 Research Article
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Advances in Relay Networks: Performance and Capacity Analysis of SpaceTime Analog Network Coding
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 232754 (2010)
Abstract
We propose modified spacetimecodingbased analog network coding (ANC) for multiplerelay network, termed spacetime 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 closedform expressions for the momentgenerating function (MGF) of the equivalent signaltonoise ratio (SNR) of the multiplerelay STANC network over independent and identically distributed (i.i.d.) Rayleigh, Nakagami, and Rician channels. Average SER of the system is evaluated using MGFbased approach. We further derive the approximate closedform expressions of ergodic capacity. These expressions are simple and enable effective evaluation of the performance and capacity of STANC system.
1. Introduction
Relay nodes are considered as the main candidates for longtermevolution(LTE) advanced standardization to provide incremental capacity growth and inbuilding 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 higherorder 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 [6–9], authors analyze the bit error rate (BER) and network capacity of physicallayer network coding (PNC) for frequency flatfading 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 pointtopoint transmission [10–12]. However, most of the research works reported in the literature so far focus only on the capacity bounds and do not provide the theoretical performance bound for symbol error rate (SER). In [7], an algorithm for PNC was proposed with certain assumptions of synchronization; symbol level, phase, and frequency. To make it more practical, the authors of [10] introduced an algorithm without those assumptions to exploit the lack of synchronization between interfering signals. In [13], the authors investigated the broadband bidirectional transmission with ANC scheme based on OFDM radio access in a frequencyselective fading channel. Most recently, a differential modulation scheme is proposed for analog network coding (ANCDM) in which the channel state information (CSI) is not required at both sources and the relay [14], but the differential scheme is about 3 dB away as compared with the coherent detection scheme.
This paper presents the novel extension of analog network coding based on spacetime coding technique (STANC) in a multihop scenario in view of further enhancing the network capacity and coverage area. In STANC, two terminals transmit their signals at the same instant to number of relays, and each relay forwards the amplified version of the signal (the sum of signals from both terminals as received at the relay) to both the terminals in the second phase. Both terminals transmit signals simultaneously as discussed in [15]. Each signal takes two time slots to reach the destination terminal with the same throughput as in ANC but with improved performance. Although our work builds on earlier work, it exploits the space diversity by involving space time coding as a new aspect. We derive the theoretical closedform expressions of momentgenerating function (MGF) for STANC system over Nakagami and Rician fading channels. Further, we analyze the performance of STANC in terms of SER using the MGFbased approach. The approximate closedform expressions of ergodic capacity over both Nakagami and Rician fading channels are derived.
This paper is organized as follows: Section 2 introduces the system model, channel model, transmission protocol, and equivalent SNR. Section 3 gives detailed derivations of closedform MGF expressions. Section 4 presents the approximate closedform expressions of ergodic capacity for both Nakagami and Rician distributions. Section 5 discusses the analytical and simulation results of the system performance. Finally, Section 6 concludes the paper.
Notations
, , , and denote the expectation operator, the magnitude of complex value, the gamma function, and confluent hypergeometric function of the second kind, respectively. describes the link from node to . , , and represent the multipath gain coefficient, Rician factor, and the Nakagami factor, respectively from th terminal to th relay. , , , and represent the transmitted symbol energy of terminal over link, equivalent signaltonoise ratio (SNR) at th terminal, total unconditional momentgenerating function of , and AWGN at terminal in the th time slot, respectively. and denote the th terminal and th relay, respectively.
2. System Model
The STANC cooperation model is shown in Figure 1. The terminals and communicate with each other through an number of relays , where . All the terminals are equipped with a single antenna and the relays operate in amplifyandforward mode.
2.1. 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 trainingbased 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 fading channels and Rician fading channels for all the links between terminals and relay. is the fading magnitude of the link between th terminal and th relay. ( and ) are gammadistributed random variables. The probability density function of for Nakagami distribution is given in [17] as
where is a gamma function. Here, we consider , where is the expectation operator. For the Nakagami distribution becomes the exponential distribution.
Using [18, equation (2.15)], the probability density function of for Rician distribution is given as:
where is the zeroth order modified Bessel function of the first kind and is the Rician fading factor for the link from th to th terminal. For , the Rician distribution becomes the Rayleigh distribution.
2.2. STANC Transmission Protocol
The signaling in the proposed protocol is explained in Table 1. represents the signal of th symbol from th terminal. For the basic analog network coding scheme (ANC) [13] as given in Table 1, the denotes th relay, (where ) and is the total number of relays. 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.

(i)
Case A: 2relay STANC

(ii)
Case B: 3relay OSTANC

(iii)
Case C: relay STANC (general condition).
Case A.
The Alamouti code for th terminal is given as
Terminals and send and , respectively, to in first time slot and and , respectively, to in second time slot. The ANC scheme exploits the interference of the simultaneously transmitted signals in wireless channels. Therefore, in third time slot, and amplify and forward the received signals to both and on two orthogonal channels (FDMAbased rather than TDMA). In the fourth time slot, and send and , respectively, to , and and , respectively, to in fifth time slot. In sixth time slot, and amplify the information and the combined signals are forwarded to and through orthogonal subcarriers.
Case B.
For simplicity, we assume OSTBC for th terminal by considering threerelay ANC scenario.
We focus only on the message sent by the source terminal to the destination terminal through , , and . In first, second, and third time slots, terminal transmits signal , , and to , , and , respectively. In fourth time slot, the relays retransmit the combined signals to both the terminals through orthogonal channels.
In fifth and sixth time slots, transmits to and to , respectively. In seventh time slot, and forward the scaled received signals to . Similarly, in eighth, ninth, and tenth time slots, and are transmitted to through and , respectively. In eleventh, twelfth, and thirteenth time slots, signals and are transmitted through and , respectively, to terminal .
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 is the total number of time slots used by number of relays.
2.3. InputOutput Equations
Case A.
The signals and are transmitted by (where ) to the relays and , respectively. The signals received at and are given by:
where and is the transmitted symbol energy at th terminal.
The th relay normalizes the received signal by a factor of and transmits the analog network coded signal to and . The amplification factor () at the th relay is given by
where is the average transmitted symbol energy at . The two terminals () receive the signal from th relay through two orthogonal channels and the general form of received signal is given as:
where at th terminal . knows and . We assume that each terminal knows the value of . Knowing all these values, the terminals and 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 , having variance .
Similarly, the signals and are transmitted by (where ) to the relays and , respectively. The signals received at th terminal through and are given by:
where , having variance . The received signals at can be written in matrix form as:
where
and , , where is a normalizing factor at the receiver. By multiplying on both side of (12), we arrive at:
Case B.
Using the TDMAbased protocol for the OSTANC system, the received signal from at in th time slot is given as
where . The relays normalize the received signal by a factor and transmit the analog network coded signal to both the terminals . The two terminals () receive the signals from the relays through three orthogonal channels, and the general form of received signal is given as:
where at th terminal . knows and . We assume that each terminal knows the value of . Knowing all these values, the terminal amd can recover the data and , respectively, and the received signals can be written in matrix form as
where , , , and are given as
, and , where is a normalizing factor at the receiver. The equivalent SISO model is obtained by premultiplying with both sides of (18).
Case C.
Similarly, the equivalent SISO model obtained for number of relays is given as:
3. Performance Analysis
3.1. STANC Equivalent SNR
At terminal , maximal ratio combining (MRC) is used to combine the signals received from relays. In general, the STANC equivalent SNR at th terminal can be expressed as:
where and . Here, and give the equivalent SNR for Cases A and B, respectively.
3.2. Average SER
The SER equations () for PSK and QAM modulation are as given below.

(1)
PSK: the average SER for PSK is given in [18] as:
(22)where .

(2)
QAM: the average SER for QAM is given in [18] as:
(23)where .
3.3. General STANC Moment Generating Function
In this section, we derive the expressions for unconditional MGFs to evaluate the aboveaverage SER for ANC communication over Nakagami, Rician, and Rayleigh fading channels. The MGF of is given as
We assume that and are independent random variables. The MGF for given is represented as:
3.3.1. For Nakagami Fading Channels
As (for ) are gammadistributed random variables, (25) can be written as
Equation (25) can also be written as
Equation (27) can be further simplified as:
where , with as real constants. The unconditional MGF is obtained by averaging (28) over and given as
where, is given in (30), and is the confluent hypergeometric function of the second kind.
3.3.2. For Rician Fading Channels
The conditional MGF for given of the SNR for STANC over Rician fading channels is given by:
where and are the Rician factor for link and , respectively. The unconditional MGF is obtained by averaging (31) over and given as
where is the confluent hypergeometric function of the second kind [20].
3.3.3. For Rayleigh Fading Channels
The unconditional MGF of the SNR for or (special case: Rayleigh) is given as
where and is an exponential integral function.
4. 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 twoway 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 twoway 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 signaltonoise ratio (TSNR). We resort to the numerical computation approach by using the Taylor expansion of with the mean of to obtain the secondorder approximation expression for given in [23] as:
where is given by (21) and and is the equivalent SNR at terminal from th relay in second phase of STANC bidirectional scheme. Now, the mean and second moment of can be given as:
The probability density function of in case of Nakagami distribution is given in [17] as:
where , is a gamma function. Here we consider , where is the expectation operator. For , the Nakagami distribution becomes the exponential distribution.
Using [18, equation (2.15)], the probability density function of for Rician distribution is given as:
where is the zeroth order modified Bessel function of the first kind and is the Rician fading factor for the link from th to th terminal. For , the Ricean distribution becomes the Rayleigh distribution. Now, by substituting (38) in (37), the mean and second moment of over Nakagami fading channels are respectively given by
where denotes the confluent hypergeometric function of the second kind. Similarly, by substituting (39) in (37) we compute the mean and second moment of over Rician fading channels as given in (42) and (43), respectively.
Consequently, the secondorder approximated ergodic capacity for amplify and forward (AF) of STANC twoway relay network systems can be obtained by substituting (40), (41) and (42), (43) in (36) for Nakagami and Rician fading channels, respectively.
5. 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 twoway relay network. Since, the SER performance at and are symmetrical, therefore, for simplicity, we compute the SER performance only at . 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 16QAM. 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.
5.1. SER Performance
Figures 2 and 3 show a good agreement between numerical analysis and simulation results for Nakagami and Rician fading channels, respectively. A negligible difference can be seen in numerical and MonteCarlo 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 datamodulated 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 16QAM, 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 STANCbased transmission compared to ANC for , the SNR () gain of approximately 10 dB is achieved in Rayleigh fading channels. It can be seen from the figure that the gain of STANC is more for lower SER when compared to ANC scheme without STC and the dependence of the SER performance on the with the number of channel links ( = no. of relays). The performance of STANC increases as increases owing to more spatial diversity gain. The figure also shows the SER curves for STANC and OSTANC by implementing Alamouti codes and OSTBC codes for and , respectively, for BPSK, QPSK, and 16QAM.
5.2. 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 and Rician , respectively, as a function of the with number of paths (Relays) for (i) analog network coding (ANC), (i.e., single source, destination terminal with single relay), (ii) analog network coding with spacetime Alamouti codes (STANC), and (iii) analog network coding with orthogonal space time OSTBC codes (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 twoway 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, . 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.
6. Conclusion
In this paper, the novel extension of analog network coding based on spacetime coding technique in multiple relay network under different fading scenarios is presented. The SER of ary QAM and PSK modulated signals are evaluated using the derived MGF expressions over Nakagami 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.
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Keywords
 Fading Channel
 Channel State Information
 Relay Network
 Ergodic Capacity
 Symbol Error Rate