Periodogram Method of Power Spectral Estimation
Richard Hayes
Introduction to Non Parametric Power Spectrum Estimation
Given a set of discrete-time data obtain an estimate of the Power Spectral Density of
the original signal.
amplitude
10
5
0
-5
0
100
200
300
time seconds
400
500
600
amplitude
5
0
Power Spectral Density (dB/Hz)
-5
0
10
20
30
40
0
0.05
0.1
0.15
0.2
50
time seconds
60
70
80
90
100
0.35
0.4
0.45
0.5
30
20
10
0
-10
0.25
0.3
Frequency (Hz)
Figure 1. Methods of Power Spectrum Estimation include Non Parametric and
Parametric methods.
Non parametric Estimates of Power Spectral Density.
Estimate the spectrum directly from the data using either the Autocorrelation of the
data or the periodogram of the data.
Bartlett
Welch

Periodogram Method
Blackman-Tukey Autocorrelation method
We can examine a signal (either random or deterministic) either in the time domain
using the auto-correlation function, or in the frequency domain using the Power
Spectral density function of the signal.
Page 1 of 7
Periodogram Method of Power Spectral Estimation
Richard Hayes
Stochastic Processes.
For a finite set of data taken from a signal x(t), x(n), n=1,2,3,…., we can write the
following expressions:
E x ( n )  
Mean:
1
N
N 1
 x ( n)  
n 0


Variance:
V x ( n )  E x ( n )      2
Autocovariance:
c xx (m)  E [ x(n)   ] [ x(n  m)   ] 
Autocorrelation Function:
rxx (m)  E{x(n) x(n  m)}
rxx ( m) 
2
N
1
 x(n ) x(n  m)
Lt
N  2 N  1 n   N
An estimate of the acf is obtained from N samples
1 N 1
rxx ( m)   x ( n ) x ( n  m)
N n 0
A second version can be expressed as
N  m 1
1
rxx (m) 
x ( n) x ( n  m) 1

N  m  1 n 0
(Eq1)
i
Power Spectral Density is the (Discrete Time) Fourier Transform of the
Autocorrelation function.
Power Spectral Density; Pxx ( ) 

N 1
m  
n 0
 rxx (m)e jm ; Pxx (k )   rxx (n)e
 j 2N nk
(Eq2)
This result is not immediately obvious. We will show it very simply below. This is the
Wiener-Kintchine Theorem. Obviously the autocorrelation function is the inverse
Fourier transform of the PSD.
Autocorrelation Function:
1
rxx (m) 
2

 jm
 Pxx ( )e d ;
rxx ( m) 

1
N
N 1
P
xx
( k )e
j 2N mk
(Eq3)
k 0
This presents one method of determining the PSD: Obtain the acf and take its FT.
1
See program for this form of the acf in footnote at end of document
Page 2 of 7
Periodogram Method of Power Spectral Estimation
Richard Hayes
Relation between The Autocorrelation Function and Power Spectral
Density.
We can write
N
1
rxx ( m)  Lt
x(n ) x(n  m)

N  2 N  1 n   N
N
1
rxx (0)  Lt
x 2 (n )   2

N  2 N  1 n   N
The acf for zero shift gives the variance which is also the mean power of the signal.
But we have from Wiener Kintchine
rxx ( m ) 
1
2

P
xx
( )e  jm d



1
1
rxx (0) 
Pxx ( )e  j 0 d 
Pxx ( )d   2

2  
2 
This tell us that the integral over all frequencies of the function, Pxx(ω), adds up to the
Mean Power of the signal then Pxx(ω) must be the power at each frequency.
Periodogram
The expression for the estimate of the Power Spectral Density obtained from Equation
N 1
1 above, Pxx ( k )   rxx ( n )e
 j 2N nk
, is called the periodogram. It is obtained directly by
n 0
calculating the acf af the data and obtaining the fourier transform of the acf.
Manipulation of Equations 1 and 2 above results in a second expression for the PSD:
2
1 N 1
Power Spectral Density: Pxx ( ) 
 x(n)e jn
N n 0
This gives us a second method of determining the PSD. Obtain the FT of the data
and sqare it.
Statistical Estimation.
We make an ESTIMATE of the PSD based on a SAMPLE of data taken from a
process. This sample is a time series. The assumption of an ergodic stationary
stochastic process should be understood.
Validity of the Estimate.
This is a statistical analysis of the confidence in the result. How good is the estimate.
How close is it to the TRUE spectrum.
The estimate is based on one set of random or noisy data taken from the process.
Other sets would provide different results. Thus the estimate itself is a random
variable. The behaviour is investigated using the MEAN and VARIANCE of the
estimate.
Page 3 of 7
Periodogram Method of Power Spectral Estimation
Richard Hayes
Bias:
It is desirable that the estimate of the PSD approach its true value as the number of
data points approaches .
i.e.
PExx ( )  Pxx ( ) ;
The estimate approaches the ttrue vale
bias  Pxx ( )  PExx ( )
If the mean of the estimate equals the true value of the estimate the the estimate is
unbiased.
Variance:
It is desirable that the variance of the estimate approach zero as the number od data
points used approaches .
Statistical Properties of the Periodogram
1
Power Spectral Density: PExx ( ) 
N
N 1
 x ( n )e
2
 jn
n 0

The Periodogram is unbiased only for large numers of data.

The periodogram is inconsistant ie successive realizations yield fluctuating
estimates and the variance does not tend to zero as N increases.
These two results are expressed more rigorously:
Bias:
 ( )  E P( ) 
m
N
 ( ) .
  sin 2fn
Consistancy: V P( )  P ( ) 1  
  N sin 2f
2
Quality Factor:
QP



2



2

E Pxx ( f )
 xx ( f ) 2


Var Pxx ( f )  xx ( f ) 2
Page 4 of 7
1
Periodogram Method of Power Spectral Estimation
Richard Hayes
The Bartlett and Welch Methods.
Improvements in the consistancy of the periodogram are obtained (at the expense of
resolution in frequency discrimination) by averaging periodograms of either
sequential or overlapped sections of the original data. This is illustrated below.
Bartlett Method.
No overlap, rectangular window on data sections.
M data
M data
M data
M data
N data points
K=N/M Sections
Bias:


E PxxB ( f )   xx ( f ), as N and M  ∞.
Therefore Unbiased asymptotically.
Variance:
  sin( 2fM )  2 
1
1
2
V P ( f )   xx ( f ) 1  
    xx ( f ) 2 (where K  N / M )
K
K
  M sin( 2f )  

B
xx

i.e. variance does not reduce to zero as N,M  ∞.
Quality Factor:
QB 
E Pxx ( f )2

 xx ( f ) 2
K
1
2
 xx ( f )
K
1
 xx ( f ) 2
K
N
 QB  K 
M
The quality factor, QB is K , the number of sections.
Page 5 of 7
Periodogram Method of Power Spectral Estimation
Richard Hayes
Welch Method.
Sectioned data windowed with other than rectangular window, possibly overlapping
data sections.
M data
M data
M data
M data
L=N/(M-ol)
Sections
M data
M data
M data
M data
N data points
Consistancy:
Bartlett:
V PW E ( f )  (9 / 8L) P 2 ( f ) where L  N /( M  ol ) .
Welch:
Welch is less than Bartlett by the factor 9/16 = 0.56.
MATLAB Laboratory Example:
Title. Periodogram method of Power Spectrum Estimation
Investigate Non-Parametric Power Spectrum Estimation using methods based on the
Periodogram.
Objectives:
1. To generate suitable test signal.
2. To obtain an estimate of the spectrum of the signal based on the
Periodogram.
3. To investigate the improvements of the estimate using the Bartlett
and Welsh methods using different section lengths and different
windows.
4. To investigate the frequency resolving characteristics of the
periodogram methods.
Tasks:
a. Generate a 216 element test signal by passing white noise through a
band-pass elliptic filter.
Page 6 of 7
Periodogram Method of Power Spectral Estimation
Richard Hayes
% Design Filter
Wp = [.2 .7], Ws = [.1 .8];
Rp=1,Rs=30;
[N,Wn]=ellipord(Wp,Ws,Rp,Rs);
[b,a]=ellip(N,Rp,Rs,Wn);
[h,w]=freqz(b,a);
plot(w/2/pi,20*log10(abs(h)),'r')
% Generate Data
n=randn(size(t));
y=filter(b,a,n);
b. Calculate the Periodogram of the data by using the MATLAB
function spectrum.
c. Compare the obtained spectrum with the frequency response of the
filter.
d. Adjust the ‘settings’ of the spectrum function to implement the
Bartlett and Welch methods for different section lengths.
i
Programming the autocorrelation function.
A function m-file file directly implementing this equation is listed here
function Rxy=myxcorr(x,y,nlags)
N=length(x);
for m=0:N-1
sum=0;
for n=0:N-m-1
sum=sum+x(n+1)*y(n+1+m)/(N-m+1);
end
Rxy(m+1)=sum;
end
Page 7 of 7