Achievable Rates and Resource Allocation Strategies for Imperfectly Known Fading Relay Channels
© J. Zhang and M. C. Gursoy. 2009
Received: 26 February 2009
Accepted: 19 October 2009
Published: 10 December 2009
Achievable rates and resource allocation strategies for imperfectly known fading relay channels are studied. It is assumed that communication starts with the network training phase in which the receivers estimate the fading coefficients. Achievable rate expressions for amplify-and-forward and decode-and-forward relaying schemes with different degrees of cooperation are obtained. We identify efficient strategies in three resource allocation problems: (1) power allocation between data and training symbols, (2) time/bandwidth allocation to the relay, and (3) power allocation between the source and relay in the presence of total power constraints. It is noted that unless the source-relay channel quality is high, cooperation is not beneficial and noncooperative direct transmission should be preferred at high signal-to-noise ratio (SNR) values when amplify-and-forward or decode-and-forward with repetition coding is employed as the cooperation strategy. On the other hand, relaying is shown to generally improve the performance at low SNRs. Additionally, transmission schemes in which the relay and source transmit in nonoverlapping intervals are seen to perform better in the low-SNR regime. Finally, it is noted that care should be exercised when operating at very low SNR levels, as energy efficiency significantly degrades below a certain SNR threshold value.
In wireless communications, deterioration in performance is experienced due to various impediments such as interference, fluctuations in power due to reflections and attenuation, and randomly-varying channel conditions caused by mobility and changing environment. Recently, cooperative wireless communication has attracted much interest as a technique that can mitigate these degradations and provide higher rates or improve the reliability through diversity gains. The relay channel was first introduced by van der Meulen in , and initial research was primarily conducted to understand the rates achieved in relay channels [2, 3]. More recently, diversity gains of cooperative transmission techniques have been studied in [4–7]. In , several cooperative protocols have been proposed, with amplify-and-forward (AF) and decode-and-forward (DF) being the two basic relaying schemes. The performance of these protocols are characterized in terms of outage events and outage probabilities. In , three different time-division AF and DF cooperative protocols with different degrees of broadcasting and receive collision are studied. Resource allocation for relay channel and networks has been addressed in several studies (see, e.g., [9–14]). In , upper and lower bounds on the outage and ergodic capacities of relay channels are obtained under the assumption that the channel side information (CSI) is available at both the transmitter and receiver. Power allocation strategies are explored in the presence of a total power constraint on the source and relay. In , under again the assumption of the availability of CSI at the receiver and transmitter, optimal dynamic resource allocation methods in relay channels are identified under total average power constraints and delay limitations by considering delay-limited capacities and outage probabilities as performance metrics. In , resource allocation schemes in relay channels are studied in the low-power regime when only the receiver has perfect CSI. Liang et al. in  investigated resource allocation strategies under separate power constraints at the source and relay nodes and showed that the optimal strategies differ depending on the channel statics and the values of the power constraints. Recently, the impact of channel state information (CSI) and power allocation on rates of transmission over fading relay channels are studied in  by Ng and Goldsmith. The authors analyzed the cases of full CSI and receiver only CSI, considered the optimum or equal power allocation between the source and relay nodes, and identified the best strategies in different cases. In general, the area has seen an explosive growth in the number of studies (see additionally, e.g., [15–17], and references therein). An excellent review of cooperative strategies from both rate and diversity improvement perspectives is provided in  in which the impacts of cooperative schemes on device architecture and higher-layer wireless networking protocols are also addressed. Recently, a special issue has been dedicated to models, theory, and codes for relaying and cooperation in communication networks in .
As noted above, studies on relaying and cooperation are numerous. However, most work has assumed that the channel conditions are perfectly known at the receiver and/or transmitter sides. Especially in mobile applications, this assumption is unwarranted as randomly varying channel conditions can be learned by the receivers only imperfectly. Moreover, the performance analysis of cooperative schemes in such scenarios is especially interesting and called for because relaying introduces additional channels and hence increases the uncertainty in the model if the channels are known only imperfectly. Recently, Wang et al. in  considered pilot-assisted transmission over wireless sensory relay networks and analyzed scaling laws achieved by the amplify-and-forward scheme in the asymptotic regimes of large nodes, large block length, and small signal-to-noise ratio (SNR) values. In this study, the channel conditions are being learned only by the relay nodes. In [21, 22], estimation of the overall source-relay-destination channel is addressed for amplify-and-forward relay channels. In , Gao et al. considered both the least squares (LSs) and minimum-mean-square error (MMSE) estimators and provided optimization formulations and guidelines for the design of training sequences and linear precoding matrices. In , under the assumption of fixed power allocation between data transmission and training, Patel and Stüber analyzed the performance of linear MMSE estimation in relay channels. In [21, 22], the training design is studied in an estimation-theoretic framework, and mean-square errors and bit error rates, rather than the achievable rates, are considered as performance metrics. To the best of our knowledge, performance analysis and resource allocation strategies have still not been sufficiently addressed for imperfectly-known relay channels in an information-theoretic context by considering rate expressions. We note that Avestimehr and Tse in  studied the outage capacity of slow fading relay channels. They showed that Bursty Amplify-Forward strategy achieves the outage capacity in the low-SNR and low outage probability regime. Interestingly, they further proved that the optimality of Bursty AF is preserved even if the receivers do not have prior knowledge of the channels.
We obtain achievable rate expressions for AF and DF relaying protocols with different degrees of cooperation, ranging from noncooperative communications to full cooperation. We provide a unified analysis that applies to both overlapped and nonoverlapped transmissions of the source and relay. We note that achievable rates are obtained by considering the ergodic scenario in which the transmitted codewords are assumed to be sufficiently long to span many fading realizations.
- (2)We identify resource allocation strategies that maximize the achievable rates. We consider three types of resource allocation problems:
power allocation between data and training symbols,
time/bandwidth allocation to the relay,
power allocation between the source and relay if there is a total power constraint in the system.
We investigate the energy efficiency in imperfectly-known relay channels by finding the bit energy requirements in the low-SNR regime.
The organization of the rest of the paper is as follows. In Section 2, we describe the channel model. Network training and data transmission phases are explained in Section 3. We obtain the achievable rate expressions in Section 4 and study the resource allocation strategies in Section 5. We discuss the energy efficiency in the low-SNR regime in Section 6. Finally, we provide conclusions in Section 6. The proofs of the achievable rate expressions are relegated to the appendix.
2. Channel Model
3. Network Training and Data Transmission
3.1. Network Training Phase
Each block transmission starts with the training phase. In the first symbol period, source transmits the pilot symbol to enable the relay and destination to estimate the channel coefficients and , respectively. The signals received by the relay and destination are
In the above formulations, , , and represent independent Gaussian random variables. Note that and are Gaussian noise samples at the destination in different time intervals, while is the Gaussian noise at the relay.
In the training process, it is assumed that the receivers employ minimum mean-square-error (MMSE) estimation. We assume that the source allocates fraction of its total power for training while the relay allocates fraction of its total power for training. As described in , the MMSE estimate of is given by
where . We denote by the estimate error which is a zero-mean complex Gaussian random variable with variance Similarly, for the fading coefficients and , we have the following estimates and estimate error variances:
With these estimates, the fading coefficients can now be expressed as
3.2. Data Transmission Phase
As discussed in the previous section, within a block of symbols, the first two symbols are allocated to network training. In the remaining duration of symbols, data transmission takes place. Throughout the paper, we consider several transmission protocols which can be classified into two categories depending on whether or not the source and relay simultaneously transmit information: nonoverlapped and overlapped transmissions. Since the practical relay node usually cannot transmit and receive data simultaneously, we assume that the relay works under half-duplex constraint. Hence, the relay first listens and then transmits. We introduce the relay transmission parameter and assume that symbols are allocated for relay transmission. Hence, can be seen as the fraction of total time or bandwidth allocated to the relay. Note that the parameter enables us to control the degree of cooperation. In nonoverlapped transmission protocol, source and relay transmit over nonoverlapping intervals. Therefore, source transmits over a duration of symbols and becomes silent as the relay transmits. On the other hand, in overlapped transmission protocol, source transmits all the time and sends symbols in each block.
We assume that the source transmits at a per-symbol power level of when the relay is silent, and when the relay is in transmission. Clearly, in nonoverlapped mode, . On the other hand, in overlapped transmission, we assume . Noting that the total power available after the transmission of the pilot symbol is , we can write
The above assumptions imply that power for data transmission is equally distributed over the symbols during the transmission periods. Hence, in nonoverlapped and overlapped modes, the symbol powers are and , respectively. Furthermore, we assume that the power of each symbol transmitted by the relay node is , which satisfies, similarly as above,
Next, we provide detailed descriptions of nonoverlapped and overlapped cooperative transmission schemes.
3.2.1. Nonoverlapped Transmission
We first consider the two simplest cooperative protocols: nonoverlapped AF where the relay amplifies the received signal and forwards it to the destination, and nonoverlapped DF with repetition coding where the relay decodes the message, reencodes it using the same codebook as the source, and forwards it. In these protocols, since the relay either amplifies the received signal or decodes it but uses the same codebook as the source when forwarding, source and relay should be allocated equal time slots in the cooperation phase. Therefore, before cooperation starts, we initially have direct transmission from the source to the destination without any aid from the relay over a duration of symbols. In this phase, source sends the -dimensional data vector and the received signal at the destination is given by
In (11) and (12), and are independent Gaussian noise vectors composed of independent and identically distributed (i.i.d.), circularly symmetric, zero-mean complex Gaussian random variables with variance , modeling the additive background noise at the transmitter in different transmission phases. Similarly, is a Gaussian noise vector at the relay, whose components are i.i.d. zero-mean Gaussian random variables with variance . For compact representation, we denote the overall source data vector by and the signal received at the destination directly from the source by where denotes the transpose operation. After completing its transmission, the source becomes silent, and the relay transmits an -dimensional symbol vector which is generated from the previously received [6, 7]. Now, the destination receives
After substituting the estimate expressions in (8) into (11)–(13), we have
For nonoverlapped transmission, we also consider DF with parallel channel coding, in which the relay uses a different codebook to encode the message. In this case, the source and relay do not have to be allocated the same duration in the cooperation phase. Therefore, source transmits over a duration of symbols while the relay transmits in the remaining duration of symbols. Clearly, the range of is now . In this case, the input-output relations are given by (12) and (13). Since there is no separate direct transmission, and in (12). Moreover, the dimensions of the vectors and are now , while and are vectors of dimension . Figure 3(b) provides a graphical description of the transmission order for nonoverlapped parallel DF scheme.
3.2.2. Overlapped Transmission
In this category, we consider a more general and complicated scenario in which the source transmits all the time. We study AF and repetition DF, in which we, similarly as in the nonoverlapped model, have unaided direct transmission from the source to the destination in the initial duration of symbols. Cooperative transmission takes place in the remaining duration of symbols. Again, we have in this setting. In these protocols, the input-output relations are expressed as follows:
Above, and , which have respective dimensions of , and , represent the source data vectors sent in direct transmission, cooperative transmission when relay is listening, and cooperative transmission when relay is transmitting, respectively. Note again that the source transmits all the time. is the relay's data vector with dimension . and are the corresponding received vectors at the destination, and is the received vector at the relay. The input vector now is defined as and we again denote . If we express the fading coefficients as in (16), we obtain the following input-output relations:
A graphical depiction of the transmission order for overlapped AF and repetition DF is given in Figure 3(c).
List of notations.
Source-destination channel fading coefficient
Relay-destination channel fading coefficient
Relay-destination channel fading coefficient
Variance of random variables
Variance of Gaussian random variables due to thermal noise
Number of symbols in each block
Fraction of total power allocated to training by the source
Fraction of total power allocated to training by the relay
Pilot symbol sent by the source
Pilot symbol sent by the relay
Additive Gaussian noise at the destination in the interval in which the source pilot symbol is sent
Additive Gaussian noise at the relay in the interval in which the source pilot symbol is sent
Additive Gaussian noise at the destination in the interval in which the relay pilot symbol is sent
Received signal at the destination in the interval in which the source pilot symbol is sent
Received signal at the relay in the interval in which the source pilot symbol is sent
Received signal at the destination in the interval in which the relay pilot symbol is sent
Power of each source symbol sent in the interval in which the relay is not transmitting
Power of each source symbol sent in the interval in which the relay is transmitting
Power of each relay symbol
Fraction of time/bandwidth allocated to the relay
4. Achievable Rates
In this section, we provide achievable rate expressions for AF and DF relaying in both nonoverlapped and overlapped transmission scenarios in a unified fashion. Achievable rate expressions are obtained by considering the estimate errors as additional sources of Gaussian noise. Since Gaussian noise is the worst uncorrelated additive noise for a Gaussian model [25, Appendix], , achievable rates given in this section can be regarded as worst-case rates.
Note that this formulation presupposes that the destination has the knowledge of . Hence, we assume that the value of is forwarded reliably from the relay to the destination over low-rate control links. In general, solving the optimization problem in (19) and obtaining the AF capacity is a difficult task. Therefore, we concentrate on finding a lower bound on the capacity. A lower bound is obtained by replacing the product of the estimate error and the transmitted signal in the input-output relations with the worst-case noise with the same correlation. Therefore, we consider in the overlapped AF scheme
as noise vectors with covariance matrices
See Appendix .
Next, we consider DF relaying scheme. In DF, there are two different coding approaches , namely, repetition coding and parallel channel coding. We first consider repetition channel coding scheme. The following result provides achievable rate expressions for both nonoverlapped and overlapped transmission scenarios.
See Appendix .
Finally, we consider DF with parallel channel coding and assume that nonoverlapped transmission scheme is adopted. From [13, Equation ( 6)], we note that an achievable rate expression is given by
Note that we do not have separate direct transmission in this relaying scheme. Using similar methods as in the proofs of Theorems 1 and 2, we obtain the following result. The proof is omitted to avoid repetition.
5. Resource Allocation Strategies
Having obtained achievable rate expressions in Section 4, we now identify resource allocation strategies that maximize these rates. We consider three resource allocation problems: power allocation between training and data symbols, time/bandwidth allocation to the relay, and power allocation between the source and relay under a total power constraint.
We first study how much power should be allocated for channel training. In nonoverlapped AF, it can be seen that appears only in in the achievable rate expression (24). Since is a monotonically increasing function of for fixed , (24) is maximized by maximizing . We can maximize by maximizing the coefficient of the random variable in (27), and the optimal is given as follows:
In Figure 8 in which SNR is low ( ), we see that the highest achievable rates are attained when there is full cooperation (i.e., when ). Note that in this figure, overlapped DF with repetition coding is considered. If overlapped AF is employed as the cooperation strategy, we have similar conclusions but it should also be noted that overlapped AF achieves smaller rates than those attained by overlapped DF with repetition coding.
(i) Cooperation employing overlapped AF or DF with repetition coding is beneficial only if the source-relay channel quality is high enough. If this is not the case or SNR is very high, noncooperative direct transmission should be employed.
(ii) Cooperation using nonoverlapped DF with parallel coding provides improvements over the performance of noncooperative direct transmission and achieves higher rates than those attained by overlapped AF and DF with repetition coding.
(iii) If the system is operating under total power constraints, all the power should be allocated to the source and hence direct transmission should be preferred over overlapped and nonoverlapped AF and overlapped and nonoverlapped DF with repetition coding.
(iv) Under total power constraints, only nonoverlapped DF with parallel coding outperforms noncooperative direct transmission when the source-relay channel is strong.
(i) Cooperation is generally beneficial.
(ii) The strengths of both the source-relay and relay-destination channels are important factors.
(iii) Nonoverlapped DF with parallel coding achieves the highest performance levels. In general, nonoverlapped transmission methods should be preferred. Also, DF provides higher gains over AF.
(iv) Under total power constraints, highest gains over noncooperative direct transmission are attained when both the source-relay and relay-destination channels are considerably stronger than the source-destination channel.
(v) Under total power constraints, noncooperative direct transmission should be preferred if the qualities of both the source-relay and relay-destination channels are comparable to that of the source-destination channel.
6. Energy Efficiency
Our analysis has shown that cooperative relaying is generally beneficial in the low-power regime, resulting in higher achievable rates when compared to direct transmission. In this section, we provide an energy efficiency perspective and remark that care should be exercised when operating at very low SNR values. The least amount of energy required to send one information bit reliably is given by where is the channel capacity in bits/symbol. (Note that is the bit energy normalized by the noise power spectral level .)In our setting, the capacity will be replaced by the achievable rate expressions and hence the resulting bit energy, denoted by , provides the least amount of normalized bit energy values in the worst-case scenario and also serves as an upper bound on the achievable bit energy levels in the channel.
is the derivative of with respect to SNR as SNR 0. The key point to prove this theorem is to show that when , the mutual information decreases as , and hence . This can be easily shown because when , in all the terms, , , , , and in Theorems 1–3, the denominator goes to a constant while the numerator decreases as . Hence, these terms diminish as . Since for small , where satisfies , we conclude that the achievable rate expressions also decrease as as vanishes.
In this paper, we have studied the imperfectly-known fading relay channels. We have assumed that the source-destination, source-relay, and relay-destination channels are not known by the corresponding receivers a priori, and transmission starts with the training phase in which the channel fading coefficients are learned with the assistance of pilot symbols, albeit imperfectly. Hence, in this setting, relaying increases the channel uncertainty in the system, and there is increased estimation cost associated with cooperation. We have investigated the performance of relaying by obtaining achievable rates for AF and DF relaying schemes. We have considered both nonoverlapped and overlapped transmission scenarios. We have controlled the degree of cooperation by varying the parameter . We have identified resource allocation strategies that maximize the achievable rate expressions. We have observed that if the source-relay channel quality is low, then cooperation is not beneficial and direct transmission should be preferred at high SNRs when amplify-and-forward or decode-and-forward with repetition coding is employed as the cooperation strategy. On the other hand, we have seen that relaying generally improves the performance at low SNRs. We have noted that DF with parallel coding provides the highest rates. Additionally, under total power constraints, we have studied power allocation between the source and relay. We have again pointed out that relaying degrades the performance at high SNRs unless DF with parallel channel coding is used and the source-relay channel quality is high. The benefits of relaying is again demonstrated at low SNRs. We have noted that nonoverlapped transmission is superior compared to overlapped one in this regime. Finally, we have considered the energy efficiency in the low-power regime and proved that the bit energy increases without bound as SNR diminishes. Hence, operation at very low SNR levels should be avoided. From the energy efficiency perspective, we have again observed that nonoverlapped transmission provides better performance. We have also noted that DF is more energy efficient than AF.
This work was supported in part by the NSF CAREER Grant CCF-0546384. The material in this paper was presented in part at the 45th Annual Allerton Conference on Communication, Control and Computing in September 2007 and in part at the 9th IEEE Workshop on Signal Processing Advances for Wireless Communications (SPAWC) in July 2008.
- van der Meulen EC: Three-terminal communication channels. Advances in Applied Probability 1971, 3(1):120-154. 10.2307/1426331MathSciNetView ArticleMATHGoogle Scholar
- Cover TM, El Gamal AA: Capacity theorems for the relay channel. IEEE Transactions on Information Theory 1979, 25(5):572-584. 10.1109/TIT.1979.1056084MathSciNetView ArticleMATHGoogle Scholar
- El Gamal AA, Aref M: The capacity of the semideterministic relay channel. IEEE Transactions on Information Theory 1982, 28(3):536. 10.1109/TIT.1982.1056502View ArticleMATHGoogle Scholar
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—part I: system description. IEEE Transactions on Communications 2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—part II: implementation aspects and performance analysis. IEEE Transactions on Communications 2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar
- Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Transactions on Information Theory 2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATHGoogle Scholar
- Laneman JN: Cooperative diversity: models, algorithms, and architectures. In Cooperation in Wireless Networks: Principles and Applications. Springer, Berlin, Germany; 2006.View ArticleGoogle Scholar
- Nabar RU, Bölcskei H, Kneubühler FW: Fading relay channels: performance limits and space-time signal design. IEEE Journal on Selected Areas in Communications 2004, 22(6):1099-1109. 10.1109/JSAC.2004.830922View ArticleGoogle Scholar
- Host-Madsen A, Zhang J: Capacity bounds and power allocation for wireless relay channels. IEEE Transactions on Information Theory 2005, 51(6):2020-2040. 10.1109/TIT.2005.847703MathSciNetView ArticleMATHGoogle Scholar
- Gunduz D, Erkip E: Opportunistic cooperation by dynamic resource allocation. IEEE Transactions on Wireless Communications 2007, 6(4):1446-1454.View ArticleGoogle Scholar
- Yao Y, Cai X, Giannakis GB: On energy efficiency and optimum resource allocation of relay transmissions in the low-power regime. IEEE Transactions on Wireless Communications 2005, 4(6):2917-2927.View ArticleGoogle Scholar
- Liang Y, Veeravalli VV, Poor HV: Resource allocation for wireless fading relay channels: max-min solution. IEEE Transactions on Information Theory 2007, 53(10):3432-3453.MathSciNetView ArticleMATHGoogle Scholar
- Liang Y, Veeravalli VV: Gaussian orthogonal relay channels: optimal resource allocation and capacity. IEEE Transactions on Information Theory 2005, 51(9):3284-3289. 10.1109/TIT.2005.853305MathSciNetView ArticleMATHGoogle Scholar
- Ng CTK, Goldsmith A: The impact of CSI and power allocation on relay channel capacity and cooperation strategies. IEEE Transactions on Wireless Communications 2008, 7(12):5380-5389.View ArticleGoogle Scholar
- Host-Madsen A: Capacity bounds for cooperative diversity. IEEE Transactions on Information Theory 2006, 52(4):1522-1544.MathSciNetView ArticleMATHGoogle Scholar
- Kramer G, Gastpar M, Gupta P: Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 2005, 51(9):3037-3063. 10.1109/TIT.2005.853304MathSciNetView ArticleMATHGoogle Scholar
- Mitran P, Ochiai H, Tarokh V: Space-time diversity enhancements using collaborative communications. IEEE Transactions on Information Theory 2005, 51(6):2041-2057. 10.1109/TIT.2005.847731MathSciNetView ArticleMATHGoogle Scholar
- Kramer G, Marić I, Yates RD: Cooperative communications. Foundations and Trends in Networking 2006, 1(3-4):271-425. 10.1561/1300000004View ArticleGoogle Scholar
- Kramer G, Berry R, El Gamal AA, et al.: Introduction to the special issue on models, theory, and codes for relaying and cooperation in communication networks. IEEE Transactions on Information Theory 2007, 53(10):3297-3301.View ArticleGoogle Scholar
- Wang B, Zhang J, Zheng L: Achievable rates and scaling laws of power-constrained wireless sensory relay networks. IEEE Transactions on Information Theory 2006, 52(9):4084-4104.MathSciNetView ArticleMATHGoogle Scholar
- Gao F, Cui T, Nallanathan A: On channel estimation and optimal training design for amplify and forward relay networks. IEEE Transactions on Wireless Communications 2008, 7(5, part 2):1907-1916.View ArticleGoogle Scholar
- Patel CS, Stüber GL: Channel estimation for amplify and forward relay based cooperation diversity systems. IEEE Transactions on Wireless Communications 2007, 6(6):2348-2356.View ArticleGoogle Scholar
- Avestimehr AS, Tse DNC: Outage capacity of the fading relay channel in the low-SNR regime. IEEE Transactions on Information Theory 2007, 53(4):1401-1415.MathSciNetView ArticleMATHGoogle Scholar
- Gursoy MC: An energy efficiency perspective on training for fading channels. Proceedings of IEEE International Symposium on Information Theory (ISIT '07), June 2007, Nice, France 1206-1210.Google Scholar
- Hassibi B, Hochwald BM: How much training is needed in multiple-antenna wireless links? IEEE Transactions on Information Theory 2003, 49(4):951-963. 10.1109/TIT.2003.809594View ArticleMATHGoogle Scholar
- Tong L, Sadler BM, Dong M: Pilot-assisted wireless transmissions. IEEE Signal Processing Magazine 2004, 21(6):12-25. 10.1109/MSP.2004.1359139View ArticleGoogle Scholar
- Verdú S: Spectral efficiency in the wideband regime. IEEE Transactions on Information Theory 2002, 48(6):1319-1343. 10.1109/TIT.2002.1003824View ArticleMathSciNetMATHGoogle Scholar
- Cover TM, Thomas JA: Elements of Information Theory. John Wiley & Sons, New York, NY, USA; 1991.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.