### Time Reversal

For any complex signal , , we have

(3.11) |

where .

*Proof: *

Arguably, should include complex conjugation. Let

(3.12) |

denote such a definition. Then in this case we have

(3.13) |

*Proof: *

(3.14) |

In the typical special case of real signals ( ), we have so that

(3.15) |

That is,

*time-reversing a real signal conjugates its spectrum*.

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Symmetry of the DTFT for Real Signals

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Linearity of the DTFT