#### Real Even (or Odd) Signals

If a signal is *even* in addition to being real, then its DTFT is
also real and even. This follows immediately from the Hermitian
symmetry of real signals, and the fact that the DTFT of any even
signal is real:

This is true since cosine is even, sine is odd, even times even is
even, even times odd is odd, and the sum over all samples of an odd
signal is zero. *I.e.*,

and

If is real and even, the following are true:

Similarly, if a signal is *odd* and real, then its DTFT is odd
and *purely imaginary*. This follows from Hermitian symmetry for
real signals, and the fact that the DTFT of any odd signal is imaginary.

where we used the fact that

and

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Proof of Aliasing Theorem

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DTFT of Real Signals