Postscript version of these notes

The discrete Fourier transform

is periodic with period 1 because all the exponentials have period 1. Moreover,

so that the periodogram satisfies

Thus the periodogram is symmetric around which is called the

Similarly the power spectral density *f*_{X} given by

is periodic with period 1 and satisfies

which is equivalent to

Here I compute the spectra of a few basic processes directly and then indirectly by a more powerful technique.

**Direct from the definition**

**White Noise**:- Since
for all non-zero
*k*we have

**MA(1)**:- For
we have
and
so that

**AR(1)**:- We have
and

**Using filters**

Any mean 0 ARMA(*p*,*q*) process can be rewritten in MA form as

and then the covariance of

Although the covariance simplifies for white noise, let us simply write

Now write the

Finally make the substitution

to see that

or

The function *A* (or )
is called the *frequency response
function* and |*A*|^{2} the *power transfer function*. The jargon *gain*
is sometimes used for |*A*|.

An ARMA(*p*,*q*) process *X* satisfies

so that if

where . At the same time so

where . Hence

For example, in the ARMA(1,1) case
we find (referring to our MA(1) calculation above that
and
so that

1999-10-13