Abstract

A Doppler global velocimetry (DGV) measurement technique with a sinusoidal laser frequency modulation is presented for measuring velocity fields in fluid flows. A cesium absorption cell is used for the conversion of the Doppler shift frequency into a change in light intensity, which can be measured by a fiber coupled avalanche photo diode array. Because of a harmonic analysis of the detector element signals, no errors due to detector offset drifts occur and no reference detector array is necessary for measuring the scattered light power. Hence, large errors such as image misalignment errors and beam split errors are eliminated. Furthermore, the measurement system is also capable of achieving high measurement rates up to the modulation frequency (100kHz) and thus opens new perspectives to multiple point investigations of instationary flows, e.g., for turbulence analysis. A fundamental measurement uncertainty analysis based on the theory of Cramér and Rao is given and validated by experimental results. The current relation between time resolution and measurement uncertainty, as well as further optimization strategies, are discussed.

© 2008 Optical Society of America

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  1. I. Röhle and C. E. Willert, “Extension of Doppler global velocimetry to periodic flows,” Meas. Sci. Technol. 12, 420-431(2001).
    [CrossRef]
  2. R. J. Adrian, “Twenty years of particle image velocimetry,” Exp. Fluids 39, 159-169 (2005).
    [CrossRef]
  3. J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8, 1379-1392 (1997).
    [CrossRef]
  4. N. J. Lawson and J. Wu, “Three-dimensional particle image velocimetry: experimental error analysis of a digital angular stereoscopic system,” Meas. Sci. Technol. 8, 1455-1464(1997).
    [CrossRef]
  5. H. Komine, “System for measuring velocity field of fluid flow utilizing a laser-Doppler spectral image converter,” U.S. patent 4,919,536 (24 April 1990).
  6. J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
    [CrossRef]
  7. G. L. Morrison and C. A. Gaharan, “Uncertainty estimates in DGV systems due to pixel location and velocity gradients,” Meas. Sci. Technol. 12, 369-377 (2001).
    [CrossRef]
  8. H. Müller, T. Lehmacher, and G. Grosche, “Profile sensor based on Doppler Global Velocimetry,” in 8th International Conference on Laser Anemometry--Advances and Applications, A. Cenedese and D. Pitrogiacomi, eds. (University of Rome “La Sapienza,” 1999), pp. 475-482.
  9. A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
    [CrossRef]
  10. R. L. McKenzie, “Measurement capabilities of planar Doppler velocimetry using pulsed lasers,” Appl. Opt. 35, 948-964 (1996).
    [CrossRef] [PubMed]
  11. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  12. H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
    [CrossRef]
  13. T. O. H. Charrett and R. P. Tatam, “Single camera three component planar velocity measurements using two-frequency planar Doppler velocimetry (2ν-PDV),” Meas. Sci. Technol. 17, 1194-1206 (2006).
    [CrossRef]
  14. T. O. H. Charrett, D. S. Nobes, and R. P. Tatam, “Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements,” Appl. Opt. 46, 4102-4116 (2007).
    [CrossRef] [PubMed]
  15. D. A. Steck, “Cesium D line data (rev. 1.6),” Los Alamos National Laboratory, http://steck.us/alkalidata (2003).
  16. O. Svelto, Principles of Lasers (Plenum, 1998).
  17. Thermochemical Properties of Inorganic Substances, 2nd ed., O. Knacke, O. Kubaschweski, and K. Hesselmann, eds. (Springer, 1991), Vol. I, pp. 543-545.
  18. R. J. Rafac and C. E. Tanner, “Measurement of the ratio of the cesium D-line transition strengths,” Phys. Rev. A 58, 1087-1097 (1998).
    [CrossRef]
  19. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

2007 (3)

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

T. O. H. Charrett, D. S. Nobes, and R. P. Tatam, “Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements,” Appl. Opt. 46, 4102-4116 (2007).
[CrossRef] [PubMed]

2006 (1)

T. O. H. Charrett and R. P. Tatam, “Single camera three component planar velocity measurements using two-frequency planar Doppler velocimetry (2ν-PDV),” Meas. Sci. Technol. 17, 1194-1206 (2006).
[CrossRef]

2005 (1)

R. J. Adrian, “Twenty years of particle image velocimetry,” Exp. Fluids 39, 159-169 (2005).
[CrossRef]

2001 (3)

J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
[CrossRef]

G. L. Morrison and C. A. Gaharan, “Uncertainty estimates in DGV systems due to pixel location and velocity gradients,” Meas. Sci. Technol. 12, 369-377 (2001).
[CrossRef]

I. Röhle and C. E. Willert, “Extension of Doppler global velocimetry to periodic flows,” Meas. Sci. Technol. 12, 420-431(2001).
[CrossRef]

1998 (1)

R. J. Rafac and C. E. Tanner, “Measurement of the ratio of the cesium D-line transition strengths,” Phys. Rev. A 58, 1087-1097 (1998).
[CrossRef]

1997 (2)

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8, 1379-1392 (1997).
[CrossRef]

N. J. Lawson and J. Wu, “Three-dimensional particle image velocimetry: experimental error analysis of a digital angular stereoscopic system,” Meas. Sci. Technol. 8, 1455-1464(1997).
[CrossRef]

1996 (1)

Adrian, R. J.

R. J. Adrian, “Twenty years of particle image velocimetry,” Exp. Fluids 39, 159-169 (2005).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

Büttner, L.

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

Charrett, T. O. H.

T. O. H. Charrett, D. S. Nobes, and R. P. Tatam, “Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements,” Appl. Opt. 46, 4102-4116 (2007).
[CrossRef] [PubMed]

T. O. H. Charrett and R. P. Tatam, “Single camera three component planar velocity measurements using two-frequency planar Doppler velocimetry (2ν-PDV),” Meas. Sci. Technol. 17, 1194-1206 (2006).
[CrossRef]

Czarske, J.

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

Eggert, M.

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

Fischer, A.

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

Gaharan, C. A.

G. L. Morrison and C. A. Gaharan, “Uncertainty estimates in DGV systems due to pixel location and velocity gradients,” Meas. Sci. Technol. 12, 369-377 (2001).
[CrossRef]

Grosche, G.

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

H. Müller, T. Lehmacher, and G. Grosche, “Profile sensor based on Doppler Global Velocimetry,” in 8th International Conference on Laser Anemometry--Advances and Applications, A. Cenedese and D. Pitrogiacomi, eds. (University of Rome “La Sapienza,” 1999), pp. 475-482.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Komine, H.

H. Komine, “System for measuring velocity field of fluid flow utilizing a laser-Doppler spectral image converter,” U.S. patent 4,919,536 (24 April 1990).

Lawson, N. J.

N. J. Lawson and J. Wu, “Three-dimensional particle image velocimetry: experimental error analysis of a digital angular stereoscopic system,” Meas. Sci. Technol. 8, 1455-1464(1997).
[CrossRef]

Lee, J. W.

J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
[CrossRef]

Lehmacher, T.

H. Müller, T. Lehmacher, and G. Grosche, “Profile sensor based on Doppler Global Velocimetry,” in 8th International Conference on Laser Anemometry--Advances and Applications, A. Cenedese and D. Pitrogiacomi, eds. (University of Rome “La Sapienza,” 1999), pp. 475-482.

McKenzie, R. L.

Meyers, J. F.

J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
[CrossRef]

Morrison, G. L.

G. L. Morrison and C. A. Gaharan, “Uncertainty estimates in DGV systems due to pixel location and velocity gradients,” Meas. Sci. Technol. 12, 369-377 (2001).
[CrossRef]

Müller, H.

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

H. Müller, T. Lehmacher, and G. Grosche, “Profile sensor based on Doppler Global Velocimetry,” in 8th International Conference on Laser Anemometry--Advances and Applications, A. Cenedese and D. Pitrogiacomi, eds. (University of Rome “La Sapienza,” 1999), pp. 475-482.

Nobes, D. S.

Rafac, R. J.

R. J. Rafac and C. E. Tanner, “Measurement of the ratio of the cesium D-line transition strengths,” Phys. Rev. A 58, 1087-1097 (1998).
[CrossRef]

Röhle, I.

I. Röhle and C. E. Willert, “Extension of Doppler global velocimetry to periodic flows,” Meas. Sci. Technol. 12, 420-431(2001).
[CrossRef]

Schwartz, R. J.

J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
[CrossRef]

Steck, D. A.

D. A. Steck, “Cesium D line data (rev. 1.6),” Los Alamos National Laboratory, http://steck.us/alkalidata (2003).

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, 1998).

Tanner, C. E.

R. J. Rafac and C. E. Tanner, “Measurement of the ratio of the cesium D-line transition strengths,” Phys. Rev. A 58, 1087-1097 (1998).
[CrossRef]

Tatam, R. P.

T. O. H. Charrett, D. S. Nobes, and R. P. Tatam, “Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements,” Appl. Opt. 46, 4102-4116 (2007).
[CrossRef] [PubMed]

T. O. H. Charrett and R. P. Tatam, “Single camera three component planar velocity measurements using two-frequency planar Doppler velocimetry (2ν-PDV),” Meas. Sci. Technol. 17, 1194-1206 (2006).
[CrossRef]

Westerweel, J.

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8, 1379-1392 (1997).
[CrossRef]

Willert, C. E.

I. Röhle and C. E. Willert, “Extension of Doppler global velocimetry to periodic flows,” Meas. Sci. Technol. 12, 420-431(2001).
[CrossRef]

Wu, J.

N. J. Lawson and J. Wu, “Three-dimensional particle image velocimetry: experimental error analysis of a digital angular stereoscopic system,” Meas. Sci. Technol. 8, 1455-1464(1997).
[CrossRef]

Appl. Opt. (2)

Exp. Fluids (2)

H. Müller, M. Eggert, J. Czarske, L. Büttner, and A. Fischer, “Single-camera Doppler global velocimetry based on frequency modulation techniques,” Exp. Fluids 43, 223-232 (2007).
[CrossRef]

R. J. Adrian, “Twenty years of particle image velocimetry,” Exp. Fluids 39, 159-169 (2005).
[CrossRef]

Meas. Sci. Technol. (7)

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8, 1379-1392 (1997).
[CrossRef]

N. J. Lawson and J. Wu, “Three-dimensional particle image velocimetry: experimental error analysis of a digital angular stereoscopic system,” Meas. Sci. Technol. 8, 1455-1464(1997).
[CrossRef]

J. F. Meyers, J. W. Lee, and R. J. Schwartz, “Characterization of measurement error sources in Doppler global velocimetry,” Meas. Sci. Technol. 12, 357-368 (2001).
[CrossRef]

G. L. Morrison and C. A. Gaharan, “Uncertainty estimates in DGV systems due to pixel location and velocity gradients,” Meas. Sci. Technol. 12, 369-377 (2001).
[CrossRef]

A. Fischer, L. Büttner, J. Czarske, M. Eggert, G. Grosche, and H. Müller, “Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation,” Meas. Sci. Technol. 18, 2529-2545 (2007).
[CrossRef]

T. O. H. Charrett and R. P. Tatam, “Single camera three component planar velocity measurements using two-frequency planar Doppler velocimetry (2ν-PDV),” Meas. Sci. Technol. 17, 1194-1206 (2006).
[CrossRef]

I. Röhle and C. E. Willert, “Extension of Doppler global velocimetry to periodic flows,” Meas. Sci. Technol. 12, 420-431(2001).
[CrossRef]

Phys. Rev. A (1)

R. J. Rafac and C. E. Tanner, “Measurement of the ratio of the cesium D-line transition strengths,” Phys. Rev. A 58, 1087-1097 (1998).
[CrossRef]

Other (7)

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

D. A. Steck, “Cesium D line data (rev. 1.6),” Los Alamos National Laboratory, http://steck.us/alkalidata (2003).

O. Svelto, Principles of Lasers (Plenum, 1998).

Thermochemical Properties of Inorganic Substances, 2nd ed., O. Knacke, O. Kubaschweski, and K. Hesselmann, eds. (Springer, 1991), Vol. I, pp. 543-545.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

H. Müller, T. Lehmacher, and G. Grosche, “Profile sensor based on Doppler Global Velocimetry,” in 8th International Conference on Laser Anemometry--Advances and Applications, A. Cenedese and D. Pitrogiacomi, eds. (University of Rome “La Sapienza,” 1999), pp. 475-482.

H. Komine, “System for measuring velocity field of fluid flow utilizing a laser-Doppler spectral image converter,” U.S. patent 4,919,536 (24 April 1990).

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Figures (12)

Fig. 1
Fig. 1

Schematic setup of the FM-DGV technique.

Fig. 2
Fig. 2

Measurement principle of the FM-DGV technique showing one frequency modulation period T m with different laser center frequencies and the resultant cell transmission signals ( f 0 = 351.7217 THz ).

Fig. 3
Fig. 3

Sample signal of one detector element in time and frequency domain (sampling frequency 1 MHz , modulation frequency 100 kHz ).

Fig. 4
Fig. 4

(a) Calculated first and second order harmonic amplitudes A 1 and A 2 , respectively, using a measured cesium absorption cell transmission curve (normalized with the signal strength at 100% cell transmission) and (b) their quotient q = A 1 / A 2 ( f 0 = 351.7217 THz ).

Fig. 5
Fig. 5

Cesium D 2 transition hyperfine structure. The arrows with dashed lines indicate the allowed transitions.

Fig. 6
Fig. 6

Calculated normalized absorption coefficients for (a) the lower and (b) the upper absorption lines at 25 ° C (solid/dashed lines: total/single absorption coefficient).

Fig. 7
Fig. 7

Measured (crosses) and calculated (solid lines) transmission of (a) the lower lines and (b) the upper lines for different cold finger temperatures (gas temperature is kept constant at 45 ° C ).

Fig. 8
Fig. 8

Estimated standard deviations of velocity for a cold finger temperature of (a)  25 ° C and (b)  35 ° C with respect to the laser center frequency based on the Cramér–Rao lower bound (CRLB) without (solid lines) and with (dashed lines) consideration of the FM-DGV signal processing (SP) algorithm (scattered light power: 1 nW ).

Fig. 9
Fig. 9

Estimated standard deviations of velocity having (a)  1 nW and (b)  40 nW scattered light power with respect to the modulation amplitude f h based on the Cramér–Rao lower bound (CRLB) without (solid lines) and with (dashed lines) consideration of the FM-DGV signal processing (SP) algorithm (for a cold finger temperature of 25 ° C ).

Fig. 10
Fig. 10

Velocity uncertainty analysis (measurement and simulation) with respect to the time resolution ( 110 MHz + f LL , 2 laser center frequency, 450 MHz modulation amplitude f h , P s = 176 pW , NEP = 35 fW / Hz , 16 ms measurement time, and 25 ° C cold finger temperature).

Fig. 11
Fig. 11

Velocity uncertainty analysis (measurement and simulation) with respect to the scattered light power ( 110 MHz + f LL , 2 laser center frequency, 450 MHz modulation amplitude f h , APD 1 : NEP = 260 fW / Hz , APD 2 : 35 fW / Hz , 16 ms measurement time, and 25 ° C cold finger temperature).

Fig. 12
Fig. 12

Velocity uncertainty comparison between FM-DGV and conventional DGV with respect to laser center frequency ( 110 MHz + f LL , 2 laser center frequency, 450 MHz modulation amplitude f h , 1 nW scattered light power P s , NEP = 35 fW / Hz , and 16 ms measurement time).

Tables (1)

Tables Icon

Table 1 Cesium Transitions at 852.3 nm

Equations (30)

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q = A 1 / A 2
f L ( t ) = f c + f h cos ( 2 π f m t ) ,
f D f c ( o i ) v c ,
f D 1.66 MHz / ( m / s ) × v o i ,
g ( f f r ) = 1 Δ f D 4 ln 2 π exp { 4 ln 2 ( f f r Δ f D ) 2 } ,
g LL ( f ) = i = 1 3 S LL , i g ( f f LL , i ) .
τ LL ( f ) = exp { n c S LL g LL ( f ) L } ,
n ˙ e = η τ ( f L ( t ) + f D ) P s λ h c ,
n e [ k ] = η τ ( f L [ k ] + f D ) n s , k = 1 , , N ,
n s = P s T a / ( h c / λ )
var ( f ˆ D ) = var ( f ˆ f ˆ ref ) var ( f ˆ ) ,
var ( n e [ k ] ) = σ n e 2 = ( η T a h c λ ) 2 × NEP 2 Δ f ,
var Q N ( f D ) 1 ( η n s ) k = 0 N 1 τ ( f L [ k ] ) ( k = 0 N 1 ( τ ( f L [ k ] ) f L [ k ] ) 2 × 1 τ ( f L [ k ] ) ) ( k = 0 N 1 τ ( f L [ k ] ) ) ( k = 0 N 1 τ ( f L [ k ] ) f L [ k ] ) 2 .
var G N ( f D ) σ n e 2 ( η n s ) 2 k = 0 N 1 τ ( f L [ k ] ) 2 ( k = 0 N 1 τ ( f L [ k ] ) 2 ) ( k = 0 N 1 ( τ ( f L [ k ] ) f L [ k ] ) 2 ) ( k = 0 N 1 ( τ ( f L [ k ] ) τ ( f L [ k ] ) f L [ k ] ) ) 2 .
var ( f D ) = var Q N ( f D ) + var G N ( f D ) .
var ( f D ) = ( q f c ) 2 [ ( q A 1 ) 2 var ( A ˆ 1 ) + ( q A 2 ) 2 var ( A ˆ 2 ) + 2 q A 1 q A 2 cov ( A ˆ 1 , A ˆ 2 ) ] .
A ˆ h = 2 N k = 0 N 1 n e [ k ] × cos ( h 2 π f m t ) , h = 1 , 2.
var ( A ˆ h ) = 2 A 0 + A 2 h N , h = 1 , 2 ,
cov ( A ˆ 1 , A ˆ 2 ) = A 1 + A 3 N .
var ( A ˆ h ) = 2 σ n e 2 N , h = 1 , 2 ,
cov ( A ˆ 1 , A ˆ 2 ) = 0.
var ( f D ) var G N ( f D ) = σ n e 2 ( η n s ) 2 N × K ,
K = 1 N k = 0 N 1 τ ( f L [ k ] ) 2 ( 1 N k = 0 N 1 τ ( f L [ k ] ) 2 ) ( 1 N k = 0 N 1 ( τ ( f L [ k ] ) f L [ k ] ) 2 ) ( 1 N k = 0 N 1 ( τ ( f L [ k ] ) τ ( f L [ k ] ) f L [ k ] ) ) 2 .
K 1 1 N k = 0 N 1 ( τ ( f L [ k ] ) f L [ k ] ) 2 .
var ( f D ) = NEP 2 2 P s 2 N T a × 2.1 × 10 17 Hz 2 .
σ v o i = NEP P s N T a × 195.2 m / s .
n e , 1 [ k ] = η d τ ( f c + f D ) n s ,
n e , 2 [ k ] = η ( 1 d ) n s , k = 1 , , N .
var Q N ( f D ) 1 N η d n s × τ ( f L ) ( τ ( f L ) f L ) 2 × ( 1 + d τ ( f L ) 1 d ) ,
var G N ( f D ) σ n e 2 N ( η d n s τ ( f L ) f L ) 2 × ( 1 + ( d τ ( f L ) 1 d ) 2 )

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