Abstract

The general analytic expression for the polynomial smoothing function of any degree for equally spaced data points is presented. In addition to the explicit formula, a simple recursion relation is also given. The determination of numerical coefficients in the convolution equation involves only integer arithmetic. These results are further used to describe in some detail the effectiveness of digital polynomial smoothing, or filtering, of sampled spectral data in their dependence on the degree K of the polynomial, the number S of smoothing passes, and the range T of points in the smoothing interval. Then it can be shown that the sharpness of the frequency cutoff increases with the degree of the polynomial, the high-frequency attenuation increases with the-number of smooths, and the cutoff of the filter moves toward lower frequencies as the range of points in the smoothing interval increases. The values of these three parameters should not be chosen entirely independently of one another, but the first two should be selected before the third.

© 1981 Optical Society of America

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  1. R. R. Ernst, "Sensitivity enhancement in magnetic resonance," in Advances in Magnetic Resonance, J. S. Waugh, ed. (Academic, New York, 1966), Vol. 2, pp. 1–135.
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  4. J. F. A. Ormsby, "Numerical filtering," in Spectral Analysis, J. A. Blackburn, ed. (Marcel Dekker, New York, 1970), pp. 67–120.
  5. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 7.
  6. T. H. Edwards and P. D. Willson, "Digital least squares smoothing of spectra," Appl. Spectrosc. 28, 541–545 (1974).
  7. P. D. Willson and T. H. Edwards, "Sampling and smoothing of spectra," Appl. Spectrosc. Rev. 12(1), 1 (1976).
  8. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 785.
  9. E. D. Tidwell, E. K. Plyler, and W. S. Benedict, "Vibration–rotation bands of N2O," J. Opt. Soc. Am. 50, 1243–1263 (1960).

1976 (1)

P. D. Willson and T. H. Edwards, "Sampling and smoothing of spectra," Appl. Spectrosc. Rev. 12(1), 1 (1976).

1974 (1)

1964 (1)

A. Savitsky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627–1639 (1964).

1960 (1)

Benedict, W. S.

Edwards, T. H.

P. D. Willson and T. H. Edwards, "Sampling and smoothing of spectra," Appl. Spectrosc. Rev. 12(1), 1 (1976).

T. H. Edwards and P. D. Willson, "Digital least squares smoothing of spectra," Appl. Spectrosc. 28, 541–545 (1974).

Ernst, R. R.

R. R. Ernst, "Sensitivity enhancement in magnetic resonance," in Advances in Magnetic Resonance, J. S. Waugh, ed. (Academic, New York, 1966), Vol. 2, pp. 1–135.

Golay, M. J. E.

A. Savitsky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627–1639 (1964).

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 7.

Ormsby, J. F. A.

J. F. A. Ormsby, "Numerical filtering," in Spectral Analysis, J. A. Blackburn, ed. (Marcel Dekker, New York, 1970), pp. 67–120.

Plyler, E. K.

Robinson, G.

E. Whittaker and G. Robinson, The Calculus of Observations (Dover, New York, 1967).

Savitsky, A.

A. Savitsky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627–1639 (1964).

Tidwell, E. D.

Whittaker, E.

E. Whittaker and G. Robinson, The Calculus of Observations (Dover, New York, 1967).

Willson, P. D.

P. D. Willson and T. H. Edwards, "Sampling and smoothing of spectra," Appl. Spectrosc. Rev. 12(1), 1 (1976).

T. H. Edwards and P. D. Willson, "Digital least squares smoothing of spectra," Appl. Spectrosc. 28, 541–545 (1974).

Anal. Chem. (1)

A. Savitsky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627–1639 (1964).

Appl. Spectrosc. (1)

Appl. Spectrosc. Rev. (1)

P. D. Willson and T. H. Edwards, "Sampling and smoothing of spectra," Appl. Spectrosc. Rev. 12(1), 1 (1976).

J. Opt. Soc. Am. (1)

Other (5)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 785.

J. F. A. Ormsby, "Numerical filtering," in Spectral Analysis, J. A. Blackburn, ed. (Marcel Dekker, New York, 1970), pp. 67–120.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 7.

R. R. Ernst, "Sensitivity enhancement in magnetic resonance," in Advances in Magnetic Resonance, J. S. Waugh, ed. (Academic, New York, 1966), Vol. 2, pp. 1–135.

E. Whittaker and G. Robinson, The Calculus of Observations (Dover, New York, 1967).

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