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70 Autonomous Mobile Robots

Let the power of the noisy range spectra be smoothed by a w-point window
function b(i) whose length is 2w + 1

w

˘
˘
P(k, l) = b(i)P(k − i, l) (2.12)
i=−w
˘
where P(k, l) is the kth power value of lth range spectra.
Smoothing is then performed by a first order recursive averaging technique:

˘
˘
˘
P(k, l) = α s P(k, l − 1) + (1 − α s )P(k, l) (2.13)
where α s is a weighting parameter (0 ≤ α s ≤ 1). First the minimum and
temporary values of the local power are initialized to P min (k,0) = P tmp (k,0) =
˘
P(k,0). Then a range bin-wise comparison is performed with the present bin l
and the previous bin l − 1.

˘
P min (k, l) = min{P min (k, l − 1), P(k, l)} (2.14)
˘
P tmp (k, l) = min{P tmp (k, l − 1), P(k, l)} (2.15)
When a predefined number of range bins have been recorded at the same
vehicle location, and the same sensor azimuth, the temporary variable, P tmp is
reinitialized as

˘
P min (k, l) = min{P tmp (k, l − 1), P(k, l)} (2.16)
˘
P tmp (k, l) = P(k, l) (2.17)

˘
Let the signal-to-noise power (SNP), P SNP (k, l) = P(k, l)/P min (k, l) be the
ratio between the local noisy power value and its derived minimum.
In the Neyman–Pearson test [32], the optimal decision (i.e., whether target
is present or absent) is made by minimizing the probability of the type II
error (see Appendix), subject to a maximum probability of type I error as
follows.
The test, based on the likelihood ratio,is

p(P SNP |H 1 ) H 1
≷ δ (2.18)
p(P SNP |H 0 ) H 0

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c002” — 2006/3/31 — 17:29 — page 70 — #30