Problem: Consider a wide sense stationary stochastic process $X(t)$, with zero mean and auto correlation function $R_X(\tau)$. Consider its transformation by a linear time invariant derivative filter, which is the first derivative of $X(t)$, that is $Y(t)=X'(t)=\dfrac{\mathrm dX(t)}{\mathrm dt}$. The frequency response function of the associated linear invariant system in time is given by $H(w)=iw$($i$ is the imaginary unit).
Using the relation between the spectral density function of the initial process and the derivative process, obtain the expression that allows to determine the auto correlation function of this derivative process as function of the auto correlation function of the initial process $X(t)$.Comment.
$$R_y(t+\tau,t)=\int_{-\infty}^{\infty}h(\lambda)E(X(t+\tau-\lambda)X(t))d\lambda=\int_{-\infty}^{\infty}h(\lambda)R_{XY}(t+\tau-\lambda,t)d\lambda$$
where $h(\lambda)$ is the inverse Fourier transform of $H(w)$.
$\mathscr{F}^{-1}(iw)=-\sqrt{2\pi}\delta(t)$
Then $$R_y(t+\tau,t)=-\sqrt{2\pi}\int_{-\infty}^{\infty}\delta(\lambda)E(X(t+\tau-\lambda)X(t))d\lambda=-\sqrt{2\pi} \int_{-\infty}^{\infty}\delta(\lambda)\int_{-\infty}^{\infty}X(t+\tau-\lambda)X(t)dt d\lambda=-\sqrt{2\pi} \int_{-\infty}^{\infty}\delta(\lambda)(X(t-\tau-\lambda)X(t)|_{-\infty}^{\infty}-\int_{-\infty}^{\infty}X'(t+\tau-\lambda)X'(t)) dt d\lambda=-\sqrt{2\pi} \int_{-\infty}^{\infty}\delta(\lambda)(X(t-\tau-\lambda)X(t)-R_y(\tau-\lambda))dtd\lambda=-\sqrt{2\pi}\int_{-\infty}^{\infty}R_Y(\tau-\lambda)\delta(\lambda)d\lambda=\sqrt{2\pi}R_Y(0)$$
Guide for computations above:
1) I used the definition of expected value as an integral
2)Computed integral by parts with respect to the variable t.
3)Replaced the integral of with respect to t recognizing it was the autocorralation function of $Y(t)=X'(t)$.
4) Used the fact X(t) is in $L_1(\lambda)$ so that $\lim_{t\to \infty}X^(t)=0$ and used the concept of wide-sense stationarity to get $R_Y(\tau-\lambda)$
5) Computed the integral with respect to the measure delta dirac.
Questions:
1)Is my approach correct?
2) I do not know if I am answering the question, since I did not establish a relationship between X(t) and Y(t) autocorrelation functions.
3) Did I prove that the autocorrelataion function of $Y(t)$ is constant and equal to $\sqrt{2\pi}R_Y(0)$?
Thanks in advnace!
Let $S_X(\omega)$ be the power spectral density of the random process $X$, which we know is obtained as the Fourier transform of the autocorrelation of $X$: $$S_X(\omega) = {\cal F}\big[ R_X(\tau)\big]$$
As $H(\omega)$ is an LTI process, we have that: $$ S_Y(\omega) = |H(\omega)|^2S_X(\omega)$$
Therefore, $$ S_Y(\omega) = |i\omega|^2S_X(\omega) = \omega^2S_X(\omega)$$
And $$\begin{align} R_Y(\tau) &= {\cal F}^{-1}\big[\omega^2S_X(\omega)\big] = {\cal F}^{-1}\big[\omega^2\big]*R_X(\tau)\\ &=-\sqrt{2\pi}\bigg(\frac{{\rm d}^2}{{\rm d}\tau^2}\delta(\tau)\bigg)*R_X(\tau) \end{align}$$
Now, knowing that $f*g^{(n)}=f^{(n)}*g$ (proof here), we obtain $$\begin{align} R_Y(\tau) &= -\sqrt{2\pi}\frac{{\rm d}^2}{{\rm d}\tau^2}R_X(\tau) \end{align}$$