What is x* when referring to the conjugate symmetry of a Fourier transform?

Question

I'm learning about symmetry properties of the discrete fourier transform. Any sequence x[n] can be expressed as: $x[n] = x_e[n] + x_o[n]$

• $x_e[n]$ is a conjugate symmetric component of the sequence ($x_e[n]=x_e^*[-n]$)
• $x_o[n]$ is a conjugate asymmetric component ($x_o[n]=-x_o^*[-n]$)

Each component is obtained using:

• $x_e[n] = \frac{1}{2}(x[n]+x^*[-n])=x_e^*[-n]$
• $x_o[n] = \frac{1}{2}(x[n]-x^*[-n])=-x_o^*[-n]$

I don't understand what the * operator is supposed to show? Is $x^*[n]$ different to $x[n]$, and if so how?

Answer 1

The star simply represent complex conjugate, that is $$ (a + bi)^* = a - bi. $$ Now you can proceed and verify all the identities given in your question.