Frequency spectrum

The frequency spectrum, including spectrum, spectral or in connection with time-dependent signals rarely frequency response, is a dependent of the frequency function. It gives the composition of a signal from its frequency dependent on the signal components.

In general, the frequency spectrum is complex. Its amount is called the amplitude spectrum, its phase angle is called phase spectrum.

The term frequency spectrum includes many different phenomena from all areas of physics like optics, acoustics, electrodynamics, or mechanics.

• The frequency of the light acting through the color stimulus than color. Natural light like sunlight consists of light waves of different frequencies.
• The frequency of a tone or sound determines its pitch. The sound of a musical instrument or human language is made up of fundamental tones of different frequencies.
• The frequency spectrum of a radio signal containing the picture and sound information.
• The frequency of a mechanical vibration is determined how many times repeatedly, the oscillation in a certain time. A complicated mechanical vibration, for example, the deflection of a seismograph during an earthquake. They are composed of vibrations of different frequencies.

The frequency spectrum of a signal can be calculated from the underlying signal by applying the Fourier transform. The representation in the frequency domain is used in physics and engineering to physical processes easier to describe than by functions of time or place.

• 2.1 Non- Periodic signal with a continuous spectrum

• 4.1 Examples

Frequency spectrum of a time signal

Due to the frequent use of the class of so-called time signals is described first. The frequency spectrum of a time signal is based on the intuition that a of the time -dependent signal x (t ) using the transformation rules of Fourier series and Fourier transform as a sum or an integral of complex exponentials of different frequencies can be put together. The complex exponential functions are called in this context " structure functions". The frequency spectrum describes with which weighting (ie strength ) received the corresponding to the respective frequency structure functions in the total signal. The calculated representation of the signal synthesis for the formulas inverse Fourier transform are illustrated. For this it is necessary to distinguish which type of signal is present.

Periodic signal with discrete spectrum

If the signal is a continuous-time periodic function with a period, as is the associated equation:

The equation describes the signal x ( t) as a sum of complex exponential oscillations of the frequencies. As spectrum of the signal x is defined as the function

At the fundamental frequency. The number is representative of n times the fundamental frequency. Exponentialschwingung complex can be described by the equation. Since the range is defined only for discrete frequencies, this is called a discrete spectrum or a line spectrum.

Non- Periodic signal with a continuous spectrum

Is the signal x ( t) is a non-periodic time-continuous function with finite signal power that is the corresponding transformation equation:

As spectrum of the signal is referred to in this case, the function

Since the spectrum is defined for all real-valued frequencies, one also speaks of a so-called continuous spectrum. The spectrum of the continuous Fourier transform can be represented as the limit of the line spectrum of the Fourier series for the border crossing of an infinitely large signal period.

Notes and other signal classes

Both frequency spectra are defined for both positive and negative frequencies. For real-valued signals x (t ), the spectra for positive and negative frequencies, however, are dependent on each other, and we have:. The asterisk indicates the complex conjugate. In general, therefore, the spectrum of negative frequencies is displayed only for complex-valued signals.

In the framework of the theory of the Fourier transform analysis formulas for other classes of signals, as defined, for example for time-discrete, value- continuous signals, that is, sampled analog signals. The terms frequency spectrum, amplitude spectrum and phase spectrum are analogously defined as a complex function and their amounts and phases. The details are shown in the article on the Fourier transform and the links contained therein.

Related to non-periodic signals, such as output signals, the concept of noise power spectral density, which also describes similar to the frequency spectrum of the spectral composition of a signal exists. The peculiarity of non-periodic power signals is that they are not fouriertransformierbar. This is evident from the fact that the associated transformation integrals diverge. Nevertheless, a connection with the concept of the Fourier transform can be produced, which is important for the metrological practice. If the signal is an ergodic development process based, it can be the power spectral density approximated thereby determine that is subjected to a partial signal of finite duration of the actually infinitely long signal to a Fourier transform. The square of the Fourier transform is then approximately proportional to the power spectral density.

Examples

Elementary signals

The spectra of elementary signals are included in the descriptions of the associated signal transformations, see Examples for a Fourier series and the Fourier transformation examples. As examples, several spectra of simple signals are displayed. The fourth example demonstrates the effect of the phase spectrum of a narrowband signal.

The amplitude spectrum of an audio signal

As an example, the amplitude spectrum of the following Geigentons should be considered

• Violin d 'in equal temperament (294 Hz)? / I

The spectrum of the Geigentons is dependent on the period of time that is chosen for analysis. Considering a signal segment, which was recorded during the strike of the strings so easily recognized not only the fundamental frequency of f0 = 294 Hz significant frequency components of the integer multiples. This can be explained by the fact that the string vibrates not only in their fundamental wave, in which the string undergoes a deflection along its entire length, but to except additional on the edge nodes at 1/ 2, 1/ 3, 2/ 3, 1 / 4, 2/ 4, 3/ 4, ... form the string length. The vibration at a multiple of the fundamental means in musical parlance overtone. The formation of the different harmonics is determined not only by the vibration of the string alone, but by the overall arrangement of the instrument ( string, sound, string pressure when painting or deflection in plucking ). In contrast to the signal section during the strike shows the signal segment, which takes into account the fade-out of the sound, no distinctive overtone.

Power density spectra of light

Light can be described as an electromagnetic wave, and is characterized by the vector of the electric field intensity and the corresponding vector of the magnetic field strength. Consider a component of the electric field strength vector of a light wave at a fixed location, then there is a signal which can be divided in the case of a signal of finite duration by using the continuous Fourier transform into its spectral components.

The phase of the electric field vector of light is all difficult to measure because of the high frequencies of light. The spectrum of visible light, for example, the wavelength range ( ) of about 380 to 780 nm comprising (see light spectrum). According to the formula with the speed of light ( ) of the vacuum, this results in a frequency range () of 384-789 × 1012 Hz, the absolute square of the Fourier transform of a light signal of finite duration, that is, its power spectral density, however, is well detected. Generally, therefore, the power spectral density of the light is indicated in connection with light. It is proportional to the intensity of light at the respective frequency.

A possibility to make the spectral composition of light visible is the use of an optical prism. In this case, one uses the fact that the light in dependence on its frequency is refracted to different degrees at the edges of the prism. Humans can detect light signals from one frequency on the basis of color, and may the power densities of the light wave differ from each other due to the perceived brightness.

In experiments it can be shown that the natural sunlight has a continuous power spectral density. In the prism test results in the typical colors of the rainbow. In contrast to the sunlight, the spectrum of a mercury vapor lamp on a power spectral density which is more like a line spectrum.

Spatial frequency spectra

Depends on the underlying signal s not t of the time, but of coordinates of the location from, then one speaks of a so-called spatial frequency spectrum. Spatial frequency spectra can be one-, two - or three-dimensional, depending on whether the one-, two - or three-dimensional structures to be analyzed. You can be both a continuous and have a discrete domain.

Examples of structures with continuous domain are

• The gray value profile along a line ( one-dimensional)
• The gray value profile of a black and white photograph ( two-dimensional)
• The intensity distribution of a physical quantity in space ( three-dimensional)

Examples of structures with discrete domain are

• The gray value sequence of discrete points along a line ( one-dimensional)
• The gray value changes on discrete points of a black and white photograph ( two-dimensional), such as pixel graphics
• The point distribution of a crystal lattice in space

Analogous to the frequency spectrum of a time function is the spatial frequency spectrum of the view based on that (z x, y ) allows the total signal S by using the transformation rules of the Fourier series and the Fourier transform as a sum or an integral of complex exponentials of the spatial frequencies, and composed.

The exponential function can be illustrated by the dependent from the location signal phase. This is shown for the case of a two-dimensional transformation in the adjacent picture for different spatial frequencies. It can be seen that, in general, the vector indicating the direction of the maximum phase change.

Non- Periodic signal with a continuous spectrum

The signal S (x, y, z) is a non-periodic time-continuous function of the three spatial coordinates X, Y and Z, that is the corresponding transformation equation:

As the spatial frequency spectrum of the signal is defined as the function

Measuring the frequency spectrum

The frequency spectrum of an electric signal can be measured with a spectrum analyzer, or signal analyzer. The spectrum is then, for example, with the aid of Fourier analysis (see also Fourier transformation ), or is determined according to the principle of the superheterodyne receiver of the timing signal. As a result of this transformation is obtained, the amplitudes of the respective frequency components A (f) as a function of frequency f and in the case of time-varying amplitude distributions distribution A ( f, T) as a function of frequency f and the time t.

Characteristic spectra

Depending on the number and the harmonics of the frequencies contained gives the spectrum of a ( one-dimensional ) audio signal a sound ( harmonic ), a sound mixture (a few discordant frequencies), a sound ( discordant ) or noise (all frequencies, statistically occurring ).

Periodic signals typically have a line spectrum, while non-periodic signals, such as pulses which have a continuous frequency spectrum.

Examples

• A pure sine wave has a frequency spectrum, only one line of their frequency.
• A square wave signal of the frequency f has a line spectrum with the frequencies f 3 · f 5 * f, ...
• With a pulse generator can seamlessly generate all harmonics up to extremely high frequencies.
• A note on a musical instrument will always be heard together with its harmonics. The set of sounding frequencies is the frequency spectrum of this tone.
• Transmits an amplitude- modulated radio transmitters ( eg, on medium wave ), language and music to 8 kHz to 1 MHz, has a frequency range of 0.992 to 1.008 MHz.
• Sends the system from electromagnetic radiation, it is called the electromagnetic spectrum.

Other meanings

In a broader sense the frequency spectrum refers to a collection of frequencies that must be seen in relation to a particular way of looking together, such as the frequency spectrum of radio and television channels; see frequency band.

A response spectrum is used for design of structures against the impact of earthquakes.