Performance Limitations of a SIMO OFDM Wireless Link Impaired by CarrierFrequency Offset, Phase Noise and Rayleigh Fading

Mehedi HasanDepartment of Electrical and Electronic Engineering

Bangladesh University of Engineering and TechnologyDhaka, Bangladesh

mehedihasan452@yahoo.com

Dr. Satya Prasad MajumderDepartment of Electrical and Electronic Engineering

Bangladesh University of Engineering and TechnologyDhaka, Bangladesh

spmajumder@eee.buet.ac.bd

Abstract—An analytical approach is developed to evaluatethe Bit Error Rate (BER) performance of an OFDM wirelesslink over a faded channel impaired by Carrier FrequencyOffset(CFO) and phase noise. The analysis is further extendedto an OFDM system with receive diversity taking into consid-eration the effects of above system impairments. The analysisis developed for SIMO OFDM to find the expression of theunconditional bit error rate of the wireless link with Rayleighfading assuming Gaussian probability density function of theCFO and phase noise. Analysis is carried out for both MaximalRatio Receiver Combining (MRRC) and selective combiningtechniques. BER performance results are evaluated numericallyfollowing the analytical approach. The results are depictedin terms of BER versus Signal to Noise Ratio (SNR) withdifferent fading variances and for different values of CFO andphase noise variances. Results show that the system BER ishighly deteriorated due to fading, CFO and phase noise. Thus,there is significant penalty in signal power at a given BER.System BER performance results are evaluated with differentreceiver diversity combining techniques. It is noticed that thereis significant improvement in system performance increasingthe receiver sensitivity due to the application of the receivediversity techniques.

Keywords-OFDM, SIMO, Maximal ratio combining, Selec-tive combining, Carrier frequency offset, Phase noise, Rayleighfading

I. INTRODUCTION

Performance of an OFDM system in presence of carrierfrequency offset (CFO) and phase noise has been analyzedin several research works during the last few years. Thebit error rate (BER) performance evaluation of an OFDMsystem in presence of CFO and phase noise is shown in[1]. The BER performance results of OFDM system overfading channels are reported in [2], [3]. OFDM system inRayleigh faded wireless channel does not yield low biterror rate. Moreover, various synchronization errors likeCFO, phase noise etc. degrades the performance. Dopplershift introduced by channel can cause frequency differencebetween the transmitter and receiver oscillator, which resultsin carrier frequency offset. Analysis of the effect of CFOon OFDM system is shown in [4], [5]. On the other hand,fluctuation of transmitter and receiver local oscillator intro-

duces phase noise in the signal. Analysis of effects of phasenoise on an OFDM system is shown in [6], [7]. Performanceof an OFDM system is highly degraded in the presence ofRayleigh fading as well as CFO and phase noise. Thus, ina high speed OFDM based wireless communication system,it is necessary to incorporate antenna diversity techniquesto achieve low bit error rate. In both cases of selectivecombining and maximal ratio combining techniques, theprobabilistic distribution of instantaneous Signal to NoiseRatio (SNR) changes with the increasing number of receiv-ing antennas. Theoretical analysis is presented in this workto show the effects of diversity techniques on the bit errorrate performance of an OFDM system.

II. SYSTEM MODEL

A. OFDM System

Fig. 1 shows the block diagram of an OFDM wirelesscommunication system. Binary data sequence is convertedform serial to parallel and mapped as MPSK modulatedsymbols. Considering the symbols in frequency domain,OFDM modulation is done using IFFT operation. It createsa time domain base band signal. Then parallel to serialconversion is done following the insertion of cyclic prefix. Inthe wireless channel, OFDM signal is subjected to Rayleighfading and AWGN noise. In the receiver side, the desireddata sequence is demodulated from the received OFDMsignal through a reverse process of the transmitter side’sblocks.

B. SIMO OFDM System

In this model OFDM signal is transmitted using only oneantenna. But, signal travelled through wireless multipathchannel is received by two or more number of antennas.In case of selective combining, signal with highest signalto noise ratio (SNR) is selected for receiver demodulation.Fig. 2 shows the block diagram of an OFDM systemwith selective combining. On the other hand, in case ofMRC technique, information of amplitude fading and phasedistortion are required to be known for each received signal.Signals are multiplied by corresponding conjugate values of

2010 Second International Conference on Communication Software and Networks

978-0-7695-3961-4/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCSN.2010.85

573

Figure 1. Block diagram of an OFDM wireless communication system

Figure 2. Block diagram of a SIMO OFDM system with selectivecombining

phase distortion and amplitude gain. The weighted receivedsignals from all the receiving branches are then summedtogether and applied to the demodulator. Fig. 3 shows theblock diagram of an OFDM system with maximal ratiocombining.

III. THEORETICAL ANALYSIS

A. SISO OFDM System

In OFDM system, MPSK modulated symbols are rep-resented as complex values. Considering these symbols infrequency domain, IFFT operation is used to achieve thetime domain signal. m th received OFDM symbol in timedomain for ideal case is,

xm(n) =N−1∑k=0

Xm(k) exp(j2πNnk) (1)

N is the number of subcarriers and Ng is the length of cyclicprefix. n ranges from 0 to N+Ng-1. Xm(k) is the MPSK

Figure 3. Block diagram of a SIMO OFDM system with maximal ratiocombining

Figure 4. Effect of CFO on OFDM signal

modulated signal considered in frequency domain.In a Rayleigh fading environment the probability densityfunction (pdf) of fading parameter (α) will be

fα(x) = {xσ2α

exp(− x2

2σ2α

) ;x ≥ 00 ; otherwise

(2)

where σ2α = E[α2] = variance of α.

Intantaneous signal to noise ratio (SNR) γ is related tofading parameter α of that instant as

γ ∝ α2

So, pdf of γ will become exponentially distributed like thefollowing equation [8]

fγ(x) = {1

Γcexp(− x

Γc) ;x ≥ 0

0 ; otherwise(3)

Where Γc = E[γ] = mean value of SNR; Γc is also related tothe SNR value of AWGN channel (Eb/N0) by the followingrelation [8]

Γc = 2σ2α(Eb/N0)

Now conditional probability (for a particular value of γ)of bit error for MPSK modulated signal in Rayleigh fadedchannel is

Pb(e/γ) =1kerfc(

√kγ sin

π

M) (4)

Here k = log2M . So, unconditional or the average BER forOFDM systems can be calculated by integrating the previousequation with respect to instantaneous SNR

BER =∫ ∞

0

Pb(x)fγ(x)dx (5)

Using the preceding equation we can get BER for an MPSKsignal transmission through a Rayleigh fading environmentwith fading variance σ2

α and for a particular value of Eb/N0.

Carrier frequency offset (CFO) is the frequency differencebetween the transmitter and receiver oscillators. It resultsfrom Doppler shift of the signal due to mobility. Fig. 4shows the effect of CFO on an OFDM signal. It introducesinterference in the down converted signal. The expression ofSignal to Interference plus Noise Ratio (SINR) of an OFDMsignal with CFO is given by [1]

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Figure 5. Effect of phase noise on OFDM signal

SINR(γ, ε) ≥ γsinc2(πε)1 + γ[.5947sin2(πε)]

|ε| ≤ .5 (6)

Where, ε is the normalized CFO of the channel

[ε = frequency offsetsubcarrier spacing ]

To obtain the preceding expression of SINR, the carrierfrequency offset ε is modeled as Gaussian distributed processwith zero mean and unit variance.

fε(ε) =1√

2πσ2ε

exp(− ε2

2σ2ε

) (7)

σ2ε is the variance of ε

Phase noise is a random process which results from thefluctuation of the transmitter and receiver local oscillators(LO) with time. It also has adverse effect on the systemperformance. Fig. 5 shows the Inter Carrier Interference(ICI) resulted from phase noise in an OFDM signal. Thesignal to noise plus interference ratio of an OFDM signalin presence of phase noise can be obtained by the followingequation [1]

SINR(γ, σ2u) ≥ γ

1 + γ{( σ2u

2N ) ∗∑N−1r=1 (1/sin2(πrN ))}

(8)

σ2u is the variance of phase noise.

By combining eqn (6) and (8) the expression of SINR in thepresence of phase noise of variance σ2

u and carrier frequencyoffset ε can be obtained [1].SINR(γ, ε, σ2

u) ≥γsinc2(πε)

1 + γ[.5947sin2(πε) + {(σ2usinc

2(πε)2N )

∑N−1r=1 (1/sin2(πrN ))}]

(9)N is the number of subcarriers in an OFDM symbol. So,conditional probability (for particular values of γ, ε and σ2

u)of bit error in presence of CFO, phase noise and fading canbe expressed as

Pb(e/[γ, ε, σ2u]) =

1kerfc(

√k ∗ SINR(γ, ε, σ2

u) sinπ

M)

(10)

In presence of combined degrading effect of CFO, phasenoise and Rayleigh fading, expression of BER becomes

BER =∫ ∞γ=0

∫ .5

ε=−.5Pb(e/[γ, ε, σ2

u])fγ(γ)fε(ε)dγdε (11)

B. SIMO OFDM System

Performance of an OFDM system can be improved byusing multiple antenna at the receiving side. Signal traveledthrough wireless multipath channel can be received by sev-eral antennas. Best signal can be selected for demodulationby comparing all the received signals. This technique isknown as selective combining. For L th order diversity(using L antennas) with selective combining, the probabilitydensity function (pdf) of SNR (γ) can be modeled by thefollowing equation [8]

fγ,L(x) =L

Γ cexp(−x

Γ c)[1− exp(−x

Γ c)]L−1 ; for x ≥ 0

(12)So, the expression of BER in presence of CFO, phase noiseand fading has to be modified as following

BER =∫ ∞γ=0

∫ .5

ε=−.5Pb(γ, ε, σ2

u)fγ,L(γ)fε(ε)dγdε (13)

On the other hand, in case of maximal ratio combining(MRC), SNR in Rayleigh fading channel becomes chi-square distributed with 2L degrees of freedom. So, pdf isgiven by [8]

fγ,L,MRC(x) =xL−1exp(−x/Γc)

(L− 1)! ∗ ΓLc; for x ≥ 0 (14)

Therefore, BER is found in presence of CFO, phase noiseand fading by the expression

BER =∫ ∞γ=0

∫ .5

ε=−.5Pb(γ, ε, σ2

u)fγ,L,MRC(γ)fε(ε)dγdε

(15)

IV. RESULTS AND DISCUSSION

In this section various performance curves of OFDMsystem are ploted using MATLAB software in presenceof CFO, phase noise and Rayleigh fading. Bit Error Rate(BER) curves are also shown for both cases of maximal ratiocombining and selective combining techniques. Comparisonbetween these two techniques are also graphically presented.System parameters used for the plots are given in thefollowing table

Number of Subcarriers (N) 1024Cyclic Prefix Length(Ng) 64Channel Type Rayleigh FadingModulation BPSK or QPSK

Description of analytical work and results are given below

575

Figure 6. BER performance comparison between OFDM systems withand without CFO and phase noise in presence of Rayleigh fading

A.

Analytical approach is developed to evaluate the effectof CFO and phase noise on the QPSK modulated OFDMsignal in a Rayleigh fading channel and results are evaluatednumerically. In fig. 6 variance of CFO and phase noise istaken as 2−3. Variance of fading parameter σ2

α is 0.9. InRayleigh fading channel, CFO and phase noise degrades theBER performance severely. It results in a BER floor whichis independent of the value of Eb/No.

B.

Performance improvement of a SIMO OFDM systemby utilizing antenna diversity with selective combining inpresence of CFO, phase noise and fading is evaluated. Infig. 7 variances of CFO and phase noise are used 4−4.Variance of fading is used 0.50. It is observed from fig.7 that increased number of receiving antenna (L) improvesthe performance very effectively. Here for BER=10−3, therequired value of Eb/N0 is 16 dB for L=2, where it is 10dB for L=4.

C.

Performance of a SIMO OFDM system by utilizing an-tenna diversity with Maximal Ratio Combining (MRC) inpresence of CFO, phase noise and fading is evaluated. In fig.8 variances of CFO and phase noise are used 4−4. Varianceof fading is used 0.50. It can be noted from fig. 8 that MRCalso improves the system BER. Here for BER=10−3, therequired value of Eb/N0 is 14 dB for L=2, where it is 6 dBfor L=4.

D.

Comparison of Maximal Ratio Combining (MRC) andSelective Combining in presence of CFO, phase noise and

Figure 7. BER performance in presence of CFO and phase noise of aSIMO OFDM system with selective combining for different number ofreceiving antennas(L)

Figure 8. BER performance in presence of CFO, phase noise and fading ofa SIMO OFDM system with MRC scheme for different number of receivingantennas(L)

fading is obtained. In fig. 9 variances of CFO and phasenoise are used as 4−4. Variance of fading is 0.50. For bothcombining techniques, antenna number L=4 and modulationis QPSK. It is clearly observed from fig. 9 that at a givenBER the MRC provides at least 2 dB improvement insensitivity than selective combining for the same numberof receiving antennas in presence of CFO, phase noise andfading. In fig. 10 it is also observed that the probability ofbeing instantaneous SNR < Mean SNR is reduced moresharply by MRC than selective combining. For MRC thisprobability approaches zero for L > 4. On the other hand incase of selection combining the probability approaches zerofor L > 8.

576

Figure 9. Comparison of performance of selective combining and MRCin an OFDM system with CFO, phase noise and fading. For L=4

Figure 10. Probability of being instantaneous SNR < mean SNR isplotted for different number of receiving antennas in both cases of selectivecombining and MRC

V. CONCLUSION

In this work it is found that the BER performance ofan OFDM system degrades severely when the channel isRayleigh faded. Considering the poor and unacceptableperformance in Rayleigh fading environment, it has beenshown that quality of service can be improved by employingdiversity schemes. Relative merits and demerits of vari-ous combining techniques are analyzed for Rayleigh fadedchannel. Effect of frequency offset and phase noise arealso considered during the analysis of diversity schemes.Analytical BER performance results are evaluated whichshow that by increasing receiver antenna number, the systemperformance can be significantly improved even in thepresence of fading, CFO and phase noise. The complexity,

cost and performance can be optimal for being commerciallyprofitable and competitive service providing. Selective com-bining technique provides performance improvement withlittle complexity. But in the absence of any error correctionscheme, the number of antennas required is very high. ForL=8 and Eb/No= 16 dB, the BER performance becomesgood in presence of CFO, phase noise and fading. On theother hand, maximal ratio combining is more effective thanselective combining. In this method, a four (L=4) receiverantenna combination provides good BER performance. Butthe complexity of the MRC method is higher than that ofselective combining. For both methods of combining, thedistribution of instantaneous SNR shifts to the right with theincrease of antenna number. That means, the probability ofgetting poor SNR over Rayleigh faded channel reduces formultiple receiver antennas. For MRC method, the number ofantennas required to keep the instantaneous SNR distributionto the right of mean SNR is four while for selectivecombining the number of antennas required to achieve thesame effect is eight. Both diversity schemes demonstratestrong capability to effectively increase the quality of serviceof OFDM signal degraded by Rayleigh fading channel. Sodiversity scheme is an integral part of the next generationof wireless communication system.

REFERENCES

[1] S. Mallick and S. P. Majumder, “Performance analysis of anOFDM system in the presence of carrier frequency offset,phase noise and timing jitter over rayleigh fading channels,”ICECE 2008, pp. 205–210, Dec. 2008.

[2] H. Steendam and M. Moeneclaey, “Analysis and optimizationof the performance of OFDM on frequency-selective time-selective fading channels,” IEEE Transactions on Communi-cations, vol. 47, pp. 1811–1819, Dec. 1999.

[3] S. Bulumulla, S. Kassam, and S. Venkatesh, “A systematicapproach to detecting OFDM signals in a fading channel,”IEEE Transactions on Communications, vol. 48, pp. 725–728,May 2000.

[4] J. Armstrong, “Analysis of new and existing methods ofreducing intercarrierinterference due to carrier frequency offsetin OFDM,” IEEE Transactions on Communications, vol. 47,pp. 365–369, Mar. 1999.

[5] C. Ibars and Y. Bar-Ness, “Inter-carrier interference cancel-lation for OFDM systems with macrodiversity and multiplefrequency offsets,” Wireless Personal Communications, vol. 26,pp. 285–304, 2003.

[6] A. Garcia Armada, “Understanding the effects of phase noisein orthogonal frequency division multiplexing (OFDM),” IEEETransactions on Broadcasting, vol. 47, pp. 153–159, Jun 2001.

[7] S. Wu and Y. Bar-Ness, “OFDM systems in the presence ofphase noise: consequences and solutions,” IEEE Transactionson Communications, vol. 52, pp. 1988– 1996, Nov. 2004.

[8] J. W. Mark and W. Zhuang, Wireless Communication andNetwork. Prentice Hall, 2002.

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