In order to evaluate the BER, we need to find the intervals for Λ(*r*_{0}) > 0 or Λ(*r*_{0}) < 0. We observe that the numerator and the denominator of the argument of the logarithm in Λ(*r*_{0}) are the sums of the bell-shaped curves, respectively. The centers of bell curves are numbered as

\begin{array}{c}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}d+1\triangleq \mu \left({A}_{0}-\sum _{i=1}^{{N}_{I}}{\left(-1\right)}^{{d}_{\left(2\right)}\left({N}_{I}+1-i\right)}{I}_{i}\right),\\ -c-1\triangleq \mu \left(-{A}_{0}+\sum _{i=1}^{{N}_{I}}{\left(-1\right)}^{{c}_{\left(2\right)}\left({N}_{I}+1-i\right)}{I}_{i}\right),\end{array}

(9)

where d,c=0,1,\dots ,{2}^{{N}_{I}}-1. Since Λ(*r*_{0}) is an odd function, the equation of Λ(*r*_{0}) = 0 always has a real root *r*_{0} = 0. It is very difficult to obtain the exact real roots except a real root *r*_{0} = 0. Therefore, we use the Jacobian logarithm to obtain the approximate real roots as follows,

\begin{array}{ll}\hfill \text{log}\left({e}^{x}+{e}^{y}\right)\phantom{\rule{1em}{0ex}}& =\text{max}\left(x,y\right)+\text{log}\left(1+{e}^{-\left|y-x\right|}\right)\phantom{\rule{2em}{0ex}}\\ \approx \text{max}\left(x,y\right).\phantom{\rule{2em}{0ex}}\end{array}

(10)

Then Λ(*r*_{0}) = 0can be written as

\underset{d}{\text{min}}\left|{r}_{0}-{\mu}^{-1}\left(d+1\right)\right|\approx \underset{c}{\text{min}}\left|{r}_{0}-{\mu}^{-1}\left(-c-1\right)\right|,

(11)

where d,c=0,1,\dots ,{2}^{{N}_{I}}-1. In order to solve (11), we have the case *C*_{
i
}, 1 ≤ *i* ≤ *N*_{
c
}, which is a set of marginal conditions, where *N*_{
c
} is the number of cases. Then the set *R*_{
i
} of real roots corresponding to *C*_{
i
} can be obtained by solving (11). We order 2{N}_{r}^{\left(i\right)}+1 real roots *p*_{
j
}(*i*) ∈ *R*_{
i
}, j=-{N}_{r}^{\left(i\right)},-{N}_{r}^{\left(i\right)}+1,\dots ,-1,0,1,\dots ,{N}_{r}^{\left(i\right)}-1,{N}_{r}^{\left(i\right)}, as follows,

{p}_{{N}_{r}^{\left(i\right)}}\left(i\right)>{p}_{{N}_{r}^{\left(i\right)}-1}\left(i\right)>\cdots >{p}_{1}\left(i\right)>{p}_{0}\left(i\right)=0>{p}_{-1}\left(i\right)=-{p}_{1}\left(i\right)>\cdots >{p}_{-{N}_{r}^{\left(i\right)}}\left(i\right)=-{p}_{{N}_{r}^{\left(i\right)}}\left(i\right).

(12)

Then, given the case *C*_{
i
}, the decision intervals for Λ(*r*_{0}) > 0 are given by

\begin{array}{c}{p}_{{N}_{r}^{\left(i\right)}}\left(i\right)\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}{r}_{0}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\hfill \\ {p}_{{N}_{r}^{\left(i\right)}-2}\left(i\right)\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}{r}_{0}\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}{p}_{{N}_{r}^{\left(i\right)}-1}\left(i\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\vdots \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\hfill \\ -{p}_{{N}_{r}^{\left(i\right)}-2}\left(i\right)\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}{r}_{0}\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}-{p}_{{N}_{r}^{\left(i\right)}-3}\left(i\right)\hfill \\ -{p}_{{N}_{r}^{\left(i\right)}}\left(i\right)\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}{r}_{0}\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}-{p}_{{N}_{r}^{\left(i\right)}-1}\left(i\right)\hfill \end{array}

(13)

For the case *C*_{1}, the BER {P}_{b}^{\left(1\right)} is calculated as follows

{P}_{b}^{\left(1\right)}=\frac{1}{{2}^{{N}_{I}}}\sum _{d=0}^{{2}^{{N}_{I}}-1}Q\left(\frac{{\mu}^{-1}\left(d+1\right)}{\sqrt{{N}_{0}/2}}\right).

(14)

For the case *C*_{
i
}, 2 ≤ *i* ≤ *N*_{
c
}, and the center -*c*-1, -{2}^{{N}_{I}}\le -c-1\le -1, we can obtain the following condition

{p}_{j-2}\left(i\right)<{p}_{j-1}\left(i\right)<{\mu}^{-1}\left(-c-1\right)<{p}_{j}\left(i\right)<{p}_{j+1}\left(i\right).

(15)

Then the conditional BER is calculated as follows

{P}_{-c-1}^{\left(i\right)}\phantom{\rule{1em}{0ex}}\approx \sum _{h=0}^{{N}_{r}^{\left(i\right)}-j}\frac{{\left(-1\right)}^{h}}{{2}^{{N}_{I}}}Q\left(\frac{{p}_{j+h}\left(i\right)-{\mu}^{-1}\left(-c-1\right)}{\sqrt{{N}_{0}/2}}\right)\phantom{\rule{2.77695pt}{0ex}}+\sum _{h=0}^{{N}_{r}^{\left(i\right)}+j-1}\frac{{\left(-1\right)}^{h}}{{2}^{{N}_{I}}}Q\left(\frac{{\mu}^{-1}\left(-c-1\right)-{p}_{j-1-h}\left(i\right)}{\sqrt{{N}_{0}/2}}\right),

(16)

where Q\left(x\right)={\int}_{x}^{\infty}\left(1/\sqrt{2\pi}\right)\text{exp}\left(-{t}^{2}/2\right)dt. Then the BER {P}_{b}^{\left(i\right)} is calculated as follows

{P}_{b}^{\left(i\right)}\phantom{\rule{1em}{0ex}}\approx \sum _{c=0}^{{2}^{{N}_{I}}-1}{P}_{-c-1}^{\left(i\right)}\phantom{\rule{1em}{0ex}},

(17)

where 2 ≤ *i* ≤ *N*_{
c
}.

**3.1. BER derivation for** *N*_{
I
}= 2

We continue our derivation for *N*_{
I
} = 2 and without loss of generality, we assume *I*_{1} ≥ *I*_{2}. Then we can obtain the real roots solving (11) as follows

\begin{array}{c}0\hfill \\ 0,\pm {I}_{2}\hfill \\ 0,\pm {I}_{2},\pm {I}_{1}\hfill \\ 0,\pm {I}_{1}\hfill \\ 0,\pm {I}_{1},\pm \left({I}_{1}+{I}_{2}\right)\hfill \\ 0,\pm {I}_{1},\pm \left({I}_{1}+{I}_{2}\right),\pm \left({I}_{1}-{I}_{2}\right)\hfill \end{array}\begin{array}{c}\hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}>{I}_{1}+{I}_{2},\hfill \\ \hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}<{I}_{1}+{I}_{2},{A}_{0}>{I}_{1},\hfill \\ \hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}>{I}_{1}-{I}_{2},{I}_{1}>{A}_{0}>{I}_{2},\hfill \\ \hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}<{I}_{1}-{I}_{2},{I}_{1}>{A}_{0}>{I}_{2},\hfill \\ \hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}>{I}_{1}-{I}_{2},{I}_{1}>{I}_{2}>{A}_{0},\hfill \\ \hfill for\phantom{\rule{2.77695pt}{0ex}}{A}_{0}<{I}_{1}-{I}_{2},{I}_{1}>{I}_{2}>{A}_{0}.\hfill \end{array}

(18)

Then, the BER can be calculated by (14) or (17).

**3.2. BER derivation for** *N*_{
I
}> 2

Next, we continue the derivation for *N*_{
I
} > 2, which is more complicated than the previous case because the number *N*_{
c
} of cases increases rapidly. To make the problem tractable, we assume the following condition

{A}_{0}>{I}_{1}>2{I}_{2}>\cdots >{2}^{{N}_{\mathsf{\text{I}}}-1}{I}_{{N}_{\mathsf{\text{I}}}}>0.

(19)

In the above condition, a practical situation, in which the interference of the CCIs is not severe, can be assumed. Then the possible marginal conditions for various real roots are given by

\begin{array}{c}{\mu}^{-1}\left(-c-1\right)-{\mu}^{-1}\left(d+1\right)<0,\\ -{\mu}^{-1}\left(-c-1\right)+{\mu}^{-1}\left(d+1\right)<0,\end{array}

(20)

where d,c=0,1,\dots ,{2}^{{N}_{I}}-1. These conditions can be written as greatly simple forms with ternary numbers, which are

t\triangleq \sum _{k=1}^{{N}_{I}}{t}_{\left(3\right)}\left(k\right){3}^{k-1}\triangleq {\left({t}_{\left(3\right)}\left({N}_{I}\right)\cdots {t}_{\left(3\right)}\left(2\right){t}_{\left(3\right)}\left(1\right)\right)}_{\left\{3,{N}_{I}\right\}},

(21)

where {\left(\bullet \right)}_{\left\{3,{N}_{I}\right\}} denotes a ternary number with *N*_{
I
} digits, and *t*_{(3)}(*k*) ∈ {0, 1, 2} is defined as the *k* th digit in the ternary representation of an integer *t*. The left-hand sides in (20) are numbered as

\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{3}^{{N}_{I}}-t\triangleq \lambda \left({A}_{0}-\sum _{i=1}^{{N}_{I}}\left({t}_{\left(3\right)}\left({N}_{I}+1-i\right)-1\right){I}_{i}\right),\\ -{3}^{{N}_{I}}+t\triangleq \lambda \left(-{A}_{0}+\sum _{i=1}^{{N}_{I}}\left({t}_{\left(3\right)}\left({N}_{I}+1-i\right)-1\right){I}_{i}\right),\end{array}

(22)

where t=0,1,\dots ,{3}^{{N}_{I}}-1. The conditions for various real roots can efficiently be represented as a table. Now, we explain the procedure to create the table. The first step is to draw a table with {\left(11\cdots 1\right)}_{\left\{3,{N}_{I}-2\right\}} rows and {\left(11\cdots 1\right)}_{\left\{3,{N}_{I}-1\right\}} columns. Fill the first row with the decimal numbers from left to right, starting with 1 and ending with {\left(11\cdots 1\right)}_{\left\{3,{N}_{I}-1\right\}}. Then copy the one-cell left-shifted version of the first row into the second row, the one-cell left-shifted version of the second row into the third row, and so on. Remove the entries below the main diagonal. On the first row, find the numbers for inequalities corresponding to *μ*^{-1}(*d*+1) < *μ*^{-1}(-*c*-1), where d=0,1,\dots ,{2}^{{N}_{I}-1}-1 and *c* = 0. Write -*c*-1 on a new additional column, following the same pattern as *d*+1 with the opposite sign, where c=0,1,\dots ,{2}^{{N}_{I}-2}-1. Finally, mark the entries at the intersection of -*c*-1 rows and *d*+1 columns. Figure 1 shows the procedure for generating the table with *N*_{
I
} = 4 interferers. The number of columns is the number of conditions for various real roots, which are numbered as 1 to 1+{\left(11\cdots 1\right)}_{\left\{3,{N}_{I}-1\right\}}. Then the set *C*_{
i
} with {N}_{c}=1+{\left(11\cdots 1\right)}_{\left\{3,{N}_{I}-1\right\}} is defined as

{C}_{i}=\left\{{\lambda}^{-1}\left(-m\left(-c-1\right)\right)<0,\right.{\lambda}^{-1}\left(n\left(-c-1\right)\right)<0\left(\right)close="\}">\left|c=0,1,\dots ,{2}^{{N}_{I}-2}-1\right.\n

(23)

where *m*(-*c*-1) is the biggest and nearest marked number from *i* including *i* and *n*(-*c*-1) is the smallest and nearest marked number from *i* excluding *i* on each row -*c*-1, where -{2}^{{N}_{I}-2}\le -c-1\le -1. If *m* does not exist, there is no corresponding marginal condition. The set *D*_{
i
} of candidates for real roots corresponding to the case *C*_{
i
} is defined as

\begin{array}{c}{D}_{i}=\left\{\pm \left({\mu}^{-1}\left({d}_{m\left(-c-1\right)}+1\right)-{\mu}^{-1}\left(-c-1\right)\right)/2,\right.\pm \left({\mu}^{-1}\left(-c-1\right)-{\mu}^{-1}\left({d}_{n\left(-c-1\right)}+1\right)\right)/2,\\ \left(\right)close="\}">0\left|c=0,1,\dots ,{2}^{{N}_{I}-2}-1\right.& ,\end{array}\n

(24)

where the indexes (*d*_{
m
}_{(-c-1)}+1) and (*d*_{
n
}_{(-c-1)}+1) of the centers of bell curves correspond to *m*(-*c*-1) and *n*(-*c*-1) on each row -*c*-1, respectively. For the set *E*_{
i
} given by

\begin{array}{c}{E}_{i}=\left\{\left(-{c}_{1}-1\right),\right.\left(-{c}_{2}-1\right)\left|\left({d}_{m\left(-{c}_{1}-1\right)}+1\right)=\left({d}_{m\left(-{c}_{2}-1\right)}+1\right)\right.,\\ \left(\right)close="\}">{c}_{1},{c}_{2}=0,1,\dots ,{2}^{{N}_{I}-2}-1,{c}_{1}\ne {c}_{2}& ,\end{array}\n

(25)

the updated set *F*_{
i
} is obtained from the set *D*_{
i
} by removing ±(*μ*^{-1}(*d*_{
m
}_{(-c-1)}+1)-*μ*^{-1}(-*c*-1))/2, except the biggest element (-*b*-1)∈*E*_{
i
}, for (-*c*-1)≠(-*b*-1), (-*c*-1)∈*E*_{
i
}, and the updated set *R*_{
i
} is obtained from the set *F*_{
i
} by removing \pm \left({\mu}^{-1}\left(-c-1\right)-{\mu}^{-1}\left({d}_{n\left(-c-1\right)}+1\right)\right)/2, except the smallest element (-*s*-1)∈*E*_{
i
}, for (-*c*-1)∈*E*_{
i
}, (-*c*-1)≠(-*s*-1).

Figure 1. Procedure of generating a table with conditions for various real roots. The number *N*_{
I
} of interferers is 4.

In summary, with the set *C*_{
i
} of marginal conditions, we obtain the set *R*_{
i
} of real roots corresponding to *C*_{
i
} by solving (11). For the case *C*_{
i
}, 1≤*i*≤*N*_{
c
}, the real roots *p*_{
j
}(*i*)∈*R*_{
i
}, j=-{N}_{r}^{\left(i\right)},\dots ,{N}_{r}^{\left(i\right)}, and the center -*c*-1, -{2}^{{N}_{I}}\le -c-1\le -1, we obtain the condition *p*_{
j
}_{-1}(*i*)<*μ*^{-1}(-*c*-1)<*p*_{
j
}(*i*). Then conditional BER {P}_{-c-1}^{\left(i\right)} is calculated by (16) and the BER {P}_{b}^{\left(i\right)} is calculated by (14) or (17).

In addition, we calculate the channel capacity. We consider the channel of (4) with possible inputs *A*_{0} or - *A*_{0}. The capacity of this channel in bit/channel use is given by [12]

C=1-\frac{1}{2}{\int}_{-\infty}^{\infty}{f}_{{R}_{0}}\left({r}_{0}|{H}_{1}\right){\text{log}}_{2}\left(1+{e}^{-\Lambda \left({r}_{0}\right)}\right)d{r}_{0}-\frac{1}{2}{\int}_{-\infty}^{\infty}{f}_{{R}_{0}}\left({r}_{0}|{H}_{0}\right){\text{log}}_{2}\left(1+{e}^{\Lambda \left({r}_{0}\right)}\right)d{r}_{0}

(26)

where by using (3) and (4), {f}_{{R}_{0}}\left({r}_{0}|{H}_{1}\right)={f}_{W}\left({r}_{0}-{A}_{0}\right) and {f}_{{R}_{0}}\left({r}_{0}|{H}_{0}\right)={f}_{W}\left({r}_{0}+{A}_{0}\right). Then the capacity is computed by (26).