Symmetry properties of periodic signals

A signal has even symmetry of it satisfies:

and odd symmetry if it satisfies

[link] shows pictures of periodic even and odd symmetric signals. If $x\left(t\right)$ is an odd symmetric periodic signal, then we must have:

This is easy to see if we choose $t_0=-T/2$ .

We also note that the product of two even signals is also even while the product of an even signal and an odd signal must be odd. Finally, the product of two odd signals must be even. For example, suppose $x_o\left(t\right)$ has odd symmetry and $x_e\left(t\right)$ has even symmetry. Their product has odd symmetry because if $y\left(t\right)=x_o\left(t\right)x_e\left(t\right)$ , then $y(-t)=x_o(-t)x_e(-t)=-y\left(t\right)$ .