The power spectrum P,(u) is defined as the squared 
amplitude spectrum 
Poux dx». 
The spectral representation of an ARMA process can easily 
be drived by evaluating the z-transformation on the complex 
unit circle |z| = 1 where z - exp(j2Iu) /10/. An estimate 
of the power spectrum of an ARMA process is obtained from 
q 
ae ra 
k=1 
u) = S 
jac +102 
k=} 
b, exp (-j2Iku) |? 
P 
  
arma 
a, exp (-j2IHku) |? 
k 
3.3 Autocorrelation Function 
The autocorrelation function R,(k) often provides the basis 
for analysing stationary random processes. The power 
spectrum and the autocorrelation function of a time series 
x(t) are related by the Wiener-Khinchin theorem. It states 
that the power spectrum P (u) is the Fourier transform of 
the autocorrelation function R_(k) 
Pu) = F { R,(k} } 
The parameters of ARMA processes are connected with the 
autocorrelation function by the Yule-Walker equations. For 
an AR(p) process they are given by 
- » a, 
AR(p) : R. (k)- ii 
"i a,  Ra(-1) + oR for k = 0 
p 
R_(k-1) for k > 0 
The extended Yule-Walker equations for ARMA processes are: 
p 
ARMA(p,q): R_(k) = « D a 
|=1 
q 
1 R,(k-1) + E b, R,, (k-1) 
1 
where R,, (k) - E (e(t):x(t-k)) for k x Q0 
R k) = 0 fork »-0 
ex( 
4. Parameter Estimation 
A time series x(t) may be described as an ARMA process of 
known order p and q 
P q 
x(t).- -:Z a, x(t-i1) tie(t) 4 I b,e(t-j) 
iz] j= 
In practical application it is a major problem to estimate 
the parameters a, and b,. Due to the non-linear relations 
of the unknown parameters b, with the unknown innovations 
e(t), the parameter estimation of the ARMA and MA models 
requires more effort than the AR model. For this reason the 
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