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In physics and electrical engineering, a **cutoff frequency**, **corner frequency**, or **break frequency** is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through.

**Physics** is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

**Electrical engineering** is a technical discipline concerned with the study, design and application of equipment, devices and systems which use electricity, electronics, and electromagnetism. It emerged as an identified activity in the latter half of the 19th century after commercialization of the electric telegraph, the telephone, and electrical power generation, distribution and use.

**Frequency response** is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant, then the frequency response also will not vary with time. Thus for LTI systems, the frequency response can be seen as applying the system's transfer function to a purely imaginary number argument representing the frequency of the sinusoidal excitation.

- Electronics
- Single-pole transfer function example
- Chebyshev filters
- Radio communications
- Waveguides
- Mathematical analysis
- See also
- References
- External links

Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example, as defined by a half-power point (a frequency for which the output of the circuit is −3 dB of the nominal passband value). Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.

In signal processing, a **filter** is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, and computer graphics.

A **communication channel** refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used to convey an information signal, for example a digital bit stream, from one or several *senders* to one or several *receivers*. A channel has a certain capacity for transmitting information, often measured by its bandwidth in Hz or its data rate in bits per second.

The **transition band**, also called the **skirt**, is a range of frequencies that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner frequency.

In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper **cutoff wavelengths**.

A **waveguide** is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

In radio engineering, an **antenna** is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

In electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a line, amplifier, or electronic filter has fallen to a given proportion of the power in the passband. Most frequently this proportion is one half the passband power, also referred to as the 3 dB point since a fall of 3 dB corresponds approximately to half power. As a voltage ratio this is a fall to of the passband voltage.^{ [1] } Other ratios besides the 3 dB point may also be relevant, for example see Chebyshev Filters below.

**Electronics** comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter.

An **electronic circuit** is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as *electronic*, rather than *electrical*, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

A **telephone line** or **telephone circuit** is a single-user circuit on a telephone communication system. This is the physical wire or other signaling medium connecting the user's telephone apparatus to the telecommunications network, and usually also implies a single telephone number for billing purposes reserved for that user. Telephone lines are used to deliver landline telephone service and Digital subscriber line (DSL) phone cable service to the premises. Telephone overhead lines are connected to the public switched telephone network.

The transfer function for the simplest low-pass filter,

In engineering, a **transfer function** of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a **transfer curve** or **characteristic curve**. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.

A **low-pass filter** (**LPF**) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a **high-cut filter**, or **treble-cut filter** in audio applications. A low-pass filter is the complement of a high-pass filter.

has a single pole at *s* = -1/α. The magnitude of this function in the *j*ω plane is

At cutoff

Hence, the cutoff frequency is given by

Where *s* is the s-plane variable, ω is angular frequency and *j* is the imaginary unit.

Sometimes other ratios are more convenient than the 3 dB point. For instance, in the case of the Chebyshev filter it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple. The amount of ripple in this class of filter can be set by the designer to any desired value, hence the ratio used could be any value.^{ [2] }

In radio communication, "skip" or "skywave" communication is a technique in which radio waves are transmitted at an angle into the sky and reflected back to Earth by layers of charged particles in the ionosphere. In this context, the term cutoff frequency means the frequency below which a radio wave fails to penetrate a layer of the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.

The cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. In fiber optics, it is more common to consider the **cutoff wavelength**, the maximum wavelength that will propagate in an optical fiber or waveguide. The cutoff frequency is found with the characteristic equation of the Helmholtz equation for electromagnetic waves, which is derived from the electromagnetic wave equation by setting the longitudinal wave number equal to zero and solving for the frequency. Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate. The following derivation assumes lossless walls. The value of c, the speed of light, should be taken to be the group velocity of light in whatever material fills the waveguide.

For a rectangular waveguide, the cutoff frequency is

where the integers are the mode numbers, and *a* and *b* the lengths of the sides of the rectangle. For TE modes, (but is not allowed), while for TM modes .

The cutoff frequency of the TM_{01} mode (next higher from dominant mode TE_{11}) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by

where is the radius of the waveguide, and is the first root of , the bessel function of the first kind of order 1.

The dominant mode TE_{11} cutoff frequency is given by

^{ [3] }

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide.^{ [4] } For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

The starting point is the wave equation (which is derived from the Maxwell equations),

which becomes a Helmholtz equation by considering only functions of the form

Substituting and evaluating the time derivative gives

The function here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field. It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse. The *z* axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form

where is the longitudinal wavenumber, resulting in

where subscript T indicates a 2-dimensional transverse Laplacian. The final step depends on the geometry of the waveguide. The easiest geometry to solve is the rectangular waveguide. In that case, the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form

Thus for the rectangular guide the Laplacian is evaluated, and we arrive at

The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry crossection with dimensions *a* and *b*:

where *n* and *m* are the two integers representing a specific eigenmode. Performing the final substitution, we obtain

which is the dispersion relation in the rectangular waveguide. The cutoff frequency is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber is zero. It is given by

The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis and the wave is thus evanescent.

- Full width at half maximum
- High-pass filter
- Miller effect
- Spatial cutoff frequency (in optical systems)
- Time constant

The **propagation constant** of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, **propagation constant **measures the change undergone by the source quantity as it propagates from one port to the next.

In radio-frequency engineering, a **transmission line** is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

**Fourier optics** is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or *superposition*, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

**Chebyshev filters** are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. The type I Chebyshev filters are called usually as just "Chebyshev filters", the type II ones are usually called "inverse Chebyshev filters".

In signal processing, a **finite impulse response** (**FIR**) **filter** is a filter whose impulse response is of *finite* duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

In physics, a **wave packet** is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

The **Butterworth filter** is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is also referred to as a **maximally flat magnitude filter**. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In digital signal processing, **upsampling**, **expansion**, and **interpolation** are terms associated with the process of resampling in a multi-rate digital signal processing system. *Upsampling* can be synonymous with *expansion*, or it can describe an entire process of *expansion* and filtering (*interpolation*). When upsampling is performed on a sequence of samples of a *signal* or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate. For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.

In microwave and radio-frequency engineering, a **stub** or **resonant stub** is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

**Sinusoidal plane-wave solutions** are particular solutions to the electromagnetic wave equation.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In many-body theory, the term **Green's function** is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation greenand annihilation operators.

The Mathieu equation is a linear second-order differential equation with periodic coefficients. The French mathematician, E. Léonard Mathieu, first introduced this family of differential equations, nowadays termed Mathieu equations, in his “Memoir on vibrations of an elliptic membrane” in 1868. "Mathieu functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude distortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration in a medium with modulated density"

The **non-radiative dielectric (NRD) waveguide ** has been introduced by Yoneyama in 1981. In Fig. 1 the cross section of NRD guide is shown: it consists of a dielectric rectangular slab of height a and width b, which is placed between two metallic parallel plates of suitable width. The structure is practically the same as the H waveguide, proposed by Tischer in 1953. Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region, for suitable frequencies, the electromagnetic field decays exponentially. Therefore, if the metallic plates are sufficiently extended, the field is practically negligible at the end of the plates and therefore the situation does not greatly differ from the ideal case in which the plates are infinitely extended. The polarization of the electric field in the required mode is mainly parallel to the conductive walls. As it is known, if the electric field is parallel to the walls, the conduction losses decrease in the metallic walls at the increasing frequency, whereas, if the field is perpendicular to the walls, losses increase at the increasing frequency. Since the NRD waveguide has been devised for its implementation at millimeter waves, the selected polarization minimizes the ohmic losses in the metallic walls.

**Filtering** in the context of large eddy simulation (LES) is a mathematical operation intended to remove a range of small scales from the solution to the Navier-Stokes equations. Because the principal difficulty in simulating turbulent flows comes from the wide range of length and time scales, this operation makes turbulent flow simulation cheaper by reducing the range of scales that must be resolved. The LES filter operation is low-pass, meaning it filters out the scales associated with high frequencies.

Fluid motion can be said to be a **two-dimensional flow** when the flow velocity at every point is parallel to a fixed plane. The velocity at any point on a given normal to that fixed plane should be constant.

In fluid dynamics, **Beltrami flows** are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

- ↑ Van Valkenburg, M. E.
*Network Analysis*(3rd ed.). pp. 383–384. ISBN 0-13-611095-9 . Retrieved 2008-06-22. - ↑ Mathaei, Young, Jones
*Microwave Filters, Impedance-Matching Networks, and Coupling Structures*, pp.85-86, McGraw-Hill 1964. - ↑ I. C. Hunter,
*Theory and Design of Microwave Filters*, p.214 IET, 2001 ISBN 0-85296-777-2. - ↑ A. Y. Modi and C. A. Balanis, "PEC-PMC Baffle Inside Circular Cross Section Waveguide for Reduction of Cut-Off Frequency," in IEEE Microwave and Wireless Components Letters, vol. 26, no. 3, pp. 171-173, March 2016. doi : 10.1109/LMWC.2016.2524529

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188 ).

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