Abstract

A state-space model (SSM) is developed for an integrated photonic architecture. This particular architecture is composed of two-port couplers and current-controllable semiconductor optical amplifiers (gains) fabricated on the same substrate. This device architecture leads to a new type of lattice filter structure. The SSM is shown to be factorizable into two matrices, one containing structural parameters of the two-port couplers, which are set during manufacturing, and the other containing the tunable gains. The SSM provides a systematic and practical approach to the analysis of the underlying filter structure, which can be easily extended to multiple-input, multiple-output optical filter structures with or without adjustable gains using two- or four-port couplers. A novel method of using the gains as loss compensation elements in addition to their tunable roles is developed.

© 2008 Optical Society of America

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  1. G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
    [CrossRef]
  2. L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).
  3. D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Pérot etalons and multilayer structures,” J. Opt. Soc. Am. A 11, 236-245 (1994).
    [CrossRef]
  4. D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
    [CrossRef]
  5. E. M. Dowling and D. L. MacFarlane, “Light wave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471-486 (1994).
    [CrossRef]
  6. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
    [CrossRef]
  7. C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
    [CrossRef]
  8. I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.
  9. I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.
  10. P. P. Vaidyanathan and S. Mitra, “A general family of multivariable digital lattice filters,” IEEE Trans. Circuits Syst. 32, 1234-1245 (1985).
    [CrossRef]
  11. A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29 (1987).
    [CrossRef]
  12. A. H. Gray, Jr., and J. D. Markel, “A normalized filter structure,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 268-277 (1975).
    [CrossRef]
  13. J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications (Prentice-Hall, 1996), 3rd ed.
  14. B. DeSchutter, “Minimal state-space realization in linear system theory: an overview,” J. Comput. Appl. Math. 121, 331-354 (2000).
    [CrossRef]
  15. P. P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice-Hall, 1993).
  16. R. A. Roberts and C. T. Mullis, Digital Signal Processing (Addison-Wesley Longman, 1987).
  17. K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1994), 2nd ed.

2006 (2)

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

2005 (1)

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

2000 (1)

B. DeSchutter, “Minimal state-space realization in linear system theory: an overview,” J. Comput. Appl. Math. 121, 331-354 (2000).
[CrossRef]

1994 (2)

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Pérot etalons and multilayer structures,” J. Opt. Soc. Am. A 11, 236-245 (1994).
[CrossRef]

E. M. Dowling and D. L. MacFarlane, “Light wave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471-486 (1994).
[CrossRef]

1987 (1)

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29 (1987).
[CrossRef]

1985 (1)

P. P. Vaidyanathan and S. Mitra, “A general family of multivariable digital lattice filters,” IEEE Trans. Circuits Syst. 32, 1234-1245 (1985).
[CrossRef]

1984 (1)

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

1975 (1)

A. H. Gray, Jr., and J. D. Markel, “A normalized filter structure,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 268-277 (1975).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29 (1987).
[CrossRef]

Christensen, M.

Constantinescu, T.

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

DeSchutter, B.

B. DeSchutter, “Minimal state-space realization in linear system theory: an overview,” J. Comput. Appl. Math. 121, 331-354 (2000).
[CrossRef]

Dowling, E. M.

E. M. Dowling and D. L. MacFarlane, “Light wave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471-486 (1994).
[CrossRef]

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Pérot etalons and multilayer structures,” J. Opt. Soc. Am. A 11, 236-245 (1994).
[CrossRef]

Evans, G.

Goodman, J. W.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

Govindan, V.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

Gray, A. H.

A. H. Gray, Jr., and J. D. Markel, “A normalized filter structure,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 268-277 (1975).
[CrossRef]

Hunt, L.

Hunt, L. R.

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

Kailath, T.

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29 (1987).
[CrossRef]

Kannan, G.

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

MacFarlane, D.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

MacFarlane, D. L.

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

E. M. Dowling and D. L. MacFarlane, “Light wave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471-486 (1994).
[CrossRef]

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Pérot etalons and multilayer structures,” J. Opt. Soc. Am. A 11, 236-245 (1994).
[CrossRef]

Madsen, C.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

Manolakis, D. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications (Prentice-Hall, 1996), 3rd ed.

Markel, J. D.

A. H. Gray, Jr., and J. D. Markel, “A normalized filter structure,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 268-277 (1975).
[CrossRef]

Mitra, S.

P. P. Vaidyanathan and S. Mitra, “A general family of multivariable digital lattice filters,” IEEE Trans. Circuits Syst. 32, 1234-1245 (1985).
[CrossRef]

Moslehi, B.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

Mullis, C. T.

R. A. Roberts and C. T. Mullis, Digital Signal Processing (Addison-Wesley Longman, 1987).

Ogata, K.

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1994), 2nd ed.

Panahi, I.

Panahi, I. M. S.

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

Proakis, J. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications (Prentice-Hall, 1996), 3rd ed.

Ramakrishna, V.

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

Roberts, R. A.

R. A. Roberts and C. T. Mullis, Digital Signal Processing (Addison-Wesley Longman, 1987).

Shaw, H. J.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

Spears, N.

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

Tong, J.

D. L. MacFarlane, N. Spears, T. Constantinescu, V. Ramakrishna, L. Hunt, J. Tong, I. Panahi, G. Kannan, G. Evans, and M. Christensen, “Composition methods for four-port couplers in photonic integrated circuitry,” J. Opt. Soc. Am. A 23, 2919-2931 (2006).
[CrossRef]

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

Tur, M.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

Vaidyanathan, P. P.

P. P. Vaidyanathan and S. Mitra, “A general family of multivariable digital lattice filters,” IEEE Trans. Circuits Syst. 32, 1234-1245 (1985).
[CrossRef]

P. P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice-Hall, 1993).

Zhao, J.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, “Active optical lattice filters,” EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

IEEE Trans. Acoust. Speech Signal Process. (1)

A. H. Gray, Jr., and J. D. Markel, “A normalized filter structure,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 268-277 (1975).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

P. P. Vaidyanathan and S. Mitra, “A general family of multivariable digital lattice filters,” IEEE Trans. Circuits Syst. 32, 1234-1245 (1985).
[CrossRef]

J. Comput. Appl. Math. (1)

B. DeSchutter, “Minimal state-space realization in linear system theory: an overview,” J. Comput. Appl. Math. 121, 331-354 (2000).
[CrossRef]

J. Lightwave Technol. (2)

G. Kannan, I. M. S. Panahi, D. L. MacFarlane, L. R. Hunt, V. Ramakrishna, T. Constantinescu, and N. Spears, “Analysis and design of active optical filter structures with two-port couplers,” J. Lightwave Technol. 24 (2006).
[CrossRef]

E. M. Dowling and D. L. MacFarlane, “Light wave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471-486 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. IEEE (1)

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909-930 (1984).
[CrossRef]

SIAM Rev. (1)

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29 (1987).
[CrossRef]

Other (7)

J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications (Prentice-Hall, 1996), 3rd ed.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, and D. MacFarlane, “Lattice filter with adjustable gains and its application in optical signal processing,” IEEE Workshop on Statistical Signal Processing, July 2005.

I. M. S. Panahi, G. Kannan, L. R. Hunt, J. Tong, D. MacFarlane, and N. Spears, “Analysis and synthesis of optical lattice filters with adjustable gains,” IEEE/LEOS--Summer Topical Conference, July 2005.

P. P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice-Hall, 1993).

R. A. Roberts and C. T. Mullis, Digital Signal Processing (Addison-Wesley Longman, 1987).

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1994), 2nd ed.

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Figures (7)

Fig. 1
Fig. 1

Cross section schematic of an active lattice filter showing grating-based couplers. The gain is provided by current controlled semiconductor optical amplifier.

Fig. 2
Fig. 2

Scanning electron micrograph pictures showing the gratings and the gains.

Fig. 3
Fig. 3

Multimirror etalon—each mirror interface reflects one fraction of the light and transmits the other fraction. The spatial delay d is modeled as the temporal delay. The mirrors actually model the gratings.

Fig. 4
Fig. 4

Signal-flow graph model of the optical interface. Each mirror interface reflects one fraction of the light ( r m = sin θ m ) and transmits the other fraction ( t m = cos θ m ). The spatial delay d is modeled as the temporal delay z 1 .

Fig. 5
Fig. 5

Node-numbering scheme for state-variable description. The dashed arrows show the numbering scheme x 1 ( n ) , x 2 ( n ) , .

Fig. 6
Fig. 6

Stability region in G 1 G 2 space (gains space) for (a)  θ 1 = 30 ° , θ 2 = 60 ° and (b)  θ 1 = 60 ° , θ 2 = 60 ° cases.

Fig. 7
Fig. 7

Development of loss compensation by splitting the gains G 1 and G 2 . g 1 and g 2 are the loss-compensating gains. γ 1 and γ 2 are the tunable gains (shown as knobs).

Equations (39)

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

x ( n + 1 ) = Ax ( n ) + B u ( n ) y ( n + 1 ) = Cx ( n ) + D u ( n ) ,
H ( z ) = Y ( z ) U ( z ) = D + C ( z I A ) 1 B .
F = [ A B C D ] .
F N = [ C N D N A N B N ] ,
[ y x 1 ( n + 1 ) x 2 ( n + 1 ) x 3 ( n + 1 ) x 4 ( n + 1 ) ] = [ 0 0 0 G 2 cos θ 2 sin θ 2 0 0 0 G 2 sin θ 2 cos θ 2 0 0 G 1 0 0 G 2 cos θ 1 G 1 sin θ 1 0 0 0 G 2 sin θ 1 G 1 cos θ 1 0 0 0 ] [ x 1 ( n ) x 2 ( n ) x 3 ( n ) x 4 ( n ) u ] .
F 2 = [ 0 0 0 G 2 cos θ 2 sin θ 2 0 0 0 G 2 sin θ 2 cos θ 2 0 0 G 1 0 0 G 2 cos θ 1 G 1 sin θ 1 0 0 0 G 2 sin θ 1 G 1 cos θ 1 0 0 0 ] .
F 2 = [ 0 0 0 cos θ 2 sin θ 2 0 0 0 sin θ 2 cos θ 2 0 0 1 0 0 cos θ 1 sin θ 1 0 0 0 sin θ 1 cos θ 1 0 0 0 ] [ G 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 0 0 0 0 0 G 2 0 0 0 0 0 1 ] .
F 2 = [ sin θ 2 cos θ 2 0 0 0 cos θ 2 sin θ 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ] [ 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 cos θ 1 sin θ 1 0 0 0 sin θ 1 cos θ 1 0 0 0 ] [ G 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 0 0 0 0 0 G 2 0 0 0 0 0 1 ] .
F ˜ 2 = [ sin θ 2 G 2 cos θ 2 0 0 0 cos θ 2 G 2 sin θ 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 sin θ 1 G 2 cos θ 1 0 0 0 G 1 cos θ 1 G 2 sin θ 1 ] ,
F ˜ 2 = [ sin θ 2 cos θ 2 0 0 0 cos θ 2 sin θ 2 0 0 0 0 0 1 0 0 0 0 0 sin θ 1 cos θ 1 0 0 0 cos θ 1 sin θ 1 ] [ 1 0 0 0 0 0 G 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 0 0 0 0 0 G 2 ] .
F ˜ 2 = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 sin θ 1 cos θ 1 0 0 0 cos θ 1 sin θ 1 ] [ sin θ 2 cos θ 2 0 0 0 cos θ 2 sin θ 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ] [ 1 0 0 0 0 0 G 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 0 0 0 0 0 G 2 ] .
F ˜ N = Θ 1 Θ 2 Θ 3 Θ N 1 Θ N Γ ˜ N .
Θ k = m [ 1 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 ( 1 ) k sin θ k cos θ k 0 0 0 0 0 cos θ k ( 1 ) k + 1 sin θ k 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ] m ,
m = { N + 1 k if   k   is   even 2 N + 1 k if k   is   odd ,
Γ ˜ N = diag ( 1 , G N , G N 1 , G N 2 , G 2 , G 1 , G 1 , G 2 , , G N 2 , G N 1 , G N ) .
H N ( z ) = D N + C N ( z I A N ) 1 B N .
F N = F o Γ ,
F o = [ A o B o C o D o ] ,
F o = [ 0 0 0 cos θ 2 sin θ 2 0 0 0 sin θ 2 cos θ 2 0 0 1 0 0 cos θ 1 sin θ 1 0 0 0 sin θ 1 cos θ 1 0 0 0 ] , Γ = [ 1 0 0 0 0 0 G 2 0 0 0 0 0 G 1 0 0 0 0 0 G 1 0 0 0 0 0 G 2 ] .
Γ = diag ( G N , G N 1 , G N 2 , G 2 , G 1 , G 1 , G 2 , , G N 2 , G N 1 , G N , 1 )
G = diag ( G N , G N 1 , G N 2 , G 2 , G 1 , G 1 , G 2 , , G N 2 , G N 1 , G N )
A N = A o G , B N = B o , C N = C o G , D N = D o .
H N ( z ) = D N + C N ( z I A N ) 1 B N = D o + C o G ( z I A o G ) 1 B 0 = D o + C o ( ( z I A o G ) G 1 ) 1 B 0 = D o + C o ( z G 1 A o ) 1 B 0 .
H o ( z ) = D o + C o ( z I A o ) 1 B 0 .
F 2 = [ 0 0 0 0.375 0.866 0 0 0 0.6495 0.5 0 0 1 0 0 0.6495 0 . 5 0 0 0 0.375 0.866 0 0 0 ] .
H 2 ( z ) = 0.866 z 2 + 0.7143 + 0.5625 z 2 z 2 + 0.7436 + 0.4871 z 2 .
R C = [ A C B C C C D C ] .
R C = [ 0 0.7436 0 0.4871 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0.0703 0 0.1407 0.8660 ] .
R = [ A o G B o C o G D o ] .
A C T = TA o G , B C = TB o , C C T = C o G .
A T PA P = Q ,
( A o G ) T PA o G P = Q G T A o T A o G I = Q Q = I G T A o T A o G .
A o T A o = [ 0 0 cos θ 1 sin θ 1 0 0 sin θ 1 cos θ 1 0 1 0 0 sin θ 2 0 0 0 ] [ 0 0 0 sin θ 2 0 0 1 0 cos θ 1 sin θ 1 0 0 sin θ 1 cos θ 1 0 0 ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 sin 2 θ 2 ] .
G T A o T A o G = [ G 2 0 0 0 0 G 1 0 0 0 0 G 1 0 0 0 0 G 2 sin 2 θ 2 ] .
Q = [ 1 G 2 2 0 0 0 0 1 G 1 2 0 0 0 0 1 G 1 2 0 0 0 0 1 G 2 2 sin 2 θ 2 ] > 0.
| G i | < 1 for     i = 1 , 2 N .
[ y x 1 ( n + 1 ) x 2 ( n + 1 ) x 3 ( n + 1 ) x 4 ( n + 1 ) ] = [ 0 0 0 g 2 2 t 2 g 2 2 r 2 0 0 0 g 2 2 r 2 g 2 2 t 2 0 0 1 0 0 g 1 2 t 1 g 1 2 r 1 0 0 0 g 1 2 r 1 g 1 2 t 1 0 0 0 ] [ γ 2 0 0 0 0 0 γ 1 0 0 0 0 0 γ 1 0 0 0 0 0 γ 2 0 0 0 0 0 1 ] [ x 1 ( n ) x 2 ( n ) x 3 ( n ) x 4 ( n ) u ] .
R = [ 0 0 0 g 2 2 t 2 g 2 2 r 2 0 0 0 g 2 2 r 2 g 2 2 t 2 0 0 1 0 0 g 1 2 t 1 g 1 2 r 1 0 0 0 g 1 2 r 1 g 1 2 t 1 0 0 0 ] .
R T R = I g m = 1 ( t m 2 + r m 2 ) 1 4 for     m = 1 , 2.

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