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

The extra attenuation at high frequencies due to the two-point average actually

or

which is the

**Frequency Warping is the Only Error**

We have

using the

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 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

Download DigitizingNewton.pdf

Download DigitizingNewton_2up.pdf

Download DigitizingNewton_4up.pdf

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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