Derive DTFT of $x[2n]$

Question

If the DTFT of discrete sequence $x[n]$ is $X(e^{j\omega})$, what is the DTFT of $g[n] = x[2n]$?

I see the textbook answer is

\begin{align*} G(e^{j\omega}) &= \frac{1}{2} \left( X(e^{j\omega/2}) + X(e^{j(\omega-2\pi)/2}) \right) \end{align*}

My start of the problem:

\begin{align*} X(e^{j\omega}) &= \sum\limits_{n=-\infty}^\infty x[n] e^{-j\omega n} \\ G(e^{j\omega}) &= \sum\limits_{n=-\infty}^\infty g[n] e^{-j\omega n} \\ G(e^{j\omega}) &= \sum\limits_{n=-\infty}^\infty x[2n] e^{-j\omega n} \\ \end{align*}

How do I derive the given textbook answer?

UPDATE:

I try the obvious $m=2n$ substitution, however I don't see how you can change the variable on the summation so that it counts by 2?

\begin{align*} G(e^{j\omega}) &= \sum\limits_{n=-\infty}^\infty x[m] e^{-j\omega m/2} \\ \end{align*}

Answer 1

HINT:

If you have a sequence

$$\hat{x}[n]=\begin{cases}x[n],&n\text{ even}\\0,&n\text{ odd}\end{cases}$$

then the DTFT of $x[2n]$ can be written as

$$\sum_{n=-\infty}^{\infty}x[2n]e^{-jn\omega}=\sum_{n=-\infty}^{\infty}\hat{x}[n]e^{-jn\omega/2}=\hat{X}(e^{j\omega/2})\tag{1}$$

You can obtain $\hat{x}[n]$ as

$$\hat{x}[n]=\frac12\left(x[n]+(-1)^nx[n]\right)\tag{2}$$

Taking the DTFT of $(2)$ to express $\hat{X}(e^{j\omega})$ in terms of $X(e^{j\omega})$, and using $(1)$ will give the desired result.