Finite-SNR Diversity-Multiplexing Tradeoff for Cooperative Transmissions with Opportunistic Network Coding

Abstract

We present a multiuser cooperative transmission scheme with opportunistic network coding (CTONC), which can improve system performance. In contrast to direct transmission and conventional cooperative transmission, the CTONC allows the relay node to decide whether or not to help do cooperation and employ network coding based on the limited feedback from the destinations. It will not help a transmission pair unless its direct transmission fails. This enables CTONC to make efficient use of the degrees of freedom of the channels. We derive and analyze the diversity-multiplexing tradeoff for the CTONC over Rayleigh fading channels at finite signal-to-noise ratios (SNRs). Theoretical analysis and numerical results show that the proposed scheme achieves better performance gain in terms of average mutual information, outage probability, and finite-SNR diversity-multiplexing tradeoff.

1. Introduction

Cooperative transmission has recently received significant attention, both from academia and industry, as a new approach to achieve spatial diversity gains, increase coverage [1], and improve reliability of energy-constrained mobile devices in wireless cellular, ad hoc and sensor networks [2, 3]. Cooperative diversity has also been widely accepted as one special effective way to combat fading over wireless channels without additional complexity of multiple-input multiple-output (MIMO) systems. In [2], the authors develop and analyze several cooperative protocols like the amplify-and-forward (AF), decode-and-forward (DF), and their selective and incremental variations. The diversity-multiplexing tradeoff formulation is first proposed by Zheng and Tse at asymptotically high SNR [4], which is a beneficial tool to investigate the role of code design on extracting the available diversity gains and spatial multiplexing gains of cooperative relay systems [5, 6]. The opportunistic relaying cooperation scheme can achieve the same diversity-multiplexing tradeoff as achieved by the distributed space-time coded cooperation scheme [7]. Narasimhan in [8] presents new definitions of diversity gain and multiplexing gain at finite SNR, due to that some practical communication systems would rather operate in low-to-moderate SNR region. In [9], the authors analyze the diversity-multiplexing tradeoff for three practical DF half-duplex cooperative protocols in a fading relay channel at finite SNRs. Liu et al. in [10] investigate the finite-SNR diversity-multiplexing tradeoff performance for the AF and DF protocols assisted by multiple relays and compare the relative outage performance of AF and DF. In this paper, we focus on the finite-SNR diversity-multiplexing tradeoff characterization for cooperative transmissions.

Recently, there have been some studies incorporating network coding [11] in cooperative communications. Network coding is a promising approach to improve spectral efficiency and reduce time-slot overhead for multiuser cooperative communications [12, 13]. Chen et al. investigate the diversity gain offered at high signal-to-noise ratios by implementing network coding at relaying nodes. In the case of user cooperation, network coding yields better diversity performance, especially when there are multiple network users in [14]. The reference [15] proposes a network-coded cooperation scheme with dynamic coding mechanism based on the observed instantaneous source-to-relay channel quality, which achieves a diversity-multiplexing tradeoff superior to conventional cooperation. In [16], the authors study the analog network coding (ANC) with AF protocol for frequency flat Rayleigh fading channels, derive a tight lower bound of outage probability in closed-form, and obtain the finite-SNR diversity-multiplexing tradeoff. However, the disadvantage of ANC scheme is noise propagation for AF relay pattern [13, 17]. The challenging problem is, from theoretical and practical point of view, how to quantify the benefits that a wireless network can enjoy by combining network coding with cooperative transmissions, or, how much performance gain can be obtained by network-coded cooperation? To the best of our knowledge, this problem is still open. Therefore, it naturally inspires our motivation.

In this paper, we consider a cooperative network with two sources, one relay and two destinations, and study the relay cooperation assuming DF cooperation as it offers the design flexibility for implementing network coding. In such system, each node is equipped with one antenna, and the finite-SNR diversity-multiplexing tradeoff is achieved across multiple nodes via the cooperation with opportunistic network coding. “Opportunistic” means that the relay decides whether or not to help do cooperation for the 𝑠−𝑑 pairs based on the limited feedbacks from the destinations (see Figure 1). It will not help forward information for the 𝑠−𝑑 pair unless the feedback indicates that its direct communication fails. The advantage is that CTONC scheme allows the encoding relay node to decide whether it employs network coding based on the limited feedbacks.

Figure 1 

Multiuser cooperative transmission scenario.

The rest of this paper is organized as follows. Section 2 describes the system model. In Section 3, the CTONC scheme is presented over Rayleigh fading channels. We derive the finite-SNR diversity-multiplexing tradeoff for the CTONC in Section 4. Numerical results are provided in Section 5, showing finite-SNR diversity-multiplexing performance, average mutual information, and outage probability. Section 6 concludes the paper.

2. System Model

We consider a cooperative network with two sources, one relay and two destinations (see Figure 1).

The source node 𝑠₁ wants to transmit data 𝑥₁ to the destination 𝑑₁, which is overheard by the relays 𝑟 and 𝑑₁. And the source node 𝑠₂ wants to transmit data 𝑥₂ to the destination 𝑑₂, which is overheard by the relay 𝑟 and the destination 𝑑₁. In the multicast case, the node 𝑠₁(𝑠₂) wants to transmit multicast data 𝑥₁(𝑥₂) to the destinations 𝑑₁, 𝑑₂, and 𝑟. The nodes 𝑑₁ and 𝑑₂ may utilize resources more efficiently. However, the multicasting is not reliable in most wireless networks. For instance, the multicasting is lack of acknowledgments in 802.11 networks, as a result, which does not ensure a complete, reliable flow of data and may experience lower quality of service. Therefore, we only consider the former case in this paper. We assume that all nodes operate in the half-duplex mode. The multiuser cooperative transmissions are based on a synchronous time-division multiplexing. While the channel gain remains constant over one timeslot, it varies independently from one timeslot to its next. Each time slot consists of an equal amount of time. For medium access, all the transmissions happen orthogonally. We focus on orthogonal transmissions by allocating nonoverlapping time slots to different transmissions throughout the paper. All channels are assumed to be flat Rayleigh-fading, and perfect channel state information (CSI) is known at the receivers. To be specific, the received signal of node 𝑣 from node 𝑢 is given by 𝑦_𝑣=ℎ_𝑢𝑣𝑥_𝑢+𝑧_𝑣, (2.1) where ℎ_𝑢𝑣 denotes the channel gain between transmitting node 𝑢 and receiving node 𝑣, which captures the effects of path-loss, shadowing, and frequency nonselective fading. We model ℎ_𝑢𝑣 as zero-mean, independent complex Gaussian random variable with variance 𝜎²_𝑢𝑣, so that the magnitude |ℎ_𝑢𝑣| is Rayleigh distributed (|ℎ_𝑢𝑣|² is exponentially distributed with mean 𝜎²_𝑢𝑣). 𝑥_𝑢 is the transmit signal of the node 𝑢, and 𝑧_𝑣 is modeled as a complex additive white Gaussian noise (AWGN) with zero-mean and variance 𝑁₀. The random noise in the form of Gaussian white noise is possibly not satisfactory but appears fluctuations of its statistical properties and exhibits randomness in the form of fractional Gaussian noise [18]. Fractional Gaussian noise is widely used in various fields, ranging from geosciences to telecommunications, which is also modeled for communication channels and internet communication [19, 20]. It will be as our future work.

For simplicity, we assume all the transmit nodes have a common transmit power, which is denoted as 𝑃. The transmit SNR is denoted as 𝜌, and then we have 𝜌=𝑃/𝑁₀. With loss of generality, all nodes are assumed to transmit their data towards their destinations at a common rate of 𝑅 bits per channel use (bit/s/Hz). Due to the nature of fading, the receivers cannot correctly decode all the data from the transmitting nodes. We denote 𝐼 as the channel average mutual information, which is a function of the fading coefficients of the channel. The event 𝐼<𝑅 is referred to as an outage event. The probability of an outage event 𝑃(𝐼<𝑅) is referred to as the outage probability of the channel.

3. CTONC Scheme

In this section, we describe the CTONC scheme that exploits limited feedbacks from the destinations, for example, a single bit indicating the success or failure of the direct transmission, which can remarkably improve spectral efficiency. The proposed scheme is described in the wireless cooperative network in details as follow.

During the first and second time slots, the node 𝑠₁ and node 𝑠₂ transmit their information towards their destinations at the rate 𝑅. Due to the broadcast nature of wireless medium, the destination 𝑑₂(𝑑₁) can hear 𝑥₁(𝑥₂), and the relay 𝑟 can hear 𝑥₁ and 𝑥₂. We assume the links 𝑠₁→𝑑₂, 𝑠₂→𝑑₁ and 𝑠_𝑖→𝑟 have their channel gains good enough to support data transmission without error. Each destination indicates the success or failure of the corresponding direct transmission by broadcasting a single bit of feedback to the source and relay, which is assumed to be reliably observed by at least the relay 𝑟. If the SNR between the source 𝑠_𝑖 and destination 𝑑_𝑖 is sufficiently high to guarantee successful direct transmission, the feedback indicates success of the direct transmission. Otherwise, the destination 𝑑_𝑖 will send a feedback informing 𝑟 of the failure of its direct transmission.

The relay takes different actions based on different status of the direct transmissions of the two 𝑠−𝑑 pairs. If the direct transmissions of both pairs succeed, the relay will do nothing. For direct transmission, the channel is modeled as 𝑦_𝑑𝑖=ℎ_{𝑠𝑖𝑑𝑖}𝑥_𝑖+𝑧_𝑑𝑖,𝑖=1,2. (3.1)

If the direct transmission of only one of the two pairs succeeds, during the next coming time slot, the relay will decode, reencode, and forward what it receives from the source of the pair whose direct transmission fails towards the corresponding destination. For cooperative diversity transmission, after the source 𝑠_𝑖 transmits the data 𝑥_𝑖, the signals received at the destination 𝑑_𝑖 and relay 𝑟 are modeled by 𝑦_𝑟=ℎ_𝑠𝑖𝑟𝑥_𝑖+𝑧_𝑟𝑦,𝑖=1or2_𝑑𝑖=ℎ_{𝑠𝑖𝑑𝑖}𝑥_𝑖+𝑧_𝑑𝑖,𝑖=1or2. (3.2) In the next time slot for the relay transmission, the received signal at the destination 𝑑_𝑖 is modeled as 𝑦_𝑑𝑖=ℎ_𝑟𝑑𝑖𝑥_𝑟+𝑧_𝑑𝑖. (3.3) After receiving the message transmitted by the sources, the relay then retransmits the same codeword in the next time slot. This constraint allows for the receiver to perform maximum ratio combining (MRC) to combine the received signals before decoding.

If the direct transmissions of both pairs fail, the relay will perform network coding (XOR operation) on the data it receives from both 𝑠₁ and 𝑠₂ before forwarding (broadcasting) the network-coded data towards destinations 𝑑₁ and 𝑑₂. The received signal at the relay 𝑟 is modeled by 𝑦_𝑟=ℎ_𝑠𝑖𝑟𝑥_𝑖+𝑧_𝑟,𝑖=1,2. (3.4) During the next coming time slot the relay forwards 𝑥_XOR=𝑥₁⊕𝑥₂ towards the two destinations. Each destination node extracts its desired data by applying XOR operation to the overhead data from the combined data. We constrain the network-coded transmit scheme to be orthogonal, so that the transmissions of the source and relay are emitted and received over parallel channels. The two parallel channels are one from the source and the other from the relay to the destination. They are modeled as, respectively, 𝑦_𝑑𝑖=ℎ_{𝑠𝑖𝑑𝑖}𝑥_𝑖+𝑧_𝑑𝑖𝑦_𝑑𝑖=ℎ_𝑟𝑑𝑖𝑥_XOR+𝑧_𝑑𝑖. (3.5) Since 𝑑₁ and 𝑑₂ have already obtained a copy of 𝑥₂ and 𝑥₁ from 𝑠₂ and 𝑠₁ by overhearing, respectively, 𝑑₁ can recover 𝑥₁ from 𝑥₁=𝑥₂⊕𝑥_XOR=𝑥₂⊕(𝑥₁⊕𝑥₂). Similarly, the 𝑥₂ can be retrieved at 𝑑₂. Clearly, with the help of network coding, the relay is used for cooperation for both 𝑠−𝑑 pairs simultaneously. The data are aggregated and combined by network coding at the relay node and can be extracted at the destination nodes.

4. Finite-SNR Diversity-Multiplexing Tradeoff

In this section, we investigate the diversity-multiplexing tradeoff of the CTONC scheme over Rayleigh fading channels. While the conventional definitions of diversity and multiplexing gains of a system refer to asymptotic quantities as the SNR approaches infinity [4], here we consider the diversity and multiplexing definitions of [8] that extend these tradeoffs to finite-SNRs. The multiplexing gain 𝑟 is defined as the ratio of the transmission data rate 𝑅 to the capacity of an AWGN channel at SNR 𝜌: 𝑅𝑟=log₂.1+𝜌 (4.1) This definition provides an indication of the sensitivity of a rate adaptation strategy as the SNR changes. Additionally, the diversity gain 𝑑(𝑟,𝜌) of a system with multiplexing gain 𝑟 at SN 𝜌 is defined by the negative slope of the log-log plot of outage probability versus SNR: 𝑑𝜌𝑟,𝜌=−𝑃_out𝑟,𝜌𝜕𝑃_out𝑟,𝜌𝜕𝜌, (4.2) where 𝑃_out(𝑟,𝜌) is the outage probability as function of the multiplexing gain and SNR, which is 𝑃_out𝑟,𝜌=𝑃𝐼<𝑟log₂1+𝜌. (4.3) The finite-SNR diversity gain provides an estimate of the additional SNR required to decrease the outage probability by a specified amount for a given multiplexing gain.

We denote 𝐼^𝐷_𝑖 as the mutual information of the direct link from 𝑠_𝑖 to 𝑑_𝑖, and then 𝐼^𝐷_𝑖=log₂1+𝜌ℎ_{𝑠𝑖𝑑𝑖}²,𝑖=1,2. (4.4)

If only the direct transmission of 𝑠−𝑑 pair 𝑖(𝑖=1or2) fails, the relay will take repetition-based decode-and-forward action for this pair. In the case of repetition coding at the relay, the mutual information between the source and destination of pair 𝑖 can be readily shown as 𝐼^noNC_𝑖=log₂1+𝜌ℎ_{𝑠𝑖𝑑𝑖}²+𝜌ℎ_𝑟𝑑𝑖²,whenlog₂1+𝜌ℎ_𝑠𝑖𝑟²>𝑅log₂1+𝜌ℎ_{𝑠𝑖𝑑𝑖}²,otherwise. (4.5)

In the case, two pairs of direct transmissions fail, the relay will perform cooperative transmission with network coding. The transmissions, from the source to the destination and from the cooperative relay to the destination, can be viewed as parallel channels [2, 9]. The mutual information between the source and destination of pair 𝑖(𝑖=1,2) is given as 𝐼^NC_𝑖=log₂1+𝜌ℎ_{𝑠𝑖𝑑𝑖}²+log₂1+𝜌ℎ_𝑟𝑑𝑖²,whenlog₂1+𝜌ℎ_𝑠𝑖𝑟²>𝑅,𝑖=1,2log₂1+𝜌ℎ_{𝑠𝑖𝑑𝑖}²,otherwise. (4.6)

To obtain diversity-multiplexing gain tradeoff for CTONC at finite SNR, we need to compute the probability of outage and its derivative. We consider 𝑠−𝑑 pair 𝑖 suffers an outage when 𝑥_𝑖 cannot be correctly received at 𝑑_𝑖. Based on (4.4)–(4.6), the probability of outage for 𝑠₁−𝑑₁ pair is given by 𝑝^ONC₁=𝑃𝐼^NC₁<𝑅,𝐼^𝐷₁<𝑅,𝐼^𝐷₂<𝑅+𝑃𝐼^noNC₁<𝑅,𝐼^𝐷₁<𝑅,𝐼^𝐷₂≥𝑅=𝑃log₂1+𝜌ℎ_𝑠1𝑑1²+log₂1+𝜌ℎ_𝑟𝑑1²<𝑅𝑃log₂1+𝜌ℎ_𝑠2𝑑2²<𝑅×𝑃log₂1+𝜌ℎ_𝑠1𝑟²≥𝑅𝑃log₂1+𝜌ℎ_𝑠2𝑟²≥𝑅+𝑃log₂1+𝜌ℎ_𝑠1𝑑1²+𝜌ℎ_𝑟𝑑1²<𝑅𝑃log₂1+𝜌ℎ_𝑠2𝑑2²≥𝑅×𝑃log₂1+𝜌ℎ_𝑠1𝑟²≥𝑅. (4.7) For 𝑠₂−𝑑₂ pair, it is similar. Here, we define the events 𝐴₁, 𝐴₂, 𝐴₃, 𝐴₄, 𝐴₅, and 𝐴₆ as follows: 𝐴₁∶log₂1+𝜌ℎ_𝑠1𝑟²𝐴≥𝑅₂∶log₂1+𝜌ℎ_𝑠2𝑟²𝐴≥𝑅₃∶log₂1+𝜌ℎ_𝑠2𝑑2²𝐴≥𝑅₄∶log₂1+𝜌ℎ_𝑠2𝑑2²𝐴<𝑅₅∶log₂1+𝜌ℎ_𝑠1𝑑1²+𝜌ℎ_𝑟𝑑1²𝐴<𝑅₆∶log₂1+𝜌ℎ_𝑠1𝑑1²+log₂1+𝜌ℎ_𝑟𝑑1²<𝑅. (4.8) Thus, the probability of outage for 𝑠₁−𝑑₁ pair is given as 𝑃^ONC₁=𝑃𝐴₆𝑃𝐴₄𝑃𝐴₁𝑃𝐴₂+𝑃𝐴₅𝑃𝐴₃𝑃𝐴₁. (4.9) Additionally, the derivative of the outage probability with respect to the SNR 𝜌 is given as follows: 𝜕𝑃^ONC₁=𝐴𝜕𝜌𝜕𝑃₁𝑃𝐴𝜕𝜌⁶𝑃𝐴₄𝑃𝐴₂+𝑃𝐴₅𝑃𝐴₃+𝑃𝐴₁𝜕𝑃𝐴₆𝑃𝐴𝜕𝜌²𝑃𝐴₄+𝜕𝑃𝐴₄𝑃𝐴𝜕𝜌²𝑃𝐴₆+𝜕𝑃𝐴₂𝑃𝐴𝜕𝜌⁶𝑃𝐴₄+𝑃𝐴₁𝜕𝑃𝐴₅𝑃𝐴𝜕𝜌³+𝜕𝑃𝐴₃𝑃𝐴𝜕𝜌⁵. (4.10)

Substituting (4.9) and (4.10) into (4.2), we can obtain the diversity gain 𝑑(𝑟,𝜌) for 𝑠₁−𝑑₁ pair. Finally, what remains is to compute these probabilities and derivatives. Expect for 𝑃(𝐴₆), they can be derived in closed-form (see the Appendix). We focus on the computation of outage probability 𝑃(𝐴₆), which unfortunately cannot be computed in closed form. Instead, it is turned to the bounding technique described in [9, 21]. Note that 𝑅 depends on 𝜌 via (4.1): 2^𝑅=1+𝜌^𝑟. (4.11) For the 𝑃(𝐴₆), there is 𝑃𝐴₆=𝑃log₂1+𝜌ℎ_𝑠1𝑑1²+log₂1+𝜌ℎ_𝑟𝑑1²ℎ<𝑅=𝑃1+𝜌_𝑠1𝑑1²1+𝜌ℎ_𝑟𝑑1²<1+𝜌^𝑟≥𝑃1+𝜌ℎ_𝑠1𝑑1²<1+𝜌^𝑎1∩1+𝜌ℎ_𝑟𝑑1²<1+𝜌^𝑎2=𝑃1+𝜌ℎ_𝑠1𝑑1²<1+𝜌^𝑎1𝑃1+𝜌ℎ_𝑟𝑑1²<1+𝜌^𝑎2=1−exp−1+𝜌^𝑎1−1𝜆₃𝜌1−exp−1+𝜌^𝑎2−1𝜆₄𝜌, (4.12) where (1+𝜌)^𝑟=(1+𝜌)^𝑎1(1+𝜌)^𝑎2. 𝜆₃𝜌=𝜎²_𝑠1𝑑1𝜌=𝛾_𝑠1𝑑1 is the mean source-destination SNR, and 𝜆₄𝜌=𝜎²_𝑟𝑑1𝜌=𝛾_𝑟𝑑1 is the mean relay destination. To obtain accurate diversity gains at finite SNRs, the lower bound in (4.12) is maximized over the exponents 𝑎₁ and 𝑎₂ for each 𝜌. A feasible point for this optimization is determined by the fact that 𝑎₁≥0 and 𝑎₂≥0. This yields the following conditions: 0≤𝑎₁≤𝑟0≤𝑎₂𝑎≤𝑟₁+𝑎₂=𝑟. (4.13)

Using efficient nonlinear programming techniques, the computational time of 𝑎₁ and 𝑎₂ is much smaller than Monte Carlo simulations for the exact outage probability [21]. Once the optimals 𝑎₁ and 𝑎₂ are found, the outage probability and its derivative can be computed. The derivative of the bound on 𝑃(𝐴₆) in (4.12) with respect to 𝜌 is given by 𝜕𝑃𝐴₆−𝜕𝜌=exp1+𝜌^𝑎1−1𝜆₃𝜌−𝑎₁1+𝜌^𝑎1−1𝜆₃𝜌+𝜆₃1+𝜌^𝑎1−1𝜆₃𝜌²×1−exp−1+𝜌^𝑎2−1𝜆₄𝜌+1−exp−1+𝜌^𝑎1−1𝜆₃𝜌×exp−1+𝜌^𝑎2−1𝜆₄𝜌−𝑎₂1+𝜌^𝑎2−1𝜆₄𝜌+𝜆₄1+𝜌^𝑎2−1𝜆₄𝜌². (4.14)

5. Numerical Results

In this section, we demonstrate the performance of the CTONC scheme. For the simplicity, we assume that the pairs 𝑠₁−𝑑₁ and 𝑠₂−𝑑₂ are symmetrical; that is, they have the same power and channel gains. This section provides some numerical results on mutual information, outage probability, and diversity-multiplexing tradeoff for the pair 𝑠₁−𝑑₁. For the pair 𝑠₂−𝑑₂, it is similar. The performance comparison of direct transmission scheme (DT), cooperative non-NC transmission scheme (CnoNC), and CTONC scheme is presented under different conditions as follows.

Figure 2 depicts the mutual information versus SNR for different schemes. We can see that both CTONC and CnoNC schemes increase the mutual information. However, the mutual information of CnoNC scheme is smaller than that of CTONC scheme when SNR 𝜌>6. The outage probabilities versus transmission rate under all three schemes are illustrated in Figure 3. It is shown that the outage probability of CTONC scheme is smaller than that of other schemes. The performance of CnoNC scheme is superior to DT scheme. As the transmission rate increases, the outage probability of the CTONC scheme increases more slowly than other schemes, meaning that the proposed scheme is less sensitive to the transmission rate variations compared with the direct transmission and cooperative transmission without network coding.

Figure 2 

Mutual information comparison of the three schemes.

Figure 3 

Outage probability comparison of the three schemes.

In Figure 4, we compare the finite-SNR diversity-multiplexing tradeoff performance of the three schemes at SNR values of 0 and 40 dB. As the multiplexing gains increase, the diversity gains degrade. The DT and CTONC schemes achieve the multiplexing gain of upper bound. The DT scheme can achieve larger diversity gains than CTONC scheme for high multiplexing gains at low SNR. However, the diversity gain of the CTONC and CnoNC schemes are superior to DT scheme for low multiplexing gains. The maximum achievable multiplexing gain of the CnoNC scheme is 0.5.

Figure 4 

Finite-SNR diversity-multiplexing comparison of the three schemes.

The outage probability versus SNR for the three schemes is illustrated at different multiplexing gains in Figure 5. We can see that the outage probability performance of CTONC scheme is optimal at multiplexing gain 𝑟=0.25 and is better than that of other schemes for high SNR at multiplexing gain 𝑟=0.5. In addition, the outage probability performance of CTONC and CnoNC schemes is always superior to the DT schemes across the whole range of SNR and at different multiplexing. Furthermore, as the multiplexing gain increases, it is clearly shown that the outage probability performance of all the three schemes degrades.

Figure 5 

Outage probability comparison of the three schemes with different multiplexing gains.

6. Conclusion

In this paper, we propose the opportunistic network coding for relay cooperative transmissions to provide a feasible method for performance improvement and study mutual information, outage probability, and the finite-SNR diversity-multiplexing tradeoff. On the whole, the proposed CTONC scheme outperforms other schemes in terms of mutual information and outage probability. For low multiplexing gains, the CTONC scheme achieves larger diversity gains than other schemes at finite SNRs. As a future work, we will extend the proposed scheme to cooperative MIMO in the context of cellular mobile systems, consider fractional Gaussian noise and further investigate performance gains.

Appendix

Outage Probability Derivation

Obviously, |ℎ_𝑢𝑣|² is exponentially distributed with parameter 𝜎⁻²_𝑢𝑣. Let 𝑋=𝜌|ℎ_𝑢𝑣|² be exponential random variable with parameter 1/𝜆𝜌, where 𝜆=𝜎²_𝑢𝑣. Thus, the PDF of 𝑋 can be expressed as 𝑝_𝑥(𝑥)=1−𝑥𝜆𝜌exp𝜆𝜌𝑥≥0. (A.1) Using the above PDF, the outage probability is given as follows: 𝑃𝐴₁=𝑃log₂1+𝜌ℎ_𝑠1𝑟²𝜌ℎ≥𝑅=1−𝑃_𝑠1𝑟²<2^𝑅−1=1−^2𝑅₀⁻¹𝑝_𝑥(𝑥)−2𝑑𝑥=exp^𝑅−1𝜆₁𝜌=exp−1+𝜌^𝑟−1𝜆₁𝜌, (A.2) where 𝜆₁=𝜎²_𝑠1𝑟. Then, the derivative 𝜕𝑃(𝐴₁)/𝜕𝜌 can be calculated by 𝜕𝑃𝐴₁−𝜕𝜌=exp1+𝜌^𝑟−1𝜆₁𝜌−𝑟1+𝜌^𝑟−1𝜆₁𝜌+𝜆₁1+𝜌^𝑟−1𝜆₁𝜌². (A.3) Based on (4.8), (A.2), and (A.3), for 𝜆₂=𝜎²_𝑠2𝑟 and 𝜆₅=𝜎²_𝑠2𝑑2, we can easily obtain 𝑃(𝐴₂), 𝜕𝑃(𝐴₂)/𝜕𝜌, 𝑃(𝐴₃), and 𝜕𝑃(𝐴₃)/𝜕𝜌, respectively.

Similarly, the probability of outage 𝑃(𝐴₄) and its derivative are given by 𝑃𝐴₄=𝑃log₂1+𝜌ℎ_𝑠2𝑑2²−<𝑅=1−exp1+𝜌^𝑟−1𝜆₅𝜌𝜕𝑃𝐴₄−𝜕𝜌=−exp1+𝜌^𝑟−1𝜆₅𝜌−𝑟1+𝜌^𝑟−1𝜆₅𝜌+𝜆₅1+𝜌^𝑟−1𝜆₅𝜌². (A.4) The 𝑃(𝐴₂) is shown as 𝑃𝐴₅=𝑃log₂1+𝜌ℎ_𝑠1𝑑1²+𝜌ℎ_𝑟𝑑1²=<𝑅1−1+1+𝜌^𝑟−1𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₃𝜌,𝜆₃=𝜆₄𝜆1−³𝜌𝜆₃𝜌−𝜆₄𝜌exp−1+𝜌^𝑟−1𝜆₃𝜌−𝜆₄𝜌𝜆₄𝜌−𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₄𝜌,𝜆₃≠𝜆₄, (A.5) where 𝜆₃=𝜎²_𝑠1𝑑1 and 𝜆₄=𝜎²_𝑟𝑑1. When 𝜆₃=𝜆₄, its derivative is given below 𝜕𝑃𝐴₅−𝜕𝜌=−exp1+𝜌^𝑟−1𝜆₃𝜌𝑟1+𝜌^𝑟−1𝜆₃𝜌−𝜆₃1+𝜌^𝑟−1𝜆₃𝜌²−1+1+𝜌^𝑟−1𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₃𝜌−𝑟1+𝜌^𝑟−1𝜆₃𝜌+𝜆₃1+𝜌^𝑟−1𝜆₃𝜌²=1+𝜌^𝑟−1𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₃𝜌𝑟1+𝜌^𝑟−1𝜆₃𝜌−𝜆₃1+𝜌^𝑟−1𝜆₃𝜌². (A.6) For 𝜆₃≠𝜆₄, there is the following: 𝜕𝑃𝐴₅=𝜆𝜕𝜌³−𝜆₄𝜆₃𝜌−𝜆₄𝜌²𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₄𝜌−𝜆₄𝜌exp−1+𝜌^𝑟−1𝜆₄𝜌−1𝜆₃𝜌−𝜆₄𝜌𝜆₃exp−1+𝜌^𝑟−1𝜆₃𝜌+𝜆₃𝜌exp−1+𝜌^𝑟−1𝜆₃𝜌×−𝑟1+𝜌^𝑟−1𝜆₃𝜌+𝜆₃1+𝜌^𝑟−1𝜆₃𝜌²+1𝜆₃𝜌−𝜆₄𝜌𝜆₄exp−1+𝜌^𝑟−1𝜆₄𝜌+𝜆₄𝜌exp−1+𝜌^𝑟−1𝜆₄𝜌−𝑟1+𝜌^𝑟−1𝜆₄𝜌+𝜆₄1+𝜌^𝑟−1𝜆₄𝜌². (A.7) Based on the above formulas, we can characterize the outage probability and finite-SNR diversity-multiplexing tradeoff performances.

Acknowledgment

This work has been supported by National S&T Major Program (no. 2010ZX03003-003-01), the National Natural Science Foundation of China (no. 61070204), and the Fundamental Research Funds for the Central Universities (BUPT2010PTB0503), the Open-ended fund program of Guangxi Key Lab of Wireless Wideband Communication & Signal Processing (no. 21106).