For the conventional TH/DS system using *N*-ary BPPM in AWGN channel, to detect the *i* th data symbol of the desired user {\hat{u}}_{i}^{\left(1\right)}, the received signal is correlated with \hat{M}\left(=\hat{N}\u22152\right) orthogonal template waveforms to obtain \hat{M} decision statistics {\left\{{r}_{m}\right\}}_{m=0}^{\hat{M}-1} as follows [3]:

{r}_{m}=\sum _{j=i{\hat{N}}_{s}}^{\left(i+1\right){\hat{N}}_{s}-1}{\int}_{j{T}_{f}}^{\left(j+1\right){T}_{f}}r\left(t\right){h}_{m}\left(t-j{T}_{f}\right)dt=\left\{\begin{array}{cc}\hfill {S}_{\mathsf{\text{conv}}}+{I}_{\mathsf{\text{conv}}}+{n}_{\mathsf{\text{conv}}},\hfill & \hfill m={m}_{i}^{\left(1\right)}\hfill \\ \hfill {I}_{\mathsf{\text{conv}}}+{n}_{\mathsf{\text{conv}}},\hfill & \hfill m\ne {m}_{i}^{\left(1\right)}\hfill \end{array}\right.

(7)

The template waveform of the *m* th correlator is given by

{h}_{m}\left(t\right)=\sqrt{\frac{{\hat{N}}_{s}}{{E}_{s}}}{a}_{j}^{\left(1\right)}p\left(t-{c}_{j}^{\left(1\right)}{T}_{c}-m\delta \right)

(8)

where {S}_{\mathsf{\text{conv}}}={\left(-1\right)}^{{n}_{i}^{\left(1\right)}}{\hat{N}}_{s}{A}_{1}R\left(0\right) is the correlator output of the desired transmitted signal. *I*_{conv} is the correlator output coming from other *K* - 1 users' signals and is called MAI. The MAI can be described as {I}_{\mathsf{\text{conv}}}=\sum _{k=2}^{K}{A}_{k}{I}_{\mathsf{\text{conv}}}^{\left(k\right)}, where {I}_{\mathsf{\text{conv}}}^{\left(k\right)} is the MAI caused by the *k* th user. *n*_{conv} is a Gaussian random variable with zero mean and variance {\sigma}_{{n}_{\mathsf{\text{conv}}}}^{2}={N}_{0}{\hat{N}}_{s}^{2}R\left(0\right)\u22152{E}_{s}. Based on the maximum likelihood decision rule for AWGN channel [5], the receiver of the desired user computes a bank of *M* correlators' outputs, {\left\{{r}_{m}\right\}}_{m=0}^{M-1} in (7), and then chooses the index corresponding to the largest absolute value of the correlator's output as the estimate of the message symbol {\hat{m}}_{i}^{\left(i\right)}:

{\hat{m}}^{\left(1\right)}=\underset{m\in \left\{0,1,\dots N\u22152-1\right\}}{argmax}\left|{r}_{m}\right|

(9)

as well as

{\hat{n}}^{\left(1\right)}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill {r}_{{\hat{m}}^{\left(1\right)}}<0\hfill \\ \hfill 0,\hfill & \hfill {r}_{{\hat{m}}^{\left(1\right)}}>0\hfill \end{array}\right.

(10)

Consider the receiver structure of the proposed TH/DS system shown in Figure 3. The output of the *m* th correlator in the *j* th frame duration is

{e}_{mj}={\int}_{j{T}_{f}}^{\left(j+1\right){T}_{f}}r\left(t\right){h}_{m\oplus {c}_{j}^{\left(1\right)}}\left(t-j{T}_{f}\right)dt=\left\{\begin{array}{cc}\hfill {S}_{j,\mathsf{\text{pro}}}+{I}_{j,\mathsf{\text{pro}}}+{n}_{j,\mathsf{\text{pro,}}}\hfill & \hfill m={m}_{i}^{\left(1\right)}\hfill \\ \hfill {I}_{j,\mathsf{\text{pro}}}+{n}_{j,\mathsf{\text{pro,}}}\hfill & \hfill m\ne {m}_{i}^{\left(1\right)}\hfill \end{array}\right.

(11)

where *n*_{j,pro}is i.i.d. Gaussian noise with zero mean and variance {\sigma}_{{n}_{j,\mathsf{\text{pro}}}}^{2}={N}_{0}{N}_{s}R\left(0\right)\u22152{E}_{s}. The template waveform of the *m* th correlator is expressed as

{h}_{m\oplus {c}_{j}^{\left(1\right)}}\left(t\right)=\sqrt{\frac{{N}_{s}}{{E}_{s}}}{a}_{j}^{\left(1\right)}p\left(t-\left(m\oplus {c}_{j}^{\left(1\right)}\right)\delta \right)

(12)

where {S}_{\mathsf{\text{conv}}}={\left(-1\right)}^{{n}_{i}^{\left(1\right)}}{\hat{N}}_{s}{A}_{1}R\left(0\right) is the {m}_{i}^{\left(1\right)}th correlator's output; {S}_{j,\mathsf{\text{pro}}}={\left(-1\right)}^{{n}_{i}^{\left(1\right)}}{A}_{1}R\left(0\right) is the desired component corresponding to the data symbol {m}_{i}^{\left(1\right)}, and {I}_{j,\mathsf{\text{pro}}}=\sum _{k=2}^{K}{A}_{k}{I}_{j,pro}^{\left(k\right)} where {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)} is the MAI from the *k* th user. Completing the combining process, the *m* th decision variable can then be acquired as

\begin{array}{ll}\hfill {r}_{m}& =\sqrt{\frac{{N}_{s}}{{E}_{s}}}\sum _{j=i{N}_{s}}^{\left(i+1\right){N}_{s}-1}{a}_{j}^{\left(1\right)}{\int}_{j{T}_{f}}^{\left(j+1\right){T}_{f}}r\left(t\right)p\left(t-j{T}_{f}-\left(m\oplus {c}_{j}^{\left(1\right)}\right)\delta \right)dt\phantom{\rule{2em}{0ex}}\\ =\left\{\begin{array}{c}\hfill {S}_{\mathsf{\text{pro}}}+{I}_{\mathsf{\text{pro}}}+{n}_{\mathsf{\text{pro}}},m={m}_{i}^{\left(1\right)}\hfill \\ \hfill {I}_{\mathsf{\text{pro}}}+{n}_{\mathsf{\text{pro}}},\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}m\ne {m}_{i}^{\left(1\right)}\hfill \end{array}\right.\phantom{\rule{2em}{0ex}}\end{array}

(13)

where *n*_{pro} is Gaussian random noise with zero mean and variance {\sigma}_{{n}_{\mathsf{\text{pro}}}}^{2}={N}_{0}{N}_{s}^{2}R\left(0\right)\u22152{E}_{s}. {S}_{\mathsf{\text{pro}}}={\left(-1\right)}^{{n}_{i}^{\left(1\right)}}{A}_{1}{N}_{s}R\left(0\right) is the desired component corresponding to the data symbol {m}_{i}^{\left(1\right)}, and *I*_{pro} is the total MAI caused by the *K* - 1 interfering users,

{I}_{\mathsf{\text{pro}}}=\sum _{j=i{N}_{s}}^{\left(i+1\right){N}_{s}-1}{I}_{j,\mathsf{\text{pro}}}=\sum _{j=i{N}_{s}}^{\left(i+1\right){N}_{s}-1}\sum _{k=2}^{K}{A}_{k}{I}_{j,\mathsf{\text{pro}}}^{\left(k\right)}

(14)

Let *τ*_{
k
}= *α*_{
k
}*T*_{
f
}+ Δ_{
k
}and Δ_{
k
}= *β*_{
k
}*T*_{
p
}+ *γ*_{
k
}, where *α*_{
k
}is the discrete uniformly distributed r.v. in {0, 1, ..., *N*_{
s
}- 1}; *β*_{
k
}is the discrete uniformly distributed r.v. in {0, 1, ..., *M* - 1}; and *γ*_{
k
}is the continuous uniformly distributed r.v. in one pulse duration, i.e., 0 ≤ *γ*_{
k
}< *T*_{
p
}[4]. Hence, we can obtain {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)} as follows:

\begin{array}{ll}\hfill {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)}& ={\int}_{j{T}_{f}}^{\left(j+1\right){T}_{f}}{a}_{j}^{\left(1\right)}\sum _{q=i{N}_{s}}^{\left(i+1\right){N}_{s}-1}{\left(-1\right)}^{{n}_{1}^{\left(k\right)}}{a}_{q}^{\left(k\right)}p\left(t-q{T}_{f}-{b}_{q}^{\left(k\right)}\delta -{\tau}_{k}\right)p\left(t-j{T}_{f}-\left(m\oplus {c}_{j}^{\left(1\right)}\right)\delta \right)dt\phantom{\rule{2em}{0ex}}\\ ={a}_{j}^{\left(1\right)}{a}_{j-{\alpha}_{k}-1}^{\left(k\right)}{\left(-1\right)}^{{n}_{\u230a\left(j-{\alpha}_{k}-1\right)\u2215{N}_{s}\u230b}^{\left(k\right)}}R\left({b}_{j-{\alpha}_{k}-1}^{\left(k\right)}{T}_{p}+{\beta}_{k}{T}_{p}+{\gamma}_{k}-\left(m\oplus {c}_{j}^{\left(1\right)}\right){T}_{p}-{T}_{f}\right)\phantom{\rule{2em}{0ex}}\\ +{a}_{j}^{\left(1\right)}{a}_{j-{\alpha}_{k}}^{\left(k\right)}{\left(-1\right)}^{{n}_{\u230a\left(j-{\alpha}_{k}\right)\u2215{N}_{s}\u230b}^{\left(k\right)}}R\left({b}_{j-{\alpha}_{k}}^{\left(k\right)}{T}_{p}+{\beta}_{k}{T}_{p}+{\gamma}_{k}-\left(m\oplus {c}_{j}^{\left(1\right)}\right){T}_{p}\right)\phantom{\rule{2em}{0ex}}\end{array}

(15)

and can be rewritten as

{I}_{j,\mathsf{\text{pro}}}^{\left(k\right)}={U}_{j}^{\left(k\right)}{\hat{R}}_{p}\left({\gamma}_{k}\right){\hat{\Gamma}}_{j}\left({\Delta}_{k}\right)+{V}_{j}^{\left(k\right)}{R}_{p}\left({\gamma}_{k}\right){\Gamma}_{j}\left({\Delta}_{k}\right)

(16)

where {\hat{R}}_{p}\left({\gamma}_{k}\right)={\int}_{{\gamma}_{k}}^{{T}_{p}}p\left(t\right)p\left(t-{\gamma}_{k}\right)dt and {R}_{p}\left({\gamma}_{k}\right)={\hat{R}}_{p}\left({T}_{p}-{\gamma}_{k}\right). {U}_{j}^{\left(k\right)} and {V}_{j}^{\left(k\right)} are the discrete uniformly distributed r.v.s in {-1, +1} because the polarity codes **a**^{(k)}and the message symbol *n*^{(k)}of the user *k* are assumed to be random and equally likely. Γ_{
j
}(Δ_{
k
}) and {\hat{\Gamma}}_{j}\left({\Delta}_{k}\right) can be expressed as follows:

{\hat{\Gamma}}_{j}\left({\Delta}_{k}\right)=\left\{\begin{array}{cc}\hfill 1;\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{0.3em}{0ex}}0\le {k}_{2}-{k}_{1}<{T}_{p}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{or}}\phantom{\rule{0.3em}{0ex}}0\le {k}_{3}-{k}_{1}<{T}_{p}\hfill \\ \hfill 0;\hfill & \hfill \mathsf{\text{otherwise}}\hfill \end{array}\right.

(17)

{\Gamma}_{j}\left({\Delta}_{k}\right)=\left\{\begin{array}{cc}\hfill 1;\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}0\le {k}_{1}-{k}_{2}<{T}_{p}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{or}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{0}}\le {k}_{1}-{k}_{3}<{T}_{p}\hfill \\ \hfill 0;\hfill & \hfill \mathsf{\text{otherwise}}\hfill \end{array}\right.

(18)

where {k}_{1}=\left(m\oplus {c}_{j}^{\left(1\right)}\right){T}_{p},{k}_{2}={\Delta}_{k}+{b}_{j-{\alpha}_{k}}^{\left(k\right)}{T}_{p}, and {k}_{3}={\Delta}_{k}+{b}_{j-{\alpha}_{k}-1}^{\left(k\right)}{T}_{p}-{T}_{f}. The *i* th data symbol of the *k* th user {m}_{i}^{\left(k\right)} is assumed to be an uniformly distributed r.v. in the range of 0\le {m}_{i}^{\left(k\right)}\le M-1 and each element {c}_{j}^{\left(k\right)} of the random TH code utilized by the *k* th user in the *j* th frame period is assumed to be an uniformly distributed r.v. with {c}_{j}^{\left(k\right)}\in G=\left\{0,1,2,\dots ,Q-1\right\}. Therefore, {b}_{j}^{\left(k\right)}={m}_{i}^{\left(k\right)}\oplus {c}_{j}^{\left(k\right)} is an uniformly distributed r.v. with {b}_{j}^{\left(k\right)}\in F. The probability density function (PDF) of {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)} conditioned on *γ*_{
k
}can be described as

{f}_{{I}_{j,\mathsf{\text{pro}}}^{\left(k\right)}|{\gamma}_{k},{U}_{j}^{\left(k\right)},{V}_{j}^{\left(k\right)}}\left(i\right)=\frac{1}{Q}{\delta}_{D}\left(i-{U}_{j}^{\left(k\right)}\hat{R}\left({\gamma}_{k}\right)\right)+\frac{1}{Q}{\delta}_{D}\left(i-{V}_{j}^{\left(k\right)}R\left({\gamma}_{k}\right)\right)+\frac{Q-2}{Q}{\delta}_{D}\left(i\right)

(19)

where *δ*_{D} is the Dirac delta function. Therefore, the conditional CF of {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)} can be obtained as obtained as

{\Phi}_{{I}_{j,pro}^{\left(k\right)}|{\gamma}_{k}}\left(\omega \right)=\frac{1}{Q}cos\left(\omega R\left({\gamma}_{k}\right)\right)+\frac{1}{Q}cos\left(\omega \hat{R}\left({\gamma}_{k}\right)\right)+\frac{Q-2}{Q}

(20)

The interferences {I}_{j,\mathsf{\text{pro}}}^{\left(k\right)} are independent of each other because each element of the user's TH code **c**^{(k)}is randomly and independently selected from the set *G*. Hence,

{\Phi}_{{I}_{\mathsf{\text{pro}}}^{\left(k\right)}|{\gamma}_{k}}\left(\omega \right)={\left[\frac{1}{Q}cos\left(\omega R\left({\gamma}_{k}\right)\right)+\frac{1}{Q}cos\left(\omega \hat{R}\left({\gamma}_{k}\right)\right)+\frac{Q-2}{Q}\right]}^{{N}_{s}}

(21)

and we then obtain

{\Phi}_{{I}_{\mathsf{\text{pro}}}^{\left(k\right)}}\left(\omega \right)=\frac{1}{{T}_{p}}{\int}_{0}^{{T}_{p}}{\left[\frac{1}{Q}cos\left(\omega R\left({\gamma}_{k}\right)\right)+\frac{1}{Q}cos\left(\omega \hat{R}\left({\gamma}_{k}\right)\right)+\frac{Q-2}{Q}\right]}^{{N}_{s}}{d}_{{\gamma}_{k}}

(22)

{\Phi}_{{I}_{\mathsf{\text{pro}}}}\left(\omega \right)=\prod _{k=2}^{K}{\Phi}_{{I}_{\mathsf{\text{pro}}}^{\left(k\right)}}\left({A}_{k}\omega \right)

(23)

It is worthy to note that the CF of the MAI component for each correlator's output of the proposed TH/DS receiver is di¤erent from that of the conventional TH/DS system which has been shown in [3].

### 3.1 Symbol error rate

Let the *i* th data symbol of the desired user be {u}_{i}^{\left(1\right)}=\left({m}_{i}^{\left(1\right)},{n}_{i}^{\left(1\right)}\right)=\left(0,0\right). According to our derived CF of the MAI component for the correlator's output of the proposed TH/DS receiver in (23), the average SER of the *N*-ary biorthogonal modulation has been expressed and calculated as [5]:

{P}_{e}=1-{\int}_{0}^{+\infty}P\left(\left|{r}_{1}\right|\le \mu ,\left|{r}_{2}\right|\le \mu ,\dots ,\left|{r}_{N\u22152-1}\right|\le \mu |\mu \right){f}_{{r}_{0}}\left(\mu \right)du

The decision statistics of the combining correlators' outputs {\left\{{r}_{m}\right\}}_{m=0}^{N\u22152-1} are assumed to be independent [2, 3, 17–21]. Hence, the SER of the proposed system is

\begin{array}{ll}\hfill {P}_{e}& =1-{\int}_{0}^{+\infty}{\left[P\left(\left|{r}_{1}\right|\le u|u\right)\right]}^{N\u22152-1}{f}_{{r}_{0}}\left(u\right)du\phantom{\rule{2em}{0ex}}\\ =1-{\int}_{0}^{+\infty}{\left[{F}_{{r}_{1}}\left(u\right)-{F}_{{r}_{1}}\left(-u\right)\right]}^{N\u22152-1}{f}_{{r}_{0}}\left(u\right)du\phantom{\rule{2em}{0ex}}\end{array}

(24)

where {F}_{{r}_{1}}\left(u\right) is the CDF of *r*_{1}. The first decision variable is *r*_{0} = *A*_{1}*N*_{
s
}*R*(0) + *I* + *n*, and the other *M* - 1 decision variables are {\left\{{r}_{m}\right\}}_{m=0}^{M-1}=I+n. Therefore, (24) can be rewritten as

{P}_{e}=1-{\int}_{0}^{+\infty}{\left[{F}_{{r}_{1}}\left(u\right)-{F}_{{r}_{1}}\left(-u\right)\right]}^{N\u22152-1}{f}_{{r}_{1}}\left(u-{A}_{1}{N}_{s}R\left(0\right)\right)du

(25)

As the MAI and AWGN are assumed to be mutually independent, we obtain the CF of *r*_{1} as {\Phi}_{{r}_{1}}\left(\omega \right)={\Phi}_{I}\left(\omega \right){\Phi}_{n}\left(\omega \right), where the CF of the AWGN is {\Phi}_{n}\left(\omega \right)={e}^{-{\sigma}_{{n}_{\mathsf{\text{pro}}}}^{2}{\omega}^{2}\u22152}. Hence, the PDF of *r*_{1} can be acquired as

{f}_{{r}_{1}}\left(u\right)=\frac{1}{\pi}{\int}_{0}^{+\infty}{\Phi}_{{r}_{1}}\left(\omega \right)cos\left(\omega u\right)d\omega

(26)

Applying the relationship between the CF and CDF [12–14], we have

{F}_{{r}_{1}}\left(u\right)=\frac{1}{2}+\frac{1}{\pi}{\int}_{0}^{+\infty}{\Phi}_{{r}_{1}}\left(\omega \right)\frac{sin\left(\omega u\right)}{\omega}d\omega

(27)

If all *M* - 1 erroneous symbols are equally likely chosen, then the corresponding BER is *P*_{
b
} = *M* · *P*_{
e
}/[2 · (*M* - 1)] [5].