- #1
Dustinsfl
- 2,217
- 5
I am integrating the characteristic equation in order to recover the PMF, but I am going to get the answer to be zero so something went wrong.
\begin{align*}
p_X[k]
&= \frac{1}{2\pi}\int_{-\pi}^{\pi}\phi_X(\omega)e^{-i\omega k}d\omega\\
&= \frac{1}{2\pi}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\\
&= \frac{i}{2\pi k}\cos(\omega)e^{-i\omega k}\bigg|_{-\pi}^{\pi} +
\frac{i}{2\pi k}\int_{-\pi}^{\pi}\sin(\omega)e^{-i\omega k}d\omega\\
&= \frac{i}{2\pi k}(e^{i\pi k} - e^{-i\pi k}) + \frac{i}{2\pi k}\Bigg[
\frac{i}{k}\sin(\omega)e^{-i\omega k}\bigg|_{-\pi}^{\pi} -
\frac{i}{k}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\Bigg]\\
&= \frac{-1}{\pi k}\bigg(\frac{e^{i\pi k} - e^{-i\pi k}}{2i}\bigg) +
\frac{1}{2\pi k^2}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\\
\bigg(\frac{k^2 - 1}{2\pi k^2}\bigg)
\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega
&= \frac{-\sin(\pi k)}{\pi k}\\
\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega
&= \frac{-2k\sin(\pi k)}{k^2 - 1}
\end{align*}
Since k is an integer, the RHS is zero.
Additionally, the \(p_X[k]\) is supposed to be defined for all integers, but if I say \(p_X[k] = \frac{-2k\sin(\pi k)}{k^2 - 1}\), k cannot be -1 or 1. This definition of the PMF would be weird anyways since \(\sin(\pi k) = 0\) anyways.
What went wrong?
\begin{align*}
p_X[k]
&= \frac{1}{2\pi}\int_{-\pi}^{\pi}\phi_X(\omega)e^{-i\omega k}d\omega\\
&= \frac{1}{2\pi}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\\
&= \frac{i}{2\pi k}\cos(\omega)e^{-i\omega k}\bigg|_{-\pi}^{\pi} +
\frac{i}{2\pi k}\int_{-\pi}^{\pi}\sin(\omega)e^{-i\omega k}d\omega\\
&= \frac{i}{2\pi k}(e^{i\pi k} - e^{-i\pi k}) + \frac{i}{2\pi k}\Bigg[
\frac{i}{k}\sin(\omega)e^{-i\omega k}\bigg|_{-\pi}^{\pi} -
\frac{i}{k}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\Bigg]\\
&= \frac{-1}{\pi k}\bigg(\frac{e^{i\pi k} - e^{-i\pi k}}{2i}\bigg) +
\frac{1}{2\pi k^2}\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega\\
\bigg(\frac{k^2 - 1}{2\pi k^2}\bigg)
\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega
&= \frac{-\sin(\pi k)}{\pi k}\\
\int_{-\pi}^{\pi}\cos(\omega)e^{-i\omega k}d\omega
&= \frac{-2k\sin(\pi k)}{k^2 - 1}
\end{align*}
Since k is an integer, the RHS is zero.
Additionally, the \(p_X[k]\) is supposed to be defined for all integers, but if I say \(p_X[k] = \frac{-2k\sin(\pi k)}{k^2 - 1}\), k cannot be -1 or 1. This definition of the PMF would be weird anyways since \(\sin(\pi k) = 0\) anyways.
What went wrong?
Last edited: