In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:

. In physics, this property is referred to as PT symmetry.

is a function of two variables it is Hermitian if

.

is a Hermitian function if and only if

• is an even function,

• is an odd function.

Motivation

Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:

• is real-valued if and only if the Fourier transform is Hermitian.

• is Hermitian if and only if the Fourier transform is real-valued.

Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal. Informally, only half of the fourier transform of a real signal is needed to lossessly represent it in frequency domain.

For the magnitude spectra (obtained from DFT), the axis of symmetry is around the Nyquist point; one half is the mirror image of the other.

• If f.

is cross-correlation is convolution.

• If both f and g.

See also