Abstract

A method for reconstructing a spectrum from a binary interferogram observed with an Fourier transform IR spectrometer is described. With this method an interferogram is quantized with a 1-bit analog-to-digital converter with a differentiator and an integrator. This method, called delta sigma modulation, features an oversampling of a signal at a rate much higher than the Nyquist sampling rate. We show experimental examples of IR spectra reconstructed by the method, which demonstrate the potential applications of the method to Fourier transform IR analysis. The results imply that the method may exceed the dynamic range over the one attainable with a conventional Fourier transform spectrometer with an analog-to-digital converter of a finite bit number.

© 1993 Optical Society of America

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References

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  1. H. Okayama, K. Minami, S. Kawata, “Super-dynamic range and super-quantization of FT–IR spectra,” Appl. Spectrosc. 47, 441–445 (1993).
    [CrossRef]
  2. K. Minami, S. Kawata, S. Minami, “Zero-crossing sampling of Fourier-transform interferograms and spectrum reconstruction using real-zero interpolation method,” Appl. Opt. 31, 6322–6327 (1992).
    [CrossRef] [PubMed]
  3. H. Inose, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
    [CrossRef]
  4. J. Candy, “A use of double integration in delta sigma modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
    [CrossRef]
  5. S. Tewksbury, R. Hallock, “Oversampled, linear predictive and noise shaping coders of order N > 1,” IEEE Trans. Circuits Syst. CAS-25, 436–447 (1978).
    [CrossRef]
  6. A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  7. Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
    [CrossRef]
  8. K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
    [CrossRef]

1993 (1)

1992 (1)

1990 (1)

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
[CrossRef]

1987 (1)

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

1985 (1)

J. Candy, “A use of double integration in delta sigma modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

1978 (1)

S. Tewksbury, R. Hallock, “Oversampled, linear predictive and noise shaping coders of order N > 1,” IEEE Trans. Circuits Syst. CAS-25, 436–447 (1978).
[CrossRef]

1963 (1)

H. Inose, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Candy, J.

J. Candy, “A use of double integration in delta sigma modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

Chao, K. C.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
[CrossRef]

Hallock, R.

S. Tewksbury, R. Hallock, “Oversampled, linear predictive and noise shaping coders of order N > 1,” IEEE Trans. Circuits Syst. CAS-25, 436–447 (1978).
[CrossRef]

Inose, H.

H. Inose, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Ishikawa, M.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Iwata, A.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Kawata, S.

Kobayashi, T.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Lee, S. H.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
[CrossRef]

Matsuya, Y.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Minami, K.

Minami, S.

Okayama, H.

Oppenheim, A.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schafer, R.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Sodini, C. G.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
[CrossRef]

Tewksbury, S.

S. Tewksbury, R. Hallock, “Oversampled, linear predictive and noise shaping coders of order N > 1,” IEEE Trans. Circuits Syst. CAS-25, 436–447 (1978).
[CrossRef]

Uchimura, K.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Yasuda, Y.

H. Inose, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Yoshitome, T.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

IEEE J. Solid State Circuits (1)

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid State Circuits SC-22, 921–929 (1987).
[CrossRef]

IEEE Trans. Circuits Syst. (2)

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Syst. 37, 309–318 (1990).
[CrossRef]

S. Tewksbury, R. Hallock, “Oversampled, linear predictive and noise shaping coders of order N > 1,” IEEE Trans. Circuits Syst. CAS-25, 436–447 (1978).
[CrossRef]

IEEE Trans. Commun. (1)

J. Candy, “A use of double integration in delta sigma modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

Proc. IEEE (1)

H. Inose, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Other (1)

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (8)

Fig. 1
Fig. 1

Principle of ΔΣ modulation.

Fig. 2
Fig. 2

ΔΣ modulator of (a) first order and (b) second order.

Fig. 3
Fig. 3

Computer simulation results: (a) interferogram around the zero path-length difference, (b) binary interferogram quantized by the second-order ΔΣ modulation, (c) Fourier spectrum of the binary interferogram in (b), (d) spectrum reconstructed from the binary interferogram in a spectrum range of interest (0–4000 cm−1), (e) spectrum reconstructed from an interferogram quantized with a 16-bit A/D converter.

Fig. 4
Fig. 4

Setup of a developed FT–IR spectrometer system equipped with a second-order ΔΣ modulator; PD, photodiode.

Fig. 5
Fig. 5

Spectrum of nichrome source reconstructed from binary interferograms (averaging 10).

Fig. 6
Fig. 6

Spectrum of a nichrome source reconstructed from a 16-bit quantized interferogram (averaging 10).

Fig. 7
Fig. 7

Absorption spectra of polypropylene film reconstructed from (a) binary interferograms measured with the developed system and (b) an interferogram quantized with a 16-bit A/D converter.

Fig. 8
Fig. 8

Example of hardware for spectrum reconstruction: A, B, input ports of the selector; SL, input selection.

Equations (9)

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Q ( z ) = 1 1 z 1 [ I ( z ) z 1 Q ( z ) ] + N ( z ) ,
Q ( z ) = I ( z ) + ( 1 z 1 ) N ( z ) .
D Δ Σ = Δ 2 1 π 0 ν N E ( ν ) d ν ,
E ( ν ) = 2 Δ 2 3 [ 1 cos ( 2 π ν M ν N ) ] ,
D Δ Σ = 10 log 10 Δ 2 2 Δ 2 3 π [ ν N M ν N 2 π sin ( 2 π M ν N ν ) ] = 30 log 10 M 10 log 10 ( 9 4 π ) ,
D A D = 6 . 0 β 4 . 7 ,
β Δ Σ = D Δ Σ + 4 . 7 6 . 0 1 . 5 log 2 M 1 . 0 .
β Δ Σ 2 2 . 5 log 2 M 3 . 3 .
β Δ Σ P ( P + 0 . 5 ) log 2 M ( 2 . 4 P 1 . 4 ) .

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