Symmetry

A discrete-time signal will often possess some form of symmetry that may be exploited in solving problems. Two symmetries of interest are as follows: A real-valued signal is said to be even if, for all indices $n$, $x\left[n\right]=x[-n]$, whereas a signal is said to be odd if, for all indices $n$, $x\left[n\right]=-x[-n]$. For complex signals the symmetries of interest are slightly different namely a complex signal is conjugate symmetric if, for all indices $n$ $x\left[n\right]=x^{\ast }[-n]$ and a signal is said to be conjugate antisymmetric if, for all indices $n$ $x\left[n\right]=-x^{\ast }[-n]$. Furthermore, any signal $x\left[n\right]$ may be decomposed into a sum of its even part, denoted by $x_e\left[n\right]$, and its odd part, denoted by $x_o\left[n\right]$. The even part can be constructed as $x_e\left[n\right]=\frac{1}{2}\left(x\right[n]+x[-n\left]\right)$, while the odd part can be constructed as $x_o\left[n\right]=\frac{1}{2}\left(x\right[n]-x[-n\left]\right)$.

The following equations summarize all these symmetries: $$\begin{array}{ccc}\text{Even}\text{Conjugate-symmetric}& :& x\left[n\right]=x[-n]x\left[n\right]=x^{\ast }[-n]\\ \text{Odd}\text{Conjugate-antisymmetric}& :& x\left[n\right]=-x[-n]x\left[n\right]=-x^{\ast }[-n]\\ \text{General:}x\left[n\right]& =& x_e\left[n\right]+x_o\left[n\right]\\ \text{with}{x}_e\left[n\right]& =& \frac{1}{2}\left(x\right[n]+x[-n\left]\right)\\ \text{and}{x}_o\left[n\right]& =& \frac{1}{2}\left(x\right[n]-x[-n\left]\right)\end{array}$$