1.Let x[n] be a random sequence generated by x=2*rand-1 and y[n]=x[n]-x[n-1]. What are the expected number of sign changes in y? What are the acf coefficients at lag 0, 1, 2, 3? Verify your analysis in Matlab. (A fairly challenging problem is to figure out what is the probability of turning points in y. Simulation shows that it’s about 73%~11/15).
2.Let x[n] be a standard normal random variable with a length of N (=10,000) and y=(x[i-1]+x[i])/2. Find the correlation and correlation coefficient in theory for y at lag 0, 1, 2, 3. Verify your results in Matlab.