In the development of Hilbert transform relationships, Prof. Oppenheim has chosen \begin{equation} \int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right)\sum_{k=-\infty}^{\infty}\delta(\omega-\theta-2\pi{}k)d\theta =X_R\left(e^{j\omega}\right) \end{equation} in his book "Discrete-Time Signal Processing, 2e" Chapter 11 pp807, Eq. 11-24. I failed to understand how he arrives at it.
Does anybody any idea how has derived, please let me know?
I tried some BUT IT DID NOT GIVE the above relationship:
1:
\begin{align*} \int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right) \sum_{k=-\infty}^{\infty}\delta(\omega-\theta-2\pi{}k)d\theta&=\frac{1}{2\pi}\int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right)2\pi\sum_{k=-\infty}^{\infty} \delta(\omega-\theta-2\pi{}k)d\theta\\& =\frac{1}{2\pi}\int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right)e^{j\theta{}n}d\theta\\& =x_e(n) \end{align*}
OR
2:
\begin{align*} \int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right) \sum_{k=-\infty}^{\infty}\delta(\omega-\theta-2\pi{}k)d\theta&=\sum_{k=-\infty}^{\infty}\int_{-\pi}^{\pi}X_R\left(e^{j\theta}\right) \delta(\omega-\theta-2\pi{}k)d\theta\\& =\sum_{k=-\infty}^{\infty}X_R\left(e^{j(\omega-2\pi{}k)}\right) \end{align*}