SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSIS - Design and Development of Medical Electronic Instrumentation - page 239

High-Resolution Methods

The main limitation of FFT-based methods is restricted spectral resolution. The highest
inherent spectral resolution (in hertz) possible with the FFT is approximately equal to the
reciprocal of the time interval (in seconds) over which data for the FFT are acquired. This
limitation, which is further complicated by leakage and the picket-fence e

ff ect, is most

noticeable when analyzing short data records.
It is important to note that short data records not only result because of the lack of data,
such as when sampling a short transient at a rate barely enough to satisfy Nyquist's criterion,
but also from data sampled from a process that varies slowly with time. Although there are
many applications in the medical

field. By ana-
lyzing the vibrations picked up from an oil-well drill, the operator can monitor the buildup
of resonance in the long pipe that carries torque to the drill bit, avoiding costly damages to
the equipment [Jangi and Jain, 1991]. Although a continuous signal from the vibration trans-
ducers is available for sampling, the vibrations on the drill assembly change rapidly, result-
ing in a limited number of data samples which represent each state of the drill bit. It is here
that high-resolution estimates would be desirable, even though the data available are limited.
A number of high-resolution spectral estimators have been proposed. These alternative
methods do not assume, as the FFT does, that the signal outside the observation window
is merely a periodic replica of what is observed through the window. For example, one of
these methods, the parametric estimator, relies on the selection of a model that suitably
represents the process that generates the signal in order to capture the true characteristics
of data outside the window. By determining the model's parameters, the theoretical PSD
implied by the model can be calculated and should represent the signal's PSD.
Many signals encountered in real-world applications are well approximated by a
rational transfer function model. For example, human speech can be characterized by the
resonances of the vocal tract that generate it. These resonances, in turn, are well repre-
sented by the poles of a digital

fi field, the best example comes from the oil fi

fi filter. Parameters for the fi filter can then be estimated, such

that the fi filter would turn a white noise input into the signal of interest, and from the fi filter's

transfer function we could easily estimate the PSD of the signal.
Various kinds of

fi filter structures exist, and they are often classi fi ed according to the type

of transfer function that they implement. An all-pole fi filter is called an autoregressive (AR)

model , an all-zero fi filter is a moving-average (MA) model , and the general case of a pole-

zero filter is called an autoregressive-moving-average (ARMA) model . Using the past

example, the model best suited for speech is then an AR model. Although high-resolution
estimators have been implemented for all these models, AR model-based estimators are the
most popular because many computationally e

fi

cient algorithms are available. A well-
behaved set of equations to determine the AR parameters with a computationally e cient

algorithm has been introduced by Marple [1987].
In the model of Figure 5.18, the AR cients a 0 , a 1 , ... , a p are estimated by

Marple's algorithm based on the input data samples x 0 , x 1 , ... , x N- 1 . The model assumes
that a white-noise source drives the

fi filter coe

filter, in which the output is regressed (thus the name
autoregressive ) through a chain of delay elements z 1 , from which p taps feed the AR
coe

fi

ciently through the
FFT, resulting in an estimate of the signal's PSD. The performance of Marple's estimator
is startling. Figure 5.19 b presents three spectral estimates obtained from a short 64-point
complex test data set suggested by Marple. Estimates obtained through the zero-padded
FFT periodogram, Welch's averaged periodogram, and Marple's method can be compared
to the theoretical spectrum of Figure 5.19 a . Only positive-frequency PSD estimates are
shown for clarity.
Notice that the closely spaced components cannot be resolved by either of the classical
methods, but they appear clearly separated in the estimate produced by Marple's method.

cients. The system's transfer function can then be computed e