To Relay or Not to Relay in Cooperative Networks

http://www.ieccr.net

To Relay or Not to Relay in Cooperative Networks: That is the Green Question Norman C. Beaulieu, Young Gil Kim, and David J. Young IEEE International Conference on Communication Technology Chengdu, China November 9, 2012

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The question of why manhole covers are typically round, at least in the U.S., was made famous by Microsoft when they began asking it as a job-interview question. Originally meant as a psychological assessment of how one approaches a question with more than one correct answer, the problem has produced a number of alternate explanations, from the pragmatic ("Manhole covers are round because manholes are round.")[6] to the philosophical. Reasons for the shape include: • A round manhole cover cannot fall through its circular opening, whereas a square manhole cover may fall in if it were inserted diagonally in the hole. (A Reuleaux triangle or other curve of constant width would also serve this purpose, but round covers are much easier to manufacture. The existence of a "lip" holding up the lid means that the underlying hole is smaller than the cover, so that other shapes might suffice.) • Round tubes are the strongest and most material-efficient shape against the compression of the earth around them, and so it is natural that the cover of a round tube assume a circular shape. • Similarly, it is easier to dig a circular hole and thus the cover is also circular. • The bearing surfaces of manhole frames and covers are machined to assure flatness and prevent them from becoming dislodged by traffic. Round castings are much easier to machine using a lathe. • Circular covers do not need to be rotated to align them when covering a circular manhole. • Human beings have a roughly circular cross-section. • A round manhole cover can be more easily moved by being rolled. • If a cover had corners and were bent that would create a protruding point that could puncture tires. • Most manhole covers are made by a few large companies. A different shape would have to be custom made. Other manhole shapes can be found, usually squares or rectangles. Nashua, New Hampshire may be unique in the U.S. for having triangular manhole covers that point in the direction of the underlying flow. The city is phasing out the triangles, which were made by a local foundry, because they are not large enough to meet modern safety standards, and larger triangles cannot be found. “Manhole Covers” Wikipedia, Wikipedia Foundation Inc., November 8, 2010


Pascal’s Triangle

AITF Wireless Communications Laboratory

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The set of numbers that form Pascal's triangle were well known before Pascal. But, Pascal developed many applications of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers. The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala between the 5th and 2nd century BC. While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029). It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. In 13th century, Yang Hui (1238–98) presented the arithmetic triangle that is the same as Pascal's triangle. Pascal's triangle is called Yang Hui's triangle in China. The "Yang Hui's triangle" was known in China in the upper half of the 11th century by the Chinese mathematician Jia Xian (1010-1070). Petrus Apianus (1495–1552) published the triangle on the frontispiece of his book on business calculations in the 16th century. This is the first record of the triangle in Europe. In Italy, it is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500– 77). Tartaglia is credited with the general formula for solving cubic polynomials, (which may be really from Scipione del Ferro but was published by Gerolamo Cardano 1545). Traité du triangle arithmétique (Treatise on Arithmetical Triangle) has been published posthumously in 1665. In the Treatise Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name. “Pascal’s Triangle” Wikipedia, Wikipedia Foundation Inc., November 15, 2010 AITF Wireless Communications 6 Laboratory












Le Chatelier's principle Any change in status quo prompts an opposing reaction in the responding system.


Hooke’s law: F=-k x The extension of a spring is in direct proportion with the load applied to it. (Strain is directly proportional to stress.)


Lenz’s law: An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux.


Outline 1. Introduction 2. Symbol Error Probabilities for Systems Using Decode-and-Forward Relaying 3. A Relay Advantage Criterion 4. Numerical Examples 5. Conclusion


Introduction


Motivation • New and improved technologies must be developed to meet the ever increasing demand for wireless user services with finite spectrum (bandwidth) resources. • Cooperative networks have promise to improve wireless communication capability, resulting in a significant increase of the capacity and diversity gain in wireless networks. – new paradigm for the development of efficient bandwidth usage


Communication without relaying Source

Destination

• Range may be limited due to channel impairments and limits on transmit energy. • Can idle users (nodes) be used to decrease symbol error probability?


Multihop Relaying Source

hop 1

R1

hop 2

R2

RK

hop 𝐾+1

Destination

• Idle users between source and destination are used to receive and retransmit the signal. • Avoids severe shadowing in long distance communication, or when the transmitted signal energy is relatively low. • Provides broader and cheaper coverage along with large spectral efficiency.


Amplify-and-Forward Relaying Source

hop 1

R1

hop 2

R2

RK

hop 𝐾+1

Destination

• Relays amplify the received signals before forwarding to next node (non-regenerative). • Once destination receives all the different replicas of the signal (direct and relayed), they are combined together. • Noise and interference are also amplified by the relays.


Decode-and-Forward Relaying Source

hop 1

R1

hop 2

R2

RK

hop 𝐾+1

Destination

• Relay receives a packet, decodes it, re-encodes it, and retransmits it (regenerative). • When relay decodes correctly, retransmitted packet is noise-free. • Extends range of network. • Link capacity is that of the minimum among all hops.


“Green”-ing Cooperative Communications Systems • Energy-efficient systems lead to reduced operating costs and reduced greenhouse gas emissions due to electricity generation from fossil fuels. • Research into making systems more “green” usually focuses on reduced-complexity transmission components, and power saving during low traffic conditions – e.g., improved transmitter efficiency, energy saving during low traffic, efficient wireless architectures and protocols, efficient base station power management. – Some work has considered combining feedback that reduces signal processing complexity with adaptively interrupting transmission. – Earlier work focused on relay selection strategies.


“Greener than Green” • We have constructed examples that show that The amount of power saved and the reduction employing relaying may degrade the quality of the in greenhouse gas emissions by not turning the message transmission rather than improve it, in relays on at all far exceeds the amounts that some practical circumstances. can be expected to be gained from optimizing • When relaying is deployed it fails to improve the components and the and system while still the transmission, powerthem has been turning on. consumed for naught. • We define relaying advantage criterion Theapower saving is nearly 100%! to investigate when relaying has advantage and when it does not.


Symbol Error Probabilities for Systems Using Decode-andForward Relaying


System Model Source 𝑅0

hop 1

𝑅0

hop 2

𝑅1

𝑅𝐾

hop 𝐾+1

Destination 𝑅𝐾+1

• Decode-and forward (DF) system with 𝐾 relays. • Half-duplex communication: in the first slot 𝑅0 , 𝑅2 , 𝑅4 , … transmit, in the second slot 𝑅1 , 𝑅3 , 𝑅5 , … transmit. • Assume no interference in any hop due to surrounding hops. • The channels between adjacent nodes are assumed independent but non-identically distributed (i.n.d.).


The State Transition Matrix • The state transition matrix (STM) is defined as Pr 𝑠0 𝑠0 Pr 𝑠1 𝑠0 ⋯ Pr 𝑠𝑀 𝑠0 Pr 𝑠0 𝑠1 Pr 𝑠1 𝑠1 ⋯ Pr 𝑠𝑀 𝑠1 𝑃= ⋮ ⋮ ⋮ Pr 𝑠0 𝑠𝑀 Pr 𝑠1 𝑠𝑀 ⋯ Pr 𝑠𝑀 𝑠𝑀 The entry at 𝑖th row and 𝑗th column is the probability of 𝑠𝑗 being detected given that 𝑠𝑖 was transmitted, i.e., the symbol transition probability Pr 𝑠𝑗 𝑠𝑖


The Symbol Error Probability for 1 Hop • Given a priori probabilities Pr(𝑠𝑚 ), the symbol error probability (SEP) is given by 𝑀

SEP = 1 − �

𝑚=1

Pr 𝑠𝑚 Pr 𝑠𝑚 𝑠𝑚

• For equally likely transmission, the SEP is 𝑀 1 1 SEP = 1 − � Pr 𝑠𝑚 𝑠𝑚 = 1 − 𝑡𝑡(𝑃) 𝑀 𝑀 𝑚=1 The SEP is a function only of the trace of the STM


STMs for DF Relaying Systems • In (𝐾 + 1)-hop DF relaying systems, there are 𝐾 + 1 STMs, 𝑃1 , 𝑃2 , … , 𝑃𝐾+1 corresponding to the 𝐾 + 1 hops. Let 𝑥𝑘 and 𝑦𝑘 be the transmitted and detected symbols at the 𝑘th hop, respectively. • The source-to-destination symbol transition property satisfies the Markov property as Pr 𝑦𝐾+1 = 𝑠𝑗 𝑥1 = 𝑠𝑖1 , x 2 = si2 , … , x K+1 = siK+1 = Pr yK+1 = sj x K+1 = siK+1

Given the symbol transmitted in the last hop, the STP does not depend on prior hops.


The SEP for Multihop DF Relaying in i.n.d. Channels • Due to the Markov property, the source-todestination STM, 𝑃𝑆𝑆 is PSD = �

𝐾+1 𝑘=1

𝑃𝑘

• Then, the SEP of multihop DF relaying systems with 𝐾 relays for equally likely transmission in any i.n.d. channels is 𝐾+1 1 SEP = 1 − 𝑡𝑡 � 𝑃𝑘 𝑀 𝑘=1


The SEP for Multihop DF Relaying in i.i.d. Channels • When the channels are identical, the 𝐾 + 1 STMs are identical, 𝑃 = 𝑃1 = 𝑃2 = ⋯ = 𝑃𝐾+1 . • The SEP calculation does not require the computation of 𝑃𝐾+1 since 𝑡𝑡 𝑃𝑘 = ∑𝑖 𝜆𝑘𝑖 , where 𝜆𝑖 are the eigenvalues of 𝑃. • The SEP for multihop DF relaying systems with 𝐾 relays in i.i.d. channels is 1 1 𝐾+1 SEP = 1 − 𝑡𝑡 𝑃 = 1 − � 𝜆𝐾+1 𝑖 𝑀 𝑀 𝑖


Circulant STMs • In a circulant matrix, every row is a right cyclic shift of the row above it. • The eigenvalues comprise the discrete Fourier Transform (DFT) of the first row of the circulant matrix

𝜖0,0

𝜖0,1 𝑀−1𝜖0,2 −⋯𝑗𝑗𝑗𝑗𝑗𝜖0,𝑀−1 𝜖0,𝑀 𝑒 𝑀 𝜆𝑖 = � 𝜖0,𝑀−1 𝜖0,0 𝑚=0𝜖0,1 ⋯ 𝜖0,𝑀−2 𝜖0,𝑀−2 𝜖0,0= 1,⋯ 𝜖0,𝑀−3 𝜆0 = 1. • Since 𝜖0,0 + 𝜖0,1 +𝜖0,𝑀−1 ⋯ + 𝜖0,𝑀−1 the eigenvalue • When the ⋮matrix is circulant⋱and symmetric ⋮ 𝑀−1 𝜖 𝜖0,1 𝜖0,2 2𝜋𝜋𝜋 𝜖0,0 0,3 ⋯ 𝜆𝑖 = �

𝑚=0

𝜖0,𝑀 cos

𝑀

.


SEP for Equidistant Modulations • Simplex, BPSK, and orthogonal FSK signals have an equidistance property. DF having relayingthis systems employing a • For multihop modulations property modulation scheme with 𝜖an equidistance property for 𝑖 ≠ 𝑗 Pr 𝑠𝑗 𝑠𝑖 = 𝑀−1 with 𝐾 relays in i.i.d. channels, the SEP is given by where 𝜖 is the single link SEP. 𝐾+1 1 𝑀 = 1 − of 1 +the𝑀single − 1 link 1 −STM 𝜖are 1 and • TheSEP eigenvalues 𝑀 𝑀−1 𝑀

1 − 𝜖 is𝜖 the with algebraicSEP. multiplicity 𝑀 − 1. where single-link 𝑀−1


SEP for Multihop DF Relaying Using Binary Modulation • The single-link STM of a binary modulation is given by 1−𝜖 𝜖

𝜖 1−𝜖

and the SEP (BEP) for multihop DF relaying using 𝐾 relays is given in the simple form SEP= 1 −

1 2

1 + 1 − 2𝜖

𝐾+1

.

SEP for Multihop DF Relaying Using QPSK Modulation http://www.ieccr.net

• For QPSK, the single-link STM can be written as 1 − 2𝜖1 − 𝜖2 𝜖1 𝜖2 𝜖1

𝜖1 1 − 2𝜖1 − 𝜖2 𝜖1 𝜖2

𝜖2 𝜖1 1 − 2𝜖1 − 𝜖2 𝜖1

𝜖1 𝜖2 𝜖1 1 − 2𝜖1 − 𝜖2

where 𝜖2 < 𝜖1 < 1. • The eigenvalues of the STM can be determined in closed-form and the SEP (BEP) for DF relaying using 𝐾 relays is again given in a simple form SEP= 1 −

1 4

1 + 1 − 4𝜖1

𝐾+1

+ 2 1 − 2𝜖1 − 2𝜖2

𝐾+1

.


SEP for Multihop DF Relaying Using 8-PSK Modulation • The single-link STM and its eigenvalues in closedform can be found in the case of 8-PSK modulation in a similar manner. • The SEP for DF relaying using 𝐾 relays for 8-PSK modulation is 1− 1 {1 + 1 − 4𝜖1 − 4𝜖3 𝐾+1 + 2 1 − 2𝜖1 − 4𝜖2 − 2𝜖3 𝐾+1 8

+2 1 − 2 𝜖1 + 𝜖2 + 𝜖3 + 𝜖4 + 2 𝜖1 − 𝜖3

+2 1 − 2 𝜖1 + 𝜖2 + 𝜖3 + 𝜖4 − 2 𝜖1 − 𝜖3

𝐾+1

𝐾+1

�.


SEP for Multihop DF Relaying Using 𝟏𝟏-QAM Modulation

• The entries of the single-link STM 𝑃𝑘 have been expressed in closed-form in the case of 16-QAM modulation and i.n.d. Rayleigh fading.

• The exact SEP for DF relaying using 𝐾 relays can then be computed using the basic formula SEP= 1 −

1

𝑀

𝑡𝑡 ∏𝐾+1 𝑘=1 𝑃𝑘 .


A Relay Advantage Criterion (“To Relay, or Not To Relay”)


A Relay Advantage Criterion • A relay advantage criterion is derived for a multihop DF relaying system employing binary modulation in i.i.d. Nakagami-𝑚 fading channels with diversity order 𝐿. • If the relay advantage criterion is satisfied, it is shown that putting more relays between the source and destination decreases the SEP given that both the source-to-destination distance and total transmitted energy are fixed. • If the relay advantage criterion is not satisfied, more relays is a waste of energy and provides no benefit!


Case 1: Fixed Node-to-Node Distance Source 𝐸𝑠

𝑑

𝑅1 𝐸𝑠

𝑑

Source 𝐸𝑠

𝑑

𝑅1 𝐸𝑠

𝑑

𝑅2 𝐸𝑠

𝑑

Source 𝐸𝑠

𝑑

𝑅1 𝐸𝑠

𝑑

𝑅2 𝐸𝑠

𝑑

Destination

𝐸𝑡 = 2𝐸𝑠 Destination

𝑑

𝑅𝐾 𝐸𝑠

𝐸𝑡 = 3𝐸𝑠 𝑑

Destination

𝐸𝑡 = (𝐾 + 1)𝐸𝑠


Case 2: Fixed Source-to-Destination Distance 𝐸𝑡 /2

𝑑SD /2

Source 𝐸𝑡 /3

Source 𝐸𝑡 /(𝐾 + 1) Source

𝐸𝑡 /2

𝑑SD /3 𝑑SD 𝐾+1

𝐸𝑡 /3

𝑑SD /3

𝑅1

𝐸𝑡 /(𝐾 + 1) 𝑅1

𝑅1

𝑑SD 𝐾+1

𝐸𝑡 /(𝐾 + 1) 𝑅2

𝑑SD 𝐾+1

𝑑SD /2 𝐸𝑡 /3 𝑅2

𝑑SD 𝐾+1

Destination

𝑑SD /3 𝐸𝑡 /(𝐾 + 1) 𝑅𝐾

𝑑SD 𝐾+1

Destination

Destination


The BEP of Binary Modulations in Nakagami-𝒎 Fading • The starting point of the development is the BEP of a binary modulation scheme for a single link with diversity order 𝐿 in Rayleigh fading channels, in a high SNR regime

where

𝑁0 𝐵𝐵𝐵 ≈ 𝑐Ω𝐸𝑠

Ω is the single link channel gain

𝐿

4, for binary PSK 𝑐 = �2, for coherent binary FSK or differential PSK 1, for noncoherent binary FSK


The BEP of Binary Modulations in Nakagami-𝒎 Fading • The BEP of a binary modulation scheme for a single link with diversity order 𝐿 in Nakagami-𝑚 fading channels can be obtained from the Rayleigh fading case using a result in [1], as 𝑚𝑚

𝑚𝑚0 𝐵𝐵𝐵 ≈ 𝑐Ω𝐸𝑠 • The BEP of multihop DF relaying systems in a high SNR regime is given in this case by BEP = 1 −

1 𝑚𝑁0 1+ 1−2 2 𝑐Ω𝐸𝑠

𝑚𝑚 𝐾+1

[1] Y.G. Kim and N.C. Beaulieu, “New results on maximal ratio combining in Nakagami-𝑚 fading channels,” Proc. IEEE Int. Conf. Commun., Ottawa, Canada, June 2012.


Relay Advantage for Fixed Node-to-Node Distance 𝑚𝑁0 0 1, the BEP goes to zero as the number of relays goes to infinity. Increasing the number of relays enhances BEP performance. • If we fix the total transmitted energy, 𝐸𝑡 = 𝐾 + 1 𝐸𝑠 , then the BEP of multihop DF relaying systems in a fixed source-to0 if 𝛿𝛿𝛿 < 1 𝑚𝑚 𝐾+1 destination distance environment goes to zero as the 𝑚𝑚 𝑚𝑁0 𝑚𝑁0 −2 lim 1 − 2 =� 𝛿 Ω to 𝑠 𝟏 if𝒎𝒎 𝛿𝛿𝛿>= 𝟏. 1 This 𝑒 𝑐ΩSD number of relays goes infinity when 𝜹𝐸− 𝐾→∞ 𝑐 𝐾+1 𝐸 SD 𝑆 1 if 𝛿𝛿𝛿 > 1 is the relay advantage criterion.


Relay Advantage for Fixed Source-toDestination Distance 𝛿 − 1 𝑚𝑚 > 1

The Relay Advantage Criterion

• Given a total transmitted energy, if the relay advantage criterion is satisfied, we can reduce the BEP by putting more relays between the source and destination. • In other words, given a target BEP, we can reduce the total transmitted energy while achieving the target BEP by using more relays.


Relay Advantage for Fixed Source-toDestination Distance 𝛿 − 1 𝑚𝑚 > 1

The Relay Advantage Criterion

• On the other hand, if the relay advantage criterion is not satisfied, the BEP increases as the number of relays between the source and destination increases. • In this case, to save energy, relaying should not be used. • In the unlikely case where 𝛿 − 1 𝑚𝑚 = 1, relaying has no effect on BEP in a high-SNR regime.


An Intuitive Explanation 𝛿 − 1 𝑚𝑚 > 1

The Relay Advantage Criterion • • •

• •

The overall source-to-destination performance will be poor if at least one of the 𝐾 + 1 hops is in a deep fade. For a small value of 𝑚𝑚, the channel is dispersive, and the probability of a deep fade is large. As the number of relays increases, the probability of at least one of the hops being in a deep fade increases more for small values of 𝑚𝑚 than for large values of 𝑚𝑚. Also, when 𝛿 is large, the transmitted power decays rapidly so it is helpful to use more relays. Thus, putting more relays between the source and the destination is advantageous when 𝒎𝒎 or 𝜹 have large values.


Numerical Examples


An Example With Fixed Node-to-Node Distance Adding more relays to extend the source-todestination distance increases the BEP given fixed node-tonode distance, even when the total transmitted energy increases linearly with the number of hops.

More relays→ worse BEP.

i.i.d. Nakagami-𝑚 fading, 16-QAM, diversity order 1, Ω1 = Ω2 = Ω3 = Ω4 = 1


An Example With Fixed Source-to-Destination Distance Since 𝛿 − 1 𝑚𝑚 < 1, putting more relays between the source and destination increases the BEP given fixed total energy.

More relays→ worse BEP.

1

𝑚 = 2 represents an interesting worst-case fading scenario. 1

i.i.d. Nakagami-2 fading, BPSK, path loss exponent 𝛿 = 2, diversity order 1, ΩSD = 1


An Example With Fixed Source-to-Destination Distance Since the relay advantage criterion 𝛿 − 1 𝑚𝑚 > 1 is satisfied, putting more relays between the source and destination decreases the BEP given fixed total energy.

1

More relays→ better BEP.

i.i.d. Nakagami-2 fading, BPSK, path loss exponent 𝛿 = 4, diversity order 1, ΩSD = 1


A Second Example With Fixed Source-toDestination Distance

Here 𝛿 − 1 𝑚𝑚 = 1, and in the highSNR regime the systems have the same SEP given fixed total energy.

More relays→ no change in SEP.

i.i.d. Rayleigh fading, QPSK, path loss exponent 𝛿 =2, diversity order 1, ΩSD = 1


A Second Example With Fixed Source-toDestination Distance Since the relay advantage criterion 𝛿 − 1 𝑚𝑚 > 1 is satisfied, putting more relays between the source and destination decreases the SEP given fixed total energy.

More relays→ better SEP.

i.i.d. Rayleigh fading, QPSK, path loss exponent 𝛿 = 4, diversity order 1, ΩSD = 1


Conclusion • Employing relays requires energy, and reducing energy consumption (and potential greenhouse gas emissions) through “green” design of individual relays offers limited advantage if relaying itself has no benefit. • To answer the question “to relay or not to relay,” we must determine if relaying has any benefit. • When relaying has no benefit, the relays should be left off, saving 100% of the relay turn-on and operation power.


Conclusion • We have defined a relaying advantage criterion applicable to DF relaying in Nakagami-𝑚 channels. The criterion depends on the severity of fading, order of diversity, and the path loss exponent. • Use of the relay advantage criterion results in “greener-than-green” design. • This research is complementary to other research being conducted on reduced-power components and systems, which have benefits when the relays are “turned on”.


References [1] Y. G. Kim and N. C. Beaulieu, “Exact closed-form solutions for the BEP of decode-and-forward cooperative systems in Nakagami-𝑚 fading channels,” IEEE Trans. Commun., vol. 59, pp. 2355–2361, Sep. 2011. [2] Y. G. Kim and N. C. Beaulieu, “New results on maximal ratio combining in Nakagami-𝑚 fading channels,” Proc. IEEE Int. Conf. Commun., Ottawa, Canada, June 2012. [3] N. C. Beaulieu and A. M. Rabiei, “Linear diversity combining on Nakagami-0.5 fading channels,” IEEE Trans. Commun., vol. 59, pp. 2742–2752, Oct. 2011.


References [4] A. Müller and J. Spiedel, “Exact symbol error probability of 𝑀PSK for multihop transmission with regenerative relays,” IEEE Commun. Lett., vol. 11, pp. 952–954, Dec. 2007. [5] K. Dhaka, R. K. Mallik and R. Schober, “Performance analysis of a multi-hop communication system with decode-and-forward relaying,” Proc. IEEE Int. Conf. Commun., Kyoto, Japan, June 2011. [6] M. O. Hasna and M-S. Alouini, “End-to-end performance of transmission systems with relays over Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 2, pp. 1126–1131, Nov. 2003.
