# General Switch-and-Stay Combing for Space Diversity over Rayleigh Fading Channels

Authors: Yawgeng A. Chau; Yao-Hua Chen
Journal: International Journal of Antennas and Propagation
 (2012)

---

## Abstract

Three multibranch switch-and-stay combining (MSSC) schemes are analyzed for Rayleigh fading channels, where different decision statistics for antenna switching (i.e., switch statistic) are used. Leta and r denote the fading factor and the received baseband signal of a diversity branch, respectively. In contrast to the traditional MSSC that uses the faded signal-to-noise ratio (SNR) of diversity branches as the corresponding switch statistic, to enhance the receiver performance, |r|, |ar|, and a new linear combination of a and |r|
are used as switch statistics of the three MSSC schemes, respectively. For performance evaluation, the bit error rate (BER) of BPSK is derived for the three MSSC schemes over both independent-and-identical distributed (i.i.d.) and independent-and-nonidentical distributed (i.n.d.) Rayleigh fading channels. To pursue optimal performance, the locally optimal switch threshold (ST) of each MSSC scheme is obtained for general i.n.d. fading channels. In addition, the locally optimal ST becomes the globally optimal ST for i.i.d. channels. Numerical results based on the analysis and simulations are presented. In contrast to the MSSC over i.i.d. fading channels, we will show that the performance of MSSC schemes can be improved by increasing the number of branches, if i.n.d. channels are considered.


---

## Body

## 1. Introduction

In a wireless system, diversity combining provides an effective way to overcome multipath fading and enhance receiver performance [1–6]. The well-known maximal-ratio-combining and equal-gain-combining schemes require numerous arithmetic operations such as adders and/or multipliers for the implementation of signal combination. On the other hand, the conventional selection combining (SC) scheme needs simultaneous and continuous monitoring and estimation of all branch SNRs. The arithmetic operations and continuous signal-to-noise ratio (SNR) estimations are time- and power-consuming, which may be impractical for some wireless communication systems [5, 7]. To further reduce the implementation complexity of a diversity system, the switch-and-stay combining (SSC) scheme has been considered [8–12]. With an SSC system, the receiver only needs to monitor and estimate the channel state of the single branch currently in use. Thus, the switch-based SSC is valuable for a mobile terminal that has a limited processing capability.For traditional switch-based diversity schemes, the faded SNR of diversity branches is usually used as the decision statistics for antenna switching (i.e., switch statistic) in a threshold testing procedure and can be categorized as the SNR-based combining scheme. In [9], the SNR-based dual-branch SSC scheme was proposed, and the bit error rate (BER) of noncoherent FSK (NCFSK) was derived. In [10], a simplified dual-branch SNR-based SSC scheme was analyzed for the NCFSK on independent and correlated Nakagami fading channels. In [11], the dual-branch SNR-based SSC scheme on general fading channels was analyzed, and the optimal switch threshold (ST) was studied. In [12], the SSC scheme addressed in [9] and [10] was extended to the case of independent multibranch channels. In [13], the dual-branch SSC scheme in correlated Rayleigh and Rician fading channels was analyzed. In [14–16], the SSC scheme was applied to wireless communication systems that involve relays. In the previous works on the SSC scheme [7–16], only the faded SNR is considered as the switch statistic in the procedure of threshold testing, where an identical ST is used. Once the SNR of the used branch is lower than a preset threshold, the receiver switches to the next branch for signal reception. In a multibranch SSC (MSSC) system, the switch-to branch of the last branch will be the first one. Although the SSC is less complex than the SC, the SNR-based SSC suffers from a considerable performance loss compared to its counterpart of SC.Recently, other statistics rather than the SNR have been considered for the SC and associated branch selections. Leta and r be the fading factor and the received baseband signal of a diversity branch, respectively. In the SC scheme proposed in [17], the largest |r| at the receiver site is selected for signal detection on i.i.d.Rayleigh fading channels. In [18], the SC by choosing the largest |ar| was analyzed for BPSK signaling also on i.i.d.Rayleigh fading channels.In this paper, we want to explore possible performance improvement compared to traditional SNR-based SSC and SC schemes, when|r| or |ar| are used as the switch statistic of MSSC schemes. In addition, as an alternative choice for generating a statistic with lower complexity than |ar|, a linear combination of a and |r| is considered as the new switch statistic. Specifically, in contrast to the traditional SNR-based SSC and SC schemes, |r|, |ar|, and the linear combination of a and |r| are used as three different new switch statistics of MSSC schemes, respectively, where the performance of BPSK on i.i.d. and i.n.d.Rayleigh fading channels is evaluated. Notice that a variable multiplier is required to generate the statistic |ar| for each branch, while only adders are used for the linear combination of a and |r|. It is well know from circuit design that an adder is much less complex than a variable multiplier. Thus, among the three MSSC schemes, the MSSC based on |ar| has the highest implementation complexity.To optimize the performance of each MSSC scheme, an optimal or locally optimal ST that minimizes the BER of BPSK is explored. For the generali.n.d. fading channels, the globally optimal ST is difficult to obtain, and thus the locally optimal ST is developed for each diversity branch. A nice result, is that, for thei.i.d.fading channels with an identical ST, the locally optimal ST is also globally optimal.In published literature [10, 12], it has been shown that the performance of MSSC schemes on traditional i.i.d.fading channels cannot be enhanced by adding more branches for diversity combining, and the diversity gain remains the same as the one of dual-branch case. In this paper, we will also show that if i.n.d.fading channels are considered, the performance of MSSC schemes can be further improved with more than two branches for combining.The remainder of the paper is organized as follows. In Section2, the MSSC models with different switch statistics for BPSK on independent Rayleigh fading channels are presented. In Section 3, the corresponding BER of BPSK is derived for the three MSSC schemes on i.i.d. and i.n.d. fading channels, respectively. In Section 4, the optimal ST of the three MSSC schemes is derived for performance optimization, where an optimal ST in closed-form is obtained for the |ar|-based MSSC. In Section 5, numerical results based on the analysis and simulations are presented. Conclusions are drawn in Section 6.


## 2. Modeling of General MSSC

### 2.1. Signal and Channel Model

ForL-branch diversity combining, let Eb be the received energy, and al denote the fading factor of branch l for l=1,2,…,L. For BPSK signaling, the received signal after phase compensation for branch l can be represented by [14]
(1)rl=alsm+nl,l=1,2,…,L,
where m=0,1, s0=-Eb, s1=Eb, and nl is the additive complex Gaussian noise with zero mean and the variance N0/2. The faded SNR of branch l is given by γl=al2Eb/N0.For independent Rayleigh fading channels, the probability density function (PDF) ofγl is given by(2)fγl(x)=1γ̅le-(x/γ̅l),x≥0,
where γ̅l=E[γl]. The corresponding cumulative distribution function (CDF) of γl is(3)Fγl(x)=1-e-(x/γ̅l),x≥0.


### 2.2. Switch Model

Throughout the paper, a discrete-time model for the switch statistic and the SNR is used. In the MSSC system, at discrete-timen, let l̂(n)=l(l=1,2,…,L) represent the event that branch l is used and λl(n) denote the general switch statistic. In the paper, the case where the switch statistic λl(n) is a stationary process is considered.For the MSSC system, the branch switching is characterized by the rule:(4)Forl̂(n-1)=l,l̂(n)={l,ifλl(n)≥ηl,l+1,ifλl(n)<ηl,l≤L-1,1,ifλl(n)<ηl,l=L,
where ηl≥0 is the designated ST. In the traditional SSC system [9–12], the faded SNR is employed as the switch statistic, that is, λl(n)=γl(n). In the following analysis, the time index n will be often omitted to simplify the notation.The multibranch switch behavior can be modeled by theL-state stationary Markov chain as illustrated in Figure 1, where state l(l=1,2,…,L) means that branch l is used for signal reception, and pi,j(i,j=1,2,…,L) denotes the state transition probability.Figure 1
TheL-state Markov chain for the switch model of the MSSC.Under the condition thatsm(m=0,1) has been transmitted, the transition probability of this Markov chain from state i to state j is given by(5)pmi,j(ηi)={Pm(λi≥ηi),j=i;i=1,2,…,L,Pm(λi<ηi),j=i+1;i=1,2,…,L-1,Pm(λi<ηi),i=L,j=1,0,otherwise.
Let πl(i=1,2,…,L) be the stationary distribution of state l of the Markov chain. With the above state transition probability, it can be shown that πl is a function of (η1,η2,…,ηL) and is given by(6)πm,l(η1,η2,…,ηL)=(∑j=1L1Pm(λj<ηj))-11Pm(λl<ηl).
Notice that if λ1,λ2,…,λL are i.i.d. with ηl=η(l=1,2,…,L), then πm,l=1/L for all l.


## 2.1. Signal and Channel Model

ForL-branch diversity combining, let Eb be the received energy, and al denote the fading factor of branch l for l=1,2,…,L. For BPSK signaling, the received signal after phase compensation for branch l can be represented by [14]
(1)rl=alsm+nl,l=1,2,…,L,
where m=0,1, s0=-Eb, s1=Eb, and nl is the additive complex Gaussian noise with zero mean and the variance N0/2. The faded SNR of branch l is given by γl=al2Eb/N0.For independent Rayleigh fading channels, the probability density function (PDF) ofγl is given by(2)fγl(x)=1γ̅le-(x/γ̅l),x≥0,
where γ̅l=E[γl]. The corresponding cumulative distribution function (CDF) of γl is(3)Fγl(x)=1-e-(x/γ̅l),x≥0.


## 2.2. Switch Model

Throughout the paper, a discrete-time model for the switch statistic and the SNR is used. In the MSSC system, at discrete-timen, let l̂(n)=l(l=1,2,…,L) represent the event that branch l is used and λl(n) denote the general switch statistic. In the paper, the case where the switch statistic λl(n) is a stationary process is considered.For the MSSC system, the branch switching is characterized by the rule:(4)Forl̂(n-1)=l,l̂(n)={l,ifλl(n)≥ηl,l+1,ifλl(n)<ηl,l≤L-1,1,ifλl(n)<ηl,l=L,
where ηl≥0 is the designated ST. In the traditional SSC system [9–12], the faded SNR is employed as the switch statistic, that is, λl(n)=γl(n). In the following analysis, the time index n will be often omitted to simplify the notation.The multibranch switch behavior can be modeled by theL-state stationary Markov chain as illustrated in Figure 1, where state l(l=1,2,…,L) means that branch l is used for signal reception, and pi,j(i,j=1,2,…,L) denotes the state transition probability.Figure 1
TheL-state Markov chain for the switch model of the MSSC.Under the condition thatsm(m=0,1) has been transmitted, the transition probability of this Markov chain from state i to state j is given by(5)pmi,j(ηi)={Pm(λi≥ηi),j=i;i=1,2,…,L,Pm(λi<ηi),j=i+1;i=1,2,…,L-1,Pm(λi<ηi),i=L,j=1,0,otherwise.
Let πl(i=1,2,…,L) be the stationary distribution of state l of the Markov chain. With the above state transition probability, it can be shown that πl is a function of (η1,η2,…,ηL) and is given by(6)πm,l(η1,η2,…,ηL)=(∑j=1L1Pm(λj<ηj))-11Pm(λl<ηl).
Notice that if λ1,λ2,…,λL are i.i.d. with ηl=η(l=1,2,…,L), then πm,l=1/L for all l.


## 3. Performance Analysis

To simplify the referring in the context, the three MSSC schemes are represented by the|r|-MSSC, |ar|-MSSC, and (a+|r|)-MSSC, respectively. Throughout the paper, equally likely signaling is considered.


### 3.1. General BER Form

LetP0(·) be the error probability under the condition that s0 has been transmitted. Then, with the stationary distribution under fixed (η1,η2,…,ηL), the average BER of BPSK is given by(7)Pe(η1,η2,…,ηL)=∑l=1Lπ0,l(η1,η2,…,ηL)Pe,0(ηl∣l),
where Pe,0(ηl∣l)=P0(error∣l̂(n-1)=l) denotes the BER conditioned on the event that branch l has been used in the previous discrete-time slot. For BPSK with equally likely signaling, the conditional BER Pe,0(ηl∣l) has the form:
(8)Pe,0(ηl∣l)=P0({errorwithbranchl}AND{λl≥ηl})+P0({errorwithbranchl+1,l≤L-1}AND{λl<ηl})+P0({errorwithbranch1,l=L}AND{λl<ηl})=P0(rl>0,λl≥ηl)+P0(r[l]L+1>0)P0(λl<ηl),
where the notations of sets and “AND” operations are simplified, [l]L means l-modulo-L for 0≤[l]L≤L-1,P0(error with branchl,λl≥ηl) is the error probability of using the old branch, and P0(error with branch[l]L+1) is the error probability of using the new switched-to branch.In the following BER analysis, we will first formulate the stationary distributionπ0,l and the BER for fixed SNRs and then derive the average BER for faded SNRs.


### 3.2.|ar|-MSSC

In the|ar|-MSSC system, the switch statistic λl=|alrl| is used. This statistic has been applied to the optimal SC [14] that yields a minimal BER among different SC schemes. Thus, it will be expected that the receiver with the |ar|-MSSC will yield the smallest BER compared to other MSSC schemes, which will be confirmed in the Section 5.Under the condition thats0 has been transmitted, the stationary distribution given by (6) for the |ar|-MSSC has the form:(9)π0,l(η1,η2,…,ηL)=(∑j=1L1P0(|ajrj|<ηj))-1×1P0(|alrl|<ηl),
where λl=|alrl|. By defining the normalized threshold η̅j=ηjEb/N0, the probability P0(|alrl|<ηl) in (9) can be written in the form:(10)P0(|ajrj|<ηj)=P0(-ηj<ajrj<ηj)=-∫0∞12erfc⁡(x+η̅jx)fγj(x)dx+∫0∞12erfc⁡(x-η̅jx)fγj(x)dx,
where erfc(·) is the complementary error function. For evaluating the integrals in (10), using integration by parts, we can evaluate the integral ∫0∞erfc(x+η̅j/x)fγj(x)dx as(11)∫0∞erfc⁡(x+η̅jx)fγj(x)dx=∫0∞erfc⁡(x+η̅jx)1γ̅lexp⁡(-xγ̅l)dx=∫0∞[ddxerfc⁡(x+η̅jx)]exp⁡(-xγ̅l)dx=-e-2η̅jπ∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx+a⋅e-2η̅jπ∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx,
where, by applying [19, equation (3.472.3)], the two integrals ∫0∞x-1/2e-[(γ̅l+1)x/γ̅l+η̅j/x]dx and ∫0∞x-3/2e-[(γ̅l+1)x/γ̅l+η̅j/x]dx can be derived and written in following closed-forms:(12)∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=πγ̅lγ̅l+1⋅e-2η̅j(γ̅l+1)/γ̅l,(13)∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=πa⋅e-2η̅j(γ̅l+1)/γ̅l.
Therefore,(14)∫0∞12erfc⁡(x+η̅jx)fγj(x)dx=12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1).
Similarly, the first integral in (10) can also be derived and written in closed-form as(15)∫0∞12erfc⁡(x-η̅jx)1γ̅lexp⁡(-xγ̅l)dx=1-e2η̅j2π∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx-η̅j⋅e2η̅j2π∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=1-12(1-γ̅lγ̅l+1)e-2η̅j(1+1/γ̅l-1),
where [19, equation (3.472.3)] is used again. Consequently, in (9), we have(16)P0(|ajrj|<ηj)=1-12(1-γ̅lγ̅l+1)e-2η̅j(1+1/γ̅l-1)-12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1).
For the |ar|-MSSC, the conditional BER Pe,0(ηl∣l) in (8) is given by(17)Pe,0(ηl∣l)=P0(rl>0,alrl≥ηl)+P0(r[l]L+1>0)P0(|alrl|<ηl),
where(18)P0(r[l]L+1>0)=∫0∞12erfc⁡(γ[l]L+1)fγ[l]L+1(x)dx=12-12γ̅[l]L+1γ̅[l]L+1+1,
and, from similar manipulations given by (11)–(13),(19)P0(rl>0,alrl≥ηl)=P0(alrl≥ηl)=∫0∞12erfc⁡(x+η̅lx)fγl(x)dx=12(1-γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l+1).
Based on (7) and (10)–(19), the BER of the |ar|-MSSC can be obtained and written in the closed-form:(20)Pe(η̅1,η̅2,…,η̅L)=12∑l=1L(1-γ̅lγ̅l+1)(∑j=1L[1-12(1+γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j-1)-12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1)]-1)-1×{γ̅lγ̅l+111+e-2η̅l(1+1+1/γ̅l)[1-12(1+γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l-1)-12(1-γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l+1)]-1}.
For i.i.d.fading channels with γ̅l=γ̅ and identical ST η̅l=η̅ for l=1,2,…,L, the BER reduces to the form:(21)Pe(η̅)=12(1-γ̅γ̅+1)×[γ̅γ̅+11+e-2η̅(1+1/γ̅+1)-12(1+γ̅γ̅+1)e-2η̅(1+1/γ̅-1)-12(1-γ̅γ̅+1)e-2η̅(1+1/γ̅+1)].
For thei.i.d. channels with η̅=0 since we always have |alrl|≥0(l=1,2,…,L) and the switch will not occur (i.e., no SSC is used), the above BER reduces to the form:(22)Pe=12-12γ̅γ̅+1.
The BER given by (22) agrees with the well-known average BER of BPSK on a Rayleigh fading channel without diversity combining [20].From (21) for the i.i.d.channels, the BER performance remains the same for any value of L≥2, and the BER cannot be improved by adding more diversity branches.


### 3.3.(a+|r|)-MSSC

To save the processing time and power in an MSSC system and maintain its performance, other combination ofa and |r| may be considered. Notice that the |ar|-MSSC given in the previous subsection uses a multiplier which is more time- and power-consuming than an adder. In the (a+|r|)-MSSC system, the switch statistic is λl=alEb+|rl|, where an adder rather than a multiplier is used and Eb is a scaling constant. In the section of numerical results, it will be shown that (a+|r|)-MSSC and |ar|-MSSC have similar performance, which is significantly better than the SNR-based SSC.With the switch statisticλl=alEb+|rl| for l=1,2,…,L, the relevant probability in (6) now is given by P0(ajEb+|rj|<ηj). For (a+|r|)-MSSC, defining another normalized ST as η̃j=ηj/N0, we have(23)P0(ajEb+|rj|<ηj)=P0(-ηj+ajEb<rj<ηj-ajEb)=P0(nj<ηj)-P0(nj<2ajEb-ηj)=∫0η̃j212erfc⁡(2x-η̃j)dx-12erfc⁡(η̃j),
where the upper limit η̃j2 of the integral comes from the intrinsic constraint associated with the set {-ηj+ajEb<rj<ηj-ajEb}. To evaluate the integral in (23), we use integration by parts to obtain(24)∫0η̃j212erfc⁡(2x-η̃j)dx=12erfc⁡(-η̃j)-e-η̃j2/γ̅j2erfc⁡(η̃j)-∫0η̃j21πxe-(2x-η̃j)2-x/γ̅ldx,
where it is straightforward to show that(25)∫0η̃j21πxe-(2x-η̃j)2-x/γ̅ldx=2γ̅l4γ̅l+1e-(-η̃j2/(4γ̅l+1))×[η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+112erfc⁡(-2η̃jγ̅l4γ̅l+1)-12erfc⁡(η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+1)].
Consequently,(26)P0(ajEb+|rj|<ηj)=1-erfc⁡(η̃j)-e-η̃j2/γ̅j2erfc⁡(η̃j)-2γ̅l4γ̅l+1e-(-η̃j2/(4γ̅l+1))×[η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+112erfc⁡(-2η̃jγ̅l4γ̅l+1)-12erfc⁡(η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+1)].For the(a+|r|)-MSSC system, the conditional BER Pe(ηl∣l) in (8) now is given by(27)Pe(ηl∣l)=Pr(nl>alEb,nl≥ηl)+P0(r[l]L+1>0)P0(αlEb+|rl|<ηl),
where(28)Pr(nl>alEb,nl≥ηl)={Pr(nl>αlEb),αlEb>ηlPr(nl>ηl),αlEb≤ηl=∫0η̃l212erfc⁡(x)fγl(x)dx+∫η̃l2∞12erfc⁡(x)fγl(x)dx=12erfc⁡(η̃l)-12γ̅lγ̅l+1⋅erfc⁡(η̃lγ̅l+1γ̅l),
and P0(r[l]L+1>0) and P0(αlEb+|rl|<ηl) have been already specified in (18) and (26), respectively.Based on (7) and (23)–(27), the BER can be derived and written in the closed-form:(29)Pe(η̃1,η̃2,…,η̃L)=(∑j=1L{((2γ̅j+1)η̃jγ̅j(4γ̅j+1))1-2+e-η̃j2/γ̅j2erfc⁡(η̃j)-γ̅j4γ̅j+1e-η̃j2/(4γ̅j+1)×[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))erfc⁡(-η̃j4γ̅j4γ̅j+1)-erfc⁡((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]}-1)-1×∑l=1L[12erfc⁡(η̃l)-12γ̅lγ̅l+1erfc⁡(η̃lγ̅l+1γ̅l)]×{[erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))]1-2+e-η̃l2/γ̅l2erfc⁡(η̃l)-γ̅l4γ̅l+1e-η̃l2/(4γ̅l+1)×[erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))][erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))]}-1+(∑j=1L{[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]1-2+e-η̃j2/γ̅j2erfc⁡(η̃j)-γ̅j4γ̅j+1e-η̃j2/(4γ̅j+1)×[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))erfc⁡(-η̃j4γ̅j4γ̅j+1)-erfc⁡((2γ̅j+1)η̃jγ̅j(4γ̅j+1))((2γ̅j+1)η̃jγ̅j(4γ̅j+1))][((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]}-1)-1×∑l=1L12(1-γ̅lγ̅l+1).
With η̃l=η̃, (29) reduces to the following form for i.i.d.fading channels:(30)Pe(η̃)=12erfc⁡(η̃)-12γ̅γ̅+1erfc⁡(γ̅+1γ̅η̃)+12(1-γ̅γ̅+1)(1-2+e-η̃j2/γ̅j2erfc⁡(η̃))-12(1-γ̅γ̅+1)γ̅4γ̅+1e-η̃j2/(4γ̅j+1)×[erfc⁡(-2η̃γ̅4γ̅+1)-erfc⁡((2γ̅+1)η̃γ̅(4γ̅+1))].From (30) for i.i.d. channels, the performance remains the same even if more than two branches are used. Again, if no SSC is activated by setting η̃=0 in (30), it is easy to recognize that (30) becomes the form given by (22) for the situation without SSC.


### 3.4.|r|-MSSC

To further reduce implementation complexity,|rl| can also be considered as the switch statistic of the MSSC, that is, λl=|rl|. The corresponding probability in the stationary distribution given by (6) for the |r|-MSSC has the form:(31)P0(|rj|<ηj)=P0(nj<ajEb+ηj)-P0(nj<ajEb-ηj)=-∫0∞12erfc⁡(x+ηjN0)fγj(x)dx+∫0∞12erfc⁡(x-ηjN0)fγj(x)dx.
To evaluate (31), following steps similar to (24) and (25), we have(32)∫0∞12erfc⁡(x±η̃j)fγj(x)dx=12erfc⁡(±η̃j)-12γ̅lγ̅l+1×e-(-η̃j2/(γ̅l+1))erfc⁡(±η̃jγ̅lγ̅l+1).
Consequently, the probability given by (31) turns into the form(33)P0(|rj|<ηj)=1-erfc⁡(η̃j)-γ̅jγ̅j+1e-(-η̃j2/(γ̅l+1))+γ̅j4γ̅j+1×e-(-η̃j2/(γ̅l+1))erfc⁡(η̃jγ̅jγ̅j+1).The conditional BERPe(ηl∣l) given by (8) now has the form(34)Pe(ηl∣l)=P0(rl>0,rl≥ηl)+P0(r[l]L+1>0)P0(|rl|<ηl),
where P0(r[l]L+1>0) has been given in (18) and based on (32):(35)P0(rl>0,rl≥ηl)=P0(rl≥ηl)=∫0∞12erfc⁡(x+η̃l)fγl(x)dx=12erfc⁡(η̃j)-12γ̅lγ̅l+1×e-(-η̃j2/(γ̅l+1))erfc⁡(η̃jγ̅lγ̅l+1).In consequence, according to (7), (33), and (34), the BER is given by the closed-form:(36)Pe(η̃1,η̃2,…,η̃L)=∑l=1L(12erfc⁡(η̃l)-e-η̃l2/(γ̅l+1)2γ̅lγ̅l+1erfc⁡(η̃lγ̅lγ̅l+1))×([(η̃jγ̅jγ̅j+1)]-1∑j=1L[1-e-η̃j2/(γ̅j+1)γ̅jγ̅j+1+e-η̃j2/(γ̅j+1)γ̅jγ̅j+1⋅erfc⁡(η̃jγ̅jγ̅j+1)-erfc⁡(η̃j)]-1)-1×[1-e-η̃l2/(γ̅l+1)γ̅lγ̅l+1+e-η̃l2/(γ̅l+1)2γ̅lγ̅l+1⋅erfc⁡(η̃lγ̅lγ̅l+1)-erfc⁡(η̃l)]-1+∑l=1L(12-12γ̅lγ̅l+1)×([(η̃jγ̅jγ̅j+1)]-1∑j=1L[1-e-η̃j2/(γ̅j+1)γ̅jγ̅j+1+e-η̃j2/(γ̅j+1)γ̅jγ̅j+1⋅erfc⁡(η̃jγ̅jγ̅j+1)-erfc⁡(η̃j)]-1)-1.
For i.i.d. fading channels with η̃l=η̃, the above BER turns into the form:(37)Pe(η̃)=(12-12γ̅γ̅+1)(1-γ̅γ̅+1e-η̃2/(γ̅+1))-12γ̅γ̅+1×[γ̅γ̅+1e-η̃2/(γ̅+1)erfc⁡(η̃1+1/γ̅)-erfc⁡(η̃)],
that is also independent of L. If η̃=0 is substituted into (37) for the i.i.d.fading channels, the BER given by (37) also reduces to the form given by (22).


## 3.1. General BER Form

LetP0(·) be the error probability under the condition that s0 has been transmitted. Then, with the stationary distribution under fixed (η1,η2,…,ηL), the average BER of BPSK is given by(7)Pe(η1,η2,…,ηL)=∑l=1Lπ0,l(η1,η2,…,ηL)Pe,0(ηl∣l),
where Pe,0(ηl∣l)=P0(error∣l̂(n-1)=l) denotes the BER conditioned on the event that branch l has been used in the previous discrete-time slot. For BPSK with equally likely signaling, the conditional BER Pe,0(ηl∣l) has the form:
(8)Pe,0(ηl∣l)=P0({errorwithbranchl}AND{λl≥ηl})+P0({errorwithbranchl+1,l≤L-1}AND{λl<ηl})+P0({errorwithbranch1,l=L}AND{λl<ηl})=P0(rl>0,λl≥ηl)+P0(r[l]L+1>0)P0(λl<ηl),
where the notations of sets and “AND” operations are simplified, [l]L means l-modulo-L for 0≤[l]L≤L-1,P0(error with branchl,λl≥ηl) is the error probability of using the old branch, and P0(error with branch[l]L+1) is the error probability of using the new switched-to branch.In the following BER analysis, we will first formulate the stationary distributionπ0,l and the BER for fixed SNRs and then derive the average BER for faded SNRs.


## 3.2.|ar|-MSSC

In the|ar|-MSSC system, the switch statistic λl=|alrl| is used. This statistic has been applied to the optimal SC [14] that yields a minimal BER among different SC schemes. Thus, it will be expected that the receiver with the |ar|-MSSC will yield the smallest BER compared to other MSSC schemes, which will be confirmed in the Section 5.Under the condition thats0 has been transmitted, the stationary distribution given by (6) for the |ar|-MSSC has the form:(9)π0,l(η1,η2,…,ηL)=(∑j=1L1P0(|ajrj|<ηj))-1×1P0(|alrl|<ηl),
where λl=|alrl|. By defining the normalized threshold η̅j=ηjEb/N0, the probability P0(|alrl|<ηl) in (9) can be written in the form:(10)P0(|ajrj|<ηj)=P0(-ηj<ajrj<ηj)=-∫0∞12erfc⁡(x+η̅jx)fγj(x)dx+∫0∞12erfc⁡(x-η̅jx)fγj(x)dx,
where erfc(·) is the complementary error function. For evaluating the integrals in (10), using integration by parts, we can evaluate the integral ∫0∞erfc(x+η̅j/x)fγj(x)dx as(11)∫0∞erfc⁡(x+η̅jx)fγj(x)dx=∫0∞erfc⁡(x+η̅jx)1γ̅lexp⁡(-xγ̅l)dx=∫0∞[ddxerfc⁡(x+η̅jx)]exp⁡(-xγ̅l)dx=-e-2η̅jπ∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx+a⋅e-2η̅jπ∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx,
where, by applying [19, equation (3.472.3)], the two integrals ∫0∞x-1/2e-[(γ̅l+1)x/γ̅l+η̅j/x]dx and ∫0∞x-3/2e-[(γ̅l+1)x/γ̅l+η̅j/x]dx can be derived and written in following closed-forms:(12)∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=πγ̅lγ̅l+1⋅e-2η̅j(γ̅l+1)/γ̅l,(13)∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=πa⋅e-2η̅j(γ̅l+1)/γ̅l.
Therefore,(14)∫0∞12erfc⁡(x+η̅jx)fγj(x)dx=12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1).
Similarly, the first integral in (10) can also be derived and written in closed-form as(15)∫0∞12erfc⁡(x-η̅jx)1γ̅lexp⁡(-xγ̅l)dx=1-e2η̅j2π∫0∞x-1/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx-η̅j⋅e2η̅j2π∫0∞x-3/2e-(((γ̅l+1)/γ̅l)x+η̅j/x)dx=1-12(1-γ̅lγ̅l+1)e-2η̅j(1+1/γ̅l-1),
where [19, equation (3.472.3)] is used again. Consequently, in (9), we have(16)P0(|ajrj|<ηj)=1-12(1-γ̅lγ̅l+1)e-2η̅j(1+1/γ̅l-1)-12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1).
For the |ar|-MSSC, the conditional BER Pe,0(ηl∣l) in (8) is given by(17)Pe,0(ηl∣l)=P0(rl>0,alrl≥ηl)+P0(r[l]L+1>0)P0(|alrl|<ηl),
where(18)P0(r[l]L+1>0)=∫0∞12erfc⁡(γ[l]L+1)fγ[l]L+1(x)dx=12-12γ̅[l]L+1γ̅[l]L+1+1,
and, from similar manipulations given by (11)–(13),(19)P0(rl>0,alrl≥ηl)=P0(alrl≥ηl)=∫0∞12erfc⁡(x+η̅lx)fγl(x)dx=12(1-γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l+1).
Based on (7) and (10)–(19), the BER of the |ar|-MSSC can be obtained and written in the closed-form:(20)Pe(η̅1,η̅2,…,η̅L)=12∑l=1L(1-γ̅lγ̅l+1)(∑j=1L[1-12(1+γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j-1)-12(1-γ̅jγ̅j+1)e-2η̅j(1+1/γ̅j+1)]-1)-1×{γ̅lγ̅l+111+e-2η̅l(1+1+1/γ̅l)[1-12(1+γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l-1)-12(1-γ̅lγ̅l+1)e-2η̅l(1+1/γ̅l+1)]-1}.
For i.i.d.fading channels with γ̅l=γ̅ and identical ST η̅l=η̅ for l=1,2,…,L, the BER reduces to the form:(21)Pe(η̅)=12(1-γ̅γ̅+1)×[γ̅γ̅+11+e-2η̅(1+1/γ̅+1)-12(1+γ̅γ̅+1)e-2η̅(1+1/γ̅-1)-12(1-γ̅γ̅+1)e-2η̅(1+1/γ̅+1)].
For thei.i.d. channels with η̅=0 since we always have |alrl|≥0(l=1,2,…,L) and the switch will not occur (i.e., no SSC is used), the above BER reduces to the form:(22)Pe=12-12γ̅γ̅+1.
The BER given by (22) agrees with the well-known average BER of BPSK on a Rayleigh fading channel without diversity combining [20].From (21) for the i.i.d.channels, the BER performance remains the same for any value of L≥2, and the BER cannot be improved by adding more diversity branches.


## 3.3.(a+|r|)-MSSC

To save the processing time and power in an MSSC system and maintain its performance, other combination ofa and |r| may be considered. Notice that the |ar|-MSSC given in the previous subsection uses a multiplier which is more time- and power-consuming than an adder. In the (a+|r|)-MSSC system, the switch statistic is λl=alEb+|rl|, where an adder rather than a multiplier is used and Eb is a scaling constant. In the section of numerical results, it will be shown that (a+|r|)-MSSC and |ar|-MSSC have similar performance, which is significantly better than the SNR-based SSC.With the switch statisticλl=alEb+|rl| for l=1,2,…,L, the relevant probability in (6) now is given by P0(ajEb+|rj|<ηj). For (a+|r|)-MSSC, defining another normalized ST as η̃j=ηj/N0, we have(23)P0(ajEb+|rj|<ηj)=P0(-ηj+ajEb<rj<ηj-ajEb)=P0(nj<ηj)-P0(nj<2ajEb-ηj)=∫0η̃j212erfc⁡(2x-η̃j)dx-12erfc⁡(η̃j),
where the upper limit η̃j2 of the integral comes from the intrinsic constraint associated with the set {-ηj+ajEb<rj<ηj-ajEb}. To evaluate the integral in (23), we use integration by parts to obtain(24)∫0η̃j212erfc⁡(2x-η̃j)dx=12erfc⁡(-η̃j)-e-η̃j2/γ̅j2erfc⁡(η̃j)-∫0η̃j21πxe-(2x-η̃j)2-x/γ̅ldx,
where it is straightforward to show that(25)∫0η̃j21πxe-(2x-η̃j)2-x/γ̅ldx=2γ̅l4γ̅l+1e-(-η̃j2/(4γ̅l+1))×[η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+112erfc⁡(-2η̃jγ̅l4γ̅l+1)-12erfc⁡(η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+1)].
Consequently,(26)P0(ajEb+|rj|<ηj)=1-erfc⁡(η̃j)-e-η̃j2/γ̅j2erfc⁡(η̃j)-2γ̅l4γ̅l+1e-(-η̃j2/(4γ̅l+1))×[η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+112erfc⁡(-2η̃jγ̅l4γ̅l+1)-12erfc⁡(η̃j2(4γ̅l+1)γ̅l-2η̃jγ̅l4γ̅l+1)].For the(a+|r|)-MSSC system, the conditional BER Pe(ηl∣l) in (8) now is given by(27)Pe(ηl∣l)=Pr(nl>alEb,nl≥ηl)+P0(r[l]L+1>0)P0(αlEb+|rl|<ηl),
where(28)Pr(nl>alEb,nl≥ηl)={Pr(nl>αlEb),αlEb>ηlPr(nl>ηl),αlEb≤ηl=∫0η̃l212erfc⁡(x)fγl(x)dx+∫η̃l2∞12erfc⁡(x)fγl(x)dx=12erfc⁡(η̃l)-12γ̅lγ̅l+1⋅erfc⁡(η̃lγ̅l+1γ̅l),
and P0(r[l]L+1>0) and P0(αlEb+|rl|<ηl) have been already specified in (18) and (26), respectively.Based on (7) and (23)–(27), the BER can be derived and written in the closed-form:(29)Pe(η̃1,η̃2,…,η̃L)=(∑j=1L{((2γ̅j+1)η̃jγ̅j(4γ̅j+1))1-2+e-η̃j2/γ̅j2erfc⁡(η̃j)-γ̅j4γ̅j+1e-η̃j2/(4γ̅j+1)×[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))erfc⁡(-η̃j4γ̅j4γ̅j+1)-erfc⁡((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]}-1)-1×∑l=1L[12erfc⁡(η̃l)-12γ̅lγ̅l+1erfc⁡(η̃lγ̅l+1γ̅l)]×{[erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))]1-2+e-η̃l2/γ̅l2erfc⁡(η̃l)-γ̅l4γ̅l+1e-η̃l2/(4γ̅l+1)×[erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))][erfc⁡(-η̃l4γ̅l4γ̅l+1)-erfc⁡((2γ̅l+1)η̃lγ̅l(4γ̅l+1))]}-1+(∑j=1L{[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]1-2+e-η̃j2/γ̅j2erfc⁡(η̃j)-γ̅j4γ̅j+1e-η̃j2/(4γ̅j+1)×[((2γ̅j+1)η̃jγ̅j(4γ̅j+1))erfc⁡(-η̃j4γ̅j4γ̅j+1)-erfc⁡((2γ̅j+1)η̃jγ̅j(4γ̅j+1))((2γ̅j+1)η̃jγ̅j(4γ̅j+1))][((2γ̅j+1)η̃jγ̅j(4γ̅j+1))]}-1)-1×∑l=1L12(1-γ̅lγ̅l+1).
With η̃l=η̃, (29) reduces to the following form for i.i.d.fading channels:(30)Pe(η̃)=12erfc⁡(η̃)-12γ̅γ̅+1erfc⁡(γ̅+1γ̅η̃)+12(1-γ̅γ̅+1)(1-2+e-η̃j2/γ̅j2erfc⁡(η̃))-12(1-γ̅γ̅+1)γ̅4γ̅+1e-η̃j2/(4γ̅j+1)×[erfc⁡(-2η̃γ̅4γ̅+1)-erfc⁡((2γ̅+1)η̃γ̅(4γ̅+1))].From (30) for i.i.d. channels, the performance remains the same even if more than two branches are used. Again, if no SSC is activated by setting η̃=0 in (30), it is easy to recognize that (30) becomes the form given by (22) for the situation without SSC.


## 3.4.|r|-MSSC

To further reduce implementation complexity,|rl| can also be considered as the switch statistic of the MSSC, that is, λl=|rl|. The corresponding probability in the stationary distribution given by (6) for the |r|-MSSC has the form:(31)P0(|rj|<ηj)=P0(nj<ajEb+ηj)-P0(nj<ajEb-ηj)=-∫0∞12erfc⁡(x+ηjN0)fγj(x)dx+∫0∞12erfc⁡(x-ηjN0)fγj(x)dx.
To evaluate (31), following steps similar to (24) and (25), we have(32)∫0∞12erfc⁡(x±η̃j)fγj(x)dx=12erfc⁡(±η̃j)-12γ̅lγ̅l+1×e-(-η̃j2/(γ̅l+1))erfc⁡(±η̃jγ̅lγ̅l+1).
Consequently, the probability given by (31) turns into the form(33)P0(|rj|<ηj)=1-erfc⁡(η̃j)-γ̅jγ̅j+1e-(-η̃j2/(γ̅l+1))+γ̅j4γ̅j+1×e-(-η̃j2/(γ̅l+1))erfc⁡(η̃jγ̅jγ̅j+1).The conditional BERPe(ηl∣l) given by (8) now has the form(34)Pe(ηl∣l)=P0(rl>0,rl≥ηl)+P0(r[l]L+1>0)P0(|rl|<ηl),
where P0(r[l]L+1>0) has been given in (18) and based on (32):(35)P0(rl>0,rl≥ηl)=P0(rl≥ηl)=∫0∞12erfc⁡(x+η̃l)fγl(x)dx=12erfc⁡(η̃j)-12γ̅lγ̅l+1×e-(-η̃j2/(γ̅l+1))erfc⁡(η̃jγ̅lγ̅l+1).In consequence, according to (7), (33), and (34), the BER is given by the closed-form:(36)Pe(η̃1,η̃2,…,η̃L)=∑l=1L(12erfc⁡(η̃l)-e-η̃l2/(γ̅l+1)2γ̅lγ̅l+1erfc⁡(η̃lγ̅lγ̅l+1))×([(η̃jγ̅jγ̅j+1)]-1∑j=1L[1-e-η̃j2/(γ̅j+1)γ̅jγ̅j+1+e-η̃j2/(γ̅j+1)γ̅jγ̅j+1⋅erfc⁡(η̃jγ̅jγ̅j+1)-erfc⁡(η̃j)]-1)-1×[1-e-η̃l2/(γ̅l+1)γ̅lγ̅l+1+e-η̃l2/(γ̅l+1)2γ̅lγ̅l+1⋅erfc⁡(η̃lγ̅lγ̅l+1)-erfc⁡(η̃l)]-1+∑l=1L(12-12γ̅lγ̅l+1)×([(η̃jγ̅jγ̅j+1)]-1∑j=1L[1-e-η̃j2/(γ̅j+1)γ̅jγ̅j+1+e-η̃j2/(γ̅j+1)γ̅jγ̅j+1⋅erfc⁡(η̃jγ̅jγ̅j+1)-erfc⁡(η̃j)]-1)-1.
For i.i.d. fading channels with η̃l=η̃, the above BER turns into the form:(37)Pe(η̃)=(12-12γ̅γ̅+1)(1-γ̅γ̅+1e-η̃2/(γ̅+1))-12γ̅γ̅+1×[γ̅γ̅+1e-η̃2/(γ̅+1)erfc⁡(η̃1+1/γ̅)-erfc⁡(η̃)],
that is also independent of L. If η̃=0 is substituted into (37) for the i.i.d.fading channels, the BER given by (37) also reduces to the form given by (22).


## 4. Performance Optimization

To minimize the average BER, the optimal ST is pursued for the three MSSC schemes in this section. To obtain the globally optimal ST for each branch, theL simultaneous equations ∂Pe(η̃1,η̃2,…,η̃L)/∂η̃l=0 (or ∂Pe(η̅1,η̅2,…,η̅L)/∂η̅l=0 if the |ar|-MSSC is used) for l=1,2,…,L have to be solved, which is difficult for a general case. Thus, we turn to the locally optimal ST. The locally optimal ST is defined as the ST that minimizes the conditional BER Pe(η̃l∣l) or Pe(η̅l∣l)  for each branch.For the MSSC implementationin a receiver, a branch is randomly chosen at the beginning of a MSSC procedure, where the STs can be designated on a timeslot-by-timeslot basis. Under this consideration, the conditioned BERPe(ηl∣l) of branch l for i.n.d. fading channels can be used as the object function to minimize for the determination of the locally optimal ηl(l=1,2,…,L).Fori.i.d. fading channels with an identical ST ηl=η for l=1,2,…,L, an interesting result is that the conditional BER is independent of l and equals the total average BER, that is, Pe(η̅l∣l)=Pe(η̅) and Pe(η̃l∣l)=Pe(η̃). In consequence, the locally optimal ST is also globally optimal for i.i.d. fading channels.


### 4.1.|ar|-MSSC

With the|ar|-MSSC, the locally optimal ST for l=1,2,…,L can be obtained by solving ∂Pe(η̅l∣l)/∂η̅l=0 that gives the equation (after some manipulations):(38)(1-γ̅[l]L+1γ̅[l]L+1+1)e2η̅l,o=(1+γ̅[l]L+1γ̅[l]L+1+1)e-2η̅l,o,
where η̅l=η̅l,o is the locally optimal ST. Solving (38) for η̅l,o, we obtain the locally optimal ST expressed in the closed-form:(39)η̅l,o=14ln(γ̅[l]L+1+1+γ̅[l]L+1γ̅[l]L+1+1-γ̅[l]L+1).
From (39), a surprising result is that the locally optimal ST of |ar|-MSSC only depends on the average SNR of the switched-to branch.As aforementioned, fori.i.d. fading channels with γ̅l=γ̅ and η̅l=η̅ for l=1,2,…,L,η̅l,o=η̅o also gives the globally optimal ST.


### 4.2.(a+|r|)-MSSC

In the(a+|r|)-MSSC system, the equation ∂Pe(η̃l∣l)/∂η̃l=0 is to be solved for the locally optimal ST for branch l(l=1,2,…,L), which leads to a complex nonlinear equation which can only be solved by a numerical method such as the one given in [21]. For the i.i.d. fading channels with an identical ST, the globally optimal ST η̃o can be derived from the nonlinear equation.


### 4.3.|r|-MSSC

For the|r|-MSSC with the conditional BER given by (34) and the normalized threshold η̃l, the equation ∂Pe(η̃l∣l)/∂η̃l=0 at the locally optimal ST η̃l=η̃l,o yields
(40)erfc⁡(η̃l,oγ̅lγ̅l+1)+γ̅[l]L+1+1γ̅[l]L+1-1=1πγ̅l+1γ̅l⋅e-γ̅lη̃l,o2/(γ̅l+1)η̃l,o,
for l=1,2,…,L. The nonlinear equation given by (40) can be easily solved by using a numerical method [21]. Again, for the i.i.d. fading channels, the globally optimal ST of the |r|-MSSC is obtained by solving (40).


## 4.1.|ar|-MSSC

With the|ar|-MSSC, the locally optimal ST for l=1,2,…,L can be obtained by solving ∂Pe(η̅l∣l)/∂η̅l=0 that gives the equation (after some manipulations):(38)(1-γ̅[l]L+1γ̅[l]L+1+1)e2η̅l,o=(1+γ̅[l]L+1γ̅[l]L+1+1)e-2η̅l,o,
where η̅l=η̅l,o is the locally optimal ST. Solving (38) for η̅l,o, we obtain the locally optimal ST expressed in the closed-form:(39)η̅l,o=14ln(γ̅[l]L+1+1+γ̅[l]L+1γ̅[l]L+1+1-γ̅[l]L+1).
From (39), a surprising result is that the locally optimal ST of |ar|-MSSC only depends on the average SNR of the switched-to branch.As aforementioned, fori.i.d. fading channels with γ̅l=γ̅ and η̅l=η̅ for l=1,2,…,L,η̅l,o=η̅o also gives the globally optimal ST.


## 4.2.(a+|r|)-MSSC

In the(a+|r|)-MSSC system, the equation ∂Pe(η̃l∣l)/∂η̃l=0 is to be solved for the locally optimal ST for branch l(l=1,2,…,L), which leads to a complex nonlinear equation which can only be solved by a numerical method such as the one given in [21]. For the i.i.d. fading channels with an identical ST, the globally optimal ST η̃o can be derived from the nonlinear equation.


## 4.3.|r|-MSSC

For the|r|-MSSC with the conditional BER given by (34) and the normalized threshold η̃l, the equation ∂Pe(η̃l∣l)/∂η̃l=0 at the locally optimal ST η̃l=η̃l,o yields
(40)erfc⁡(η̃l,oγ̅lγ̅l+1)+γ̅[l]L+1+1γ̅[l]L+1-1=1πγ̅l+1γ̅l⋅e-γ̅lη̃l,o2/(γ̅l+1)η̃l,o,
for l=1,2,…,L. The nonlinear equation given by (40) can be easily solved by using a numerical method [21]. Again, for the i.i.d. fading channels, the globally optimal ST of the |r|-MSSC is obtained by solving (40).


## 5. Numerical Results

In this section, numerical results of BER based on the analysis are presented, and the results from Monte Carlo simulation are also supplied.In Figure2, the BERs of the three MSSC schemes on i.i.d. fading channels for L=2 are illustrated. As expected in Section 3, among the MSSC schemes, the |ar|-MSSC yields the best performance. The (a+|r|)-MSSC and |ar|-MSSC schemes have similar performance, and both of them have a much smaller BER than the |r|-MSSC scheme. Furthermore, the |ar|-MSSC and (a+|r|)-MSSC schemes even have a smaller BER than SNR-based SC scheme for some ranges of γ̅ (e.g., γ̅≤6dB in Figure 2), where, unlike the MSSC, the SC requires power-consuming multibranch monitoring.Figure 2
Performance comparisons for different dual-branch MSSC schemes oni.i.d. channels.The BER of the SNR-based SSC is also plotted in Figure2, where all the three MSSC schemes outperform the traditional SNR-based SSC and the power gain for the |ar|-MSSC or (a+|r|)-MSSC over the SNR-based SSC could be more than 2 dB for larger values of γ̅.The BER comparison oni.n.d. fading channels is illustrated in Figure 3, where corresponding locally optimal STs are used for the MSSC schemes, and two scenarios of the average SNRs are considered. Basically, three different average SNRs (γ̅1,γ̅2,γ̅3) are used, where the relations γ̅2=γ̅1/4 and γ̅3=γ̅1/8 are set. Thus, γ̅1>γ̅2>γ̅3. For the dual-branch case, two branches with γ̅2 and γ̅3 are used. For the triple-branch case, another branch with γ̅1 is added. Notice that the new added branch has the largest average SNR. From numerical results, we find that the performance of MSSC schemes can be enhanced if newly added branches have a larger average SNR than the used branches, which can be observed from Figure 3. On the other hand, if newly added branches do not have a larger average SNR, the resultant performance with more branches may be worse.Figure 3
Performance comparisons for different MSSC schemes oni.n.d. channels.The BER of the SNR-based SC is also plotted in Figure3 for performance comparisons. Based on these comparisons, the performance of both the |ar|-MSSC and (a+|r|)-MSSC schemes approaches and may be even better than that of the SNR-based SC scheme for some ranges of smaller average SNRs. For example, in the dual-branch case in Figure 3, both the |ar|-MSSC and (a+|r|)-MSSC schemes outperform the SNR-based SC scheme for γ̅1≤14dB (which is similar to the phenomenon on i.i.d. fading channels).


## 6. Conclusions

Three MSSC schemes with new switch statistics have been developed and evaluated for bothi.i.d. and i.n.d. fading channels, where the |ar|-MSSC yields the best receiver performance but also has the highest implementation complexity. The (a+|r|)-MSSC and |ar|-MSSC schemes have similar performance. To generate the switch statistic, the (a+|r|)-MSSC uses an adder instead of the multiplier required by the |ar|-MSSC, which makes the (a+|r|)-MSSC faster and more power-saving than the |ar|-MSSC. All the three new MSSC schemes outperform the traditional SNR-based SSC. The |ar|-MSSC and (a+|r|)-MSSC may even have a lower BER than the traditional SNR-based SC that requires simultaneous multibranch monitoring. Furthermore, the manipulation procedure used for the performance evaluation of BPSK can be easily extended to other signaling methods such as DPSK or FSK.



---

*Source: 717494-2012-05-08.xml*