DFT for Time Series

Question 1

Let Xo,…,Xn-1 be a time series

Xo^?

Answer 1

Xo^ = 1/√n Σ Xt e^(-2πi0t)
= 1/√n Σ Xt
= √n * Xt~
-sum from t=0 to t=n-1 and Xt~ is the sample mean

Question 2

Let Xo,…,Xn-1 be a time series

Var(X)?

Answer 2

Var(X) = 1/n Σ |Xt|²
-sum from t=0 to t=n-1
-by conservation of energy:
Var(x) = 1/n Σ |Xj^|²
-sum from j=0 to j=n-1
-by definition:
Var(X) = 1/n Σ I(fj)
≈ ∫ I(fj) df
-integral between 0 and 1
-so variance of X is the integral of the spectral density over all frequencies

Question 3

Let Xo,…,Xn-1 be a time series

Variance contributed by range [a,b]?

Answer 3

1/n Σ I(fj) ≈ ∫ I(fj) df

• sum over fj in interval [a,b]
• integrate from a to b
• this quantifies the variance contributed by the frequency range [a,b]

Question 4

Spectral Density in Terms of Sample Autocovariance

General Case

Answer 4

-if gk is the lag-k ample autocovariance of X then:

I(fj) = √n * gk

Question 5

Spectral Density in Terms of Sample Autocovariance

Periodic X

Answer 5

I(fj) = go + 2Σgkcos(2πfj*k)
-sum k between 1 and n/2
-if n is even there is an additional term:
+ gn/2 * (-1)^j

Question 6

Power Spectrum

Answer 6

• the spectral density I is an estimate for the power 𝐼(f) = γ0 + 2 Σ γk*cos(2πfk)
• sum from k= 1 to k=∞

Question 7

White Noise

Spectral Density

Answer 7

-let Xt ~ N(0,σ²) i.i.d., then:
Xj^ = 1/√n Σ Xte^(-2πifj*t)
-this can be written as a sum of real and imaginary parts
-so Xj^ is (complex) normally distributed with:
E[Re(Xj^)] = 0
E[Im(Xj^)] = 0

Question 8

White Noise

Var[Re(Xj^)]

Answer 8

Var[Re(Xj^)] = σ² if j is 0 or n/2 OR σ²/2 otherwise

Question 9

White Noise

Var[Im(Xj^)]

Answer 9

Var[Im(Xj^)] = 0 if j=0 or n/2 OR σ²/2 otherwise

Question 10

White Noise

Cov[Re(Xj^),Im(Xj^)]

Answer 10

Cov[Re(Xj^),Im(Xj^)] = 0

Question 11

White Noise

Correlation

Answer 11

Xj^ and Xk^ are uncorrelated for j≠k

Question 12

White Noise

E[I(fj)]

Answer 12

E[I(fj)] = σ²

Question 13

AR(p)

DFT of Xt

Answer 13

Xj^ = εj^ / [1 - Σ αke^(-2πifj*k)]

-sum from k=1 to k=p

Question 14

AR(p)

Spectral Density

Answer 14

I(fj) = |Xj^|²
= |εj^|² / |1 - Σ αke^(-2πifj*k)|²
-sum from k=1 to k=p

Question 15

AR(p)

E[I(fj)]

Answer 15

E[I(fj)] = σε² / |1 - Σ αke^(-2πifj*k)|²

-sum from k=1 to k=p