## Multi-rate synchronous optical undersampling of several bandwidth-limited signals |

Optics Express, Vol. 18, Issue 16, pp. 16929-16945 (2010)

http://dx.doi.org/10.1364/OE.18.016929

Acrobat PDF (1276 KB)

### Abstract

We demonstrate experimentally an optical system for undersampling several bandwidth-limited signals with carrier frequencies that are not known apriori and can be located within a broad frequency region of 0–20 GHz. The system is based on undersampling synchronously at three different rates. The optical undersampling down-converts the entire system bandwidth into a low frequency region called baseband. The synchronous sampling at several rates enables to accurately reconstruct signals even in cases in which different signals overlap in the baseband region of all sampling channels. Reconstruction of three simultaneously generated chirped signals, each with a bandwidth of about 200 MHz, was experimentally demonstrated.

© 2010 Optical Society of America

## 1. Introduction

1. P. W. Joudawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical Down-Sampling of Wide-Band Microwave Signals,” J. Lightwave Technol. **21**, 3116–3124 (2004). [CrossRef]

3. J. Kim, M. J. Park, M. H. Perrott, and F. X. Kärtner, Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution, Opt. Express16, 16509–16515 (2008). [CrossRef] [PubMed]

6. M. Mishali and Y. Eldar, “Blind multiband signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. **57**, 993–1009 (2009). [CrossRef]

7. A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach, and L. Singer, “Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals,” J. Lightwave Technol. **27**, 1027–1033 (2009). [CrossRef]

8. M. Fleyer, A. Linden, M. Horowitz, and A. Rosenthal, “Multirate Synchronous Sampling of Sparse Multiband Signals,” IEEE Trans. Signal Process. **58**, 1144–1156 (2010). [CrossRef]

7. A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach, and L. Singer, “Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals,” J. Lightwave Technol. **27**, 1027–1033 (2009). [CrossRef]

7. A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach, and L. Singer, “Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals,” J. Lightwave Technol. **27**, 1027–1033 (2009). [CrossRef]

*µs*) equals to about 57 dB. This SNDR value corresponds to a resolution of 9.2 effective number of bits (ENOB).

## 2. Synchronous multirate sampling scheme

1. P. W. Joudawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical Down-Sampling of Wide-Band Microwave Signals,” J. Lightwave Technol. **21**, 3116–3124 (2004). [CrossRef]

3. J. Kim, M. J. Park, M. H. Perrott, and F. X. Kärtner, Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution, Opt. Express16, 16509–16515 (2008). [CrossRef] [PubMed]

*F*

_{1}= 3.8 GHz,

*F*

_{2}= 3.5 GHz and

*F*

_{3}= 4.0 GHz by an RF signal. The rates

*F*were chosen to fulfill

_{i}*F*=

_{i}*M*Δ

_{i}*f*, where

*M*are positive integers and Δ

_{i}*f*= 100 MHz is the bandwidth of a frequency cell in the reconstruction algorithm [8

8. M. Fleyer, A. Linden, M. Horowitz, and A. Rosenthal, “Multirate Synchronous Sampling of Sparse Multiband Signals,” IEEE Trans. Signal Process. **58**, 1144–1156 (2010). [CrossRef]

**27**, 1027–1033 (2009). [CrossRef]

**27**, 1027–1033 (2009). [CrossRef]

9. J. F. Gravel and J. Wight, “On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO,” IEEE Trans. Microwave Theory Tech. **54**, 153–159 (2006). [CrossRef]

**27**, 1027–1033 (2009). [CrossRef]

*Vπ*of the modulator equals 4.6 V and the maximum power of the RF signal at the modulator input was about 0 dBm. The bias voltage of the modulator was adjusted to obtain the minimum second order distortion. Thus, we operated the modulator in its linear operating region.

## 3. Synchronous MRS signal reconstruction algorithm

8. M. Fleyer, A. Linden, M. Horowitz, and A. Rosenthal, “Multirate Synchronous Sampling of Sparse Multiband Signals,” IEEE Trans. Signal Process. **58**, 1144–1156 (2010). [CrossRef]

*x*(

*t*) modulates simultaneously

*P*optical pulse-trains with frequencies

*F*=

_{i}*M*Δ

_{i}*f*, 1 ≤

*i*≤

*P*, where

*M*are integers. The resulting signals

_{i}*x*(

_{i}*t*) contain replicas of the original signal spectrum

*X*(

*f*) shifted by an integer multiples of

*F*[2

_{i}2. 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]

*i*-th channel equals [

*−F*/2,

_{i}*F*/2]. These baseband signals are electronically sampled for each channel synchronously at a rate

_{i}*F*≥ max

_{s}_{i}{

*F*}.

_{i}*X*(

*f*) is connected to the sampled signals spectra

*X*(

_{i}*f*) through the system of linear equations [8

**58**, 1144–1156 (2010). [CrossRef]

**x**̂(

*β*) represents the Fourier transform of the sampled signals, the vector

**x**(

*β*) corresponds to the original signal Fourier transform and

**Q**is a sampling matrix. Vectors

**x**̂(

*β*) and

**x**(

*β*) are given with a resolution Δ

*f*. The variable 0 ≤

*β*0 Δ

*f*gives the location within the frequency cells that construct the spectrum vectors. Each column of the matrix

**Q**corresponds to a frequency cell of the original signal spectrum and each row corresponds to a frequency cell of the down-converted signal. In undersampling the number of rows in the matrix

**Q**is smaller than the number of columns. Therefore, a unique reconstruction of the original signal can not be obtained unless more assumptions on the signal are added. Although in modern military environment the available frequency spectrum is heavily used, at a given instant of time only a small portion of the spectrum is occupied. Therefore, in electronic warfare systems it is required to sample and reconstruct multi-band sparse signals.

**27**, 1027–1033 (2009). [CrossRef]

**Q**. If the resulting reduced matrix has a full column rank, then the search is complete and the signal can be reconstructed via the pseudo-inverse of the sampling matrix. If the reduced sampling matrix is not full column rank, we assume that the signal obtained after the reduction step is block-sparse. Therefore, we look for a solution that fulfills the reduced system of equations, but also contains the smallest number of bands. The pursuit algorithm that we use to find such solution is a variation of the well-known Orthogonal Matching Pursuit (OMP) modified to search for blocks instead of columns. Numerical simulations indicate that a very high success rate that is close to 100% is obtained when the sum of the sampling rates in all channels is approximately 8 times higher than the total signal frequency support. This success rate is higher than obtained by using previously published methods for undersampling assuming that the number of sampling channels is limited to 3 [8

**58**, 1144–1156 (2010). [CrossRef]

## 4. Non ideal synchronous multi-rate sampling

**58**, 1144–1156 (2010). [CrossRef]

*x*(

*t*) is undersampled by multiplying the signal with a low-rate pulse train. The resulting signal is then detected, amplified and sampled by an electronic A/D converter. The signal at the input of the

*i*-th (

*i*= 1..3) electronic A/D,

*x*(

_{i}*t*) is given by

*p*(

_{i}*t*),

*τ*and

_{i}*h*(

_{i}*t*) are the pulse shape of the

*i*-th channel, the delay of the

*i*-th pulse train at

*t*= 0 and the impulse response function of the detector and the electrical amplifier of the

*i*-th channel, respectively. Although the electrical signals at the output of the PLL-DROs are synchronized, a time delay

*τ*must be added due to a slightly different propagation times of the signals in different channels. The time delay

_{i}*τ*is added since the electronic A/D starts sampling in a time offset with respect to the pulse trains. In deriving Eq. (2) we neglect the transfer function of the modulator. This function is required to obtain the correct spectrum amplitude at each frequency; however, since the same modulator is used to modulate all three pulse trains, it does not affect the stability of the reconstruction.

*X*(

*f*),

*X*(

_{i}*f*),

*H*(

_{i}*f*) and

*P*(

_{i}*f*) are the Fourier transform of the original signal

*x*(

*t*), the baseband signal

*x*(

_{i}*t*), the transfer function

*h*(

_{i}*t*) and a pulse profile

*p*(

_{i}*t*), respectively.

*τ*,

_{i}*τ*,

*p*(

_{i}*t*) and

*h*(

_{i}*t*) should be known. These parameters change the sampling matrix that was given in [8

**58**, 1144–1156 (2010). [CrossRef]

**x**̂(

*β*) and the original signal spectrum

**x**(

*β*):

**x**̂(

*β*) and the matrix

**Q**are obtained by concatenating vectors

*M*×

_{i}*M*matrix

**Q**

_{i}are given by

*P*(

_{i}*f*) is the Fourier transform of the

*i*-th channel pulse, and matrices

**Q**

_{i}correspond to a single channel sampling matrix and the matrix

**Q**is the sampling matrix for three sampling channels used in our system.

### 4.1. Measurement of the impulse response h(t)

*h*(

_{i}*t*) of the

*i*-th sampling channel is determined by the delay of the optical channels between the modulator and the detectors, the detector response function and the response function of the electrical amplifiers and filters. We measured the Fourier transform of

*h*(

_{i}*t*) −

*H*(

_{i}*f*) by supplying a sinusoidal wave with a controllable frequency to the sampling system and to additional (fourth) port of the electrical A/D. The frequency of the sinusoidal wave varied between 5 MHz and 2 GHz with a step size of 5 MHz. The transfer function

*H*(

_{i}*f*) was measured for each channel

*i*(

*i*= 1 … 3). Figure 2 shows the amplitude and the phase of

*H*(

_{i}*f*) for three channels. The same results were also obtained by using a network analyzer.

### 4.2. Minimizing the relative delays between τ

*τ*. Delays of different optical pulse trains

_{i}*τ*can be made approximately equal by adding a tunable electrical phase shifter (

_{i}*φ*) to each pulse generator as shown in Fig. 1. The tuning of the phase shifters is performed by measuring the pulse trains at the entrance of the modulator by using a sampling oscilloscope with a 50 GHz bandwidth optical input. Figure 3 shows three optical pulse trains when the phase shifters are adjusted to obtain the best overlap between them. We have verified that the relative delays between the pulse trains did not change over a very long time duration of several days. Therefore, after the calibration process we could assume in our reconstruction algorithm that

*τ*= 0.

_{i}### 4.3. Extracting parameters of the optical pulsed trains

*nF*, where

_{i}*n*≥ 0 is an integer number. While the amplitude coefficients of the pulse spectrum

*P*(

_{i}*nF*) can be directly measured by using an RF spectrum analyzer, we retrieved the phases of the pulse coefficients

_{i}*P*(

_{i}*nF*) indirectly by applying an optimization procedure to Eq. (4). We supply the system with an RF multi-band signal with known frequency bands that were measured by using RF spectrum analyzer. Since the signal frequencies are known we retain in the matrix

_{i}**Q**, given in Eq. (4), only columns that correspond to those frequencies. Our system is not ideal and the noise is added to measurements. Therefore, Eq. (4) is not fulfilled with equality and we choose to find a least square solution to this equation. This solution

**x**

^{*}

_{S}is obtained by multiplying both sides of Eq. (4) by a pseudo-inverse of the matrix

**Q**

_{S}, denoted by

**Q**

^{†}

_{S}[15]:

**Q**

^{†}

_{S}depends on pulses parameters

*P*(

_{i}*nF*) that we want to retrieve. We estimate these parameters by optimizing Eq. (7) to make the calculated baseband signals most consistent with the measured once. We use a square norm error criteria as a cost function for our optimization problem and substitute

_{i}**x**

^{*}

_{S}back to Eq. (4). The resulting error equals:

*P*(

_{i}*nF*);

_{i}*i*= 1 … 3 and the initial A/D sampling time

*τ*.

*f*= 100 MHz.

*τ*. The optimized pulses coefficients amplitudes and phases were

*P*

_{1}(

*2F*

_{1}) = 1.08exp(

*j*0.53),

*P*

_{1}(3

*F*

_{1}) = 0.93exp(

*j*0.14) for the 3.8 GHz channel,

*P*

_{2}(2

*F*

_{2}) = 1.20exp(

*j*0.77),

*P*

_{2}(3

*F*

_{2}) = 0.95exp(

*j*0.52) for the 3.5 GHz channel, and

*P*

_{3}(2

*F*

_{3}) = 0.65exp(

*j*0.89),

*P*

_{3}(3

*F*

_{3}) = 0.48exp(

*j*3.18) for the 4.0 GHz channel.

**Q**

_{S}will be small and the optimization can be performed in a reasonable time. Additional signals may be added to enhance the optimization accuracy.

#### 4.3.1. Extracting the time offset of the electronic A/D

*τ*in Eq. (2). The time offset

*τ*changes the sampling matrix in Eq. (4) and therefore it should be extracted. The optical pulse trains are synchronized to a reference signal with a frequency of 100 MHz. Therefore, we sampled this reference signal simultaneously with the other three channels by adding another low-rate A/D converter (fourth channel). At the calibration process the signals that are used to extract the parameters of the optical pulses are supplied to the system. The phase of the reference signal was also measured at the beginning of the sampling that is used for the calibration. Then, the value of

*τ*was extracted by solving the optimization problem described in the previous subsection. The extraction of the delay

*τ*should be performed only once. After completing the calibration the value of

*τ*was updated by comparing the phase of the reference 100 MHz signal to the phase of the reference signal that was measured in the calibration process.

## 5. Experimental results

**58**, 1144–1156 (2010). [CrossRef]

*µs*that corresponds to a frequency resolution of 122 kHz. To reduce software runtime the frequency cell width was chosen to be Δ

*f*= 4 GHz=840 = 4.7619 MHz. This is consistent with our sampling rates requirement:

*F*

_{1}= 3.8 GHz = 798 · Δ

*f*,

*F*

_{2}= 3.5 GHz = 735 · Δ

*f*and

*F*

_{3}= 4.0 GHz = 840 · Δ

*f*. After inverting the matrix and performing the signal reconstruction the frequency resolution of the reconstructed signal was increased to that of the sampled data (122 kHz) by setting

*β*=

_{n}*n*Δ

*f*/39, 0 ≤

*n*Δ 39, in Eq. (22) (Appendix A) where

*n*is an integer number. To decrease the reconstruction error and to increase the robustness of the system to variations in system parameters we apply a simple algorithm that enhances the consistency of the sampling equations. The algorithm is described in Appendix B.

**27**, 1027–1033 (2009). [CrossRef]

### 5.1. Measuring the system performance and the sampling jitter

10. G. C. Valley, “Photonic analog-to-digital converters,” Opt. Express **15**, 1955–1982 (2007). [CrossRef] [PubMed]

*µs*. The SNDR depends on the resolution bandwidth. Assuming a window duration of 10

*µs*the SNDR for a 100 kHz resolution bandwidth equals about 57 dB. This result corresponds to 9.2 effective bits. Assuming a window duration of 0.1

*µs*the resolution bandwidth equals 10 MHz and the SNDR equals 41.7 dB which corresponds to about 6.6 effective bits. One of the causes of noise added by the down-conversion is the timing jitter of the optical pulse-trains. Assuming that the input signal is a sinusoidal wave with a frequency

*f*, the signal-to-noise ratio of the baseband signals is theoretically limited by the aperture jitter [12, 13]:

_{c}*L*(

*f*) of three optical pulse-trains.

*L*(

*f*) over the specified bandwidth [13]:

*f*

_{0}and

*f*

_{1}define the integration region. The initial frequency

*f*

_{0}is connected to the sampling window duration,

*T*, by

_{W}*f*

_{0}= 1/4

*T*[14

_{W}14. M. Rodwell, D. Bloom, and K. Weingarten, “Subpicosecond Laser Timing Stabilization,” IEEE J. Quantum Electron. **25**, 817–827 (1989). [CrossRef]

*µs*and therefore the initial integration frequency is

*f*

_{0}= 30.5 kHz. The jitter in a frequency region of 31 kHz – 1 MHz equals 11.5, 13.1 and 17.1 fs for the 3.8, 3.5 and 4.0 GHz sampling channels, respectively. Using the measured phase noise and Eq. (10) we can estimate the SNR that is caused by the jitter. Assuming a sinusoidal input wave with a frequency of 20 GHz the jitter induced SNR equals 56.8 dB, 55.7 dB and 53.6 dB for the 3.8, 3.5 and 4.0 GHz channel, respectively. These results imply that for a sampling window duration of about 10

*µs*and a signal with a 20 GHz carrier frequency, the noise induced by the jitter is expected to be similar to the noise level measured in our system for the 11.1 GHz sine wave signal. By assuming a shorter window duration (larger resolution bandwidth) the amplitude noise is expected to be the dominant noise in the system.

## 6. Conclusions

**27**, 1027–1033 (2009). [CrossRef]

*µs*that corresponds to a resolution bandwidth of 100 kHz.

## 7. Appendix A

*x*(

*t*) and the down-converted signal

*x*(

_{i}*t*).

*X*(

*f*) and the down-converted signals

*X*(

_{i}*f*) is given by

*P*(

_{i}*f*) is the Fourier transform of a single pulse

*p*(

_{i}*t*),

*H*(

_{i}*f*) is a Fourier transform of

*h*(

_{i}*t*) and

*δ*(

*f*) is the Dirac delta function.

*X̃*(

*f*) =

*X*(

*f*)exp(

*j*2

*πfτ*) we obtain

*f*is a baseband frequency 0 ≤

*f*<

*F*.

_{i}**58**, 1144–1156 (2010). [CrossRef]

*k*and scalar

*β*, 0 ≤

*β*< Δ

*f*, so that for any 0 ≤

*f*≤

*F*,

_{i}*f*=

*k*Δ

*f*+

*β*. Equation (13) can be written as

*f*for all the sampling channels we are able to construct a system of linear equations that connect the original and the down-converted spectra. By defining

*M*= [

*F*

_{max}/Δ

*f*] to be the number of cells in the support of the original signal

*X*(

*f*) Eq. (17) becomes

*M*×

_{i}*M*matrix

**Q**

_{i}whose elements are given by:

**x**

_{i}(

*β*) and

**x**(

*β*) are given by

*i*-th channel:

*P*sampling channels a single system of linear equations can be obtained:

**x**̂(

*β*) and the matrix

**Q**are obtained by concatenating vectors

## 8. Appendix B

**58**, 1144–1156 (2010). [CrossRef]

**Q**

_{S}a sampling matrix with columns corresponding to these locations. This matrix is constructed by retaining corresponding columns in a matrix

**Q**given in Eq. (22).

**x**̂(

*β*)−

**Q**

_{S}

**x**

_{S}(

*β*). The solution vector

**x**

_{S}is found by multiplying

**x**̂(

*β*) by a pseudo-inverse matrix

**Q**

^{†}

_{S}[15].

*α*,

_{i}*i*= 1,2,3 …

*P*that multiply each sampling channel. These coefficients compensate errors in the baseband spectrum. Assuming

*α*

_{i}**b**

_{i}(

*β*) be a compensated baseband signal vector, where

**M**=

**Q**

_{S}

**Q**

^{†}

_{S}−

**I**. We define matrices

**M**

_{i}with the number of columns equal to the length of the baseband vectors

**b**

_{i}such that:

**M**= [

**M**

_{1}∣

**M**

_{2}… ∣

**M**

_{P}],

**M**

_{i}in a row provides the matrix

**M**. It is easy to verify that the minimization problem (23) can be converted to the following eigenvalue problem:

**V**is equal to a stacked matrix [

**M**

_{1}

**b**

_{1}∣

**M**

_{2}

**b**

_{2}… ∣

**M**

_{P}

**b**

_{P}].

**a**corresponding to a smallest eigenvalue

*λ*contains coefficients

*α*that are used as a gain for each channel sequence. The reconstructed signal

_{i}**x**

_{S}is obtained by multiplying a baseband vector

**Q**

^{†}

_{S}.

## References and links

1. | P. W. Joudawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical Down-Sampling of Wide-Band Microwave Signals,” J. Lightwave Technol. |

2. | A. Zeitouny, A. Feldser, and M. Horowitz, “Optical sampling of narrowband microwave signals using pulses generated by electroabsorption modulators,” Opt. Commun. |

3. | J. Kim, M. J. Park, M. H. Perrott, and F. X. Kärtner, Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution, Opt. Express16, 16509–16515 (2008). [CrossRef] [PubMed] |

4. | P. Feng and Y. Bresler, “Spectrum-blind minimum-rate sampling and reconstruction of multiband signals,” in |

5. | R. Venkantaramani and Y. Bresler, “Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals,” IEEE Trans. Signal Process. |

6. | M. Mishali and Y. Eldar, “Blind multiband signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. |

7. | A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach, and L. Singer, “Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals,” J. Lightwave Technol. |

8. | M. Fleyer, A. Linden, M. Horowitz, and A. Rosenthal, “Multirate Synchronous Sampling of Sparse Multiband Signals,” IEEE Trans. Signal Process. |

9. | J. F. Gravel and J. Wight, “On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO,” IEEE Trans. Microwave Theory Tech. |

10. | G. C. Valley, “Photonic analog-to-digital converters,” Opt. Express |

11. | R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. |

12. | B. Le, T. W. Rondeau, J. H. Reed, and C. W. Bosti, “Analog-to-digital converters,” Signal Process. Mag. |

13. | M. Shinagawa, Y. Akazawa, and T. Wakimoto, Jitter Analysis of High-Speed Sampling Systems, IEEE J. Solid-State Circuits 25, 220–224 (1990). |

14. | M. Rodwell, D. Bloom, and K. Weingarten, “Subpicosecond Laser Timing Stabilization,” IEEE J. Quantum Electron. |

15. | R. Penrose,“A generalized inverse for matrices,” in Proc. Cambridge Philosophical Society, Cambridge 51, 406–413 (1955). |

**OCIS Codes**

(000.0000) General : General

**ToC Category:**

Signal processing

**History**

Original Manuscript: May 3, 2010

Revised Manuscript: July 1, 2010

Manuscript Accepted: July 1, 2010

Published: July 26, 2010

**Citation**

M. Fleyer, M. Horowitz, A. Feldtser, and V. Smulakovsky, "Multi-rate synchronous optical undersampling of several
bandwidth-limited signals," Opt. Express **18**, 16929-16945 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16929

Sort: Year | Journal | Reset

### References

- P. W. Joudawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, J. C. Twichell, "Optical Down-Sampling of Wide-Band Microwave Signals," J. Lightwave Technol. 21, 3116-3124 (2004). [CrossRef]
- 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]
- J. Kim, M. J. Park, M. H. Perrott and F. X. Kärtner, "Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution," Opt. Express 16, 16509-16515 (2008). [CrossRef] [PubMed]
- P. Feng and Y. Bresler, "Spectrum-blind minimum-rate sampling and reconstruction of multiband signals," in Proc. IEEE Int. Conf. ASSP, vol. 3, pp. 1688-1691, May 1996.
- R. Venkantaramani and Y. Bresler, "Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals," IEEE Trans. Signal Process. 49, 2301-2313 (2001). [CrossRef]
- M. Mishali and Y. Eldar, "Blind multiband signal reconstruction: compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009). [CrossRef]
- A. Feldster, Y. P. Shapira, M. Horowitz, A. Rosenthal, S. Zach and L. Singer, "Optical Under-Sampling and Reconstruction of Several Bandwidth-Limited Signals," J. Lightwave Technol. 27, 1027-1033 (2009). [CrossRef]
- M. Fleyer, A. Linden, M. Horowitz and A. Rosenthal, "Multirate Synchronous Sampling of Sparse Multiband Signals," IEEE Trans. Signal Process. 58, 1144-1156 (2010). [CrossRef]
- J. F. Gravel and J. Wight, "On the Conception and Analysis of a 12-GHz PushPush Phase-Locked DRO," IEEE Trans. Microwave Theory Tech. 54, 153-159 (2006). [CrossRef]
- G. C. Valley, "Photonic analog-to-digital converters," Opt. Express 15, 1955-1982 (2007). [CrossRef] [PubMed]
- R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999). [CrossRef]
- B. Le, T. W. Rondeau, J. H. Reed and C. W. Bosti, "Analog-to-digital converters," Signal Process. Mag. 69,69-77 (2005).
- M. Shinagawa, Y. Akazawa and T. Wakimoto, "Jitter Analysis of High-Speed Sampling Systems," IEEE J. Solid-State Circuits 25,220-224 (1990).
- M. Rodwell, D. Bloom and K. Weingarten, "Subpicosecond Laser Timing Stabilization," IEEE J. Quantum Electron. 25, 817-827 (1989). [CrossRef]
- R. Penrose, "A generalized inverse for matrices," in Proc. Cambridge Philosophical Society, Cambridge 51, 406-413 (1955).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.