doi:10.1155/2010/232754 Research Article Advances in Relay Networks: Performance and Capacity Analysis of Space-Time Analog Network Coding

We propose modified space-time-coding-based analog network coding (ANC) for multiple-relay network, termed space-time analog network coding (STANC). We present the performance and capacity analysis of the proposed network in terms of SER and ergodic capacity. We derive the closed-form expressions for the moment-generating function (MGF) of the equivalent signal-to-noise ratio (SNR) of the multiple-relay STANC network over independent and identically distributed (i.i.d.) Rayleigh, Nakagami, and Rician channels. Average SER of the system is evaluated using MGF-based approach. We further derive the approximate closed-form expressions of ergodic capacity. These expressions are simple and enable effective evaluation of the performance and capacity of STANC system.


Introduction
Relay nodes are considered as the main candidates for longterm-evolution-(LTE-) advanced standardization to provide incremental capacity growth and in-building coverage [1]. IEEE 802.16j standardization is developed for cooperative techniques to describe different modes of operation and frame structures [2]. In addition to the various advancements in the field of relay networks, it has also become vital to apply channel coding techniques to achieve high capacity and coding gain. Methods for implementing code diversity in fading scenario using higher-order coded modulation schemes are reported in [3,4]. The idea of network coding (NC) was first introduced by Ahlswede et al. [5] to enhance the capacity of the wired networks. The essential idea of NC was further extended to wireless networks to explore the broadcast nature of wireless medium and to yield significant performance improvements in the wireless networks [5]. In [6][7][8][9], authors analyze the bit error rate (BER) and network capacity of physical-layer network coding (PNC) for frequency flat-fading channel and report that the capacity of bidirectional communication in a cooperative transmission (CT) increases by employing PNC. In PNC, the logical operation, that is XOR, is used by relay to map received signal into a digital bit stream, so that the interference becomes part of the arithmetic operation in network coding. In analog network coding (ANC), the relay simply amplifies and broadcasts the received signal without any processing.
The interference is always considered as a most harmful factor for wireless transmission. Wireless networks attempt to avoid scheduling multiple transmissions at the same time in order to prevent interference. Analog network coding (ANC), recognized as a variant of PNC, follows the opposite approach, which strategically allows the selected senders to interfere in order to exploits interference instead of avoiding it. The concept of ANC was first proposed by the author of [10], in which the terminals are allowed to simultaneously transmit their signals. The wireless channel naturally combines the signals, and the relay amplifies and forwards the resultant signal instead of forwarding combined packets as in digital network coding. The potential of analog network coding is recognized in information theory where it almost doubles the canonical relay network capacity in comparison with the conventional point-to-point transmission Notations. E(·), | · |, Γ(·), and ψ(·, ·, ·) denote the expectation operator, the magnitude of complex value, the gamma function, and confluent hypergeometric function of the second kind, respectively. X → Y describes the link from node X to Y . α i j , K i j , and m represent the multipath gain coefficient, Rician-K factor, and the Nakagami-m factor, respectively from ith terminal to jth relay. E x , γ i , M γi (s), and n xk represent the transmitted symbol energy of terminal X over X → Y link, equivalent signal-to-noise ratio (SNR) at ith terminal, total unconditional moment-generating function of γ i , and AWGN at terminal X in the qth time slot, respectively. T i and R j denote the ith terminal and jth relay, respectively.

System Model
The STANC cooperation model is shown in Figure 1. The terminals T 1 and T 2 communicate with each other through an L number of relays R j , where j = 1, 2, . . . , L. All the terminals are equipped with a single antenna and the relays operate in amplify-and-forward mode.

Channel Model.
It is assumed that the channel fading coefficients will remain the same for the two phases. It is also assumed that the transmitter (source terminal) has no channel information, but only the receiver (destination terminal) has perfect channel state information (CSI), which can be acquired by applying the training-based channel estimation schemes as reported in [16], where the entire channel link from source terminal to destination terminal is only estimated at the destination terminal. Here, we assume identical and independently distributed (i.i.d) Nakagami-m fading channels and Rician fading channels for all the links between terminals and relay. h i j is the fading magnitude of the link between ith terminal and jth relay. α i j = |h i j | 2 (i = {1, 2} and j = {1, 2, . . . , L}) are gamma-distributed random variables. The probability density function of α i j for Nakagami-m distribution is given in [17] as where Γ(·) is a gamma function. Here, we consider E(|h i j | 2 ) = 1, where E(·) is the expectation operator. For m = 1 the Nakagami-m distribution becomes the exponential distribution.
Using [18, equation (2.15)], the probability density function of α i j for Rician distribution is given as: where I 0 (·) is the zero-th order modified Bessel function of the first kind and K i j is the Rician fading factor for the link from ith to jth terminal. For K i j = 0, the Rician distribution becomes the Rayleigh distribution.  [13] as given in Table 1, the R j denotes jth relay, (where j = {1, 2, . . . , L}) and L is the total number of relays. L = 1 implies that the system uses the basic ANC scheme without STC. As the number of relays increases, the system starts using STANC based on STC codes and orthogonal STANC (OSTANC) based on OSTBC codes. Further, we compare the basic ANC scheme with STANC and OSTANC in terms of the three cases listed below.
Case A. The Alamouti code for ith terminal is given as Terminals T 1 and T 2 send x 11 and x 21 , respectively, to R 1 in first time slot and x 12 and x 22 , respectively, to R 2 in second time slot. The ANC scheme exploits the interference of the simultaneously transmitted signals in wireless channels. Therefore, in third time slot, R 1 and R 2 amplify and forward the received signals to both T 1 and T 2 on two orthogonal channels (FDMA-based rather than TDMA). In the fourth time slot, T 1 and T 2 send −x * 12 and −x * 22 , respectively, to R 1 , and x * 11 and x * 21 , respectively, to R 2 in fifth time slot. In sixth time slot, R 1 and R 2 amplify the information and the combined signals are forwarded to T 1 and T 2 through orthogonal subcarriers.
Case B. For simplicity, we assume 4 × 3 OSTBC for ith terminal by considering three-relay ANC scenario.
We focus only on the message sent by the source terminal T 1 to the destination terminal T 2 through R 1 , R 2 , and R 3 . In first, second, and third time slots, terminal T 1 transmits signal x 11 , x 12 , and x 13 to R 1 , R 2 , and R 3 , respectively. In fourth time slot, the relays retransmit the combined signals to both the terminals through orthogonal channels. In fifth and sixth time slots, T 1 transmits −x * 12 to R 1 and x * 11 to R 2 , respectively. In seventh time slot, R 1 and R 2 forward the scaled received signals to T 2 . Similarly, in eighth, ninth, and tenth time slots, x * 13 and −x * 11 are transmitted to T 2 through R 1 and R 3 , respectively. In eleventh, twelfth, and thirteenth time slots, signals x * 13 and −x * 12 are transmitted through R 2 and R 3 , respectively, to terminal T 2 .
Case C. In orthogonal transmission, the number of time slots increases with the increase in the number of transmitted symbols. The general form of OSTBC is given in [19] as: where t is the total number of time slots used by L number of relays.

Input-Output Equations
Case A. The signals x i1 and x i2 are transmitted by T i (where i ∈ {1, 2}) to the relays R 1 and R 2 , respectively. The signals received at R 1 and R 2 are given by: where n rj ∼ CN (0, N 0 ) and E i is the transmitted symbol energy at ith terminal.
The jth relay normalizes the received signal by a factor of E(|y rj | 2 ) and transmits the analog network coded signal to T 1 and T 2 . The amplification factor (β j ) at the jth relay is given by where E rj is the average transmitted symbol energy at R j . The two terminals T i (i ∈ {1, 2}) receive the signal from jth relay through two orthogonal channels and the general form of received signal is given as: where and h i j . We assume that each terminal knows the value of β j . Knowing all these values, the terminals T 1 and T 2 can recover the data by removing their self information as given below: Hence, the recovered signals from both the relays can be written as: where

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Similarly, the signals −x * i2 and x * i1 are transmitted by T i (where i ∈ {1, 2}) to the relays R 1 and R 2 , respectively. The signals received at ith terminal through R 1 and R 2 are given by: where n * i j = β j h i j n i j + n i , having variance (β 2 j |h i j | 2 + 1) N 0 . The received signals at T 2 can be written in matrix form as: where and where ω i j = (β 2 j |h i j | 2 + 1) is a normalizing factor at the receiver. By multiplying H H on both side of (12), we arrive at: Case B. Using the TDMA-based protocol for the OSTANC system, the received signal from T i at R j in qth time slot is given as where n j ∼ CN(0, N 0 ). The relays R j normalize the received signal by a factor E(|y i j | 2 ) and transmit the analog network coded signal to both the terminals T i . The two terminals T i (i ∈ {1, 2}) receive the signals from the relays through three orthogonal channels, and the general form of received signal is given as: where n i j ∼ CN (0, N 0 ) at ith terminal T i . T i knows x i j and h i j . We assume that each terminal knows the value of β j . Knowing all these values, the terminal T 1 amd T 2 can recover the data x 2 j and x 1 j , respectively, and the received signals can be written in matrix form as where Y, H, X, andŃ are given as is a normalizing factor at the receiver. The equivalent SISO model is obtained by premultiplying with H H both sides of (18).
Case C. Similarly, the equivalent SISO model obtained for L number of relays is given as:

STANC Equivalent SNR.
At terminal T i , maximal ratio combining (MRC) is used to combine the signals received from L relays. In general, the STANC equivalent SNR at ith terminal can be expressed as: where γ = E/N 0 and E = E 1 = E 2 . Here, L = 2 and L = 3 give the equivalent SNR for Cases A and B, respectively.

Average SER.
The SER equations (P e ) for M-PSK and M-QAM modulation are as given below.
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The unconditional MGF is obtained by averaging (28) over α 2 j and given as M γ2 (s) where, C v, j is given in (30), and ψ(·, ·, ·) is the confluent hypergeometric function of the second kind.

For Rician Fading Channels.
The conditional MGF for given α 2 j of the SNR for STANC over Rician fading channels is given by: where K 1 j and K 2 j are the Rician factor for link T 1 → R j and T 2 → R j , respectively. The unconditional MGF is obtained by averaging (31) over α 2j and given as where ψ(·, ·, ·) is the confluent hypergeometric function of the second kind [20].

For Rayleigh Fading Channels.
The unconditional MGF of the SNR for K = 0 or m = 1 (special case: Rayleigh) is given as where k j = (1/β 2 j (1 + γs)) and Ei(·) is an exponential integral function.

Ergodic Capacity Analysis
The ergodic capacity is the expectation of the information rate over the channel distribution between the source and destination and is defined in [21] as which is upper bounded by Jensen's Inequality in [22] and is given as: It is more important to determine the capacity of STANC two-way channels in a multipath fading environment. In this context, we propose that the STANC channel ergodic capacity can be accurately approximated by a Gaussian random variable, for the case where the channel state information (CSI) is not available at the transmitter but the receiver has the perfect channel state information. We present the ergodic capacity analysis of STANC two-way relay network based on the Gaussian approximation [13] under Nakagami and Rician fading channels. The implication of this Gaussian approximation is that only the capacity mean and variance are required to obtain an accurate approximation even for a lower number of relays, for various fading scenarios, and with a wide range of transmitting signal-to-noise ratio (TSNR). We resort to the numerical computation approach by using the Taylor expansion of ln[1 + E{γ i j }] with the mean of E{γ i j } to obtain the second-order approximation expression for C erg given in [23] as: where γ 2 is given by (21) and γ i j = (γβ 2 j |h 1 j | 2 |h 2 j | 2 /(β 2 j h i j | 2 + 1)), and γ i j is the equivalent SNR at terminal T i from jth relay in second phase of STANC bidirectional scheme. Now, the mean and second moment of γ i j can be given as: The probability density function of α i j in case of Nakagamim distribution is given in [17] as: where α i j = |h i j | 2 , Γ(·) is a gamma function. Here we consider E(|h i j | 2 ) = 1, where E(·) is the expectation operator. For m = 1, the Nakagami-m distribution becomes the exponential distribution.
Using [18, equation (2.15)], the probability density function of α i j for Rician distribution is given as: where I 0 (·) is the zero-th order modified Bessel function of the first kind and K i j is the Rician fading factor for the link from ith to jth terminal. For K i j = 0, the Ricean distribution becomes the Rayleigh distribution. Now, by substituting (38) in (37), the mean and second moment of γ i j over Nakagamim fading channels are respectively given by where ψ(·, ·, ·) denotes the confluent hypergeometric function of the second kind. Similarly, by substituting (39) in (37) EURASIP Journal on Wireless Communications and Networking 7 we compute the mean and second moment of γ i j over Rician-K fading channels as given in (42) and (43), respectively.

Results and Discussions
Numerical study results illustrate the performance and capacity analysis of our proposed STANC system. The exact SER and ergodic capacity performance are analytically derived in Section 3 and 4, respectively, and are drawn by using Mathematica software and shown together with the results obtained from Monte Carlo simulations. We obtained the analytical results for SER performance by substituting (29) and (32) in (22) and (23) for Nakagami and Rician fading channels, respectively. Similarly, we obtain analytical results for ergodic capacity as discussed in Section 4 for Nakagami and Rician fading channels. All the SER and capacity results are drawn w.r.t. the average SNR. We initially present the SER performance of the two sources in the two-way relay network. Since, the SER performance at T 1 and T 2 are symmetrical, therefore, for simplicity, we compute the SER performance only at T 2 . We assume that the channel coefficients remain constant during the time slots for transmission from terminals to relay and relay to terminals. Perfect channel estimation is assumed at the receiver, but transmitter does not know the channel conditions in terms of CSI. The uncorrelated propagation channel coefficients are Rayleigh, Nakagami, and Rician distributed. We consider the three types of ideal coherent modulation/demodulation schemes that is: BPSK, QPSK, and 16-QAM. The analysis is performed in terms of symbol error rate (SER). As a future work, this work can be extended to analyze the performance of the proposed STANC networks by taking path loss and shadowing effect into consideration.  Figures 2 and 3 show a good agreement between numerical analysis and simulation results for Nakagami m = 2, 3 and Rician K = 1, 5, 10 fading channels, respectively. A negligible difference can be seen in numerical and Monte-Carlo simulation results because at relatively high SNR values it was not possible to compute the numerical integration, since the crossing points no longer exist. At such points, numerical integration errors start to appear. These curves present the performance of the system with 2 relays operating in AF mode. In this case, at the destination end, each received signal carry data-modulated symbol which is exploited through STANC to obtain spatialdiversity gain resulting in improved performance. The figures also illustrate that the performance of STANC improves due to the channel coding gain in a block frequency flatfading channel. It can be seen from Figures 2, 3, and 4 that the average SER exhibits decreasing coding gain as the constellation size increases for example, BPSK, QPSK, and 16-QAM, while full diversity is always achieved for sufficiently higher SNR. The coding gain can be certainly improved by invoking error control coding (ECC) at the expense of rate loss. Figure 4 shows the performance improvement achieved in case of STANC-based transmission compared to ANC for SER = 10 −2 , the SNR (E b /N 0 ) gain of approximately 10 dB is achieved in Rayleigh fading channels. It can be seen from the figure that the E b /N 0 gain of STANC is more for lower SER when compared to ANC scheme without STC and the dependence of the SER performance on the E b /N 0 with the number of channel links L (L = no. of relays). The performance of STANC increases as L increases owing to more spatial diversity gain. The figure also shows the SER curves for STANC and OSTANC by implementing Alamouti codes 2 × 2 and OSTBC codes 3 × 4 for L = 2 and L = 3, respectively, for BPSK, QPSK, and 16QAM.

Ergodic Capacity.
Analytically, the upper bound is derived from Jensen's inequality approach for Nakagami and Rician fading as in [22], which discussed only the Rayleigh fading channel. The ergodic capacity can be upper bounded by (35) because of concavity and with (40) and (42), this upper bound can be evaluated for Nakagami and Rician fading channels, respectively. The capacity given in (36) is evaluated using Monte Carlo numerical computational method to investigate the accuracy of the Gaussian approximations in a Nakagami and Rician fading environment. It can be seen from Figures 5 and 6 that the approximate ergodic capacity expressions of (36) are very stringent. The  difference is negligible with the increase in the number of relays and in practical SNR regions. With a fewer number of relays, the simple upper bound is not so tight. Figures 5 and 6 also illustrate the achievable ergodic capacity over Nakagami m = 2, 3 and Rician k = 1, 5, 10, respectively, as a function of the Eb/N 0 with number of paths (Relays) L = 1, 2, 3 for (i) analog network coding (ANC), (i.e., single source, destination terminal with single relay), (ii) analog network coding with space-time Alamouti codes 2 × 2 (STANC), and (iii) analog network coding with orthogonal space time OSTBC codes 4 × 3 (OSTANC). It can be observed that STANC is defined as 2 terminals communicating with each other through 2 relays by using Alamouti codes and 3, 4 relays by using OSTBC codes in the case of OSTANC. Both the source and destination terminals fully exploit the two-way transmission scheme of ANC and transmit their signals simultaneously to the relays in the first stage and receive the interfered signal from the relays in the second stage. Figures 5 and 6 show that the ergodic capacity of analog network coding (ANC), that is, single source, destination terminal with single relay is considerably increased in comparison with STANC and OSTANC scheme. This is because of the orthogonal partitioning of system resources. In ANC, the total available transmit power is divided only among the source terminals and the single relay whereas in STANC relaying, the total power is divided among the source terminals and also increasing number of relay terminals that is, L = 2, 3. It can also be observed from the figures that the performance of the networks involving more number of relays that is, STANC and OSTANC is considerably better than ANC due to the gain obtained through diversity combining.

Conclusion
In this paper, the novel extension of analog network coding based on space-time coding technique in multiple relay network under different fading scenarios is presented. The SER of M-ary QAM and PSK modulated signals are evaluated using the derived MGF expressions over Nakagamim and Rician fading channels. The approximate closedform expressions of ergodic capacity are derived for STANC system. Numerical results indicate that all the approximate analytical expressions derived are very tight and can be effectively used for performance and capacity analysis of the proposed STANC system by arbitrarily changing different parameters of interest and in any SNR region. The analysis shows that the STANC system has significant performance improvement compared to ANC due to the diversity of combining. Reduction in ergodic capacity performance is also observed due to the orthogonal partitioning of system resources.