Bilinear transform Guide, Meaning , Facts, Information and Description

In digital signal processing, the bilinear transform converts a transfer function of a linear, time-invariant filter in the continuous-time domain and to a transfer function of a linear, shift-invariant filter in the discrete-time domain. It maps positions on the unit circle, , in the z-plane to the axis, , in the s-plane.

The transform preserves stability and maps every point of the frequency response of the continuous-time filter, to a corresponding point in the frequency response of the discrete-time filter, although to a somewhat different frequency, as shown in the Frequency Warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but perhaps at a somewhat different frequency. This is barely noticable at low frequencies but quite evident at frequencies close to the Nyquist frequency.

The bilinear transform is accomplished by substituting into

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

Table of contents

1 Background 2 Example 3 Warping

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.

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