Abstract

Compressive sensing (CS) of sparse gigahertz-band RF signals using microwave photonics may achieve better performances with smaller size, weight, and power than electronic CS or conventional Nyquist rate sampling. The critical element in a CS system is the device that produces the CS measurement matrix (MM). We show that passive speckle patterns in multimode waveguides potentially provide excellent MMs for CS. We measure and calculate the MM for a multimode fiber and perform simulations using this MM in a CS system. We show that the speckle MM exhibits the sharp phase transition and coherence properties needed for CS and that these properties are similar to those of a sub-Gaussian MM with the same mean and standard deviation. We calculate the MM for a multimode planar waveguide and find dimensions of the planar guide that give a speckle MM with a performance similar to that of the multimode fiber. The CS simulations show that all measured and calculated speckle MMs exhibit a robust performance with equal amplitude signals that are sparse in time, in frequency, and in wavelets (Haar wavelet transform). The planar waveguide results indicate a path to a microwave photonic integrated circuit for measuring sparse gigahertz-band RF signals using CS.

© 2016 Optical Society of America

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References

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Y. Chen, X. Yu, H. Chi, S. Zheng, X. Zhang, X. Jin, and M. Galili, Opt. Commun. 338, 428 (2015).
[Crossref]

B. T. Bosworth, J. R. Stroud, D. N. Tran, T. D. Tran, S. Chin, and M. A. Foster, Opt. Lett. 40, 3045 (2015).
[Crossref]

2013 (4)

2012 (2)

2011 (3)

J. M. Nichols and F. Bucholtz, Opt. Express 19, 7339 (2011).
[Crossref]

H. Nan, Y. Gu, and H. Zhang, IEEE Photon. Technol. Lett. 23, 67 (2011).
[Crossref]

A. Juditsky and A. Nemirovski, Math. Program. 127, 57 (2011).
[Crossref]

2010 (3)

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

M. Mishali and Y. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[Crossref]

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 14 (2010).

2009 (1)

D. Donoho and J. Tanner, Philos. Trans. R. Soc. London Ser. A 367, 4273 (2009).
[Crossref]

2008 (2)

I. Loris, Comput. Phys. Commun. 179, 895 (2008). (We also used Section 1.1.3 “Lasso” at http://scikit-learn.org/stable/modules/linear_model.html .)
[Crossref]

E. J. Candès and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).
[Crossref]

2007 (1)

R. G. Baraniuk, IEEE Signal Process. Mag. 24(4), 118 (2007).
[Crossref]

2006 (1)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[Crossref]

1973 (1)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

1971 (1)

Baraniuk, R. G.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

R. G. Baraniuk, IEEE Signal Process. Mag. 24(4), 118 (2007).
[Crossref]

Bosworth, B. T.

Bucholtz, F.

Candès, E. J.

E. J. Candès and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).
[Crossref]

Cao, H.

Cevher, V.

J. T. Parker, V. Cevher, and P. Schniter, “Compressive sensing under matrix uncertainties: an approximate message passing approach,” in 45th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California (IEEE, 2011), pp. 804–808.

Chen, Y.

Chi, H.

Chin, S.

Donoho, D.

D. Donoho and J. Tanner, Philos. Trans. R. Soc. London Ser. A 367, 4273 (2009).
[Crossref]

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[Crossref]

Duarte, M. F.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

Eldar, Y.

M. Mishali and Y. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[Crossref]

Foster, M. A.

Galili, M.

Y. Chen, X. Yu, H. Chi, S. Zheng, X. Zhang, X. Jin, and M. Galili, Opt. Commun. 338, 428 (2015).
[Crossref]

Gloge, D.

Gu, Y.

H. Nan, Y. Gu, and H. Zhang, IEEE Photon. Technol. Lett. 23, 67 (2011).
[Crossref]

Jei, Y.

Jin, X.

Juditsky, A.

A. Juditsky and A. Nemirovski, Math. Program. 127, 57 (2011).
[Crossref]

Laska, J. N.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

Loris, I.

I. Loris, Comput. Phys. Commun. 179, 895 (2008). (We also used Section 1.1.3 “Lasso” at http://scikit-learn.org/stable/modules/linear_model.html .)
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Waveguides (Academic, 1974).

Mei, Y.

Mishali, M.

M. Mishali and Y. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[Crossref]

Nan, H.

H. Nan, Y. Gu, and H. Zhang, IEEE Photon. Technol. Lett. 23, 67 (2011).
[Crossref]

Nemirovski, A.

A. Juditsky and A. Nemirovski, Math. Program. 127, 57 (2011).
[Crossref]

Nichols, J. M.

Parker, J. T.

J. T. Parker, V. Cevher, and P. Schniter, “Compressive sensing under matrix uncertainties: an approximate message passing approach,” in 45th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California (IEEE, 2011), pp. 804–808.

Popoff, S. M.

Redding, B.

Romberg, J. K.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

Schniter, P.

J. T. Parker, V. Cevher, and P. Schniter, “Compressive sensing under matrix uncertainties: an approximate message passing approach,” in 45th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California (IEEE, 2011), pp. 804–808.

Sefler, G. A.

G. C. Valley, G. A. Sefler, and T. J. Shaw, Proc. SPIE 8645, 86450P (2013).
[Crossref]

G. C. Valley, G. A. Sefler, and T. J. Shaw, Opt. Lett. 37, 4675 (2012).
[Crossref]

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 14 (2010).

Shaw, T. J.

G. C. Valley, G. A. Sefler, and T. J. Shaw, Proc. SPIE 8645, 86450P (2013).
[Crossref]

G. C. Valley, G. A. Sefler, and T. J. Shaw, Opt. Lett. 37, 4675 (2012).
[Crossref]

Stroud, J. R.

Tanner, J.

D. Donoho and J. Tanner, Philos. Trans. R. Soc. London Ser. A 367, 4273 (2009).
[Crossref]

Tran, D. N.

Tran, T. D.

Tropp, J. A.

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

Valley, G. C.

G. C. Valley, G. A. Sefler, and T. J. Shaw, Proc. SPIE 8645, 86450P (2013).
[Crossref]

G. C. Valley, G. A. Sefler, and T. J. Shaw, Opt. Lett. 37, 4675 (2012).
[Crossref]

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 14 (2010).

Wakin, M. B.

E. J. Candès and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).
[Crossref]

Wang, D.

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

Yu, X.

Y. Chen, X. Yu, H. Chi, S. Zheng, X. Zhang, X. Jin, and M. Galili, Opt. Commun. 338, 428 (2015).
[Crossref]

Zhang, H.

H. Nan, Y. Gu, and H. Zhang, IEEE Photon. Technol. Lett. 23, 67 (2011).
[Crossref]

Zhang, X.

Zheng, S.

Appl. Opt. (1)

Comput. Phys. Commun. (1)

I. Loris, Comput. Phys. Commun. 179, 895 (2008). (We also used Section 1.1.3 “Lasso” at http://scikit-learn.org/stable/modules/linear_model.html .)
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

M. Mishali and Y. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (1)

H. Nan, Y. Gu, and H. Zhang, IEEE Photon. Technol. Lett. 23, 67 (2011).
[Crossref]

IEEE Signal Process. Mag. (2)

E. J. Candès and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).
[Crossref]

R. G. Baraniuk, IEEE Signal Process. Mag. 24(4), 118 (2007).
[Crossref]

IEEE Trans. Inf. Theory (2)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[Crossref]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
[Crossref]

Math. Program. (1)

A. Juditsky and A. Nemirovski, Math. Program. 127, 57 (2011).
[Crossref]

Opt. Commun. (1)

Y. Chen, X. Yu, H. Chi, S. Zheng, X. Zhang, X. Jin, and M. Galili, Opt. Commun. 338, 428 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Philos. Trans. R. Soc. London Ser. A (1)

D. Donoho and J. Tanner, Philos. Trans. R. Soc. London Ser. A 367, 4273 (2009).
[Crossref]

Proc. SPIE (2)

G. C. Valley, G. A. Sefler, and T. J. Shaw, Proc. SPIE 8645, 86450P (2013).
[Crossref]

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 14 (2010).

Other (2)

J. T. Parker, V. Cevher, and P. Schniter, “Compressive sensing under matrix uncertainties: an approximate message passing approach,” in 45th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California (IEEE, 2011), pp. 804–808.

D. Marcuse, Theory of Dielectric Waveguides (Academic, 1974).

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Figures (10)

Fig. 1.
Fig. 1. Compressive sensing system for measuring sparse RF signals using a multimode waveguide to implement the measurement matrix.
Fig. 2.
Fig. 2. Speckle patterns at end of 1 m long, 105 μm diameter, 0.22 NA multimode for λ = 1539.44    nm (left) and 1539.52 nm (right).
Fig. 3.
Fig. 3. Measured intensity as a function of wavelength at 4 locations within the output plane of a 1 m, 105 μm, 0.22 NA step-index fiber.
Fig. 4.
Fig. 4. Probability of signal recovery as a function of small dimension of the measurement matrix for 100 trials for K = 2 , 4, 8, and 16 and sparse signals in the identity, discrete cosine, and Haar wavelet transforms.
Fig. 5.
Fig. 5. Mutual coherence between rows for speckle MM (left) and Gaussian random-number MM (right).
Fig. 6.
Fig. 6. Probability of recovery as a function of sparsity K and number of measurements M for the speckle MM (left) and the Gaussian random-number MM (right).
Fig. 7.
Fig. 7. Probability of recovery as a function of sparsity K and the number of measurements M for the measured speckle MM with a signal composed of K sinusoids (left) and a signal that is K sparse under the Haar wavelet transform (right).
Fig. 8.
Fig. 8. Speckle patterns for planar guides. (a) 10 cm long by width 5 to 30 μm at λ = 1.537 and 1.53701 mm. (b) 25.4 μm wide by 1 mm to 1 m long for 50 wavelengths ( Δλ = 0.01    nm ).
Fig. 9.
Fig. 9. Probability of recovery as a function of sparsity K and the small dimension of the MM M for the calculated fiber MM (left) and the calculated planar waveguide MM (right).
Fig. 10.
Fig. 10. Two independent calibrations of the multimode fiber separated in time by more than 1 h.

Equations (2)

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y = Φ x = Φ Ψ 1 s ,
C i j = Φ ( i ) · Φ ( j ) / [ | Φ ( i ) | | Φ ( j ) | ] ,

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