Abstract

We investigate the temporal intensity noise characteristics of analog liquid-crystal-based spatial light modulators and how they affect the device’s achievable discrete numeric accuracies in an optical computing system. First we present an analytical development that defines the concept of precision in analog computing systems, then we define a noise metric and a precision-optimal quantizer for determining the discrete numeric characteristics of the devices. Second we present an experimental discussion in which a low-noise test facility constructed for this investigation is described, and the noise characteristics of three commercially available liquid-crystal-based modulators are measured and analyzed. The accuracy implications of this measured noise are then discussed within the context of the analytical model for each modulator.

© 1994 Optical Society of America

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  1. F. Koyama, “Intensity noise and polarization study of GaAlAs–GaAs surface emitting lasers,” IEEE J. Quantum Electron. 27, 1410–1416(1991).
    [CrossRef]
  2. N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
    [CrossRef]
  3. S. G. Batsell, “Accuracy limitations in optical linear algebraic processors,” Ph.D. dissertation (Dept. of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1990).
  4. D. A. Gregory, T. D. Hudson, J. C. Kirsch, “Measurement of spatial light modulator parameters,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1297, 176–185 (1990).
  5. M. G. Robinson, K. M. Johnson, “Noise analysis of polarization-based optoelectronic connectionist machine,” Appl. Opt. 31, 263–272 (1992).
    [CrossRef] [PubMed]
  6. T. Kajiyama, H. Kikuchi, A. Takahara, “Polymer/(liquid crystal) composite systems for novel electro-optic effects,” in Liquid Crystal Materials, Devices, and Applications, P. S. Drzaic, U. Efron, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1665, 20–31 (1992).
  7. S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
    [CrossRef]
  8. J. W. Goodman, L. M. Woody, “Method for performing complex-valued linear operations on complex-valued data using incoherent light,” Appl. Opt. 16, 2611–2622 (1977).
    [CrossRef] [PubMed]
  9. S. G. Batsell, T. L. Long, J. Walkup, T. T. Krile, “Noise limitations in optical algebra processors,” Appl. Opt. 29, 2084–2090 (1990).
    [CrossRef] [PubMed]
  10. S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, London, 1992), Chap. 3.
    [CrossRef]
  11. Personal communication with D. Timucin of the Optical Systems Laboratory, Department of Electrical Engineering, Texas Tech University, Lubbock, Texas 79406-3102, who is working on similar issues.
  12. H. W. Ottis, Noise Reduction Techniques in Electronic Systems (Wiley, New York, 1976), Chaps. 3–6.
  13. P. Norton, P. Yao, Window 3.0 Power Programming Techniques (Bantam, New York, 1990), Chap. 6.
  14. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  15. W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).
  16. T. D. Hudson, D. A. Gregory, “Optically-addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
    [CrossRef]
  17. D. B. Taber, J. A. Davis, L. A. Holloway, O. Almagor, “Optically controlled Fabry–Perot interferometer using a liquid crystal light valve,” Appl. Opt. 29, 2623–2631 (1990).
    [CrossRef] [PubMed]
  18. J. Grinberg, A. D. Jacobson, “Transmission characteristics of a twisted nematic liquid-crystal layer,” J. Opt. Soc. Am. 66, 1003–1009 (1976).
    [CrossRef]
  19. K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optimal spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
    [CrossRef]
  20. J. C. Kirsch, “Optical Modulation Characteristics and Applications of Liquid Crystal Televisions,” U.S. Army Tech. Rep. RD-WS-92-6 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1992)
  21. Meadowlark Optics Polarization Optics Catalog and Handbook (Meadowlark Optics, Longmont, Colo., 1993), pp. 10–14.
  22. Almost periodic signals result from the combination of periodic or complex periodic signals whose ratio of individual fundamental periods is not a rational number; hence no resultant fundamental period exists. Obviously this can easily arise in practice.

1992 (2)

M. G. Robinson, K. M. Johnson, “Noise analysis of polarization-based optoelectronic connectionist machine,” Appl. Opt. 31, 263–272 (1992).
[CrossRef] [PubMed]

S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
[CrossRef]

1991 (3)

F. Koyama, “Intensity noise and polarization study of GaAlAs–GaAs surface emitting lasers,” IEEE J. Quantum Electron. 27, 1410–1416(1991).
[CrossRef]

N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
[CrossRef]

T. D. Hudson, D. A. Gregory, “Optically-addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

1990 (3)

1978 (2)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

1977 (1)

1976 (1)

Almagor, O.

Batsell, S. G.

S. G. Batsell, T. L. Long, J. Walkup, T. T. Krile, “Noise limitations in optical algebra processors,” Appl. Opt. 29, 2084–2090 (1990).
[CrossRef] [PubMed]

S. G. Batsell, “Accuracy limitations in optical linear algebraic processors,” Ph.D. dissertation (Dept. of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1990).

Bleha, W. P.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Brown, H. B.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Casasent, D.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, London, 1992), Chap. 3.
[CrossRef]

Davis, J. A.

Fujiwara, K.

S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
[CrossRef]

Giugni, S.

S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
[CrossRef]

Goodman, J. W.

Gregory, D. A.

T. D. Hudson, D. A. Gregory, “Optically-addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

D. A. Gregory, T. D. Hudson, J. C. Kirsch, “Measurement of spatial light modulator parameters,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1297, 176–185 (1990).

Grinberg, J.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

J. Grinberg, A. D. Jacobson, “Transmission characteristics of a twisted nematic liquid-crystal layer,” J. Opt. Soc. Am. 66, 1003–1009 (1976).
[CrossRef]

Hakim, N. Z.

N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
[CrossRef]

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Holloway, L. A.

Hudson, T. D.

T. D. Hudson, D. A. Gregory, “Optically-addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

D. A. Gregory, T. D. Hudson, J. C. Kirsch, “Measurement of spatial light modulator parameters,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1297, 176–185 (1990).

Jacobson, A. D.

Johnson, K. M.

Kajiyama, T.

T. Kajiyama, H. Kikuchi, A. Takahara, “Polymer/(liquid crystal) composite systems for novel electro-optic effects,” in Liquid Crystal Materials, Devices, and Applications, P. S. Drzaic, U. Efron, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1665, 20–31 (1992).

Kawashima, K.

S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
[CrossRef]

Kikuchi, H.

T. Kajiyama, H. Kikuchi, A. Takahara, “Polymer/(liquid crystal) composite systems for novel electro-optic effects,” in Liquid Crystal Materials, Devices, and Applications, P. S. Drzaic, U. Efron, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1665, 20–31 (1992).

Kirsch, J. C.

D. A. Gregory, T. D. Hudson, J. C. Kirsch, “Measurement of spatial light modulator parameters,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1297, 176–185 (1990).

J. C. Kirsch, “Optical Modulation Characteristics and Applications of Liquid Crystal Televisions,” U.S. Army Tech. Rep. RD-WS-92-6 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1992)

Koyama, F.

F. Koyama, “Intensity noise and polarization study of GaAlAs–GaAs surface emitting lasers,” IEEE J. Quantum Electron. 27, 1410–1416(1991).
[CrossRef]

Krile, T. T.

Lipton, L. T.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Long, T. L.

Lu, K.

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optimal spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Markevitch, B. V.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Norton, P.

P. Norton, P. Yao, Window 3.0 Power Programming Techniques (Bantam, New York, 1990), Chap. 6.

Ottis, H. W.

H. W. Ottis, Noise Reduction Techniques in Electronic Systems (Wiley, New York, 1976), Chaps. 3–6.

Rief, P. G.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Robinson, M. G.

Saleh, B. E. A.

N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
[CrossRef]

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optimal spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Taber, D. B.

Takahara, A.

T. Kajiyama, H. Kikuchi, A. Takahara, “Polymer/(liquid crystal) composite systems for novel electro-optic effects,” in Liquid Crystal Materials, Devices, and Applications, P. S. Drzaic, U. Efron, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1665, 20–31 (1992).

Teich, M. C.

N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
[CrossRef]

Timucin, D.

Personal communication with D. Timucin of the Optical Systems Laboratory, Department of Electrical Engineering, Texas Tech University, Lubbock, Texas 79406-3102, who is working on similar issues.

Walkup, J.

Weiner-Avnear, E.

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Woody, L. M.

Yao, P.

P. Norton, P. Yao, Window 3.0 Power Programming Techniques (Bantam, New York, 1990), Chap. 6.

Appl Phys. Lett. (1)

S. Giugni, K. Kawashima, K. Fujiwara, “New self-electro-optic effect device using two wavelengths in InGaAs/AlGaAs multiple quantum wells,” Appl Phys. Lett. 61, 376–382 (1992).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

F. Koyama, “Intensity noise and polarization study of GaAlAs–GaAs surface emitting lasers,” IEEE J. Quantum Electron. 27, 1410–1416(1991).
[CrossRef]

J. Lightwave Technol. (1)

N. Z. Hakim, B. E. A. Saleh, M. C. Teich, “Signal-to-noise ratio for lightwave systems using avalanche photodiodes,” J. Lightwave Technol. 9, 318–320 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optimal spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

W. P. Bleha, L. T. Lipton, E. Weiner-Avnear, J. Grinberg, P. G. Rief, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Opt. Laser Technol. (1)

T. D. Hudson, D. A. Gregory, “Optically-addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other (10)

T. Kajiyama, H. Kikuchi, A. Takahara, “Polymer/(liquid crystal) composite systems for novel electro-optic effects,” in Liquid Crystal Materials, Devices, and Applications, P. S. Drzaic, U. Efron, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1665, 20–31 (1992).

J. C. Kirsch, “Optical Modulation Characteristics and Applications of Liquid Crystal Televisions,” U.S. Army Tech. Rep. RD-WS-92-6 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1992)

Meadowlark Optics Polarization Optics Catalog and Handbook (Meadowlark Optics, Longmont, Colo., 1993), pp. 10–14.

Almost periodic signals result from the combination of periodic or complex periodic signals whose ratio of individual fundamental periods is not a rational number; hence no resultant fundamental period exists. Obviously this can easily arise in practice.

S. G. Batsell, “Accuracy limitations in optical linear algebraic processors,” Ph.D. dissertation (Dept. of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1990).

D. A. Gregory, T. D. Hudson, J. C. Kirsch, “Measurement of spatial light modulator parameters,” in Hybrid Image and Signal Processing II, D. P. Casasent, A. G. Tescher, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1297, 176–185 (1990).

S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, London, 1992), Chap. 3.
[CrossRef]

Personal communication with D. Timucin of the Optical Systems Laboratory, Department of Electrical Engineering, Texas Tech University, Lubbock, Texas 79406-3102, who is working on similar issues.

H. W. Ottis, Noise Reduction Techniques in Electronic Systems (Wiley, New York, 1976), Chaps. 3–6.

P. Norton, P. Yao, Window 3.0 Power Programming Techniques (Bantam, New York, 1990), Chap. 6.

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Figures (22)

Fig. 1
Fig. 1

Lumped parameter devices in a one-channel OMVM: NDF, neutral density filter; LLF’s, laser line filters; Pol’s, polarizers; SLM, spatial light modulator.

Fig. 2
Fig. 2

Spectra Physics 124 He–Ne output driven by SMPS: (a) PSD, (b) histogram.

Fig. 3
Fig. 3

Quantized monotonically increasing signal space showing L signal states and L − 1 decision regions. The average BERF is the sum of all areas of intersection along this range divided by L. Shown in the upper right is a single signal state that depicts the bit-decision levels, the BERF, and the BERF-based dispersion.

Fig. 4
Fig. 4

PSD of Spectra Physics 145 He–Ne output driven by a linear power supply.

Fig. 5
Fig. 5

PSD of the dark detector circuit output.

Fig. 6
Fig. 6

Histograms of He–Ne output when the detector is driven by a linear power supply (LPS) and the dark detector circuit output.

Fig. 7
Fig. 7

LCLV 4050 experimental setup. VNDF, variable neutral density filter; PD’s, photodetectors; NPBS, nonpolarizing beam splitter.

Fig. 8
Fig. 8

PSD of Lexel 95 Ar+ output when operating in a light control mode.

Fig. 9
Fig. 9

LCLV 4050 output mean, LDR, BERF-based dispersion, and SNR over the experimentally selected monotonic operating range.

Fig. 10
Fig. 10

Temporal signals for the LCLV 4050 drive and output: (a) worst noise case, (b) best noise case.

Fig. 11
Fig. 11

PSD of the LCLV 4050 output: (a) worst noise case, (b) best noise case.

Fig. 12
Fig. 12

Histograms for the LCLV 4050: best and worst noise cases.

Fig. 13
Fig. 13

LCTV 08TA-OA experimental setup. PD’s, photodetectors; DMM, digital multimeter.

Fig. 14
Fig. 14

LCTV 08TA-OA output mean, LDR, BERF-based dispersion, and SNR over the experimentally selected monotonic operating range.

Fig. 15
Fig. 15

Temporal signals for the LCTV 08TA-OA output: (a) worst noise case, (b) best noise case.

Fig. 16
Fig. 16

PSD of the LCTV 08TA-OA output: (a) worst noise case, (b) best noise case.

Fig. 17
Fig. 17

Histograms for the LCTV 08TA-OA: best and worst noise cases.

Fig. 18
Fig. 18

LCVR LVR0.7 experimental setup.

Fig. 19
Fig. 19

LCVR LVR0.7 output mean, LDR, BERF-based dispersion, and SNR over the experimentally selected monotonic operating range.

Fig. 20
Fig. 20

Temporal signals for the LVR LVR0.7 drive and output: (a) worst noise case, (b) best noise case.

Fig. 21
Fig. 21

PSD of the LCVR LVR0.7 output: (a) worst noise case, (b) best noise case.

Fig. 22
Fig. 22

Histograms for the LCVR LVR0.7: best and worst noise cases.

Tables (1)

Tables Icon

Table 1 Performance Measures for the Test Modulatorsa

Equations (24)

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

v d ( t ) = L d I { I m ( t ) } + L d i { i d n ( t ) } .
I m ( t ) = I s ( t ) T m ( t ) .
I s ( t ) = L s { i s ( t ) } + H s { i s ( t ) } = ˙ L s { i s ( t ) } + L s { i s n ( t ) } + H s { i s ( t ) } .
T m ( t ) = L m { v m ( t ) } + H m { v m ( t ) } = L m { v m rms } + L m { v m n ( t ) } + H m { v m ( t ) } ,
v d ( t ) = G [ I s ( t ) T m ( t ) ] + G ( L m { v m n ( t ) } [ I s ( t ) + L s { i s n ( t ) } + H s { i s ( t ) } ] ) + G ( H m { v m ( t ) } [ I s ( t ) + L s { i s n ( t ) } + H s { i s ( t ) } ] ) + G ( T m ( t ) [ L s { i s n ( t ) } + H s { i s ( t ) } ] ) + R f [ i d n ( t ) ] .
R m ( t ) = L m { ρ w ( t ) v m ( t ) } + H m { ρ w ( t ) v m ( t ) } = L m { ( [ ρ w ( t ) v m ( t ) ] 2 ) 1 / 2 } + L m { ρ w ( t ) v m n ( t ) } + H m { ρ w ( t ) v m ( t ) } ,
ρ w ( t ) = R p [ I w ( t ) ] = R p [ L s { i w ( t ) } + H s { i w ( t ) } ] = R p [ L s { i w ( t ) } + L s { i w n ( t ) } + H s { i w ( t ) } ] .
v d ( t ) = G [ I s ( t ) R m ( t ) ] + G [ L m { v m n ( t ) } ( L m { R p I w ( t ) + R p L s { i w n ( t ) } + R p H s { i w ( t ) } } ) [ I s ( t ) + L s { i s n ( t ) } + H s { i s ( t ) } ] + G [ H m { v m ( t ) } ( H m { R p I w ( t ) + R p L s { i w n ( t ) } + R p H s { i w ( t ) } } ) [ I s ( t ) + L s { i s n ( t ) } + H s { i s ( t ) } ] + G ( R m ( t ) [ L s { i s n ( t ) } + H s { i s ( t ) } ] ) + R f [ i d n ( t ) ] .
R f [ i d n ( t ) ] = R f ( i d n ( t ) + δ d [ i d EMI ( t ) ] ) + δ w [ v d EMI ( t ) ] ,
ɛ ¯ 1 L k = 1 L [ d k f v k ( v ) d v + - d k f v k + 1 ( v ) d v ] ,
N N ± [ ɛ ¯ 2 ( N - 1 ) LSB ]
P N - ɛ ¯ ( N - 1 ) .
ɛ 2 - v ( ɛ ) f v d ( v d ) d v 2 v ( ɛ ) f v d ( v d ) d v .
ζ ( ɛ ) v ( ɛ ) - v ( ɛ ) .
f v ( v ) = i f v i ( v ) ,
Γ ( ɛ ) = max { ζ k ( ɛ ) } k LDR .
L ( ɛ ) = LDR / Γ ( ɛ ) .
W ( k ) = [ a 0 - a 1 cos ( 2 π N k ) + a 2 cos ( 2 π N 2 k ) - a 3 cos ( 2 π N 3 k ) for k = 0 , , N - 1 ,
v d n ( t ) = G ( H m { v m ( t ) } [ H m { R p I w ( t ) } ] [ I s ( t ) ] )
v d n ( t ) = γ [ 1 + α sin ( 2 π f o t ) + β sin ( 4 π f o t ) ] ,
v d n ( t ) = G ( H m { v m ( t ) } [ I s ( t ) ] ) ,
v ( t ) = γ [ 1 + α exp ( - t / τ 1 ) ] [ u ( t ) - u ( t - t o ) ] + γ β [ 1 - exp ( - t / τ 2 ) ] [ u ( t - t o ) - u ( t - t p ) ] .
v d n ( t ) = G ( H m { v m ( t ) } [ I s ( t ) + H s { i s ( t ) } ] ) + G ( T m ( t ) [ H s { i s ( t ) } ] ) + R f [ i d n ( t ) ] ,
v d n ( t ) = γ [ 1 + α exp ( - t / τ ) ] [ u ( t ) - u ( t - t p ) ] .

Metrics