Bilinear transformation (signal processing)

The bilinear transformation , in the English language as Tustin's method (dt. "Tustin method") is designated, in the signal processing a transformation - a conversion mode in Mathematics - between the continuous-time and discrete-time display of system functions. It plays a role in digital signal processing and control theory , as it creates a relationship in the system description between analog, continuous systems and digital, discrete systems.

motivation

In signal processing and control engineering, bilinear transformation makes it possible to convert time-continuous transfer functions G (s) from linear, time-invariant systems into time-discrete transfer functions H [z] with similar behavior. The transformation can take place in both directions. The transfer function G (s) can describe an analog filter , for example , and H [z] represents a time-discrete transfer function derived from the analog filter, which describes an equivalent digital filter .

The description of the system functions of continuous-time systems takes place in the so-called s-level and their analysis takes place by means of the Laplace transformation . In the case of time-discrete systems, the representation takes place in the so-called z-plane and the analysis takes place by means of the Z-transformation . A possible transformation of systems between the s- and z-plane is in the form of the bilinear transformation. Compared to other methods such as the impulse invariance method and the matched Z transformation, the bilinear transformation offers the advantage of avoiding aliasing effects in the time-discrete system. The associated disadvantage is a non-linear distortion in the transition of the transfer functions from G (s) to H [z].

description

Assignment of the s and z planes in the bilinear transformation. The colored lines correspond to exemplary assignments between the two complex levels

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

In the bilinear transformation, the time- continuous system function G (s) corresponds to the time-discrete system function H [z] through the substitution of the variable s in the form:

${\displaystyle s={\frac {2}{T}}\cdot {\frac {z-1}{z+1}}={\frac {2}{T}}\cdot {\frac {1-z^{-1}}{1+z^{-1}}}}$

which does ______________ mean:

${\displaystyle H(z)=G\left[{\frac {2}{T}}\cdot {\frac {z-1}{z+1}}\right]}$

The parameter T represents the temporal sampling interval (period duration). The reciprocal value is referred to as the sampling rate . The reverse assignment results with s = σ + jΩ to: ${\displaystyle f_{s}={\tfrac {1}{T}}}$

${\displaystyle z={\frac {1+Ts/2}{1-Ts/2}}={\frac {1+\sigma T/2+j\Omega T/2}{1-\sigma T/2-j\Omega T/2}}}$

If the real part of s is set equal to 0 (s = jΩ) we get:

${\displaystyle z={\frac {1+j\Omega T/2}{1-j\Omega T/2}}}$

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

Frequency distortion

Relationship of the continuous frequency axis Ω to the unit circle in the z-plane with angle ω.

Due to the fact that the continuous frequency range -∞ ≤ Ω ≤ ∞ of the s-plane is mapped to the angle -π ≤ ω ≤ π on the unit circle of the z-plane, the transformation from the time-continuous to the time-discrete frequency variable must be non-linear. In order to derive the relationship between the frequency axis Ω in the s-plane and the unit circle with angle ω in the z-plane, z is substituted with e ^jω :

${\displaystyle z=e^{j\omega }={\frac {1+j\Omega T/2}{1-j\Omega T/2}}}$.

This corresponds exactly to the picture of the frequency axis of the s-plane. From this it can now be determined with the help of the bilinear transformation : ${\displaystyle s}$

${\displaystyle s={\frac {2}{T}}{\frac {e^{j\omega }-1}{e^{j\omega }+1}}=\sigma +j\Omega ={\frac {2}{T}}\left[{\frac {2\cdot e^{-j{\frac {\omega }{2}}}\cdot j\sin({\frac {\omega }{2}})}{2\cdot e^{-j{\frac {\omega }{2}}}\cdot \cos({\frac {\omega }{2}})}}\right]={\frac {2j}{T}}\tan \left({\frac {\omega }{2}}\right)}$.

With σ = 0 this leads to the relationship:

${\displaystyle \Omega ={\frac {2}{T}}\tan \left({\frac {\omega }{2}}\right)}$

or on the curve ω (Ω) shown on the right in the figure:

${\displaystyle \omega =2\arctan \left({\frac {\Omega T}{2}}\right)}$.

The bilinear transformation avoids aliasing effects by “compressing” the entire imaginary axis Ω onto the unit circle in the z-plane. The resulting non-linear compression of the frequency axis represents a frequency distortion and must be taken into account when designing a filter, for example, if analog (continuous-time) filters such as elliptical filters are to be implemented as time-discrete, digital IIR filters . In these cases, a predistortion of the continuous transfer function G (s) of the filter is necessary, in addition to observing the Nyquist bandwidth, in order to obtain the appropriate time-discrete transfer function H [z] after the bilinear transformation .

literature

• Alan V. Oppenheim, Ronald W. Schafer: Discrete-time signal processing . 3. Edition. Oldenbourg, Munich 1999, ISBN 3-486-24145-1 .