Harman Patil (Editor)

Hermitian function

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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 ) = f ( − x )

(where the ∗ 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 ) = 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

  • 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 ⋆ g = f ∗ g .

    • Where the ⋆ is cross-correlation, and ∗ is convolution.

  • If both f and g are Hermitian, then f ⋆ g = g ⋆ f .

    • References

    Hermitian function Wikipedia
    (Text) CC BY-SA