2π ≠ 1

Chap 13: Introduction to Spectral Analysis

Spectral analysis has many important uses, e.g. using the residual

series to check on the white noise assumption of the error series.

Its use leads to a much broader alternative than the use of the acf

The spectral density, S(f), is defined by

Notice that S(f) is constant for white noise.

Some basic definitions,

t=1:96; cos1=cos(2*pi*t*4/96); cos2=cos(2*pi*(t*14/96+.3))

2*cos1 + 3*cos 2

The cosine model may be written in linear form,

Historical approximation for time series data,

The Fourier frequencies.

fj = j/n

The periodogram.

Height of periodogram shows relative strength of cosine-sine pairs at

frequencies, f = j/n,, in overall behavior of series, {Yj}.

An identity, a decomposition

Signal plus noise model. random signal case

A, B, W ~ IN(0,1)

Hidden frequencies.

f unknown

look for maxima of periodogram

The Spectral Representation.

If

A j , B j ~ IN(0, σ j 2 )

then

Rewrite

Cp. representing discrete r.v. by cdf.

The sample spectral density.

Inverse relation

The spectral density.

An Example