![Bandwidth](https://upload.wikimedia.org/wikipedia/commons/f/f6/Bandwidth.svg)
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
at which energy flowing through the system begins to be reduced (
attenuated or reflected) rather than passing through.
Typically in electronic systems such as
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
and
communication channel
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 for informa ...
s, cutoff frequency applies to an edge in a
lowpass
A low-pass filter 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 ...
,
highpass
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
,
bandpass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-po ...
, or
band-stop
In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a b ...
characteristic – a frequency characterizing a boundary between a
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
and a
stopband. It is sometimes taken to be the point in the filter response where a
transition band 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 ...
and passband meet, for example, as defined by a
half-power point
The half-power point is the point at which the output power has dropped to half of its peak value; that is, at a level of approximately -3 dB.
In filters, optical filters, and electronic amplifiers, the half-power point is also known as hal ...
(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 the case of a
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
or an
antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.
Electronics
In
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, cutoff frequency or corner frequency is the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
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
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
. 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. Other ratios besides the 3 dB point may also be relevant, for example see below. Far from the cutoff frequency in the transition band, the rate of increase of attenuation (
roll-off
Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband. It is most typically app ...
) with logarithm of frequency is
asymptotic to a constant. For a
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
network, the roll-off is −20 dB per
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
(−6 dB per
octave.)
Single-pole transfer function example
The
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
for the simplest
low-pass filter
A low-pass filter 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 filt ...
,
has a single
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
at . The magnitude of this function in the plane is
At cutoff
Hence, the cutoff frequency is given by
Where is the
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
variable, is
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
Chebyshev filters
Sometimes other ratios are more convenient than the 3 dB point. For instance, in the case of the
Chebyshev filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
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.
Radio communications
In
radio communication,
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'' refers to the
maximum usable frequency
In radio transmission maximum usable frequency (MUF) is the highest radio frequency that can be used for transmission between two points via reflection from the ionosphere (skywave or "skip" propagation) at a specified time, independent of transm ...
, the frequency above which a radio wave fails to reflect off the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.
Waveguides
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
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
that will propagate in an
optical fiber
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass ( silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a mea ...
or
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
. 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
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
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
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, 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
are the mode numbers for the rectangle's sides of length
and
respectively. 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
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
of the first kind of order 1.
The dominant mode TE
11 cutoff frequency is given by
However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide.
For a
single-mode optical fiber
In fiber-optic communication, a single-mode optical fiber (SMF), also known as fundamental- or mono-mode, is an optical fiber designed to carry only a single mode of light - the transverse mode. Modes are the possible solutions of the Helmholtz ...
, the cutoff wavelength is the wavelength at which the
normalized frequency is approximately equal to 2.405.
Mathematical analysis
The starting point is the wave equation (which is derived from the
Maxwell equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
T ...
),
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
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
, 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 cross-section with dimensions and :
where and are the two integers representing a specific eigenmode. Performing the final substitution, we obtain
which is the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
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.
See also
*
Full width at half maximum
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
*
High-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency ...
*
Miller effect In electronics, the Miller effect accounts for the increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of the effect of capacitance between the input and output terminals. The virtually increased inpu ...
*
Spatial cutoff frequency (in optical systems)
*
Time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
References
*
{{refend
External links
Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution*
ttp://mathworld.wolfram.com/BesselFunction.html Mathematical definition of and information about the Bessel functions
Filter theory