Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 37

which for most intentions and purposes can be viewed as a linear filter with
a finite number of poles and zeroes in the z -plane. The inverse z -transform
provides the resulting difference equation:

q p

x ( n )= b k w ( n − k ) − a k x ( n − k ) (2.85)

k =0 k =1

which is a linear model of the signal x ( n ). Under the assumption that the
sequence w ( n ) is zero-mean white noise with autocorrelation

2 ] δ ( m )

γ ww ( m )= E [ | w ( n ) | (2.86)

the power density spectrum of the observed data is

2

2 ] |

B ( f ) |

Γ xx ( f )= E [ | w ( n ) | (2.87)

| A ( f ) |

2

To compute this, we first require an estimate of the model parameters { a k }

and , given N samples, and then use Equation 2.86 to obtain the power

spectrum estimate.
There are several methods for computing the model parameters beginning
with the autoregressive moving average (ARMA) model of order ( p, q ). If one
sets q = 0 and b 0 = 1, the process reduces to an autoregressive (AR) process
of order p . Alternatively, if we set A ( z ) = 1 and have B ( z )= H ( z ), the output
x ( n ) is referred to as a moving average (MA) process of order q . The AR
model by far is the most popular of the three, mainly because it is useful for
representing spectra with narrow peaks, and it is easily implemented requiring
only the solution of linear equations. For sinusoidal signals, the spectral peaks
in the AR estimate are proportional to the square of the signal power.

{ b k }

2.4.3.1 Autoregressive Model
The AR model parameters can be estimated using several methods such as
the Yule-Walker, Burg (1968), covariance, and modified covariance methods.
These have been implemented in many software packages, for example, MAT-
LAB ∗ Signal Processing Toolbox and are now easily accessible. The Yule-
Walker method uses a biased form of the autocorrelation estimate to ensure a
positive semidefinite autocorrelation matrix. The Burg method on the other
hand is a form of order-recursive least squares lattice method that estimates
parameters by minimizing forward and backward errors of the linear system.
The key advantages of the Burg method are that it provides high-frequency
resolution, gives a stable AR model, and is computationally ecient. There
are however several disadvantages such as spectral line splitting at high signal
to noise ratios and the introduction of spurious peaks at high model orders.

∗ See http://www.mathworks.com/products/matlab/