Given:
(1) Complex Symmetric Sequence has the property:
$$x[n]=x^{*}[-n]$$
What’s the general formula for the Complex Symmetric Component of the sequence x[n], in terms of x[n] and x*[n]?
I'm guessing it's this:
$$x_{complex\ symetric}[n] =\frac{1}{2}\left(x\left[n\right]+x^*\left[-n\right]\right)$$
How to prove this equation is true or false?
A Conjugate Symmetric Sequence (CSS) is defined as:
$$x[n] = x^{*}[-n]$$
A Anti-Conjugate Symmetric Sequence (CAS) is defined as:
$$x[n] = - x^{*}[-n]$$
It can be shown that any sequence can be written as the sum of an CSS sequence and an CAS sequence as follows:
$$x[n] = x_{css}[n] + x_{cas}[n] $$
$$x[n] = (0.5)(x[n] + X^{*}[-n]) + (0.5)(x[n] - X^{*}[-n])$$
$$x[n] = x[n] $$
we can see that CSS Sequence is equal to:
$$x_{css}[n] = (0.5) (x[n] + x^{*}[n])$$