Abstract

The emerging field of compressive sampling has potentially powerful implications for the design of analog-to-digital sampling systems. In particular, the mathematics of compressive sampling suggests that one can recover a signal at a smaller sampling interval than is dictated by the rate at which the samples are digitized. In a recent work the authors presented an all-photonic implementation of such a system and experimentally demonstrated the basic operating principles. This paper offers a more in-depth study of the system, including a more detailed description of the hardware, issues involved in real-time implementation, and how choice of signal model and model fidelity can influence the reconstruction.

© 2012 Optical Society of America

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References

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  1. 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, 520–544 (2010).
    [CrossRef]
  2. J. M. Nichols and F. Bucholtz, “Beating Nyquist with light: a compressively sampled photonic link,” Opt. Express 19, 7339–7348 (2011).
    [CrossRef]
  3. Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
    [CrossRef]
  4. 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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.
  5. M. Mishali and Y. C. Eldar, “Xampling: analog data compression,” in Proceedings of the 2010 Data Compression Conference (IEEE, 2010), pp. 366–375.
  6. 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, 375–391 (2010).
    [CrossRef]
  7. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
    [CrossRef]
  8. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  9. R. G. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag.24(4), 118–124 (2007).
    [CrossRef]
  10. J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
    [CrossRef]
  11. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [CrossRef]
  12. J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
    [CrossRef]
  13. D. L. Donoho and J. Tanner, “Precise undersampling theorems,” Proc. IEEE 98, 913–924 (2010).
    [CrossRef]
  14. C. E. Shannon, “Communication in the presence of noise,” Proc. IEEE 86, 447–457 (1998), reprinted from Proc. IRE 37, 10–21 (1949).
    [CrossRef]
  15. E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
    [CrossRef]
  16. E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
    [CrossRef]
  17. D. L. Donoho and J. Tanner, “Exponential bounds implying construction of compressed sensing matrices, error-correcting codes, and neighborly polytopes by random sampling,” IEEE Trans. Inf. Theory 56, 2002–2016 (2010).
    [CrossRef]
  18. D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
    [CrossRef]
  19. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
    [CrossRef]
  20. 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, 586–597 (2007).
    [CrossRef]
  21. R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Tech. Rep. (Technion–Israel Institute of Technology, 2008).
  22. A. Harms, W. U. Bajwa, and R. Calderbank, “Beating Nyquist through correlations: a constrained random demodulator for sampling of sparse bandlimited signals,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2011), pp. 5968–5971.

2011 (3)

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[CrossRef]

J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
[CrossRef]

J. M. Nichols and F. Bucholtz, “Beating Nyquist with light: a compressively sampled photonic link,” Opt. Express 19, 7339–7348 (2011).
[CrossRef]

2010 (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, 520–544 (2010).
[CrossRef]

D. L. Donoho and J. Tanner, “Exponential bounds implying construction of compressed sensing matrices, error-correcting codes, and neighborly polytopes by random sampling,” IEEE Trans. Inf. Theory 56, 2002–2016 (2010).
[CrossRef]

D. L. Donoho and J. Tanner, “Precise undersampling theorems,” Proc. IEEE 98, 913–924 (2010).
[CrossRef]

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, 375–391 (2010).
[CrossRef]

2009 (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

2008 (3)

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

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

2007 (1)

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, 586–597 (2007).
[CrossRef]

2006 (3)

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

2005 (1)

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

1998 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IEEE 86, 447–457 (1998), reprinted from Proc. IRE 37, 10–21 (1949).
[CrossRef]

Bajwa, W. U.

A. Harms, W. U. Bajwa, and R. Calderbank, “Beating Nyquist through correlations: a constrained random demodulator for sampling of sparse bandlimited signals,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2011), pp. 5968–5971.

Baraniuk, R.

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

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, 520–544 (2010).
[CrossRef]

R. G. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag.24(4), 118–124 (2007).
[CrossRef]

Blanchard, J. D.

J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Bucholtz, F.

Calderbank, A. R.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[CrossRef]

Calderbank, R.

A. Harms, W. U. Bajwa, and R. Calderbank, “Beating Nyquist through correlations: a constrained random demodulator for sampling of sparse bandlimited signals,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2011), pp. 5968–5971.

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, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

Cartis, C.

J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
[CrossRef]

Chi, Y.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[CrossRef]

Donoho, D. L.

D. L. Donoho and J. Tanner, “Precise undersampling theorems,” Proc. IEEE 98, 913–924 (2010).
[CrossRef]

D. L. Donoho and J. Tanner, “Exponential bounds implying construction of compressed sensing matrices, error-correcting codes, and neighborly polytopes by random sampling,” IEEE Trans. Inf. Theory 56, 2002–2016 (2010).
[CrossRef]

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

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

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (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, 520–544 (2010).
[CrossRef]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Tech. Rep. (Technion–Israel Institute of Technology, 2008).

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, 375–391 (2010).
[CrossRef]

M. Mishali and Y. C. Eldar, “Xampling: analog data compression,” in Proceedings of the 2010 Data Compression Conference (IEEE, 2010), pp. 366–375.

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, 586–597 (2007).
[CrossRef]

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

Harms, A.

A. Harms, W. U. Bajwa, and R. Calderbank, “Beating Nyquist through correlations: a constrained random demodulator for sampling of sparse bandlimited signals,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2011), pp. 5968–5971.

Laska, J.

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

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, 520–544 (2010).
[CrossRef]

Massoud, Y.

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

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, 375–391 (2010).
[CrossRef]

M. Mishali and Y. C. Eldar, “Xampling: analog data compression,” in Proceedings of the 2010 Data Compression Conference (IEEE, 2010), pp. 366–375.

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

Nichols, J. M.

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, 586–597 (2007).
[CrossRef]

Pezeshki, A.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[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 the IEEE International Symposium on Circuits and Systems (IEEE,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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

Romberg, J.

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[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, 520–544 (2010).
[CrossRef]

Rubinstein, R.

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Tech. Rep. (Technion–Israel Institute of Technology, 2008).

Scharf, L. L.

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IEEE 86, 447–457 (1998), reprinted from Proc. IRE 37, 10–21 (1949).
[CrossRef]

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

Tanner, J.

J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
[CrossRef]

D. L. Donoho and J. Tanner, “Exponential bounds implying construction of compressed sensing matrices, error-correcting codes, and neighborly polytopes by random sampling,” IEEE Trans. Inf. Theory 56, 2002–2016 (2010).
[CrossRef]

D. L. Donoho and J. Tanner, “Precise undersampling theorems,” Proc. IEEE 98, 913–924 (2010).
[CrossRef]

Tao, T.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

Temlyakov, V. N.

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
[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, 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, 586–597 (2007).
[CrossRef]

Zibulevsky, M.

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Tech. Rep. (Technion–Israel Institute of Technology, 2008).

C. R. Math. (1)

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

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

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, 586–597 (2007).
[CrossRef]

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, 375–391 (2010).
[CrossRef]

IEEE Signal Process. Mag. (2)

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

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

IEEE Trans. Inf. Theory (6)

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

D. L. Donoho and J. Tanner, “Exponential bounds implying construction of compressed sensing matrices, error-correcting codes, and neighborly polytopes by random sampling,” IEEE Trans. Inf. Theory 56, 2002–2016 (2010).
[CrossRef]

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
[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, 520–544 (2010).
[CrossRef]

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

IEEE Trans. Signal Process. (1)

Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process. 59, 2182–2195 (2011).
[CrossRef]

Opt. Express (1)

Proc. IEEE (2)

D. L. Donoho and J. Tanner, “Precise undersampling theorems,” Proc. IEEE 98, 913–924 (2010).
[CrossRef]

C. E. Shannon, “Communication in the presence of noise,” Proc. IEEE 86, 447–457 (1998), reprinted from Proc. IRE 37, 10–21 (1949).
[CrossRef]

SIAM Rev. (2)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

J. D. Blanchard, C. Cartis, and J. Tanner, “Compressed sensing: how sharp is the restricted isometry property,” SIAM Rev. 53, 105–125 (2011).
[CrossRef]

Other (5)

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 the IEEE International Symposium on Circuits and Systems (IEEE,2008), pp. 1480–1483.

M. Mishali and Y. C. Eldar, “Xampling: analog data compression,” in Proceedings of the 2010 Data Compression Conference (IEEE, 2010), pp. 366–375.

R. G. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag.24(4), 118–124 (2007).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Tech. Rep. (Technion–Israel Institute of Technology, 2008).

A. Harms, W. U. Bajwa, and R. Calderbank, “Beating Nyquist through correlations: a constrained random demodulator for sampling of sparse bandlimited signals,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2011), pp. 5968–5971.

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Figures (8)

Fig. 1.
Fig. 1.

Conditions under which we expect to solve Eq. (1) given a known random projection Φ and basis Ψ and the estimator [Eq. (2)]. Shown are the curves for N=1000, N=2000, N. Accurate solutions to the inverse problem can be obtained with 99% probability for combinations ρ, δ that lie below the given curves.

Fig. 2.
Fig. 2.

Detailed block diagram of the hardware acquisition portion of the photonic compressive sampling system employing amplitude modulators. PRBS, pseudorandom bit sequence; RFA, radio-frequency (RF) amplifier; DFB, distributed feedback laser; MZM, Mach–Zehnder modulator; 3 dB, 3 dB RF power divider; PD, photodetector; 10 dB, 10 dB attenuator; LPF, low-pass filter.

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

Normalized error for optical CS ADC described in this paper as a function of ρ and δ for N=2000. The transition boundary is that predicted by the polytope theory of Donoho and Tanner [17].

Fig. 5.
Fig. 5.

Coherence as a function of how the PRBS is constructed. The coherence ranges from 1<Mμ(A)M (for this set of calculations we set M=N). In this case values are significantly closer to unity than to M, suggesting that indeed the optical CS ADC is likely to meet the criteria for good recovery.

Fig. 6.
Fig. 6.

Compressed samples (dots) acquired at (a) 1Gsamples/s and (c) 333 MHz (δ=0.1 and δ=0.033, respectively), superimposed on the uncompressed 10Gsample/s waveform present at channel 4 of the digitizer (see Fig. 2). (b), (d) Reconstructed signal x^ superimposed on a replica of the actual input signal x(t) (channel 2 of the digitizer in Fig. 2).

Fig. 7.
Fig. 7.

Expected system performance averaged over 100 separate experimental runs. Shown is the expected error E[x^x22/x22] as a function of the sparsity-promoting term τ. Also shown are the number of support failures, defined as the number of times the recovery algorithm failed to identify the correct basis support. For this example the system was using 363 samples, collected at 910Msamples/s, to reconstruct 4000 samples at 10.0Gsamples/s.

Fig. 8.
Fig. 8.

Error as a function of basis mismatch. When the tone directly aligns with a basis vector, excellent recovery is achieved. However, if the tone lies at the midpoint of two vectors, the error increases significantly.

Equations (10)

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

y=Φx+η,
x^:minxx1subject toyΦx22<ξ
y=ΦΨθ+η,θ^:minθ{θ1subject toyΦΨθ22<ξ},
μ(A)max1i,jNij|AiTAj|.
y(t)=h(t)*(α(B+Csin(βVR(t)))x(t)+Asin(βVR(t)))+η(t),
y=DHR1+DHR2x+η,
R1=Asin(βVR),R2=diag{α[B+Csin(βVR)]}
H=[h(1)00000h(2)h(1)0000h(W)h(W1)h(1)0000h(W)h(W1)h(1)]
θ^:minθ{yΦΨθ22+τθ1},
x(n)=k=1Ksin(2πfknΔNy+ϕk)n=1N,

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