The traditional combining schemes, MRC as well as SC and EG, are based on Bayesian theory. DST as a generalization of the Bayesian theory has unique merits in uncertainty processing, based on which a novel combining scheme called DSC is proposed in [10].

### A DSC

DSC refers to the modulation constellation set *U* as the frame of discernment with mutually exclusive and exhaustive hypotheses. Focal element set (FES) *S*_{
m
} is a subset *S*_{
m
} ⊂ *U*, in which the number of elements is denoted by *m*, e.g. *S*_{1} = {*s*_{
α
} } or *S*_{2} = {*s*_{
α
} , *s*_{
β
} } or *S*_{3} = {*s*_{
α
} , *s*_{
β
} , *s*_{
γ
} }..., where *α* ≠ *β* ≠ *γ* and *α*, *β*, *γ* = 1, 2,..., *M*. Set *S*_{
m
} reflects the uncertainty of decision judgements. For example, *S*_{2} = {*s*_{
α
} , *s*_{
β
} } contains more uncertainty than *S*_{1} = {*s*_{
α
} }, which implies that the transmitted symbol may be *s*_{
α
} or *s*_{
β
} , but there is no convincing evidence for deciding which one must be the transmitted symbol. In wireless communication systems, the transmitted signals suffer from multipath fading channels and interferences, and the received signals thus contain much uncertainty. Therefore, it is reasonable to use FES *S*_{
m
} to characterize the uncertain decisions. In the proposed DSC scheme [10], the uncertain decision propositions *S*_{
m
} consist of the adjacent constellation points, since it is usually difficult to ensure which one is the transmitted symbol between the adjacent constellation points.

Basic probability assignment (BPA) denoted by Mas(*S*_{
m
} ) characterizes the confidence reposed in the transmitted signal being contained in set *S*^{m}. Two methods for BPA calculations are proposed for equiprobable and non-equiprobable sources, respectively. One is based on the distance from the received signal to the decision candidate set, i.e. the nearer-distance-more-confidence rule, and the other is based on the posterior probability of the transmitted signals, both of which are introduced in detail as follows:

*(1) distance-based BPA calculations*: The nearer-distance-more-confidence principle is used for BPA calculations in [10], which is based on the distance between the received signal and the decision candidate set consisting of adjacent constellation points with the assumption that the source bits are equiprobable. The corresponding Mas_{D} (*S*_{
m
} |*y* ^{(t)}) function is expressed as

Ma{s}_{D}\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)=\frac{{R}_{D}^{\left(t\right)}-{\left|{y}^{\left(t\right)}-{h}^{\left(t\right)}\cdot \frac{{\sum}_{{s}_{\alpha}\in {S}_{m}}{s}_{\alpha}}{m}\right|}^{2}}{\left({\sum}_{m=1}^{P}N\left({S}_{m}\right)-1\right){R}_{D}^{\left(t\right)}},\phantom{\rule{1em}{0ex}}m=1,2,\dots ,P;\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t=1,2,\dots ,\stackrel{\u0304}{T},

(3)

where *N*(*S*_{
m
} ) denotes the total number of the set *S*_{
m
} containing *m* adjacent constellation points, *P* is a key issue concerned with the trade-off between performance and complexity, and

{R}_{D}^{\left(t\right)}=\sum _{m=1}^{P}\sum _{{S}_{m}}{\left|{y}^{\left(t\right)}-{h}^{\left(t\right)}\cdot \frac{{\sum}_{{s}_{\alpha}\in {S}_{m}}{s}_{\alpha}}{m}\right|}^{2}

is a normalization coefficient, satisfying

\sum _{m=1}^{P}\sum _{{S}_{m}}Ma{s}_{D}\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)=1,\phantom{\rule{1em}{0ex}}m=1,2,\dots ,M,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\le P\le M.

(4)

From (3) it is obvious that the nearer it is from the received signal *y*^{(t)}to the decision candidate set *S*_{
m
} , the more confidence (larger MasD *Sm* _*y*(*t*)) is placed in the set.

*(2) a posterior probability-based BPA calculations*: When the source bits are non-equiprobable (NEP), of which the priori probability is available to the receiver, ML become suboptimal and MAP is the optimal method. In view of this, BPA calculations can thus be performed based on the posterior probability of the transmitted signals as

Ma{s}_{APP}\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)=\frac{{\left({\prod}_{{s}_{\alpha}\in {S}_{m}}Pr\left({s}_{\alpha}\right)\right)}^{-N\left({S}_{m}\right)}f\left(\left(\right)close="|">{y}^{\left(t\right)}{S}_{m}\right)}{}{R}_{APP}^{\left(t\right)}\n ,\n

(5)

where *f y*^{(t)}*|S*_{
m
} is likelihood function,

f\left(\left(\right)close="|">{y}^{\left(t\right)}{S}_{m}\right)\n \n =\n \n \n 1\n \n \n \n \n 2\n \pi \n \n \n \sigma \n \n \n 2\n \n \n \n \n \n \n \n e\n x\n p\n \n \n \n -\n \n \n \n \n \n \n \n \n y\n \n \n \n (\n \n t\n \n )\n \n \n \n -\n \n \n h\n \n \n \n (\n \n t\n \n )\n \n \n \n \u22c5\n \n \n \n \n \u2211\n \n \n \n \n s\n \n \n \alpha \n \n \n \u2208\n \n \n S\n \n \n m\n \n \n \n \n \n \n s\n \n \n \alpha \n \n \n \n \n m\n \n \n \n \n \n \n 2\n \n \n \n \n 2\n \n \n \sigma \n \n \n 2\n \n \n \n \n \n \n

(6)

with *σ*^{2} denoting the AWGN noise power. The normalization coefficient {R}_{APP}^{\left(t\right)} is expressed as

{R}_{APP}^{\left(t\right)}=\sum _{m=1}^{P}\sum _{{S}_{m}}{\left(\prod _{{s}_{\alpha}\in {S}_{m}}Pr\left({s}_{\alpha}\right)\right)}^{-N\left({S}_{m}\right)}f\left(\left(\right)close="|">{y}^{\left(t\right)}{S}_{m}\right)\n ,\n

whereby the summation of 5 is unity as like 4.

If not specially pointed, the Mas (·) function has two expressions MasD (·) and MasAPP (·) as the above mentioned, both of which denote the soft information BPA but obtained by diverse calculation methods. For simplicity, only Mas (·) is used in the following context.

In addition, DST contains two new measure of "belief" or "credibility" that are foreign to Bayesian theory. These are the notions of support and plausibility [16], respectively. The support for the transmitted signal being in the set *S*_{
m
} is defined as the total BPA of all subsets implying the *S*_{
m
} set. Thus,

Spt\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)=\sum _{{S}_{{m}^{\prime}}\subseteq {S}_{m}}Mas\left({S}_{{m}^{\prime}}\left|{y}^{\left(t\right)}\right.\right).

(7)

The support is a kind of loose lower limit to the uncertainty. On the other hand, a loose upper limit to the uncertainty is the plausibility. This is defined, for the *S*_{
m
} set, as the total BPA of all subsets that do not contradict the *S*_{
m
} set. In other words,

Pls\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)=\sum _{{S}_{{m}^{\prime}}\cap {S}_{m}\ne \varphi}Mas\left({S}_{{m}^{\prime}}\left|{y}^{\left(t\right)}\right.\right).

(8)

As a result, it can be inferred that the belief of the transmitted signal contained in set *S*_{
m
} lies in the interval [Spt *S*_{
m
} *|y*(*t*), Pls *S*_{
m
} *|y*(*t*)], which represents the uncertain propositions. The smaller the interval is, the clearer the evidence is to support the corresponding propositions. The more detailed explanations about the support and the plausibility functions refer to Shafer's original work on DST in [17].

As the approach above mentioned, the similar belief interval as [Spt *S*_{
m
} |*y*^{(t)}), Pls(*S*_{
m
} |(*y*^{(t)})] can be achieved for each {y}^{\left(t\right)},t=1,2,\dots ,\stackrel{\u0304}{T}. The interval is gradually reduced along with making more use of the received signals as follows

\begin{array}{ll}\hfill Spt\left({S}_{m}\right)& =\underset{1\le t\le \stackrel{\u0304}{T}}{\mathrm{sup}}\left\{Spt\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)\right\},\phantom{\rule{2em}{0ex}}\\ \hfill Pls\left({S}_{m}\right)& =\underset{1\le t\le \stackrel{\u0304}{T}}{\mathrm{inf}}\left\{Pls\left({S}_{m}\left|{y}^{\left(t\right)}\right.\right)\right\},\phantom{\rule{2em}{0ex}}\\ m=1,2,\dots ,P.\phantom{\rule{2em}{0ex}}\\ \end{array}

(9)

At this time, Spt(*S*_{
m
} ) and Pls(*S*_{
m
} ) are two measures of the aggregate belief in the transmitted signal being contained in set *S*_{
m
} , which are achieved after combining multiple information sources by (9). These two measures of the aggregate belief need to be further merged before decision-making, since it is beneficial to make more reliable decisions by taking full advantage of them. The proposed DSC merges Spt(*S*_{
m
} ) and Pls(*S*_{
m
} ) in terms of the Dempster's rule [18], which is a generalization of Bayes' rule and is justified under many situations. The aggregation can be expressed as

Spt\text{\_}Pls\left({S}_{m}\right)=\frac{Spt\left({S}_{m}\right)Pls\left({S}_{m}\right)}{1-Spt\left({S}_{m}\right)\left(1-Pls\left({S}_{m}\right)\right)},\phantom{\rule{1em}{0ex}}m=1,2,\dots ,P,

(10)

where Spt_Pls(*S*_{
m
} ) is regarded as the reliable belief in the transmitted signal that is included in set *S*_{
m
} and is applied to assist in making decisions. However, *S*_{
m
} is still a set containing *m* adjacent constellation points with *m* = 1, 2,..., *P*. The ultimate goal of the proposed scheme is to correctly judge which point of the constellation is the transmitted signal, thus the decision statistics are defined as

De\left({s}_{\alpha}\right)=\sum _{{s}_{\alpha}\in {S}_{m}}\frac{Spt\text{\_}Pls\left({S}_{m}\right)}{m},\alpha =1,2,\dots ,M,

(11)

where the summation is carried out among all the sets (*S*_{
m
} ) that contains the constellation *s*_{
α
} . Finally, the resulting decision is written as \u015d=arg\underset{{s}_{\alpha}\in U}{\mathrm{max}}De\left({s}_{\alpha}\right).

### B MRC

MRC receiver is deemed as the optimal since it results in a maximum likelihood receiver [8] when the source bits are equiprobable. If the same signal is transmitted \stackrel{\u0304}{T} times, the corresponding channel fading coefficients, received signals and noise variables are concatenated as \stackrel{\u0303}{H}={\left[{h}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}{h}^{\left(2\right)}\cdots {h}^{\left(\stackrel{\u0304}{T}\right)}\right]}^{T}, \stackrel{\u0303}{y}={\left[{y}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}{y}^{\left(2\right)}\cdots {y}^{\left(\stackrel{\u0304}{T}\right)}\right]}^{T}, \stackrel{\u0303}{n}={\left[{n}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}{n}^{\left(2\right)}\cdots {n}^{\left(\stackrel{\u0304}{T}\right)}\right]}^{T}, respectively. \stackrel{\u0304}{T} transmissions for the signal *x* can thus be written in matrix expression as \stackrel{\u0303}{y}=x\stackrel{\u0303}{H}+\stackrel{\u0303}{n}, to which model the MRC scheme is applied, and the resulting decision statistics can be expressed as

x=\frac{{\stackrel{\u0303}{H}}^{H}\stackrel{\u0303}{y}}{\left|\right|\stackrel{\u0303}{H}|{|}^{2}}=x+\frac{1}{\left|\right|\stackrel{\u0303}{H}|{|}^{2}}{\stackrel{\u0303}{H}}^{H}\stackrel{\u0303}{n},

(12)

where \widehat{x} is a Gaussian random variable with *x* mean and *σ*^{2} variance.

If the source bits are equiprobable, the ML rule is equivalent to the minimum distance rule. The decision result of MRC is accordingly written as

\u015d=arg\underset{{s}_{\alpha}\in U}{max}\frac{1}{\sqrt{2\pi {\sigma}^{2}}}exp\left(-\frac{{\left(\widehat{x}-{s}_{\alpha}\right)}^{2}}{2{\sigma}^{2}}\right)=arg\underset{{s}_{\alpha}\in U}{min}{\left(\widehat{x}-{s}_{\alpha}\right)}^{2}.

(13)

Otherwise, if the source bits are non-equiprobable and the priori probability of the source signals Pr(*s*_{
α
} ), *α* = 1, 2,..., *M* , is available to the receiver, the decision result according to the maximum posterior probability rule can thus be achieved as

\u015d=arg\underset{{s}_{\alpha}\in U}{max}\frac{Pr\left({s}_{\alpha}\right)}{\sqrt{2\pi {\sigma}^{2}}}exp\left(-\frac{{\left(\widehat{x}-{s}_{\alpha}\right)}^{2}}{2{\sigma}^{2}}\right).

(14)