Derivative calculation can be carried out with all 2D and 3D data objects,
regardless if they contain equidistant or discrete data points. The order
of the derivative is variable and can be adjusted by the user. The algorithm
used for calculation of the derivative might also be adjusted.
The derivative for each data point of the 2D data object is calculated
from the average differential quotients of two adjacent data points to
the data point of interest. This procedure is applied to all data points
of the 2D data object. It is assumed, that data points are available in
ascending order regarding to the x-axis of the 2D data object.
The derivative is calculated from both sides of a data point P2 as follows:
There are two adjacent data points P1(x1;y1)
and P3(x3;y3) next to the data
point P2(x2;y2).
The differential quotients must be calculated to obtain the slopes S21 between points P2
and P1 and S32 between the data points P3 and P2
as follows:
and
The derivative D2 of data
point P2 is calculated as average
of the slopes S21 and S32:
Caution when applying this algorithm to discrete
2D data objects!
The distance between data points is neglected,
which might have relevant effects when the derivative is applied to discrete
2D data objects.
The higher order coefficients a1
to aM of the Savitzky-Golay
smoothing algorithm are used for computation of numerical derivatives
of 2D data objects. The derivative for each data point of the 2D data
object is derived from the convolution gi.
It has to be multiplied by m! to obtain the mth
order derivative coefficients.
Tip: For derivative calculations the
order M of the polynomials should be equal or greater than 4.
This value indicates the order of the derivative function applied to
the current 2D data object. An integer value greater than 0 must be entered
into this text field.
The smoothing window parameter is only used with the Savitzky-Golay
derivative algorithm. An odd number of data points around each data point
of the spectrum will be taken into account for the derivative calculation.
For details, please refer to the Savtizky-Golay
documentation. The number of data points can be selected from the
drop down combo box by clicking the icon at the right side of the parameter field.
Tip: With increasing derivative order
the number of window points must be increased accordingly. If the number
of window points is too small, a math error message is displayed on calculation.
This is a flag indicating, whether the differential quotient algorithm
(default) or the Savitzky-Golay algorithm is used for derivative calculation.
The flag might be toggled by clicking the icon at the right side of the parameter field
and selecting a new value from the list.
No
Differential quotient algorithm
Yes
Savitzky-Golay algorithm
References
K. H. Norris and P. C. Williams, Cereal Chem, 61 #2 (1984), 158
Madden, Anal. Chem., 50 #9 (1978), 1383
Steiner et al., Anal. Chem., 44 #11 (1972), 1906
Savitzky A., and Golay, M.J.E. 1964, Analytical Chemistry, vol. 36,
pp. 1627â1639.
Tip: For derivative calculations the
order M of the polynomials should be equal or greater than 4. 
icon at the right side of the parameter field.
