Page 332 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

TIME-OF-FLIGHT ESTIMATION OF AN ACOUSTIC TONE BURST 321

The literature roughly mentions three concepts to determine the ToF,
i.e. thresholding, data fitting (regression) and ML (maximum likelihood)
estimation (Heijden van der et al., 2003). Many variants of these con-
cepts have been proposed. This section only considers the main repre-
sentatives of each concept:

. Comparing the envelope of the wave against a threshold that is
proportional to the magnitude of the waveform.
. Fitting a one-sided parabola to the foot of the envelope of the
waveform.
. Conventional matched filtering.
. Extended matched filtering based on a covariance model of the signal.

The first one is a heuristic method that does not optimize any criterion
function. The second one is a regression method, and as such a repre-
sentative of data fitting. The last two methods are ML estimators. The
difference between these is that the latter uses an explicit model of
multiple echoes. In the former case, such a model is missing.
The section first describes the methods. Next, the optimization of the
design parameters using the data set is explained, and the evaluation is
reported. The data set and the MATLAB listings of the various methods
can be found on the accompanying website.

9.2.1 Models of the observed waveform

The moment of time at which a transmission begins is well defined since
it is triggered under full control of the sensory system. The measurement
of the moment of arrival is much more involved. Due to the narrow
bandwidth of the transducers the received waveform starts slowly. A low
SNR makes the moment of arrival indeterminate. Therefore, the design
of a ToF estimator requires the availability of a model describing the
arriving waveform. This waveform w(t) consists of three parts: the
nominal response a h(t t) to the transmitted wave; the interfering
echoes a r(t t); and the noise v(t):

wðtÞ¼ a hðt tÞþ a rðt tÞþ vðtÞ ð9:1Þ

We assume that the waveform is transmitted at time t ¼ 0, so that t is
the ToF. (9.1) simply states that the observed waveform equals the