Page 340 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

TIME-OF-FLIGHT ESTIMATION OF AN ACOUSTIC TONE BURST 329

the dependence of t on z is now captured in a concise model, i.e. a single
covariance matrix:

T
2
2
C zjt ¼ ðhðtÞh ðtÞþ C rjt Þþ I ð9:13Þ
a
v
This matrix completes the covariance model of the measurements. In the
sequel, we assume a Gaussian conditional density for z. Strictly speak-
ing, this holds true only if sufficient echoes are present since in that case
the central limit theorem applies.

Maximum likelihood estimation of the time-of-flight
With the measurements modelled as a zero mean, Gaussian random
vector with the covariance matrix given in (9.13), the likelihood function
for t becomes:

1 1
1
T
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp z C z ð9:14Þ
pðzjtÞ¼ q zjt
K 2
ð2 Þ jC zjt j
The maximization of this probability with respect to t yields the max-
imum likelihood estimate; see Section 3.1.4. Unfortunately, this solution
is not practical because it involves the inversion of the matrix C zjt . The
size of C zjt is N N where N is the number of samples of the registration
4
(which can easily be in the order of 10 ).
Principal component analysis

Economical solutions are attainable by using PCA techniques (Section
7.1.1). If the registration period is sufficiently large, the determinant
jC zjt j will not depend on t. With that, we can safely ignore this factor.
What remains is the maximization of the argument of the exponential:

def T 1
ðzjtÞ¼ z C z ð9:15Þ
zjt
The functional (zjt) is a scaled version of the log-likelihood function.
The first computational savings can be achieved if we apply a principal
component analysis to C zjt . This matrix can be decomposed as follows:

N 1
X
T
C zjt ¼ n ðtÞu n ðtÞu ðtÞ ð9:16Þ
n
n¼0