The 2010 International Conference on Advanced Technologies for Communications

Adaptive Modulation for Distributed Switch-and-Stay Combining with Single Relay Thanh Tran-Thien and Tuan Do-Hong

Vo Nguyen Quoc Bao

Telecommunication Dept. Ho Chi Minh City University of Technology, Vietnam Email: [email protected], [email protected]

School of Telecommunications Posts and Telecommunications Institute of Technology Email: [email protected]

Abstract—In this letter, we investigate the performance of distributed switch-and-stay network, an appropriate solution for low-complexity cooperative relaying applications, under adaptive transmission. Under Rayleigh fading channels, we are able to derive the closed-form expression of the occurrence probability, outage probability, bit error probability and achievable spectral efficiency of the system. Numerous simulations are performed to verify the analytic results showing that by applying adaptive modulation, the achievable spectral efficiency of the system is much improved. Index Terms—switch and stay combining, amplify-andforward, Rayleigh fading, outage probability, spectral efficiency, bit error rate, adaptive transmission.

I. I NTRODUCTION For communication over wireless fading channels, cooperative systems have been shown to offer tremendous spatial diversity gain in capacity and in coverage as compared to direct transmission since it is impossible to install multi-antenna on wireless agents [1]. However, this gain is obtained at the cost of lower spectral efficiency and higher complexity and implementation cost since at least two time-slots are needed to transmit one data packet and combining technique, e.g. maximal ratio combining, selection combining, is expected at the destination [2]. To overcome such problems while still maintaining the spatial diversity, the concept of switched diversity is applied for cooperative networks promising a significant reduction in the system complexity as well as an improvement in the system spectral efficiency as compared to conventional cooperative networks [3], [4], [5], [6]. In particular, the concept of distributed switch-and-stay (SSC) is introduced to single relay cooperative network in [3] and [5]. It is then extended to two relays networks in [4]. Recently, to offer more spatial diversity, distributed switch-and-stay network in conjunction with relay selection is proposed in [6]. To further improve spectral efficiency, in this paper, we propose to apply the adaptive modulation on distributed SSC network with a single relay. The basic idea is that based on the limited feedback signal, the source adapts its transmitted constellation size according to channel conditions efficiently achieving a higher data rate for a given target bit error rate (BER). There has been a lot research, such as that presented in [7], [8], [9], [10], on the performance analysis of cooperative systems in conjunction with adaptive modulation.

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In particular, the performance analysis of repetition-based amplify-and-forward cooperative networks with five mode 𝑀 QAM over Rayleigh fading channels is introduced in [7]. In [8], Hwang et. al consider an incremental relaying protocol based on an amplify-and-forward transmission in conjunction with the best relay selection scheme over non-identically distributed relay channels. The numerical results show that it can provide a certain improvement of the spectral efficiency and outage probability while satisfying the desired BER. Beside amplify-and-forward, the adaptive modulation is also applied for repetition-based decode-and-forward network in [9] where both optimal and fixed switching are investigated. More recently, the work in [10] is devoted to analyze the performance of adaptive discrete modulation for opportunistic amplify-and-forward relaying networks. Based on the probability density function (PDF) of the end-to-end SNR which is expressed under a mathematically tractable form, the important metrics of the network (in terms of outage probability, spectral efficiency and bit error rate) are thoroughly derived. In this paper, we favor an elegant approach to study performance of distributed SSC with a single relay selection under adaptive modulation. The importance of our contribution is twofold. First, we propose adaptive modulation for distributed switch-and-stay networks. We then try to derive the important performance metrics of the systems: occurrence probability, outage probability, spectral efficiency and bit error probability over Rayleigh fading channels. After detailing out all the measure, we provide qualitative numerical results to verify the correctness of the analytical results. The rest of this paper is organized as follows. In Sect. II, we introduce the model under study and describe the proposed protocol. Sect. III shows the formulas allowing for evaluation of outage probability and achievable spectral efficiency of the system. In Sect. IV, we contrast the simulations and the results yielded by theory. Furthermore, some discussions on the behavior of the proposed system at low and high SNR regime are provided. Finally, the paper is closed in Sect. V. II. S YSTEM M ODEL We employ the same distributed switch-and-stay network with a signal relay as in [3]. In particular, suppose that the network under consideration consists of one source (𝒮), one amplify-and-forward relay (ℛ) and one destination (𝒟)

316

R

and (1b), the equivalent instantaneous of the 𝒮 → ℛ → 𝒟 link is [1] 𝛾𝒮ℛ 𝛾𝒮ℛ (4) 𝛾ℛ = 𝛾𝒮ℛ + 𝛾𝒮ℛ + 1

H*  T ?

Making use the fact that in medium and high SNR regimes, dual-hop amplify-and-forward relaying is dominated by the weakest hop, therefore, (4) can be well-approximated by [11], [12]

D

S H  T ?

𝛾ℛ ≈ min{𝛾𝒮ℛ , 𝛾ℛ𝒟 }

Since 𝛾𝒮ℛ and 𝛾ℛ𝒟 are exponentially distributed, 𝛾ℛ is again an exponential random variable [13]. The probability density function (PDF) and cumulative probability function (CDF) of 𝛾ℛ is therefore written, respectively, as follows:

Feedback channel Fig. 1.

Distributed switch-and-stay network under adaptive modulation

operating over Rayleigh fading channels as shown in Fig. 1. All nodes operate in half-duplex mode and the source signal reaches to the destination either by the direct link (𝒮 → 𝒟) or the relaying link (𝒮 → ℛ → 𝒟) according to the rule of switch-and-stay combining. In each transmission slot, only one link is active. At the destination, the active link is driven by comparing the received instantaneous signal-to-noise ratio (SNR) with the given threshold, 𝑇 . The switching will not occur as long as the instantaneous SNR on the currently active link remains above 𝑇 . Under the slow-fading assumption, the fading envelope is assumed to be constant over at least two consecutive transmission slots, therefore, the switching process becomes effective during the following transmission slot with the help of a limited feedback sent to the source and the relay. Since the relaying link is in active, the transmission slot is divided into two-sub time-slots. During the first sub-slot, the source transmits its signal, 𝑠, with the average transmit power 𝒫𝑠 to the relay. In the second sub-slot, the relay amplifies and then forwards the received signals towards the destination with the average transmit power 𝒫𝑟 . Let ℎ𝒜ℬ be the link coefficients between transmitted node 𝒜 and reception node ℬ, the received signals at the relay and the destination are mathematically modeled by the set of equations as follows: √ (1a) 𝑟𝒮ℛ = 𝒫𝑠 ℎ𝒮ℛ 𝑠 + 𝑛𝒮ℛ √ 𝑟ℛ𝒟 = 𝒫𝑟 ℎℛ𝒟 𝒢𝑟𝒮ℛ + 𝑛ℛ𝒟 (1b) where 𝑛𝒮ℛ and 𝑛ℛ𝒟 are the additive white Gaussian noise (AWGN) samples (having have zero mean and equal variance 𝒩0 ) at ℛ and 𝒟, respectively. The amplifying factor, 𝒢, is defined as √ 𝒫𝑟 (2) 𝒢= 2 𝒫𝑠 ∣ℎ𝒮ℛ ∣ + 𝒩0 Since the direct link is in active, the received signal at the destination is given 𝑟𝒮𝒟 = 𝒫𝑠 ℎ𝒮𝒟 𝑠 + 𝑛𝒮𝒟

(5)

𝑓𝛾ℛ (𝛾) =

1 − 𝛾¯𝛾 𝑒 ℛ 𝛾¯ℛ

∫𝛾 𝐹𝛾ℛ (𝛾) =

𝑓𝛾ℛ (𝛾)𝑑𝛾 = 1 − 𝑒

(6a) − 𝛾¯𝛾

ℛ

(6b)

0 ¯ℛ𝒟 𝒮ℛ 𝛾 where 𝛾¯ℛ = 𝛾¯𝛾¯𝒮ℛ +¯ 𝛾ℛ𝒟 [13, and 𝛾¯ℛ𝒟 = 𝐸{𝛾ℛ𝒟 }. 𝐸{.}

Eq. (6.82)] with 𝛾¯𝒮ℛ = 𝐸{𝛾𝒮ℛ } is the expectation operator. For the direct link, under a flat Rayleigh fading channel, the PDF and CDF of 𝛾𝒟 can be expressed as 𝑓𝛾𝒟 (𝛾) =

1 − 𝛾¯𝛾 𝑒 𝒟 𝛾¯𝒟 − 𝛾¯𝛾

𝐹𝛾𝒟 (𝛾) = 1 − 𝑒

𝒟

(7a) (7b)

where 𝛾¯𝒮𝒟 = 𝐸{𝛾𝒮𝒟 }. According the operation mode of adaptive discrete modulation, we partition the entire effective received SNR range into 𝐾 non-overlapping intervals, determined by a set of boundary values denoted as 𝛾𝑇0 = 0 < 𝛾𝑇1 < ⋅ ⋅ ⋅ < 𝛾𝑇𝑘 < ⋅ ⋅ ⋅ < 𝛾𝑇𝐾 = +∞. Since the end-to-end instantaneous received SNR, 𝛾Σ , is in the interval, [𝛾𝑇𝑘−1 𝛾𝑇𝑘 ), the destination will choose modulation scheme 𝑀𝑘 -QAM among the 𝐾 possible modulation ones to maintain the bit error probability of each scheme below the target bit error rate, BER𝑇 . Afterthat, the destination feedbacks this information to the source through a feedback channel. To facilitate the analysis, in this paper, we assume that the feedback channel is error free with no delay. Furthermore, to avoid deep fades, no data are sent since the end-to-end instantaneous received SNR is lower than 𝛾𝑇1 . To determine the boundary values, we start with the bit error rate of 𝑀 -QAM with Gray coding over additive white Gaussian noise channels [14, Table 6.1], namely ) (√ (8) 𝛽𝑘 𝛾Σ 𝑃bQAM (𝑘, 𝛾Σ ) ≈ 𝛼𝑘 𝑄 {

with

(3)

Let 𝛾𝒮𝒟 = 𝒫𝑠 ∣ℎ𝒮𝒟 ∣2 /𝒩0 , 𝛾𝒮ℛ = 𝒫𝑠 ∣ℎ𝒮ℛ ∣2 /𝒩0 and 𝛾ℛ𝒟 = 𝒫𝑟 ∣ℎℛ𝒟 ∣2 /𝒩0 denote the instantaneous SNRs for 𝒮 → 𝒟, 𝒮 → ℛ and ℛ → 𝒟 links, respectively. From (1a)

317

𝛼𝑘 = { 𝛽𝑘 =

1

, 𝑚𝑘 = 1, 2

4/𝑚𝑘

, 𝑚𝑘 ≥ 3 , 𝑚𝑘 = 1, 2

2/𝑚𝑘 𝑚𝑘

3/(2

− 1)

, 𝑚𝑘 ≥ 3

(9a)

(9b)

The boundary value relative to each mode can be found straightforwardly by inverting (8) and setting to the SNR required to satisfy the target BER, BER𝑇 , as follows: [ ( )]2 1 BER𝑇 𝑘 −1 𝛾𝑇 = 𝑄 , 𝑘 = 1, . . . , 𝐾 − 1. (10) 𝛽𝑘 𝛼𝑘 Before investigating the performance of DSSC network under adaptive modulation, we first study the percentage of time (or equivalent the connection probability) that the direct and relaying link are connected to the destination. Recalling that the switching will occur if the SNR of the currently connected link falls below the switching threshold, 𝑇 . Mathematically, we have [15, eq. (9.326)] =

𝑝𝒟

=

Pr(𝛾ℛ < 𝑇 ) Pr(𝛾𝒟 < 𝑇 ) + Pr(𝛾ℛ < 𝑇 ) 𝐹𝛾ℛ (𝑇 ) 𝐹𝛾𝒟 (𝑇 ) + 𝐹𝛾𝐷 (𝑇 )

(11a)

and =

𝑝ℛ

=

Pr(𝛾𝒟 < 𝑇 ) Pr(𝛾𝒟 < 𝑇 ) + Pr(𝛾ℛ < 𝑇 ) 𝐹𝛾𝒟 (𝑇 ) 𝐹𝛾𝒟 (𝑇 ) + 𝐹𝛾ℛ (𝑇 )

where Pr(𝛾𝑍 ≤ 𝑇 ) = 1−Pr(𝛾𝑍 > 𝑇 ) = 𝐹𝛾𝑍 (𝑇 ) and Pr(𝑥 < 𝛾𝑍 ≤ 𝛾𝑇𝑘 ) = 𝐹𝛾𝑍 (𝛾𝑇𝑘 ) − 𝐹𝛾𝑍 (𝑥) with 𝑥 ∈ {𝑇, 𝛾𝑇𝑘−1 }. B. Outage Probability To avoid deep fades, adaptive modulation distributed switchand-stay systems stop transmission since the end-to-end SNR is in between [𝛾𝑇0 𝛾𝑇1 ), thereby suffer a probability of outage. Mathematically speaking, the outage probability of the systems, defined as the probability that the end-to-end SNR falls below 𝛾𝑇1 , is equal to 𝜋1 , namely OP =𝜋1

Recalling that 𝑇 could be smaller and equal or greater than 𝛾𝑇1 leading to two cases that need to be considered separately. To this end, the system outage probability is ⎧ ⎨ 𝜋 (1) , 𝛾𝑇1 < 𝑇 1 (15) OP = ⎩ 𝜋1(2) , 𝛾𝑇0 < 𝑇 ≤ 𝛾𝑇1 with (1)

III. P ERFORMANCE A NALYSIS A. Occurence Probability of each mode For a given arbitrary value of 𝑇 and taking into account all possible cases of 𝑇 relative to [𝛾𝑇𝑘−1 𝛾𝑇𝑘 ), the occurrence probability of mode 𝑘 with 𝑘 = 1, . . . , 𝐾 can be generally expressed as1 [16] ⎧ (1) 𝜋 , 𝛾𝑇𝑘 < 𝑇   ⎨ 𝑘 (2) 𝜋𝑘 , 𝛾𝑇𝑘−1 < 𝑇 ≤ 𝛾𝑇𝑘 𝜋𝑘 = (12)   (3) 𝑘−1 ⎩ 𝜋 , 𝑇 <𝛾 where (1)

𝜋𝑘

[ ] 𝑝𝒟 Pr(𝛾𝒟 ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾ℛ ≤ 𝛾𝑇𝑘 ) (13a) [ ] +𝑝ℛ Pr(𝛾ℛ ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾𝒟 ≤ 𝛾𝑇𝑘 )

=

[ (2) 𝜋𝑘

Pr(𝛾𝒟 ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾ℛ ≤ 𝛾𝑇𝑘 )+

= 𝑝𝒟 +𝑝ℛ

=

𝑝𝒟 +𝑝ℛ

1 It

]

(13b) Pr(𝑇 < 𝛾𝒟 ≤ 𝛾𝑇𝑘 ) [ ] Pr(𝛾ℛ ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾𝒟 ≤ 𝛾𝑇𝑘 )+

[ (3) 𝜋𝑘

𝑇

Pr(𝑇 < 𝛾ℛ ≤

𝛾𝑇𝑘 )

=

𝜋1

(11b)

where 𝐹𝛾𝑍 (𝑇 ) with 𝑍 ∈ {𝒟, ℛ} can be trivially obtained by evaluating 𝐹𝛾𝑍 (𝛾) at 𝛾 = 𝑇 , respectively.

𝑘

(14)

[ ] 𝑝𝒟 Pr(𝛾𝒟 ≤ 𝑇 ) Pr(𝛾𝑇0 < 𝛾ℛ ≤ 𝛾𝑇1 ) (16a) [ ] +𝑝ℛ Pr(𝛾ℛ ≤ 𝑇 ) Pr(𝛾𝑇0 < 𝛾𝒟 ≤ 𝛾𝑇1 ) [

(2) 𝜋1

=

] Pr(𝛾𝒟 ≤ 𝑇 ) Pr(𝛾𝑇0 < 𝛾ℛ ≤ 𝛾𝑇1 )+ (16b) 𝑝𝒟 Pr(𝑇 < 𝛾𝒟 ≤ 𝛾𝑇1 ) [ ] Pr(𝛾ℛ ≤ 𝑇 ) Pr(𝛾𝑇0 < 𝛾𝒟 ≤ 𝛾𝑇1 )+ +𝑝ℛ Pr(𝑇 < 𝛾ℛ ≤ 𝛾𝑇1 )

C. Bit Error Rate The average BER for the adaptive modulation DSSC networks can be calculated as [17] ∑𝐾 𝑘=2 𝑚𝑘 BER𝑘 BER = ∑ (17) 𝐾 𝑘=2 𝑚𝑘 𝜋𝑘 in which BER𝑘 denotes the average BER in a specific region of [𝛾𝑇𝑘 𝛾𝑇𝑘+1 ), given by ⎧ (1)  BER𝑘 , 𝛾𝑇𝑘 < 𝑇   ⎨ (2) BER𝑘 = (18) BER𝑘 , 𝛾𝑇𝑘−1 < 𝑇 ≤ 𝛾𝑇𝑘   (3)  ⎩ BER , 𝑇 < 𝛾 𝑘−1 𝑘 𝑇 where (1)

Pr(𝛾𝒟 ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾ℛ ≤ 𝛾𝑇𝑘 )

𝑘 𝑘 BER𝑘 = 𝑝𝒟 Pr(𝛾𝒟 ≤ 𝑇 )ℐℛ (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) + 𝑝ℛ ℐ𝒟 (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 )

]

(13c) + Pr(𝛾𝑇𝑘−1 < 𝛾𝒟 ≤ 𝛾𝑇𝑘 ) [ ] Pr(𝛾ℛ ≤ 𝑇 ) Pr(𝛾𝑇𝑘−1 < 𝛾𝒟 ≤ 𝛾𝑇𝑘 ) + Pr(𝛾𝑇𝑘−1 < 𝛾ℛ ≤ 𝛾𝑇𝑘 )

(2)

BER𝑘

(3)

BER𝑘

(3)

is noted that the case of 𝑇 < 𝛾𝑇𝑘−1 , i.e., 𝜋1 , will not exist for 𝑘 = 1.

318

=

[ ] 𝑘 𝑘 𝑝𝒟 Pr(𝛾𝒟 ≤ 𝑇 )ℐℛ (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) + ℐ𝒟 (𝑇, 𝛾𝑇𝑘 ) [ ] 𝑘 𝑘 (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) + ℐℛ (𝑇, 𝛾𝑇𝑘 ) +𝑝ℛ Pr(𝛾ℛ ≤ 𝑇 )ℐ𝒟

=

[ ] 𝑘 𝑘 𝑝𝒟 Pr(𝛾𝒟 ≤ 𝑇 )ℐℛ (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) + ℐ𝒟 (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) [ ] 𝑘 𝑘 (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) + ℐℛ (𝛾𝑇𝑘−1 , 𝛾𝑇𝑘 ) +𝑝ℛ Pr(𝛾ℛ ≤ 𝑇 )ℐ𝒟

𝑘

ℐ𝑍𝑘 (𝑥, 𝛾𝑇𝑘 )

∫𝛾𝑇 =

𝛼𝑘 𝑄 𝑥

=

𝛼𝑘

{

(√ ) 1 − 𝛾 𝛽𝑘 𝛾 𝑒 𝛾¯𝑍 𝑑𝛾 𝛾¯𝑍

(21)

[ )− 12 ( ( ) )]}𝛾𝑇𝑘 ) 1 √ 1 (1 𝛽 𝛾 ) √𝛽 (𝛽 (√ )( 𝛾 1 𝛽 1 1 𝑘 𝑘 𝑘 𝑘 − Γ , − + , + 𝛾 𝑄 𝛽𝑘 𝛾 1 − 𝑒 𝛾¯𝑍 − Γ 2 𝜋 2 2 2𝜋 2 𝛾¯𝑍 2 2 𝛾¯𝑍 𝑥

Furthermore, ℐ𝑍𝑘 (𝑥, 𝛾𝑇𝑘 ) is defined as follows: ∫ 𝛾𝑇𝑘 ℐ𝑍𝑘 (𝑥, 𝛾𝑇𝑘 ) = 𝑃bQAM (𝑘, 𝛾)𝑓𝑍 (𝛾)𝑑𝛾

0

10

(20)

𝑥

−1

10 Outage Probability

Substituting (8) on (20) and after some steps of manipulations yields the closed-form of ℐ𝑍𝑘 (𝑥, 𝛾𝑇𝑘 ) as shown in (21) at the top of this page. IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we perform MATLAB-based computer simulations to verify the theoretical analysis. In all simulations, the variance of the channel fading between any two nodes is set as 𝐸{∣ℎ𝒮𝒟 ∣2 } = 1, 𝐸{∣ℎ𝒮ℛ ∣2 } = 2 and 𝐸{∣ℎℛ𝒟 ∣2 } = 3 and five-mode adaptive 𝑀 -QAM is employed [7], [9], [10]. Furthermore, the transmit power of the source is assumed to be equal that of the relay, i.e., 𝒫𝑠 = 𝒫𝑟 .

−6

BER =10 T

−2

10

BERT=10−3

−3

10

Analysis Simulation −4

10

0

5

10

15 20 Average SNR [dB]

25

30

Fig. 3. Outage probability for distributed SSC under adaptive modulation, 𝑇 = 10. 1 0.9

section. Furthermore, with adaptive systems, the higher the required target BER is, the higher outage probability the system suffers. For example, with the same probability of outage, 10−6 , the system with BER𝑇 = 10−6 gains around 8 dB as compared to that with BER𝑇 = 10−3 .

0.8 Occurence Probability

π

1

0.7

π2

0.6

π

3

π4

0.5

π5 Simulation

0.4

4.5 0.3

3.5

0

5

Fig. 2.

10

15

20 25 30 Average SNR [dB]

35

40

45

Spectral Efficiency

0.1 0

Analysis Simulation

4

0.2

50

Occurrence probability of each mode, 𝑇 = 10.

We first investigate the occurrence probability of each mode. It can be observed from Fig. 2 that mode 1 and mode 5 are dominated in low and high SNR regimes, respectively. Therefore, it is concluded that adaptive modulation will be in-effective at high SNR since the highest modulation constellation is used most of the time In Fig. 3, we study the outage probability of adaptive modulation DSSC systems as well as investigate the impact of the target BER on the system outage probability. It can be seen that there is an excellent match between the analytical and simulated curves, validating the analysis in the previous

3 BERT=10−3

2.5 2

BERT=10−6

1.5 1 0.5 0

0

5

10

15 20 Average SNR [dB]

25

30

Fig. 4. Spectral efficiency for distributed SSC under adaptive modulation, 𝑇 = 10.

In Fig. 4, we investigate the most interesting measure of any adaptive systems, achievable spectral efficiency. As expected,

319

the spectral efficiency of the system increases as the average SNR increases in medium range of SNR. It is obvious to observe that since adaptive modulation is used, we can trade the system spectral efficiency for better error performance and vice versa, e.g. for the same spectral efficiency, 5 dB transmit power can be saved since the target BER is reduced from 10−6 to 10−3 . −3

10

Analysis Simulation −3

Bit Error Probability

BERT=10

−4

10

−5

10

BERT=10−4

−6

10

0

5

10

15 20 Average SNR [dB]

25

30

Fig. 5. Bit error rate for distributed SSC under adaptive modulation, 𝑇 = 10.

In Fig. 5, we study the average BER of the proposed scheme where the target BER is set to 10−3 and 10−4 . It can be observed that under the fixed boundary values, the average BER is always well below the target BER and hence satisfies the QoS requirement. It is worth remarking that there is still a gap between the obtained BER and the target BER implying that the spectral efficiency of the system could be further improved by optimizing the boundary values. However, due to page limitation, we leave this problem for our next study. V. C ONCLUSION In this paper, for the first time, we have derived analytic results for the occurrence probability, outage probability, average bit error probability and spectral efficiency of distributed switch-and-stay network under adaptive modulation over Rayleigh fading channels. The analytic results are verified by some selected simulation results showing that an excellent agreement between them is observed.

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