## yt = 2yt-1 + wt
## so 2yt-1 = yt - wt
## and yt-1 = .5yt - .5wt. 

set.seed(184) 
alpha = .5 
sigma = .5 
par(mfrow=c(1,2)) 
n = 100 
w = rnorm(n,mean=0,sd=sigma)
x = rep(0,n)
x[1] = w[1] 
for(i in 2:n) x[i] = alpha*x[i-1] + w[i] 
plot(1:n,x,type="l") 
y = rep(0,n) 
y[n] = w[1]
for(i in 2:n) y[n+1-i] = alpha*y[n+2-i] - alpha * w[n+2-i]
plot(1:n,y,type="l")  

## Verify that yt = 2yt-1 + wt. 
y[1:3]
w[1:3]
2*y[1]+w[2] 
2*y[2]+w[3] 

## For the periodogram, see the bottom of p172. 
library(astsa) 
x = soi 
n = length(x) 
P = Mod(2*fft(x)/n)^2;  Fr = 0:(n-1)/n
plot(Fr, P, type="o", xlab="frequency", ylab="scaled periodogram")