Even and Odd Functions

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Even and Odd Functions

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Even and Odd Functions

Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.



Definition: A function $f(n)$ is said to be even if $f(-n)=f(n)$. An even function is alsosymmetric, but the term symmetric applies also to functions symmetric about a point other than $0$.



Definition: A function $f(n)$ is said to be oddif $f(-n)=-f(n)$. An odd function is also calledantisymmetric.

Note that every odd function $f(n)$ must satisfy $f(0)=0$. Moreover, for any $x\ with $N$ even, we also have $x(N/2)=0$ since $x(N/2)=-x(-N/2)=-x(-N/2+N)=-x(N/2)$, i.e., $N/2$ and $-N/2$ index the same point.



Theorem: Every function $f(n)$ can be decomposed into a sum of its even part$f_e(n)$ and odd part $f_o(n)$, where

\



Proof: In the above definitions, $f_e$ is even and $f_o$ is odd by construction. Summing, we have

\



Theorem: The product of even functions is even, the product of odd functions is even, and the product of an even times an odd function is odd.

Proof: Readily shown.

Since even times even is even, odd times odd is even, and even times odd is odd, we can think of even as $(+)$ and odd as $(-)$: $(+)\, $(-)\, and $(+)\.



Example: $\ is an even signal since $\.



Example: $\ is an odd signal since $\.



Example: $\ is odd (even times odd).



Example: $\ is even (odd times odd).



Theorem: The sum of all the samples of an odd signal $x_o$ in ${\ is zero.

Proof: This is readily shown by writing the sum as $x_o(0) + [x_o(1) + x_o(-1)]
+ \, where the last term only occurs when $N$ is even. Each term so written is zero for an odd signal $x_o$.



Example: For all DFT sinusoidal frequencies $\,

\

More generally,

\

for any even signal $x_e$ and odd signal $x_o$ in ${\.

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