Bilinear transform on arvau

Wikipedia / Bilinear transform

The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used for converting a transfer function of a linear, time-invariant (LTI) filter in the continuous-time domain (often named an analog filter of a linear, shift-invariant filter in the discrete-time domain (often named a digital filter although there are analog filters constructed with switched capacitors, in the s-plane to the unit circle, in the z-plane. Other bilinear transforms can be used for warping the frequency response with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response 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. The change in frequency is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.

Discrete-time approximation

The bilinear transform is a first-order Padé approximant of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of

is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; can be performed.

The inverse of this mapping (and its first-order bilinear approximation) is

That is

Stability and minimum-phase property preserved

A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.

Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.

Transformation of a General LTI System

A general LTI system The order of the transfer function is the greater of and (in practice this is most likely as the transfer function must be proper where is defined as either or otherwise if using frequency warping Multiplying the numerator and denominator by the largest power of present, It can be seen here that after the transformation, the degree of the numerator and denominator are both .

The roots of the numerator and denominator polynomials, and , are the zeros and poles of the system. The bilinear transform is a one-to-one mapping yielding some of the discretized transfer function's zeros and poles and As described above, the degree of the numerator and denominator are now both , in other words there is now an equal number of zeros and poles. The multiplication by means the additional zeros or poles are

Example

As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function

the formula above; after some reworking, we get the following filter representation:

The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used for implementing a real-time digital filter.

Transformation for a general first-order continuous-time filter

It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function

using the bilinear transform (without prewarping any frequency specification) requires the substitution of

where

, then

This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:

Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in

The difference equation (using the Direct form I) is

General second-order biquad transformation

A similar process can be used for a general second-order filter with the given transfer function

This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:

Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation. This results in

The difference equation (using the Direct form I) is

Frequency warping

To determine the frequency response of a continuous-time filter, the transfer function axis of the s) to the unit circle of the z), but it is not is mapped to.

. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is

and the inverse mapping is

), then the features are mapped to a slightly.

One can see that the entire continuous frequency range

is mapped onto the fundamental frequency interval

This effect of the bilinear transform is called frequency warping for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.

, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function. This is a modified version of Tustin's transform shown above.

However, note that this transform becomes the original transform

The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.