STAT 804: Lecture 22 Notes

Properties of the Periodogram

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 Nyquist or folding frequency. (The value is always 1/2 in cycles per point but usually it would be converted to cycles per time unit like year or day or whatever.)

Similarly the power spectral density given by

is periodic with period 1 and satisfies

which is equivalent to

Spectra of Some Basic Processes

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 X is

Although the covariance simplifies for white noise, let us simply write for the covariance in this double sum so that the calculation will apply to any stationary . Then plug this double sum into the definition of to get

Now write the h in the complex exponential in the form and bring the sum over h to the inside to get

Finally make the substitution k=h+s-r in the inside sum and define

to see that

or

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

The Spectrum of an ARMA(p,q)

An ARMA(p,q) process X satisfies

so that if Y is the process then

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