This document discusses non-parametric power spectrum estimation methods. It introduces applications of power spectrum estimation such as feature extraction and Wiener filtering. The periodogram method is described as directly calculating the discrete Fourier transform of the data, but it has high variance. Improved methods average multiple periodograms, such as Welch's method, or apply windows to the data or periodograms, like Bartlett's and Blackman-Turkey methods. These methods trade off bias and variance to provide better resolution and convergence to the true power spectrum as the data length increases.

1. Non-Parametric
Power Spectrum Estimation
Methods
Eric Hui
SYDE 770 Course Project
November 28, 2002
1

2. Introduction

Applications of Power Spectrum Estimation
(PSE):
Wiener Filter
 Feature Extraction


Non-parametric PSE does NOT assume any
data-generating process or model (e.g.
autoregressive model).
2

3. Motivation

Ideal autocorrelation:
N
 1

rx (k ) = lim 
x ( n + k ) x ( n) 
∑N
N →∞ 2 N + 1
n=−



Actual autocorrelation:
1
ˆ
rx (k ) =
N

N −1− k
∑ x ( n + k ) x ( n)
n =0
Limited (finite length of) data due to:


Availability of data
Assumption of stationary
3

4. Periodogram Method
x(n)
1
ˆ
rx (k ) =
N
N −1− k
∑ x ( n + k ) x ( n)
n =0
1 ∞
=
∑ x N ( n + k ) x N ( n)
N n = −∞
1
= x N (k ) ∗ x N (−k )
N
0
N
n
xN(n)
DTFT
0
N
1
jω 2
Pper (e ) =
X N (e )
N
jω
4
n
redefined
as

5. Periodogram Method
1

ˆx (k )} = E  ∑ x(n + k ) x(n)
E{ r
 N n =0

1 N −1− k
=
∑ E{ x(n + k ) x(n)}
N n =0
w∆ (k ) =
N −1− k
N −1− k
∑ rx (k ) =
n =0
N −k
rx (k )
N
0
DTFT
{
}
N
k
DTFT
1
=
N
-N
N −k
N
1  sin( 1 Nω ) 
2
W∆ (e ) = 
 sin( 1 ω ) 

N
2

jω
2
1 1  sin( 1 Nω ) 
jω
2
ˆ

 ∗ Px (e jω )
E Pper (e ) =
2π N  sin( 1 ω ) 
2


5
0
k
2

6. “Good” Method?

Necessary conditions for mean-square
convergence:

Asymptotically Unbiased
ˆ
lim E P (e jw ) = P (e jw )
N →∞
{
}
PSD
PSD
as N ↑
k

Zero Variance
ˆ
lim Var P (e jw ) = 0
N →∞
{
}
PSD
k
PSD
as N ↑
k
6
k

7. Evaluation of Methods

Resolution

How much “blurring” effect is there on the power
spectrum?
PSD

Bias (Asymptotic)

k
Does the estimation approach the true value with
more data (i.e. as N increases)?
as N ↑
PSD

PSD
Variance

k
k
Does the amount of deviation from the true value
depend on the data length (i.e. N)?
as N ↑
PSD
PSD
k
7
k

8. Different PSE Methods

Periodogram Method

W∆ (e jω ) =
Average the Periodogram estimate of non-overlapping
sub-intervals of x(n).
1  sin( 1 Nω ) 
2


N  sin( 1 ω ) 
2


Average the Modified Periodogram estimate of overlapping
PSD
sub-intervals of x(n).
as N ↑
2
k
0
Welch’s Method


Apply non-rectangular window to x(n) to get xN(n).
Bartlett’s Method


Apply rectangular window to x(n) to get xN(n).
N k
0
Modified Periodogram Method


-N
N −k
N
DTFT

w∆ (k ) =
PSD
Blackman-Turkey Method

Apply non-triangular window to r(x).
k
8
k

9. Application: Feature Extraction
4
2.5
x 10
2
PSD
1.5
1
0.5
0
0.1
linearize
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Linearized PSD Slope (Horizontal)
50
0
-50
repeat for whole image
-100
-150
-200
-250
0
0.1
9

10. Questions or Comments?

…
10