Abstract

A heterodyne interferometer for highly sensitive vibration measurements in the range 100kHz − 1.3GHz is presented. The interferometer measures absolute amplitude and phase. The signal processing of the setup is analyzed and described in detail to optimize noise suppression. A noise floor of 7.1fm/Hz1/2 at 21MHz was achieved experimentally where the bandwidth is the inverse of all time needed for filter settling and signal sampling. To demonstrate the interferometer, measurements up to 220MHz were performed on arrays of capacitive micromachined ultrasonic transducers (CMUTs). The measurements provided detailed information e.g. about the frequency response, vibration patterns and array uniformity. Such measurements are highly valuable in the design process of ultrasonic transducers.

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References

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  1. J. Knuuttila, P. Tikka, and M. Salomaa, “Scanning Michelson interferometer for imaging surface acoustic wave fields,” Opt. Lett.25, 613–615 (2000).
    [CrossRef]
  2. J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
    [CrossRef]
  3. G. Fattinger and P. Tikka, “Modified Mach-Zender laser interferometer for probing bulk acoustic waves,” Appl. Phys. Lett.79, 290–292 (2001).
    [CrossRef]
  4. J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum.56, 543–546 (1985).
    [CrossRef]
  5. H. Martinussen, A. Aksnes, and H. E. Engan, “Wide frequency range measurements of absolute phase and amplitude of vibrations in micro- and nanostructures by optical interferometry,” Opt. Express15, 11370–11384 (2007).
    [CrossRef] [PubMed]
  6. K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett.92, 063502 (2008).
    [CrossRef]
  7. T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
    [CrossRef]
  8. T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
    [CrossRef]
  9. J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
    [CrossRef]
  10. E. Leirset and A. Aksnes, “Optical vibration measurements of cross coupling effects in capacitive micromachined ultrasonic transducer arrays,” Proc. SPIE8082, 80823N (2011)
    [CrossRef]
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    [CrossRef]
  12. R. L. Whitman and A. Korpel, “Probing of acoustic surface perturbations by coherent light,” Appl. Opt.8, 1567–1576 (1969).
    [CrossRef] [PubMed]
  13. J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
    [CrossRef]
  14. J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Prentice Hall, 1996), 3rd ed.
  15. J. W. Nilsson and S. A. Riedel, Electric Circuits (Prentice Hall, 2001), 6th ed.

2012

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

2011

E. Leirset and A. Aksnes, “Optical vibration measurements of cross coupling effects in capacitive micromachined ultrasonic transducer arrays,” Proc. SPIE8082, 80823N (2011)
[CrossRef]

2008

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett.92, 063502 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

2007

2006

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

2001

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

G. Fattinger and P. Tikka, “Modified Mach-Zender laser interferometer for probing bulk acoustic waves,” Appl. Phys. Lett.79, 290–292 (2001).
[CrossRef]

2000

J. Knuuttila, P. Tikka, and M. Salomaa, “Scanning Michelson interferometer for imaging surface acoustic wave fields,” Opt. Lett.25, 613–615 (2000).
[CrossRef]

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

1985

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum.56, 543–546 (1985).
[CrossRef]

1969

Aksnes, A.

E. Leirset and A. Aksnes, “Optical vibration measurements of cross coupling effects in capacitive micromachined ultrasonic transducer arrays,” Proc. SPIE8082, 80823N (2011)
[CrossRef]

H. Martinussen, A. Aksnes, and H. E. Engan, “Wide frequency range measurements of absolute phase and amplitude of vibrations in micro- and nanostructures by optical interferometry,” Opt. Express15, 11370–11384 (2007).
[CrossRef] [PubMed]

Ballandras, S.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

Barber, B.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

Breivik, L.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Due-Hansen, J.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Engan, H. E.

Fattinger, G.

G. Fattinger and P. Tikka, “Modified Mach-Zender laser interferometer for probing bulk acoustic waves,” Appl. Phys. Lett.79, 290–292 (2001).
[CrossRef]

Fujikura, T.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

Gammel, P.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

Gopani, S.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

Graebner, J.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

Greywall, D.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

Hurley, D.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Jensen, G.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Kaivola, M.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett.92, 063502 (2008).
[CrossRef]

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

Knuuttila, J.

Kokkonen, K.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett.92, 063502 (2008).
[CrossRef]

Korpel, A.

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

Leirset, E.

E. Leirset and A. Aksnes, “Optical vibration measurements of cross coupling effects in capacitive micromachined ultrasonic transducer arrays,” Proc. SPIE8082, 80823N (2011)
[CrossRef]

Manolakis, D. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Prentice Hall, 1996), 3rd ed.

Martinussen, H.

Masson, J.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

Matsuda, O.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Midtbø, K.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Monchalin, J.-P.

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum.56, 543–546 (1985).
[CrossRef]

Muroya, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Nilsson, J. W.

J. W. Nilsson and S. A. Riedel, Electric Circuits (Prentice Hall, 2001), 6th ed.

Poppe, E.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Proakis, J. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Prentice Hall, 1996), 3rd ed.

Profunser, D.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

Riedel, S. A.

J. W. Nilsson and S. A. Riedel, Electric Circuits (Prentice Hall, 2001), 6th ed.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).
[CrossRef]

Salomaa, M.

Schjølberg-Henriksen, K.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Sugawara, Y.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Summanwar, A.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Tachizaki, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).
[CrossRef]

Tikka, P.

G. Fattinger and P. Tikka, “Modified Mach-Zender laser interferometer for probing bulk acoustic waves,” Appl. Phys. Lett.79, 290–292 (2001).
[CrossRef]

J. Knuuttila, P. Tikka, and M. Salomaa, “Scanning Michelson interferometer for imaging surface acoustic wave fields,” Opt. Lett.25, 613–615 (2000).
[CrossRef]

Wang, D.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Whitman, R. L.

Wright, O.

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

J. Graebner, B. Barber, P. Gammel, D. Greywall, and S. Gopani, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett.78, 159–161 (2001).
[CrossRef]

G. Fattinger and P. Tikka, “Modified Mach-Zender laser interferometer for probing bulk acoustic waves,” Appl. Phys. Lett.79, 290–292 (2001).
[CrossRef]

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett.92, 063502 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. Profunser, O. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett.93, 261101 (2008).
[CrossRef]

J. Micromech. Microeng.

J. Due-Hansen, K. Midtbø, E. Poppe, A. Summanwar, G. Jensen, L. Breivik, D. Wang, and K. Schjølberg-Henriksen, “Fabrication process for CMUT arrays with polysilicon electrodes, nanometre precision cavity gaps and through-silicon vias,”J. Micromech. Microeng.22, 074009 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

E. Leirset and A. Aksnes, “Optical vibration measurements of cross coupling effects in capacitive micromachined ultrasonic transducer arrays,” Proc. SPIE8082, 80823N (2011)
[CrossRef]

Rev. Sci. Instrum.

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum.56, 543–546 (1985).
[CrossRef]

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. Hurley, and O. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum.77, 043713 (2006).
[CrossRef]

Other

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).
[CrossRef]

J. G. Proakis and D. G. Manolakis, Digital Signal Processing (Prentice Hall, 1996), 3rd ed.

J. W. Nilsson and S. A. Riedel, Electric Circuits (Prentice Hall, 2001), 6th ed.

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Figures (13)

Fig. 1
Fig. 1

Sketch of the optical setup.

Fig. 2
Fig. 2

Sketch of the electrical setup for extracting the complex amplitudes RI and RN in Eq. (5)

Fig. 3
Fig. 3

Sketch of parts of the detector signal spectrum. The black solid arrows indicate the RI and RN signals, and the stippled arrows the local oscillator signals defined in Eqs. (6) and (7).

Fig. 4
Fig. 4

Block diagram of the signal processing following the mixers in the lock-in amplifiers. The upper branch is the path of the down mixed RN-signal and the lower branch is the path of the down mixed RI-signal.

Fig. 5
Fig. 5

Cross sectional view of a CMUT with a via going to the back side of the wafer. The cross section is along the dashed line in Fig. 6. Note that the figure is not to scale.

Fig. 6
Fig. 6

The etched out cavities in a cell consists of four circular CMUTs with connection bridges. Note that only the darkest areas are in contact with the top plate. Figure 5 illustrates the cross section along the dashed line.

Fig. 7
Fig. 7

Measured vibration amplitude at the center of a CMUT as a function of RMS excitation AC voltage. The measurement was performed with an MBW of 1Hz. The red line is at 7.1fm and indicates the root mean square of the measured amplitudes where the excitation voltage is less than 3μV.

Fig. 8
Fig. 8

Vibration pattern of four excited CMUT cells. The plots are shown with both linear and logarithmic color scale. The two black circles indicate the points where the frequency response measurements in Fig. 9 were made.

Fig. 9
Fig. 9

Vibration amplitude and RN as a function of excitation frequency. (a) illustrates the measured vibration amplitude from the entire measured frequency range, (b) illustrates a close-up of the fundamental resonance, and (c) illustrates RN which is a measure of the detected signal level. The amplitudes are measured at the points indicated by black circles in Fig. 8, and the legend also indicates the coordinates of these points. The measurement was performed with an MBW of 4.3Hz.

Fig. 10
Fig. 10

Vibration amplitude and phase of a CMUT cell at different frequencies. The x-axis is common for all the plots, and is only shown in (d)

Fig. 11
Fig. 11

Vibration amplitude at the center of all CMUTs in the array for two excitation frequencies. The x– and y-axes and the color scale are identical for both plots. Note that the color scale is logarithmic.

Fig. 12
Fig. 12

Vibration amplitude and phase of a CMUT cell near the lower edge of the array. (a) plots the amplitude with logarithmic color scale and (b) plots the phase.

Fig. 13
Fig. 13

Plots of the ENBW as a function of Fs for τ equal to 1ms and 3ms, Ta = 1s, and q = 3. The solid lines are numerical calculations of Eq. (20), and the dashed lines are from the approximation in Eq. (21) using only the terms k = −1, 0, 1 in the series.

Tables (3)

Tables Icon

Table 1 AOM specifications from the manufacturer

Tables Icon

Table 2 AOM configurations enabling a frequency shift up to 650MHz allowing twice the measuring frequency.

Tables Icon

Table 3 Solutions to Eq. (24), the ENBW of HLIA(F) alone, and the figure of merit Tw × ENBW

Equations (27)

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

F 1 = V a 1 V a 1 + V a 2 F m + F 0
F 2 = ± ( V a 2 V a 1 + V a 2 F m F 0 ) .
d ( t ) = a cos ( 2 π F a t + ϕ a )
I D ( t ) = α P avg [ 1 + C ( cos [ 2 π F m t ϕ 1 + ϕ 2 ] + 2 π a λ sin [ 2 π ( F a F m ) t + ϕ 1 ϕ 2 + ϕ a ] + 2 π a λ sin [ 2 π ( F a + F m ) t ϕ 1 + ϕ 2 + ϕ a ] ) ] .
a e i ϕ a = λ 2 π ( R I R N ) *
F mix , I = 1 2 ( F a + Δ F ) + F L I A
F mix , N = 1 2 ( F a Δ F ) F LIA
i s = 2 q α P avg E N B W
a min = λ π C q E N B W α P avg
2 | H L I A ( F s ) | 2 1
M B W 1 T a
E N B W 2 T a .
X ( F ) = 1 F s n = x ( T s n ) e i 2 π n F F s
H L I A ( F ) = X I f ( F ) X Im ( F ) = X N f ( F ) X N m ( F ) = ( 1 1 + i 2 π F τ ) p
X I s ( F ) = l = X I f ( F F s l ) .
H avg ( F ) = X savg ( F ) X r ( F ) = 1 M a sin ( π M a F F s ) sin ( π F F s ) e i π ( M a 1 ) F F s .
X avg ( F ) = l = 0 M a 1 X savg ( F F s M a l ) .
A ( F ) = λ 2 π X N f * ( 0 ) [ l = 0 M a 1 H avg ( F F s M a l ) k = X Im ( F F s M a l F s k ) H L I A ( F F s M a l F s k ) ] * .
N a ( F ) = N Im | λ 2 π X N f ( 0 ) | 2 [ l = 0 M a 1 | H avg ( F F s M a l ) | 2 k = | H L I A ( F F s M a l F s k ) | 2 ] .
E N B W = 2 F = F s 2 F s 2 | H avg ( F ) | 2 k = | H L I A ( F F s k ) | 2 d F .
E N B W 2 k = | H L I A ( F s k ) | 2 F = F s 2 F s 2 | H avg ( F ) | 2 d F = 2 T a k = | H L I A ( F s k ) | 2 .
2 | H L I A ( F s ) | 2 1 .
h L I A ( t ) = t p 1 e t τ τ p ( p 1 ) ! u ( t )
t = T w h L I A ( t ) d t = 1 1000
T m = T a + T w
M B W = 1 T a + T w 1 2 E N B W .
T a = 1 M B W T w .

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