Conjugation and Reversal

Theorem: For any

,
Proof:
Theorem: For any

,
Proof: Making the change of summation variable

, we get
Theorem: For any

,
Proof:
Corollary:
For any

,
Proof: Picking up the previous proof at the third formula, remembering that

is real,
when

is real.
Thus,
conjugation in the
frequency domain corresponds to
reversal in the time domain.
Another way to say it is that
negating spectral phase flips the signal around backwards in
time.
Corollary:
For any

,
Proof: This follows from the previous two cases.
Definition: The property

is called
Hermitian symmetry
or ``conjugate symmetry.'' If

, it may be called
skew-Hermitian.
Another way to state the preceding corollary is
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