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Inmathematical analysis, aHermitian function is acomplex function with the property that itscomplex conjugate is equal to the original function with the variable changed insign:

${\displaystyle f^{\ast }(x)=f(-x)}$

(where the${\displaystyle {}^{\ast }}$ indicates the complex conjugate) for all${\displaystyle x}$ in the domain of${\displaystyle f}$. Inphysics, this property is referred to asPT symmetry.

This definition extends also to functions of two or more variables, e.g., in the case that${\displaystyle f}$ is a function of two variables it is Hermitian if

${\displaystyle f^{\ast }(x_1,x_2)=f(-x_1,-x_2)}$

for all pairs${\displaystyle (x_1,x_2)}$ in the domain of${\displaystyle f}$.

From this definition it follows immediately that:${\displaystyle f}$ is a Hermitian functionif and only if

• the real part of${\displaystyle f}$ is aneven function,
• the imaginary part of${\displaystyle f}$ is anodd function.

Hermitian functions appear frequently in mathematics, physics, andsignal processing. For example, the following two statements follow from basic properties of the Fourier transform:^{[citation needed]}

• The function${\displaystyle f}$ is real-valued if and only if theFourier transform of${\displaystyle f}$ is Hermitian.
• The function${\displaystyle f}$ is Hermitian if and only if theFourier transform of${\displaystyle f}$ 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 thediscrete 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 fromDFT), the axis of symmetry is around theNyquist point; one half is the mirror image of the other.

• Iff is Hermitian, then${\displaystyle f\star g=f\ast g}$.

Where the${\displaystyle \star }$ iscross-correlation, and${\displaystyle \ast }$ isconvolution.

• If bothf andg are Hermitian, then${\displaystyle f\star g=g\star f}$.

• Complex conjugate – Fundamental operation on complex numbers
• Even and odd functions – Functions such that f(–x) equals f(x) or –f(x)

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