5. Differentiation

The symbolic Differentiationof functions is a topic that is introduced in all elementary Calculus courses. The numerical differentiation of digitized signals is an application of this concept that has many uses in analytical signal processing. The first derivative of a signal is the rate of change of y with x, that is, dy/dx, which is interpreted as the slope of the tangent to the signal at each point. Assuming that the x-interval between adjacent points is constant, the simplest algorithm for computing a first derivative is:

(for 1< j <n-1).

where X'_j and Y'_j are the X and Y values of the j^th point of the derivative, n = number of points in the signal, and X is the difference between the X values of adjacent data points. A commonly used variation of this algorithm computes the average slope between three adjacent points:

(for 2 < j <n-1).

The second derivative is the derivative of the derivative: it is a measure of the curvature of the signal, that is, the rate of change of the slope of the signal. It can be calculated by applying the first derivative calculation twice in succession. The simplest algorithm for direct computation of the second derivative in one step is

(for 2 < j <n-1).

Similarly, higher derivative orders can be computed using the appropriate sequence of coefficients: for example +1, -2, +2, -1 for the third derivative and +1, -4, +6, -4, +1 for the 4^th  derivative, although these derivatives can also be computed simply by taking successive lower order derivatives.