Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

### Trapezoidal Rule for Numerical Integration

The velocity can be written as In particular, • This approximation replaces a one-sample integral by the area under the trapezoid having vertices • In other words, is approximated by a straight line between time and • This is a first-order approximation of in contrast to the zero-order approximation used by forward and backward Euler schemes
• We will see that the commonly used bilinear transform is equivalent
• Model is exact if driving force is piecewise linear, having a constant slope over each sampling interval
• (Backward Euler is similarly exact for a piecewise-constant driving force)

Bilinear Transform as Compensated BE/FE

In Newton's law , look at the Backward Euler (BE) approximation of the time-derivative: We see there is a 1/2 sample delay in the first-order difference on the right. This misaligns the force and subsequent velocity by half a sample. A very simple delay compensation is to use a two-point average on the left: The extra attenuation at high frequencies due to the two-point average actually helps. Taking the transform: or which is the bilinear transform of : Frequency Warping is the Only Error

We have using the bilinear transform (trapezoidal integration in the time domain)

Let's look along the unit circle in the plane: Since the exact formula is , we can push all of the error into a frequency warping: • Frequency-warping is the only error over the unit circle when using the bilinear transform
• What started out as different gain errors on the left and right became the correct gains at warped frequency locations
• Frequency-warping implications should also be considered in the time domain

Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search