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

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. Let 𝑎 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.

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. Let 𝑎 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 general i.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 the i.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 Section 2, 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

For 𝐿-branch diversity combining, let 𝐸_𝑏 be the received energy, and 𝑎_𝑙 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] 𝑟_𝑙=𝑎_𝑙𝑠_𝑚+𝑛_𝑙,𝑙=1,2,…,𝐿, (1) where 𝑚=0,1, 𝑠₀=−𝐸_𝑏, 𝑠₁=𝐸_𝑏, and 𝑛_𝑙 is the additive complex Gaussian noise with zero mean and the variance 𝑁₀/2. The faded SNR of branch 𝑙 is given by 𝛾_𝑙=𝑎²_𝑙𝐸_𝑏/𝑁₀.

For independent Rayleigh fading channels, the probability density function (PDF) of 𝛾_𝑙 is given by𝑓_𝛾𝑙(𝑥)=1𝛾_𝑙𝑒^{−(𝑥/𝛾𝑙)},𝑥≥0, (2) where 𝛾_𝑙=𝐸[𝛾_𝑙]. The corresponding cumulative distribution function (CDF) of 𝛾_𝑙 is𝐹_𝛾𝑙(𝑥)=1−𝑒^{−(𝑥/𝛾𝑙)},𝑥≥0. (3)

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-time 𝑛, let 𝑙(̂𝑛)=𝑙(𝑙=1,2,…,𝐿) represent the event that branch 𝑙 is used and 𝜆_𝑙(𝑛) denote the general switch statistic. In the paper, the case where the switch statistic 𝜆_𝑙(𝑛) is a stationary process is considered.

For the MSSC system, the branch switching is characterized by the rule:For𝑙̂(𝑛−1)=𝑙,𝑙̂(𝑛)=𝑙,if𝜆_𝑙(𝑛)≥𝜂_𝑙,𝑙+1,if𝜆_𝑙(𝑛)<𝜂_𝑙,𝑙≤𝐿−1,1,if𝜆_𝑙(𝑛)<𝜂_𝑙,𝑙=𝐿, (4) where 𝜂_𝑙≥0 is the designated ST. In the traditional SSC system [9–12], the faded SNR is employed as the switch statistic, that is, 𝜆_𝑙(𝑛)=𝛾_𝑙(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 the 𝐿-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 𝑝^𝑖,𝑗(𝑖,𝑗=1,2,…,𝐿) denotes the state transition probability.

Figure 1 

The 𝐿-state Markov chain for the switch model of the MSSC.

Under the condition that 𝑠_𝑚(𝑚=0,1) has been transmitted, the transition probability of this Markov chain from state 𝑖 to state 𝑗 is given by𝑝_𝑚^𝑖,𝑗𝜂_𝑖=𝑃_𝑚𝜆_𝑖≥𝜂_𝑖𝑃,𝑗=𝑖;𝑖=1,2,…,𝐿,_𝑚𝜆_𝑖<𝜂_𝑖𝑃,𝑗=𝑖+1;𝑖=1,2,…,𝐿−1,_𝑚𝜆_𝑖<𝜂_𝑖,𝑖=𝐿,𝑗=1,0,otherwise. (5) Let 𝜋_𝑙(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 𝜋_𝑙 is a function of (𝜂₁,𝜂₂,…,𝜂_𝐿) and is given by𝜋_𝑚,𝑙𝜂₁,𝜂₂,…,𝜂_𝐿=^𝐿_𝑗=11𝑃_𝑚𝜆_𝑗<𝜂_𝑗⁻¹1𝑃_𝑚𝜆_𝑙<𝜂_𝑙. (6) Notice that if 𝜆₁,𝜆₂,…,𝜆_𝐿 are i.i.d. with 𝜂_𝑙=𝜂(𝑙=1,2,…,𝐿), then 𝜋_𝑚,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

Let 𝑃₀(⋅) be the error probability under the condition that 𝑠₀ has been transmitted. Then, with the stationary distribution under fixed (𝜂₁,𝜂₂,…,𝜂_𝐿), the average BER of BPSK is given by𝑃_𝑒𝜂₁,𝜂₂,…,𝜂_𝐿=^𝐿_𝑙=1𝜋_0,𝑙𝜂₁,𝜂₂,…,𝜂_𝐿𝑃_𝑒,0𝜂_𝑙∣𝑙, (7) where 𝑃_𝑒,0(𝜂_𝑙∣𝑙)=𝑃₀(error∣𝑙(̂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 𝑃_𝑒,0(𝜂_𝑙∣𝑙) has the form: 𝑃_𝑒,0𝜂_𝑙∣𝑙=𝑃₀{errorwithbranch𝑙}AND𝜆_𝑙≥𝜂_𝑙+𝑃₀{errorwithbranch𝑙+1,𝑙≤𝐿−1}AND𝜆_𝑙<𝜂_𝑙+𝑃₀{errorwithbranch1,𝑙=𝐿}AND𝜆_𝑙<𝜂_𝑙=𝑃₀𝑟_𝑙>0,𝜆_𝑙≥𝜂_𝑙+𝑃₀𝑟_[𝑙]𝐿+1>0𝑃₀𝜆_𝑙<𝜂_𝑙, (8) where the notations of sets and “AND” operations are simplified, [𝑙]_𝐿 means 𝑙-modulo-𝐿 for 0≤[𝑙]_𝐿≤𝐿−1,𝑃₀(errorwithbranch𝑙,𝜆_l≥𝜂_l) is the error probability of using the old branch, and 𝑃₀(errorwithbranch[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 distribution 𝜋_0,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 𝜆_𝑙=|𝑎_𝑙𝑟_𝑙| 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 that 𝑠₀ has been transmitted, the stationary distribution given by (6) for the |𝑎𝑟|-MSSC has the form:𝜋_0,𝑙𝜂₁,𝜂₂,…,𝜂_𝐿=^𝐿_𝑗=11𝑃₀𝑎_𝑗𝑟_𝑗<𝜂_𝑗⁻¹×1𝑃₀𝑎_𝑙𝑟_𝑙<𝜂_𝑙, (9) where 𝜆_𝑙=|𝑎_𝑙𝑟_𝑙|. By defining the normalized threshold 𝜂_𝑗=𝜂_𝑗𝐸_𝑏/𝑁₀, the probability 𝑃₀(|𝑎_𝑙𝑟_𝑙|<𝜂_𝑙) in (9) can be written in the form:𝑃₀𝑎_𝑗𝑟_𝑗<𝜂_𝑗=𝑃₀−𝜂_𝑗<𝑎_𝑗𝑟_𝑗<𝜂_𝑗=−^∞₀12erfc√𝑥+𝜂_𝑗√𝑥𝑓_𝛾𝑗(𝑥)+𝑑𝑥_∞012erfc√𝑥−𝜂_𝑗√𝑥𝑓_𝛾𝑗(𝑥)𝑑𝑥, (10) where erfc(⋅) is the complementary error function. For evaluating the integrals in (10), using integration by parts, we can evaluate the integral ∫^∞₀erfc(√𝑥+𝜂_𝑗/√𝑥)𝑓_𝛾𝑗(𝑥)𝑑𝑥 aserfc√^∞₀𝑥+𝜂_𝑗√𝑥𝑓_𝛾𝑗(𝑥)=𝑑𝑥_∞0erfc√𝑥+𝜂_𝑗√𝑥1𝛾_𝑙−𝑥exp𝛾_𝑙𝑑𝑥=^∞₀𝑑√𝑑𝑥erfc𝑥+𝜂_𝑗√𝑥−𝑥exp𝛾_𝑙𝑒𝑑𝑥=−^−2𝜂𝑗√𝜋^∞₀𝑥^−1/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}+𝑑𝑥𝑎⋅𝑒^−2𝜂𝑗√𝜋^∞₀𝑥^−3/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}𝑑𝑥, (11) where, by applying [19, equation (3.472.3)], the two integrals ∫^∞₀𝑥^−1/2𝑒^{−[(𝛾𝑙+1)𝑥/𝛾𝑙+𝜂𝑗/𝑥]}𝑑𝑥 and ∫^∞₀𝑥^−3/2𝑒^{−[(𝛾𝑙+1)𝑥/𝛾𝑙+𝜂𝑗/𝑥]}𝑑𝑥 can be derived and written in following closed-forms:^∞₀𝑥^−1/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}𝑑𝑥=𝜋𝛾_𝑙𝛾_𝑙+1⋅𝑒^{−2𝜂𝑗(𝛾𝑙+1)/𝛾𝑙}, (12)^∞₀𝑥^−3/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}√𝑑𝑥=𝜋𝑎⋅𝑒^{−2𝜂𝑗(𝛾𝑙+1)/𝛾𝑙}. (13) Therefore,^∞₀12erfc√𝑥+𝜂_𝑗√𝑥𝑓_𝛾𝑗(𝑥)=1𝑑𝑥21−𝛾_𝑗𝛾_𝑗𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑗+1)}. (14) Similarly, the first integral in (10) can also be derived and written in closed-form as^∞₀12erfc√𝑥−𝜂_𝑗√𝑥1𝛾_𝑙−𝑥exp𝛾_𝑙𝑒𝑑𝑥=1−^2𝜂𝑗2√𝜋^∞₀𝑥^−1/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}−𝑑𝑥𝜂_𝑗⋅𝑒^2𝜂𝑗2√𝜋^∞₀𝑥^−3/2𝑒^{−(((𝛾𝑙+1)/𝛾𝑙)𝑥+𝜂𝑗/𝑥)}1𝑑𝑥=1−21−𝛾_𝑙𝛾_𝑙𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑙−1)}, (15) where [19, equation (3.472.3)] is used again. Consequently, in (9), we have𝑃₀𝑎_𝑗𝑟_𝑗<𝜂_𝑗1=1−21−𝛾_𝑙𝛾_𝑙𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑙−1)}−121−𝛾_𝑗𝛾_𝑗𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑗+1)}. (16) For the |𝑎𝑟|-MSSC, the conditional BER 𝑃_𝑒,0(𝜂_𝑙∣𝑙) in (8) is given by𝑃_𝑒,0𝜂_𝑙∣𝑙=𝑃₀𝑟_𝑙>0,𝑎_𝑙𝑟_𝑙≥𝜂_𝑙+𝑃₀𝑟_[𝑙]𝐿+1>0𝑃₀𝑎_𝑙𝑟_𝑙<𝜂_𝑙, (17) where𝑃₀𝑟_[𝑙]𝐿+1>0=^∞₀12erfc𝛾_[𝑙]𝐿+1𝑓_{𝛾𝐿[𝑙]+1}(𝑥)=1𝑑𝑥2−12𝛾_[𝑙]𝐿+1𝛾_[𝑙]𝐿+1,+1 (18) and, from similar manipulations given by (11)–(13),𝑃₀𝑟_𝑙>0,𝑎_𝑙𝑟_𝑙≥𝜂_𝑙=𝑃₀𝑎_𝑙𝑟_𝑙≥𝜂_𝑙=^∞₀12erfc√𝑥+𝜂_𝑙√𝑥𝑓_𝛾𝑙(𝑥)=1𝑑𝑥21−𝛾_𝑙𝛾_𝑙𝑒+1^{−2𝜂𝑙(1+1/𝛾𝑙+1)}. (19) Based on (7) and (10)–(19), the BER of the |𝑎𝑟|-MSSC can be obtained and written in the closed-form:𝑃_𝑒𝜂₁,𝜂₂,…,𝜂_𝐿=12^𝐿_𝑙=11−𝛾_𝑙𝛾_𝑙+1^𝐿_𝑗=111−21+𝛾_𝑗𝛾_𝑗𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑗−1)}−121−𝛾_𝑗𝛾_𝑗𝑒+1^{−2𝜂𝑗(1+1/𝛾𝑗+1)−1−1}×1+𝑒^{−2𝜂𝑙(1+1+1/𝛾𝑙)}11−21+𝛾_𝑙𝛾_𝑙𝑒+1^{−2𝜂𝑙(1+1/𝛾𝑙−1)}−121−𝛾_𝑙𝛾_𝑙𝑒+1^{−2𝜂𝑙(1+1/𝛾𝑙+1)−1}. (20) For i.i.d. fading channels with 𝛾_𝑙=𝛾 and identical ST 𝜂_𝑙=𝜂 for 𝑙=1,2,…,𝐿, the BER reduces to the form:𝑃_𝑒𝜂=121−𝛾×𝛾+11+𝑒^{−2𝜂(1+1/𝛾+1)}−121+𝛾𝑒𝛾+1^{−2𝜂(1+1/𝛾−1)}−121−𝛾𝑒𝛾+1^{−2𝜂(1+1/𝛾+1)}. (21) For the i.i.d. channels with 𝜂=0 since we always have |𝑎_𝑙𝑟_𝑙|≥0(𝑙=1,2,…,𝐿) and the switch will not occur (i.e., no SSC is used), the above BER reduces to the form:𝑃_𝑒=12−12𝛾𝛾+1. (22) 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 of 𝑎 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 𝜆_𝑙=𝑎_𝑙𝐸_𝑏+|𝑟_𝑙|, where an adder rather than a multiplier is used and 𝐸_𝑏 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 statistic 𝜆_𝑙=𝑎_𝑙𝐸_𝑏+|𝑟_𝑙| for 𝑙=1,2,…,𝐿, the relevant probability in (6) now is given by 𝑃₀(𝑎_𝑗𝐸_𝑏+|𝑟_𝑗|<𝜂_𝑗). For (𝑎+|𝑟|)-MSSC, defining another normalized ST as 𝜂_𝑗=𝜂_𝑗/𝑁₀, we have𝑃₀𝑎_𝑗𝐸_𝑏+𝑟_𝑗<𝜂_𝑗=𝑃₀−𝜂_𝑗+𝑎_𝑗𝐸_𝑏<𝑟_𝑗<𝜂_𝑗−𝑎_𝑗𝐸_𝑏=𝑃₀𝑛_𝑗<𝜂_𝑗−𝑃₀𝑛_𝑗<2𝑎_𝑗𝐸_𝑏−𝜂_𝑗=²𝜂_𝑗012erfc2√𝑥−𝜂_𝑗1𝑑𝑥−2erfc𝜂_𝑗, (23) where the upper limit 𝜂²_j of the integral comes from the intrinsic constraint associated with the set {−𝜂_𝑗+𝑎_𝑗𝐸_𝑏<𝑟_𝑗<𝜂_𝑗−𝑎_𝑗𝐸_𝑏}. To evaluate the integral in (23), we use integration by parts to obtain²𝜂_𝑗012erfc2√𝑥−𝜂_𝑗=1𝑑𝑥2erfc−𝜂_𝑗−𝑒⁻𝜂²_{𝑗/𝛾𝑗}2erfc𝜂_𝑗−²𝜂_𝑗01√𝑒𝜋𝑥^{√−(2𝑥−}𝜂_𝑗)²_{−𝑥/𝛾𝑙}𝑑𝑥, (24) where it is straightforward to show that²𝜂_𝑗01√𝑒𝜋𝑥^{√−(2𝑥−}𝜂_𝑗)²_{−𝑥/𝛾𝑙}𝑑𝑥=2𝛾_𝑙4𝛾_𝑙𝑒+1⁻⁽⁻𝜂²_{𝑗/(4𝛾𝑙+1))}×12erfc−2𝜂_𝑗𝛾_𝑙4𝛾_𝑙−1+12erfc𝜂²_𝑗4𝛾_𝑙+1𝛾_𝑙−2𝜂_𝑗𝛾_𝑙4𝛾_𝑙.+1 (25) Consequently,𝑃₀𝑎_𝑗𝐸_𝑏+𝑟_𝑗<𝜂_𝑗=1−erfc𝜂_𝑗−𝑒⁻𝜂²_{𝑗/𝛾𝑗}2erfc𝜂_𝑗−2𝛾_𝑙4𝛾_𝑙𝑒+1⁻⁽⁻𝜂²_{𝑗/(4𝛾𝑙+1))}×12erfc−2𝜂_𝑗𝛾_𝑙4𝛾_𝑙−1+12erfc𝜂²_𝑗4𝛾_𝑙+1𝛾_𝑙−2𝜂_𝑗𝛾_𝑙4𝛾_𝑙.+1 (26)

For the (𝑎+|𝑟|)-MSSC system, the conditional BER 𝑃_𝑒(𝜂_𝑙∣𝑙) in (8) now is given by𝑃_𝑒𝜂_𝑙∣𝑙=Pr𝑛_𝑙>𝑎_𝑙𝐸_𝑏,𝑛_𝑙≥𝜂_𝑙+𝑃₀𝑟_[𝑙]𝐿+1>0𝑃₀𝛼_𝑙𝐸_𝑏+𝑟_𝑙<𝜂_𝑙, (27) wherePr𝑛_𝑙>𝑎_𝑙𝐸_𝑏,𝑛_𝑙≥𝜂_𝑙=Pr𝑛_𝑙>𝛼_𝑙𝐸_𝑏,𝛼_𝑙𝐸_𝑏>𝜂_𝑙Pr𝑛_𝑙>𝜂_𝑙,𝛼_𝑙𝐸_𝑏≤𝜂_𝑙=²𝜂_𝑙012erfc(𝑥)𝑓_𝛾𝑙(𝑥)𝑑𝑥+^∞𝜂²_𝑙12erfc(𝑥)𝑓_𝛾𝑙(𝑥)=1𝑑𝑥2erfc𝜂_𝑙−12𝛾_𝑙𝛾_𝑙+1⋅erfc𝜂_𝑙𝛾_𝑙+1𝛾_𝑙, (28) and 𝑃₀(𝑟_[𝑙]𝐿+1>0) and 𝑃₀(𝛼_𝑙𝐸_𝑏+|𝑟_𝑙|<𝜂_𝑙) 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:𝑃_𝑒𝜂₁,𝜂₂,…,𝜂_𝐿=^𝐿_𝑗=11−2+𝑒⁻𝜂²_{𝑗/𝛾𝑗}2erfc𝜂_𝑗−𝛾_𝑗4𝛾_𝑗𝑒+1⁻𝜂²_{𝑗/(4𝛾𝑗+1)}×erfc−𝜂_𝑗4𝛾_𝑗4𝛾_𝑗2+1−erfc𝛾_𝑗+1𝜂_𝑗𝛾_𝑗4𝛾_𝑗+1⁻¹⁻¹×^𝐿_𝑙=112erfc𝜂_𝑙−12𝛾_𝑙𝛾_𝑙+1erfc𝜂_𝑙𝛾_𝑙+1𝛾_𝑙×1−2+𝑒⁻𝜂²_{𝑙/𝛾𝑙}2erfc𝜂_𝑙−𝛾_𝑙4𝛾_𝑙𝑒+1⁻𝜂²_{𝑙/(4𝛾𝑙+1)}×erfc−𝜂_𝑙4𝛾_𝑙4𝛾_𝑙+1−erfc(2𝛾_𝑙+1)𝜂_𝑙𝛾_𝑙(4𝛾_𝑙+1)⁻¹+^𝐿_𝑗=11−2+𝑒⁻𝜂²_{𝑗/𝛾𝑗}2erfc𝜂_𝑗−𝛾_𝑗4𝛾_𝑗𝑒+1⁻𝜂²_{𝑗/(4𝛾𝑗+1)}×erfc−𝜂_𝑗4𝛾_𝑗4𝛾_𝑗+1−erfc(2𝛾_𝑗+1)𝜂_𝑗𝛾_𝑗(4𝛾_𝑗+1)⁻¹⁻¹×^𝐿_𝑙=1121−𝛾_𝑙𝛾_𝑙.+1 (29) With 𝜂_𝑙=𝜂, (29) reduces to the following form for i.i.d. fading channels:𝑃_𝑒=1𝜂2−1erfc𝜂2𝛾𝛾+1erfc𝛾+1𝛾+1𝜂21−𝛾𝛾+11−2+𝑒⁻𝜂²_{𝑗/𝛾𝑗}2−1erfc𝜂21−𝛾𝛾+1𝛾4𝑒𝛾+1⁻𝜂²_{𝑗/(4𝛾𝑗+1)}×erfc−2𝜂𝛾42𝛾+1−erfc𝛾+1𝜂𝛾4.𝛾+1 (30)

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 𝜂=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, |𝑟_𝑙| can also be considered as the switch statistic of the MSSC, that is, 𝜆_𝑙=|𝑟_𝑙|. The corresponding probability in the stationary distribution given by (6) for the |𝑟|-MSSC has the form:𝑃₀𝑟_𝑗<𝜂_𝑗=𝑃₀𝑛_𝑗<𝑎_𝑗𝐸_𝑏+𝜂_𝑗−𝑃₀𝑛_𝑗<𝑎_𝑗𝐸_𝑏−𝜂_𝑗=−^∞₀12erfc√𝜂𝑥+^𝑗𝑁₀𝑓_𝛾𝑗(𝑥)+𝑑𝑥_∞012erfc√𝜂𝑥−^𝑗𝑁₀𝑓_𝛾𝑗(𝑥)𝑑𝑥. (31) To evaluate (31), following steps similar to (24) and (25), we have^∞₀12erfc√𝑥±𝜂_𝑗𝑓_𝛾𝑗(𝑥)=1𝑑𝑥2erfc±𝜂_𝑗−12𝛾_𝑙𝛾_𝑙+1×𝑒⁻⁽⁻𝜂²_{𝑗/(𝛾𝑙+1))}erfc±𝜂_𝑗𝛾_𝑙𝛾_𝑙.+1 (32) Consequently, the probability given by (31) turns into the form𝑃₀𝑟_𝑗<𝜂_𝑗=1−erfc𝜂_𝑗−𝛾_𝑗𝛾_𝑗𝑒+1⁻⁽⁻𝜂²_{𝑗/(𝛾𝑙+1))}+𝛾_𝑗4𝛾_𝑗+1×𝑒⁻⁽⁻𝜂²_{𝑗/(𝛾𝑙+1))}erfc𝜂_𝑗𝛾_𝑗𝛾_𝑗.+1 (33)

The conditional BER 𝑃_𝑒(𝜂_𝑙∣𝑙) given by (8) now has the form𝑃_𝑒𝜂_𝑙∣𝑙=𝑃₀𝑟_𝑙>0,𝑟_𝑙≥𝜂_𝑙+𝑃₀𝑟_[𝑙]𝐿+1>0𝑃₀𝑟_𝑙<𝜂_𝑙, (34) where 𝑃₀(𝑟_[𝑙]𝐿+1>0) has been given in (18) and based on (32):𝑃₀𝑟_𝑙>0,𝑟_𝑙≥𝜂_𝑙=𝑃₀𝑟_𝑙≥𝜂_𝑙=^∞₀12erfc√𝑥+𝜂_𝑙𝑓_𝛾𝑙(𝑥)=1𝑑𝑥2erfc𝜂_𝑗−12𝛾_𝑙𝛾_𝑙+1×𝑒⁻⁽⁻𝜂²_{𝑗/(𝛾𝑙+1))}erfc𝜂_𝑗𝛾_𝑙𝛾_𝑙.+1 (35)

In consequence, according to (7), (33), and (34), the BER is given by the closed-form:𝑃_𝑒𝜂₁,𝜂₂,…,𝜂_𝐿=^𝐿_𝑙=112erfc𝜂_𝑙−𝑒⁻𝜂²_{𝑙/(𝛾𝑙+1)}2𝛾_𝑙𝛾_𝑙+1erfc𝜂_𝑙𝛾_𝑙𝛾_𝑙×+1_𝐿𝑗=11−𝑒⁻𝜂²_{𝑗/(𝛾𝑗+1)}𝛾_𝑗𝛾_𝑗+1+𝑒⁻𝜂²_{𝑗/(𝛾𝑗+1)}𝛾_𝑗𝛾_𝑗+1⋅erfc𝜂_𝑗𝛾_𝑗𝛾_𝑗+1−erfc𝜂_𝑗⁻¹⁻¹×1−𝑒⁻𝜂²_{𝑙/(𝛾𝑙+1)}𝛾_𝑙𝛾_𝑙+𝑒+1⁻𝜂²_{𝑙/(𝛾𝑙+1)}2𝛾_𝑙𝛾_𝑙+1⋅erfc𝜂_𝑙𝛾_𝑙𝛾_𝑙+1−erfc𝜂_𝑙⁻¹+^𝐿_𝑙=112−12𝛾_𝑙𝛾_𝑙×+1_𝐿𝑗=11−𝑒⁻𝜂²_{𝑗/(𝛾𝑗+1)}𝛾_𝑗𝛾_𝑗+1+𝑒⁻𝜂²_{𝑗/(𝛾𝑗+1)}𝛾_𝑗𝛾_𝑗+1⋅erfc𝜂_𝑗𝛾_𝑗𝛾_𝑗+1−erfc𝜂_𝑗⁻¹⁻¹. (36) For i.i.d. fading channels with 𝜂_𝑙=𝜂, the above BER turns into the form:𝑃_𝑒=1𝜂2−12𝛾𝛾+11−𝛾𝑒𝛾+1⁻𝜂²_/(𝛾+1)−12𝛾×𝛾+1𝛾𝑒𝛾+1⁻𝜂²_/(𝛾+1)erfc𝜂1+1/𝛾−erfc𝜂, (37) that is also independent of 𝐿. If 𝜂=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, the 𝐿 simultaneous equations 𝜕𝑃_𝑒(𝜂₁,𝜂₂,…,𝜂_𝐿)/𝜕𝜂_𝑙=0 (or 𝜕𝑃_𝑒(𝜂₁,𝜂₂,…,𝜂_𝐿)/𝜕𝜂_𝑙=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 𝑃_𝑒(𝜂_𝑙∣𝑙) or 𝑃_𝑒(𝜂_𝑙∣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 BER 𝑃_𝑒(𝜂_𝑙∣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 𝜂_𝑙(𝑙=1,2,…,𝐿).

For i.i.d. fading channels with an identical ST 𝜂_𝑙=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, 𝑃_𝑒(𝜂_𝑙∣𝑙)=𝑃_𝑒(𝜂) and 𝑃_𝑒(𝜂_𝑙∣𝑙)=𝑃_𝑒(𝜂). 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 𝜕𝑃_𝑒(𝜂_𝑙∣𝑙)/𝜕𝜂_𝑙=0 that gives the equation (after some manipulations):1−𝛾_[𝑙]𝐿+1𝛾_[𝑙]𝐿+1𝑒+1^{2𝜂𝑙,𝑜}=1+𝛾_[𝑙]𝐿+1𝛾_[𝑙]𝐿+1𝑒+1^{−2𝜂𝑙,𝑜}, (38) where 𝜂_𝑙=𝜂_𝑙,𝑜 is the locally optimal ST. Solving (38) for 𝜂_𝑙,𝑜, we obtain the locally optimal ST expressed in the closed-form:𝜂_𝑙,𝑜=14𝑙𝑛𝛾_[𝑙]𝐿+1+1+𝛾_[𝑙]𝐿+1𝛾_[𝑙]𝐿+1+1−𝛾_[𝑙]𝐿+1. (39) 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, for i.i.d. fading channels with 𝛾_𝑙=𝛾 and 𝜂_𝑙=𝜂 for 𝑙=1,2,…,𝐿,𝜂_𝑙,𝑜=𝜂_𝑜 also gives the globally optimal ST.

4.2. (𝑎+|𝑟|)-MSSC

In the (𝑎+|𝑟|)-MSSC system, the equation 𝜕𝑃_𝑒(𝜂_𝑙∣𝑙)/𝜕𝜂_𝑙=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 𝜂_𝑜 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 𝜂_𝑙, the equation 𝜕𝑃_𝑒(𝜂_𝑙∣𝑙)/𝜕𝜂_𝑙=0 at the locally optimal ST 𝜂_𝑙=𝜂_𝑙,𝑜 yields erfc𝜂_𝑙,𝑜𝛾_𝑙𝛾_𝑙++1𝛾_[𝑙]𝐿+1+1𝛾_[𝑙]𝐿+1=1−1√𝜋𝛾_𝑙+1𝛾_𝑙⋅𝑒^−𝛾𝑙𝜂²_{𝑙,𝑜/(𝛾𝑙+1)}𝜂_𝑙,𝑜, (40) 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 Figure 2, 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 𝛾 (e.g., 𝛾≤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 on i.i.d. channels.

The BER of the SNR-based SSC is also plotted in Figure 2, 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 𝛾.

The BER comparison on i.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 (𝛾₁,𝛾₂,𝛾₃) are used, where the relations 𝛾₂=𝛾₁/4 and 𝛾₃=𝛾₁/8 are set. Thus, 𝛾₁>𝛾₂>𝛾₃. For the dual-branch case, two branches with 𝛾₂ and 𝛾₃ are used. For the triple-branch case, another branch with 𝛾₁ 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 on i.n.d. channels.

The BER of the SNR-based SC is also plotted in Figure 3 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 𝛾₁≤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 both i.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.

Acknowledgment

This paper was supported by the National Science Council, Taiwan, under Grants nos. NSC99-2221-E-155-033, NSC100-2220-E-155-001, and NSC100-2220-E-155-005.