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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References

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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.
    [Crossref]
  2. E. Whittaker and G. Robinson, The Calculus of Observations (Dover, New York, 1967).
  3. A. Savitsky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
    [Crossref]
  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).
    [Crossref]
  7. P. D. Willson and T. H. Edwards, “Sampling and smoothing of spectra,” Appl. Spectrosc. Rev. 12(1), 1 (1976).
    [Crossref]
  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).
    [Crossref]

1976 (1)

P. D. Willson and T. H. Edwards, “Sampling and smoothing of spectra,” Appl. Spectrosc. Rev. 12(1), 1 (1976).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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.
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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).
[Crossref]

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).
[Crossref]

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.
[Crossref]

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

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Figures (7)

Fig. 1
Fig. 1

Transfer functions for polynomial smoothing filters of 11th degree and different ranges of 19, 31, and 65 points with one smoothing pass.

Fig. 2
Fig. 2

Transfer functions for three equivalent filters of seventh degree and appropriate ranges with 1, 4, and 8 smoothing passes.

Fig. 3
Fig. 3

Transfer functions for three equivalent four-pass smoothing filters of appropriate ranges, illustrating the effect of changing the polynomial degree.

Fig. 4
Fig. 4

Suggested values of the ratio T/FWHH for smoothing of Gaussian lines as a function of the number of smooths and polynomial degrees.

Fig. 5
Fig. 5

Suggested values of the ratio T/FWHH for smoothing of Lorentzian lines as a function of the number of smooths and polynomial degrees.

Fig. 6
Fig. 6

Polynomial smoothing of N2O spectral data (see text). The numbers refer to the degree, number of points, and number of passes of four equivalent filters selected by use of Fig. 5 for lines with 40 points per FWHH.

Fig. 7
Fig. 7

Polynomial smoothing of N2O spectral data (see text) by filters of 95-point range, 4 passes, and different degrees. The data are the same as for Fig. 6.

Tables (2)

Tables Icon

Table 1 Suggested Values of the Range T for Smoothing Gaussian Lines of FWHH = 30 Points

Tables Icon

Table 2 Suggested Values of the Range T for Smoothing Lorentzian Lines of FWHH = 30 Points

Equations (13)

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Y 1 ( i ) = m = - J J h ( K ) ( J ; m ) Y 0 ( i - m ) .
h ( K ) ( J ; m ) = k = 0 K P ( k ) ( J ; 0 ) P ( k ) ( J ; m ) / C ( k , J ) .
P ( k ) ( J ; m ) = q = 0 k ( - 1 ) k + q ( k + q ) ( 2 q ) ( J + m ) ( q ) ( q ! ) 2 ( 2 J ) ( q ) ,
x ( n ) = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) .
h ( 3 ) ( J ; m ) = 3 [ ( 3 J 2 + 3 J - 1 ) - 5 m 2 ] ( 2 J + 3 ) ( 2 J + 1 ) ( 2 J - 1 ) ,
h ( 5 ) ( J ; m ) = 15 4 [ ( 15 J 4 + 30 J 3 - 35 J 2 - 50 J + 12 ) - 35 ( 2 J 2 + 2 J - 3 ) m 2 + 63 m 4 ] ( 2 J + 5 ) ( 2 J + 3 ) ( 2 J + 1 ) ( 2 J - 1 ) ( 2 J - 3 ) ,
k = 0 K P ( k ) ( J ; m ) P ( k ) ( J ; m ) / C ( k , J ) = 1 C ( K , J ) a K ( J ) a K + 1 ( J ) × P ( K + 1 ) ( J ; m ) P ( K ) ( J ; m ) - P ( K ) ( J ; m ) P ( K + 1 ) ( J ; m ) m - m ,
h ( K - 1 ) ( J ; m ) = h ( K ) ( J ; m ) = f 0 ( J , K ) 1 2 m P ( K ) ( J ; m ) .
f 0 ( J ; K ) = ( - 1 ) ( K - 1 ) / 2 1 × 3 × 5 × × K 2 × 4 × × ( K - 1 ) × 2 J ( 2 J - 2 ) ( 2 J - K + 1 ) ( 2 J + 1 ) ( 2 J + 3 ) ( 2 J + K ) .
F ( K ) ( J ; m ) = ( 2 J ) ( K ) P ( K ) ( J ; m ) .
G ( 0 ) = 1 , G ( 1 ) = 1 , 2 G ( 2 ) = 3 ( 2 m ) 2 G ( 1 ) + ( 1 - T 2 ) G ( 0 ) , 3 G ( 3 ) = 5 G ( 2 ) + 2 ( 4 - T 2 ) G ( 1 ) , K G ( K ) = ( 2 K - 1 ) a 2 G ( K - 1 ) + ( K - 1 ) [ ( K - 1 ) 2 - T 2 ] G ( K - 2 ) .
h ( K - 1 ) ( T ; m ) = h ( K ) ( T ; m ) = g 0 ( T ; K ) G ( K ) ( T ; m ) ,
g 0 ( T ; K ) = ( - 1 ) ( K - 1 ) / 2 1 × 3 × 5 × K 2 × 4 × × ( K - 1 ) × 1 ( T - K + 1 ) ( T - 2 ) T ( T + 2 ) ( T + K - 1 ) .