Achievable data rates and power allocation for frequencyselective fading relay channels with imperfect channel estimation
 2.2k Downloads
 2 Citations
Abstract
In this article, we investigate the informationtheoretical performance of a cooperative orthogonal frequency division multiplexing (OFDM) system with imperfect channel estimation. Assuming the deployment of trainingaided channel estimators, we derive a lower bound on the achievable rate for the cooperative OFDM system with amplifyandforward relaying over frequencyselective Rayleigh fading channels. The bound is later utilized to allocate power among the training and data transmission phases. Numerical results demonstrate that the proposed power allocation scheme brings between 5 and 19% improvement depending on the level of signaltonoise ratio and relay locations.
Keywords
Achievable rate Amplifyandforward relaying Channel estimation Power allocation OFDM1 Introduction
Cooperative transmission has been proposed as a powerful method to overcome the degrading effects of fading in wireless channels [1, 2, 3]. Exploiting the broadcasting nature of the wireless channel, cooperative transmission builds upon the idea of a number of nodes helping each other through relaying. It extracts spatial diversity advantages in a distributed manner and brings significant improvements in link reliability, spectral efficiency, and coverage area. Two popular relaying schemes are decodeandforward (DF) and amplifyandforward (AF), which are sometimes referred to as regenerative and nonregenerative relaying, respectively. In AF relaying, the relay node retransmits a scaled version of the received message without any attempt to decode it. In DF relaying, the relay node decodes the received message, reencodes, and transmits to the destination.
Informationtheoretical aspects of cooperative communications have been investigated by several authors [4, 5, 6, 7, 8]. Gastpar and Vetterli [4] have examined the asymptotic capacity as the number of relay nodes goes to infinity. In their derivations, they have assumed arbitrarily complex network coding over Gaussian relay channels and ignored the effects of fading. Wong et al. [7] have derived upper and lower bounds on the capacity for both deterministic (i.e., fixed channel coefficients) and Rayleigh fading channels. Optimum resource allocation has been proposed in [9, 10] to optimize the capacity of AF networks. Specifically, Maric and Yates [9] have investigated power and bandwidth allocations for a large number of relay nodes assuming that the channel state information is available at the transmitter. Deng and Haimovich [10] have developed power allocation strategies to optimize the outage performance for a singlerelay AF cooperative system. Zheng and Gursoy [11] have derived achievable rates for AF and DF relaying with imperfect channel estimation.
A common assumption in the aforementioned works is frequencyflat fading channel model. Although this model is sufficient to model narrowband systems, it becomes unrealistic for broadband communication systems where the transmission bandwidth is larger than the coherence bandwidth of the channel. This, in return, results in a frequencyselective channel, which causes intersymbol interference (ISI) at the receiver. A widely used approach to overcome the degrading effects of ISI is orthogonal frequency division multiplexing (OFDM). OFDM has already been adopted by various industry standards such as IEEE 802.11 (WiFi) and 802.16 (WiMax). Currently, there has been a growing interest in the application of OFDM to cooperative communication systems [12, 13, 14, 15, 16]. In [12], a space–time cooperative protocol with the transmitter and receiver architecture, frame structure, and synchronization algorithms are designed for an OFDM relay system. In [13], power loading is considered in the frequency and time domains to maximize an instantaneous rate, assuming channel knowledge is available at the transmitter. In [14], equalization methods for cooperative diversity schemes over frequencyselective channels have been investigated. Ma et al. [15] have proposed a marginadaptive bit and power loading approach for an OFDM singlerelay system. Ibrahimi and Liang [16] have investigated joint power allocation among the source, relays, and OFDM subchannels for coherent reception. These works are based on the assumption that perfect channel knowledge is available at the receiver and/or transmitter. In practice, the channel coefficients need to be estimated and made available to the receiver. Recent research efforts have focused on the analysis and design of OFDM relay systems with imperfect channel estimations. Amin and Uysal [17] have investigated bit and power loading for an AF OFDM relay systems using bit error rate as the performance measure. Wang et al. [18] have considered the resource allocation and relay selection in a DF orthogonal frequency division multiple accessbased downlink network. However, few of the current works address the achievable rates for an OFDM system with imperfect channel estimation.
In this article, we study the achievable rate for a singlerelay OFDM system with AF relaying and a trainingaided channel estimator at the receivers. We assume no knowledge of channel state information at the transmitter side realizing an openloop scheme. Minimum mean square error (MMSE) estimators are applied to obtain the channel estimates. In the derivation of the achievable rate of the OFDM relay system, the channel estimation errors are considered together with the noise forwarded from the relay node and noise at the destination. Since the statistical distributions of the channel estimation errors are difficult to characterize, a closedform analytical expression for the achievable rate is mathematically intractable. We, therefore, resort to find a lower bound on the achievable rate. Then, we use this bound to allocate power between the training and data transmission phases.
The rest of the article is organized as follows: Section 2 introduces the system model and describes the training and data transmission phases. In Section 2, we derive a lower bound on the achievable rate, assuming the distintegrated estimation of sourcetorelay and relaytodestination links. Section 2 presents a power allocation scheme by maximizing the derived bound. In Section 2, we investigate the problem of cascaded channel estimation for the overall relaying link. Numerical results are provided in Section 2. Finally, Section 2 concludes the article.
Notation
Matrices and column vectors are denoted by uppercase and lowercase boldface characters, respectively (e.g., A, a). The transpose of A is denoted by A^{T}, and the conjugate and transpose of A by A^{H}. A vector s of length N is denoted by s=[s(1),s(2),…,s(N)]. I_{ K } denotes a K×K identity matrix and 0 stands for an allzero matrix of appropriate dimensions. X(i,j)denotes the (i,j)th element in matrix X. The i th diagonal element in diagonal matrix D is denoted by D(i). $\mathbb{E}[\xb7]$ is the expectation operator, and log(·) represents a logarithm of base 2. The notation $\mathbf{n}\sim \mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{\Sigma})$ means that n is a circularly symmetric complex Gaussian (CSCG) random vector with zero mean and covariance matrix Σ. Matrix V_{AB} denotes a Q×L_{AB} matrix whose (k,m)th element is given by V_{AB}(k,m)=exp(−j 2Π(k−1)(m−1)/Q), 1≤k≤Q, 1≤m≤L_{AB}.
2 System model
The underlying channels are modeled as frequencyselective Rayleigh fading with a uniform delay profile. To overcome the ISI in frequencyselective channels, we apply the OFDM scheme to the relay system, which converts the frequencyselective fading channel into a number of parallel frequencyflat channels free of ISI. An aggregate channel model consisting of both longterm path loss and shortterm fading effects is considered. The path loss is proportional to d^{−γ}, where d is the propagation distance and γ is the path loss exponent. By normalizing the path loss in the sourcetodestination (S→D) link to be unity, the relative gains from sourcetorelay (S→R), and from the relaytodestination (R→D) links are defined, respectively, as K_{SR}=(d_{SD}/d_{SR})^{ γ }and K_{RD}=(d_{SD}/d_{RD})^{ γ }[19]. The channel impulse responses (CIRs) for S →R, R → D, and S → D links are given, respectively, by h_{SR} = [h_{SR} (1), h_{SR} (2), …, h_{SR} (L_{SR})], h_{RD} = [h_{RD} (1), h_{RD} (2),…,h_{RD} (L_{RD})], and h_{SD} = [h_{SD} (1), h_{SD} (2),…,h_{SD} (L_{SD})]. The entries in h_{SR},h_{RD}, and h_{SD}are independent identically distributed (i.i.d) zeromean CSCG random variables with variances of 1/L_{SR}, 1/L_{RD}, and 1/L_{SD}, respectively. The underlying channels are modeled as quasistatic Rayleigh fading, whereas the CIRs remain constant in the duration of one OFDM block and change to independent values that hold for another block.
To avoid the interblock interference for the OFDM system, a cyclic prefix of length max(L_{SD},L_{SR},L_{RD})−1 is applied. After the cyclic prefix is removed at the receivers, the length of the OFDM block is denoted by Q (which is also the number of subcarriers in one OFDM block). We assume that both relay and destination nodes are equipped with channel estimators. The relay node obtains an estimate of the CIR from the S → R link through training symbols and feedforwards this information to the destination. The relay node also transmits “clean” training symbols so that the CIR for the R →D link can be obtained at the destination.^{a} In [20], it is proven that the minimum length of training symbols required for a noncooperative OFDM system equals the channel length, and the optimal placement is that the training symbols are periodically inserted in each OFDM block. In this article, we adopt a similar channel training strategy. The number of training symbols is chosen as the maximum channel length among the links, i.e., N=max(L_{SD},L_{SR},L_{RD}), where N is the number of training symbols. The number of subcarriers in an OFDM block is chosen as Q=(M + 1)N, with M≥1being an integer. The training symbols are placed periodically at positions i_{ ℓ }=1 + (ℓ−1)(M + 1),ℓ=1,…,N in the OFDM block.
Let the vectors x_{S}=[x_{S}(1),x_{S}(2),…,x_{S}(N)]^{ T }and x_{ R }=[x_{ R }(1),x_{ R }(2),…,x_{ R }(N)]^{ T } denote, respectively, the training symbols transmitted from the source and relay nodes. The data symbols are collected in vector y=[y(1),y(2),…,y(MN)]^{ T }. With the training symbols periodically inserted, an OFDM block transmitted from the source node is expressed as [x_{S}(1),y(1),y(2),…,y(M),x_{S}(2),y(M + 1),…,y(2M),…,x_{S}(N),y((N−1)M + 1)…,y(MN)]. Let P_{S} and P_{ R } denote, respectively, the available power at the source and relay nodes. Assuming that the training symbols are independent of the data symbols, we define ${P}_{\mathrm{S}}=\frac{1}{(M+1)N}({\mathbf{x}}_{\mathrm{S}}^{H}{\mathbf{x}}_{\mathrm{S}}+\mathbb{E}[{\mathbf{y}}^{H}\mathbf{y}\left]\right)$, and ${P}_{R}=\frac{1}{(M+1)N}({\mathbf{x}}_{\mathrm{R}}^{H}{\mathbf{x}}_{\mathrm{R}}+\mathbb{E}[{\mathbf{w}}_{\mathrm{R}}^{H}{\mathbf{w}}_{\mathrm{R}}\left]\right)$, where ${\mathbf{w}}_{\mathrm{R}}^{H}$ is the signal vector forwarded from the relay node. The power allocated in training and data transmission phases at the source and relay nodes can individually be written as ${\mathbf{x}}_{\mathrm{S}}^{H}{\mathbf{x}}_{\mathrm{S}}/N={\alpha}_{\mathrm{t}}{P}_{\mathrm{S}},\mathbb{E}\left[{\mathbf{y}}^{H}\mathbf{y}\right]/\left(\mathit{\text{MN}}\right)={\alpha}_{\mathrm{d}}{P}_{\mathrm{S}},{\mathbf{x}}_{\mathrm{R}}^{H}{\mathbf{x}}_{\mathrm{R}}/N={\beta}_{\mathrm{t}}{P}_{\mathrm{R}}$, and $\mathbb{E}\left[{\mathbf{w}}_{\mathrm{R}}^{H}{\mathbf{w}}_{\mathrm{R}}\right]/\left(\mathit{\text{MN}}\right)={\beta}_{\mathrm{d}}{P}_{\mathrm{R}}$, where α_{t},α_{d},β_{t}, and β_{d} are, respectively, the power allocation factors deployed at the source and the relay node, and they are related by α_{t} + M α_{d}=M + 1 and β_{t} + M β_{d}=M + 1.
2.1 Training phase
2.2 Data transmission phase
3 Lower bound on the achievable rate
4 Power allocation
5 Lower bound on the achievable rate and power allocation for the case of cascaded channel estimation
So far, we have assumed that both relay and destination nodes are equipped with channel estimators and the channel estimates in S→R and R→D links are obtained, respectively, by the training symbols sent at the source and the relay nodes. In this section, we assume that only the destination is equipped with a channel estimator. Therefore, it is the duty of the destination to obtain an estimate of the overall relaying S→R→D link using the training symbols sent from the source. In describing this alternative scheme, we try to use the same variables, whenever possible, as in prior sections, or we use $\stackrel{\u030c}{(\xb7)}$ whenever necessary.
Inserting the training symbols periodically (similar placement as in prior discussion), the OFDM block transmitted at the source node is written as $\mathbf{d}=\left[{x}_{\mathrm{S}}\left(1\right),y\left(1\right),y\left(2\right),\dots ,y\left(M\right),{x}_{\mathrm{S}}\left(2\right),y(M\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1),\dots ,y\left(2M\right),\dots ,{x}_{\mathrm{S}}\left(N\right),y\left((\u01471)M\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right),\dots ,y\left(\mathrm{M\u0147}\right)\right]$, where ${x}_{\mathrm{S}}\left(i\right),i=1,\dots ,\u0147$ are the training symbols. The power constraint at the source is ${P}_{\mathrm{S}}=\frac{1}{(M+1)\u0147}({\mathbf{x}}_{\mathrm{S}}^{H}{\mathbf{x}}_{\mathrm{S}}+\mathbb{E}[{\mathbf{y}}^{H}\mathbf{y}\left]\right)$, ${\mathbf{x}}_{\mathrm{S}}^{H}{\mathbf{x}}_{\mathrm{S}}/\u0147={\alpha}_{\mathrm{t}}{P}_{\mathrm{S}},\mathbb{E}\left[{\mathbf{y}}^{H}\mathbf{y}\right]/\left(\mathrm{M\u0147}\right)={\alpha}_{\mathrm{d}}{P}_{\mathrm{S}}$. To remove interblock interference, a cyclic prefix is added at the beginning of the transmitted vectors at the source node. Let d_{wcp}denote the transmitted vector with cyclic prefix, the length of which is $\stackrel{\u030c}{Q}+{L}_{\text{cp}}$, where $\stackrel{\u030c}{Q}=(M+1)\u0147$ is the total length of training symbols and data information (length of vector D), L_{cp} is the length of the cyclic prefix, and “wcp” stands for “with cyclic prefix.” The received signal at the relay can be expressed as $\sqrt{{K}_{\text{SR}}}{\mathbf{h}}_{\text{SR}}\otimes {\mathbf{d}}_{\text{wcp}}$, in addition to the noise at relay, where ⊗denotes the operation of convolution. Scaling the received signal by factor $\stackrel{\u030c}{A}=\sqrt{\frac{{P}_{r}}{(\stackrel{\u030c}{Q}+{L}_{\text{cp}})\left({K}_{\text{SR}}(\u0147{\alpha}_{t}+\mathrm{M\u0147}{\alpha}_{d}){P}_{S}+{\sigma}^{2}\right)}}$, the relay forwards the signal to the destination. The received signal at the destination is, therefore, $\stackrel{\u030c}{A}\sqrt{{K}_{\text{SR}}{K}_{\text{RD}}}{\mathbf{h}}_{\text{RD}}\otimes {\mathbf{h}}_{\text{SR}}\otimes {\mathbf{d}}_{\text{wcp}}$ and subject to the noise forwarded by the relay and noise at the destination.
There seems no analytical solutions to the power allocation factors α_{t} and α_{d} to maximize (36). The solutions can be obtained by maximizing the bound numerically.
6 Numerical results
In this section, we present numerical results to elaborate the derived bound on the achievable rate and the possible improvements through the proposed power allocation schemes. Unless specified otherwise, the plots are obtained for the disintegrated channel estimation scheme described in Section 2. For simplicity, the angle between links S →R and R →D in the relay system is chosen as θ=60°, and the relay is located at equal distance from the source and the destination, except for the case where the effect of relay locations is considered. The following power allocation schemes for the training and data transmission are considered:

The proposed power allocation (ProposedPA) strategy, with the values of ${\stackrel{\u0306}{\alpha}}_{\mathrm{t}}$ and ${\stackrel{\u0306}{\beta}}_{\mathrm{t}}$ given, respectively, in (28) and (29).

Uniform power allocation (UniformPA) with α_{t}=β_{t}=1.

Numerical power allocation (NumericalPA), with the values of α_{t}and β_{t}obtained by a numerically exhaustive search to maximize the lower bound on the trainingbased achievable rate in (23).
6.1 Performance of power allocation schemes
Comparison of the power allocation factors for ProposedPA and NumericalPA
SNR in dB  ProposedPA${\stackrel{\u0306}{\alpha}}_{\text{t}}$  NumericalPAα_{t}  ProposedPA${\stackrel{\u0306}{\beta}}_{\text{t}}$  NumericalPAβ_{t} 

0  7.42  7.36  6.6  6.31 
6  6.15  6.31  5.43  5.26 
12  5.74  5.26  5.07  5.26 
18  5.63  5.26  4.97  5.26 
24  5.60  5.26  4.95  5.26 
30  5.60  5.26  4.94  5.26 
To demonstrate the performance in the case of cascaded channel estimation described in Section 2, we obtain the power allocation factors by numerically maximizing the bound derived in (36) and plot the maximized bound. As shown in Figure 2, the curve is marked by “ProposedPA (cascaded estimation).” In this case, the number of training symbols is N=max(L_{SD},L_{SR} + L_{RD})−1=5, and the OFDM block length is chosen as 205. The bound in (36) with uniform power allocation α_{ t }=1 is also included, marked by “UniformPA (cascaded estimation).” It can been observed that, with appropriate power allocation factors, the lower bounds on the achievable rates remain almost the same, regardless of whether the channel gains in the relaying link are estimated as two separate channels or as one overall channel.
In the following, we demonstrate the impact of some practical parameters such as OFDM block length, relay location, and channel lengths on the bound of the achievable rate.
6.2 OFDM block length
6.3 Relay location
6.4 Channel lengths
In Figure 5, we investigate the effect of channel lengths on the achievable rate. We consider the following three scenarios:

Scenario 1: L_{SD} = 30, L_{SR} = 3, L_{RD} = 3

Scenario 2: L_{SD} = 3, L_{SR} = 30, L_{RD} = 3

Scenario 3: L_{SD} = 3, L_{SR} = 3, L_{RD} = 30
For these channel configurations, the number of pilot symbols increased to N=max(L_{RD},L_{RD},L_{RD})=30. We choose the block length Q=30(M + 1)=150with M=4, and SNR =10 dB. It can be observed that the system achieves almost the same bounds for Scenarios 2 and 3, and a little less for Scenario 1. These can be explained through the values of B_{1} and B_{2}. One can observe from (24) and (25) that B_{1}and B_{2} capture, respectively, the equivalent SNR in the direct link and in the relaying link with imperfect channel gains. In Scenarios 2 and 3, B_{1} and B_{2} yield identical values, resulting in a similar performance for the mutual information. On the other hand, for Scenario 1, the value ofB_{1} drops by 89% while B_{2} increases by 32%. The larger drop in B_{1}results in a little more decrease of the bound in Scenario 1.
6.5 Channel estimate and power allocation
7 Conclusion
In this article, we have investigated the achievable rates for a singlerelay OFDM system with imperfect channel estimation. We first obtained a lower bound on the achievable rate and then used this bound to optimally allocate power between the training and data transmission phases. Since the optimum solution does not yield a closedform expression, we proposed a suboptimal scheme by sequentially maximizing the terms in the integrand of the lower bound. Monte Carlo simulations demonstrate that the proposed power allocation scheme brings improvements of 5–19% depending on the SNR and relay locations in the bound on achievable rate, depending on the relay location and level of SNR.
8 Endnote
^{a} In Section 2, we will further consider an alternative scheme in which only the destination is equipped with a channel estimator and therefore the overall cascaded channel is estimated.
Notes
Supplementary material
References
 1.Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—Part I: system description. IEEE Trans. Commun 2003, 51: 19271938. 10.1109/TCOMM.2003.818096CrossRefGoogle Scholar
 2.Azarian K, El Gamal H, Schniter P: On the achievable diversitymultiplexing tradeoff in halfduplex cooperative channels. IEEE Trans. Inf. Theory 2005, 51: 41524172. 10.1109/TIT.2005.858920MathSciNetCrossRefzbMATHGoogle Scholar
 3.Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans. Inf. Theory 2004, 50: 30623080. 10.1109/TIT.2004.838089MathSciNetCrossRefzbMATHGoogle Scholar
 4.Gastpar M, Vetterli M: On the capacity of wireless networks: the relay case. New York: ; June 2002.zbMATHGoogle Scholar
 5.Nabar RU, Bölcskei H, Kneubühler FW: Fading relay channel: performance limits and spacetime signal design. IEEE J. Sel. Areas Commun 2004, 22: 10991109. 10.1109/JSAC.2004.830922CrossRefGoogle Scholar
 6.El Gamal A, Zahedi S: Capacity of a class of relay channels with orthogonal components. IEEE Trans. Inf. Theory 2005, 51(5):18151817. 10.1109/TIT.2005.846438MathSciNetCrossRefzbMATHGoogle Scholar
 7.Wong B, Zhang J, HostMadsen A: On the capacity of MIMO relay channels. IEEE Trans. Inf. Theory 2005, 51(1):2943.CrossRefMathSciNetzbMATHGoogle Scholar
 8.Böckei H, Nabar RU, Oyman O, Paulraj AJ: Capacity scaling laws in MIMO relay networks. IEEE Trans. Wirel. Commun 2006, 5: 14331443.CrossRefGoogle Scholar
 9.Marić I, Yates RD: bandwidth and power allocaiton for cooperative startegies in Gaussian realy networks. In The 38th Asilomar Conference on Signals, Systems and Computers. (Pacific Grove, CA, USA: ; November 2004:19071911.Google Scholar
 10.Deng X, Haimovich AM: Power allocation for cooperative relaying in wireless networks. IEEE Commun. Lett 2005, 9(11):994996. 10.1109/LCOMM.2005.11012CrossRefGoogle Scholar
 11.Zhang J, Gursoy MC: Achievable rates and resource allocation strategies for imperfectlyknown fading relay channels. EURASIP J. Wirel. Commun. Netw 2009., 2009:Google Scholar
 12.Shin OS, Chan A, Kung HT, Tarokh V: Design of an OFDM cooperative diversity system. IEEE Trans. Veh. Technol 2007, 56(4):22032215.CrossRefGoogle Scholar
 13.Hammerstrom I, Wittneben A: Power allocation schemes for amplifyandforward MIMOOFDM relay links. IEEE Trans. Wirel. Commun 2007, 8(8):45074512.Google Scholar
 14.Mheidat H, Uysal M, AlDhahir N: Equalization techniques for distributed spacetime block codes with amplifyandforward relaying. IEEE Trans. Signal Process 2007, 55(5):18391852.MathSciNetCrossRefGoogle Scholar
 15.Ma Y, Yi N, Tafazolli R: Bit and power loading for OFDMbased threenode relaying communications. IEEE Trans. Signal Process 2008, 56(7):32363247.MathSciNetCrossRefGoogle Scholar
 16.Ibrahimi M, Liang B: Efficient power allocation in cooperative OFDM system with channel variation. In Proceedings of the IEEE International Conference on Communication (ICC). (Beijing, China: ; May 2008:30223028.Google Scholar
 17.Amin O, Uysal M: Optimal bit and power loading for amplifyandforward cooperative OFDM systems. IEEE Trans. Wirel. Commun 2011, 10(3):772781.CrossRefGoogle Scholar
 18.Wang JH, Lin YB, Su YT: Resource allocation and relay selection for cooperative OFDMA networks with imperfect channel information. In Proceedings of the IEEE Signal Processing Workshop in Signal Advances in Wireless Communication (SPAWC). (San Francisco: ; June 2011:286290.Google Scholar
 19.Ochiai H, Mitran P, Tarokh V: Variablerate twophase collaborative communication protocols for wireless networks. IEEE Trans. Inf. Theory 2006, 52(9):42994313.MathSciNetCrossRefzbMATHGoogle Scholar
 20.Adireddy S, Tong L, Viswanathan H: Optimal placement of training for frequencyselective blockfading channels. IEEE Trans. Inf. Theory 2002, 48(8):23382353. 10.1109/TIT.2002.800466MathSciNetCrossRefzbMATHGoogle Scholar
 21.Kay SM: Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Upper Saddle River, NJ 07458: PrenticeHall, Inc.; 1993.zbMATHGoogle Scholar
 22.Hassibi B, Hochwald BM: How much training is needed in multipleantenna wireless links? IEEE Trans, Inf. Theory 2003, 49(4):951963. 10.1109/TIT.2003.809594CrossRefzbMATHGoogle Scholar
 23.Cover TM, Thomas JA: Elements of Information Theory. New York: Wiley; 1991.CrossRefzbMATHGoogle Scholar
 24.Lamahewa TA, Sadeghi P, Zhou X: On lower bounding the information capacity of amplify and forward wireless relay channels with channel estimation errors. IEEE Trans. Wirel. Commun 2011, 10(7):20752079.CrossRefGoogle Scholar
Copyright information
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.