In this section, some detailed analysis and mathematical deduction are done to demonstrate the PDMA-PM. Here, we take the binary modulation as an example, and analyze its multiuser separation, demodulation and BER performance. Especially, we assume that all of the polarization states allocated to different users are available at the receiver.

Figure 6 depicts the simple block diagram of the transmitter. Consider there are *N* users in the system, the transmitting signal for the *n* th user is derived as

{s}_{n}\left(t\right)={P}_{in}g\left(t\right)

(4)

where *P*_{
in
} (*i* = 0, 1, *n* = 1, 2, ..., *N*) shows the PS representing bit 1 or bit 0 for the *n* th user, and *P*_{
in
} = [cos *ε*_{
in
} , sin *ε*_{
in
} exp(*j δ*_{
in
} )]^{T}, where cos *ε*_{0n}+ cos *ε*_{1n}= π*/* 2, *δ*_{0n}-*δ*_{1n}= *± π* . This is the sufficient and necessary condition for a pair of orthogonal polarization. *g*(*t*) is the transmitting waveform for all the users. Then the received signal *r*(*t*) is the sum of *N* users, i.e., r\left(t\right)={\sum}_{n=1}^{N}{s}_{n}\left(t\right)+n\left(t\right),\phantom{\rule{2.77695pt}{0ex}}n\left(t\right) is the additive white gaussian noise (AWGN) with zero mean and the variance of *σ*^{2}.

Figure 7 shows the block diagram of the receiver, where NPE is the estimation of the noise power, **E**_{0} and **E**_{1} are the oblique projection operators constructed by the polarization states of 0 and 1, respectively, to cancel the interference [20–23]. **H**_{0} and **H**_{1} are the operators of polarization filtering. *D* denotes the decision process, here decision is the comparison of the output value between the two branches. Detailed introduction is as follows.

### 3.1. Multiuser separation in PDMA-PM

Since there are *N* signals embedded in the received signal, in order to suppress the unwanted *N* - 1 signals, PF is adopted here. The received signal can be rearranged in the matrix form

r\left(t\right)=\left[{P}_{i1},{P}_{i2},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{P}_{iN}\right]g\left(t\right)

(5)

where *P*_{0n}, *P*_{1n}is a pair of orthogonal polarization.

Suppose the first user is the desired signal. In order to suppress other *N* - 1 signals, PF can be used. The authors proposed oblique projection PF (OPPF) technique in [20–23], we just briefly introduce the OPPF here.

For simplicity, consider there are two users (*N* = 2). We can construct a filter with the operator written as

E=U{\left({U}^{H}{P}_{S}^{\perp}U\right)}^{-1}{U}^{H}{P}_{S}^{\perp}=\frac{1}{\gamma}\left[\begin{array}{c}\hfill A,\phantom{\rule{1em}{0ex}}B\hfill \\ \hfill C,\phantom{\rule{1em}{0ex}}D\hfill \end{array}\right]

(6)

where **U** and **S** are the polarization states of the desired signal and interference, respectively. **P**_{
S
}^{⊥} is the orthogonal projection operator onto the complementary of **S**. Suppose the first and the second users are operating at the same time, then the elements of **E** are

\begin{array}{c}\gamma ={\left(cos{\epsilon}_{i1}sin{\epsilon}_{i2}\right)}^{2}+{\left(sin{\epsilon}_{i1}cos{\epsilon}_{i2}\right)}^{2}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-2sin{\epsilon}_{i1}sin{\epsilon}_{i2}cos{\epsilon}_{i1}cos{\epsilon}_{i2}cos\left({\delta}_{i2}-{\delta}_{i1}\right)\end{array}

(7)

A={\left(sin{\epsilon}_{i2}cos{\epsilon}_{i1}\right)}^{2}-sin{\epsilon}_{i2}cos{\epsilon}_{i2}sin{\epsilon}_{i1}cos{\epsilon}_{i1}{e}^{j\left({\delta}_{i2}-{\delta}_{i1}\right)}

(8)

B={cos}^{2}{\epsilon}_{i2}sin{\epsilon}_{i1}cos{\epsilon}_{i1}{e}^{-j{\delta}_{i1}}-{cos}^{2}{\epsilon}_{i1}sin{\epsilon}_{i2}cos{\epsilon}_{i2}{e}^{-j{\delta}_{i2}}

(9)

C={sin}^{2}{\epsilon}_{i2}sin{\epsilon}_{i1}cos{\epsilon}_{i1}{e}^{j{\delta}_{i1}}-{sin}^{2}{\epsilon}_{i1}sin{\epsilon}_{i2}cos{\epsilon}_{i2}{e}^{j{\delta}_{i2}}

(10)

D={\left(sin{\epsilon}_{i1}cos{\epsilon}_{i2}\right)}^{2}-sin{\epsilon}_{i2}cos{\epsilon}_{i2}sin{\epsilon}_{i1}cos{\epsilon}_{i1}{e}^{j\left({\delta}_{i1}-{\delta}_{i2}\right)}

(11)

respectively.

It is easy to prove that

Er\left(t\right)={P}_{i1}g\left(t\right)={s}_{1}\left(t\right)

(12)

since **E** *P*_{i 1}= *P*_{i 1}and **E** *P*_{i 2}= 0. This shows the interference can be suppressed totally by OPPF.

It is easy to find that if both the two user's PSs are available, then the OPPF can perfectly separate them. While if there is uncertainty with the other user's PS, i.e., there is estimation error in the PS of the other user, then the OPPF cannot separate them totally. In order to deeply understand the performance of the OPPF, we give the SIR analysis here.

The signal-to-interference (SIR) ratio of the received signal SIR_{i 1}is

\mathsf{\text{SI}}{\mathsf{\text{R}}}_{i1}=20\mathsf{\text{lg}}\frac{\left|{E}_{i1}\right|}{\left|{E}_{i2}\right|}

(13)

Since there is estimation error on polarized angle of interference, the OPPF operator is not an accurate one. Suppose the parameters in formulae (8-11) are *A*_{1}, *B*_{1}, *C*_{1} and *D*_{1}, the SIR after OPPF is

\mathsf{\text{SI}}{\mathsf{\text{R}}}_{o}=20\mathsf{\text{lg}}\frac{\left|{E}_{i1}\left(cos{\epsilon}_{i1}{B}_{1}+sin{\epsilon}_{i1}{C}_{1}\right)\right|}{\left|{E}_{i2}\left(cos{\epsilon}_{i2}{B}_{1}+sin{\epsilon}_{i2}{C}_{1}\right)\right|}

(14)

The gain of SIR is

\Delta \mathsf{\text{SIR}}=\mathsf{\text{SI}}{\mathsf{\text{R}}}_{o}-\mathsf{\text{SI}}{\mathsf{\text{R}}}_{i}=20\mathsf{\text{lg}}\frac{|sin\left({\epsilon}_{i1}-{\epsilon}_{i2}-\Delta {\epsilon}_{i2}\right)|}{|sin\left(\Delta \epsilon -i2|\right)}

(15)

The formula listed above shows that the gain of SIR ratio has relationship with the difference between polarized angles of the target signal and interference i.e., *ε*_{i 1}- *ε*_{i 2}, and the error deviation on polarized angle of interference, i.e., Δ*ε*_{i 2}. The larger the *ε*_{i 1}- ε_{i 2}is, the larger the ΔSIR can be obtained. In order to get good ΔSIR performance, the moderately large difference between the polarized angles of the target signal and interference should be designed.

If the polarization of interference is not available at the receiver, another form of OPPF can be written as

E=U{\left({S}^{H}{{R}_{xx}}^{\u2020}U\right)}^{-1}{U}^{H}{{R}_{xx}}^{\u2020}

(16)

Where

{R}_{xx}={R}_{rr}-{\sigma}^{2}E=\left[U,S\right]\phantom{\rule{2.77695pt}{0ex}}\left[\begin{array}{c}\hfill {R}_{UU}\phantom{\rule{1em}{0ex}}0\hfill \\ \hfill 0\phantom{\rule{1em}{0ex}}{R}_{SS}\hfill \end{array}\right]\phantom{\rule{2.77695pt}{0ex}}{\left[U,\phantom{\rule{2.77695pt}{0ex}}S\right]}^{H}

(17)

and

\begin{array}{c}{R}_{rr}=E\left\{r{r}^{H}\right\}\\ =E\left\{[{\displaystyle \sum _{n=1}^{N}{s}_{n}}+n]{[{\displaystyle \sum _{n=1}^{N}{s}_{n}}+n\text{]}}^{H}\right\}\\ =E\{(x+n){(}^{x+n)}H\}\\ ={R}_{xx}+{\sigma}^{2}E\end{array}

(18)

As shown, if the noise power is known exactly, then the OPPF can also separate the two users' signals. Here, we give the SIR analysis if the noise variance is not exactly estimated. The similar analysis of SIR enhancement can be obtained as

\begin{array}{lll}\hfill \Delta SIR\phantom{\rule{2.77695pt}{0ex}}& =SI{R}_{\mathsf{\text{o}}}\phantom{\rule{2.77695pt}{0ex}}-SI{R}_{\mathsf{\text{i}}}\phantom{\rule{2em}{0ex}}& \hfill \\ =20lg\frac{|\frac{\Delta {\sigma}^{2}}{{\sigma}^{2}}-\frac{{E}_{i2}^{2}}{{\sigma}^{2}}{sin}^{2}\left({\epsilon}_{i1}-{\epsilon}_{i2}\right)|}{|\frac{\Delta {\sigma}^{2}}{{\sigma}^{2}}cos\left({\epsilon}_{i1}-{\epsilon}_{i2}\right)|}\phantom{\rule{2em}{0ex}}& \hfill \\ \hfill \end{array}

(19)

Therefore, **E**_{0} is the oblique projection operator (OOP) which can remain the component of the polarization representing 0 while suppressing the interference due to the property of the OOP, and **E**_{1} can remain the component of the polarization representing 1 while suppressing the interference. As shown in Figure 5, the input signal (received signal) passes through both **E**_{0} and **E**_{1}, then the interference is canceled. This is the multiuser separation process. **E**_{0} and **E**_{1} for the first user can be designed

as

{E}_{0}={P}_{01}{\left({{P}_{01}}^{H}{{R}_{xx}}^{\u2020}{P}_{01}\right)}^{-1}{P}_{01}^{H}{{R}_{xx}}^{\u2020}

(20)

as

{E}_{1}={P}_{11}{\left({{P}_{11}}^{H}{{R}_{xx}}^{\u2020}{P}_{11}\right)}^{-1}{P}_{11}^{H}{{R}_{xx}}^{\u2020}

(21)

### 3.2. Demodulation in PDMA-PM

After the separation process, the interference is suppressed, and the next step is to demodulate the bit information. Suppose bit 1 is transmitted, i.e., *P*_{11}. It is easy to obtain

{E}_{1}\left({P}_{11}+{P}_{i2}\right)={P}_{11}

(22)

This shows that the polarization of the desired bit is invariable after **E**_{1}, while **E**_{0} is constructed by the polarization representing 0, then after **E**_{0}, the output is another vector even if *P*_{i 2}is suppressed, since

{E}_{0}\left({P}_{11}+{P}_{i2}\right)\ne {P}_{11}

(23)

**H**_{0} and **H**_{1} are the operators of PF, and they meet the conditions

{H}_{0}\bullet {P}_{01}=0\phantom{\rule{1em}{0ex}}{H}_{0}\bullet {P}_{11}=1

(24)

{H}_{1}\bullet {P}_{11}=0\phantom{\rule{1em}{0ex}}{H}_{1}\bullet {P}_{01}=1

(25)

where *·* denotes the dot product. These conditions show **H**_{0} is the orthogonal complementary of the polarization representing bit 0, and the same for **H**_{1}.

If bit 1 is transmitted, then after **E**_{1} and **H**_{1}, the output value will be zero since *P*_{11} is invariable after **E**_{1}, and it is suppressed after **H**_{1} due to the property. While the output after **E**_{0} and **H**_{0} is not zero since **H**_{0} is not orthogonal to the output vector after **E**_{0} when bit 1 is transmitted. Figure 8 shows the output value after **E**_{0} and **H**_{0} when bit 1 is transmitted, the absolute value is large than 0. If the transmitting energy is large, i.e., the signal-to-noise ratio (SNR) is high, then the absolute value is far away from 0 if they are not orthogonal.

Hence, we can select the minimum output value as the desired bit, i.e., when the output after **E**_{1} and **H**_{1} is the minimum value, then we can decide that 1 is transmitted, the same analysis for bit 0.

### 3.3. Impact of noise

For AWGN, its polarization is called non-polarized which means it holds all of PS, i.e., covers the whole Poincare-Sphere surface. In the analysis of demodulation, we use OPPF to separate multi user signals and extract bit information. From the signal processing point of view, the OPPF is a kind of notch filter that can set one or several notches for some PS, then those PS are nulled by these notches. From this sense, the noise component which holds these PS are suppressed at the same time which shows that this method is capable of denoising. But the answer is no, since according to the oblique projection theory [35], AWGN can be even amplified after oblique projection which can be demonstrated in Figure 9.

In this figure, subspace **H** is the desired subspace and **S** is the interfering subspace that need to null. Suppose they are not orthogonal then oblique projection can be used to cancel any components in **S** while keeping everything in **H**. Since AWGN lies in every subspace (it is full rank and rank is infinity), the component of AWGN in subspace **S** is suppressed while some other components lie in other subspaces and will be amplified after oblique projection as shown in Figure 9. We can determine its lower bound and upper bound by calculating the principal angles of the subspaces [35]. When subspaces **H** and **S** are orthogonal, then AWGN is not amplified since it is an orthogonal projection, and this is the lower bound. If the angle between **H** and **S** is zero which means they are the same subspace, then everything is nulled but noise, and this is the upper bound.

From the above analysis, if the principal angles of polarization subspace for multiple users are small, then the impact of noise will be significant. Even though we can set infinite PS for multi users, but from the noise impact point of view, this will lead to performance deterioration since the principal angles of large number of users are small.

### 3.4. BER performance of PDMA-PM

When bit 1 is transmitted, the error decision is made when the output of the first branch *y*_{
a
} is smaller than the second branch *y*_{
b
} , i.e., the output after **H**_{0} is smaller than that of **H**_{1}, then the probability of error is

{P}_{e}=Q\left(\sqrt{\frac{4{E}_{b}}{\parallel {H}_{0}^{T}{H}_{0}\parallel {\sigma}^{2}}}\right)

(26)

where *m* = *y*_{
a
} *- y*_{
b
} , and *E*_{
b
} is the energy of one bit. The coefficient of \left|\right|{H}_{0}^{T}{H}_{0}\left|\right| is due to the rule of oblique projection amplifying the noise which has been analyzed in the above section.

Simulation results of 5 users, 10 users 30 users and 60 users in binary modulation system are done, as shown in Figure 10. From the simulation results, we can see that the BER performance of large number of users (30 and 60) is much worse than few users (5 and 10). This is due to when there are a lot of users, the principal angles will be very small which can result in noise being largely amplified after oblique projection (demodulation).

### 3.5. Relationship with classic signal processing

From the signal processing point of view, to solve the problems of detection and estimation, the minimum variance unbiased (MVU) estimation, least squares (LS) estimation, and minimum mean square error (MMSE) estimation are the most important criterions to evaluate the performance of the estimator. For the optimal detection techniques, the classic representatives are matched filter (MF) and maximum likelihood (ML) estimation or maximum a posterior probability (MAP) which is equivalent with ML when different bit has equal probability. For AWGN channel, these techniques are optimal techniques, based on different assumptions and detailed algorithms under different application scenarios.

In the classical work on oblique projection [35] and our previous publications [20, 21], Behrens and Scharf and we have indicated that the OPP and our proposed OPPF scheme are both MVU estimation. When interference dominates noise, OPP will converge to LS. Due to the noise amplified phenomenon, the noise variance is different after oblique projection. For the signal detection, Behrens and Scharf showed that OPP is a generalized likelihood ratio (GLR) detector which is a generalized result on invariant detectors. From above analysis, the proposed OPPF is not a new thing, and it is still in line with the classical signal processing. For Cramer-Rao lower bound analysis, there are no obvious differences between those techniques. Moreover this is not our contribution, since Behrens and Scharf have already proved this in their pioneering work. The only difference is the noise amplification.

### 3.6. Comparison with classic modulation schemes

For phase-shift-keying (PSK), frequency-shift-keying (FSK), and amplitude-shift-keying (ASK) based modulation schemes, the common methodology is to use phase, frequency, and amplitude difference or their relationship to carry bit information. The combination of PSK and ASK can form quadrature amplitude modulation (QAM). These bit carriers are all from the characteristics or parameters of waveform, and they are highly determined by the signal waveform. Hence the waveform design is critical for these modulation schemes. For the proposed PM, the basic methodology is PS of EM waves, and it is independent of signal's parameters. Moreover, the transmitter and receiver design is quite different from them; for the classic PSK, ASK, FSK, and QAM schemes, it can be a baseband transmission, while for our proposed PM scheme, it must be an RF system, and the modulation process of PM scheme is easier than those schemes, since power splitter and phase shifter are enough for implementing bit modulation, as shown in Figure 6. The demodulation process for the proposed scheme is still an open issue, and we believe that the proposed OPPF for demodulation is not the only scheme. There must be other better schemes for PS based demodulation.

For performance analysis, since the proposed scheme is based on oblique projection which is sensitive to additive noise and highly dependent with the number of users which can be shown in Figures 9 and 10, the anti-noise performance is worse than those schemes. From this sense, the capacity of the network is limited.