# Frequency response

The frequency response is the relationship between the input and output signal of a linear time-invariant system (LZI system) with a sinusoidal excitation in terms of amplitude and phase. It is therefore a complex function of frequency .

The output signal has the same frequency as the input signal due to the linear behavior of the system. However, the two signals differ in amplitude and phase . The ratio of the amplitudes of the input signal and the output signal as a function of the frequency is the amplitude response , sometimes also called the magnitude frequency response . The difference in phase between the input signal and the output signal depending on the frequency is the phase response .

The frequency response can also be determined from the Fourier transform of the impulse response of the system.

## General

The frequency response describes the relationship between sinusoidal oscillations at the input and output of a system ( transmission element ) as a function of the frequency  f or the angular frequency  ω .

The system has the following properties:

Frequency response of a PT1 element :
The output amplitude is smaller at higher frequencies.
Bode diagram :
amplitude and phase frequency response of a passive low-pass filter or PT1 element

Such a system has a harmonic input signal

${\ displaystyle x (t) = {\ hat {x}} \ sin (\ omega t + \ phi _ {x}) \;}$

a harmonic output signal:

${\ displaystyle y (t) = {\ hat {y}} (\ omega) \ sin (\ omega t + \ phi _ {y} (\ omega)) \;}$.

Due to the linearity, the angular frequency is not influenced. Only the amplitude ( → ) and phase ( → ) are changed. ${\ displaystyle \ omega \;}$${\ displaystyle {\ hat {x}} \;}$${\ displaystyle {\ hat {y}} \;}$${\ displaystyle \ phi _ {x} \;}$${\ displaystyle \ phi _ {y} \;}$

The amplitude-frequency response is the ratio

${\ displaystyle A (\ omega) = {\ frac {{\ hat {y}} (\ omega)} {\ hat {x}}}}$.

The phase frequency response is the phase difference

${\ displaystyle \ phi (\ omega) = \ phi _ {y} (\ omega) - \ phi _ {x} \;}$.

## Graphical representation

### Bode diagram

The Bode diagram is used for a clear representation of the frequency response (see illustration). The amplitude-frequency response and the phase-frequency response are shown in one graph each. The majority of the axes are logarithmically divided (except for the phase shift), which makes the diagram easier to use. For example, the multiplication of two frequency responses is a simple addition of the distance, and the inversion of a frequency response results from reflection on the f or ω axis in the diagram.

### Locus

An alternative graphic representation of the frequency response is its locus . In contrast to the Bode diagram, this vector image contains both pieces of information: The vector length corresponds to the amplitude ratio, its argument φ is the phase shift.

This locus is also called the Nyquist diagram . With the idea that only the tips of frozen pointers are connected to the locus in the (complex) plane, the frequency response can be made clear without knowledge of the complex mathematics and the mathematical transformations from the time to the frequency domain.

## Fourier transform

LZI systems with a finite number of internal degrees of freedom are described by the linear differential equation of the nth order in the time domain (time as a variable):

${\ displaystyle y ^ {(n)} + a_ {n-1} y ^ {(n-1)} + \ ldots + a_ {1} y ^ {(1)} + a_ {0} y = b_ { m} x ^ {(m)} + \ ldots + b_ {1} x ^ {(1)} + b_ {0} x}$.

The application of the Fourier transformation to the differential equation leads to the frequency response as an image function in the complex number plane.

Frequency response is the quotient of the Fourier transforms of the output signal and the input signal: ${\ displaystyle H (\ mathrm {j} \ omega)}$${\ displaystyle Y (\ mathrm {j} \ omega)}$${\ displaystyle X (\ mathrm {j} \ omega)}$

${\ displaystyle H (\ mathrm {j} \ omega) = {\ frac {Y (\ mathrm {j} \ omega)} {X (\ mathrm {j} \ omega)}} = {\ frac {b_ {m } (\ mathrm {j} \ omega) ^ {m} + \ ldots + b_ {1} (\ mathrm {j} \ omega) + b_ {0}} {(\ mathrm {j} \ omega) ^ {n } + a_ {n-1} (\ mathrm {j} \ omega) ^ {n-1} + \ ldots + a_ {1} (\ mathrm {j} \ omega) + a_ {0}}}}$.

The inverse Fourier transform of the frequency response is the weight function or impulse response:

${\ displaystyle g (t) = {\ frac {1} {2 \ pi}} \ int _ {- \ infty} ^ {\ infty} H (\ mathrm {j} \ omega) e ^ {\ mathrm {j } \ omega t} \ mathrm {d} \ omega}$.

Notation of the frequency response:

• with real and imaginary part
${\ displaystyle H (\ mathrm {j} \ omega) = \ operatorname {Re} H (\ mathrm {j} \ omega) + \ mathrm {j} \, \ operatorname {Im} H (\ mathrm {j} \ omega)}$ .
• with amount and phase
${\ displaystyle H (\ mathrm {j} \ omega) = \ left | H (\ mathrm {j} \ omega) \ right | e ^ {\ mathrm {j} \ varphi (\ mathrm {j} \ omega)} }$.
${\ displaystyle \ left | H (\ mathrm {j} \ omega) \ right | = {\ sqrt {(\ operatorname {Re} H (\ mathrm {j} \ omega)) ^ {2} + (\ operatorname { Im} H (\ mathrm {j} \ omega)) ^ {2}}}}$     amount
${\ displaystyle \ varphi (\ mathrm {j} \ omega) = \ arctan \ left ({\ frac {\ operatorname {Im} H (\ mathrm {j} \ omega)} {\ operatorname {Re} H (\ mathrm {j} \ omega)}} \ right)}$     phase

## Relationship with the transfer function

see main article: Transfer function

With in , the Laplace transfer function changes into the frequency response . ${\ displaystyle \ sigma = 0}$${\ displaystyle s = \ sigma + j \ omega}$${\ displaystyle F (s)}$${\ displaystyle F (\ omega)}$

The frequency response therefore does not describe any transition processes (transient processes due to time constants). Nor is it suitable for describing unstable emerging systems.

The Laplace transfer function is more general in these aspects due to the additional parameter . ${\ displaystyle \ sigma}$

## Experimental determination of the frequency response

The importance of the frequency response for LZI systems is based on the simplicity of its experimental extraction. For this purpose, the system is stimulated with a signal generator with different frequencies and the system response is measured.

In systems with a rapid transient response after a (small) change in frequency, the measurement can be carried out using a wobble generator , as is the case, for example, in communications technology . The wobble generator is a special signal generator that continuously changes its output frequency.

Frequency response determination with signal generator and time-synchronous measurement

However, if after each frequency excitation one has to wait a certain time until the amplitude of the system response no longer changes, then the process with the help of a signal generator is more time-consuming.

In this case it is easier to stimulate the system with all frequencies of interest simultaneously and to determine the frequency response, for example, by measuring the impulse response .

In any case, the experimental frequency response determination requires a time-synchronous measurement of the input signal x and the output signal y of the system.

## Word meaning in a broader sense

In a more general sense, “frequency response” can also mean another frequency-dependent property of a physical system, such as power consumption, temperature or radiated power as a function of frequency. More common than z. B. "Frequency response of a service" is however the expression "frequency dependence of a service". According to one source, “frequency response” in the language of control engineers also denotes the known frequency spectrum of special, non-periodic excitation signals.