Frequency response
The frequency response is the correlation between input and output of a linear time invariant system with respect to the amplitude and phase. It is the Fourier transform of the impulse response of the system. The frequency response is a complex function of frequency. It indicates the ratio between input signal and output signal for sinusoidal signals.
The output signal has the same due to the nonlinear behavior of the system frequency as the input signal. The two signals are different in amplitude and in phase. The ratio of the amplitudes of input signal and output signal in response to the frequency of the amplitude response. The difference in phase between the input signal and output signal in dependence on the frequency of phase transition.
General
In an LTI system, the frequency response describing the relationship between the sinusoidal oscillations at the input and output of the transmission element as a function of frequency ω or f of the circular frequency.
Such a system has in a harmonic input signal
A harmonious output:
Due to the linearity of the angular frequency is not affected. Only amplitude ( → ) and phase (→ ) can be changed. Amplitude-frequency response, the ratio
Phase- frequency response is the phase difference
Graphical representation of
Bode plot
To illustrate the representation of the frequency response, the Bode diagram is used ( see figure). Each in a graph of the amplitude frequency response and phase - frequency response is shown. The axes are mainly divided logarithmic (except for the phase shift ), which facilitates the use of the diagram. For example, the multiplication of two frequency responses of a simple line addition and inversion of a frequency response is obtained by reflection in the f- or ω - axis in the chart.
Locus
An alternative view showing the frequency response is its locus. This image contains pointer in contrast to the Bode diagram both pieces of information: the vector length corresponds to the amplitude ratio, its argument is the phase shift φ.
The locus used in control engineering of the frequency response is also called Nyquist diagram. With the idea that in the (complex) plane, only the tips of frozen hands, the circumferential vibrations as circular movements represent, are connected to the locus of the frequency response can be used without knowledge of the complex mathematics and mathematical transformations of the time - in made visible the frequency range be.
Fourier transformation
LTI systems with finitely many internal degrees of freedom are described by the linear differential equation of order n in the time domain (time as a variable ):
The application of the Fourier transform of the equation leads to the frequency response of an image function in the complex plane.
Frequency response is the ratio of the Fourier transform of the output signal and the input signal:
Inverse Fourier transform of the frequency response is the weight function or impulse response:
Spellings of the frequency response:
- With real and imaginary parts
- With magnitude and phase
Related to the transfer function
→ Main article: transfer function
The importance of the frequency response of LTI systems based on the simplicity of its experimental extraction (for example, in communications technology means sweep ). But he closes transients is not one. In the theoretical treatment of the system of this case, with the transfer function, which includes the frequency response detected.
With the transfer function goes to the frequency response.
Word meaning in a broader sense
In a more general sense, with ' response ', another characteristic of a physical system to be meant, such as the power, temperature, or the radiation intensity as a function of frequency. Often referred to as eg ' frequency response of a power ', however, is the expression ' frequency dependence of power '. According to one source called ' frequency response ' in the parlance of the control engineer the well-known frequency spectrum of special non-periodic excitation signals.