Abstract

We demonstrate experimentally an optical system for undersampling several bandwidth-limited signals with carrier frequencies that are not known apriori and can be located within a broad frequency region of 0–20 GHz. The system is based on undersampling synchronously at three different rates. The optical undersampling down-converts the entire system bandwidth into a low frequency region called baseband. The synchronous sampling at several rates enables to accurately reconstruct signals even in cases in which different signals overlap in the baseband region of all sampling channels. Reconstruction of three simultaneously generated chirped signals, each with a bandwidth of about 200 MHz, was experimentally demonstrated.

© 2010 Optical Society of America

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  1. P. W. Joudawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, J. C. Twichell, "Optical Down-Sampling of Wide-Band Microwave Signals," J. Lightwave Technol. 21, 3116-3124 (2004).
    [CrossRef]
  2. A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
    [CrossRef]
  3. J. Kim, M. J. Park, M. H. Perrott and F. X. Kärtner, "Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution," Opt. Express 16, 16509-16515 (2008).
    [CrossRef] [PubMed]
  4. P. Feng and Y. Bresler, "Spectrum-blind minimum-rate sampling and reconstruction of multiband signals," in Proc. IEEE Int. Conf. ASSP, vol. 3, pp. 1688-1691, May 1996.
  5. R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001).
    [CrossRef]
  6. M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
    [CrossRef]
  7. A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach and L. Singer, "Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals," J. Lightwave Technol. 27, 1027-1033 (2009).
    [CrossRef]
  8. M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
    [CrossRef]
  9. J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006).
    [CrossRef]
  10. G. C. Valley, "Photonic analog-to-digital converters," Opt. Express 15, 1955-1982 (2007).
    [CrossRef] [PubMed]
  11. R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
    [CrossRef]
  12. B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).
  13. M. Shinagawa, Y. Akazawa and T. Wakimoto, "Jitter Analysis of High-Speed Sampling Systems," IEEE J. Solid-State Circuits 25,220-224 (1990).
  14. M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
    [CrossRef]
  15. R. Penrose, "A generalized inverse for matrices," in Proc. Cambridge Philosophical Society, Cambridge 51, 406-413 (1955).

2010 (1)

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

2009 (2)

M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach and L. Singer, "Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals," J. Lightwave Technol. 27, 1027-1033 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (1)

J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006).
[CrossRef]

2005 (2)

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
[CrossRef]

2004 (1)

2001 (1)

R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001).
[CrossRef]

1999 (1)

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

1989 (1)

M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
[CrossRef]

1955 (1)

R. Penrose, "A generalized inverse for matrices," in Proc. Cambridge Philosophical Society, Cambridge 51, 406-413 (1955).

Bloom, D.

M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
[CrossRef]

Bosti, C. W.

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

Bresler, Y.

R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001).
[CrossRef]

Eldar, Y.

M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

Feldser, A.

A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
[CrossRef]

Feldster, A.

Fleyer, M.

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

Gravel, J. F.

J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006).
[CrossRef]

Hargreaves, J. J.

Horowitz, M.

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach and L. Singer, "Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals," J. Lightwave Technol. 27, 1027-1033 (2009).
[CrossRef]

A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
[CrossRef]

Joudawlkis, P. W.

Kärtner, F. X.

Kim, J.

Le, B.

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

Linden, A.

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

Mishali, M.

M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

Park, M. J.

Penrose, R.

R. Penrose, "A generalized inverse for matrices," in Proc. Cambridge Philosophical Society, Cambridge 51, 406-413 (1955).

Perrott, M. H.

Reed, J. H.

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

Rodwell, M.

M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
[CrossRef]

Rondeau, T. W.

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

Rosenthal, A.

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach and L. Singer, "Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals," J. Lightwave Technol. 27, 1027-1033 (2009).
[CrossRef]

Shapira, Y. P.

Singer, L.

Titi, G. W.

Twichell, J. C.

Valley, G. C.

Venkantaramani, R.

R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001).
[CrossRef]

Walden, R. H.

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

Weingarten, K.

M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
[CrossRef]

Wight, J.

J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006).
[CrossRef]

Younger, R. D.

Zach, S.

Zeitouny, A.

A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
[CrossRef]

Philosophical Society, Cambridge (1)

R. Penrose, "A generalized inverse for matrices," in Proc. Cambridge Philosophical Society, Cambridge 51, 406-413 (1955).

IEEE J. Quantum Electron. (1)

M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006).
[CrossRef]

IEEE Trans. Signal Process. (3)

R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001).
[CrossRef]

M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Commun. (1)

A. Zeitouny, A. Feldser and M. Horowitz, "Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators," Opt. Commun. 256, 248-255 (2005).
[CrossRef]

Opt. Express (2)

Signal Process. Mag. (1)

B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).

Other (2)

M. Shinagawa, Y. Akazawa and T. Wakimoto, "Jitter Analysis of High-Speed Sampling Systems," IEEE J. Solid-State Circuits 25,220-224 (1990).

P. Feng and Y. Bresler, "Spectrum-blind minimum-rate sampling and reconstruction of multiband signals," in Proc. IEEE Int. Conf. ASSP, vol. 3, pp. 1688-1691, May 1996.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic description of (a) optical sampling system and (b) optical pulse generator (OPG). The synchronous multirate sampling is performed simultaneously at three different rates: F 1 = 3.8 GHz, F 2 = 3.5 GHz and F 3 = 4.0 GHz. A 100 MHz reference signal is used to synchronize between the different channels. OPG consists of a phase-locked loop dielectric resonator oscillators (PLL-DROs), a comb generator, an Electro-Absorption (EA) modulator, and a continuous wave (CW) laser. The optical wavelengths of the OPGs were 1535.04 nm, 1536.61 nm and 1544.53 nm. MUX and DEMUX are optical add-drop modules that combine and demultiplex three optical pulse trains. EDFA is an erbium-doped fiber amplifier, MODULATOR is a LiNbO3 Mach-Zehnder modulator, φ is an electrical phase shifter, D is a detector, LPF is an electrical lowpass filter with a bandwidth of 2 GHz and AMP is an electrical amplifier.

Fig. 2.
Fig. 2.

Transfer function Hi (f) of three sampling channels measured in the frequency region 5 MHz–2 GHz with a resolution of 5 MHz.

Fig. 3.
Fig. 3.

Three optical pulse-trains measured at the modulator input by using a sampling oscilloscope with an optical bandwidth of 50 GHz. Delays between the pulse-trains were adjusted to obtain the best overlap between pulses.

Fig. 4.
Fig. 4.

Reconstructed spectrum of three signals and a zoom on each signal. The signal located around 11.25 GHz overlaps with other signals in the baseband region of all channels.

Fig. 5.
Fig. 5.

Baseband spectra of three sampling channels calculated from the reconstructed signal (blue curves) and compared to the spectrum of the measured baseband signals.

Fig. 6.
Fig. 6.

Comparison between the reconstructed spectrum (blue curve) and the spectrum measured by the RF spectrum analyzer (red dashed curve).

Fig. 7.
Fig. 7.

Reconstructed spectrum of three signals and a zoom on each signal (blue curve). Since no aliasing occurs at the baseband of the 3.8 GHz channel, the original spectrum can be also directly calculated from the measured spectrum of that channel. An excellent agreement is obtained between the reconstructed and directly calculated spectra.

Fig. 8.
Fig. 8.

Baseband spectra of three sampling channels calculated from the reconstructed signal (blue curves) and compared to the spectrum of the measured baseband signals.

Fig. 9.
Fig. 9.

Baseband spectrum of a signal at the 3.8 GHz sampling channel measured using the RF spectrum analyzer (green curve) and compared to the Fourier transform of the sampled baseband signal (blue curve). The input signal frequency equals 11.1 GHz and its power equals 6.5 dBm. The resolution bandwidth of the spectrum analyzer equals 100 kHz. The acquisition window of the sampling equals 8.19 µs that corresponds to a frequency resolution of about 122 kHz.

Fig. 10.
Fig. 10.

Phase noise of three optical trains. The integrated jitter in the frequency region, 30.5kHz – 1 MHz equals: 11.5, 13.1 and 17.1 fs for 3.8, 3.5 and 4.0 GHz sampling channels, respectively.

Equations (31)

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x ̂ ( β ) = Q x ( β ) ,
x i ( t ) = h i ( t ) * ( x ( t ) Σ n p i ( t n F i τ i τ ) ) ,
X i ( f ) = F i H i ( f ) exp [ j 2 π f ( τ i + τ ) ] Σ n P i ( n F i ) X ( f n F i ) exp [ j 2 π ( f n F i ) ( τ i + τ ) ] ,
x ̂ ( β ) = Q x ( β ) ,
x ̂ ( β ) = ( W H 1 ( β ) x 1 ( β ) W H 2 ( β ) x 2 ( β ) W H 3 ( β ) x 3 ( β ) ) , Q = ( Q 1 W τ 1 Q 2 W τ 2 Q 3 W τ 3 ) .
W τ i = diag k { exp ( j 2 π k Δ f τ i ) }
W H i ( β ) = diag k { H i 1 ( k Δ f + β ) exp [ j 2 π ( k Δ f + β ) ( τ i + τ ) ] exp ( j 2 π β τ i ) } F i .
x S * ( β ) = Q S x ̂ ( β ) .
ε 2 ( β ) = ( I Q S Q S ) x ̂ ( β ) 2 .
ENOB = ( SNDR ( dB ) 1.76 ) 6.02 .
SNR [ dB ] = 20 log 10 ( 2 π f c Δ t ) ,
σ J = 1 2 π f c 2 f 0 f 1 L ( f ) d f ,
X i ( f ) F i = H i ( f ) { X ( f ) * Σ n P i ( n F i ) δ ( f n F i ) exp [ j 2 π n F i ( τ i + τ ) ] } =
= H i ( f ) { Σ n P i ( n F i ) X ( f n F i ) exp [ j 2 π n F i ( τ i + τ ) ] } ,
X i ( f ) F i = H i ( f ) exp [ j 2 π f ( τ i + τ ) ] Σ n P i ( n F i ) X ˜ ( f n F i ) exp [ j 2 π ( f n F i ) τ i ] ,
X i ( k Δ f + β ) H i 1 ( k Δ f + β ) exp [ j 2 π ( k Δ f + β ) ( τ i + τ ) ] exp ( j 2 π β τ i ) F i =
= Σ n P i ( n F i ) X ˜ ( ( k n M i ) Δ f + β ) exp [ j 2 π ( k n M i ) Δ f τ i ] .
X i k ( β ) = X i ( k Δ f + β )
X k ( β ) = X ( k Δ f + β )
W τ i = diag k { exp ( j 2 π k Δ f τ i ) }
W H i ( β ) = diag k { H i 1 ( k Δ f + β ) exp [ j 2 π ( k Δ f + β ) ( τ i + τ ) ] exp ( j 2 π β τ i ) } F i .
[ W H i ( β ) ] k , k X i k ( β ) = Σ n P i ( n F i ) X k n M i ( β ) ( W τ i ) k n M i , k n M i .
[ W H i ( β ) ] k , k X i k ( β ) = Σ l = 0 M 1 X l ( β ) ( W τ i ) l , l Σ n P i ( n F i ) δ [ l ( k n M i ) ] .
( Q i ) k + 1 , l + 1 = Σ n P i ( n F i ) δ [ l ( k + n M i ) ] .
[ x i ( β ) ] k j = X i k j ( β ) , 1 k j M i
[ x ( β ) ] l = X l ( β ) , 1 l M .
W H i ( β ) x i ( β ) = Q i W τ i x ( β ) .
x ̂ ( β ) = Q x ( β ) ,
x ̂ ( β ) = ( W H 1 ( β ) x 1 ( β ) W H 2 ( β ) x 2 ( β ) W H P ( β ) x P ( β ) ) , Q = ( Q 1 W τ 1 Q 2 W τ 2 Q P W τ P ) .
min x ( β ) , α 1 , . . . , α P , Σ i α i 2 > 0 Q S x ( β ) [ α 1 b 1 ( β ) α P b P ( β ) ] 2 ,
V T Va = λ a ,

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