Transfer function

In engineering systems theory, the transfer function or system function describes mathematically the relationship between the input and output signals of a linear dynamic system in an image space .

A dynamic system can, for example, be a mechanical structure, an electrical network or another biological, physical or economic process. With the help of the transfer function (as an alternative to the calculation in the time domain ) the output signal, i. H. the response of the system can be determined more easily than by solving differential equations. In addition: Subsystems that are graphically arranged in a signal flow plan can be transformed and combined with the help of transfer functions using simple calculation rules.

For continuous systems the image space is given by the Laplace transform . One axis is the Fourier frequency parameter iω. The transfer function is therefore also related to the frequency response of a system.

For discrete systems the image space is given by the z-transformation .

General

Under a system is meant in the system theory abstract a process which converts a signal or transmits . The signal fed to it is then called the input signal and the resulting signal is called the output signal . How the signal is converted or how these two signals are related to one another is described mathematically by the transfer function.

The transfer function describes the dynamic behavior of a system over time. It can be used to calculate how any input signal is converted by the system or which output signal it produces. It describes the dynamic behavior of the system completely and independently of the specific signals. The transfer function only depicts the mathematical system behavior, but not the individual components of the system. Conversely, the details of the implementation cannot be read directly from the transfer function.

Transfer functions are used in engineering wherever changes in signals - whether intentional or unintentional - are described or calculated. They are mostly used in the analysis of SISO systems, typically in signal processing , control and communication engineering , as well as coding theory . All systems that can be represented by linear differential or difference equations can be modeled mathematically in this way. The process that changes the signal can often be approximately described by a linear model . Then the theory of the LZI systems can be used, they are easily accessible analytically and well researched theoretically.

Since LZI systems only change the amplitude and the phase angle of the frequency components of the signal, the description in the frequency domain is usually more practical and also more compact. The description of the time behavior of an LZI system can be done in the continuous case by linear differential equations. It can be transferred to the frequency domain using the Laplace transformation . Conversely, the inverse Laplace transformation can be used to reconstruct the time behavior from the transfer function.

In discrete systems, such as B. most digital technical systems (e.g. digital filters ), the behavior of the system is only defined at certain times . Such systems can be described in the time domain by linear difference equations and transferred to the image domain using the z-transformation .

As a link between continuous and time-discrete transfer functions, various transformations such as the bilinear transformation or the impulse invariance transformation are available in order to be able to convert transfer functions between these two forms, taking certain restrictions into account.

There are two ways of obtaining the transfer function of a system:

System analysis: If the internal structure of the system is known, it can be modeled mathematically and its behavior calculated from it.
System identification : With known output and input signals Y and X, which can either be measured or specified, the transfer function is obtained by forming the quotient . ${\ displaystyle {\ tfrac {Y} {X}}}$

example

A simple example of a desired signal change is a low-pass filter : It filters high frequencies from an input signal and only leaves the lower frequency components in the output signal. An unintended change is e.g. B. the distortion during transmission through a channel (e.g. a copper cable, a fiber optic cable or a radio link). Here one would basically wish that the channel would not change the signal. However, it does so because it is not ideal in reality. Such distortions must then be compensated either at the transmitter or at the receiver.

Basics

definition

For continuous systems that are linear and time-invariant (i.e. the system shows the same behavior at all times, given the same input), the transfer function is defined as

{\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {{\ mathcal {L}} \ {y (t) \}} {{\ mathcal { L}} \ {u (t) \}}}}

or alternatively in operator notation

{\ displaystyle Y (s) = G (s) \ cdot U (s)}

The functions Y (s) and U (s) are the Laplace transforms of the output or input signal. G (s) is the quotient of these two quantities and thus describes the system. (The two-sided Laplace transform plays a subordinate role in real technical systems, since these are causal .)

For discrete-time LZI systems, such as those used e.g. B. are used in digital signal processing , the definition is similar, only that the z-transforms are used here:

{\ displaystyle G (z) = {\ frac {Y (z)} {X (z)}} = {\ frac {{\ mathcal {Z}} \ {y [k] \}} {{\ mathcal { Z}} \ {x [k] \}}}}

Derivation via the system equations (system analysis)

If the internal structure of the system is known, the time behavior can be described by the associated system equation. In the case of continuous systems, these are differential equations ; in the case of time-discrete systems, differential equations . If these are still linear equations, the associated system is also linear and at the same time also time-invariant - an LZI system.

Instead of describing the behavior of the system in the time domain, it can instead also be transferred to the associated frequency domain and further analyzed there. With the help of the transformed equation, a solution can usually be found more easily and thus the system response for any input signal or the transfer function can be determined.

For continuous systems, the Laplace transformation is used as standard, for discrete time systems the z-transformation. Such a relationship between time and image functions is called correspondence . Since the analytical determination of these transformations is complex and the same ones often occur again and again, so-called correspondence tables exist in which frequently used transformations can be looked up.

The initial values of the system equations represent the internal state of the system at the beginning, e.g. B. the internal energy storage. In most cases the initial state is of no interest to the system analysis and it is assumed that all initial values are zero, i.e. This means that the internal energy storage of the system is empty.

Signal processing (system identification)

In signal processing, there is usually the desire to convert a given input signal into a specific output signal or to change the spectrum of the input signal in a specific way. I.e. In contrast to the system analysis, the reaction of the system is known, but not how it works.

In this case the system equation (in the time domain as well as in the frequency domain) is unknown and it is determined from the input and output signal.

In a continuous system, the input and output signals are mapped in the frequency range:

{\ displaystyle X (s) = {\ mathcal {L}} \ left \ {x (t) \ right \} \ {\ stackrel {\ mathrm {def}} {=}} \ \ int _ {- \ infty } ^ {\ infty} x (t) e ^ {- st} \, \ mathrm {d} t}

{\ displaystyle Y (s) = {\ mathcal {L}} \ left \ {y (t) \ right \} \ {\ stackrel {\ mathrm {def}} {=}} \ \ int _ {- \ infty } ^ {\ infty} y (t) e ^ {- st} \, \ mathrm {d} t}

The output signal then depends on the input signal via the transfer function:

{\ displaystyle Y (s) = G (s) \ cdot X (s)}

And by moving you get the same:

{\ displaystyle G (s) = {\ frac {Y (s)} {X (s)}}}

.

The method works in an equivalent manner in discrete-time systems in that the z-transform of the signals is used here.

Forms of representation

The transfer function can be given either as a mathematical formula or as graphical curves. With the formal representation one usually chooses between the polynomial representation , its product representation or the partial fraction decomposition .

The graph is called the Bode diagram and consists of a description of the amplitude gain and phase shift experienced by the input signal.

Form of representation	Notation in the frequency domain
polynomial	${\ displaystyle G (s) = {\ frac {b_ {m} s ^ {m} + b_ {m-1} s ^ {m-1} + \ dotsb + b_ {1} s + b_ {0}} {a_ {n} s ^ {n} + a_ {n-1} s ^ {n-1} + \ dotsb + a_ {1} s + a_ {0}}}}$
Pole zeros	${\ displaystyle G (s) = k \ cdot {\ frac {(s-s_ {0,1}) (s-s_ {0,2}) \ dotsm (s-s_ {0, m})} {( s-s _ {\ infty, 1}) (s-s _ {\ infty, 2}) \ dotsm (s-s _ {\ infty, n})}}}$
Partial fraction	${\ displaystyle G (s) = {\ frac {A_ {1}} {s-s _ {\ infty, 1}}} + {\ frac {A_ {2}} {s-s _ {\ infty, 2}} } + \ dotsb + {\ frac {A_ {n}} {s-s _ {\ infty, n}}}}$

The poles and zeros of the function can be read out very easily in the product display. The representation in partial fractions is particularly suitable for the inverse transformation into the time domain.

Examples

System analysis

Continuous LZI system

A system is described by the following DGL:

{\ displaystyle y '' (t) + a_ {1} y '(t) + a_ {0} y (t) = b_ {1} x' (t) + b_ {0} x (t)}

Here are real-valued constants. ${\ displaystyle a_ {n}, b_ {m}}$

The Laplace transform of the differential equation is

{\ displaystyle {\ begin {aligned} {\ mathcal {L}} {\ {y '' (t) \}} + a_ {1} {\ mathcal {L}} {\ {y '(t) \} } + a_ {0} {\ mathcal {L}} {\ {y (t) \}} & = b_ {1} {\ mathcal {L}} {\ {x '(t) \}} + b_ { 0} {\ mathcal {L}} {\ {x (t) \}} \\\ Leftrightarrow \ quad (s ^ {2} Y (s) -sy_ {0} -y_ {1}) + a_ {1 } (sY (s) -y_ {0}) + a_ {0} Y (s) & = b_ {1} (sX (s) -x_ {0}) + b_ {0} X (s) \ end { aligned}}}

Let all initial values and . Inserted you get: ${\ displaystyle y ^ {(k)} = 0 \; k = 0 \ dots n-1}$ ${\ displaystyle x (0 ^ {-}) = 0}$

{\ displaystyle Y (s) (s ^ {2} + a_ {1} s + a_ {0}) = X (s) (b_ {1} s + b_ {0})}

According to the definition, the transfer function is the quotient Y / X, if one divides accordingly on both sides, one obtains:

{\ displaystyle G (s): = {\ frac {Y (s)} {X (s)}} = {\ frac {b_ {1} s + b_ {0}} {s ^ {2} + a_ { 1} s + a_ {0}}}}

Discrete-time LZI system

Similar to the continuous system above, the system function of a discrete LZI system is described by the following difference equation:

{\ displaystyle y [k] + a_ {1} y [k-1] + a_ {0} y [k-2] = x [k] + b_ {0} x [k-1] + b_ {1} x [k-2]}

Here are real-valued constants. ${\ displaystyle a_ {n}, b_ {m}}$

The z-transform of the difference equation then reads

{\ displaystyle {\ begin {aligned} Y (z) + a_ {1} Y (z) z ^ {- 1} + a_ {0} Y (z) z ^ {- 2} & = X (z) + b_ {0} X (z) z ^ {- 1} + b_ {1} X (z) z ^ {- 2} \\ Y (z) (1 + a_ {1} z ^ {- 1} + a_ {0} z ^ {- 2}) & = X (z) (1 + b_ {0} z ^ {- 1} + b_ {1} z ^ {- 2}) \ end {aligned}}}

The transfer function is obtained by reshaping

{\ displaystyle G (z) = {\ frac {Y (z)} {X (z)}} = {\ frac {1 + b_ {0} z ^ {- 1} + b_ {1} z ^ {- 2}} {1 + a_ {1} z ^ {- 1} + a_ {0} z ^ {- 2}}}}

Frequently used transfer functions

In signal processing and communications technology:

In control engineering:

literature

Wikibooks: Introduction to Systems Theory / Transfer Function - Learning and Teaching Materials

Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 .

Fernando Puente León, Uwe Kiencke, Holger Jäkel: Signals and Systems . 5th edition. Oldenbourg Verlag, Munich 2011, ISBN 978-3-486-59748-6 .

Jan Lunze: Control engineering 1: System theory basics, analysis and design of single-loop controls. 7th edition. Springer, 2008, ISBN 978-3-540-68907-2 .

Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Europa-Lehrmittel, 2019, ISBN 978-3-8085-5869-0 .

Individual evidence

↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 101 .
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 7 .
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 6 .
↑ John G. Proakis, Masoud Salehi: Communication systems engineering . 2nd Edition. Prentice Hall, Upper Saddle River, NJ 2002, ISBN 0-13-095007-6 , pp. 626 (English).
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 326 .
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 102 .
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 100 .
↑ Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 303 .
^ Douglas K. Lindner: Signals and Systems . McGraw-Hill, ISBN 0-07-116489-8 , pp. 294 f.

[1] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 101 .

[2] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 7 .

[3] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 6 .

[proakis1-4] John G. Proakis, Masoud Salehi: Communication systems engineering . 2nd Edition. Prentice Hall, Upper Saddle River, NJ 2002, ISBN 0-13-095007-6 , pp. 626 (English).

[5] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 326 .

[6] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 102 .

[7] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 100 .

[8] Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 , p. 303 .

[9] Douglas K. Lindner: Signals and Systems . McGraw-Hill, ISBN 0-07-116489-8 , pp. 294 f.