Spectral Analysis - Signal Processing of Random Physiological Signals - page 172

5. Smoothing : To obtain a fairly good estimator, statisticians would insist on some

“Smoothing” function, since the Spectral Density Function or Periodogram
is a raw power spectrum estimate that contains high-frequency components.
Smoothing is a technique equivalent to low-pass filtering to eliminate the high-
frequency components from the spectrum for the final smoothed spectral density
estimate.

15.6 ADVANCED TOPIC
15.6.1 Brief Description of Autoregressive Spectral Estimation
The concept of smoothing serves as a transition to the Autoregressive (AR) Spectral
Estimation Method. The AR Spectral Estimator, also known as the Maximum Entropy
Spectral Estimator, is becoming more popular in the engineering community as an
accurate estimator of the spectral density function.
To perform the AR Spectral Estimation, it is necessary to do the following pro-
cedural steps in order.

1. Fit AR model to data

2. Calculate AR coefficients

3. Enter the AR coefficients into formula for spectrum of AR process.

4. Estimate spectrum of AR process

The AR model of order p is represented by (15.24):

= α + α + α +····+ α + (15.24)

x t 1 x t −1 2 x t −2 p x t − p z t 0

The order of the model is chosen using Akaike's Information Criterion (AIC).
The coefficients are then calculated from (15.25).

N −1
t 1 ( x t

− x )( x t −1 − x )

=

α ˆ = (15.25)

N −1
t

k

1 ( x t −

x ) 2

=

where x is the mean of the sample record, N is the total number of data points, x ( t )is
the point at time t , and ˆ α k is the approximate estimator of α .