# Here we generate a causal AR(2) where the roots are complex, thus giving rise to 
# pseudo periods

# Since it is causal, lambda must be less than one (this ensures the roots of the characteristic function lie
# outside the unit circle). 
# omega must lie between 0 and pi

ar2complex = function(lambda,omega,n1){

       n = (n1+200)

         gen = rnorm(n)

            x = rep(0,n)
                  
         for(j in c(3:n)){

        x[j] = (2*lambda*cos(omega)*x[j-1] - (lambda**2)*x[j-2] + gen[j])

         }

         x1 = x[-c(1:200)]  
           
         return(x1)

     }

# Experiment with keeping omega fixed but letting lambda get closer to one.

# test

test = ar2complex(lambda=0.8,omega = pi/3,n1=200)

acf(test)

# To plot the periodogram and corresponding spectral density

Periodogram <- abs(fft(test)/200)**2
frequency = c(0:199)/200
plot(frequency, Periodogram,type="o")

library("astsa")

lambda = 0.8
omega = pi/3

# note that on the plot below 0.5 corresponds to pi

par(mfrow=c(2,1))

arma.spec( ar = c(2*lambda*cos(omega), -lambda**2), log = "no", main = "Autoregressive")
plot(frequency[c(1:100)], Periodogram[c(1:100)],type="o")