$2)$ A causal filter H$($z$)$ is excited by x$($n$)$ which is a white noise signal of zero mean and unit variance. Its overall output is y$($n$).$ The filter input$-$output difference equation is given by$.(15$ pts$)$ y$($n$)=$ x$($n$)0\mathrm{.}8$ y$($n$-1)2$a$)$ Calculate the mean value $\backslash$mu y of the overall output y$($n$).2$b$)$ Calculate the autocorrelation ryy$($m$)$ of the output of H$1($z$),$ namely y$($n$),$ in closed form. Show all your work to obtain ryy$($m$).2$c$)$ Give numerical values for ryy $(0),$ ryy$(1),$ ryy$(2).2$d$)$ Calculate the variance of y$($n$).$ Provide a numerical value. $2$e$)$ Give the power spectral density $($PSD$)$ of y$($n$).$

$3)$We need to design a digital Butterworth filter by analog filter approximation. The filter is to be designed using the bilinear transformation. Determine the order of the filter and the s$-$domain poles. The design has the following specifications: Sampling frequency $70$ kHz$,$ passband edge $=7$ kHz$,$ stopband edge $=21$ kHz$,$ gain at passband edge $=-2$ dB$,$ and gain at stopband edge $=-14$ dB$.3$a$)$ Calculate the order of the Butterworth filter. Show you calculations. $3$b$)$ Calculate the s$-$domain poles of the Butterworth filter. Show you calculations. $($dont use any code to solve these problems$)$