## Estimating the optimal sampling rate using wavelet transform: an application to optical turbulence |

Optics Express, Vol. 21, Issue 13, pp. 15230-15236 (2013)

http://dx.doi.org/10.1364/OE.21.015230

Acrobat PDF (1090 KB)

### Abstract

Sampling rate and frequency content determination for optical quantities related to light propagation through turbulence are paramount experimental topics. Some papers about estimating properties of the optical turbulence seem to use *ad hoc* assumptions to set the sampling frequency used; this chosen sampling rate is assumed good enough to perform a proper measurement. On the other hand, other authors estimate the optimal sampling rate via fast Fourier transform of data series associated to the experiment. When possible, with the help of analytical models, cut-off frequencies, or frequency content, can be determined; yet, these approaches require prior knowledge of the optical turbulence. The aim of this paper is to propose an alternative, practical, experimental method to estimate a proper sampling rate. By means of the discrete wavelet transform, this approach can prevent any loss of information and, at the same time, avoid oversampling. Moreover, it is independent of the statistical model imposed on the turbulence.

© 2013 OSA

## 1. Introduction

*frozen turbulence hypothesis*and the Obukhov-Kolmogorov (OK) model, he showed that the phase and amplitude frequency spectra of a wave propagating through turbulent media span up to a frequency that linearly depends on the mean transverse flow velocity. Further extensions were obtained [2

2. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. **64**, 59–67 (1974) [CrossRef] .

5. R. L. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. **64**, 592–598 (1974) [CrossRef] .

2. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. **64**, 59–67 (1974) [CrossRef] .

18. L. P. Poggio, M. Furger, A. H. Prévôt, W. K. Graber, and E. L. Andreas, “Scintillometer Wind Measurements over Complex Terrain,” J. Atmos. Oceanic Technol. **17**, 17–26 (2000) [CrossRef] .

19. G. Potvin, D. Dion, and J. L. Forand, “Wind effects on scintillation decorrelation times,” Opt. Eng. **44**, 016001 (2005) [CrossRef] .

16. J. A. Anguita and J. E. Cisternas, “Influence of turbulence strength on temporal correlation of scintillation,” Opt. Lett. **36**, 1725–1727 (2011) [CrossRef] [PubMed] .

## 2. Wavelet method

23. M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. **24**, 395–457 (1992) [CrossRef] .

*et al.*[ 24

24. L. Hudgins, C. A. Friehe, and M. E. Mayer, “Wavelet transform and atmospheric turbulence,” Phys. Rev. Lett. **71**, 3279–3282 (1993) [CrossRef] [PubMed] .

*discrete wavelet transform*(DWT, or Mallat Algorithm) decomposes a signal into low-and high-frequency components by convolution with low- and high-pass filters, respectively. These filters are generated from a special function of compact support, called

*the mother wavelet*—see [20] for definitions. This procedure divides the signal in two parts: the global features are kept by low-frequency components,

*approximation coefficients*, whereas the local features are retained by the high-frequency ones,

*detail coefficients*. The second decomposition level takes the

*approximation coefficients*and starts the convolution again. This tree conforms the multiresolution analysis [22]. The decomposition can be done

*N*times being 2

*the length of the signal. The DWT can be defined as a matrix product where*

^{N}*S*represents the signal as a column vector,

*W̃*is a matrix containing all dilations and translations of the mother wavelet, and

*W*is the wavelet transform. The last one is a column matrix composed of all wavelet coefficient

*C*(

_{J}*k*) from all level decompositions (

*J*) and all times (

*k*). Since the DWT is an ortonormal transform, it permits to establish an energy preserving condition [21] This condition makes the DWT particularly useful in estimating the normalized energy content per frequency bands: the

*wavelet energy spectrum*(WES). This provides a fast and practical visualization of the frequency content; that is, an estimation of the optimal sampling rate. The scale band frequency employed in the wavelet decomposition for the WES is defined by a dyadic scaling of the sampling rate where

*J*goes from

*J*

_{min}to −1, with

*J*

_{min}determined by the length of the signal to be analysed (

*J*

_{min}= −

*N*). In general, each band is composed of frequencies from 2

^{(}

^{J}^{−1)}

*f*to 2

_{s}*—for example, the*

^{J}f_{s}*J*= −1 band contains frequencies from 2

^{−2}

*f*to 2

_{s}^{−1}

*f*. The WES is obtained by evaluating the following equation: where

_{s}*k*is the sampling time index. In general the DWT is a versatile tool for extracting features and information from any given signal. For instance, the detail coefficients can be examined by bands to detect transient events, or specific denoising algorithms can be applied to eliminate spurious noise. Also, the approximation coefficients can be used to find trends—see Sec. 3.

*power spectral density*(PSD) is the Fourier analogue to the WES. Usually, the determination of the sampling rate through the PSD is bound to an arbitrary criterion, e.g.: locating the frequency at which 90% of the power is contained, and then estimating the sampling rate as twice this frequency (Nyquist). Alternatively, the inverse of the cross-temporal correlation’s half-life time can provide another estimate. Although arbitrary, these criteria are still very powerful tools. And the Fourier transform also provides a strong analytical framework. Unfortunately, The results derived by using it are only valid for stationary series. The optical turbulence (like many natural phenomena) is prominently non-stationary; therefore, a criterion based on the wavelet energy spectrum should be more robust. Naturally, it presents a significant fraction of energy located in those bands where the turbulent phenomenon is more active. We particularly argue that if the highest band (

*J*= −1) is quite different from zero, some activity may be missing—the spectrum may appear slashed. That is, there are unaccounted for features at higher frequencies disregarded by the actual sampling rate: the recorded data is undersampled.

## 3. Experiment & discussion

*e*

^{2}diameter of 3mm) through artificial turbulence (Fig. 1). For the purpose of having isotropic turbulence at stable conditions we employ a device similar to the one described in [26

26. O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization”, Appl. Opt. **45**, 4888–4897 (2006) [CrossRef] [PubMed] .

*turbulator*: two air fluxes at different temperatures collide in a chamber producing an isotropic mix between hot and cold sources. The beam propagates along forty centimeters of turbulence, with an estimated inner-scale of 6mm [25

25. D. G. Pérez, A. Fernandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” SPIE Proc. **8535**(2012) [CrossRef] .

## 4. Conclusions

*f*; and the lowest and highest bands, associated to mechanical and electronic noise, should have a negligible signature in the spectrum—compare the temporal evolution of the WES in Fig. 2 for 6 and 12kHz. Under these conditions we can obtain a practical estimation of the optimal sampling rate without losing any information regarding the original dynamics. The advantage of this method is twofold; it permits to isolate noise from signal and be applied indistinctly to both stationary and non-stationary series. Furthermore, this procedure is independent from any theoretical model or ad hoc hypotheses regarding the optical turbulence. Finally, even though this approach is highly qualitative, it has proven to be fast and effective; therefore, our future objective is to improve it by using more complex wavelet transforms such as the wavelet packets [27

_{s}27. S. Blanco, A. Figliola, R. Quian Quiroga, O. A. Rosso, and E. Serrano, “Time-frequency analysis of electroencephalogram series. III. Wavelet packets and information cost function,” Phys. Rev. E **57**, 932–940 (1998) [CrossRef] .

## Acknowledgments

## References and links

1. | V. I. Tatarskĭ, |

2. | H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. |

3. | A. Ishimaru, |

4. | L. C. Andrews and R. L. Phillips, |

5. | R. L. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. |

6. | S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. |

7. | D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. |

8. | G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A |

9. | L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. |

10. | A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Commun. |

11. | V. P. Lukin and V. V. Pokasov, “Optical wave phase fluctuations,” Appl. Opt. |

12. | N. Ben-Yosef and E. Goldner, “Sample size influence on optical scintillation analysis. Analytical treatment of the higher-order irradiance moments,” Appl. Opt. |

13. | F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “G.S.M.: a Grating Scale Monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Sup. |

14. | F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt. |

15. | H. T. Yura and D. A. Kozlowski, “Low Earth orbit satellite-to-ground optical scintillation: comparison of experimental observations and theoretical predictions,” Opt. Lett. |

16. | J. A. Anguita and J. E. Cisternas, “Influence of turbulence strength on temporal correlation of scintillation,” Opt. Lett. |

17. | L. Kral, I. Prochazka, and K. Hamal, “Optical signal path delay fluctuations caused by atmospheric turbulence,” Opt. Lett. |

18. | L. P. Poggio, M. Furger, A. H. Prévôt, W. K. Graber, and E. L. Andreas, “Scintillometer Wind Measurements over Complex Terrain,” J. Atmos. Oceanic Technol. |

19. | G. Potvin, D. Dion, and J. L. Forand, “Wind effects on scintillation decorrelation times,” Opt. Eng. |

20. | S. Mallat, |

21. | D. Percival and A. Walden, |

22. | C. K. Chui, |

23. | M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. |

24. | L. Hudgins, C. A. Friehe, and M. E. Mayer, “Wavelet transform and atmospheric turbulence,” Phys. Rev. Lett. |

25. | D. G. Pérez, A. Fernandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” SPIE Proc. |

26. | O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization”, Appl. Opt. |

27. | S. Blanco, A. Figliola, R. Quian Quiroga, O. A. Rosso, and E. Serrano, “Time-frequency analysis of electroencephalogram series. III. Wavelet packets and information cost function,” Phys. Rev. E |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.7060) Atmospheric and oceanic optics : Turbulence

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 2, 2013

Revised Manuscript: June 3, 2013

Manuscript Accepted: June 11, 2013

Published: June 18, 2013

**Citation**

Gustavo Funes, Ángel Fernández, Darío G. Pérez, Luciano Zunino, and Eduardo Serrano, "Estimating the optimal sampling rate using wavelet transform: an application to optical turbulence," Opt. Express **21**, 15230-15236 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15230

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### References

- V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere(Nauka Press, Moscow, 1967).
- H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am.64, 59–67 (1974). [CrossRef]
- A. Ishimaru, Wave Propagation and Scattering in Random Media(IEEE Press & Oxford University Press, 1997).
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE, 1998).
- R. L. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am.64, 592–598 (1974). [CrossRef]
- S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am.61, 1285–1292 (1971). [CrossRef]
- D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am.67, 390–393 (1977). [CrossRef]
- G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A11, 358–367 (1994). [CrossRef]
- L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt.16, 2242–2251 (1977). [CrossRef] [PubMed]
- A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Commun.214, 9–14 (2002). [CrossRef]
- V. P. Lukin and V. V. Pokasov, “Optical wave phase fluctuations,” Appl. Opt.20, 121–135 (1981). [CrossRef] [PubMed]
- N. Ben-Yosef and E. Goldner, “Sample size influence on optical scintillation analysis. Analytical treatment of the higher-order irradiance moments,” Appl. Opt.27, 2167–2171 (1988). [CrossRef] [PubMed]
- F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “G.S.M.: a Grating Scale Monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Sup.108, 173–180 (1994).
- F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt.45, 3534–3543 (2006). [CrossRef] [PubMed]
- H. T. Yura and D. A. Kozlowski, “Low Earth orbit satellite-to-ground optical scintillation: comparison of experimental observations and theoretical predictions,” Opt. Lett.36, 2507–2509 (2011). [CrossRef] [PubMed]
- J. A. Anguita and J. E. Cisternas, “Influence of turbulence strength on temporal correlation of scintillation,” Opt. Lett.36, 1725–1727 (2011). [CrossRef] [PubMed]
- L. Kral, I. Prochazka, and K. Hamal, “Optical signal path delay fluctuations caused by atmospheric turbulence,” Opt. Lett.30, 1767–1769 (2005). [CrossRef] [PubMed]
- L. P. Poggio, M. Furger, A. H. Prévôt, W. K. Graber, and E. L. Andreas, “Scintillometer Wind Measurements over Complex Terrain,” J. Atmos. Oceanic Technol.17, 17–26 (2000). [CrossRef]
- G. Potvin, D. Dion, and J. L. Forand, “Wind effects on scintillation decorrelation times,” Opt. Eng.44, 016001 (2005). [CrossRef]
- S. Mallat, A Wavelet Tour of Signal Processing(Academic Press, Elsevier, 2009).
- D. Percival and A. Walden, Wavelet Methods for Time Series Analysis, Cambridge Series In Statistical And Probabilistic Mathematics (Cambridge University Press, 2006).
- C. K. Chui, An Introduction to Wavelets(Academic Press, 1992).
- M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech.24, 395–457 (1992). [CrossRef]
- L. Hudgins, C. A. Friehe, and M. E. Mayer, “Wavelet transform and atmospheric turbulence,” Phys. Rev. Lett.71, 3279–3282 (1993). [CrossRef] [PubMed]
- D. G. Pérez, A. Fernandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” SPIE Proc.8535(2012). [CrossRef]
- O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization”, Appl. Opt.45, 4888–4897 (2006). [CrossRef] [PubMed]
- S. Blanco, A. Figliola, R. Quian Quiroga, O. A. Rosso, and E. Serrano, “Time-frequency analysis of electroencephalogram series. III. Wavelet packets and information cost function,” Phys. Rev. E57, 932–940 (1998). [CrossRef]

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