Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 35

frequency resolution, and leakage. Furthermore, the periodogram is not a con-
sistent estimate of the power density spectrum. In view of this, better esti-
mation techniques such as nonparametric and parametric estimation models
were devised to circumvent the problems.

2.4.2 Nonparametric Estimation Models
The classical nonparametric methods that have been popularly used for esti-
mation of power density spectrum include the Bartlett, Welch, and Blackman
and Tukey methods. These methods do not make any assumption on the
distribution of the digital signal and hence are nonparametric.
The Bartlett method reduces the variance observed in the periodogram by
averaging the periodograms. This is achieved in three steps:

1. Let the digital sequence x ( n ) have N samples and divide them into

K nonoverlapping uniform segments, each of length M .

2. For each segment i , compute its periodogram using

x i ( n )e −j 2 πfn

2

M − 1

1
M

P xx ( f )= (2.77)

n =0

3. Finally, compute the Bartlett power density spectrum as the average

of the K periodograms as follows:

K

P xx ( f )= 1 P xx ( f ) (2.78)

K

i =1

It is known that the variance of the power spectrum estimate using the
Bartlett method is reduced by a factor of K . Bartlett's averaging method
was later modified by Welch, who allowed the K segments to overlap and
computed a modified periodogram, which used just a selected part of the
segment. The Welch method can be summarized as follows:

1. Let the digital sequence x ( n ) have N samples. Divide the N samples

into K segments each of length M defined as follows:

x i ( n )= x ( n + iO ) (2.79)

where n =0 , 1 ,...,M 1. Here iO is an offset

indicating the starting point of the ith sequence.

− 1 and i =0 , 1 ,...,K −

2. For each segment i , compute a modified periodogram using

x i ( n ) w ( n )e −j 2 πfn

2

M− 1

1
MU

P xx ( f )=

(2.80)

n =0