In order to consider decoding errors at the relays within the detector at the destination, a suitable model describing the overall transmission including the decoding reliabilities of the relays is required. On the one hand, this model should be accurate enough to actually improve the detection at the destination, on the other hand it should be simple enough to avoid an excessive increase in the complexity of the detector or in the signaling overhead.

### 4.1 Equivalent transmission model for 1st hop transmission

Based on the ideas presented in [14]**,**[15]**], decoding errors at the relays can be described using binary symmetric channels (BSCs) with a certain crossover probability. According to this description, the relay information word** **b**
_{
m,n
} **in (** 4) is modeled as

{\mathbf{b}}_{m,n}={\text{BSC}}_{m,n}\left\{{\mathbf{b}}_{m},{q}_{m,n}\right\}\phantom{\rule{0.3em}{0ex}},

(11)

where *q*
_{
m,n
} **is the bit error probability of the estimate**
{\widehat{\mathbf{b}}}_{m}^{{R}_{n}} **at relay** *R*
_{
n
} **regarding the source information word** **b**
_{
m
}
**. This crossover probability is zero for perfect decoding at the relay and increases as the relay’s decoding reliability decreases. Using this description, an equivalent transmission model for the transmission from the sources over the first hop to the relays, including decoding at the relays, can be derived. Figure**
5 depicts such an equivalent joint model for the transmission of **b**
_{
m
} **via the relays** *R*
_{
1
} **up to** *R*
_{
n
}
**. The correlation between the source information word** **b**
_{
m
} **and the relay information words** **b**
_{
m,n
} **is described by BSC**
_{
m,1} **up to BSC**
_{
m,N
} **with error probabilities** *q*
_{
m,1} **up to** *q*
_{
m,n
}
**. These error probabilities** *q*
_{
m,n
} **are given by**

{q}_{m,n}=\frac{{d}_{\mathrm{H}}({\mathbf{b}}_{m},{\widehat{\mathbf{b}}}_{m}^{{R}_{n}})}{{L}_{\mathrm{b}}}\phantom{\rule{0.3em}{0ex}},

(12)

where *d*
_{
H
}
**(·) denotes the Hamming distance and** *L*
_{
b
} **is the length of the information sequence** **b**
_{
m
}
**. Obviously, the calculation (** 12) would require perfect knowledge of **b**
_{
m
} **at the relays which is not the case in practical systems. However, an estimation of** *q*
_{
m,n
} **using the LLRs**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{{R}_{n}} **of the information bits generated by the MUD at the relay is possible [**[20]**]. Denoting this estimate**
{\widehat{q}}_{m,n}
**, it holds**

{\widehat{q}}_{m,n}=\mathrm{E}\left\{\frac{1}{1+{e}^{\left|{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{{R}_{n}}\right|}}\right\}\approx \frac{1}{{L}_{\mathrm{b}}}\phantom{\rule{1em}{0ex}}\sum _{i=1}^{{L}_{\mathrm{b}}}\frac{1}{1+{e}^{\left|{\Lambda}_{{\mathrm{b}}_{m},i}^{{R}_{n}}\right|}}\phantom{\rule{1em}{0ex}},

(13)

where the expectation can be approximated by the time average due to the ergodic theorem. Note that{\widehat{q}}_{m,n}\ne 0 **even if the decoding at the relay was correct. Hence, in case of successful decoding at the relay ACK is signaled to the destination, while unsuccessful decoding leads to the signaling of a NAK in form of**
{\widehat{q}}_{m,n}
**. The principle of this signaling is depicted in Figure**
6, where CRC_{
m,n
} **denotes the CRC check at relay** *R*
_{
n
} **regarding** **b**
_{
m
}
**. For a more detailed discussion of the signaling refer to [**[16]**].**

### 4.2 RAID scheme

Based on the presented equivalent transmission model, the new RAID is proposed. This detection scheme takes the decoding success (CRC_{
m,n
}
**=ACK/NAK) of the relays as well as the error probabilities**
{\widehat{q}}_{m,n} **into account in order to improve the detection quality compared to the common detection scheme discussed in Section** 3.1. In the following, the components of the RAID scheme, i.e., the relay grouping, the detection process and the weighted combining are discussed in detail.

#### 4.2.1 Relay grouping

To address the second issue of the common detection strategy, i.e., the loss of information about the individual relay signals **x**
_{
m,n
}
**, a user specific separation of the correct relays and all erroneous relays is introduced. Since all correct relays have transmitted the same code word** **c**
_{
m,n
}
**=** **c**
_{
m
}
**, their LLRs can be combined after relay specific de-interleaving. All erroneous relays, however, may have transmitted pairwise different code words and, hence, are all processed separately.**

Based on the decoding success (CRC_{
m,n
}
**=ACK/NAK) with respect to one specific source** *S*
_{
m
}
**, each relay** *R*
_{
n
} **is assigned to one of two disjoint groups, the group of relays which have correctly decoded the source message, i.e.,** **b**
_{
m,n
}
**=** **b**
_{
m
}
**, and the group of relays which have not correctly decoded the source message, i.e.,** **b**
_{
m,n
}
**≠** **b**
_{
m
}
**. For the sake of notational simplicity, the set**
{\mathcal{R}}_{m} **of indices of the correct relays w.r.t. source** *S*
_{
m
} **and the set**
{\stackrel{\u0304}{\mathcal{R}}}_{m} **of indices of erroneous relays w.r.t. source** *S*
_{
m
} **are introduced**

\begin{array}{ll}{\mathcal{R}}_{m}& =\left\{n\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}{q}_{m,n}=0,1\le n\le N\}\phantom{\rule{2em}{0ex}}\end{array}

(14a)

\begin{array}{ll}{\stackrel{\u0304}{\mathcal{R}}}_{m}& =\left\{n\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}{q}_{m,n}\ne 0,1\le n\le N\}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}

(14b)

Obviously, the union of both sets is the set of the indices of all relays, i.e.,{\mathcal{R}}_{m}\cup {\stackrel{\u0304}{\mathcal{R}}}_{m}=\{1\phantom{\rule{0.3em}{0ex}},\dots N\}
**. Furthermore, two indexing functions** *ρ*
_{
m
} **and**
{\stackrel{\u0304}{\rho}}_{m} **are defined, such that**

\begin{array}{lll}{\mathcal{R}}_{m}& =\left\{{\rho}_{m}\right(1),\dots ,{\rho}_{m}({I}_{m}\left)\right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}& {I}_{m}=\left|{\mathcal{R}}_{m}\right|\phantom{\rule{2em}{0ex}}\end{array}

(15)

\begin{array}{lll}{\stackrel{\u0304}{\mathcal{R}}}_{m}& =\left\{{\stackrel{\u0304}{\rho}}_{m}\right(1),\dots ,{\stackrel{\u0304}{\rho}}_{m}({K}_{m}\left)\right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}& {K}_{m}=\left|{\stackrel{\u0304}{\mathcal{R}}}_{m}\right|\phantom{\rule{2em}{0ex}}\end{array}

(16)

with *ρ*
_{
m
}
**(1)<** *ρ*
_{
m
}
**(2)⋯<** *ρ*
_{
m
}
**(** *I*
_{
m
}
**) and**
{\stackrel{\u0304}{\rho}}_{m}\left(1\right)<{\stackrel{\u0304}{\rho}}_{m}\left(2\right)<\cdots <{\stackrel{\u0304}{\rho}}_{m}\left({K}_{m}\right)
**. This means** *ρ*
_{
m
}
**(1) up to** *ρ*
_{
m
}
**(** *I*
_{
m
}
**) represent the** *I*
_{
m
} **indices of the correct relays w.r.t** *S*
_{
m
} **and**
{\stackrel{\u0304}{\rho}}_{m}\left(1\right) **up to**
{\stackrel{\u0304}{\rho}}_{m}\left({K}_{m}\right) **represent the** *K*
_{
m
} **indices of the erroneous relays w.r.t** *S*
_{
m
}
**, i.e.,**

\begin{array}{lc}{\mathbf{b}}_{m,{\rho}_{m}\left(i\right)}& ={\mathbf{b}}_{m},\phantom{\rule{1em}{0ex}}1\le i\le {I}_{m}\end{array}

(17a)

\begin{array}{lc}{\mathbf{b}}_{m,{\stackrel{\u0304}{\rho}}_{m}\left(k\right)}& \ne {\mathbf{b}}_{m},\phantom{\rule{1em}{0ex}}1\le k\le {K}_{m}.\end{array}

(17b)

Figure 7 shows the part of the overall proposed detector which is relevant to the detection of **b**
_{
m
}
**. The LLRs from the IC are grouped based on the decoding success (ACK/NAK) at the relays. Since the correct relays have transmitted the same code word** **c**
_{
m
}
**, their LLRs are summed up after relay specific de-interleaving, similar to the common detection scheme, i.e.,**

{\mathit{\Lambda}}_{{\mathrm{c}}_{m,{\rho}_{m}}^{\prime}}^{\text{IC}}=\sum _{i=1}^{{I}_{m}}{\Pi}_{\mathrm{r},{\rho}_{m}\left(i\right)}^{\text{-}1}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{c}}_{m,{\rho}_{m}\left(i\right)}^{\mathrm{\prime \prime}}}^{\text{IC}}\right)\phantom{\rule{0.3em}{0ex}},

(18)

and are then jointly de-interleaved by the user specific interleaver and jointly decoded (bottom part). The erroneous relays, however, have transmitted different code words and are, therefore, processed and decoded separately (top part). The goal of this first stage of the detection is the best possible estimation of the relay information words **b**
_{
m,n
} **and not of the source information words** **b**
_{
m
}
**. The estimation of the source information words is exclusively performed in the second stage of the detector.**

Finally, after the last iteration, the *K*
_{
m
}
**+1 decoders**
\mathcal{D} **deliver LLRs**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m,{\rho}_{m}}} **for the information words of the correct relays and LLRs**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m,\stackrel{\u0304}{\rho}\left(k\right)}} **for the** *K*
_{
m
} **information words of the erroneous relays. The explicit decoding, hard decision and subsequent re-encoding at the relays ensures that all relays actually transmitted a valid code word which is fundamental for the validity of the joint equivalent transmission model.**

#### 4.2.2 Weighted combining

Based on the estimates for the relay information words **b**
_{
m,n
}
**, now an overall estimate for the source information word** **b**
_{
m
} **should be determined. This estimate should not only include the LLRs from the correct relays, but also the LLRs from the erroneous relays as, depending on the error probabilities**
{\widehat{q}}_{m,n}
**, the relay information of the erroneous relays is still correlated to the source information.**

The question arises, how to obtain an estimate{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n} **for the source information** **b**
_{
m
} **given the estimate**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}} **for a specific relay information word** **b**
_{
m,n
} **and the corresponding error probability**
{\widehat{q}}_{m,n}
**. How does the BSC modeling of the first hop transmission translate to the LLRs for the relay information word and the source information word? Figure**
8 illustrates this relationship, where the function\mathcal{W}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}(\xb7) **needs to be found, such that**

{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n}=\mathcal{W}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}},{\widehat{q}}_{m,n}\right),

(19)

with{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n} **denoting the estimate for the source information** **b**
_{
m
} **taking only relay** *R*
_{
n
} **into account. In order to find this function, the estimate**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n} **for an arbitrary element** *b*
_{
m
} **of** **b**
_{
m
} **is written as**

\begin{array}{ll}{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n}& =log\phantom{\rule{0.3em}{0ex}}\left(\frac{p({b}_{m}=0,y)}{p({b}_{m}=1,y)}\right)\phantom{\rule{2em}{0ex}}\\ =log\phantom{\rule{0.3em}{0ex}}\left(\frac{P({b}_{m}=0|y\left)\phantom{\rule{0.3em}{0ex}}p\right(y)}{P({b}_{m}=1|y\left)\phantom{\rule{0.3em}{0ex}}p\right(y)}\right)\phantom{\rule{2em}{0ex}}\\ =log\phantom{\rule{0.3em}{0ex}}\left(\frac{P({b}_{m}=0|y)}{P({b}_{m}=1|y)}\right)\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}

(20)

By using the law of total probabilities [21]**], the probabilities of the source information** *b*
_{
m
} **can be written w.r.t. the probabilities of the relay information** *b*
_{
m,n
} **as**

\begin{array}{ll}P({b}_{m}=0|y)& =P({b}_{m}=0|{b}_{m,n}=0,y\left)\phantom{\rule{0.3em}{0ex}}P\right({b}_{m,n}=0\left|y\right)\phantom{\rule{2em}{0ex}}\\ +P({b}_{m}=0|{b}_{m,n}=1,y\left)\phantom{\rule{0.3em}{0ex}}P\right({b}_{m,n}=1\left|y\right)\phantom{\rule{2em}{0ex}}\end{array}

(21a)

\begin{array}{ll}P({b}_{m}=1|y)& =P({b}_{m}=1|{b}_{m,n}=1,y\left)\phantom{\rule{0.3em}{0ex}}P\right({b}_{m,n}=1\left|y\right)\phantom{\rule{2em}{0ex}}\\ +P({b}_{m}=1|{b}_{m,n}=0,y\left)\phantom{\rule{0.3em}{0ex}}P\right({b}_{m,n}=0\left|y\right)\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}

(21b)

The probabilities of *b*
_{
m
} **given** *b*
_{
m,n
} **solely depend on the crossover probability**
{\widehat{q}}_{m,n} **of the BSC,**

\begin{array}{ll}P({b}_{m}=0|{b}_{m,n}=0)& =P({b}_{m}=1|{b}_{m,n}=1)\phantom{\rule{2em}{0ex}}\\ =1-{\widehat{q}}_{m,n}\phantom{\rule{2em}{0ex}}\end{array}

(22a)

\begin{array}{ll}P({b}_{m}=1|{b}_{m,n}=0)& =P({b}_{m}=0|{b}_{m,n}=1)\phantom{\rule{2em}{0ex}}\\ ={\widehat{q}}_{m,n}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{2em}{0ex}}\end{array}

(22b)

such that the estimate{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n} **can be rewritten as (25a). Expressing the probabilities by LLRs [**[22]**]**

\begin{array}{ll}P({b}_{m,n}=0|y)& =\frac{{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{2em}{0ex}}\end{array}

(23)

\begin{array}{ll}P({b}_{m,n}=1|y)& =\frac{1}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{2em}{0ex}}\end{array}

(24)

leads to (25b) and after some algebraic manipulations to (25c). Thus, the desired function\mathcal{W}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}(\xb7) **is finally found to be**

\begin{array}{ll}\phantom{\rule{-12.0pt}{0ex}}{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{n}& =log\phantom{\rule{0.3em}{0ex}}\left(\frac{P({b}_{m,n}=0|y\left)\phantom{\rule{0.3em}{0ex}}\right(1-{\widehat{q}}_{m,n})+P({b}_{m,n}=1\left|y\right)\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}{P({b}_{m,n}=1|y\left)\phantom{\rule{0.3em}{0ex}}\right(1-{\widehat{q}}_{m,n})+P({b}_{m,n}=0\left|y\right)\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(25a)

\begin{array}{l}=log\phantom{\rule{0.3em}{0ex}}\left(\frac{\frac{{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+\frac{1}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}{\frac{1}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+\frac{{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}{1+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}}}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(25b)

\begin{array}{l}=log\phantom{\rule{0.3em}{0ex}}\left(\frac{{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+{e}^{-{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}{{e}^{-{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(25c)

\begin{array}{l}\mathcal{W}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}},{\widehat{q}}_{m,n}\right)\phantom{\rule{2em}{0ex}}\\ =& log\phantom{\rule{0.3em}{0ex}}\left(\frac{{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+{e}^{-{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}{{e}^{-{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}(1-{\widehat{q}}_{m,n})+{e}^{{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}/2}\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{m,n}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{2em}{0ex}}\end{array}

(26)

where\mathcal{W}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}},{\widehat{q}}_{m,n}\right) **is a shorthand notation for applying (** 25c) to every element of{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}
**.**

In (25c) the error probability{\widehat{q}}_{m,n} **of the BSC obviously leads to a weighting of the estimates of the relay information word given by**
{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}}
**. For completely uncorrelated** **b**
_{
m,n
} **and** **b**
_{
m
}
**, i.e.,**
{\widehat{q}}_{m,n}=0.5
**, the relay transmitted no information regarding** **b**
_{
m
} **and, hence,** *Λ* b_{
m
}
*n*=0. However, as{\widehat{q}}_{m,n} **decreases,** *Λ* b_{
m
}
*n* tends to{\mathit{\Lambda}}_{{\mathrm{b}}_{m,n}} **giving an estimate of** **b**
_{
m
} **with respect to the information from relay** *R*
_{
n
}
**.**

Since all transmit channels are statistically independent, the observations from all relays can be summed up resulting in the estimate

\begin{array}{ll}{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}& ={\mathit{\Lambda}}_{{\mathrm{b}}_{m,{\rho}_{m}}}+\sum _{k=1}^{{K}_{m}}\mathcal{W}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{b}}_{m},{\stackrel{\u0304}{\rho}}_{m}\left(k\right)},{\widehat{q}}_{m,n}\right)\phantom{\rule{2em}{0ex}}\end{array}

(27)

\begin{array}{l}={\mathit{\Lambda}}_{{\mathrm{b}}_{m,{\rho}_{m}}}+\sum _{k=1}^{{K}_{m}}{\mathit{\Lambda}}_{{\mathrm{b}}_{m}}^{{\stackrel{\u0304}{\rho}}_{m}\left(k\right)}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{2em}{0ex}}\end{array}

(28)

where{\mathit{\Lambda}}_{{\mathrm{b}}_{m,{\rho}_{m}}} **is the unweighted estimate from the correct relays. Finally, hard quantization leads to the overall estimate**
{\widehat{\mathbf{b}}}_{m}=\mathcal{Q}\phantom{\rule{0.3em}{0ex}}\left({\mathit{\Lambda}}_{{\mathrm{b}}_{m}}\right) **for the source message** **b**
_{
m
}.

### 4.3 Pseudo code

In order to facilitate the comprehension of the proposed RAID scheme, a pseudo code for the overall detection process is given in Algorithm 1. The algorithm requires the set of all CRC check results {CRC_{
m,n
}
**}, the set of all error probabilities**
\left\{{\widehat{q}}_{m,n}\right\}
**, the set of all channel impulse responses from the relays to the destination {** **g**
_{
n
}
**}, the sets of all relay specific and user specific interleavers {** *Π*
_{
r,
n
}
**} and {** *Π*
_{
n
}
**} and the noise variance**
{\sigma}_{\mathrm{n}}^{2}.