SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSIS - Design and Development of Medical Electronic Instrumentation - page 243

order, but an unreasonably “peaky” spectrum is often obtained before ill-conditioning can
be detected.

Array Signal Processing

The greatest interest in high-resolution spectral estimators has been generated in the eld of

array signal processing. Here, a number of sensors are placed at various locations in space
to detect traveling waves. For example, in seismology, a number of sensors capable of
detecting the shock waves of a tremor or earthquake are spread over a certain area. As the
shock waves travel under the sensor array, signals from each sensor can be sampled along
time, producing a data record which also contains information regarding the spatial charac-
teristics of the waves (because the sensor locations are known). The processing of resulting
spatiotemporal data is meant to estimate the number, vector velocity (speed and direction),
and waveshape of the overlapping traveling waves in the presence of interference and noise.
Array signal processing has been applied successfully to biomedical diagnosis and has been
used to track weak electrical potentials from the brain, nerves, and muscles. Other applica-
tions involve image reconstruction from projections, such as MRI and medical tomography.
The most common form of traveling wave is the plane wave. In its simplest form, a
plane wave is a sinusoidal wave that not only propagates through time t but also through
space. In the direction of propagation r , this wave can be represented by

fi

r

g ( t , r ) A sin 2 π ft v

1/s), and v the
velocity (in m/s or any other suitable velocity units) at which the wave propagates through
space.
If one such simple plane wave is sampled discretely along time and space, we would
obtain a record similar to that presented on the left side of Figure 5.20 a . As you may well
see, at any given time the spatial sampling of the wave will also form into a sinusoid with
frequency k 1 . The spatial frequency (in 1/m) of such a simple plane wave, called the
wavenumber , is given by k

where A is the amplitude of the wave, f its temporal frequency (in hertz

f / v . Its physical meaning indicates that at a distance r from
the origin, the phase of the wave accumulates by 2 kr radians.

The two-dimensional spectrum of the plane wave in our example would be an impulse

π

δ (the spectrum of a sinusoid) located in the frequency-wavenumber ( f-k ) plane at f 1 , k 1 .

Through this type of spectral analysis we can infer not only the components of the waveform
but also their velocity, because the slope at which the components are found is equal to
their propagation velocity. In this case, v 1 (m/s) f 1 (1/s)/ k 1 (1/m).

erent frequency and propagation velocity
(Figure 5.20 b ) to the original component, we obtain a plane wave (Figure 5.20 c ) that
regardless of its simplicity, can hardly be recognized in the space-time domain. However,
the two-dimensional frequency-wavenumber spectrum of the signal clearly resolves the
components and their propagation velocities.
The two-dimensional spectrum can be computed with ease knowing that the two-
dimensional DFT is computable as a sequence of one-dimensional DFTs of the columns
of the data array, followed by a sequence of one-dimensional DFTs of the rows of this new
array, or vice versa. As such, the simplest two-dimensional PSD estimator is implemented
through the FFT. In practice, however, due to the limited number of spatial samples
(because only a few sensors are normally used), high-resolution estimators must be used.
Since enough samples x 0 , x 1 , ... , x N 1 can usually be obtained from each of the R sensors
through time, a hybrid two-dimensional spectral estimator can be implemented by combining

By adding a second component with a di ff