[ieee 2009 ieee 69th vehicular technology conference spring - barcelona, spain...

Nonlinear Amplification Effects on OFDM ErrorRate Performance in Fading Environment

Natalia Y. ErmolovaDepartment of Communications and Networking

Helsinki University of TechnologyFIN-02015, TKK, Finland

Email: natalia.ermolova@tkk.fi

Olav TirkkonenDepartment of Communications and Networking

Helsinki University of TechnologyFIN-02015, TKK, FinlandEmail: olav.tirkkonen@tkk.fi

Abstract—In this paper, we study the impact of nonlinearamplification on average error rate of OFDM in Nakagami-mfading. We consider the conventional receiver (separate detectionof each subcarrier) wuthout any countermeasures against thenonlinearity to assess the pure joint effect of fading and nonlinearamplification. We present an approximate technique that allowsto evaluate the average error rate in Nakagami-m fading in aclosed form.

I. INTRODUCTIONNonlinear distortions caused by high power amplifiers

(HPA) at the transmitters degrade reliability of the OFDMtransmission and thus their assessment and mitigation aretopics of research and practical interest.Nonlinearly distorted OFDM can be analysed on the basis

of the Bussgangs theory [1]– [2]. Originally derived formemoryless systems [1], this theory can be extended (withmodifications including memory effects) for some groups ofmemory systems, e.g. for Wiener-Hammerstein structures. Inmemoryless systems, nonlinear amplification results in anattenuation of the useful signal and its corrupting by additivenonlinear noise. The useful signal and noise are uncorrelated.Nonlinear noise is generated at the transmitter and fades

along with the useful signal in a multipath channel. On thebasis of this fact it was reported in [3] that in multipathchannels nonlinear distortions do not affect strongly the errorrate performance because it degrades the mainly owing tosubcarriers with deep fading that at the same time mitigatenonlinear noise. This conclusion is confirmed in [3] by anexample of OFDM propagation over a fixed two-path channel.While agreeing with the observation [3] that nonlinear noise

is mitigated when the signal propagates over a multipath, weat the same time observe that nonlinear amplification resultsalso in an attenuation of the useful signal as well as that theaverage bit (symbol) error rate (BER) is a fair characteristicthat is commonly used for assessment of performances andcomparison of communication systems operating in fadingchannels [4].The problem of analytical evaluation of the average BER

has been successfully solved under the assumption of linearamplification for many types of fading statistics (see, e.g. [5]).

This paper was supported by the Academy of Finland and Finnish FundingAgency for Technology and Innovation.

In the case of nonlinear amplification, the specific form of theBER expression obtained on the basis of Bussgang’s theorymakes the problem of analytical evaluation very challenging.The probabilty density functions (PDF) of the signal-to-noiseratio (SNR) under nonlinear amplification for Rayleigh andRice fading have been derived in [6]. These formulas, however,have not allowed obtaining closed-form BER expressions andan approximate expression has been given in [6].Recently [7]– [8], it was reported that under maximum-

likelihood sequence detection, clipping leads to diversity andperformance gain in frequency selective fading. But althoughsub-optimal receiver algorithms have been suggested in [7]–[8], their complexity is still high.In this paper, we study the joint effect of fading severity and

nonlinear amplification on OFDM average error rate. We con-sider conventional detection where each subcarrier is detectedseparetely. To study the effect of the fading severity it is conve-nient to consider the generalized Nakagami-m distribution. Avalue of the Nakagami m parameter (1/2 ≤m <∞) inverslyrepresents the fading severity. We prove that the effect of anonlinear amplifier on the average BER depends on the fadingseverity (the value of the Nakagami-m parameter) and transmitSNR. Analyzing the statistical behavior of the effective SNR inNakagami-m nonlinear fading channels, we suggest to approx-imate the statistical distribution of the effective SNR by thebeta-distribution. This approximation allows obtaining closed-form expressions for the average BER. Our simulation resultsconfirm a good accuracy of the suggested approximation.

II. AVERAGE BER IN NAKAGAMI-m FADING UNDERNONLINEAR AMPLIFICATION

A. Conditional BER Expression under Nonlinear Amplifica-tionBussgangs theory provides a framework for error rate eval-

uation. If an OFDM signal with N subcarriers propagates overa multipath channel with the channel gains Hk, k = 1, . . . ,N ,then the expression for the bit-error rate at the k-th subcarrierfor a nonlinearly distorted (by a memoryless HPA) signalunder separate detection of each subcarrier is [2]:

Pnonlk = aM · erfcÃs

|Hk|2βM |Hk|2 + γM

!(1)

978-1-4244-2517-4/09/$20.00 ©2009 IEEE 1

where erfc (.) is the complementary error function and aM ,βM and γM are parameters dependent on the modulation-detection combination. The parameter βM is affected by thenonlinear noise power and γM depends on that of additivechannel noise, i.e. the transmit SNR, γM ∝ 1

SNR .

B. Comparison with Nonlinear Amplification in AWGN Chan-nelLet E{|Hk|2} = μ < 1 (where E denotes the expectation).

Then the following Lemma holds.Lemma 1: In any fading channel, the average BER under

nonlinear amplification is worse than that in an AWGN chan-nel.Proof:Since erfc(.)is a convex function, on the basis of Jensen’s

inequality [9] we obtain that

Paver = E{1/NNXk=1

Pbk} ≥ aM · erfcµr

μ

βMμ+ γM

¶(2)

Since erfc(.) is a decreasing function

aM · erfc(r

μ

βMμ+ γM) > aM · erfc

µr1

βM + γM

¶=

PAWGN (3)

The inequality (3) holds for γM > 0, i.e. if channel noise doesexist.Corollary: The irreducible error floor (owing to nonlinear

amplification only) in a fading channel is that in an AWGNone:

Pfloorfad = PfloorAWGN = aM · erfcµr

1

βM

¶(4)

In absence of channel noise (γM = 0) we observe that theinequality (3) transforms into an equality.

C. Comparison with Linear Amplification in Nakagami-mFadingThe following lemma is valid.Lemma 2: In a Nakagami-m fading channel, more is the

transmit SNR and/or less is the fading severity, stronger is theeffect of nonlinear amplification (i.e. the difference betweenthe average BER under nonlinear and linear amplifications).Proof: First we consider the effect of transmit SNR increas-

ing. We rewrite (1) in an equivalent form in order to the SNRwould present explicitly:

Pnonlk(x) = aM · erfc⎛⎝s |α|2 · SNR · x

β0M · SNR · x+ γ0M

⎞⎠ (5)

where α is the attenuation factor [2], β0M and γ0M are thecorresponding parameters in (1) multiplied by SNR · |α|2and x = |Hk|2. The difference between nonlinear and linearamplification in terms of the average BER is

BERavernonl −BERaverlin =

Z ∞0

(Pnonlk(x)− Plink(x))×p(x,m)dx (6)

where p(x,m) denotes the gamma PDF [10]:

p(x) = (m

μ)m · x

m − 1Γ(m)

· exp(−mμx) (7)

In (7), Γ(m) is the gamma-function.The function Plink(x) in (6) describes a BER expression

for the case of linear amplification [2]:

Plink(x) = aM · erfcÃs

SNR · xγMlin

!(8)

We observe that the difference between arguments of erfc-functions in (8) and (5) is an increasing function of SNR thatmeans that such is the function f(x) = Pnonlk(x)−Plink(x)in (6) because erfc(.) is a decreasing function.Analysing the effect of the fading severity m on (6), we

consider the difference between values of (6) for m = m1and m =m2 such that m1 > m2, i.e.:Z ∞

0

f(x) · (p(x,m1)− p(x,m2))dx (9)

It is esay to show that the second factor in the integrand in(9) is an increasing function of (m1 −m2) that means thatsuch (9) is.In Fig. 1, we present results of comparison of the average

BER for 4-QAM OFDM with linear and nonlinear amplifiersin Nakagami m fading. In the case of linear amplification, ananalytical estimate can be obtained on the basis of [11]:

BERaverlin =

Z ∞0

Plink(x) · p(x,m)dx = 1−

2

µμ · SNR

π ·m · γMlin

¶1/2Γ(m+ 1/2)

Γ(m)×

2F1

µ1/2,m+ 1/2; 3/2;− SNR · μ

γMlin·m¶

(10)

where 2F1() is the Gauss hypergeometric function.As the nonlinear model of the HPA we use that of travelling

wave tube amplifier (TWTA) [2]. We recall that a workingpoint of an HPA defines the severity of nonlinear distortions.Commonly the working point of an HPA is expressed in termsof output back-off (OBO)that is the difference (in dB)betweenthe output saturation level and average power. The graphs inFig. 1 are shown for OBO = 2.45dB. Such an OBO valuecorresponds to rather severe nonlinear distortions. It is seenthat the results presented in Fig. 1 totally agree with thestatements of Lemma 1 and Lemma 2.

D. Approximate Technique for Analytical Evaluationof Average BER in Nakagami-m fadingWe try to assess the statistical distribution of the random

variablet =

x

βMx+ γM(11)

under the condition that x is subject to the gamma-distribution(7) in a form allowing to obtain a closed form expression forthe average BER.

2

We observe that t ∈ [0, 1/βM ] if x ∈ [0,∞). Additionally,if the first term of the denominator in (11) is dominant (i.e.channel noise is small compared with nonlinear noise), thePDF of t approaches δ(1/βM ) (where δ(.) is the Dirac δ-function). In the opposite case, where nonlinear noise is smallcompared with channel noise, the PDF of t approaches thegamma PDF. These observations allow us to try to approximatethe PDF of βM t by the beta PDF [10] owing to its shapes fordifferent combinations of the parameters (see Fig. 2):

pβ(x) =Γ(a+ b)

Γ(a)Γ(b)xa−1(1− x)b−1, x ∈ [0, 1] . (12)

It is worth noting that the approximation (12) allows toevaluate the average BER analytically by using [11].As the criterion of equivalence of the real distribution of the

variable βM t to (12)we use the equalities between first tworaw moments of both distributions. The first two raw momentsof βM t can be evaluated on the basis of [12]:

μ1 = βM

Z ∞0

x

βMx+ γM· p(x,m)dx =

mm+1

µγM

μ · βM

¶mU

µm+ 1,m+ 1,

m · γMμ · βM

¶(13)

μ2 = βM2

Z ∞0

x2

(βMx+ γM )2· p(x,m)dx = (m+ 1)×

mm+1

µγM

μ · βM

¶mU

µm+ 2,m+ 1,

m · γMμ · βM

¶(14)

In (13)-(14), U(.) is the Tricomi confluent hypergeometricfunction [11].Equating μ1 and μ2 to the corresponding raw moments of

the beta-distribution (12) [10] we define its parameters:

a = μ1

μ2μ1− 1

μ1 − μ2μ1

,

b = (1− μ1)

μ2μ1− 1

μ1 − μ2μ1

(15)

Thus the distribution of t can be approximated as

pt(x) ≈ βM · pβ(βMx) (16)

with a and b defined by (15).The approximation (16) provides a closed-form approximate

expression for the average BER under nonlinear amplificationin Nakagami-m fading obtained by using [11]:

BERaver ≈ aM

Z 1/βM

0

erfc(√x) · pt(x)dx = aM−

2aM · Γ(a+ b)Γ¡a+ 1

2

¢√πβMΓ(a)Γ

¡a+ b+ 1

2

¢×2F2

µa+

1

2,1

2; a+ b+

1

2,3

2;− 1

βM

¶(17)

where 2F2() is a hypergeometric function [11].In Fig. 3 and Fig. 4 we present estimates of the average

BER obtained for 4-QAM OFDM by numerical averaging

0 5 10 15 20 25 3010

−10

10−8

10−6

10−4

10−2

100

Transmit SNR, dB

Ave

rage

BE

R

lin.,AWGNlin., m=0.7lin., m=1.7lin., m=2.7nonl., AWGNnonl., m=0.7nonl., m=1.7nonl., m=2.7

Fig. 1. Average bit error rates of 4-QAM OFDM in linear and nonlinear(TWTA, OBO=4.41 dB) Nakagami-m fading channels.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

x

p(x)

Fig. 2. Shapes of the beta PDF for different combinations of the parameters.

(5) over the distribution (7) as well as those evaluated onthe basis of (17). The curves in Fig.3–Fig. 4 are given fordifferent OBO values and Nakagami-m parameters. It is seenthat the approximation (17) provides an acceptable accuracy,which, however, depends on values of m, SNR and OBO. Thecurves in Fig. 3–Fig. 4 indicate that the less is the transmitSNR or/and more severe are nonlinear distortions or/and lessis fading severity, the better is the accuracy of the proposedtechnique.

III. CONCLUSIONIn this paper, we consider OFDM operating in a fading

environment under the conventional (separate) detection ofeach subcarrier. We take into account nonlinear effects of the

3

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR, dB

Ave

rage

BE

R

m=1

m=2

Fig. 3. Average bit error rates of 4-QAM OFDM in nonlinear Nakagami-mfading channels. OBO=2.45 dB. Marked lines represent analytical estimates(17).

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR, dB

Ave

rage

BE

R

m=1

m=2

m=2.5

Fig. 4. Average bit error rates of 4-QAM OFDM in nonlinear Nakagami-mfading channels. OBO=4.41 dB. Marked lines represent analytical estimates(17).

transmitter HPA and analyse the effect of nonlinear amplifi-cation on the average bit error rate.We prove that although nonlinear noise generated at the

transmitter fades along with the information signal, the av-erage BER in any fading channel is worse than that in anAWGN. This means, particularly, that, as in a linear fadingchannel, power loading transforming a fading channel intoan AWGN one (i.e. a technique equalizing the received SNR[13] ) improves the system BER performance under nonlinearamplification.

For Nakagami-m fading, we propose an approximate tech-nique based on an approximation of the statistical distributionof the effective SNR by the beta-distribution. This approxima-tion allows to obtain a closed-form expression for the averageBER (formula (17)). It is worth noting that all the specialfunctions needed for evaluation of (17) are implemented inmodern software packages (e.g. in Mathematica). The ac-curacy of the proposed method depends on the fading andnonlinear severities and values of the transmit SNR. The less isthe transmit SNR or/and more severe are nonlinear distortionsor/and less is the fading severity, more accurate estimates areobtained on the basis of the proposed technique.We also prove that in Nakagami-m fading, more is the

transmit SNR and/or less is the fading severity, stronger is theeffect of nonlinear amplification, i.e. the difference betweenthe average BER under nonlinear and linear amplifications.

REFERENCES[1] J. J. Bussgang,“Crosscorrelation functions of amplitude-distorted Gau-

sian signals,” Res. Lab. Electron., M.I.T., Cambridge, MA, Techn. Rep.216, 1952.

[2] D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization ofnonlinear distortion effects in OFDM systems, ”IEEE Trans. Commun.,vol.48, pp.1755 – 1764, Oct. 2000.

[3] K. R. Panta and J. Armstrong,“Effects of clipping on the errorperformance of OFDM in frequency selective fading channels,” IEEETrans. Wireless Commun., vol. 3, pp.668 –671, March. 2004.

[4] M. K Simon and M.-S. Alouini, Digital communication over fadingchannels, Wiley, New York, 2005.

[5] P. S. Bithas, N. S. Sagias, and P. T. Mathiopoulus, “GSC diversityreceivers over generalized–gamma fading channels, ”IEEE Commun.Lett.,vol. 11, pp.964 – 966, Dec.2007.

[6] P. Banelli, “Theoretical analysis and performance of OFDM signalsin nonlinear fading channels, ”IEEE Trans. Wireless Commun. ,vol. 2,pp.284 – 293, March 2003.

[7] H. Ochiai, “Performance of optimal and suboptimal detection foruncoded OFDM system with deliberate clipping and filtering, ”IEEEGlobeCom 2003, pp.1618 – 1622,Dec. 2003.

[8] F. Peng and W. E. Ryan, “MLSD bounds and receiver design for clippedOFDM channels, ”IEEE Trans. Wireless Commun.,vol. 7, pp.3568 –3578, Sept. 2008.

[9] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.[10] A. Papoulis, Probability, Random Variables and Stochastic Processes,

McGraw-Hill, 1994.[11] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and

Series, vol. 2, Gordon and Breach Science Publishers, 1986.[12] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and

Series, vol. 1, Gordon and Breach Science Publishers, 1986.[13] N. Y. Ermolova and B. Makarevitch, “On subchannel inversion for im-

provement of power efficiency in OFDM-based systems, ”IET Commun.,vol. 1, pp.1263 – 1266, Dec. 2007.

4