#########################################
# R Function for Taking a PPS Sample    #
#                                       #
# Randomized Systematic PPS Method      #
#                                       #
# Input: x=(x_1,x_2,...,x_N)            #
#        n: sample size                 #
# Output: s, the set of sampled units   #
#                                       #
# Note: Require x_i>0 and x_i/X < 1/n   #
#       where X= sum_1^N x_i            #         
#                                       #
# Written by Changbao Wu, July, 2006    #
#########################################

syspps=function(x,n){ 
## 
##Population is first randomized!
##
N=length(x)
U=sample(N,N)
xx=x[U]
z=rep(0,N)
for(i in 1:N) z[i]=n*sum(xx[1:i])/sum(x)
r=runif(1)
s=numeric()
for(i in 1:N){
if(z[i]>=r){
  s=c(s,U[i])
  r=r+1
           }
             }
return(s[order(s)])
       }