Chapter 3: Spectral analysis of discrete time stationary processs

Question 1

State the spectral representation theorem

Answer 1

There ists an orthogonal process {Z(f)} defined on [-1/2,1/2] such that
Xt = int_-1/2^1/2 {exp(i 2π f t △t) dZ(f)} for all t.

Question 2

By othogonal increments dZ*(f’) dZ(f) = …

Answer 2

d S(I)(f) or S(f) df for f'=f
0 otherwise

Question 3

What are the properties of {Z(f)}?

Answer 3

• expectation: E{dZ(f)} = 0 for all |f|<1/2
• variance: var{dZ(f)} = E{|dZ(f)|^2} = dS(I)(f) = S(f) df for all |f|<1/2
• uncorrelated: cov{dZ(f’),dZ(f)} = E{dZ(f’)dZ(f)} - E{dZ(f’)} E{dZ(f)} = E{dZ*(f’)dZ(f)} for all |f|<1/2

Question 4

Write the Fourier transform pair of S(f) and s_tau

Answer 4

S(f) = sum_τ=-∞ ^∞ sτ exp(- i 2π f τ) 
sτ = int_-1/2^1/2 S(f) exp(i 2π f τ) df = int_-1/2^1/2 exp(i 2π f τ) d S(I)(f)

Question 5

Define the spectral density funcion S(f) and state its 4 properties

Answer 5

S(f) df is the average contribution (over all realizations) to the power from components with frequencies in a small interval about f.

• S(I)(f) = int_-1/2 ^f S(f’)df’
• 0 <= S(I)(f) <= sigma^2, and S(I)(f) increasing, ie. S(f) >= 0
• S(I)(-1/2) = 0, S(I)(f) = sigma^2, int_-1/2^1/2 S(f) df = sigma^2
• f < f’ => S(I)(f) <= S(I)(f’), S(-f)=S(f)

Question 6

What is the Nyquist frequency version of the spectral representation theorem (for discrete and continuous time processes) ?

Answer 6

for discrete {Xt}:
Xt = int_-fn ^fn {exp(-2π f t △t) dZ(f)} for fn=1/(2△t) called the Nyquist frequency
for continuous Xt:
Xt = int_-∞ ^∞ {exp(-2 π f t △t) dZ(f)} for △t → 0

Question 7

Define Linear Time-Invariant digital filter and its spectral density

Answer 7

An LTI is a transformation yt = L{xt} with

• scale-preservation
• superposition
• time-invariance

Question 8

State the 4 classifications of spectra S(I)(f) an their characteristics

Answer 8

• purely continuous
• purely discrete
• mixed: discrete + non-white continuous
• discrete: discrete + white continuous noise

Question 9

What is the sdf of a LTI yt,f = L{ξt,f} where ξt,f = exp(i 2πft) ?

Answer 9

Sy(f) = |G(f)|^2 Sx(f)
where G(f) = sum_u=-∞ ^∞ g(u) exp(- i 2π f u) for |f|<1/2

for AR(p), MA(q) or ARMA(p,q) replace z by exp(- i 2π f) in G(z) and Sx(f) = |G(f)|^2 Sε(f)

Question 10

What are the two methods for aliasing?

Answer 10

• method 1: Sxt (f) discrete and Sx(t) (f) continuous
  Sxt (f) = sum_k=-∞ ^∞ Sx(t) (f+k/△t) for |f|<1/2
• method 2: folding method using graphs (fold in the tails outside of [-fn,fn])