Bilinear transform

The bilinear transformation, in English-speaking countries also known as Tustin 's method (Eng. " Tustin method" ), is in signal processing, a transformation - a conversion type in Mathematics - between the continuous-time and discrete-time representation of system functions. She plays in the digital signal processing and control theory a role because it creates a reference to the system or between analog, digital and continuous systems, discrete systems.

Motivation

In the signal processing and control theory is by bilinear transformation, the possibility of continuous-time transfer functions G (s) to convert from linear, time-invariant systems in discrete-time transfer function H [ z] with similar behavior. The transformation can be carried out in both directions. The transfer function G (s), for example, describe an analog filter H and [Z] is a derivative of the analog filters, discrete-time transfer function, which describes an equivalent digital filter.

The description of the system functions of the continuous-time systems is in the so-called S- layer, and its analysis is carried out by means of Laplace transform. Wherein the discrete-time systems, the display is in the so-called z-plane and the analysis is carried out by means of the Z-transform. A possible transformation systems between the s- and z-plane is in the form of the bi-linear transformation. The bilinear transformation provides over other methods such as the Impulsinvarianzmethode and Matched Z transform the advantage of avoiding aliasing in the discrete-time system. The disadvantage associated with it is a non-linear distortion in the transition of the transfer functions of G (s) to H [z ].

Description

The bilinear transformation is a conformal mapping and an application of the Möbius transformation. It assigns to each point s = σ j · Ω, as shown clearly to a certain point in the complex z-plane and vice versa in the complex S plane in the adjacent figure for different values ​​of σ and Ω graphically. Z | | For example, the values ​​on the imaginary axis Ω, shown in red on the unit circle = 1 are displayed in the z-plane. All points in the left-hand s-plane with negative real value are displayed in the z-plane to points within the red marked unit circle - this fact is essential for stability analysis of linear systems, as stable systems with poles in the left-hand s-plane in discrete-time systems with poles pass within the unit circle.

The continuous-time system function G (s) corresponds with the bilinear transform using the discrete-time system function H [ z] of the substition of the variable s in the form:

Which means:

The parameter T represents the sampling time ( period ) dar. The reciprocal is referred to as the sampling rate. The reverse mapping is obtained with s = σ j · Ω to:

If the real part of s is set equal to 0 ( s = j · Ω ) we have:

The absolute value of z is then for all values ​​of Ω is equal to 1 ( | z | = 1), which corresponds to the image of the imaginary axis of the s-plane on the unit circle in the z plane.

Frequency distortion

By the fact that the continuous frequency range - ∞ ≤ Ω ≤ ∞ the s-plane is mapped to the angular - π ≤ ω ≤ π on the unit circle of the z-plane, the transformation of the continuous-time to discrete-time variable frequency need not be linear. The relation between the frequency Ω axis in the s-plane, and the unit circle with angle ω in the z-plane to derive, z is substituted with ejω:

This corresponds indeed to the picture of the frequency axis of the s-plane. From this it can be calculated using the bilinear transformation s to determine:

With σ = 0, this leads to the equation:

Or on the course shown on the left in the figure ω ( Ω ):

Bilinear transformation avoids aliasing by " compression " of the entire imaginary axis Ω on the unit circle in the z plane. The resulting non-linear compression of the frequency axis represents a frequency distortion and needs, for example, be observed in the context of filter draft when analog ( continuous-time ) filter as elliptic filters are to be implemented as discrete-time, digital IIR filters. In these cases a pre-distortion of the continuous transmission function G (s ) of the filter is necessary, in addition to compliance with the Nyquist bandwidth in order to obtain the right time-discrete transfer function H [z] according to the bilinear transformation.