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Symmetry Properties of Periodic Signals

Module by: Carlos E. Davila. E-mail the author

A signal has even symmetry of it satisfies:

x ( t ) = x ( - t ) x ( t ) = x ( - t ) (1)

and odd symmetry if it satisfies

x ( t ) = - x ( - t ) x ( t ) = - x ( - t ) (2)

Figure 1 shows pictures of periodic even and odd symmetric signals. If x(t)x(t) is an odd symmetric periodic signal, then we must have:

∫ t 0 t 0 + T x ( t ) d t = 0 ∫ t 0 t 0 + T x ( t ) d t = 0 (3)

This is easy to see if we choose t0=-T/2t0=-T/2.

Figure 1: (a) Even-symmetric, and (b) odd-symmetric periodic signals. Note that the integral over any period of an odd-symmetric periodic signal is zero.
Figure 1 (even_odd_sig.png)

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 xo(t)xo(t) has odd symmetry and xe(t)xe(t) has even symmetry. Their product has odd symmetry because if y(t)=xo(t)xe(t)y(t)=xo(t)xe(t), then y(-t)=xo(-t)xe(-t)=-y(t)y(-t)=xo(-t)xe(-t)=-y(t).

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