Cutoff frequency

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 approximately −3.01 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 or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.

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.

[1] Other ratios besides the 3 dB point may also be relevant, for example see § Chebyshev filters below.

Far from the cutoff frequency in the transition band, the rate of increase of attenuation (roll-off) with logarithm of frequency is asymptotic to a constant.

For a [[Low-pass filter#:~:text=information: Electronic filter-,First-order passive,-[edit]|first-order]] network, the roll-off is −20 dB per decade (approximately −6 dB per octave.)

The transfer function for the simplest low-pass filter,

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

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, 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, 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.

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.

are the mode numbers for the rectangle's sides of length

The cutoff frequency of the TM01 mode (next higher from dominant mode TE11) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by

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.

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

It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse.

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

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 a and b:

In this case, the field decays exponentially along the waveguide axis and the wave is thus evanescent.