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,
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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Symmetry
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