Finite Extension or Periodic Functions—Fourier Coefficients - Vision with Direction - page 66

because the Fourier basis ψ m is an orthogonal basis set yielding unique FCs. This
property of the FCs, which is only valid for real functions f , is called Hermitian sym-
metry . Because of this, the Fourier coefficients of a real signal contains redundancy.
If we know a coefficient, say F ( ω 17 )=0 . 5+ i 0 . 3, then the mirrored coefficient is
locked to its conjugate, i.e., F ( ω − 17 )=0 . 5

i 0 . 3 .
The coefficient F ( ω m ) (as well as F ∗ ( ω −m ) ) is complex and can be written in
terms of its real and imaginary parts

−

F ( ω m )= ( F ( ω m )) + i ( F ( ω m )) (5.33)

so that

F ∗ ( ω −m )= ( F ∗ ( ω −m )) + i ( F ∗ ( ω −m )) =

( F ( ω −m )) − i ( F ( ω −m ))

Thus, Eq. (5.32) can be rewritten as

( F ( ω m )) = ( F ( ω −m )) , and ( F ( ω m )) = − ( F ( ω −m )) . (5.34)

We summarize our finding on the redundancy of the FC coefficents of f as follows.

Theorem 5.4. The Fourier coefficients of a real signal have Hermitian symmetry:

F ( ω m )= F ∗ ( ω −m ) (5.35)

so that the real and imaginary parts of their FCs are even and odd, respectively,
whereas their magnitudes are even.