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Hermitian function

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Title: Hermitian function  
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Hermitian function

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)}

(where the overbar indicates the complex conjugate) 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: f is a Hermitian function if and only if

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.

See also

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