Hermitian function

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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:

$f(-x)=\overline {f(x)}$

for all $x$ in the domain of $f$ .

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

$f(-x_{1},-x_{2})=\overline {f(x_{1},x_{2})}$

for all pairs $(x_{1},x_{2})$ in the domain of $f$ .

From this definition it follows immediately that, if $f$ is a Hermitian function, then

the real part of $f$ is an even function
the imaginary part of $f$ 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:

The function $f$ is real-valued if and only if the Fourier transform of $f$ is Hermitian.

The function $f$ is Hermitian if and only if the Fourier transform of $f$ 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 the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal.

If f is Hermitian, then $f\star g=f*g$

Where the $\star$ is cross-correlation, and $*$ is convolution.

If both f and g are Hermitian, then $f\star g=g\star f$ , which in general is not true.

Hermitian function

Motivation

See also