Abstract

We report the successful demonstration of a compressively sampled photonic link. The system takes advantage of recent theoretical developments in compressive sampling to enable signal recovery beyond the Nyquist limit of the digitizer. This rather remarkable result requires that (1) the signal being recovered has a sparse (low-dimensional) representation and (2) the digitized samples be incoherent with this representation. We describe an all-photonic system architecture that meets these requirements and then show that 1GHz harmonic signals can be faithfully reconstructed even when digitizing at 500MS/s, well below the Nyquist rate.

© 2011 OSA

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References

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  1. J. Campmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [CrossRef]
  2. C. H. Lee, Microwave Photonics (CRC Press, 2007).
  3. J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
    [CrossRef]
  4. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
    [CrossRef]
  5. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
    [CrossRef]
  6. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
    [CrossRef]
  7. O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
    [CrossRef]
  8. D. Yang, H. Li, G. Peterson, and A. Fathy, “Compressed Sensing Based UWB Receiver: Hardware Compressing and FPGA Reconstruction,” Proceedings of the 43rd Conference on Information Sciences and Systems (CISS) (2009).
  9. M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
    [CrossRef]
  10. M. Mishali and Y. C. Eldar, “Xampling: Analog Data Compression,” vol. http://doi.ieeecomputersociety.org/10.1109/DCC.2010.39 of Proceedings of the 2010 Data Compression Conference, pp. 366–375 (2010).
  11. E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
    [CrossRef]
  12. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
    [CrossRef]
  13. D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
    [CrossRef]
  14. E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [CrossRef]
  15. J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
    [CrossRef]
  16. R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24, 118–124 (2007).
    [CrossRef]
  17. J. M. Nichols, M. Currie, F. Buholtz, and W. A. Link, “Bayesian Estimation of Weak Material Dispersion: Theory and Experiment,” Opt. Express 18(3), 2076–2089 (2010).
    [CrossRef] [PubMed]
  18. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
    [CrossRef]

2010 (3)

M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
[CrossRef]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

J. M. Nichols, M. Currie, F. Buholtz, and W. A. Link, “Bayesian Estimation of Weak Material Dispersion: Theory and Experiment,” Opt. Express 18(3), 2076–2089 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (4)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

2007 (4)

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24, 118–124 (2007).
[CrossRef]

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

C. H. Lee, Microwave Photonics (CRC Press, 2007).

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
[CrossRef]

2006 (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

2005 (1)

E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[CrossRef]

Baraniuk, R. G.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24, 118–124 (2007).
[CrossRef]

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Buholtz, F.

Campmany, J.

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

Candes, E. J.

E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[CrossRef]

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Currie, M.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Donoho, D. L.

D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

Duarte, M. F.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Eldar, Y. C.

M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
[CrossRef]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[CrossRef]

Figueiredo, M. A. T.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
[CrossRef]

Gazit, S.

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Kelly, K. F.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Laska, J. N.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Lee, C. H.

C. H. Lee, Microwave Photonics (CRC Press, 2007).

Link, W. A.

Mishali, M.

M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
[CrossRef]

Mittleman, D. M.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Nichols, J. M.

Novak, D.

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

Nowak, R. D.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
[CrossRef]

Romberg, J.

J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Romberg, J. K.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

Segev, M.

Silberburg, Y.

O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Szameit, A.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[CrossRef]

Tropp, J. A.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Wright, S. J.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (2)

M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010).
[CrossRef]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007).
[CrossRef]

IEEE Signal Process. Mag. (4)

E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24, 118–124 (2007).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

IEEE Trans. Inf. Theory (4)

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals,” IEEE Trans. Inf. Theory 56(1), 520–544 (2010).
[CrossRef]

E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

Nat. Photonics (1)

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

Opt. Express (2)

Other (3)

M. Mishali and Y. C. Eldar, “Xampling: Analog Data Compression,” vol. http://doi.ieeecomputersociety.org/10.1109/DCC.2010.39 of Proceedings of the 2010 Data Compression Conference, pp. 366–375 (2010).

D. Yang, H. Li, G. Peterson, and A. Fathy, “Compressed Sensing Based UWB Receiver: Hardware Compressing and FPGA Reconstruction,” Proceedings of the 43rd Conference on Information Sciences and Systems (CISS) (2009).

C. H. Lee, Microwave Photonics (CRC Press, 2007).

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Figures (5)

Fig. 1
Fig. 1

(a) Overview of the approach. Both the signal voltage and the pseudo-random bit sequence (PRBS) voltage applied to the photonic compressive-sampling system modulate optical power and yield a mixing term in the output voltage. (b) Detailed system layout of the compressively sampled, photonic link. Electro-optical components are shown as shaded boxes. PRBS = pseudo-random bit sequence, DFB = Distributed-feedback laser, RFA = RF Amplifier, MZM = Mach-Zehnder modulator, 3-dB = RF power divider, PD = photodetector, LPF = low-pass filter, and 10-dB=10 dB attenuator. The compressed samples y(t) are recorded on Channel 4. The other channels were recorded for calibrating the system and testing the fidelity of our signal model.

Fig. 2
Fig. 2

(a) Frequency domain and (b) Time domain representation of the filter, sampled at the 10GHz rate.

Fig. 3
Fig. 3

Acquired vs. predicted compressed measurements .

Fig. 4
Fig. 4

(a) The reconstructed cosine basis coefficients θ and (b) the associated signal reconstruction .

Fig. 5
Fig. 5

(a) The N = 1002 cosine basis coefficients θ spanning 0 – 5GHz, recovered from only M = 51 data points collected at 500MS/s.

Equations (13)

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

x i = j = 1 N W i j θ j ,
y = Φ x = ΦW θ ,
θ ^ : min θ { y ΦW θ 2 2 + τ θ 1 }
P o = ( P i / 2 ) ( 1 + sin ( ψ ) )
v out = ( a + b sin ( ψ ) )
v out ( t ) = ( a + b sin ( ϕ ( t ) ) ) ( c + d sin ( γ ( t ) ) )
v out ( t ) = a d × sin ( γ ( t ) ) + [ b c + b d × sin ( γ ( t ) ) ] sin ( ϕ ( t ) )
y ( t ) = A sin ( π × v PRBS ( t ) / V π PRBS ) + [ B + C sin ( π + v PRBS ( t ) / V π PRBS ) ] × sin ( π x ( t ) / V π s )
y ˜ = y A sin ( π × v PRBS ( t ) / V π PRBS ) = π V π s i g [ B + C sin ( π × v PRBS ( t ) / V π PRBS ) ] x .
R = diag ( π V π s i g [ B + C sin ( π × v PRBS ( t ) / V π PRBS ) ] ) .
H = [ h ( 1 ) 0 0 0 0 0 h ( 2 ) h ( 1 ) 0 0 0 0 h ( L ) h ( L 1 ) h ( 1 ) 0 0 0 h ( L ) h ( L 1 ) h ( 1 ) 0 0 0 h ( L ) h ( L 1 ) h ( 1 ) ]
D i j = δ ( i j / M ) i = 1 M , j = 1 N
y ˜ = DHRW θ .

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