tutorial (tut8) for data analysis I added

tuturials/gw_tutorials.tex

@@ -197,6 +197,27 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 \end{enumerate}
 
 \subsection{Tutorial 8: GW data analysis}
+\begin{enumerate}
+	\item Show that, for stationary random processes, power spectral density is the
+	Fourier transform of the autocorrelation function. 
+	\item  Show that, for stationary random processes 
+	$$\left<\tilde{x}^{*}(f^{'})\tilde{x}(f)\right>=\frac{1}{2}S_x(f)\delta(f-f^{'})$$.
+	\item  Noise time series $x(t)$ sampled at regular intervals $t_j$ to produce $N$ samples $x_j=x(t_j)$.  
+	Let's consider this to be a  Gaussian noise (with zero mean and variance $\sigma^2$) i.e the probability distribution of $\{x_j\}$ is 
+	$$p_x\left(\{x_j\}\right)= \left(\frac{1}{\sqrt{2\pi\sigma}}\right)^N \mathrm{exp}\left\lbrace-\frac{1}{2\sigma^2}\sum_{j=0}^{N-1} x_j^2\right\rbrace$$. Derive the continuum limit of  $p_x\left(\{x_j\}\right)$, i.e show that 
+	$$p_x\left[x(t)\right]\propto\mathrm{exp}\left\{\frac{1}{2}4\int_0^{\infty}\frac{|\tilde{x}(f)|^2}{S_x} df\right\} $$.
+	\item Now consider colored Gaussian noise $y(t) = \int K(t-t^{\prime})x(t^{\prime}) dt^{\prime}$. Show that 
+	$$p_y\left[y(t)\right]\propto\mathrm{exp}\left\{\frac{1}{2}4\int_0^{\infty}\frac{|\tilde{y}(f)|^2}{S_y} df\right\} $$, where $\tilde{y}(f) = \tilde{K}(f)\tilde{x}(f)$
+	
+	\item Assume a signal of the form $h(t) = A g(t)$. Find the value of A that maximizes the matched filter inner product.
+	\item Assume a signal of the form $g(t) = cos [\phi + \phi_0]$. Find the value of $\phi_0$
+	that maximizes the matched filter inner product. 
+	\item Calculate the mean and variance of the matched filter statistic in the absence and presence of a signal of the form $h(t) = A g(t)$.
+\end{enumerate}
+
+
+
+\subsection{Tutorial 9: GW data analysis}
 
 In the case a known signal $h(t)$ buried in stationary Gaussian noise, the optimal technique for 
 signal extraction is the \emph{matched filtering}, which is a noise-weighted correlation of the data