Abstract

Power-spectral-density measurements of any sampled signal are typically restricted by both acquisition rate and frequency response limitations of instruments, which can be particularly prohibitive for video-based measurements. We have developed a new method called intensity modulation spectral analysis that circumvents these limitations, dramatically extending the effective detection bandwidth. We demonstrate this by video tracking an optically trapped microsphere while oscillating an LED illumination source. This approach allows us to quantify fluctuations of the microsphere at frequencies over 10 times higher than the Nyquist frequency, mimicking a significantly higher frame rate.

© 2009 Optical Society of America

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  1. L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
    [CrossRef] [PubMed]
  2. C. V. Gabel and H. C. Berg, Proc. Natl. Acad. Sci. USA 100, 8748 (2003).
    [CrossRef] [PubMed]
  3. E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
    [CrossRef] [PubMed]
  4. M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
    [CrossRef]
  5. F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
    [CrossRef]
  6. M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
    [CrossRef]
  7. K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
    [CrossRef] [PubMed]
  8. F. R. Verdun, T. L. Ricca, and A. G. Marshall, Appl. Spectrosc. 42, 199 (1988).
    [CrossRef]
  9. M. H. El-Shafey, in Proceedings of the International Conference on Electrical, Electronic, and Computer Engineering (IEEE, 2004), pp. 210-213.
  10. W. P. Wong and K. Halvorsen, Opt. Express 14, 12517 (2006).
    [CrossRef] [PubMed]
  11. C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.
  12. R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
    [CrossRef]
  13. R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
    [CrossRef] [PubMed]
  14. T. Savin and P. S. Doyle, Biophys. J. 88, 623 (2005).
    [CrossRef]
  15. T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 (2005).
    [CrossRef]
  16. Note that 1/4πN2∫∣H̃0(ω)∣2dω=1/2WN2.
  17. Determination of ϕ to eliminate terms t3 and t4 can be done by calculating ∣H̃(ω)∣2, setting the sum of the ϕ-dependent terms to zero, and solving for the specific case where W=n2π/ω′.
  18. For example, measurement sensitivity can be improved by removing the dc offset (i.e., set B=0). For video imaging, this can be accomplished by simulating positive and negative light (e.g., by using two colors or polarizations) or, for low-light applications, by properly synchronizing two camera frames to each illumination cycle. At the sensor level, IMSA can be implemented by using a camera capable of modulating the incoming signal , or by having two collection wells for each pixel to simulate positive and negative light.
  19. A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
    [CrossRef] [PubMed]
  20. P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
    [CrossRef]
  21. D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
    [CrossRef]
  22. J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298-310 (1996).
    [CrossRef]
  23. E. Toprak and P. R. Selvin, Annu. Rev. Biophys. Biomol. Struct. 36, 349 (2007).
    [CrossRef] [PubMed]

2007 (2)

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

E. Toprak and P. R. Selvin, Annu. Rev. Biophys. Biomol. Struct. 36, 349 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (3)

T. Savin and P. S. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 (2005).
[CrossRef]

C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.

2004 (4)

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
[CrossRef]

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

M. H. El-Shafey, in Proceedings of the International Conference on Electrical, Electronic, and Computer Engineering (IEEE, 2004), pp. 210-213.

2003 (1)

C. V. Gabel and H. C. Berg, Proc. Natl. Acad. Sci. USA 100, 8748 (2003).
[CrossRef] [PubMed]

2002 (2)

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

2001 (1)

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

1997 (1)

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

1996 (2)

R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
[CrossRef] [PubMed]

J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

1994 (1)

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef] [PubMed]

1991 (1)

R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
[CrossRef]

1988 (1)

Beck, M.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

Berg, H. C.

C. V. Gabel and H. C. Berg, Proc. Natl. Acad. Sci. USA 100, 8748 (2003).
[CrossRef] [PubMed]

Berg-Sørensen, K.

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

Block, S. M.

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef] [PubMed]

Brown, S.

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

Ciobanu, R.

C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.

Clark, M.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Cluzel, P.

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Crocker, J. C.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Crowe, J. A.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Dmochowski, P. R.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Donciu, C.

C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.

Doyle, P. S.

T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 (2005).
[CrossRef]

T. Savin and P. S. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

Dreyer, J. K.

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

El-Shafey, M. H.

M. H. El-Shafey, in Proceedings of the International Conference on Electrical, Electronic, and Computer Engineering (IEEE, 2004), pp. 210-213.

Emonet, T.

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Gabel, C. V.

C. V. Gabel and H. C. Berg, Proc. Natl. Acad. Sci. USA 100, 8748 (2003).
[CrossRef] [PubMed]

Gisler, T.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

Gittes, F.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Grego, S.

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

Grier, D. G.

M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
[CrossRef]

J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Grootjans, R.

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

Guyot-Sionnest, P.

M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
[CrossRef]

Halvorsen, K.

Hayes-Gill, B. R.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Kaplan, P. D.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

Kinosita, K.

R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
[CrossRef] [PubMed]

Korobkova, E.

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Kuijk, M.

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

MacKintosh, F. C.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Marshall, A. G.

Mitchell, A. C.

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

Miyata, H.

R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
[CrossRef] [PubMed]

Morgan, C. G.

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

Morgan, S. P.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Murray, J. G.

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

Oddershede, L.

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

Olmsted, P. D.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Pelton, M.

M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
[CrossRef]

Prud'homme, R. K.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

Ricca, T. L.

Savin, T.

T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 (2005).
[CrossRef]

T. Savin and P. S. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

Schmidt, C. F.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Schnurr, B.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Scott, N. L.

R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
[CrossRef]

Selvin, P. R.

E. Toprak and P. R. Selvin, Annu. Rev. Biophys. Biomol. Struct. 36, 349 (2007).
[CrossRef] [PubMed]

Shimizu, T. S.

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Somekh, M. G.

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

Stiens, J.

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

Svoboda, K.

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef] [PubMed]

Temneanu, C.

C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.

Thota, D.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

Toprak, E.

E. Toprak and P. R. Selvin, Annu. Rev. Biophys. Biomol. Struct. 36, 349 (2007).
[CrossRef] [PubMed]

Valentine, M. T.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

van der Tempel, W.

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

Van Nieuwenhove, D.

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

Vaughan, R. G.

R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
[CrossRef]

Verdun, F. R.

Vilar, J. M. G.

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Wall, J. E.

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

Weitz, D. A.

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

White, D. R.

R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
[CrossRef]

Wong, W. P.

Yasuda, R.

R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
[CrossRef] [PubMed]

Annu. Rev. Biophys. Biomol. Struct. (2)

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef] [PubMed]

E. Toprak and P. R. Selvin, Annu. Rev. Biophys. Biomol. Struct. 36, 349 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

M. Pelton, D. G. Grier, and P. Guyot-Sionnest, Appl. Phys. Lett. 85, 819 (2004).
[CrossRef]

Appl. Spectrosc. (1)

Biophys. J. (2)

L. Oddershede, J. K. Dreyer, S. Grego, S. Brown, and K. Berg-Sørensen, Biophys. J. 83, 3152 (2002).
[CrossRef] [PubMed]

T. Savin and P. S. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

Electron. Lett. (1)

P. R. Dmochowski, B. R. Hayes-Gill, M. Clark, J. A. Crowe, M. G. Somekh, and S. P. Morgan, Electron. Lett. 40, 1403 (2004).
[CrossRef]

IEEE Sens. J. (1)

D. Van Nieuwenhove, W. van der Tempel, R. Grootjans, J. Stiens, and M. Kuijk, IEEE Sens. J. 7, 317 (2007).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. G. Vaughan, N. L. Scott, and D. R. White, IEEE Trans. Signal Process. 39, 1973 (1991).
[CrossRef]

J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

J. Microsc. (1)

A. C. Mitchell, J. E. Wall, J. G. Murray, and C. G. Morgan, J. Microsc. 206, 225 (2002).
[CrossRef] [PubMed]

J. Mol. Biol. (1)

R. Yasuda, H. Miyata, and K. Kinosita, Jr, J. Mol. Biol. 263, 227 (1996).
[CrossRef] [PubMed]

Nature (1)

E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, and P. Cluzel, Nature 428, 574 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. E (2)

M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz, Phys. Rev. E 64, 061506 (2001).
[CrossRef]

T. Savin and P. S. Doyle, Phys. Rev. E 71, 041106 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

C. V. Gabel and H. C. Berg, Proc. Natl. Acad. Sci. USA 100, 8748 (2003).
[CrossRef] [PubMed]

Other (5)

C. Donciu, C. Temneanu, and R. Ciobanu, in Proceedings of the IEEE International Conference on Virtual Environments, Human-Computer Interfaces, and Measurement Systems (IEEE, 2005), pp. 50-53.

Note that 1/4πN2∫∣H̃0(ω)∣2dω=1/2WN2.

Determination of ϕ to eliminate terms t3 and t4 can be done by calculating ∣H̃(ω)∣2, setting the sum of the ϕ-dependent terms to zero, and solving for the specific case where W=n2π/ω′.

For example, measurement sensitivity can be improved by removing the dc offset (i.e., set B=0). For video imaging, this can be accomplished by simulating positive and negative light (e.g., by using two colors or polarizations) or, for low-light applications, by properly synchronizing two camera frames to each illumination cycle. At the sensor level, IMSA can be implemented by using a camera capable of modulating the incoming signal , or by having two collection wells for each pixel to simulate positive and negative light.

M. H. El-Shafey, in Proceedings of the International Conference on Electrical, Electronic, and Computer Engineering (IEEE, 2004), pp. 210-213.

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Figures (2)

Fig. 1
Fig. 1

Graphical representation of IMSA showing (a) unnormalized impulse responses H ( t ) = L ( t ) H 0 ( t ) for ϕ = 0 and B = 0 [solid line; see Eq. (5)] and H ( t ) = H 0 ( t ) [dotted line; see Eq. (3)] and (b) the squared magnitude of their Fourier transforms, which filter P ( ω ) in terms t 1 and t 2 of Eq. (6).

Fig. 2
Fig. 2

Measurement of the power spectrum using IMSA for two trapped beads with different spring constants (data points). The expected power spectra are superimposed (solid curves), showing good agreement well beyond the Nyquist frequency (vertical dashed line). Error bars represent statistical error.

Equations (13)

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

X m ( t ) = ( X H ) ( t ) X ( t ) H ( t t ) d t .
var [ X m ] = 1 2 π P ( ω ) H ̃ ( ω ) 2 d ω ,
H 0 ( t ) = { 1 W W 2 < t W 2 0 elsewhere } .
H ̃ 0 ( ω ) 2 = ( sin ( ω W 2 ) ω W 2 ) 2 .
H ( t ) = L ( t ) H 0 ( t ) N ,
var [ X m ] = 1 4 π N 2 P ( ω ) H ̃ 0 ( ω ω ) 2 d ω [ t 1 ]
+ B 2 2 π N 2 P ( ω ) H ̃ 0 ( ω ) 2 d ω [ t 2 ]
+ B sin ( ϕ ) π N 2 P ( ω ) H ̃ 0 ( ω ) H ̃ 0 ( ω ω ) d ω [ t 3 ]
cos ( 2 ϕ ) 4 π N 2 P ( ω ) H ̃ 0 ( ω + ω ) H ̃ 0 ( ω ω ) d ω [ t 4 ] .
P ( ω ) = 2 W N 2 var [ X m ] ( ω ) when B = 0 ,
P ( ω ) = 2 W B 2 ( var [ X m ] ( ω ) var [ X m ] ( 0 ) )
when B 0 ,
ϕ = ( 1 ) n sin 1 ( B 2 + 1 2 B ) .

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