Phase noise is a topic of theoretical and practical interest in electronic circuits. Although progress has been made in the characterization of its description, there are still considerable gaps in its effects especially on multi-carrier spreading systems. In this paper, we investigate the impact of a local oscillator phase noise on the multi-carrier 2 dimensional (2D) spreading systems based on a combination of orthogonal frequency division multiplexing (OFDM) and code division multiple access

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... (CDMA) and known as OFDM-CDMA. The contribution of this paper is multifold. First, we use some properties of random matrix and free probability theory to give a simplified expression of signal to interference and noise ratio (SINR) obtained after equalization and despreading. This expression is independent of the actual value of the spreading codes and depends mainly on the complex amplitudes of estimated channel coefficients. Secondly, we use this expression to derive new weighting functions which are very interesting for the radio frequency (RF) engineers when they design the frequency synthesizer. Therefore, based on these asymptotic results, we adapt a new method to predict the bit error rate (BER) at the output of the channel decoder by using an effective SINR value. We show by simulations the validity of our models and that at a given BER, the required signal to noise ratio (SNR) may easily increase due to the carrier phase noise. Index Terms---Multi-carrier spreading systems, large system analysis, phase noise, SINR, EESM. I. INTRODUCTION Recently, orthogonal frequency and code division multiplexing (OFCDM) access technology has been investigated for the next generation of mobile communication systems [1][2]. It is a combination of orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA). The interest of the 2D spreading systems known as OFDM-CDMA is to optimize the trade-off between diversity gain and multiple access interference (MAI). To achieve high spectrum efficiency, OFDM-CDMA systems will implement a large number of subcarriers and, as a consequence, will be highly sensitive to the carrier phase noise (CPN) [3] . The effect Author manuscript, published in "IEEE Transactions on Communications (2010) TCOM-08-0122 (Index: A)" 2 of the CPN could be similar to that caused by a constant carrier frequency offset (CFO) i.e. CPN introduces a phase rotation on the received symbols and generates inter-carrier interference (ICI). However, in contrast to the CFO, the CPN is a random process and needs a whole apart study. In the literature, several works could be found on the CFO effects in the mutli-carrier based systems [3][4][5]. We will restrict our study in this paper to the CPN effects and results in OFDM-CDMA systems. In a multi-carrier transmission scheme like OFDM-CDMA, the CPN creates two effects: common phase error (CPE) and ICI. The CPE alters equivalently different carriers by causing sub-carrier phase rotation while ICI introduces interference to any sub-carrier from all other sub-carriers. In literature, the CPN effect has been presented in some papers for OFDM [3][6][7] and MC-CDMA systems [8]. Nevertheless, for MC-CDMA systems, the analytical results of [8] have assumed independent identically distributed (i.i.d.) spreading codes. Moreover, their results are only suitable for additive white Gaussian noise (AWGN) and small angle approximation of the CPN. In [9] , the performance of OFDM-CDMA systems is derived in terms of signal to interference and noise ratio (SINR) degradation at the detector output and the channel coding is not accounted for. Moreover, a small angle approximation is assumed. To the authors' knowledge, no contribution has been presented without the assumption of a small angle approximation. In this paper, we propose a new approach to present the degradation caused by the CPN on the performance of the OFDM-CDMA systems. Our contribution extends the work of [9] which was done for a CPN in the MC-CDMA system. It is also based on our previous work done for different synchronization errors [10] [11] [12] [13] . Contrarily to the previous works, we consider in this paper a CPN with small angle approximation but also with small variation approximation. The contribution of our work can be summarized as follows. 1) Using some properties of random matrix (RM) and free probability (FP) theories, we develop a new analytical expression of the SINR in an asymptotic regime. This SINR formula is independent of the actual values of the spreading codes while taking into account their orthogonality. It depends only on the complex amplitudes of the channel coefficients, the average transmitted power of the interfering users and the affected power of the reference user. 2) Using asymptotic SINRs, we are able to derive SINR degradations and new weighting functions, such as proposed by Stott for OFDM system [6] . These functions will give a better understanding of the CPN effect on the system performance. Indeed, using these functions, we show that the low frequency components of the CPN power spectral density (PSD) have more impact on the ICI while high frequency components have more impact on the inter-band interference (IBI). As a consequence, using these functions, RF engineers will adapt easily the oscillator and the phase locked-loop (PLL) specifications [14] .