Abstract

Because optical systems have a huge bandwidth and are capable of generating low-noise short pulses, they are ideal for undersampling multiband signals that are located within a very broad frequency range. We propose a new scheme for reconstructing multiband signals that occupy a small part of a given broad frequency range under the constraint of a small number of sampling channels. The scheme, which we call multirate sampling (MRS), entails gathering samples at several different rates whose sum is significantly lower than the Nyquist sampling rate. The number of channels does not depend on any characteristics of a signal. In order to be implemented with simplified hardware, the reconstruction method does not rely on the synchronization between different sampling channels. Also, because the method does not solve a system of linear equations, it avoids one source of lack of robustness of previously published undersampling schemes. Our simulations indicate that our MRS scheme is robust both to different signal types and to relatively high noise levels. The scheme can be implemented easily with optical sampling systems.

© 2008 Optical Society of America

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References

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  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]
  2. H. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37-52 (1967).
    [CrossRef]
  3. A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432-1436 (1953).
    [CrossRef]
  4. R. Venkantaramani and Y. Bresler, “Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals,” IEEE Trans. Signal Process. 49, 2301-2313 (2001).
    [CrossRef]
  5. Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process. 56, 2334-2345 (2008).
    [CrossRef]
  6. M. Mishali and Y. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” http://www.arxiv.org/abs/0709.1563.
  7. P. Feng and Y. Bresler, “Spectrum-blind minimum-rate sampling and reconstruction of multiband signals,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), Vol. 1, pp. 701-704.
    [CrossRef]
  8. Y. P. Lin and P. P. Vaidyanathan, “Periodically uniform sampling of bandpass signals,” J. Chem. Phys. 45, 340-351 (1998).
  9. C. Herley and W. Wong, “Minimum rate sampling and reconstruction of signals with arbitrary frequency support,” IEEE Trans. Inf. Theory 45, 1555-1564 (1999).
    [CrossRef]
  10. I. Stewart and D. Tall, The Foundations of Mathematics (Oxford U. Press, 1977).

2008

Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process. 56, 2334-2345 (2008).
[CrossRef]

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]

2001

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

C. Herley and W. Wong, “Minimum rate sampling and reconstruction of signals with arbitrary frequency support,” IEEE Trans. Inf. Theory 45, 1555-1564 (1999).
[CrossRef]

1998

Y. P. Lin and P. P. Vaidyanathan, “Periodically uniform sampling of bandpass signals,” J. Chem. Phys. 45, 340-351 (1998).

1967

H. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37-52 (1967).
[CrossRef]

1953

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432-1436 (1953).
[CrossRef]

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]

P. Feng and Y. Bresler, “Spectrum-blind minimum-rate sampling and reconstruction of multiband signals,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), Vol. 1, pp. 701-704.
[CrossRef]

Do, M. N.

Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process. 56, 2334-2345 (2008).
[CrossRef]

Eldar, Y.

M. Mishali and Y. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” http://www.arxiv.org/abs/0709.1563.

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]

Feng, P.

P. Feng and Y. Bresler, “Spectrum-blind minimum-rate sampling and reconstruction of multiband signals,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), Vol. 1, pp. 701-704.
[CrossRef]

Herley, C.

C. Herley and W. Wong, “Minimum rate sampling and reconstruction of signals with arbitrary frequency support,” IEEE Trans. Inf. Theory 45, 1555-1564 (1999).
[CrossRef]

Horowitz, M.

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]

Kohlenberg, A.

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432-1436 (1953).
[CrossRef]

Landau, H.

H. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37-52 (1967).
[CrossRef]

Lin, Y. P.

Y. P. Lin and P. P. Vaidyanathan, “Periodically uniform sampling of bandpass signals,” J. Chem. Phys. 45, 340-351 (1998).

Lu, Y. M.

Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process. 56, 2334-2345 (2008).
[CrossRef]

Mishali, M.

M. Mishali and Y. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” http://www.arxiv.org/abs/0709.1563.

Stewart, I.

I. Stewart and D. Tall, The Foundations of Mathematics (Oxford U. Press, 1977).

Tall, D.

I. Stewart and D. Tall, The Foundations of Mathematics (Oxford U. Press, 1977).

Vaidyanathan, P. P.

Y. P. Lin and P. P. Vaidyanathan, “Periodically uniform sampling of bandpass signals,” J. Chem. Phys. 45, 340-351 (1998).

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]

Wong, W.

C. Herley and W. Wong, “Minimum rate sampling and reconstruction of signals with arbitrary frequency support,” IEEE Trans. Inf. Theory 45, 1555-1564 (1999).
[CrossRef]

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]

Acta Math.

H. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37-52 (1967).
[CrossRef]

IEEE Trans. Inf. Theory

C. Herley and W. Wong, “Minimum rate sampling and reconstruction of signals with arbitrary frequency support,” IEEE Trans. Inf. Theory 45, 1555-1564 (1999).
[CrossRef]

IEEE Trans. Signal Process.

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

Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process. 56, 2334-2345 (2008).
[CrossRef]

J. Appl. Phys.

A. Kohlenberg, “Exact interpolation of band-limited functions,” J. Appl. Phys. 24, 1432-1436 (1953).
[CrossRef]

J. Chem. Phys.

Y. P. Lin and P. P. Vaidyanathan, “Periodically uniform sampling of bandpass signals,” J. Chem. Phys. 45, 340-351 (1998).

Opt. Commun.

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]

Other

M. Mishali and Y. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” http://www.arxiv.org/abs/0709.1563.

P. Feng and Y. Bresler, “Spectrum-blind minimum-rate sampling and reconstruction of multiband signals,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), Vol. 1, pp. 701-704.
[CrossRef]

I. Stewart and D. Tall, The Foundations of Mathematics (Oxford U. Press, 1977).

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Figures (7)

Fig. 1
Fig. 1

Illustration of the spectrum of a sparse one-band real signal (a) and the spectrum of its samples that are obtained for the sampling rates F 1 (b) and F 2 (c). At f 0 , the signal is unaliased at the sampling rate F 1 but is aliased at the sampling rate F 2 .

Fig. 2
Fig. 2

Illustration demonstrating how support consistency is checked. The input of the algorithm is the sampled signals whose spectra X 1 ( f ) and X 2 ( f ) are shown Figs. 1b, 1c, respectively; their respective indicator functions I 1 ( f ) and I 2 ( f ) are shown in Figs. 2a, 2b. Figure 2c shows the indicator function I ( f ) = I 1 ( f ) I 2 ( f ) . In Figs. 2d, 2e, we check whether the subset U = { U 2 } P { U } is support consistent. Figures 2d, 2e show the indicator functions for the downconversion of U 2 at rates F 1 and F 2 : I U 2 1 ( f ) and I U 2 2 ( f ) , respectively. The dashed lines illustrate U 2 , U 2 , and their downconversions. It is evident that the functions I 1 ( f ) and I U 2 1 ( f ) are not equal. Hence, U = { U 2 } is not a support-consistent combination.

Fig. 3
Fig. 3

Success percentage for the first set of simulations with F 0 = 1 GHz as a function of the Nyquist rate. In Fig. 3a, the percentage of a correct band detection is shown by the squares. The full reconstruction percentage is shown by circles. The open circles and squares represent the results obtained when the assumed maximum number of positive bands equals three. The dark circles and squares represent the cases in which the maximum assumed positive band number equals four. The full reconstruction percentages were the same for both choices of the maximum number of bands, and thus the open and dark circles are lndistinghishable in this figure. Figure 3b shows the band-detection percentage (solid curve) and reconstruction percentages (dashed curve) in the case that both the maximum number of originating and assumed positive bands equals four.

Fig. 4
Fig. 4

Run time for the second set of simulations as a function of the Nyquist rate. The results in the case of four input positive bands with an assumed number of positive bands equal to four is shown by the solid curve. The results in the case of three input positive bands are shown with the dotted curve in the case of three assumed positive bands and with the dashed curve in the case of four assumed positive bands.

Fig. 5
Fig. 5

Success percentage for the first set of simulations as a function of the sum of the sampling rates divided by the Landau rate. As in Fig. 3, in Fig. 5a, the percentage of a correct band detection is shown by the squares. The full reconstruction percentage is shown by circles. The open circles and squares represent the results obtained when the assumed maximum number of positive bands equals three. The dark circles and squares represent the cases in which the maximum assumed positive band number equals four. Figure 5b shows the band-detection percentage (solid curve) and reconstruction percentages (dashed curve) in the case where both the maximum number of originating and assumed positive bands equals four.

Fig. 6
Fig. 6

Run time for the first set of simulations as a function of the sum of the sampling rates divided by the Landau rate in the cases of signals with four and three positive bands. The results in the case of four input positive bands with an assumed number of positive bands equal to four is shown by the solid curve. The results in the case of three input positive bands is shown by the dotted curve in the case of three assumed positive bands and by the dashed curve in the case of four assumed positive bands.

Fig. 7
Fig. 7

Success percentage for the third set of simulations with F 0 = 1 GHz and F nyq = 20 GHz as a function of standard deviation σ of the added noise. The figure shows the band-detection percentage (solid curve) and reconstruction percentages (dashed curve) in the case where both the maximum number of originating and assumed positive bands equals four.

Equations (29)

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X ( f ) = x ( t ) exp ( 2 π i f t ) .
X ( f ) = n = 1 N [ S n ( f ) + S ¯ n ( f ) ] ,
x i ( t ) = x ( t + Δ i ) n = δ ( t n F i ) ,
X i ( f ) = F i n = X ( f + n F i ) exp [ 2 π i ( f + n F i ) Δ i ] .
D i ( f ) = min [ f mod F i , ( F i f ) mod F i ] .
{ ( m = [ α i + m F i , β i + m F i ] ) ( m = [ β i + m F i , α i + m F i ] ) } [ 0 , F nyq 2 ] ,
I i ( f ) = { 1 for all f [ 0 , F nyq 2 ] such that for all ε > 0 , f ε f + ε X i ( f ) 2 d f > 0 0 otherwise . }
I ( f ) = i = 1 P I i ( f ) , f [ 0 , F nyq 2 ] .
I k ( f ) = { 1 if f U k 0 otherwise . }
I ( f ) = k = 1 K I k ( f ) .
I k i ( f ) = I [ 0 , F i 2 ] ( f ) H ( n = n = I k ( f + n F i ) + I k ( f + n F i ) ) .
I [ 0 , F i 2 ] ( f ) = { 1 if f [ 0 , F i 2 ] 0 otherwise . }
H ( f ) = { 0 if f 0 1 if f > 0 . }
I U i ( f ) = U k U I k i ( f ) , f [ 0 , F i 2 ] .
E 1 ( U ) = i = 1 P 0 F i 2 I U i ( f ) I i ( f ) d f .
E 2 ( U ) = i 1 i 2 Σ U i 1 Σ U i 2 ( X i 1 ( f ) F i 1 X i 2 ( f ) F i 2 ) 2 d f .
E 3 ( U ) = i 1 i 2 λ ( Σ U i 1 Σ U i 2 ) ,
X U ( f ) = 1 r ( f ) i = 1 P X i ( f ) I Σ U i ( f ) F i n .
arg [ X i 1 ( f ) X i 2 ( f ) ] = 2 π f ( Δ i 1 Δ i 2 ) + 2 π k ,
for some integer k .
I i ( f ) = { 1 if f [ 0 , F nyq 2 ] and 1 2 ξ f ξ f ξ X α ( f ) d f > T 0 otherwise . }
E 1 ( U ) < a min U [ E 1 ( U ) ] + b ,
E ̂ 3 ( U ) = i 1 i 2 0 F nyq 2 X i 1 ( f ) F i 1 2 W i 1 , i 2 ( f , U ) d f ,
μ i 1 , i 2 k ( U ) = V k i 1 , i 2 X i 1 ( f ) F i 1 X i 2 ( f ) F i 2 d f V k i 1 , i 2 X i 1 ( f ) + X i 2 ( f ) d f .
W i 1 , i 2 ( f ) = k exp [ ρ μ i 1 , i 2 k ( U ) ] I V k i 1 , i 2 ( f ) ,
E tot ( U ) = E 1 ( U ) min U { E 1 ( U ) } E 2 ( U ) min U { E 2 ( U ) } + E ̂ 3 ( U ) min U { E ̂ 3 ( U ) }
n = 1 N ( K n ) .
S n ( f ) = { A n cos [ π ( f f n ) B n ] if 2 f f m B n < 1 0 otherwise , }
B m X U ( f ) X ( f ) < max i ( σ i ) B m .

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