### 5.1. The BER of PSPF

The BER performance of the proposed PSPF scheme with discriminator detection is evaluated using the characteristic function (CF) approach; see [15]. In the analysis, the variances of the fading processes *g*_{
AR
} (*t*), *g*_{
BR
} (*t*), *g*_{
AB
} (*t*), *g*_{
BA
} (*t*), *g*_{
RA
} (*t*), and *g*_{
RB
} (*t*) in (1) to (6) are denoted as {\sigma}_{{g}_{AR}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{g}_{BR}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{g}_{AB}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{g}_{B\mathsf{\text{A}}}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{g}_{R\mathsf{\text{A}}}}^{2}, and {\sigma}_{{g}_{RB}}^{2}, respectively, with {\sigma}_{{g}_{AR}}^{2}={\sigma}_{{g}_{R\mathsf{\text{A}}}}^{2},{\sigma}_{{g}_{BR}}^{2}={\sigma}_{{g}_{RB}}^{2}, and {\sigma}_{{g}_{AR}}^{2}={\sigma}_{{g}_{B\mathsf{\text{A}}}}^{2}. On the other hand, the variances of the noise processes *n*_{R, 1}(*t*), *n*_{B, 1}(*t*), *n*_{R, 2}(*t*), *n*_{A,2}(*t*), *n*_{A,3}(*t*), and *n*_{B,3}(*t*) in these equations are {\sigma}_{{n}_{R,1}}^{2},\phantom{\rule{0.3em}{0ex}}{\sigma}_{{n}_{B,1}}^{2},\phantom{\rule{0.3em}{0ex}}{\sigma}_{{n}_{R,2}}^{2},\phantom{\rule{0.3em}{0ex}}{\sigma}_{{n}_{B,2}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{{n}_{A,3}}^{2}, and {\sigma}_{{n}_{B,3}}^{2}, respectively, with {\sigma}_{{n}_{R,1}}^{2}={\sigma}_{{n}_{B,1}}^{2}={\sigma}_{{n}_{R,2}}^{2}={\sigma}_{{n}_{B,2}}^{2}={N}_{0}{B}_{12} and {\sigma}_{{n}_{A,3}}^{2}={\sigma}_{{n}_{B,3}}^{2}={N}_{0}{B}_{3}, where *N*_{0} is the noise power spectral density (PSD), *B*_{12} the bandwidth of the receive LPFs in Phases I and II, and *B*_{3} the bandwidth of the receive LPF in Phase III. In this investigation, *B*_{12} is always set to the 99% bandwidth of *x*_{
A
} (*t*) and *x*_{
B
} (*t*), while *B*_{3} is either the same as *B*_{12}, or set to the 99% bandwidth of the relay signal *x*_{
R
} (*t*). Given the nature of the symbol-by-symbol detectors described in the previous section, we take the liberty to drop the symbol index k in *d*_{
A, k
} and *d*_{
B, k
} in the performance analysis.

First, it is observed that the terms *D*_{2} in (21) is a quadratic forms of complex Gaussian variables \left({y}_{A,2},{\u1e8f}_{A,2}\right) when conditioned on {\dot{\theta}}_{B}; refer to the Appendix for the statistical relationships between different parameters in the general channel model

y\left(t\right)=g\left(t\right){\mathsf{\text{e}}}^{j\theta \left(t\right)}+n\left(t\right)=a\left(t\right){\mathsf{\text{e}}}^{j\psi \left(t\right)},

where *g*(*t*) and *n*(*t*) are, respectively, CN\left(0,{\sigma}_{g}^{2}\right) and CN\left(0,{\sigma}_{n}^{2}\right),\theta \left(t\right) is the signal phase, and *a*(*t*) and *ψ*(*t*) are respectively the amplitude and phase of *y*(*t*). Without loss of generality, we assume *d*_{
B, k
} = +1 and hence {\dot{\theta}}_{B}\left(t\right)=\pi h\u2215T. By substituting \theta ={\dot{\theta}}_{B} into (A5) and (A8), and with **F** in (A10) set to the \left[\begin{array}{cc}0\hfill & -j\hfill \\ j\hfill & 0\hfill \end{array}\right] matrix in (21), we can find the two poles of the CF of *D*_{2} as following:

{p}_{1}=-\frac{1}{2{\alpha}_{A,2}{\beta}_{A,2}\left(1+{\rho}_{A,2}\right)}<0,\phantom{\rule{1em}{0ex}}{p}_{2}=+\frac{1}{2{\alpha}_{A,2}{\beta}_{A,2}\left(1-{\rho}_{A,2}\right)}>0,

(28)

where *α*_{A,2}, *β*_{A,2}, *ρ*_{A,2}are determined from (A10) under the conditions \dot{\theta}=\pi h\u2215T,\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{g}^{2}={\sigma}_{{g}_{B\mathsf{\text{A}}}}^{2}, and {\sigma}_{n}^{2}={N}_{0}{B}_{12};{B}_{12} the bandwidth of the receive filter in Phases I and II.

How about the term *D*_{3} in (21)? This term can be rewritten as {D}_{3}=2{a}_{A,3}^{2}{\dot{\Psi}}_{A,3}=2\left({a}_{A,3}^{2}{\dot{\psi}}_{A,3}-{a}_{A,3}^{2}{\dot{\theta}}_{A}\right), or as

{D}_{3}=\left({y}_{A,3}^{*}{\u1e8f}_{A,3}^{*}\right)\left(\begin{array}{cc}-2{\dot{\theta}}_{A}\hfill & -j\hfill \\ j\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}{y}_{A,3}\hfill \\ {\u1e8f}_{A,3}\hfill \end{array}\right),

(29)

which is once again a quadratic form of complex Gaussian variables. This quadratic form, however, depends on a number of parameters. First is the data phase derivation {\dot{\theta}}_{A}. Second, it depends on the forwarded phase derivative {\dot{\theta}}_{R}={\dot{\psi}}_{R,1}+{\dot{\psi}}_{R,2}, which in turns depends on both {\dot{\psi}}_{R,1} and {\dot{\psi}}_{R,2}; refer to (16). Of course, {\dot{\psi}}_{R,1} depends on {\dot{\theta}}_{A}, while {\dot{\psi}}_{R,2} depends on {\dot{\theta}}_{B}, refer to (14) and (15). Note that *D*_{2} and *D*_{3} are statistically independent. Conditioned on {\dot{\psi}}_{R,1},\phantom{\rule{2.77695pt}{0ex}}{\dot{\psi}}_{R,2},\phantom{\rule{2.77695pt}{0ex}}{\dot{\theta}}_{A},\phantom{\rule{2.77695pt}{0ex}}{\dot{\theta}}_{B}=\pi h\u2215T, and F=\left(\begin{array}{cc}-2{\dot{\theta}}_{A}\hfill & -j\hfill \\ j\hfill & 0\hfill \end{array}\right), we can determine from (A10) the poles of the CF of *D*_{3} as

\begin{array}{c}{Q}_{1}=\frac{\left({\chi}_{A,3}^{2}-{\dot{\theta}}_{A}{\alpha}_{A,3}^{2}\right)-\sqrt{{\alpha}_{A,3}^{2}\left({\dot{\theta}}_{A}^{2}{\alpha}_{A,3}^{2}-,2{\dot{\theta}}_{A}{\chi}_{A,3}^{2}+{\beta}_{A,3}^{2}\right)}}{2\left(1-{\rho}_{A,3}^{2}\right){\alpha}_{A,3}^{2}{\beta}_{A,3}^{2}}<0,\\ {Q}_{2}=\frac{\left({\chi}_{A,3}^{2}-{\dot{\theta}}_{A}{\alpha}_{A,3}^{2}\right)+\sqrt{{\alpha}_{A,3}^{2}\left({\dot{\theta}}_{A}^{2}{\alpha}_{A,3}^{2}-2{\dot{\theta}}_{A}{\chi}_{A,3}^{2}+{\beta}_{A,3}^{2}\right)}}{2\left(1-{\rho}_{A,3}^{2}\right){\alpha}_{A,3}^{2}{\beta}_{A,3}^{2}}>0.\end{array}

(30)

where {\alpha}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{\beta}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{p}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{\chi}_{A,3}^{2} are determined from (A10) under the conditions \dot{\theta}={\dot{\psi}}_{R,1}+{\dot{\psi}}_{R,2}, {\sigma}_{g}^{2}={\sigma}_{{g}_{R\mathsf{\text{A}}}}^{2}, and {\sigma}_{n}^{2}={N}_{0}{B}_{3}; B_{3} the bandwidth of the receive filter in Phase III.

Recall that we assume *d*_{
B
} = +1 and hence {\dot{\theta}}_{B}\left(t\right)=\pi h\u2215T. In this case, the detector makes a wrong decision when *D* < 0. Since the characteristic function of *D* is {\varphi}_{D}\left(s\right)=\left({p}_{1}{p}_{2}\right)\left({Q}_{1}{Q}_{2}\right)/\left\{\left(s-{p}_{1}\right)\left(s-{p}_{2}\right)\left(s-{Q}_{1}\right)\left(s-{Q}_{2}\right)\right\},

the probability that *D* < 0 is the sum of residues of -*ϕ*_{
D
}(*s*)/*s* at the right plane poles *p*_{2} and *Q*_{2}, yielding

\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A},{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\psi}}_{R,1},{\dot{\psi}}_{R,2}\right]=\frac{-{p}_{1}}{{p}_{2}-{p}_{1}}\cdot \frac{{Q}_{1}{Q}_{2}}{\left({p}_{2}-{Q}_{1}\right)\left({p}_{2}-{Q}_{2}\right)}+\frac{-{Q}_{1}}{{Q}_{2}-{Q}_{1}}\cdot \frac{{p}_{1}{p}_{2}}{\left({Q}_{2}-{p}_{1}\right)\left({Q}_{2}-{p}_{2}\right)}.

(31)

Finally, since {\dot{\psi}}_{R,1} and {\dot{\psi}}_{R,2} are random variables given {\dot{\theta}}_{A} and {\dot{\theta}}_{B}, respectively, the unconditional error probability can be expressed in semi-analytical form as

\begin{array}{c}{P}_{b}=\frac{1}{2}\sum _{{d}_{A}=-1}^{+1}\underset{{\dot{\psi}}_{R,1=-\infty}}{\overset{\infty}{\int}}\underset{{\dot{\psi}}_{R,2=-\infty}}{\overset{\infty}{\int}}\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T,{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\psi}}_{{R}_{2}1},{\dot{\psi}}_{{R}_{2}2}\right]\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}p\left({\dot{\psi}}_{R,1}|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T\right)p\left({\dot{\psi}}_{R,2}|{\dot{\theta}}_{B}=\pi h\u2215T\right)d{\dot{\psi}}_{R,1}d{\dot{\psi}}_{R,2},\end{array}

(32)

where the marginal probability density functions (PDF) p\left({\dot{\psi}}_{R,1}|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T\right) and p\left({\dot{\psi}}_{R,2}|{\dot{\theta}}_{B}=\pi h\u2215T\right) can be determined from (A5) to (A6) in the Appendix.

### 5.2. BER of 3-DPF and A4-DPF Signals

The two multi-level DPF signals broadcasted by the relay in (11) and (12) are constructed from decisions made by the relay about Users *A* and *B*'s data. Although different from (10), the exact BER analysis of these signals can still be determined via the characteristic function approach. This stems from the fact that the decision variable *D* of these DPF schemes are again quadratic forms of complex Gaussian variables when conditioned on the data phase derivatives {\dot{\theta}}_{A} and {\dot{\theta}}_{B}, as well as their decoded versions {\dot{\widehat{\theta}}}_{A}and {\dot{\widehat{\theta}}}_{B} at the relay. Specifically, the poles of the CF of *D*_{2} are identical to those in the PSPF case, and can be found in (28). As for the poles of the CF of *D*_{3}, we should first replace the term \dot{\theta} in the Appendix by {\dot{\theta}}_{R}={w}_{A}{\dot{\widehat{\theta}}}_{A}+{w}_{B}{\dot{\widehat{\theta}}}_{B} and then modify the **F** matrix in (A10) to

\mathbf{F}=\left(\begin{array}{ll}-2\frac{{w}_{A}}{{w}_{B}}{\dot{\theta}}_{A}\hfill & \frac{-j}{{w}_{B}}\\ \frac{j}{{w}_{B}}\hfill & 0\hfill \end{array}\right).

(33)

The resultant poles are found to be

\begin{array}{c}{Z}_{1}=\frac{\left({\chi}_{A,3}^{2}-{w}_{A}{\dot{\theta}}_{A}{\alpha}_{A,3}^{2}\right)-\sqrt{{\alpha}_{A,3}^{2}\left({\left({w}_{A}{\dot{\theta}}_{A}\right)}^{2}{\alpha}_{A,3}^{2}-2{w}_{A}{\dot{\theta}}_{A}{\chi}_{A,3}^{2}+{\beta}_{A,3}^{2}\right)}}{2\left(1-{\rho}_{A,3}^{2}\right){\alpha}_{A,3}^{2}{\beta}_{A,3}^{2}}.{w}_{B}<0,\\ {Z}_{2}=\frac{\left({\chi}_{A,3}^{2}-{w}_{A}{\dot{\theta}}_{A}{\alpha}_{A,3}^{2}\right)+\sqrt{{\alpha}_{A,3}^{2}\left({\left({w}_{A}{\dot{\theta}}_{A}\right)}^{2}{\alpha}_{A,3}^{2}-2{w}_{A}{\dot{\theta}}_{A}{\chi}_{A,3}^{2}+{\beta}_{A,3}^{2}\right)}}{2\left(1-{\rho}_{A,3}^{2}\right){\alpha}_{A,3}^{2}{\beta}_{A,3}^{2}}.{w}_{B}>0,\end{array}

(34)

Where {\alpha}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{\beta}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{\rho}_{A,3},\phantom{\rule{2.77695pt}{0ex}}{\chi}_{A,3}^{2} are determined from (A10) under the conditions \dot{\theta}={w}_{A}{\dot{\widehat{\theta}}}_{A}+{w}_{B}{\dot{\widehat{\theta}}}_{B}, {\sigma}_{g}^{2}={\sigma}_{{g}_{R\mathsf{\text{A}}}}^{2}, and {\sigma}_{n}^{2}={N}_{0}{B}_{3};{B}_{3} the bandwidth of the receive filter in Phase III. As in the case of PSPF, the conditional BER is expressed in the form

\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A},{\dot{\theta}}_{B}=\frac{\pi h}{T},{\dot{\widehat{\theta}}}_{A},{\dot{\widehat{\theta}}}_{B}\right]=\frac{-{p}_{1}}{{p}_{2}-{p}_{1}}\cdot \frac{{z}_{1}{z}_{2}}{\left({p}_{2}-{Z}_{1}\right)\left({p}_{2}-{Z}_{2}\right)}+\frac{-{Z}_{1}}{{z}_{2}-{Z}_{1}}\cdot \frac{{p}_{1}{p}_{2}}{\left({Z}_{2}-{p}_{1}\right)\left({Z}_{2}-{p}_{2}\right)}.

(35)

The only difference between (35) and (31) is that the former is conditioned on the hard decisions {\dot{\widehat{\theta}}}_{A} and {\dot{\widehat{\theta}}}_{B} made at the relay, while the latter is based on the soft decisions {\dot{\psi}}_{R,1} and {\dot{\psi}}_{R,1}. If we let *P*_{
e, A
} and *P*_{
e, B
} be the probabilities that the relay makes a wrong decision about *A* and *B*'s data, respectively, then the unconditional BER is

\begin{array}{c}{P}_{b}=\frac{1}{2{N}_{w}}\sum _{{d}_{A}=-1}^{+1}\sum _{\left\{{w}_{A},{w}_{B}\right\}}\left\{\left(1-{P}_{e,A}\right)\left(1-{P}_{e,B}\right)\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T,{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\widehat{\theta}}}_{A}={\dot{\theta}}_{A},{\dot{\widehat{\theta}}}_{B}={\dot{\theta}}_{B}\right]\right.\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}+\left(1-{P}_{e,A}\right){P}_{e,B}\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T,{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\widehat{\theta}}}_{A}={\dot{\theta}}_{A},{\dot{\widehat{\theta}}}_{B}=-{\dot{\theta}}_{B}\right]\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}+{P}_{e,A}\left(1-{P}_{e,B}\right)\mathsf{\text{Pr}}\left[D<0|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T,{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\widehat{\theta}}}_{A}=-{\dot{\theta}}_{A},{\dot{\widehat{\theta}}}_{B}={\dot{\theta}}_{B}\right]\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\left(\right)close="\}">+{P}_{e,A}{P}_{e,B}\mathsf{\text{Pr}}\left[D0|{\dot{\theta}}_{A}=\pi h{d}_{A}\u2215T,{\dot{\theta}}_{B}=\pi h\u2215T,{\dot{\widehat{\theta}}}_{A}=-{\dot{\theta}}_{A},{\dot{\widehat{\theta}}}_{B}=-{\dot{\theta}}_{B}\right]\end{array}\n

(36)

where *N*_{
w
} = 1 for 3-DPF and *N*_{
w
} = 2 for A4-DPF, and the inner summation is over the two different permutations of *w*_{
A
} and *w*_{
B
} in (13). It should be pointed out the error probabilities *P*_{
e, A
} and *P*_{
e, B
} can be determined by integrating the marginal pdf in (A6) from -∞ to 0 when the data bit is a + 1, or from 0 to +∞ when the data bit is a -1. The end result is of the form [15, 21]

{P}_{e,i}=\frac{1}{2}\left(1-\left|{\rho}_{i}\right|\right);\phantom{\rule{1em}{0ex}}i=A,B,

(37)

where |*ρ*_{
A
} | and |*ρ*_{
B
} | are |*ρ*| in (A5) obtained under the conditions {\sigma}_{g}^{2}={\sigma}_{{g}_{AR}}^{2},\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{n}^{2}={N}_{0}{B}_{12} and {\sigma}_{g}^{2}={\sigma}_{{g}_{BR}}^{2},{\sigma}_{n}^{2}={N}_{0}{B}_{12}, respectively.