Abstract

A fully integrated 4-bit true time delay device using polymer optical switches and waveguide delay lines is demonstrated. The fabricated device, which contains five 2×2 thermo-optic switches, has a maximum power consumption of 143  mW and a switching time less than 3  ms. The rf phase error, which is affected by the optical switch cross talk, is also theoretically analyzed and proved to be negligible by experimental results.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
    [CrossRef]
  2. Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
    [CrossRef]
  3. K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).
  4. J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
    [CrossRef]
  5. B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
    [CrossRef]
  6. Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
    [CrossRef]
  7. H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating deltaβ," IEEE J. Quantum Electron. 12, 396-401 (1976).
    [CrossRef]
  8. X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
    [CrossRef]
  9. Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
    [CrossRef]
  10. X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
    [CrossRef]
  11. A. A. Szep, "Polymer-based integrated optical waveguide true time delay for wide-band phased array antennas," Ph.D. dissertation (University of Southern California, 2002), pp. 17, 22-36.
  12. G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
    [CrossRef]

2006 (2)

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

2005 (1)

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

2004 (2)

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
[CrossRef]

2003 (2)

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

1998 (1)

Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
[CrossRef]

1995 (1)

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

1991 (1)

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

1976 (1)

H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating deltaβ," IEEE J. Quantum Electron. 12, 396-401 (1976).
[CrossRef]

Bernstein, N.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Chen, M. Y.

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

Chen, R. T.

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

Chen, Y.

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

Choi, H. J.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

DeNatale, J. F.

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

Hacker, J. B.

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

Han, S. G.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Horikawa, K.

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

Howley, B.

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

Hunziker, W.

Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
[CrossRef]

Jiang, Y.

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

Kim, B. G.

J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
[CrossRef]

Kim, J. M.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Kitoh, T.

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

Kogelnik, H.

H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating deltaβ," IEEE J. Quantum Electron. 12, 396-401 (1976).
[CrossRef]

Lai, Q.

Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
[CrossRef]

Lee, B. S.

J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
[CrossRef]

Lee, H. J.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Lee, J. J.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Melchior, H.

Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
[CrossRef]

Mihailovich, R. E.

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

Newberg, I.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Ng, W.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Noh, Y. O.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Ogawa, H.

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

Ogawa, I.

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

Rebeiz, G. M.

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

Schmidt, R. V.

H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating deltaβ," IEEE J. Quantum Electron. 12, 396-401 (1976).
[CrossRef]

Shi, Z.

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

Shin, J. D.

J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
[CrossRef]

Szep, A. A.

A. A. Szep, "Polymer-based integrated optical waveguide true time delay for wide-band phased array antennas," Ph.D. dissertation (University of Southern California, 2002), pp. 17, 22-36.

Tan, G. L.

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

Tangonan, L.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Walston, A.

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Wang, X.

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

Won, Y. H.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Yang, M. S.

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Zhao, F.

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating deltaβ," IEEE J. Quantum Electron. 12, 396-401 (1976).
[CrossRef]

IEEE MTT-S Int. Microwave Symp. Dig. (1)

K. Horikawa, I. Ogawa, H. Ogawa, and T. Kitoh, "Photonic switched true time delay beam forming network integrated on silica waveguide circuits," IEEE MTT-S Int. Microwave Symp. Dig. 1, 65-68 (1995).

IEEE Photon. Technol. Lett. (6)

J. D. Shin, B. S. Lee, and B. G. Kim, "Optical true time-delay feeder for X-band phased array antennas composed of 2 × 2 optical MEMS switches and fiber delay lines," IEEE Photon. Technol. Lett. 16, 1364-1365 (2004).
[CrossRef]

B. Howley, Y. Chen, X. Wang, and R. T. Chen, "2-bit reconfigurable true time delay lines using 2 × 2 polymer waveguide switches," IEEE Photon. Technol. Lett. 17, 1944-1946 (2005).
[CrossRef]

Q. Lai, W. Hunziker, and H. Melchior, "Low-power compact 2 × 2 thermooptic silica-on-silicon waveguide switch with fast response," IEEE Photon. Technol. Lett. 10, 681-683 (1998).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Crosstalk minimized polymeric 2 × 2 optical switch," IEEE Photon. Technol. Lett. 11, 16-18 (2006).
[CrossRef]

Y. O. Noh, J. M. Kim, M. S. Yang, H. J. Choi, H. J. Lee, Y. H. Won, and S. G. Han, "Thermo-optic 2 × 2 asymmetric digital optical switches with zero-voltage operation state," IEEE Photon. Technol. Lett. 16, 446-448 (2004).
[CrossRef]

Z. Shi, Y. Jiang, B. Howley, Y. Chen, F. Zhao, and R. T. Chen, "Continuously delay-time tunable-waveguide hologram module for x-band phased-array antenna," IEEE Photon. Technol. Lett. 15, 972-974 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. L. Tan, R. E. Mihailovich, J. B. Hacker, J. F. DeNatale, and G. M. Rebeiz, "Low-loss 2- and 4-bit TTD MEMS phase shifters based on SP4T switches," IEEE Trans. Microwave Theory Tech. 51, 297-304 (2003).
[CrossRef]

J. Lightwave Technol. (2)

X. Wang, B. Howley, M. Y. Chen, and R. T. Chen, "Polarization-independent all-wave polymer based TIR thermo-optic switch," J. Lightwave Technol. 24, 1558-1565 (2006).
[CrossRef]

W. Ng, A. Walston, L. Tangonan, J. J. Lee, I. Newberg, and N. Bernstein, "The first demonstration of an optically steered microwave phased array antenna using true-time-delay," J. Lightwave Technol. 9, 1124-1131 (1991).
[CrossRef]

Other (1)

A. A. Szep, "Polymer-based integrated optical waveguide true time delay for wide-band phased array antennas," Ph.D. dissertation (University of Southern California, 2002), pp. 17, 22-36.

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

(Color online) (a) Schematic of the 4-bit TTD device using TIR optical switches. (b) Schematic of the TIR optical switches.

Fig. 2
Fig. 2

(Color online) Chip die of the 4-bit TTD device with an enlarged view of the optical switch.

Fig. 3
Fig. 3

Switching speed measurement of the 4-bit TTD device.

Fig. 4
Fig. 4

(Color online) Four-bit TTD device, the last stage using (a) 2 × 1 combiner, (b) 2 × 2 optical switch.

Fig. 5
Fig. 5

Illustration of the rf signal phase error in the 4-bit TTD device.

Fig. 6
Fig. 6

(Color online) Simulated rf phase error caused by the optical switch cross talk.

Fig. 7
Fig. 7

(Color online) Simulation and experimental results of the rf phase error.

Tables (1)

Tables Icon

Table 1 Delay Configuration, Power Consumption, and RF Performance

Equations (6)

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

a : b = 10 ( CT / 10 ) : 1 .
E = E 0 e j ω ( C 1 e j ϕ 1 + C 2 e j ϕ 2 + + C 16 e j ϕ 16 ) .
I = E 0 ( C 1 e j ϕ 1 + C 2 e j ϕ 2 + + C 16 e j ϕ 16 ) × E 0 ( C 1 e j ϕ 1 + C 2 e j ϕ 2 + + C 16 e j ϕ 16 ) .
X i = j = 1 16 C i C j cos ( ϕ i ϕ j ) .
ϕ error , i = tan 1 j = 1 , j i 16 X j sin ( ϕ j ϕ i ) X i + j = 1 , j i 16 X j cos ( ϕ j ϕ i ) .
ϕ error , i = tan 1 b 2 j = 1 10 sin ( ϕ j ϕ i ) a 2 + b 2 j = 1 10 X j cos ( ϕ j ϕ i ) .

Metrics