Abstract

We describe a method for analyzing frequency-chirped sinusoidal signals using a complex heterodyne, sometimes also known as complex demodulation on the digitized waveform. This method allows one to use prior knowledge of the signal to reduce the effective bandwidth of the signal. The method can be used to extract a frequency-chirped signal even when it is sampled well below the Nyquist criterion. Accordingly, the method facilitates the use of less-expensive data acquisition and signal processing hardware than has traditionally been used for these applications. This technique is particularly useful for high-precision (parts in 109) interferometer applications in which there exists a differential acceleration between the two arms (commonly found in absolute gravity meters or gradiometers).

© 2006 Optical Society of America

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References

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  1. T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
    [CrossRef]
  2. I. Murata, "A transportable apparatus for absolute measurement of gravity," Bull. Earthquake Res. Inst., Univ. Tokyo 53, 49-130 (1979).
  3. P. R. Parker, E. L. Canuteson, and M. A. Zumberge, "Optical fiber gravity meter," U.S. patent 5,637,797 (10 June 1997).
  4. G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
    [CrossRef]
  5. C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).
  6. P. Bloomfield, Fourier Analysis of Time Series (Wiley, 2000), p. 97.
    [CrossRef]
  7. T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
    [CrossRef]
  8. K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
    [CrossRef]

2005

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

2003

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

1995

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

1993

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

1979

I. Murata, "A transportable apparatus for absolute measurement of gravity," Bull. Earthquake Res. Inst., Univ. Tokyo 53, 49-130 (1979).

1976

C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).

Ando, M.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Barbato, G.

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Bingham, C.

C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).

Bloomfield, P.

P. Bloomfield, Fourier Analysis of Time Series (Wiley, 2000), p. 97.
[CrossRef]

Canuteson, E. L.

P. R. Parker, E. L. Canuteson, and M. A. Zumberge, "Optical fiber gravity meter," U.S. patent 5,637,797 (10 June 1997).

D'Agostino, G.

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Danzmann, K.

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Desogus, S.

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Faller, J. E.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

Germak, A.

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Godfrey, J. D.

C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).

Hilt, R.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

Kanda, N.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Klopping, F.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

Murata, I.

I. Murata, "A transportable apparatus for absolute measurement of gravity," Bull. Earthquake Res. Inst., Univ. Tokyo 53, 49-130 (1979).

Niebauer, T. M.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Origlia, C.

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Parker, P. R.

P. R. Parker, E. L. Canuteson, and M. A. Zumberge, "Optical fiber gravity meter," U.S. patent 5,637,797 (10 June 1997).

Rudiger, A.

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Sasagawa, G. S.

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

Schilling, R.

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Schnupp, L.

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Soida, K.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Tagoshi, H.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Tatsumi, D.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Tsubono, K.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

Tukey, J. W.

C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).

Winkler, W.

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Zumberge, M. A.

P. R. Parker, E. L. Canuteson, and M. A. Zumberge, "Optical fiber gravity meter," U.S. patent 5,637,797 (10 June 1997).

Bull. Earthquake Res. Inst., Univ. Tokyo

I. Murata, "A transportable apparatus for absolute measurement of gravity," Bull. Earthquake Res. Inst., Univ. Tokyo 53, 49-130 (1979).

Class. Quantum Grav.

K. Soida, M. Ando, N. Kanda, H. Tagoshi, D. Tatsumi, K. Tsubono, and the TAMA Collaboration, "Search for continuous gravitational waves from the SN1987A remnant using TAMA300 data," Class. Quantum Grav. 20, S645-S654 (2003).
[CrossRef]

IEEE Trans. Audio Electroacoust.

C. Bingham, J. D. Godfrey, and J. W. Tukey, "Modern techniques of power spectrum estimation," IEEE Trans. Audio Electroacoust. AU-15, 56-66 (1976).

Metrologia

T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, "A new generation of absolute gravimeters," Metrologia 32, 159-180 (1995).
[CrossRef]

G. D'Agostino, A. Germak, S. Desogus, C. Origlia, and G. Barbato, "A method to estimate the time-position coordinates of a free-falling test-mass in absolute gravimetry," Metrologia 42, 233-238 (2005).
[CrossRef]

Phys. Rev. D

T. M. Niebauer, A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, and K. Danzmann, "Pulsar search using data compression with the Garching Gravitational Wave Detector," Phys. Rev. D 47, 3106-3123 (1993).
[CrossRef]

Other

P. Bloomfield, Fourier Analysis of Time Series (Wiley, 2000), p. 97.
[CrossRef]

P. R. Parker, E. L. Canuteson, and M. A. Zumberge, "Optical fiber gravity meter," U.S. patent 5,637,797 (10 June 1997).

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Figures (9)

Fig. 1
Fig. 1

(Color online) Schematic diagram of an absolute gravimeter with a Michelson-type interferometer.

Fig. 2
Fig. 2

Interference signal from an absolute gravity meter [see Eq. (1)].

Fig. 3
Fig. 3

(Color online) Power spectrum of a chirped sinusoid with an amplitude of 10 V peak to peak and frequency varying from 0 to 6 MHz over 0.2 s sampled at 20 MHz. The resolution bandwidth of the FFT is 5 Hz. Only positive frequencies are shown.

Fig. 4
Fig. 4

(Color online) Amplitude spectrum of a 10 V peak-to-peak chirped sinusoid with chirped frequency from dc to 6 MHz in 0.2 s sampled at 20 MHz and then multiplied by a complex heterodyne with chirp from dc to 5.94 MHz (60 kHz smaller than the original signal chirp). The resolution bandwidth of the FFT is 5 Hz. Only positive frequencies are shown.

Fig. 5
Fig. 5

(Color online) Amplitude spectrum of a 10 V peak-to-peak chirped sinusoid with chirped frequency from dc to 6 MHz in 0.2 s sampled at 20 MHz and then multiplied by a complex heterodyne with chirp from 20 kHz to 5.94 MHz (60 kHz smaller than original signal chirp). The resolution bandwidth of the FFT is 5 Hz. Only positive frequencies are shown.

Fig. 6
Fig. 6

(Color online) Undersampled waveform after demodulation with a chirped complex heterodyne.

Fig. 7
Fig. 7

Chirp error caused by 1 ns of time jitter as a function of the number of samples.

Fig. 8
Fig. 8

Chirp error caused by 1 % amplitude noise.

Fig. 9
Fig. 9

(Color online) Hardware-generated chirped sinusoid undersampled at 50 kHz.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

V ( t ) = A sin [ 4 π λ ( x 0 + ν 0 t + 1 2 g t 2 ) ] ,
S ( t ) = A cos ( ω 0 t + 1 2 α 0 t 2 ) ,
δ chirp chirp = 10 N δ t jitter T obs .
δ chirp chirp = 140 × 10 9 N δ A A .

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