# We define the even and odd parts of a sequence (x(n), mathcal{E}{x(n)}) and (mathcal{O}{x(n)}) respectively, as [begin{aligned}mathcal{E}{x(n)}

## Question:

We define the even and odd parts of a sequence \(x(n), \mathcal{E}\{x(n)\}\) and \(\mathcal{O}\{x(n)\}\) respectively, as

\[\begin{aligned}\mathcal{E}\{x(n)\} & =\frac{x(n)+x(-n)}{2} \\\mathcal{O}\{x(n)\} & =\frac{x(n)-x(-n)}{2}\end{aligned}\]

Show that

\[\sum_{n=-\infty}^{\infty} x^{2}(n)=\sum_{n=-\infty}^{\infty} \mathcal{E}\{x(n)\}^{2}+\sum_{n=-\infty}^{\infty} \mathcal{O}\{x(n)\}^{2}\]

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**Related Book For**

## Digital Signal Processing System Analysis And Design

**ISBN:** 9780521887755

2nd Edition

**Authors:** Paulo S. R. Diniz, Eduardo A. B. Da Silva , Sergio L. Netto