Interpolation, extrapolation

Interpolation

Interpolation is a type of estimation, a function that allow to estimate the value (x, y) between two known data points (x₁, y₁ ; x₂, y_2).

A common interpolation method used on ECU is the linear interpolation. The interpolation is more precise on high resolutions maps (multiples points)

Linear interpolation

The formula :

$y=y_{1}+\left(x-x_{1}\right){\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$

where:

• x₁, y₁ are the first known coordinates
• x₂, y₂ are the second known coordinates
• x is the point with want at which we do the interpolation
• y is the interpolated value

Using linear interpolation on 3D maps:

1. Interpolate the row above and below
2. Interpolate the two interpolated values together

Here is a example, consider the following map:

Example map (load, engine speed)
Load /RPM	100	150
1000	10	20
2000	20	40

To get the value at load : 125 and engine speed : 1500, we will need to do a linear interpolation since it is not in the map.

The table would look like this after step 1:

Example map (load, engine speed)
Load /RPM	100	125	150
1000	10	15	20
2000	20	30	40

Now we interpolate 15 and 30 together to get the value at 1500 rpm (step 2):

Example map (load, engine speed)
Load /RPM	100	125	150
1000	10	15	20
1500		22.5
2000	20	30	40

At load : 125 and engine speed at 1500, we get the value 22.5 with linear interpolation

Extrapolation

ECU does not work with extrapolation. Extrapolation is a an estimation of values that are outside the scope of a map data. There are multiple proposed methods for extrapolation : linear, polynomial, conic, etc.

Extrapolation is not reliable and need to be used with care (if used).