Applications of Level Crossing Theory to ClippingNoise Characterization in Filtered OFDM Signals

Jacek Ilow and Parvathy VenkatasubramanianDalhousie University

Department of Electrical and Computer EngineeringHalifax, NS, B3J 2X4, Canada.

j.ilow@dal.ca and parvathy@dal.ca

Abstract— This paper applies the level crossing rate (LCR) andaverage fade duration (AFD) analysis to characterize clippingnoise in filtered OFDM signals. Because orthogonal frequencydivision multiplexing (OFDM) signals exhibit complex Gaussianprocess behavior, well established results for the Rayleigh enve-lope of the correlated Gaussian process are used to derive theLCR and AFD statistics of the OFDM signal at the output ofa square-root raised cosine shaping filter. The LCR and AFDinformation is then used to determine the statistics of the OFDMsignal that passes through a soft-limiter, approximating at thebaseband high power amplifier (HPA) memory-less nonlinearity.Simulation results corroborate theoretical derivations for theLCR and AFD statistics in real OFDM signals. The resultsobtained are important in predicting the clipping noise impacton the in-band and adjacent channel interference.

I. INTRODUCTION

Orthogonal frequency-division multiplexing (OFDM) is anattractive modulation method for channels with a non-flatfrequency response, as it does not require complex equaliz-ers [1]. The major drawback to OFDM is its high peak-to-average power ratio (PAPR), making it extremely sensitiveto non-linear distortions in the high power amplifier (HPA)of a transmitter. This high PAPR problem results from thecentral limit theorem (CLT) as follows: as a summationof a large number of sub-carriers with random quadratureamplitude modulated (QAM) symbols, the resulting complexOFDM signal is Gaussian distributed over each of its real andimaginary components. The magnitude of a complex Gaussiansignal is Rayleigh distributed, so the envelope of the OFDMsignal has a large dynamic range, theoretically achieving largevalues with finite probability.

Many methods for reducing the PAPR have been proposed,with clipping and filtering perhaps being the simplest approachto suppressing PAPR [2], [3]. Excessive envelope valuestend to occur infrequently, allowing for amplitude clippingto constitute a reasonably effective solution. Clipping thesepeak envelope values leads to spectral distortion, both in-bandand out-of-band. This paper aims at characterizing the clippingdistortions in OFDM signals by providing time characteristicsas opposed to more traditional approaches based on proba-bility distributions. Specifically, the paper develops closed-form approximation formulas for the level crossing rate (LCR)and the average fade duration (AFD) for the envelope of thecontinuous OFDM signal at the output of the shaping transmit

filter. In our baseband model of the transmitter, the shapingfilter is followed by a soft-limiter approximating a non-linearHPA. In the soft-limiter model, if the signal envelope exceedscertain threshold, it is limited, giving rise to a clipping noise.As a result, level crossing theory provides an alternative tocharacterizing clipping noise. Our formulas are exact for thecase of Gaussian processes and, since OFDM signals exhibitGaussian-like behavior, these formulas allow for predictingclipping noise characteristics quite well.

The LCR of a random process conveys useful informationabout the underlying process, and is of interest in diverseengineering fields [4], [5]. In wireless communications, it isrelated to the system characteristics such as handoff, outageprobability, fading rate, average duration of fades and the effectof diversity on fading. The AFD is especially well understoodamong communications engineers, as it determines the numberof symbols affected by a deep fade [6], [7]. As a result, eventhough from the perspective of clipping noise characterizationit would be more practical to consider Average Peak Duration(APD), i.e., time when envelope exceeds the threshold, byobserving that APD and AFD are complementary events wewill use more standard terminology for the convenience of thereader.

The LCR formula was originally derived by Rice in terms ofthe joint probability density function (PDF) of the underlyingprocess and its time derivative [8]. The Rice’s formulas toanalyze an OFDM signal in a similar context to the one in thispaper has been partially exploited in [9], [10] and [11]. In com-parison to the results presented in [9] where the distribution ofthe PAPR in strictly band-limited OFDM signals is studied, theinvestigations in this paper address a broader class of OFDMsignals shaped by the a square-root raised cosine (RRC)shaping filter at the transmitter. The contribution of this paperis in specializing the LCR theory to the complex Gaussianprocesses with the autocorrelation function determined by theRRC filter impulse response.

The analytical results are thoroughly validated by workinginitially with Gaussian processes, for which our solutions arederived from Rice formulas, and then verified by means ofsimulation for the case of OFDM signals which are wellapproximated by Gaussian processes only for large number ofsub-carriers. The results presented in this paper for LCR andAFD offer a lot of insight into the statistical behavior of the

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real OFDM signals. It is expected that results obtained couldbe used to the spectral and bit error rate (BER) characterizationof the clipping process in OFDM signals similarly as in [10]

II. CROSSING RATE AND FADE DURATION

The basic results in LCR and AFD for complex Gaussiansignals were derived by Rice in his fundamental work [8], andwere discussed further in [5]. In this Section, we reproducefollowing [4] and [5] some of these results for completeness.This is because we rely quite heavily on these formulas toderive our specialized results in the case of the filtered OFDMsignal approximated as a complex Gaussian process.

The LCR at envelope level R is defined as the rate at whicha magnitude of the complex signal s(t) crosses the thresholdR in a positive (or negative) direction. Denoting the signalenvelope and its time derivative by r and

·r, respectively, the

average level crossing rate NR, is given by:

NR =∫ ∞

0

·rf

(r = R, r

)dr (1)

where f(r = R, r) is the joint PDF of r at R and r at thesame instant.

The AFD is defined as the average time that the signalenvelope remains below a specified threshold R after crossingthat level in a downward direction and is given by:

TR = Prob(r ≤ R

)/NR (2)

We visualize graphically the LCR and AFD concepts in Fig. 1.Using the notation from this figure, NR = E(L)

T0and TR =

E(ti)T0

where E(·) is the expectation operator.

Rt11 2 3 4 L

tT0

t2 t3 t4 tL

r signal envelope

Fig. 1. Illustration of LCR and AFD.

In case where the underlying process s(t) = x(t)+j ·y(t) is acomplex Gaussian with x(t) and y(t) being real valued, zero-mean, Gaussian processes, r(t) = |s(t)| is Rayleigh distributedwith PDF:

f(r) =r

σ2exp

(− r2

2σ2

)(3)

and r is Gaussian distributed with PDF:

f(r) =

√1

2πσ2exp

(− r2

2σ2

)(4)

where σ2 is the variance of the real (imaginary) componentof the s(t) and σ2 = 2 · σ2 is the variance of the differen-tiated envelope. When the auto-correlation function for bothprocesses x(t) and y(t) is R(τ), (1) transforms into:

NR =

√β

2π· f(R) =

√β

2π· R

σ2exp

(− R2

2σ2

)(5)

where

β = − d2

dτ2R(τ)

∣∣∣τ=0

= −R(0) (6)

Also, for the Rayleigh process r(t), the AFD is given by

TR =√

2π

β· σ2

Rexp

( R2

2σ2− 1

)(7)

It is worth observing that the ADF and LCR are related tocumulative distribution function F (R) of r(t) as

TR =F (R)NR

(8)

where F (R) is the probability that r(t) is less or equal to thelevel R, i.e.,

F (R) =∫ R

0

f(r)dr (9)

III. OFDM SIGNAL

A. Baseband OFDM System

Figure 2 shows the baseband-equivalent functional blockdiagram for the OFDM transmitter used in this paper [1].

hT (t)

s(t)

Transmit filter HPA

Xm0

Xm1

XmN−1 xm

N−1

xm0

t

QAM...symbols

xm1

· · ·xm

0

xm1

xmN−1

R

rin

rout

P/S...IDFT

Fig. 2. A simplified block diagram of the baseband-equivalent OFDMtransmitter.

The input to the transmitter is a m-th frame ofN QAM symbols from the finite alphabet of size M(Xm

0 ,Xm1 , · · · ,Xm

N−1). (The number of symbols in a frameN is the same as the number of sub-carriers in the sys-tem.) An OFDM transmitted signal is the sum of N sub-carriers modulated with QAM symbols Xm

i , each with equalbandwidth and frequency separation ∆f , where 1/∆f = Tis the duration of the OFDM block. The continuous-timemathematical representation of a OFDM multicarrier signalat the baseband is given by:

xm(t) =N−1∑k=0

Xmk ej2π∆fktw(t) (10)

where m is the OFDM block number of duration T , Nis the number of sub-carriers, fk = ∆f · k is subcarrierfrequency, and w(t) is a rectangular window (nominally ofheight one over the interval [0, T ]). The OFDM signal xm(t)is a complex signal. In practice, baseband OFDM signals aretypically generated using an inverse discrete Fourier transform

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(IDFT) by working with the discrete-time version of (10):

xm[n] ≡ xm(t)|t=nTs= xm(nTs)

=N−1∑k=0

Xmk ej2πfknTsw(nTs)

=N−1∑k=0

Xmk ej2π kn

N w[n]

(11)

where Ts is the sampling period in the time domain such thatTs = T

N . The window w[n] is the discrete equivalent of w(t).The values xm[n], n = 0, 1, · · · , N − 1, are referred to asOFDM symbols. Careful examination of (11) reveals that infact OFDM symbols xm[n] are related to QAM symbols Xm

k

through IDFT.

B. Continuous-Time OFDM Signal

Before the OFDM signal goes through HPA basebandequivalent, OFDM symbols xm[n] are filtered with the shaping(transmit) filter to form the continuous time signal s(t).The transmit filter hTx(t) in cascade with the receive filterhRx(t) satisfies the Nyquist condition for zero inter-symboldinterference (ISI) at the OFDM symbol rate. In principle,one distinguishes between continuous-time and discrete-timesignals but in simulations to get the signal going to the HPA,we work with the up-sampled version of OFDM symbolsfiltered by the discrete version of the square root raised cosine(RRC) pulses. In this case, we work with the RRC hTx(n)pulses that are symmetric hRx(n) = hTx(n) = hTx(−n).The convolution of the transmit and receive filters p(t) =hTx(t) � hRx(t) is given in the continuous time domain asraised cosine pulses:

p(t) = sinc(t

T) ·

( cos(πα tT )

1 − 4α2( tT )2

)(12)

with the excess bandwidth or roll-off factor α. Assuminguncorrelated QAM symbols in the system, OFDM symbols xm

l

with l = 0, 1, · · · , N −1 are independent and their correlationfunction in the continuous time domain is the delta functionRx(τ) = δ(τ). In order to obtain the autocorrelation functionRs(τ) of a signal s(t) at the output of the transmit filter, weobserve that

Rs(τ) = Rx(τ) � p(τ) = p(τ) (13)

where p(t) is as in (12). In this paper, our focus is on thetransmitter side, and the receive filter is only mentioned toarrive at the relation in (13)

IV. LCR AND AFD IN OFDM SIGNALS

In order to predict time characteristics of clipping in thelinear soft-limiter as in Fig. 2, we assume initially that theOFDM signal is well modelled through Gaussian process sothat its envelope LCR and AFD can be evaluated as in (5)and (7), respectively. In order to make use of these formulas,one has to evaluate β as in (6) based on Rs(τ) as in (13). The

original derivations for β when shaping with RRC pulses arepresented in the appendix giving:

β = R(0) = p(0) = 8α2 − π2

6

[(1 + α3) + (1 − α3)

](14)

V. SIMULATION RESULTS

Figure 3 shows the AFD at the output of the RRC shapingfilter as a function of a threshold R calculated using resultsfrom (7) and (14) for the continuous time Gaussian processes.Corresponding results for LCR are plotted in Fig. 4. In bothfigures, R is expressed in dB when normalized with respectto the variance of the real Gaussian processes and when theroll-off factor α of the RRC filter is used as a parameter.These figures corroborate our visual examination of filteredGaussian signal envelopes that lower α’s result in longer AFD(shorter APD) and smaller number of level crossings. Thesecalculations and signal examination are intuitive as pulses withsmaller α are more “peaked”.

1 2 3 4 5 6 7 8 9 10

100

101

102

Average time below the threshold

R [dB]

AF

D

a=0.05a=0.5a=0.95

Fig. 3. AFD as a function of threshold R in dB and roll-off α as a parameter.

1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

Average level crossing rate

R [dB]

LCR

a=0.05a=0.5a=0.95

Fig. 4. LCR as a function of threshold R in dB and roll-off α as a parameter.

To verify how well our theoretical formulas fit in the case ofOFDM signals, we ran Monte Carlo simulations on the up-sampled (by a factor of 8) and RRC filtered Gaussian andOFDM discrete-time signals with α = 0.05, 0.5, 0.95, N =32 and M = 16. The results obtained from data records close

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TABLE I

THEORETICAL RESULTS AND MONTE CARLO SIMULATIONS FOR THE

AVERAGE AFD.

R = 1dBα = 0.05 α = 0.5 α = 0.95

Theory 1.08 1.0103 0.8878Gaussian 1.126 1.041 0.9193OFDM 1.1156 1.0317 0.9117

R = 9dBα = 0.05 α = 0.5 α = 0.95

Theory 25.52 23.89 20.75Gaussian 28.03 25.57 23.23OFDM 26.93 24.75 22.18

TABLE II

THEORETICAL RESULTS AND MONTE CARLO SIMULATIONS FOR THE

AVERAGE LCR.

R = 1dBα = 0.05 α = 0.5 α = 0.95

Theory 0.4329 0.4624 0.5321Gaussian 0.4171 0.4507 0.5109OFDM 0.4205 0.4517 0.5091

R = 9dBα = 0.05 α = 0.5 α = 0.95

Theory 0.0385 0.0411 0.0472Gaussian 0.0352 0.0391 0.0423OFDM 0.0368 0.0399 0.0445

to thousand OFDM symbols are presented in Tables I andII, respectively, and show very good agreement between thesimulated and theoretical results. The agreements are closefor different values of the up-sampling factor which is chosenusually to be higher then 4 in which case the PAPR of thecontinuous OFDM signal is well approximated by the PAPRof the up-sampled version of the discrete signal [12].

VI. CONCLUSION

Closed-form expressions for the level crossing rate andaverage fade duration are presented for the filtered OFDMsignals with uncorrelated QAM input symbols. When codingis introduced or other kinds of correlation between QAMsymbols, the OFDM symbols will not have the autocorrelationRx(τ) given by the delta function and modifications in (13)need to be taken into account when calculating LCR and AFDparameters of s(t).

APPENDIX

The objective here is to find the second derivative of araised cosine pulse p(t) as in (12) at zero. Rather then findingthe second order derivative directly, we calculate its limit atzero by using Taylor series expansions and basic trigonometricfunctions.

The autocorrelation p(t) is re-written as:

p(t) =sin(πt

T )cos(παtT )

(πtT )[1 − ( 2αt

T )2](15)

where α is the roll-off factor of the function. To simplify

notation, we assume T = 1. Using a trigonometric identity,

sinA cos B =sin(A + B) + sin(A − B)

2(16)

and substituting (16) into (15) we obtain:

p(t) =sin(πt(1 + α)) + sin(πt(1 − α))

2πt[1 − (2αt)2](17)

Using the Taylor series expansion for sin(x) = x − x3

3! +x5

5! − · · · , (17) is more conveniently written in the form

p(t) =(2πt) − (πt)3

3! [(1 + α)3 + (1 − α)3 + · · · ](2πt)(1 − (2αt)2)

(18)

where we are disregarding terms with powers of t greaterthan 3, as when calculating second derivative these termswhen evaluated at zero are vanishing. Application of geometricseries aids in further simplifying (18) into:

p(t) =

(2πt − (πt)3

3! [(1 + α)3 + (1 − α)3])(∑∞

0 (c · t)2n

)

(2πt)(19)

where c = 2α.Expanding the product of the geometric and Taylor series

and limiting (19) to contain only second order polynomialterms, we get:

p(t) =

(2πt − (πt)3

3![(1 + α)3 + (1 − α)3]

)(1 + c2t2 + · · · + c2nt2n

)(2πt)

(20)Taking double derivative of the filter function 20) at the origin,we obtain the final expression:

p(t)|t=0 = 8α2 − π2

6

[(1 + α3) + (1 − α3)

](21)

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