# Reverse Link Outage Probabilities of Multicarrier CDMA Systems with Beamforming in the Presence of Carrier Frequency Offset

- Xiaoyu Hu
^{1}and - Yu-Dong Yao
^{1}Email author

**2008**:218740

https://doi.org/10.1155/2008/218740

© X. Hu and Y.-D. Yao. 2008

**Received: **30 April 2007

**Accepted: **25 September 2007

**Published: **21 October 2007

## Abstract

The outage probability of reverse link multicarrier (MC) code-division multiple access (CDMA) systems with beamforming in the presence of carrier frequency offset (CFO) is studied. A conventional uniform linear array (ULA) beamformer is utilized. An independent Nakagami fading channel is assumed for each subcarrier of all users. The outage probability is first investigated under a scenario where perfect beamforming is assumed. A closed form expression of the outage probability is derived. The impact of different types of beamforming impairments on the outage probability is then evaluated, including direction-of-arrival (DOA) estimation errors, angle spreads, and mutual couplings. Numerical results show that the outage probability improves significantly as the number of antenna elements increases. The effect of CFO on the outage probability is reduced significantly when the beamforming technique is employed. Also, it is seen that small beamforming impairments (DOA estimation errors and angle spreads) only affect the outage probability very slightly, and the mutual coupling between adjacent antenna elements does not affect the outage probability noticeably.

## Keywords

## 1. Introduction

Future wireless communication systems demand high-datarate multimedia transmissions in diverse mobile environments. The underlying wideband nature makes the overall system vulnerable to the hostile frequency-selective multipath fading. Code-division multiple access (CDMA) has received tremendous attentions because it offers various attractive features such as high spectrum efficiency, narrow-band interference rejection, and soft capacity [1, 2]. Recently, the multicarrier (MC) CDMA system, which is a combination of orthogonal frequency division multiplexing (OFDM) and CDMA, has gained significant interests as a powerful candidate for future wireless broadband communications [3]. Multicarrier CDMA inherits distinct advantages from both OFDM and CDMA. By dividing the full available bandwidth into a large number of small orthogonal narrow bands or subcarriers each having bandwidth much less than the channel coherent bandwidth, the transmission over each subcarrier will experience frequency nonselective fading. Also, it can be interpreted as CDMA with spreading taking place in the frequency domain rather than temporal domain, achieving enhanced frequency diversity. MC-CDMA is basically a multicarrier transmission scheme and its receiver is vulnerable to carrier frequency offset (CFO) which is due to the mismatch in frequencies between the local oscillators in the transmitter and the receiver.

Antenna array techniques are used to reduce interference to meet increased capacity requirements without sacrificing the frequency spectrum [4, 5], which can be realized through space diversity, beamforming, and spatial multiplexing [6]. In this paper, the use of conventional uniform linear array (ULA) beamformer [16] is to provide performance improvements in MC-CDMA systems, especially with the consideration of CFO.

The outage probability is an important performance measure in the design of wireless communication systems, which represents the probability of unsatisfactory reception over an intended coverage area. The performance in terms of the bit error rate (BER) for MC-CDMA systems has been investigated in a number of literatures, either assuming perfect carrier frequency synchronization [8, 9], or with CFO [10–13]. There have been several papers studying the outage probability performance in various CDMA systems [14, 15, 18]. However, MC-CDMA systems have not been examined in such studies.

In this paper, the reverse link of an MC-CDMA system with the beamforming technique in the presence of CFO is considered, and we concentrate the analysis on the outage probability performance. A Nakagami fading channel is assumed throughout the paper. Based on a newly developed simplified beamforming model [18], a closed-form expression is derived for the outage probability when perfect beamforming is considered. The impact of CFO and beamforming is modeled in signal and interference expressions. Furthermore, the effect of various beamforming impairments is examined, including direction-of-arrival (DOA) estimation errors, angle spreads, and mutual couplings. To summarize, this paper differs from previous research mainly in two aspects: first, we develop signal and interference models to characterize the beamforming gain and CFO in MC-CDMA systems; second, outage probabilities are derived for MC-CDMA systems with either perfect or imperfect beamforming in the presence of CFO.

The remainder of the paper is organized as follows. The system model is described in Section 2. The outage probability for MC-CDMA with beamforming in the presence of CFO is presented in Section 3. The effect of impairments in beamforming is investigated in Section 4. Numerical results are presented and discussed in Section 5. Conclusions are given in Section 6.

## 2. System Model

### 2.1. Beamforming

### 2.2. Simplified Beamforming Model

### 2.3. MC-CDMA

where is the channel fading gain on the th subcarrier of the th user and is assumed to be independent for different and , is the Nakagami- fading parameter which ranges from to , , and is a gamma function.

where is the power spectral density (PSD) of the additive white Gaussian noise (AWGN).

## 3. Outage Probability Analysis

Due to the complexity of the actual beamforming pattern, a closed-form expression to evaluate the average outage probability in (22) could not be derived. While, a numerical approach can be used to evaluate (22), the computation complexity of calculating above multi-dimensional integration is significant when the number of users presented in the system is large.

## 4. Outage Probability with Imperfect Beam Forming

In practice, a variety of beamforming impairments, such as DOA estimation errors, spatial spreads, and mutual couplings, exist in the system. However, the outage probability analysis in previous section is just based on perfect beamforming. In this section, we will evaluate the outage probability by considering those beamforming impairments. All impairments will affect the shape of the beam pattern and antenna gain. We need to point out that in the simplified beamforming model, only parameters and need to be modified according to the change of the beam pattern due to impairments. The outage probability can still be obtained through (24) but with revised parameters and accordingly.

### 4.1. Effect of DOA Estimation Errors

### 4.2. Effect of Angle Spreads

respectively, where is the expectation with respect to all the RVs , and . and are the mean of RV and , respectively.

### 4.3. Effect of Mutual Couplings

Substitute (31) into (2) , the modified and can be obtained.

## 5. Numerical Results

The actual beam pattern is used for the desired user, while for the interference users, the simplified beam pattern described in Section 2.2 is used.

Figure 6 gives the outage probability under different Nakagami fading parameter . Again, the SNR is set to 10 dB. The figure shows that the outage probability decreases as the parameter increases. That is because that the better channel environment the system experiences, the larger the parameter . Better channel conditions definitely improve the system performance.

In the following, a small CFO is assumed in the system, that is, ; the SNR is set to 10 dB and all users experience Nakagami fading ( ) over each subcarrier. Figure 7 shows the effect of DOA estimation errors. The DOA error is assumed to follow a uniform distribution with a standard deviation , and is given in (27). It can be seen from Figure 7 that the DOA estimation error does not impact much on the outage probability when the error is within the half null-to-null beam width . When a larger DOA estimation error is present, that is, the case of in Figure 7, it leads to a significant increase of the outage probability.

Figure 9 illustrates the impact of the mutual coupling among antenna elements on the outage probability. From Figure 9, only a very small change of the outage probability is observed when the mutual coupling exists in the system. This is because the distance between adjacent antenna elements is which is large enough to eliminate any noticeable coupling.

## 6. Conclusion

The outage probability of reverse link MC-CDMA systems with beamforming in the presence of CFO over Nakagami fading channels is evaluated in this paper. A simplified beamforming model is utilized to reduce the complexity of the analysis. A closed-form expression of the outage probability is obtained to examine the effect of CFO and beamforming. First, the outage probability is evaluated when perfect beamforming is assumed. It can be concluded that the outage probability improves significantly as the number of antenna elements increases; second, the outage probability is investigated when different types of beamforming impairments are present in the system. It is seen that small DOA estimation errors and angle spreads have only a slight impact on the outage probability of the system; however, as those impairments become large, the outage probability deteriorates significantly. Also it is observed that the outage probability changes very slightly when there is mutual coupling in the antenna array.

## Declarations

### Acknowledgment

This work has been supported in part by NSF through Grants CNS-0452235 and CNS-0435297.

## Authors’ Affiliations

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