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# Synthesis with Even and Odd Functions

A signal has even symmetry if , which means the signal is symmetric about the vertical axis. A signal has odd symmetry if , which means the signal to the left of the vertical axis is the inverted mirror image of the signal to the right of the vertical axis. This Demonstration shows that any arbitrary signal can be synthesized by adding an even and an odd signal. Slide the time offset to change the input signal to .
A function is even if and odd if . For example, the functions and are even, while and are odd. Any signal can be synthesized from an even and an odd signal , where
, which is even because , and
, which is odd because .
The proof is straightforward:

.
The synthesis of signals from even and odd components is useful in applications to Fourier transforms.

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