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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University