Abstract

We introduce and simulate a technique enabling to utilize the polarization dimension in direct-detection optical transmission, supporting polarization multiplexing (POL-MUX) over direct-detection (DD) methods previously demonstrated for a single polarization such as direct-detection OFDM. POL-MUX is currently precluded in self-coherent DD with remotely transmitted pilot, as signal x pilot components may randomly fade out. We propose POL-MUX transmission of advanced modulation formats, such as 16-QAM and higher, by means of a novel low-complexity photonic integrated optical front-end and adaptive 3x2 MIMO DSP. The principle of operation is as follows: an additional X x Y cross-polarizations signal is generated, providing three projections onto an over-complete frame of three dependent vectors. This enables to resiliently reconstruct the received state of polarization even when the remotely transmitted pilot fades along one of the received polarization axes.

© 2013 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Asymmetric direct detection of orthogonal offset carriers assisted polarization multiplexed single-sideband signals

Xueyang Li, Maurice O’Sullivan, Zhenping Xing, Md Samiul Alam, Thang Hoang, Meng Xiang, Mingyue Zhu, Jinsong Zhang, Eslam Elfiky, and David V. Plant
Opt. Express 28(3) 3226-3236 (2020)

Direct detection of polarization multiplexed single sideband signals with orthogonal offset carriers

Yixiao Zhu, Mingxuan Jiang, and Fan Zhang
Opt. Express 26(12) 15887-15898 (2018)

Spectrally efficient polarization multiplexed direct-detection OFDM system without frequency gap

Chia-Chien Wei, Wei-Siang Zeng, and Chun-Ting Lin
Opt. Express 24(2) 1823-1828 (2016)

References

  • View by:
  • |
  • |
  • |

  1. B. J. C. Schmidt, Z. Zan, L. B. Du, and A. J. Lowery, “120 Gbit/s Over 500-km Using Single-Band Polarization-Multiplexed Self-Coherent Optical OFDM,” J. Lightwave Technol. 28(4), 328–335 (2010).
    [Crossref]
  2. D. Hsu, C. Wei, H. Chen, C. Song, I. Lu, and J. Chen, “74. 4% SSII Cancellation in an EAM-based OFDM-IMDD Transmission System” in OFC 2013, 2013, p. OM2C.7.
  3. J. Leibrich, A. Ali, and W. Rosenkranz, “Decision Feedback Compensation of Transmitter / Receiver Nonlinearity for DD-OFDM” in European Conference of Optical Communication (ECOC) (2011), p. We.8.A.5.
    [Crossref]
  4. D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108 Gb/s OFDMA-PON With Polarization Multiplexing and Direct Detection,” J. Lightwave Technol. 28(4), 484–493 (2010).
    [Crossref]
  5. C. C. Wei, C.-T. Lin, C.-Y. Wang, and F.-M. Wu, “A Novel Polarization Division Multiplexed OFDM System with a Direct-detection BLAST-Aided Receiver” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013(2013), p. JTh2A.49.
    [Crossref]
  6. N. Cvijetic, N. Prasad, M. Cvijetic, and T. Wang, “Cvijetic Efficient and Robust MIMO DSP Equalization in POLMUX OFDM transmission with direct-detection” in ECOC 2011(2011).
  7. N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
    [Crossref]
  8. J. Rahn, G. Goldfarb, H.-S. Tsai, W. Chen, S. Chu, B. Little, J. Hryniewicz, F. Johnson, C. Wenlu, T. Butrie, J. Zhang, M. Ziari, J. Tang, A. Nilsson, S. Grubb, I. Lyubomirsky, J. Stewart, R. Nagarajan, F. Kish, and D. F. Welch, “Low-Power, Polarization Tracked 45.6 GB/s per Wavelength PM-DQPSK Receiver in a 10-Channel Integrated Module” in OFC 2010(2010), p. OThE2.
  9. M. Nazarathy and A. Agmon, “Doubling Direct-detection Data Rate by Polarization Multiplexing of 16-QAM without a Polarization Controller” in ECOC 2013(2013), p. Mo.4.C.4.
  10. A. Agmon, M. Nazarathy, D. M. Marom, S. Ben-Ezra, A. Tolmachev, R. Killey, P. Bayvel, L. Meder, M. Hübner, W. Meredith, G. Vickers, P. C. Schindler, R. Schmogrow, D. Hillerkuss, W. Freude, and J. Leuthold, “Bi-directional Ultra-dense Polarization-muxed/diverse OFDM/WDM PON with Laserless Colorless 1Gb/s ONUs Based on Si PICs and <417 MHz mixed-signal ICs” in OFC 2013(2013), p. OTh3A.6.
  11. O. Christensen, Frames and Bases. Birkhauser, 2008.
  12. B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
    [Crossref]
  13. B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
    [Crossref]
  14. Z. Dong, X. Li, J. Yu, and J. Yu, “Generation and transmission of 8 × 112-Gb/s WDM PDM-16QAM on a 25-GHz grid with simplified heterodyne detection,” Opt. Express 21(2), 1773–1778 (2013).
    [Crossref] [PubMed]

2013 (1)

2011 (2)

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

2010 (3)

Cvijetic, N.

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108 Gb/s OFDMA-PON With Polarization Multiplexing and Direct Detection,” J. Lightwave Technol. 28(4), 484–493 (2010).
[Crossref]

Dong, Z.

Du, L. B.

Filsinger, V.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Hu, J.

Koch, B.

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Li, X.

Lowery, A. J.

Mirvoda, V.

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Noe, R.

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

Noé, R.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Prasad, N.

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

Puntsri, K.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Qian, D.

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108 Gb/s OFDMA-PON With Polarization Multiplexing and Direct Detection,” J. Lightwave Technol. 28(4), 484–493 (2010).
[Crossref]

Sandel, D.

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

Schmidt, B. J. C.

Wang, T.

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108 Gb/s OFDMA-PON With Polarization Multiplexing and Direct Detection,” J. Lightwave Technol. 28(4), 484–493 (2010).
[Crossref]

Yu, J.

Zan, Z.

Electron. Lett. (1)

B. Koch, R. Noe, V. Mirvoda, and D. Sandel, “100-krad/s Endless Polarisation Tracking with Miniaturised Module Card,” Electron. Lett. 47(14), 813–814 (2011).
[Crossref]

IEEE Photon. Technol. Lett. (2)

N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011).
[Crossref]

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010).
[Crossref]

J. Lightwave Technol. (2)

Opt. Express (1)

Other (8)

J. Rahn, G. Goldfarb, H.-S. Tsai, W. Chen, S. Chu, B. Little, J. Hryniewicz, F. Johnson, C. Wenlu, T. Butrie, J. Zhang, M. Ziari, J. Tang, A. Nilsson, S. Grubb, I. Lyubomirsky, J. Stewart, R. Nagarajan, F. Kish, and D. F. Welch, “Low-Power, Polarization Tracked 45.6 GB/s per Wavelength PM-DQPSK Receiver in a 10-Channel Integrated Module” in OFC 2010(2010), p. OThE2.

M. Nazarathy and A. Agmon, “Doubling Direct-detection Data Rate by Polarization Multiplexing of 16-QAM without a Polarization Controller” in ECOC 2013(2013), p. Mo.4.C.4.

A. Agmon, M. Nazarathy, D. M. Marom, S. Ben-Ezra, A. Tolmachev, R. Killey, P. Bayvel, L. Meder, M. Hübner, W. Meredith, G. Vickers, P. C. Schindler, R. Schmogrow, D. Hillerkuss, W. Freude, and J. Leuthold, “Bi-directional Ultra-dense Polarization-muxed/diverse OFDM/WDM PON with Laserless Colorless 1Gb/s ONUs Based on Si PICs and <417 MHz mixed-signal ICs” in OFC 2013(2013), p. OTh3A.6.

O. Christensen, Frames and Bases. Birkhauser, 2008.

D. Hsu, C. Wei, H. Chen, C. Song, I. Lu, and J. Chen, “74. 4% SSII Cancellation in an EAM-based OFDM-IMDD Transmission System” in OFC 2013, 2013, p. OM2C.7.

J. Leibrich, A. Ali, and W. Rosenkranz, “Decision Feedback Compensation of Transmitter / Receiver Nonlinearity for DD-OFDM” in European Conference of Optical Communication (ECOC) (2011), p. We.8.A.5.
[Crossref]

C. C. Wei, C.-T. Lin, C.-Y. Wang, and F.-M. Wu, “A Novel Polarization Division Multiplexed OFDM System with a Direct-detection BLAST-Aided Receiver” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013(2013), p. JTh2A.49.
[Crossref]

N. Cvijetic, N. Prasad, M. Cvijetic, and T. Wang, “Cvijetic Efficient and Robust MIMO DSP Equalization in POLMUX OFDM transmission with direct-detection” in ECOC 2011(2011).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Received spectra for a self-heterodyne link with pilot and spectral gap. (a). Spectrum of the received optical field, comprising a Pilot (P) and modulated Data (D), partitioned into two spectral slots (the transmitted field spectrum is similar). The fields are taken here as scalar, for simplicity, but more generally the P and D terms may be resolved in X and Y polarization components.(b). Spectrum of the directly-detected photo-current, comprising three mixing terms. The desired term is PxD, falling in the [W,2W] spectral band. The PxP term is at DC, whereas the DxD spurious term is at baseband, [-W,W]. The spectral guardband ensures that the spurious PxP and DxD terms are spectrally disjoint from the desired PxD term, which may be electrically filtered out and demodulated to baseband.
Fig. 2
Fig. 2 Self-heterodyne receiver options. (a) Conventional “reference” polarization-multiplexed direct-detection (laserless) receiver with transmitted optical pilot and spectral guard-band, suited to detect polarization diversity signals. The polarization is separated into its orthogonal components which are electrically demodulated to baseband and A/D converted. After some pre-DSP, the two polarization signals are 2x2 MIMO processed in order to demultiplex the two polarizations, followed by some post-DSP. This Rx functions intermittently in the POL-MUX mode, due to the pilot polarization fading impairment. However, this receiver may be adapted to be reliably used in the POL-DIV mode, by transmitting the same data on both polarizations and replacing the 2x2 MIMO by a 2x1 coherent “maximal ratio” combiner of the two polarizations. (b): Novel self-heterodyne POL-MUX receiver. Robustness to the polarization fading impairment is achieved by adding a third “X x Y” redundant measurement of the state of polarization. The 2x2 MIMO processing of the Rx in (a) is now replaced by 3x2 MIMO processing in order to recover the two independently transmitted polarization components. Abbreviations: OS: Optical Splitter, ODC: Optical Directional Coupler, EQZ: Equalizer
Fig. 3
Fig. 3 Frames graphical representations. Assuming that the pilot is received linearly polarized, we plot here in 2 the triplets of frame vectors, for various settings of the pilot polarization angle, θ p π 8 , in the range 0 θ p π/2 . The angle θ p parameterizing each frame is indicated by a black marker arrow pointing inward on the circle circumference. For the extreme angles θ p =0,π/2 , corresponding to the respective frames in the upper-left and lower right corners (surrounded by rounded squares), the triplet of frame vectors degenerates to a pair of vectors. Actually, at these pilot angles, there occurs fading of the vertical (horizontal) POL component, V ˜ =0( U ˜ =0) , such that the corresponding blue (red) vector vanishes, but the missing vector is supplanted by the green vector corresponding to the W ˜ measurement, which is always perpendicular to the surviving vector, such that there are left two orthogonal vectors, the green one and the red (blue) one, forming in these extreme cases an orthogonal base (whereas in all other cases the processing is based on projecting the vector of the three measurements onto a proper frame of three analysis vectors). The “best” frame, is the one at θ p =π/4 , as depicted in the center of the array, providing the highest processing gain. An interesting geometric property is that the angle θ p , as indicated by the black marker and the angle made by the green vector, corresponding to the new measurement, W ˜ , are symmetrical around a ray at π/4 .
Fig. 4
Fig. 4 Simulated short-reach, 117 km, optically amplified POL-MUX OFDM link. Link parameters are marked in the figure. Independent OFDM signals are transmitted onto the two POLs with equal powers of 2 dBm each, 5 dBm total power, which equals the total pilot power, which is also equi-partitioned between the two orthogononal pilot POL components, each at 2 dBm, as the pilot is launched at an angle of 45º relative to the Tx principal axes. The polarization transformation in the fiber (not explicitly shown in the figure) is modeled as a rotation of the SOP by an angle α. Both optical amplifier noise and Rx thermal noise are accounted for. The assumed link parameters imply that the receiver operates in the beat-noise limited regime, i.e., thermal noise is negligible relative to the SOA induced signal x ASE noise. An optical filter is further assumed at Rx input, bringing the ASE x ASE noise term to a negligible level.
Fig. 5
Fig. 5 Received constellations for the Rx of Fig. 2(b) on the link of Fig. 4, for a received LO SOP aligned with the X-axis and the Tx data launched with equal powers in its X,Y POLs. (a,b,c): Input ports X,Y, XxY of the 3x2 MIMO equalizer, as generated by the three optical front-end measurements. The PxD signal is seen to totally fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs(detected at a ± 45 relative to the P direction). Nevertheless, the third branch signal (c), (itself a diverse mixture of the transmitted X, Y POLs) enables recovery of the two transmitted X and Y 16-QAM constellations (e,f), by adaptive 3x2 MIMO. The three vectors frame degenerates in this case to an orthogonal base of two vectors, as shown in (d).
Fig. 6
Fig. 6 Received constellations for the Rx of Fig. 2(b) on the link of Fig. 4, for a received LO SOP as specified in (d) for the set of LHS sub-figures, namely received at + 22.5 with respect to the X-axis and in (k) for the set of RHS sub-figures ( + 45 ). (a,b,c): Input ports X,Y, XxY of the 3x2 MIMO equalizer, as generated by the three optical front-end measurements. The PxD signal is seen to partially fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs. Nevertheless, the power of the third branch signal (c), is maintained, preventing noise enhancement while recovering the transmitted X and Y 16-QAM constellations (e,f), by adaptive 3x2 MIMO.
Fig. 7
Fig. 7 Received constellations for a naïve 2x2 MIMO Rx (Fig. 2(a)) over the link of Fig. 4, conventionally processing just the 2 PBS output signals, lacking the proposed XxY cross-POL signal. Pilot SOPs with regard to PBS X axis are: + 22.5 in sub-figures (a-d) incurring moderate fading conditions as in Fig. 6. Cases (e-h) correspond to no fading, as the pilot SOP is received at + 45 . These constellations are to be compared with the improved ones of Figs. 6,7 for the novel 3x2 Rx.(a,b): Input ports X,Y of the 2x2 MIMO equalizer, as generated by the two optical front-end measurements. The PxD signal is seen to partially fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs. (d,e): adaptive 2x2 MIMO equalizer outputs for the X and Y ports. The measured SNR is 18.4/15.5 dB for X/Y respectively, lower than that attained with the novel 3x2 receiver at the same pilot SOP, as shown in Fig. 6(e), 6(f), (SNR there is 20.4/18.3 dB). Evidently, even moderate fading conditions incur significant SNR degradation for the naïve 2x2 Rx. (e-h): constellations for received SOP at + 45 . Compared to the SNR for proposed 3x2 Rx in Fig. 6(l), 6(m), the SNR attained with the conventional 2x2 Rx is lower by 1.0 dB, even in this case which is most favorable for conventional 2x2 detection.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

d X|Y (t)= d ˜ X|Y (t) e j2π( ν p +1.5W )t
p X|Y (t)= p ˜ X|Y e j2π ν p t .
u(t)= | d ˜ X (t)+ p ˜ X | BPF 2 Re{ d ˜ X (t) p ˜ X * e j3πWt } v(t)= | d ˜ Y (t)+ p ˜ Y | BPF 2 Re{ d ˜ Y (t) p ˜ Y * e j3πWt }
U ˜ [t]= p ˜ X * d ˜ X = p 0 cos θ p d ˜ X ; V ˜ [t]= p ˜ Y * d ˜ Y = p 0 e j ϕ p sin θ p
[ p ˜ X , p ˜ Y ]= p 0 [ cos θ p , e j ϕ p sin θ p ]
U ˜ [t]= p ˜ X * d ˜ 0 = p 0 cos θ p d ˜ 0 ; V ˜ [t]= p ˜ Y * d ˜ 0 = p 0 e j ϕ p sin θ p d ˜ 0
r ˜ [t] c ˜ u U ˜ [t]+ c ˜ v V ˜ [t]= c ˜ u p 0 cos θ p d ˜ 0 + c ˜ v p 0 e j ϕ p sin θ p d ˜ 0 = r ˜ [t]= p 0 d ˜ 0 ( cos 2 θ p + sin 2 θ p )= p 0 d ˜ 0
w(t)= 1 4 { | E ˜ X×Y + (t) | 2 | E ˜ X×Y (t) | 2 } BPF =Re { E ˜ X E ˜ Y * } BPF =Re {( d ˜ X e j3πWt + p ˜ X ) ( d ˜ Y e j3πWt + p ˜ Y ) * } BPF =Re {( d ˜ X e j3πWt + p ˜ X ) ( d ˜ Y e j3πWt + p ˜ Y ) * } BPF =Re{ d ˜ X p ˜ Y * e j3πWt }+Re{ d ˜ Y p ˜ X * e j3πWt }=Re{[ d ˜ X p ˜ Y * + d ˜ Y p ˜ X * ] e j3πWt }
W ˜ [t]= p ˜ Y * d ˜ X [t]+ p ˜ X * d ˜ Y [t]
[ U ˜ W ˜ V ˜ ] R ˜ = [ p ˜ X * 0 p ˜ Y * p ˜ X * 0 p ˜ Y * ] P [ d ˜ X d ˜ Y ] d = [ cos θ p 0 e j ϕ p sin θ p cos θ p 0 e j ϕ p sin θ p ] P [ d ˜ X d ˜ Y ] d ˜
P [2×3] = ( P [2×3] P [3×2] ) 1 P [2×3]
G [2×2] P [2×3] P [3×2] = p 0 [ 1 e j ϕ p cos θ p sin θ p e j ϕ p cos θ p sin θ p 1 ]
G [2×2] 1 ( P [2×3] P [3×2] ) 1 = 8 p 0 1 7+cos[4 θ p ] [ 1 e j ϕ p cos θ p sin θ p e j ϕ p cos θ p sin θ p 1 ]
P [2×3] = ( P [2×3] P [3×2] ) 1 P [2×3] = 8 p 0 1 7+cos[4 θ p ] [ cos θ p e j ϕ p sin 3 θ p e j2 ϕ p cos θ p sin 2 θ p e j ϕ p cos 2 θ p sin θ p cos 3 θ p e j ϕ p sin θ p ]
d ^ ˜ [2×1] = P [2×3] R ˜ [3×1] = U ˜ | p (1) + W ˜ | p (2) + V ˜ | p (3)
d ^ ˜ [2×1] = P [2×3] R [3×1] = P [2×3] P [3×2] d ˜ [2×1] = 1 [2×2] d ˜ [2×1] = d ˜ [2×1]
W ˜ = p ˜ Y * d ˜ X [t]+ p ˜ X * d ˜ Y [t] | p ˜ Y =0 = p X * d ˜ Y = p 0 d ˜ Y .
d ^ ˜ [2×1] = p 0 1 [ 1 0 0 0 1 0 ] P [2×3] [ U ˜ W ˜ 0 ] R ˜ = p 0 1 [ U ˜ W ˜ ]= p 0 1 [ p 0 d ˜ X p 0 d ˜ Y ]=[ d ˜ X d ˜ Y ]
p (1) |=[ cos θ p ,0 ], p (2) |=[ sin θ p ,cos θ p ], p (3) |=[ 0,sin θ p ]
d ˜ [ d ˜ X d ˜ Y ]=J d ˜ Tx =[ J XX J XY J YX J YY ][ d ˜ X Tx d ˜ Y Tx ].
d ^ ˜ Tx = J 1 d ^ ˜ = J [2×2] 1 P [2×3] R [3×1] = C [2×3] R [3×1]
C [2×3] J [2×2] 1 P [2×3] ,

Metrics