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. 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 from
DFT), the axis of symmetry is around the
Nyquist point; one half is the mirror image of the other. • 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 ==