Since the key concept of the PDMA-PM is based on the use of PS, we are revisiting the polarization in EM waves in this subsection. When representing a completely polarized EM wave in a right-handed xyz Cartesian coordinate system with the z coordinate representing the direction of propagation and the xy coordinates be orthogonal basis defined by a pair of horizontally (H) and vertically (V) polarized unit vectors, the signal by ignoring the absolute phase can be described as the following equation 19

E ( t ) = E H ( t ) E V ( t ) = E cos ε exp ( j ω t ) E sin ε exp [ j ( ω t + δ ) ]

where ε is the amplitude relationship (or polarized angle) between the two components of the signal, δ represents the difference of them in phase (or phase difference in polarized angle), in addition, ω is the center frequencies of the signal, and E denotes the envelope.

The PS is determined by the amplitude and phase relationship between the two components received by the ODPA. Then the two parameters can be derived as

ε = arctan | E v ( t ) | | E H ( t ) |

δ = arg { E H ( t ) } - arg { E V ( t ) }

where arg{x} indicates the argument of vector x. After being radiated by the ODPA, the EM wave may produce the so-called cross-polarization (XP) component whose polarization is orthogonal to the original transmitting polarization, and this is the effect of depolarization 22 . The degree of depolarization is described by cross-polarization discrimination (XPD) which indicates the ratio of the original transmitted polarization component power and cross-polarization component power at the same location point. Depolarization effect can be eliminated by introducing depolarization compensation technology 32 . There are also many methods for estimation of PS (see, for example, 33 34 ). We assume that the polarized state is already known at the receiver in this paper. Since we are considering the scenario of LOS, then the PS is not changed at a high probability. The phenomenon of PS changes due to the reflecting, scattering, and diffracting by objects is outside of scope in this paper. Therefore, we only consider its application in LOS environment.

Polarized angle and phase difference in polarized angle determine the polarized state of the signal as follows: (1) If δ = 0, π, the polarization is linear. (2) δ = - π 2 , ε = π 4 indicates the right-handed circular polarization, and δ = - π 2 , ε = π 4 shows the left-handed circular polarization. (3) If δ ∈ (0, 90°) and ε ∈ (0, 90°), then the polarization is left-handed elliptic. When δ ∈ (-90°, 0) and ε ∈ (0, 90°), the polarization is right-handed elliptic.

We can also denote the PS on the Poincare spherical surface as shown in Figure 2, where the north pole (green point) is the left-handed circular polarization, and the south pole (blue point) represents the right-handed circular polarization, then the equator (red circle) indicates the linear polarization. The surface of the northern semi-sphere (gray color but excluding the north pole and equator) denotes the left-handed elliptic polarization, while the surface of the southern semi-sphere (yellow color but excluding the south pole and equator) denotes the right-handed elliptic polarization. It establishes the one-to-one mapping between the polarized state and each point on the Poincare-spherical surface, and there are infinite PSs intuitively.

In Figure 3, the orthogonal polarization and the cross polarization are illustrated. Two points on the surface of the sphere are potential orthogonal polarization representations if the line joining them passes through the center of the sphere, for example the pairs (P1, P2), (Q1, Q2), (M1, M2), and (L, R), as shown in Figure 3. Contrarily the two points on the surface of the sphere are cross polarization representations if they are symmetrical about the surface of equator, such as pairs (P1, Q1), (P2, Q2) and (L, R). It can be found that the orthogonal polarization and the cross polarization are the same for circular polarization.

As depicted in Figures 2 and 3, bit information can be carried by the PS, and the PS is determined by the amplitude and phase of the two components radiated by ODPA. Hence, each user can use certain PS when it wants to transmit information to the receiver. For example, circular polarization is allocated to some user, if the polarization is a left-handed circular one, then bit 1 is transmitted, and the right-handed circular polarization carries bit 0. This binary modulation scheme can be extended by exploiting different polarizations. We can also use the vertical polarization and the horizontal polarization or +45° linear polarization and -45° linear polarization to carry binary information. We can see that the transmitter complexity is simple.

Figure 4 shows a circle with the center of the sphere as its center, since the orthogonal polarization and cross polarization are on the same circle. For simplicity, circular polarization representations are on this circle. Then we can allocate different polarizations for different users. Taking User4 in Figure 2 for example, linear polarization is allocated to User4, and the bit information can be carried by the two orthogonal linear PS (yellow pints) as shown in the figure. Other users can also use the linear polarization by properly adjusting the polarized angles. It can be found that it is an orthogonal modulation scheme for binary modulation in PDMA-PM systems since the PSs for bits 1 and 0 are orthogonal, while different users hold non-orthogonal polarizations with each other.

Binary modulation can be easily extended to quaternary modulation by introducing cross-polarization. Figure 5 illustrates the constellation of the quaternary modulation. For User1, four blue points represent bits 00, 01, 11, 10, respectively, and the polarization for 00 and 11 is a pair of orthogonal polarization, the same for 01 and 10. The polarization for 00 and 10 is a pair of cross-polarization, the same for 01 and 11. Different users can be separated by rotating the rectangle with different angles.

Other high-order modulation schemes can be extended in a straightforward way. It is easy to find that we can also combine the PDMA with conventional FDMA and OFDMA, for example, some users share the same frequency band and other users share another frequency band. The multiple access technique of PDMA-FDMA is easy to implement.

Since the polarization is nothing with the specific waveform, bandwidth, rate, then the proposed PDMA-PM scheme is not strict with the waveform design, band-width limitation and other restrictions. Hence, it is expected to improve the spectrum efficiency, while the perfect implementation relies on the precise polarization measurement.

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 ( t ) = P i 1 , P i 2 , … , P i N g ( t )

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 21 22 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 ( U H P S ⊥ U ) - 1 U H P S ⊥ = 1 γ A , B C , D

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

γ = ( cos ε i 1 sin ε i 2 ) 2 + ( sin ε i 1 cos ε i 2 ) 2 - 2 sin ε i 1 sin ε i 2 cos ε i 1 cos ε i 2 cos ( δ i 2 - δ i 1 )

A = ( sin ε i 2 cos ε i 1 ) 2 - sin ε i 2 cos ε i 2 sin ε i 1 cos ε i 1 e j ( δ i 2 - δ i 1 )

B = cos 2 ε i 2 sin ε i 1 cos ε i 1 e - j δ i 1 - cos 2 ε i 1 sin ε i 2 cos ε i 2 e - j δ i 2

C = sin 2 ε i 2 sin ε i 1 cos ε i 1 e j δ i 1 - sin 2 ε i 1 sin ε i 2 cos ε i 2 e j δ i 2

D = ( sin ε i 1 cos ε i 2 ) 2 - sin ε i 2 cos ε i 2 sin ε i 1 cos ε i 1 e j ( δ i 1 - δ i 2 )

respectively.

It is easy to prove that

E r ( t ) = P i 1 g ( t ) = s 1 ( t )

since E P _i1 = P _i1 and E P _i2 = 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 _i1 is

SI R i 1 = 2 0 lg | E i 1 | | E i 2 |

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

SI R o = 2 0 lg | E i 1 ( cos ε i 1 B 1 + sin ε i 1 C 1 ) | | E i 2 ( cos ε i 2 B 1 + sin ε i 2 C 1 ) |

The gain of SIR is

Δ SIR = SI R o - SI R i = 2 0 lg | sin ( ε i 1 - ε i 2 - Δ ε i 2 ) | | sin ( Δ ε - i 2 | )

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 _i1 - ε _i2, and the error deviation on polarized angle of interference, i.e., Δε _i2. The larger the ε _i1 - ε _i2 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 ( S H R x x † U ) - 1 U H R x x †

Where

R x x = R r r - σ 2 E = [ U , S ] R U U 0 0 R S S [ U , S ] H

and

R r r = E { r r H } = E { [ ∑ n = 1 N s n + n ] [ ∑ n = 1 N s n + n ] H } = E { ( x + n ) ( x + n ) H } = R x x + σ 2 E

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

Δ S I R = S I R o - S I R i (1) = 2 0 l g | Δ σ 2 σ 2 - E i 2 2 σ 2 sin 2 ( ε i 1 - ε i 2 ) | | Δ σ 2 σ 2 cos ( ε i 1 - ε i 2 ) | (2) (3)

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 0 1 ( P 0 1 H R x x † P 0 1 ) - 1 P 0 1 H R x x †

as

E 1 = P 1 1 ( P 1 1 H R x x † P 1 1 ) - 1 P 1 1 H R x x †

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 ( P 1 1 + P i 2 ) = P 1 1

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 _i2 is suppressed, since

E 0 ( P 1 1 + P i 2 ) ≠ P 1 1

H ₀ and H ₁ are the operators of PF, and they meet the conditions

H 0 ∙ P 0 1 = 0 H 0 ∙ P 1 1 = 1

H 1 ∙ P 1 1 = 0 H 1 ∙ P 0 1 = 1

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.

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.

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 4 E b ∥ H 0 T H 0 ∥ σ 2

where m = y_a - y_b , and E_b is the energy of one bit. The coefficient of | | H 0 T H 0 | | 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).

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.

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.