriers like, e.g., WiMAX, DVB and DAB, since for thesekind of systems N is much larger and the typically permissible is considerably lower.

IV. CONCLUSIONS

The wide application of orthogonal frequency divisionmultiplexing (OFDM) in wireless systems justifies a carefulinvestigation of its performance limiting factors. The influenceof imperfect oscillators, i.e., phase noise (PN), is identified asone of the major impairments of OFDM, especially when low-cost and high-frequency systems are considered. Therefore, thedistribution of the inter-carrier interference (ICI) due to PN isstudied in this paper.

In most of the previous contributions, the ICI was assumedto be zero-mean complex Gaussian distributed. In this paper,however, it was shown that this assumption is not valid andthe limit distribution of the ICI term for large number ofsubcarriers is derived. This distribution is shown to exhibitthicker tails than the Gaussian distribution with the samemean and variance. In an analysis of the tail probabilitiesthese finding were confirmed and it was shown that bit-errorprobabilities are severely underestimated when the Gaussianapproximation for the ICI term is used.

Results from simulation studies show the validity of thelimit distribution, obtained under the assumption of a largenumber of subcarries, already for a modest number of sub-carriers. Furthermore, they show that for small values of thePN variance, the limit distribution very well resembles the ICIdistribution. Additionally, it is shown that the tail probabilitiesare severely underestimated by the corresponding Gaussiandistribution. Finally, results from a BEP-study for a IEEE802.11a-like system show that the derived limit distribution issuitable to obtain the correct BEP of a PN-impaired system,this in contrast to an approach using the Gaussian approxima-tion.

The results in this paper can be used by system designers tobetter specify the level of tolerable PN, for an OFDM systemto achieve a certain bit-error probability. This paper shows thatapplying the Gaussian approximation would lead to a seriousunderspecification of the LO.

APPENDIX

In this Appendix we will provide the proof for Theorem 3.1.We first reduce the proof of Theorem 3.1 to four keyconvergence statements (52), (59), (65) and (66). After this,we prove these four statements separately.

Proof of Theorem 3.1 subject to (52), (59), (65) and (66).We recall that XN converges in probability to X , and writeXN

P X , when, for every > 0,lim

NP(|XN X | > ) = 0. (43)

We will make frequent use of the fact that if XN and YNconverge in probability to X and Y , then also XN + YNconverges to X + Y in probability.

We start by rewriting the sumN1n=0

ein

j=0 jei

2lnN . (44)

For this, we will use partial summation, which states that forany two sequences of numbers {an}mn=1 and {bn}mn=1, wehave

m1n=0

an(bn+1 bn)

= ambm a0b0 m1n=0

(an+1 an)bn+1. (45)

We apply this to

an = ein

j=0 j , bn+1 bn = ei 2lnN , (46)

so that

bn =n1j=0

ei2ljN . (47)

For this choice, we can compute that b0 = bN = 0, and

bn =ei

2lnN 1

ei2lN 1 . (48)

Therefore, we arrive at

N1n=0

ein

j=0 jei

2lnN

=1

ei2lN 1

N1n=0

[ei

n+1 1]einj=0 j [1 ei 2l(n+1)N ].

(49)

Now, n+1 is small, since it has variance 2 = 2

N . Therefore,we can expand ein+1 1 in+1 to arrive at

N1n=0

ein

j=0 jei

2lnN

iei

2lN 1

N1n=0

n+1ein

j=0 j[1 ei 2l(n+1)N ]. (50)


Using (50) we define an approximation of given by

=N1l=1

isl

N [ei2lN 1]

N1n=0

n+1ein

j=0 j[1 ei 2l(n+1)N ].

(51)Indeed, below we will prove that, N , which is defined tobe the difference between and , converges to zero inprobability, i.e.,

N = P 0, (52)in probability. Therefore, to prove the claim, it now sufficesto prove that P .

When we use the periodicity of ei 2lnN and ei 2lN , we canmore conveniently rewrite as

0


When n1 < n2, we have that n2+1 is independent ofn1+1e

in1

j=0 jei

n2j=0

j , so that the expected value equals

zero. Therefore, we arrive at

E

[N1n=0

n+1ein

j=0 j2] = N1

n=0

E[2n+1

]= 2, (70)

so that we obtain

E

K|l|N/2

isl

N [ei2lN 1]

N1n=0

n+1ein

j=0 j[1 ei 2l(n+1)N ]2

]

2s2

K|l|N/2

1

N2|ei 2lN 1|2 . (71)

It is not hard to see that when |l| N4 , we have that

|ei 2lN 1| 1 cos

(2lN

) 1. (72)Furthermore, using the fact that | sin(x)| 2|x| for all|x| 2 , we obtain that for |l| N4

|ei 2lN 1| sin

(2lN

) 4|l|N . (73)We conclude that

|ei 2lN 1| min{ |l|

N, 1}

. (74)

As a consequence, we obtain that

E

K|l|N/2

isl

N [ei2lN 1]

N1n=0

n+1ein

j=0 j[1 ei 2l(n+1)N ]2

]

2s2

K|l|N/2

1min{|l|2, N2} , (75)

so that, uniformly in N , the variance of the summands with|l| > K is bounded by a constant times K1. Therefore, whenK = KN tends to infinity, the variance of this term tends tozero, so that this term tends to zero in probability by (67). Anidentical computation shows that

E

KN|l|


Similarly to (70), we obtain that

E

[N1n=0

[ei

n+1 1 in+1

]ein

j=0 j2]

=N1n=0

E

[ein+1 1 in+1]2]

N1n=0

E[4n+1

]=

34

N, (86)

where we use that E[4n+1

]= 3

4

N2 . We arrive at

E[2N ] 342s

N

0 0.

Proof of (59). This again follows from a second momentcalculation. We can interchange the order of summation whenwe have a finite sum, but for an infinite sum that is not soclear. However, by a similar computation as in the proof of(65), we see that

10

eiBt(S(t) SKN (t))dBtP 0, (88)

whereSKN (t) =

0KN

sl2l

10

eiBt [1 ei2lt]dBt P 0. (90)

Noting that

= 10

eiBtS(t)dBt+

|l|>KN

sl2l

10

eiBt [1ei2lt]dBt

10

eiBt(S(t) SKN (t))dBt (91)

then completes the proof.

Proof of (66). We use that j = [B j+1

NB j

N

], where

{Bt}t0 is a standard Brownian motion. Then,N1n=0

n+1ein

j=0 jS

(n+ 1N

)

= N1n=0

eiB n+1

N S

(n+ 1N

)[Bn+2

NBn+1

N

]

= N

n=1

eiB n

N S( nN

) [Bn+1

NB n

N

]. (92)

The above is the usual approximation to the stochastic integral

10

eiBtS(t)dBt. (93)

Since, for every t, the integrand eiBtS(t) has a finite secondmoment, and since the process t eiBtS(t) is predictable,the sum in (92) converges in probability to the integral in(93). See, e.g., [24, Section 13.8] for convergence results ofstochastic integrals.

ACKNOWLEDGEMENTSThe authors are grateful to the anonymous referees for their

valuable comments.

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[7] T. Pollet, M. van Bladel, and M. Moeneclaey, BER sensitivity ofOFDM systems to carrier frequency offset and Wiener phase noise,IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 191193, Feb./Mar./Apr.1995.

[8] P. Robertson and S. Kaiser, Analysis of the effects of phase-noise inorthogonal frequency division multiplex (OFDM) systems, in Proc.IEEE ICC, June 1995, vol. 3, pp. 16521657.

[9] L. Tomba, Analysis of phase noise effects in OFDM modems, IEEETrans. Commun., vol. 46, no. 5, pp. 580583, May 1998.

[10] A. G. Armada, Understanding the effects of phase noise in orthogonalfrequency division multiplexing (OFDM), IEEE Trans. Broadcasting,vol. 47, no. 2, pp. 153159, June 2001.

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[12] H. Nikookar and R. Prasad, On the sensitivity of multicarrier trans-mission over multipath channels to phase noise and frequency offset,in Proc. 7th IEEE International Symposium Personal, Indoor MobileRadio Commun., Oct. 1996, vol. 1, pp. 6872.

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[14] L. Piazzo and P. Mandarini, Analysis of phase noise effects in OFDMmodems, IEEE Trans. Commun., vol. 50, no. 10, pp. 16961705, Oct.2002.

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[16] V. S. Abhayawardhana and I. J. Wassell, Common phase errorcorrection with feedback for OFDM in wireless communication, inProc. IEEE Globecom, Nov. 2002, vol. 1, pp. 651655.

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Tim C. W. Schenk (S01-M07) was born in Heerle,The Netherlands, on June 28, 1978. He received theM.Sc. and Ph.D. degree in electrical engineeringfrom Eindhoven University of Technology (TU/e),Eindhoven, The Netherlands, in 2002 and 2006,respectively. The research for this Ph.D. thesis wasfocussed on the influence and digital compensationof RF impairments in multiple antenna and multi-carrier systems. From 2002 to 2004 he was with theWireless Systems Research group of Agere Systemsin Nieuwegein, The Netherlands. From 2004 to 2006

he was a research assistant in the Radiocommunication group of TU/e. Duringthe first part of 2006 he was a visiting researcher at CEA-LETI in Grenoble,France. In October 2006 he joined the Connectivity Systems and Networksdepartment of Philips Research Laboratories, Eindhoven, The Netherlands asa research scientist.

Dr. Schenk is a member of the Royal Institute of Engineers (KIVI-NIRIA)in The Netherlands and the Dutch Electronics and Radio Society (NERG).

Remco W. van der Hofstad was born in Eindhoven,The Netherlands, on May 3, 1971. He receivedhis M.Sc. degree with honors at the Departmentof Mathematics of the University of Utrecht in1993, where he subsequently continued with hisPh.D. He received his Ph.D. degree in 1997 onresearch in statistical physics concerning randompolymers. In the following year, he was a post-docat McMaster University in Hamilton, Canada, andworked at Microsoft Research in Redmond, U.S.A.In 1998, he obtained an assistant professor position

at Delft University of Technology, and in 2002 an associate professorshipat Eindhoven University of Technology, where, in January 2005, he waspromoted to full professor in probability. Since January 2004, he also actsas one of the scientific advisors in the Random Spatial Structures programin the European research institute in probability and statistics EURANDOM,which is located on the campus of Eindhoven University of Technology. Hisresearch is in applications of probability, mostly in statistical physics andin telecommunications. His research interests include large deviation theory,self-interacting stochastic processes and random walks, random graphs andcomplex networks, percolation and interacting particle systems.

Erik R. Fledderus received the M.Sc. (cum laude)and Ph.D. degrees in Applied Mathematics fromTwente University of Technology, Enschede, TheNetherlands, in 1993 and 1997, respectively. From1997 onwards, he was with KPN Research (nowTNO Information and Communication Technology),Leidschendam, The Netherlands, and from 2003 heis part-time professor with the Radiocommunica-tions group of Eindhoven University of Technologyin the area of wireless communications networks.His research interests and publications are in the

areas of design and simulation of radio networks, smart antennas andpropagation modelling. At TNO Information and Communication Technology,Prof. Fledderus is manager of the Next Generation ICT infrastructuresprogram.

Peter F. M. Smulders (S82-M82-SM98) grad-uated from Eindhoven University of Technology in1985. In 1985 he joined the Propagation and Electro-magnetic Compatibility Department of the ResearchNeher Laboratories of the Netherlands PTT. Duringthat time he was doing research in the field ofcompromising emanation from civil data processingequipment. In 1988 he moved to the EindhovenUniversity of Technology as a staff member of theTelecommunication Division. Next to his lecturingduties he performed a Ph.D. research in the field of

60 GHz broadband wireless LANs. His current work addresses the feasibilityof low-cost, low power small sized wireless LAN technology operating in the60 GHz frequency band. With this low-cost technology it should be possibleto serve an unprecedented maximum user data rate in the order of Gbps.

He is senior member of IEEE and his research interest that covers 60 GHzphysical and higher layers is reflected in numerous IEEE publications. Hewas involved in various research projects addressing 60 GHz antennas andinterworking (ACTS, MEDIAN) and 60 GHz propagation (MinEZ, BroadbandRadio@Hand). Currently, he is also addressing baseband design in the contextof 60 GHz radio (Freeband, Wicomm). As project manager of the SiGi Spotproject his future research activities will range from 60 GHz physical layerdesign to associated network aspects.

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