Transfer function

The transfer function or system function mathematically describes the engineering system theory, the relationship between the input and output of a dynamic system in the frequency domain. A dynamic system can be, for example, a mechanical structure, an electrical network or any other biological, physical or economic process. Using the transfer function can be determined for an arbitrary input signal, the output signal, i.e. the response of the system. Typical of the system behavior is the time-delayed reaction of the output signal against the input signal.

• 4.1 System Analysis
• 4.2 Transfer Functions frequently used

Generally

Under a system refers to the systems theory Abstract a process for converting or transmitting a signal. The signal supplied to it is then referred to the input signal and the resulting signal is output. As the signal is converted to each other and how these two signals in the ratio is mathematically described by the transfer function.

The transfer function is a function of the complex frequency, i.e., it describes the change finds each individual frequency of an input signal. With it can be calculated, as any input signal is converted by the system and which output signal it produces. Describes the behavior of the system completely, and independently of the signals without image the individual components of the system. Conversely, the details of the realization of the transfer function are not directly readable.

The application of the transfer functions for calculating the response of a system is not limited to electrical circuits. All systems can be represented by linear differential or difference equations can be analyzed in this way.

Transfer functions used in the engineering sciences in all applications where the changes of signals - are described or calculated - whether intentionally or unintentionally. They are mostly used in the analysis of SISO systems, typically in the signal processing, control and communication equipment as well as the coding theory. Often it is the process that changes the signal, approximately described by a linear model. Then you can fall back on the theory of LTI systems, they are analytically explored easily accessible and theoretically well.

As LTI systems change only the amplitude and the phase angle of the frequency components of the signal, the description in the frequency domain is usually convenient to use and more compact. The description of the time behavior of an LTI system can be done by linear differential equations in the continuous case. Of the two-sided Laplacian can be converted to the frequency domain. Conversely, can be reconstructed by the inverse Laplace transform of the transfer function of the re- timing.

For discrete systems, as most digital systems, for example, technical (e.g., digital filtering ), the behavior of the system is defined only at specific times. Such systems may be described in the time domain using linear differential equations, and are transferred by using the z-transform in the frequency domain.

As a link between continuous and discrete-time transfer functions are various transformations, such as the bilinear transformation or the transformation Impulsinvarianz available to convert transfer functions, subject to certain restrictions, between these two forms can.

To obtain the transfer function of a system, there are two options:

Example

A simple example of a desired signal change is a low pass filter: It filters the high frequencies from an input signal out and leaves in the output signal, only the lower frequency components. Unintentional change is, for example, the distortion in the transmission through a channel (for example, a copper cable, a fiber optic cable or a radio link ). Here one would wish in principle that the channel does not change the signal. It does so, however, since it is not ideal in practice. Such distortions must be compensated either by the sender or the receiver.

Basics

Definition

For continuous systems, which are linear and time invariant (i.e., the system is at any time - the same input - the same action), the transfer function is defined as

Y ( s ) or function U (s) is the Laplace transform of the two-sided output and the input signal. G (s) is the ratio of these two quantities and describes the system therethrough. For causal signals to set the left-hand part of the signal to zero, this corresponds to the one-sided Laplacian transform.

For LTI discrete-time systems, such as are used, for example in the digital signal processing, the definition is similar, except that in this case the z-transform is used:

Derivation of the system equations ( systems analysis )

When the internal structure of the system is known, the time response is described by the corresponding system equation. In the case of continuous systems, these are differential equations with discrete-time systems, difference equations. If it's still are linear equations, the associated system is also linear and time-invariant at the same time - a LTI system.

Instead the behavior of the system is now to be described in the time domain, it may instead be converted into the associated frequency range, and are further analyzed. Using the transformed equation a solution can be found mostly lighter and therefore overall system response for any input signal and the transfer function can be determined.

For continuous systems are used by default, the Laplace transform for discrete-time systems, z-transform. Such a relationship between time and image function is called correspondence. Since the analytical determination of these transformations is complex and often occur repeatedly the same, there are so-called correspondence tables in which frequently used transformations can be looked up.

The initial values ​​of the system equations represent the internal state of the system is at the beginning, for example, the internal energy store. In most cases, the initial state is uninteresting for system analysis and it is assumed that all initial values ​​are zero, that is, the internal energy of the system memory are empty.

Signal processing ( system identification )

In the signal processing, there is usually a desire to convert a given input signal to a specific output signal, or to change the spectrum of the input signal in a certain way. That is, although in contrast to the system of analysis of the response of the system is known, but not the function.

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

In a continuous system is formed to the input and output signal in the frequency range from:

Then the output signal depends on the input signal via the transfer function:

And by rearranging one obtains selbige:

The method works in equivalent discrete-time systems by using this, the z-transform of the signals.

Forms of representation

The transfer function can be expressed as a mathematical formula, or as a graphic curve, either. The formal representation is usually chosen between the polynomial representation, its product presentation or partial fractions.

The graphic representation is called Bode diagram and is composed of the description of the amplitude gain and phase shift experienced by the input signal.

In the product presentation can very easily be the poles and zeros of the function read. The illustration in partial fractions is particularly suitable for the back-transformation into the time domain.

Examples

Systems analysis

Continuous LTI system

A system is described by the following differential equation:

Here are real-valued constants.

Is the Laplace transform of the equation is

Here are all initial values ​​and. Used is obtained:

According to definition, the transfer function of the quotient of Y / X, is shared respectively on both sides, one obtains:

Discrete-time LTI system

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

Here are real-valued constants.

The z-transform of the difference equation is then

Rearranging we obtain the transfer function

Transfer functions frequently used

In the signal processing and communications technology:

• Butterworth filter,
• Bessel filter,
• Cauer filter,
• Chebyshev filter,
• Gaussian filter,
• Raised cosine filter.

In control engineering:

• P-element
• I-element
• D element
• PT1
• PT2 element
• Dead-time element