# Performance degradation of OFDM and MC-CDMA to carrier phase jitter

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2 Fig. 1. Conceptuel block diagram of OFDM system. The demodulation is done using a fast Fourier transform (FFT) to give the samples : z i,m = 1 l=0 Es,l a i,l e k(m l) j2π e j(t) + w i,m (4) w i,m is the contribution of additive white noise. By taking into consideration different interference terms, equation (4) can be simplified as follows : z i,m = l=0 Es,l a i,l I i,l,m + w i,m (5) I i,l,m = 1 k(m l) j2π N e F e j(t) (6) In the sum described in equation (5), we can distinguish the case of l = m and the case of l m, yielding the samples: z i,m = E s,m a i,m I i,m,m + l=0 et l m Es,l a i,l I i,l,m +w i,m (7) The first term in equation (7) is referred to the useful component. This contribution can be further decomposed into an average useful component E(I i,m,m ) and a zero mean fluctuation I i,m,m E(I i,m,m ) or self interference (SI). The second contribution (l m) is the intersymbol interference (ISI), caused by other symbols. The last contribution is the additive noise term of average N 0. B. MC-CDMA system The conceptual bloc diagram of a downlink MC-CDMA system is shown in Fig.2. The data symbols a i,l transmitted to user l during the ith symbol interval are multiplied with corresponding chip c inc+n,l, n is the chip index and N c is the spreading factor. In fact, in a multiuser scenario, each user is assigned a unique spreading sequence. The sequences are considered orthogonal and consisting of user-dependent Walh-Hadamard sequences multiplied with a complex value random scrambling sequence that is common to all active users [3], [6]. The resulting samples are modulated on the orthogonal carriers (OFDM) using an IFFT of length. To avoid intersymbol interference, a guard interval of ν samples can be introduced. The obtain signal is, then, fed to the transmit pulse p(t). The resulting signal to be sent over the channel is given by : s(t) = N u l=1 i= k= ν N 1 s i,l = NF N c s i(nf+ν)+k,lp(t (i( +ν)+k)t τ c ) k= ν n=0 (8) kn j2π a i,l c inc+n,le (9) In expression (8), T is the symbol period and τ c is the time delay corresponding to phase clock transmitter. The signal is disturbed by additive white gaussian noise and a carrier phase jitter, l (t). This jitter, was first modeled as a zero mean random process with jitter variance σ 2 l [2], [5]. This random jitter has an unknown behavior. The sensitivity of MC systems performance to this random jitter has been investigated in [3]. Here, we propose to model the carrier phase jitter by gaussian and rayleigh laws and to see the effect of such jitter on downlink MC-CDMA and uplink MC-CDMA systems performance. At the receiver, the signal is applied to a filter matched to the transmit filter then sampled. The ν samples of guard interval are disregarded from the obtain samples. At the output of the FFT bloc used for demodulation, samples are multiplied with

3 Fig. 2. Conceptuel block diagram of a downlink MC-CDMA system. g l i,n coefficients of the equalizer and with the chip c i N c+n,l of considered user, then summed yielding: NF + N u z i,l = δ i,i a i,l NF + N P i = l =1 1 c in N c c+n,l i N c+n,l c g l i,n [ 1 n,n =0 e k(n n j2π ) ] e j l(t) + w i,l (10) w i,l is the contribution of the additive white noise. We can simplify and rewrite expression (10) as follows : NF z i,l = a i,l I i,i,l,l + a i,li i,i,l,l + N P + i = I i,i,l,l = 1 N c δ i,i u 1 l =0 et l l n,n =0 i = et i i a i,l I i,i,l,l + w i,l (11) c in c+n,l c i N c+n,l gl i,n N e j2π k(n n ) e j l(t) (12) The quantity I i,i,l,l represents the contribution of the symbol a i,l to the output of the receiver of the lth user during the i th MC-CDMA block. In equation (12), the first contribution is the useful component. This contribution can be further decomposed into an average useful component E(I i,i,l,l ) and a zero mean fluctuation I i,i,l,l E(I i,i,l,l ) or self interference (SI). The second contribution (i i and l = l) is the intersymbol interference (ISI), caused by other symbols from the same user. The third contribution (l l) denotes the multiuser interference (MUI). The last contribution is the additive noise term. III. PERFORMANCE DEGRADATION In this section, we compute the degradation (in decibels) of the signal to noise ratio (SNR) when carrier phase jitter is present for OFDM and MC-CDMA systems. The performance degradation of these systems is studied with small random process, gaussian process and rayleigh process for phase jitter. If SNR(0) is the signal to noise ratio in the absence of carrier phase jitter, then, the degradation is given by : SNR() SN R(0) (13) At this point, we introduce the term E(e j ) as the average of the carrier phase jitter modeled by small random process, gaussian or rayleigh distributions. For a jitter phase modeled by a small random process without any specification on the law followed by this jitter, we can write : e j(t) = 1 + j(t) (14) If we consider zero-mean jitter, then the term E(e j ) is simplified as : E(e j(t) ) = 1 (15)

5 Degradation(dB) Rayleigh model for OFDM 10 4 Rayleigh model for MC CDMA Gaussian model for OFDM Gaussian model for MC CDMA Small random model for OFDM Small random model for MC CDMA Jitter variance Fig. 3. Comparison of performance degradation for OFDM and MC-CDMA systems in presence of phase jitter with SNR(0)=30dB and assuming small random, gaussian and rayleigh jitter models carrier phase errors, we point out that OFDM and MC-CDMA have essentially the same sensitivities to phase jitter. And, by comparing small random jitter, gaussian jitter and rayleigh jitter models, we conclude that these waveforms perform better with rayleigh model. REFERENCES [1] M. Moeneclaey, The effect of synchronization errors on the performance of Orthogonal Frequency Division Multiplexed (OFDM) systems, Proc.COST 254 Emerging techniques for communication terminals, Toulouse, France, Jul97, pp [2] H. Steendam, M. Moeneclaey and H.Sari, The effect of carrier phase jitter on the Performances of Orthogonal Frequency Division Multiple Access systems, IEEE Trans. on Communications, Vol. 46, No. 4, April 98, p.456,459. [3] H. Steendam, M. Moeneclaey, Sensitivity of Multicarrier Systems to Synchronization Errors, International Symposium on Signals, Systems and Electronics ISSSE 01, Tokyo, July 24-27, 2001 (invited paper), pp [4] S.Hara, R.Prasad, Overview of Multicarrier CDMA, IEEE com.magazine, Dec 1997, Vol. 35, No. 12, pp [5] Nabila Soudani, Ridha Bouallegue, Sensitivity of OFDM/OFDMA systems to jitter phase, 3rd International Conference Sciences of Electronics, Technologies of Information and Telecommunications, SETIT 2005, Mars [6] E.A. Sourour, MNakagawa, Performance of orthogonal Multicarrier CDMA in a multipath fading channel, IEEE Trans. on comm., vol. 44, no. 3, Mars 96, pp [7] T.Pollet, M. Van Bladel, M. Moeneclaey, BER Sensitivity of OFDM systems to carrier frequency offset and wiener phase noise, IEEE Trans. on comm., vol. 43, no. 2/3/4, Feb/Mar/Apr 93, pp [8] H. Steendam, M. Moeneclaey, Sensitivity of Orthogonal Frequency Division Multiplexed systems to carrier and clok synchronisation errors, Signal Processing, Vol. 80, No. 7, 2000, pp

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