Conjugation and Reversal

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Conjugation and Reversal

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Conjugation and Reversal



Theorem: For any $x\,

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Proof:

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Theorem: For any $x\,

\

Proof: Making the change of summation variable $m\, we get

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Theorem: For any $x\,

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Proof:

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Corollary: For any $x\,

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Proof: Picking up the previous proof at the third formula, remembering that $x$ is real,

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when $x(n)$ 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 $x\,

\

Proof: This follows from the previous two cases.



Definition: The property $X(-k)=\ is called Hermitian symmetryor “conjugate symmetry.” If $X(-k)=-\, it may be calledskew-Hermitian.

Another way to state the preceding corollary is

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