## Beating Nyquist with light: a compressively sampled photonic link |

Optics Express, Vol. 19, Issue 8, pp. 7339-7348 (2011)

http://dx.doi.org/10.1364/OE.19.007339

Acrobat PDF (770 KB)

### 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

## 1. Introduction

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

*f*= 1/Δ which, according to the Shannon-Nyquist sampling theorem, guarantees exact recovery of signals possessing frequency content in the interval 0 ≤

_{ADC}*f*≤

*f*= 1/2Δ. However many signals do not possess significant energy outside of

_{Nyquist}*K*discrete frequencies and are said to be

*K*-sparse in the frequency domain. The emerging field of compressive sampling (CS) has suggested that

*random projections*of such signals, digitized at the much slower frequency

*f*∼

_{ADC}*O*(

*K*log(

*f*

_{Nyquist}*/K*)) <

*f*, can be used to recovered the original signal at the 1/Δ rate [3

_{Nyquist}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]

*et al.*describes an experimental imager capable of producing sub-wavelength resolution in the reconstructed image [6

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]

*et al.*also takes advantage of CS to experimentally demonstrate a significant reduction in the number of measurements required in pseudothermal “Ghost Imaging” [7

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

*et al.*[3

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]

*et al.*[8] suggested using a bank of distributed amplifiers to perform the downsampling. However to-date, the only experimental device we are aware of comes from Mishali and Eldar [9

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]

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).

## 2. Review of compressive sampling

*N*pieces of information by collecting

*M << N*linear projections of the signal of interest and applying nonlinear reconstruction algorithms. The fundamentals of this surprising result were developed in the works of Candes & Tao [11

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

*et al.*[12

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]

*x*(

*t*) at a discrete set of temporally sampled points

**x ≡**

*x*=

_{i}*x*((

*i –*1)Δ)

*i*= 1···

*N*where Δ is the sampling interval at which we hope to eventually reconstruct the signal. In compressive sampling, Δ is potentially much smaller than the time interval

*δ*at which the modulated signal is actually digitized. It is common to model the signal as a linear combination of a set of

*N*basis vectors

*W*

_{i j}*i, j*= 1

*··· N*i.e. or in matrix form,

**x**=

**W**. Using this representation the signal is completely specified by the coefficients

*θ**θ*. Compressive sampling takes advantage of situations where the signal model

_{j}**W**is chosen such that

*K*<<

*N*basis vectors provide an accurate representation of the signal. Such a signal is said to be

*K*–sparse in

**W**and the recovery problem becomes one of measuring or estimating the

*θ*. That a signal can often be described with only a few coefficients is not surprising and underlies data compression methods. However in data compression one records all

_{j}*N*observations

*x*and

_{i}*then*solves for

*θ*, storing only the

_{j}*K*coefficients. Compressive sampling circumvents the compression step by directly recording only

*M linear projections*of the signal where

*K < M << N*. That is we record and

*then*recover

*θ*. The matrix

_{j}**Φ**is an

*M × N*matrix that takes the unknown

*N –*vector

**W**

*θ*and projects it onto the

*M*samples

**y**. If

**W**is a standard ortho-basis (as will be the case in this work), and

**Φ**consists of random entries drawn from a suitable probability distribution, then the rows of

**Φ**and columns of

**W**are incoherent and the matrix product

**Φ**

*W*satisfies the restricted isometry property (RIP) [14

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

**is possible from the**

*θ**M–*vector

**y**by solving the minimization problem giving the signal estimate

**x̂**=

**W**. That is, we find the coefficients such that the reconstructed data matches our observations in a least-squares sense, and also possesses the smallest possible

θ ^ *L*

_{1}norm,

**y**at a coarse sampling interval

**x**. The challenge is to design hardware that can perform the function of the projection matrix

**Φ**. Our system realizes this projection through a combination of random optical modulation, low-pass filtering, and electrical digitization at frequency 1

*/δ*.

## 3. System architecture

*et al.*[3

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]

**Φ**with random

*±*1 entries (i.e. a PRBS). This was accomplished in their work by simulating a circuit that multiplied the incoming data by a stream of random

*±*1 digits, spaced Δ apart in time and then sampling the result at the slow time-scale

*δ*. Our system also randomly modulates the signal at the fast time-scale, but does so optically. Additionally, the system proposed in [3

**56**(1), 520–544 (2010). [CrossRef]

*δ*apart, carries information about the signal at the desired fast time-scale Δ. This behavior is necessary to satisfy the RIP property.

^{10}− 1 sequence length. An Analog Signal Generator (Agilent E8257B) provided the clock for the Pattern Generator. The signal itself was produced by a Vector Signal Generator (Agilent E8267C). A distributed-feedback (DFB) laser (EM4 EM253-080-0049) provided light to single-mode optical fiber and two Mach-Zehnder modulators (MZM). The first MZM (JDS Uniphase 21041423) used the signal voltage to modulate the light intensity while the second MZM (JDS Uniphase OC 192 10024180) modulated the light a second time, this time with the PRBS voltage. In both cases, the MZMs were held at optical quadrature manually. Modulated light was then detected using a photodetector (Discovery Semiconductor DSC 30S).

## 4. Signal model

*V*, input optical power

_{i}*P*and output optical power

_{i}*P*is well-known [2]: where ψ =

_{o}*πV*is the optical phase shift in the MZM due to the applied voltage

_{i}/V_{π}*V*and

_{i}*V*is the “half-wave voltage”. At the photodetector, the output voltage is, to a very good approximation, linearly proportional to the optical power. An expression for output voltage more appropriate for an actual (nonideal) device is where the coefficients

_{π}*a*and

*b*are determined through experimental calibration. In the case of two modulators in series, the overall output is just the product of two expressions of the type shown in Eq. (5). Let

*ϕ*denote the phase shift in the first MZM due to the signal voltage and let

*γ*correspond to the phase shift in the second MZM due to the PRBS voltage. Then the output voltage of the photodetector, including PRBS and signal voltages is where

*ϕ*(

*t*) =

*π*×

*v*(

_{sig}*t*)

*/V*,

_{π sig}*γ*(

*t*) =

*π*×

*v*(

_{PRBS}*t*)

*/V*, and

_{π PRBS}*v*(

_{sig}*t*) and

*v*(

_{PRBS}*t*) are, respectively, the signal and PRBS voltages, and

*V*and

_{π sig}*V*are the half-wave voltages for the signal and PRBS modulators, respectively. Note that

_{π PRBS}*v*(

_{sig}*t*) is the signal we wish to recover. As a practical matter, the photodetector output is AC-coupled and, hence the term in Eq. (6) proportional to

*ac*is not present at the output. Then the output voltage can be written or, denoting the input and output voltages

*x*(

*t*) and

*y*(

*t*) respectively, where

*A*=

*ad,B*=

*bc,C*=

*bd*are constants to be estimated. The two half-wave voltages were measured and found to be

*V*= 7.97

_{π PRBS}*V*and

*V*= 4.20

_{π sig}*V*. The other three parameters were estimated from a single set of

*N*= 1002 observations

**y**using a Bayesian approach. Starting with vague priors on each of the variables, we used a simple Markov Chain Monte Carlo algorithm with Gibbs sampling to generate values from each parameter’s posterior distribution [17

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]

*a posterior*values as the final estimate. These estimated values were

*A*= −0.26

*V*,

*B*= 0.27

*V*, and

*C*= −0.49

*V*.

**x**, however for this type of optical link, the signals are small enough that we can make the approximation sin(

*π*

**x**/

*V*) ≈

_{π sig}*π*

**x**/

*V*. Additionally, the first term on the right hand side of Eq. (8) is known and can be subtracted from the observations

_{π sig}**y**. Thus, our signal model becomes The problem can now be put in the form of Eq. (2) by describing both the random modulation and filtering processes as linear matrix operations. The random modulation can be modeled as a multiplication of the input

**x**with the

*N × N*matrix

*y*(

*t*) with the filter’s impulse response function

*h*(

*n*)

*n*= 1

*···N*. In order to mimic the behavior of the sample-and-hold device, the filter can be selected such that the impulse response decays quickly relative to the coarse sampling interval

*δ*. In discrete terms this means we only need to retain

*L << N*coefficients to accurately predict the action of the filter on the signal. The randomly modulated, filtered signal is therefore given by the product

**HRx**where The final step is to model the downsampling effect of the digitizer which simply takes every

*N/M*samples. This matrix is given explicitly by

**Φ**=

**DHR**and serves as a model for the modulation, filtering, and sampling steps. Upon substituting the sparse signal representation,

**x**=

**W**, our overall compressed signal model becomes The random construction of

*θ***Φ**ensures that the RIP property is satisfied [3

**56**(1), 520–544 (2010). [CrossRef]

**is indeed sparse in**

*θ***W**, Eq. (3) can now be solved for

**even though**

*θ**M < N*.

**Φ**

*,*

**W**, and the observed data

*y*(

*t*). In each of the algorithms the key parameter is the sparsity promoting constant τ. We have found that proper selection of this parameter is essential for good results. However, once the value has been selected it can be fixed for subsequent experiments. In this work we use the “Gradient Projection for Sparse Reconstruction” algorithm of Figueiredo

*et al.*[18

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]

*N*∼

*O*(

*ksamples*)) the algorithm is extremely fast, taking

*<<*1

*s*to recover

**on a single 3GHz processor. For our experiments the value τ = 0.0075 was chosen as the value that minimized the reconstruction error.**

*θ*## 5. Experimental results

*MHz*. This represents a compression factor of 10

*e*9/0.833

*e*9 = 12. Ideally one would recover the signal at the PRBS clock rate of 2.5MS/s (the highest frequency at which signal information is present). However, in order to accurately model the low-pass filter’s impulse response, the 10GS/s recovery time-scale was needed. A better choice of low-pass filter in future designs will perhaps eliminate this wasted bandwidth.

**H**. Both the frequency and time-domain descriptions of the filter are shown in Fig. (2).

*e*9) giving

*N*= 1002 observations

*x*((

*i –*1)Δ)

*i*= 1

*···N*. We also acquired

*M*= 83 compressed samples

*ỹ*((

*j –*1)

*δ*)

*j*= 1

*···M*with sampling interval

*δ*= 1/0.833

*e*9

*s*. Figure 3 compares the model to the acquired data.

*N*= 1002 points, despite only using the

*M*= 83 points shown in Figure 3. While there are clearly non-zero coefficients other than the one associated with the 1GHz signal, these are small in magnitude.

*δ*= 1/500

*e*6 seconds apart, we were able to recover 1002 samples of the 1.1GHz tone at sampling interval Δ = 1/10

*e*9 seconds. The results of this recovery are shown in the frequency domain in Fig. 5.

## 6. Discussion

## Acknowledgments

## References and links

1. | J. Campmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics |

2. | C. H. Lee, |

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 |

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. |

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. |

6. | S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express |

7. | O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. |

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. |

10. | M. Mishali and Y. C. Eldar, “Xampling: Analog Data Compression,” vol. http://doi.ieeecomputersociety.org/10.1109/DCC.2010.39 of |

11. | E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory |

12. | E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory |

13. | D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory |

14. | E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. |

15. | J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. |

16. | R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. |

17. | J. M. Nichols, M. Currie, F. Buholtz, and W. A. Link, “Bayesian Estimation of Weak Material Dispersion: Theory and Experiment,” Opt. Express |

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. |

**OCIS Codes**

(000.3870) General : Mathematics

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 21, 2011

Revised Manuscript: March 7, 2011

Manuscript Accepted: March 7, 2011

Published: April 1, 2011

**Citation**

J. M. Nichols and F. Bucholtz, "Beating Nyquist with light: a compressively sampled photonic link," Opt. Express **19**, 7339-7348 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7339

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### References

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

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