Basic Mathematics

Basic Mathematics

The basic mathematics function group provides functions for fundamental mathematical calculations. The following section describes methods for numeric differentiation and integration, and for the calculation of the floating root mean square.

Differentiate

The differentiate function calculates the numerical derivative from the values that are time-discrete in the computer. DIAdem calculates the derivative either over a forward difference quotient or over a central difference quotient.

Calculation of the Forward Difference Quotient of The First Order

Up to Version 2011, DIAdem calculated the derivative only over the first order forward difference quotient according to the following formula

With this method the result channels are one value shorter than the input channels. The first result channel contains the average of two subsequent values from the x-channel. The second result channel contains the forward difference quotients of two successive value pairs from the input channels. If two subsequent x-values are identical, the y-result value is NoValue.

Calculation over the Second Order Central Difference Quotient

From Version 2012, DIAdem calculates the derivative optionally over the second order central difference quotient according to the following formula

N is the number of values in the channel. This procedure needs equidistant and strictly monotonic increasing x-values.

The necessary number of interpolation points for the calculation of the central difference quotient is not available at the edges. DIAdem calculates the second order forward difference quotient for the first two points of the channel according to the following formula

DIAdem calculates the second order backward difference quotient for the last two points of the channel according to the following formula

When the central difference quotient is calculated, the original channel length remains and DIAdem only creates a result channel with the y-values. The x-values remain unchanged. This method is used, for example, for crash evaluations and is described in the regulations SAE J1727 and ECE-R94.

Integrate

The integrate function uses either the trapezoidal rule or the Simpson rule to calculate the integral. In each partial interval, DIAdem approximates the surface below the y-values. DIAdem saves the summated sections in the result channel up to the corresponding point on the x-axis. The last value in the result channel receives the integral of the entire x-range.

Up to Version 2012, DIAdem used the trapezoidal rule without an integration constant for calculating the integration. If the integration constant is not enabled, the x-values need not be monotonic increasing. If a negative difference exists between two successive x-values, DIAdem subtracts the partial integral. If one of the input channels contains a NoValue, the result for this and the subsequent value is NoValue. The integration is calculated with the following formula

From Version 2014 onwards, DIAdem uses the trapezoidal rule with the integration constant as an option in order to calculate the integration. If the integration constant is enabled, the x-values must be equidistant and monotonic increasing and the y-values must not contain NoValues. This type of integration corresponds to the trapezoidal integration in LabVIEW and uses the following formula. IC stands for integration constant

From Version 2014 onwards, DIAdem uses the Simpson rule as an option in order to calculate the integration. The x-values must be equidistant and monotonic increasing and the y-values must not contain NoValues. This type of integration corresponds to the Simpson integration in LabVIEW and uses the following formula. IC stands for integration constant and IE for integration end value

Calculate RMS

The calculate RMS function calculates the floating root mean square (RMS). The RMS values are always positive and provide a measure for the energy of a signal. The following formula calculates the time continuous RMS in the time interval from t1 to t2:

For the time-discrete signal values in the computer, the following formula calculates the RMS:

DIAdem weights the first and the last signal value with 0.5 and uses the following formula to calculate the RMS:

Mathematical Functions

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