Abstract

We demonstrate an optical mixing system for measuring properties of sparse radio frequency (RF) signals using compressive sensing (CS). Two types of sparse RF signals are investigated: (1) a signal that consists of a few 0.4 ns pulses in a 26.8 ns window and (2) a signal that consists of a few sinusoids at different frequencies. The RF is modulated onto the intensity of a repetitively pulsed, wavelength-chirped optical field, and time-wavelength-space mapping is used to map the optical field onto a 118-pixel, one-dimensional spatial light modulator (SLM). The SLM pixels are programmed with a pseudo-random bit sequence (PRBS) to form one row of the CS measurement matrix, and the optical throughput is integrated with a photodiode to obtain one value of the CS measurement vector. Then the PRBS is changed to form the second row of the mixing matrix and a second value of the measurement vector is obtained. This process is performed 118 times so that we can vary the dimensions of the CS measurement matrix from 1×118 to 118×118 (square). We use the penalized 1 norm method with stopping parameter λ (also called basis pursuit denoising) to recover pulsed or sinusoidal RF signals as a function of the small dimension of the measurement matrix and stopping parameter. For a square matrix, we also find that penalized 1 norm recovery performs better than conventional recovery using matrix inversion.

© 2012 Optical Society of America

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  1. R. G. Baraniuk, IEEE Signal Process. Mag. 24, 118(2007).
    [CrossRef]
  2. J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, IEEE Trans. Inf. Theory 56, 520 (2010).
    [CrossRef]
  3. M. Mishali and Y. Eldar, IEEE J. Sel. Top. Signal Process 4, 375 (2010).
    [CrossRef]
  4. G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 77970F (2010).
    [CrossRef]
  5. I. Loris, Comput. Phys. Commun. 179, 895 (2008).
    [CrossRef]
  6. P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (2011).
    [CrossRef]
  7. S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.
  8. A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
    [CrossRef]
  9. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
    [CrossRef]
  10. G. C. Valley and T. J. Shaw, in Applications of Digital Signal Processing, C. Cuadrado-Laborde, ed. (Intech, 2011), pp. 169–190.

2011 (1)

P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (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, 77970F (2010).
[CrossRef]

2008 (2)

I. Loris, Comput. Phys. Commun. 179, 895 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

2007 (1)

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

2000 (1)

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

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]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

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

Chang-Hussain, C.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

Dinu, R.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

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]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

Eldar, Y.

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

Gill, P. R.

P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (2011).
[CrossRef]

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[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]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

Loris, I.

I. Loris, Comput. Phys. Commun. 179, 895 (2008).
[CrossRef]

Mishali, M.

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

Molnar, A.

P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (2011).
[CrossRef]

Nuccio, S. R.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

Parekh, D.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

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]

Sefler, G. A.

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 77970F (2010).
[CrossRef]

Shamee, B.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

Shaw, T. J.

G. C. Valley and T. J. Shaw, in Applications of Digital Signal Processing, C. Cuadrado-Laborde, ed. (Intech, 2011), pp. 169–190.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

Ting, S.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

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 and G. A. Sefler, Proc. SPIE 7797, 77970F (2010).
[CrossRef]

G. C. Valley and T. J. Shaw, in Applications of Digital Signal Processing, C. Cuadrado-Laborde, ed. (Intech, 2011), pp. 169–190.

Wang, A.

P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (2011).
[CrossRef]

Weiner, A. M.

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Wilner, A. E.

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

Comput. Phys. Commun. (1)

I. Loris, Comput. Phys. Commun. 179, 895 (2008).
[CrossRef]

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

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

IEEE Signal Process. Mag. (2)

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

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 25, 83 (2008).
[CrossRef]

IEEE Trans. Inf. Theory (1)

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

IEEE Trans. Signal Process (1)

P. R. Gill, A. Wang, and A. Molnar, IEEE Trans. Signal Process 59, 4595 (2011).
[CrossRef]

Proc. SPIE (1)

G. C. Valley and G. A. Sefler, Proc. SPIE 7797, 77970F (2010).
[CrossRef]

Rev. Sci. Instrum. (1)

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Other (2)

S. R. Nuccio, R. Dinu, B. Shamee, D. Parekh, C. Chang-Hussain, and A. E. Wilner, Proceedings of Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC) (2011), p. 1.

G. C. Valley and T. J. Shaw, in Applications of Digital Signal Processing, C. Cuadrado-Laborde, ed. (Intech, 2011), pp. 169–190.

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Figures (5)

Fig. 1.
Fig. 1.

Optical time domain mixer showing the mode-locked laser (MLL), dispersion compensating fiber (DCF), Mach–Zehnder modulator (MZM), grating, spatial light modulator, photodiode (PD), and analog-to-digital converter (ADC).

Fig. 2.
Fig. 2.

Amplitude of input (blue line) and recovered signal (black dots) as a function of SLM pixel for a N=118 by M mixing matrix with M=20, 40 and λ=0.5.

Fig. 3.
Fig. 3.

Mean square error for N=118 as a function of (a) small dimension of the mixing matrix M with the stopping parameter λ as a parameter and (b) λ for values of M.

Fig. 4.
Fig. 4.

Amplitude of input (solid blue line) and recovered (black dots) signal as a function of pixel for a 39×39 mixing matrix. Recovered using inverse of measurement matrix (left). Recovered using penalized 1 norm (right) with λ=0.01.

Fig. 5.
Fig. 5.

Input (blue) and recovered (black, dashed) amplitude as a function of SLM pixel for a sparse RF signal composed of sinusoids at 71.42 and 214.16 MHz (left: N=118, M=40, λ=0.001, right: N=118, M=118, λ=0.001).

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y=Φx+n=ΦΨ1s+n=Θs+n,
s(λ)=argmins(λs1+1/2yΘs22),

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