Page 334 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

TIME-OF-FLIGHT ESTIMATION OF AN ACOUSTIC TONE BURST 323

becomes random too, and the echoes are seen as disturbing noise with
non-stationary properties.
The observed waveform z ¼ [ z 0 ... z N 1 ] T is a sampled version
of w(t):

z n ¼ wðn Þ ð9:3Þ

is the sampling period. N is the number of samples. Hence, N is the
registration period. With that, the noise v(t) manifests itself as a random
2
vector v, with zero mean and covariance matrix C v ¼ I.
v

9.2.2 Heuristic methods for determining the ToF

Some applications require cheap solutions that are suitable for direct
implementation using dedicated hardware, such as instrumental elec-
tronics. For that reason, a popular method to determine the ToF is
simply thresholding the observed waveform at a level T. The estimated
t
ToF ^ t thres is the moment at which the waveform crosses a threshold
level T.
t
Due to the slow rising of the nominal response, the moment ^ t thres of
level crossing appears just after the true t, thus causing a bias. Such a
bias can be compensated afterwards. The threshold level T should be
chosen above the noise level. The threshold operation is simple to
realize, but a disadvantage is that the bias depends on the magnitude
of the waveform. Therefore, an improvement is to define the threshold
level relative to the maximum of the waveform, that is T ¼ max (w(t)).
is a constant set to, for instance, 30%.
The observed waveform can be written as g(t) cos (2 ft þ ’) þ n(t)
where g(t) is the envelope. The carrier cos (2 ft þ ’) of the waveform
causes a resolution error equal to 1/f. Therefore, rather than applying
the threshold operation directly, it is better to apply it to the envelope.
A simple, but inaccurate method to get the envelope is to rectify the
waveform and to apply a low-pass filter to the result. The optimal method,
however, is much more involved and uses quadrature filtering. A simpler
approximation is as follows. First, the waveform is band-filtered to
reduce the noise. Next, the filtered signal is phase-shifted over 90 to

obtain the quadrature component q(t) ¼ g(t) sin (2 ft þ ’) þ n q (t).
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
g
Finally, the envelope is estimated using ^ g(t) ¼ w 2 (t) þ q (t).
band-filtered
Figure 9.6 provides an example.