Hermitian functionIn 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: (where the indicates the complex conjugate) for all in the domain of . In physics, this property is referred to as PT symmetry. This definition extends also to functions of two or more variables, e.g., in the case that is a function of two variables it is Hermitian if for all pairs in the domain of . From this definition it follows immediately that: is a Hermitian function if and only if
MotivationHermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:[citation needed]
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.
Where the is cross-correlation, and is convolution.
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