Abstract

We demonstrate a new photonically assisted reconfigurable radio-frequency waveform generator. The setup is based on phase modulating a multi-wavelength pulse source and subsequent compression in a dispersive medium. Under the appropriate conditions, we show that the photodetected electrical signal is broadband and coherent. Specifically, we show that this system allows for the synthesis of a reconfigurable finite-impulse-response filter where the number of filter taps is given by the number of wavelengths available from the multi-wavelength source and the reconfiguration is determined simply by their power and wavelength separation. We also show that this technique allows for time-multiplexing the synthesized waveforms, thus leading to an effective switching speed fixed by the clock rate. In particular, we show transitions between synthesized waveforms with a frequency content > 60 GHz in periods shorter than 100 ps.

© 2009 OSA

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2009 (5)

2008 (5)

2007 (5)

2006 (6)

S. Yang and X. Bao, “Generating a high-extinction-ratio pulse from a phase-modulated optical signal with a dispersion-imbalanced nonlinear loop mirror,” Opt. Lett. 31(8), 1032–1034 (2006).
[CrossRef] [PubMed]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[CrossRef]

V. Torres-Company, J. Lancis, and P. Andrés, “Arbitrary waveform generator based on all-incoherent pulse shaping,” IEEE Photon. Technol. Lett. 18(24), 2626–2628 (2006).
[CrossRef]

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

X. Yi and R. A. Minasian, “Noise mitigation in spectrum sliced microwave photonic signal processors,” J. Lightwave Technol. 24(12), 4959–4965 (2006).
[CrossRef]

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic fitlers,” J. Lightwave Technol. 24(1), 201–229 (2006).
[CrossRef]

2005 (4)

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Reconfigurable generation of high-repetition rate pulse sequences based on time-domain phase-only filtering,” Opt. Lett. 30(23), 3228–3230 (2005).
[CrossRef] [PubMed]

D. E. Leaird, Z. Jiang, and A. M. Weiner, “Experimental investigation of security issues in OCDMA: a code-switching scheme,” Electron. Lett. 41(14), 817–819 (2005).
[CrossRef]

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(8), 1746–1748 (2005).
[CrossRef]

2004 (2)

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited pulses,” IEEE Photon. Technol. Lett. 16(4), 1059–1061 (2004).
[CrossRef]

J. van Howe, J. Hansryd, and C. Xu, “Multiwavelength pulse generator using time-lens compression,” Opt. Lett. 29(13), 1470–1472 (2004).
[CrossRef] [PubMed]

2003 (1)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

2002 (2)

T. Yilmaz, C. M. DePriest, T. Turpin, J. H. Abeles, and P. J. Delfyett., “Toward a photonic arbitrary waveform generator using a modelocked external cavity semiconductor laser,” IEEE Photon. Technol. Lett. 14(11), 1608–1610 (2002).
[CrossRef]

J. D. McKinney, D. E. Leaird, and A. M. Weiner, “Millimeter-wave arbitrary waveform generation with a direct space-to-time pulse shaper,” Opt. Lett. 27(15), 1345–1347 (2002).
[CrossRef] [PubMed]

1999 (1)

J. Capmany, D. Pastor, and B. Ortega, “New and flexible fiber-optic delay line filters using chirped fibre Bragg gratings and laser arrays,” IEEE Trans. Microw. Theory Tech. 47(7), 1321–1326 (1999).
[CrossRef]

1998 (1)

T. A. Cusick, S. Iezekiel, and R. E. Miles; “All-optical microwave filter design employing a genetic algorithm,” IEEE Photon. Technol. Lett. 10(8), 1156–1158 (1998).
[CrossRef]

1996 (2)

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2(3), 709–719 (1996).
[CrossRef]

T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80 Gb/s highly exctinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. 2(3), 643–649 (1996).
[CrossRef]

1995 (1)

1988 (1)

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

1983 (1)

Abeles, J. H.

T. Yilmaz, C. M. DePriest, T. Turpin, J. H. Abeles, and P. J. Delfyett., “Toward a photonic arbitrary waveform generator using a modelocked external cavity semiconductor laser,” IEEE Photon. Technol. Lett. 14(11), 1608–1610 (2002).
[CrossRef]

Amano, K.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

Andrés, P.

Anzai, S.

Azaña, J.

Bao, X.

Berger, N. K.

Cao, J.

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[CrossRef]

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic fitlers,” J. Lightwave Technol. 24(1), 201–229 (2006).
[CrossRef]

J. Capmany, D. Pastor, and B. Ortega, “New and flexible fiber-optic delay line filters using chirped fibre Bragg gratings and laser arrays,” IEEE Trans. Microw. Theory Tech. 47(7), 1321–1326 (1999).
[CrossRef]

S. Sales, J. Capmany, J. Martí, and D. Pastor, “Solutions to the synthesis problem of optical delay line filters,” Opt. Lett. 20(23), 2438–2440 (1995).
[CrossRef] [PubMed]

Caraquitena, J.

Chen, L. R.

Chou, J.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Cusick, T. A.

T. A. Cusick, S. Iezekiel, and R. E. Miles; “All-optical microwave filter design employing a genetic algorithm,” IEEE Photon. Technol. Lett. 10(8), 1156–1158 (1998).
[CrossRef]

Dai, Y.

Delfyett, P. J.

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

T. Yilmaz, C. M. DePriest, T. Turpin, J. H. Abeles, and P. J. Delfyett., “Toward a photonic arbitrary waveform generator using a modelocked external cavity semiconductor laser,” IEEE Photon. Technol. Lett. 14(11), 1608–1610 (2002).
[CrossRef]

DePriest, C. M.

T. Yilmaz, C. M. DePriest, T. Turpin, J. H. Abeles, and P. J. Delfyett., “Toward a photonic arbitrary waveform generator using a modelocked external cavity semiconductor laser,” IEEE Photon. Technol. Lett. 14(11), 1608–1610 (2002).
[CrossRef]

Dorrer, C.

Fainman, Y.

Fernández-Pousa, C. R.

Fischer, B.

Fontaine, N. K.

Fukushima, Y.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

Futami, F.

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited pulses,” IEEE Photon. Technol. Lett. 16(4), 1059–1061 (2004).
[CrossRef]

Gee, S.

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

Geisler, D. J.

Han, Y.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Hansryd, J.

Heritage, J. P.

Hirooka, T.

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited pulses,” IEEE Photon. Technol. Lett. 16(4), 1059–1061 (2004).
[CrossRef]

Huang, C.-B.

C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Time-multiplexed photonically enabled radio-frequency arbitrary waveform generation with 100 ps transitions,” Opt. Lett. 32(22), 3242–3244 (2007).
[CrossRef] [PubMed]

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

Iezekiel, S.

T. A. Cusick, S. Iezekiel, and R. E. Miles; “All-optical microwave filter design employing a genetic algorithm,” IEEE Photon. Technol. Lett. 10(8), 1156–1158 (1998).
[CrossRef]

Jalali, B.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Jannson, T.

Jiang, W.

Jiang, Z.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

D. E. Leaird, Z. Jiang, and A. M. Weiner, “Experimental investigation of security issues in OCDMA: a code-switching scheme,” Electron. Lett. 41(14), 817–819 (2005).
[CrossRef]

Jurokawa, T.

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

Karalar, A.

Kawanishi, S.

T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(8), 1746–1748 (2005).
[CrossRef]

Kobayashi, T.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

Kodate, K.

Kolner, B. H.

Komai, Y.

Komukai, T.

T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(8), 1746–1748 (2005).
[CrossRef]

Lancis, J.

Leaird, D. E.

C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Time-multiplexed photonically enabled radio-frequency arbitrary waveform generation with 100 ps transitions,” Opt. Lett. 32(22), 3242–3244 (2007).
[CrossRef] [PubMed]

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

D. E. Leaird, Z. Jiang, and A. M. Weiner, “Experimental investigation of security issues in OCDMA: a code-switching scheme,” Electron. Lett. 41(14), 817–819 (2005).
[CrossRef]

J. D. McKinney, D. E. Leaird, and A. M. Weiner, “Millimeter-wave arbitrary waveform generation with a direct space-to-time pulse shaper,” Opt. Lett. 27(15), 1345–1347 (2002).
[CrossRef] [PubMed]

Levit, B.

Liu, Y.

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2(3), 709–719 (1996).
[CrossRef]

Mandai, K.

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

Martí, J.

McKinney, J. D.

Mieno, M.

Miles, R. E.

T. A. Cusick, S. Iezekiel, and R. E. Miles; “All-optical microwave filter design employing a genetic algorithm,” IEEE Photon. Technol. Lett. 10(8), 1156–1158 (1998).
[CrossRef]

Minasian, R. A.

X. Yi and R. A. Minasian, “Noise mitigation in spectrum sliced microwave photonic signal processors,” J. Lightwave Technol. 24(12), 4959–4965 (2006).
[CrossRef]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[CrossRef]

Miyamoto, D.

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

Miyazaki, T.

Morimoto, A.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

Nagatsuma, T.

T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80 Gb/s highly exctinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. 2(3), 643–649 (1996).
[CrossRef]

Nakazawa, M.

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited pulses,” IEEE Photon. Technol. Lett. 16(4), 1059–1061 (2004).
[CrossRef]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[CrossRef]

Okamoto, K.

Ortega, B.

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic fitlers,” J. Lightwave Technol. 24(1), 201–229 (2006).
[CrossRef]

J. Capmany, D. Pastor, and B. Ortega, “New and flexible fiber-optic delay line filters using chirped fibre Bragg gratings and laser arrays,” IEEE Trans. Microw. Theory Tech. 47(7), 1321–1326 (1999).
[CrossRef]

Otsuji, T.

T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80 Gb/s highly exctinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. 2(3), 643–649 (1996).
[CrossRef]

Ozharar, S.

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

Park, S.-G.

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2(3), 709–719 (1996).
[CrossRef]

Pastor, D.

Quinlan, F.

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

Sales, S.

Sano, E.

T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80 Gb/s highly exctinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. 2(3), 643–649 (1996).
[CrossRef]

Saperstein, R. E.

Scott, R. P.

Shioda, T.

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

Sueta, T.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
[CrossRef]

Takeda, S.

D. Miyamoto, K. Mandai, T. Jurokawa, S. Takeda, T. Shioda, and H. Tsuda, “Waveform-controllable optical pulse generation using an optical pulse synthesizer,” IEEE Photon. Technol. Lett. 18(5), 721–723 (2006).
[CrossRef]

Torres-Company, V.

Tsuda, H.

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Appl. Opt. (1)

Electron. Lett. (1)

D. E. Leaird, Z. Jiang, and A. M. Weiner, “Experimental investigation of security issues in OCDMA: a code-switching scheme,” Electron. Lett. 41(14), 817–819 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988).
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IEEE J. Sel. Top. Quantum Electron. (2)

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[CrossRef]

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2(3), 709–719 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (8)

T. Yilmaz, C. M. DePriest, T. Turpin, J. H. Abeles, and P. J. Delfyett., “Toward a photonic arbitrary waveform generator using a modelocked external cavity semiconductor laser,” IEEE Photon. Technol. Lett. 14(11), 1608–1610 (2002).
[CrossRef]

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
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[CrossRef]

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[CrossRef]

S. Ozharar, F. Quinlan, S. Gee, and P. J. Delfyett, “Demonstration of endless phase modulation for arbitrary waveform generation,” IEEE Photon. Technol. Lett. 17(12), 2739–2741 (2005).
[CrossRef]

T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(8), 1746–1748 (2005).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Cartoon of the physical process behind the proposed setup: generation of a multiwavelength pulse train where the power and delay can be controlled in the optical domain. (b) Particular proposal in this work.

Fig. 2
Fig. 2

Scheme of the experimental setup. PC: polarization controller; EDFA; Erbium-doped fiber amplifier; OSA: optical spectrum analyzer; PPG: pulse pattern generator.

Fig. 3
Fig. 3

Single pulse characteristics. Simulation results show (a) the compressed intensity pulse with 11.9 ps duration assuming a 55 GHz bandwidth (at 3dB) photodiode; (b) corresponding RF spectrum in blue solid line and filter response in green dash-dotted line; and (c) expected optical spectrum for 0.7 π rad of modulation index. Measurement results show (d) a compressed pulse with 10.9 ps width; (e) pulse train at 6 GHz and (f) optical spectrum.

Fig. 4
Fig. 4

Illustration of the AWG technique in the static regime. (a)-(d) screen images of the scope for each of the individual taps. (e)-(h) are the corresponding optical spectra in false color. (i) screen image of the optical spectrum when the four lasers are on. (j) and (l) show the achieved electrical waveform in different scale. (k) shows the superposition of the raw data from (a)-(d) and (j).

Fig. 5
Fig. 5

Triplet and intensity decreasing burst examples. (a) and (e) show the raw data with the superimposed intensities for each tap. (b) and (f) show the screen images of the optical spectra when all the lasers are on and (c-d) and (g-h) show the screen images of the scope traces in different scales.

Fig. 6
Fig. 6

Experimental setup for time-multiplexing of RF waveforms. The clock rate is set to 12 GHz.

Fig. 7
Fig. 7

Time-multiplexing of two different waveforms at a 12 GHz rate. (a) and (b) are the data and data bar gates measured in the optical domain. (c) and (d) the synthesized compressed waveforms coming from EOM1 and EOM2, respectively. (e) and (f) are a zoomed version of (c) and (d). (g) and (h) show the multiplexing possibilities when all the lasers are on.

Fig. 8
Fig. 8

Time-multiplexing of two different waveforms at a 12 GHz rate with the switching code 1000. (a) and (b) are the data and data bar gates measured in the optical domain. (c) and (d) the synthesized waveforms from EOM1 and EOM2, respectively. (e) and (f) are a zoomed version of (c) and (d). (g) and (h) show the multiplexing possibilities when all the lasers are on.

Equations (12)

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U i n ( t ) = m ( t ) n = 0 N 1 P n exp [ j { ω n t + ϕ n ( t ) } ] .
U o u t ( t ' ) = n = 0 N 1 α n a n ( t ' t n ) exp [ j ω n t ' ] ,
t n = Φ 2 ( ω n ω 0 ) .
I o u t ( t ' ) = n = 0 | α n | 2 | a n ( t ' t n ) | 2 + n l n α n * α l a n * ( t ' t n ) a l ( t ' t l ) exp [ j ( ω n ω l ) t ' ] .
| ω n + 1 ω n | / 2 π > B + σ α , f ,
a n ( t ' ) = exp [ j t ' 2 / 2 Φ 2 ] ψ ( t ) exp [ j ϕ n ( t ) ] exp [ j ( K 1 / Φ 2 ) t 2 / 2 ] exp [ j t t ' / Φ 2 ] d t .
a n ( t ' ) = exp [ j t ' 2 / 2 Φ 2 ] ψ ( t ) exp [ j ϕ n ( t ) ] exp [ j t t ' / Φ 2 ] d t .
i o u t ( t ' ) { n = 0 P n I 0 ( t ' t n ) } h P D ( t ' ) ,
I 0 ( t ' ) = | ψ ( t ) exp ( j t t ' / Φ 2 ) d t | 2 .
i ˜ o u t ( f ) H P D ( f ) { n = 0 N 1 P n exp ( j 2 π f t n ) } I ˜ 0 ( f ) .
t n = ( P + δ ) T ,
N max Δ Ω Δ ω min = Δ Ω 2 π ( σ α , f + B ) ,

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