Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 36

for i =0 , 1 ,...,K 1 and w ( n ) is a window function. The normaliza-

tion factor U is computed via

−

M− 1

1
M

w ( n ) 2 U = (2.81)

n =0

3. Finally, compute the Welch power density spectrum as the average of

the K modified periodograms as follows:

K

P xx ( f )= 1
K

P xx ( f )

(2.82)

i =1

The introduction of the window function results in the variance of the power
spectrum density being dependant on the type of window used. In the Black-
man and Tukey approach, this problem is addressed differently by first win-
dowing the sample autocorrelation sequence and then obtaining the power
spectrum from the Fourier transform. The Blackman-Tukey power spectrum
estimate is given by

M − 1

P BT r xx ( m ) w ( n )e −j 2 πfm

xx ( f )=

(2.83)

i = − ( M− 1)

where the window function is 2 M − 1 in length and is zero for the case

| M .

It is has been shown that the Welch and Blackman-Tukey power spectrum
estimates are more accurate than the Bartlett estimate (Proakis and Dimitris,
1996). Computationally, the Bartlett method requires the fewest multiplica-
tions although the Welch method requires the most. However, the difference
in accuracy, quality, and computational performance is very small such that
any method may be selected depending on the application requirements.

m |≥

2.4.3 Parametric Estimation Models
The nonparametric methods described previously still suffer from problems of
leakage due to the window functions used and the assumption that the auto-
correlation is zero for values of signal samples outside the signal length N .
One way to improve the power spectrum estimates is to assume a model distri-
bution of the generating data and extrapolate the values of the autocorrelation
for values of the signal outside the length N . The model used is based on the
output of a linear system having the form

k =0 b k z −k

q

H ( z )= B ( z )
A ( z )

= (2.84)

k =1

p

1+ a k z −k