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# Bilinear transform

The bilinear transform in digital signal processing takes a transfer function in the continuous domain and transforms it to the sampled or -domain, to obtain a function . This transform preserves stability and is computed by substituting
where is the sample time of the discrete system. The inverse transform is
The bilinear transform is a special case of a conformal mapping, namely, the Möbius transformation defined as

## Background

An analog filter is stable if the poles of its transfer function fall in the negative real half of the complex plane. A digital filter is stable if the poles of its transfer function fall inside the unit circle in the complex plane. The bilinear transform maps the negative real half of the complex plane to the interior of the unit circle. This way, filters designed in the continuous domain can be easily converted to the sampled domain while preserving their stability.

## Example

As an example take a simple RC-filter. This filter has a transfer function

If we wish to simulate this filter in a digital simulation, we can apply the bilinear transform by substituting for the formula above, after some reworking, we get the following filter representation:

## Warping

When transforming a continuous transfer function to a discrete transfer function, one must take two frequencies into acount, namely a continuous frequency and a discrete frequency . The corresponding circular pulsations are

and
By looking at the bilinear substitution equation
one can see that the entire continuous frequency range
is mapped onto the fundamental frequency interval

The continuous pulsation corresponds to the discrete pulsation and the continuous pulsations correspond to the discrete pulsations . One can find the relation between the continuous and the discrete pulsations by substituting and in the formula
yielding
so
and
We notice that there is a nonlinear relationship between and . This effect of the bilinear transform is called warping.

The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency characteristic, such as observed with the impulse invariant method. It is necessary, however, to pre-warp the given specifications of the continuous system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete system. Note that the warping also occurs in the phase characteristic, as expected.