# LGF_DifferenceQuotientFC (FC / V3.0.1)

Author: Siemens Digital Industry

## Short description

This function numerically differentiates a signal sampled equidistantly in time. For example, the velocity can be calculated from a measured locus curve, or the acceleration can be calculated from the measured velocity. In order to minimize the effects of a scattering measurement signal, this algorithm uses a compensating polynomial.

The function calculates the differentiated values acyclically.

The function reads an array that is differentiated. N-4 smoothed measured values can be calculated from N measured values. The output array contains the value 0 in the index (0,1,N-1,N). However, replacement values can be calculated.

### Block Interface

| LGF_DifferenceQuotientFC (FC) |
--- | --- | ---
LReal |  |  | deltaT |  | Ret_Val |  |  | Void
|  |  |  |  | error |  |  | Bool
|  |  |  |  | status |  |  | Word
Array[*] of LReal |  |  | values |  |  | Array[*] of LReal
Array[*] of LReal |  |  | derivatedValues |  |  | Array[*] of LReal

### Input parameter

Identifier | Data type | Description
--- | --- | ---
deltaT | LReal | Equidistant distance between two measured values. (e.g. 1s)

### Output parameter

Identifier | Data type | Description
--- | --- | ---
Ret_Val | Void | Void - Function has no return value
error | Bool | FALSE: No error TRUE: An error occurred during the execution of the FB
status | Word | 16#0000-16#7FFF: Status of the FB 16#8000-16#FFFF: Error identification (see following Table)

### In/Out parameter

Identifier | Data type | Description
--- | --- | ---
values | Array[*] of LReal | Values that will be included in the differentiation.
derivatedValues | Array[*] of LReal | The differentiated value range.

### Status & Error codes

Code / Value | Identifier / Description
--- | ---
16#0000 | STATUS_FINISHED_NO_ERROR Status: Execution finished without errors
16#8200 | ERR_DELTA_T Error: Delta time `deltaT` must not be zero.
16#8400 | ERR_ARRAYS_DIFFERENT Error: The Array sizes are not equal. The arrays `values` and `derivatedValues` must have the same size.
16#8401 | ERR_NOT_ENOUGH_VALUES Error: Not enough values. The block requires five (5) values to calculate a differntiated value. Transfer additional values with a positive edge on the `insert` input.

## Functional description

To calculate the difference quotient of a scattering signal, a third-degree compensation polynomial is first placed through the measured values. This polynomial is then differentiated. With this method, even a distorted input signal can be sensibly differentiated.

The difference quotient is calculated with the following formula:

y^{\prime}(n) = \frac{y(n-2) - 8y(n-1) + 8y(n+1) - y(n+2)}{12 \cdot deltaT}

deltaT: equidistant distance between two measured values (e.g. 1s).

The function (FC) can calculate N-4 differentiated and smoothed measured values from N measured values. The output array would be assigned with 0 in the index (0,1,N-1,N). However, the following formalisms can be used to calculate substitute values:

y^{\prime}(n-2) = \frac{-125(y(n-2) + 136y(n-1) + 48y(n) - 88y(n+1) + 29y(n+2)}{84 \cdot deltaT} y^{\prime}(n-1) = \frac{-38(y(n-2) - 2y(n-1) + 24y(n) + 26y(n+1) - 10y(n+2)}{84 \cdot deltaT} y^{\prime}(n+1) = \frac{10(y(n-2) - 26y(n-1) - 24y(n) + 2y(n+1) + 38y(n+2)}{84 \cdot deltaT} y^{\prime}(n+2) = \frac{-29(y(n-2) + 88y(n-1) - 48y(n) - 136y(n+1) + 125y(n+2)}{84 \cdot deltaT}

## Change log

Version & Date | Change description
--- | ---
| 01.00.00 | Siemens Industry Online Support
| 23.11.2018 | First released version
| 01.00.01 | Simatic Systems Support
| 15.11.2019 | Regions, comments and constants are added
| 03.00.00 | Simatic Systems Support
| 23.04.2020 | Set version to V3.0.0, harmonize the version of the whole library
| 03.00.01 | Simatic Systems Support
| 06.04.2021 | Insert documentation