Abstract

A photonic approach to instantaneously identify frequency components of microwave signals with multiple tones is conceived and practically demonstrated. A mathematical model was first developed to predict the behavior of the system. Then the system operation was tested in practice. The system employs a double mixing technique that enables high-frequency measurement without the need for any high-frequency RF component or broadband photodetector. The system operation was demonstrated over a frequency range of 0.1–40 GHz. Frequency measurement of two simultaneous RF tones is demonstrated; however, the system has the potential to be expanded to measure a larger number of simultaneous RF tones. It also has the potential to operate over a wider frequency range.

© 2013 Optical Society of America

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References

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  1. J. B. Tsui, “Simultaneous signal detector for an instantaneous frequency measurement receiver,” U.S. patent4,336,541 (22June1982).
  2. J. B. Y. Tsui, “Simultaneous signal detection for IFM receivers by detecting intermodulation products,” U.S. patent4,426,648 (17January1984).
  3. S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.
  4. T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.
  5. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1, 319–330 (2007).
    [CrossRef]
  6. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27, 314–335 (2009).
    [CrossRef]
  7. A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24, 4628–4641 (2006).
    [CrossRef]
  8. R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microwave Theor. Tech. 54, 832–846 (2006).
    [CrossRef]
  9. H. Emami, N. Sarkhosh, E. Lopez, and A. Mitchell, “Photonic feed for sinuous antenna,” J. Lightwave Technol. 30, 2725–2732 (2012).
    [CrossRef]
  10. L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photon. Technol. Lett. 21, 642–644 (2009).
    [CrossRef]
  11. B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
    [CrossRef]
  12. N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
    [CrossRef]
  13. H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
    [CrossRef]
  14. L. A. Bui and A. Mitchell, “All optical instantaneous frequency measurement incorporating optical Hilbert transformer,” in Proceedings of 2012 MWP International Topical Meeting on Microwave Photonics (MWP) (Institute of Electrical and Electronics Engineers, 2008), pp. 323–326.
  15. H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
    [CrossRef]
  16. N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

2012 (1)

2010 (1)

B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
[CrossRef]

2009 (2)

L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photon. Technol. Lett. 21, 642–644 (2009).
[CrossRef]

J. Yao, “Microwave photonics,” J. Lightwave Technol. 27, 314–335 (2009).
[CrossRef]

2008 (2)

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

2007 (2)

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1, 319–330 (2007).
[CrossRef]

2006 (2)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microwave Theor. Tech. 54, 832–846 (2006).
[CrossRef]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24, 4628–4641 (2006).
[CrossRef]

Amagai, J.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Baraniuk, R.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Bui, L. A.

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

L. A. Bui and A. Mitchell, “All optical instantaneous frequency measurement incorporating optical Hilbert transformer,” in Proceedings of 2012 MWP International Topical Meeting on Microwave Photonics (MWP) (Institute of Electrical and Electronics Engineers, 2008), pp. 323–326.

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1, 319–330 (2007).
[CrossRef]

Emami, H.

H. Emami, N. Sarkhosh, E. Lopez, and A. Mitchell, “Photonic feed for sinuous antenna,” J. Lightwave Technol. 30, 2725–2732 (2012).
[CrossRef]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

Gilbert, A.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Izutsu, M.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Kawanishi, T.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Kirolos, S.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

Kiuchi, H.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Laska, J.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Lopez, E.

Marti, J.

B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
[CrossRef]

Massoud, Y.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Mengual, T.

B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
[CrossRef]

Minasian, R. A.

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microwave Theor. Tech. 54, 832–846 (2006).
[CrossRef]

Mitchell, A.

H. Emami, N. Sarkhosh, E. Lopez, and A. Mitchell, “Photonic feed for sinuous antenna,” J. Lightwave Technol. 30, 2725–2732 (2012).
[CrossRef]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

L. A. Bui and A. Mitchell, “All optical instantaneous frequency measurement incorporating optical Hilbert transformer,” in Proceedings of 2012 MWP International Topical Meeting on Microwave Photonics (MWP) (Institute of Electrical and Electronics Engineers, 2008), pp. 323–326.

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

Nejati, H.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

Nguyen, L. V. T.

L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photon. Technol. Lett. 21, 642–644 (2009).
[CrossRef]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1, 319–330 (2007).
[CrossRef]

Pfetsch, S.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Ragheb, T.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

Sakamato, T.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Sarkhosh, N.

H. Emami, N. Sarkhosh, E. Lopez, and A. Mitchell, “Photonic feed for sinuous antenna,” J. Lightwave Technol. 30, 2725–2732 (2012).
[CrossRef]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Wideband RF photonic in-phase and quadrature-phase generation,” Opt. Lett. 33, 98–100 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

Seeds, A. J.

Strauss, M.

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

Tsuchiya, M.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Tsui, J. B.

J. B. Tsui, “Simultaneous signal detector for an instantaneous frequency measurement receiver,” U.S. patent4,336,541 (22June1982).

Tsui, J. B. Y.

J. B. Y. Tsui, “Simultaneous signal detection for IFM receivers by detecting intermodulation products,” U.S. patent4,426,648 (17January1984).

Vidal, B.

B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
[CrossRef]

Williams, K. J.

Yamada, M.

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

Yao, J.

IEEE Photon. Technol. Lett. (2)

L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photon. Technol. Lett. 21, 642–644 (2009).
[CrossRef]

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Reduced cost microwave photonic instantaneous frequency measurement system,” IEEE Photon. Technol. Lett. 20, 1521–1523 (2008).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (3)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microwave Theor. Tech. 54, 832–846 (2006).
[CrossRef]

B. Vidal, T. Mengual, and J. Marti, “Photonic technique for the measurement of frequency and power of multiple microwave signals,” IEEE Trans. Microwave Theor. Tech. 58, 3103–3108 (2010).
[CrossRef]

H. Kiuchi, T. Kawanishi, M. Yamada, T. Sakamato, M. Tsuchiya, J. Amagai, and M. Izutsu, “High extinction ration Mach-Zehnder modulator applied to a highly stable optical signal generator,” IEEE Trans. Microwave Theor. Tech. 55, 1964–1972 (2007).
[CrossRef]

J. Lightwave Technol. (3)

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1, 319–330 (2007).
[CrossRef]

Opt. Lett. (1)

Other (6)

N. Sarkhosh, H. Emami, L. A. Bui, and A. Mitchell, “Photonic instantaneous frequency measurement using non-linear optical mixing,” in Proceedings of 2008 IEEE-MTT-S International Microwave Symposium Digest (MTT) (Institute of Electrical and Electronics Engineers, 2008), pp. 599–601.

L. A. Bui and A. Mitchell, “All optical instantaneous frequency measurement incorporating optical Hilbert transformer,” in Proceedings of 2012 MWP International Topical Meeting on Microwave Photonics (MWP) (Institute of Electrical and Electronics Engineers, 2008), pp. 323–326.

J. B. Tsui, “Simultaneous signal detector for an instantaneous frequency measurement receiver,” U.S. patent4,336,541 (22June1982).

J. B. Y. Tsui, “Simultaneous signal detection for IFM receivers by detecting intermodulation products,” U.S. patent4,426,648 (17January1984).

S. Pfetsch, T. Ragheb, J. Laska, H. Nejati, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud, “On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion,” in Proceedings of 2008 IEEE-ISCAS International Symposium on Circuits and Systems, (Institute of Electrical and Electronics Engineers, Seattle, Washington, 2008), pp. 1480–1483.

T. Ragheb, J. Laska, H. Nejati, S. Kirolos, R. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Proceedings of 2008 MWSCAS Midwest Symposium on Circuits and Systems (MWSCAS) (Institute of Electrical and Electronics Engineers, 2008), pp. 37–40.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Basic IFM system. (b) Dual-tone IFM system.

Fig. 2.
Fig. 2.

Proposed IFM setup.

Fig. 3.
Fig. 3.

Measured frequency as a function of input frequency.

Fig. 4.
Fig. 4.

Maximum frequency measurement error.

Fig. 5.
Fig. 5.

Block diagram of a possible extension of the dual-tone IFM system to a quadruple-tone system.

Equations (29)

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

{VDC1=1/8cosΩ1τ+1/8cosΩ2τVDC2=1/8sinΩ1τ+1/8sinΩ2τ.
V1=rGPoZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]+8a2(1+2a2)M1[M1+2a2M22]cosΩ1τ+8a2(1+2a2)M2[M2+2a2M12]cosΩ2τ},
V2=rGP1ZL{2(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]16a2(1+2a2)M1[M1+2a2M22]sinΩ1τsinΩ1τ16a2(1+2a2)M2[M2+2a2M12]sinΩ2τsinΩ2τ},
G=12GLPFGEDFALAWGLMZM2LcircLFBG,
a=π2ZinPRF/2Vπ,
E1(t)=E(t)LMZMcos[π(Vin+VB)2Vπ],
E1(t)=LMZMEoejωtcos{π[Vo(cosΩ1t+cosΩ2t)+VB]2Vπ}.
E1(t)=LMZMEoejωt[cosπ(VocosΩ1t+VocosΩ2t)2Vπ1cosπ4sinπ(VocosΩ1t+VocosΩ2t)2Vπ1sinπ4],
E1(t)=b1Eoejωt×{[cos(acosΩ1t)cos(acosΩ2t)sin(acosΩ1t)sin(acosΩ2t)][sin(acosΩ1t)cos(acosΩ2t)+cos(acosΩ1t)sin(acosΩ2t)]},
E1(t)=b1Eoejωt{[(J0(a)+2n=1(1)nJ2n(a)cos2nΩ1t)×(J0(a)+2n=1(1)nJ2n(a)cos2nΩ2t)(2n=1(1)nJ2n1(a)cos(2n1)Ω1t)×(2n=1(1)nJ2n1(a)cos(2n1)Ω2t)][(2n=1(1)nJ2n1(a)cos(2n1)Ω1t)×(J0(a)+2n=1(1)nJ2n(a)cos2nΩ2t)+(J0(a)+2n=1(1)nJ2n(a)cos2nΩ1t)×(2n=1(1)nJ2n1(a)cos(2n1)Ω2t)]}.
E1(t)=b1Eoejωt[(J02(a)4J12(a)cosΩ1tcosΩ2t)2J0(a)J1(a)(cosΩ1t+cosΩ2t)].
E2(t)=b2Eoejωt[(J02(a)4J12(a)cosΩ1tcosΩ2t)2J0(a)J1(a)(cosΩ1t+cosΩ2t)],
Vin(t)=Vo[M(Ω1)cosΩ1(tτ)+M(Ω1)cosΩ2(tτ)],
E3(t)=b3Eoejωt[J02(a)4J12(a)cosΩ1tcosΩ2t2J0(a)J1(a)(cosΩ1t+cosΩ2t)]×[J0(aM1)J0(aM2)4J1(aM1)J1(aM2)cosΩ1(tτ)cosΩ2(tτ)2J1(aM1)J0(aM2)cosΩ1(tτ)2J1(aM2)J0(aM1)cosΩ2(tτ)],
J0(a)1,J1(aM1)1,J0(aM2)1,J1(a)a,J1(aM1)aM1,J1(aM2)aM2.
E3(t)=b3Eoejωt[14a2cosΩ1tcosΩ2t2a(cosΩ1t+cosΩ2t)]×{14a2M1M2cosΩ1(tτ)cosΩ2(tτ)2a[M1cosΩ1(tτ)+M2cosΩ2(tτ)]}.
E3(t)=b4Eoejωt[14a2cosΩ1tcosΩ2t2a(cosΩ1t+cosΩ2t)]×{14a2M1M2cosΩ1(tτ)cosΩ2(tτ)2a[M1cosΩ1(tτ)+M2cosΩ2(tτ)]},
I1=rE3×E3*,
I1(t)=rb42Eo2[14a2cosΩ1tcosΩ2t2a(cosΩ1t+cosΩ2t)]2×{14a2M1M2cosΩ1(tτ)cosΩ2(tτ)2a[M1cosΩ1(tτ)+M2cosΩ2(tτ)]}2.
Vin,LPF1(t)=rb42Eo2ZL[14a2cosΩ1tcosΩ2t2a(cosΩ1t+cosΩ2t)]2×{14a2M1M2cosΩ1(tτ)cosΩ2(tτ)2a[M1cosΩ1(tτ)+M2cosΩ2(tτ)]}2,
V1=rGEo2ZL{(1+4a4+4a2)×[1+4a4M12M22+2a2(M12+M22)]+8a2(1+2a2)M1[M1+2a2M22]cosΩ1τ+8a2(1+2a2)M2[M2+2a2M12]cosΩ2τ+2a4M12(2a2+1)(2a2M22+1)cos2Ω1τ+2a4M22(2a2+1)(2a2M12+1)cos2Ω1τ+2a8M12M22×(cos2(Ω1+Ω2)τ+cos2(Ω1Ω2)τ)+8a6M1M2(M1cos(2Ω1+Ω2)τ+M2cos(Ω1+2Ω2)τM1cos(2Ω1Ω2)τ+M2cos(Ω12Ω2)τ)},
V1=rGEo2ZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]+8a2(1+2a2)M1[M1+2a2M22]cosΩ1τ+8a2(1+2a2)M2[M2+2a2M12]cosΩ2τ}.
V1=rGPoZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]+8a2(1+2a2)M1[M1+2a2M22]cosΩ1τ+8a2(1+2a2)M2[M2+2a2M12]cosΩ2τ},
G=12GLPFGEDFALAWGLMZM2LcircLFBG.
{E1(t)=E(t)LMZMcos[π(VinVB)2Vπ]E1(t)=E(t)LMZMcos[π(Vin+VB)2Vπ],
Vλ1=rGP1ZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]8a2(1+2a2)M1[M1+2a2M22]cosΩ1(τ+τ)8a2(1+2a2)M2[M2+2a2M12]cosΩ2(τ+τ)},
Vλ2=rGP1ZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]8a2(1+2a2)M1[M1+2a2M22]cosΩ1(ττ)8a2(1+2a2)M2[M2+2a2M12]cosΩ2(ττ)}.
V2=Vλ1+V21=rGP1ZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]8a2(1+2a2)M1[M1+2a2M22]cosΩ1(τ+τ)8a2(1+2a2)M2[M2+2a2M12]cosΩ2(τ+τ)}+rGP1ZL{(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]8a2(1+2a2)M1[M1+2a2M22]cosΩ1(ττ)8a2(1+2a2)M2[M2+2a2M12]cosΩ2(ττ)}.
V2=rGP1ZL{2(1+4a4+4a2)[1+4a4M12M22+2a2(M12+M22)]16a2(1+2a2)M1[M1+2a2M22]sinΩ1τsinΩ1τ16a2(1+2a2)M2[M2+2a2M12]sinΩ2τsinΩ2τ}.

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