### 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,

*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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Linearity