Biomedical Engineering Reference
In-Depth Information
we have some peak model, x = f(tt
0
), which maximizes at t = t
0
, and
describes what the peak lineshape, i.e., the arrival time distribution, would
be. This function could be obtained either empirically or derived from laws
of physics/chemistry, but the bottom line is that it captures most of the
characteristics of a typical peak in the spectrum. We are going to use t
0
to
label the position of the window.
Thus, for the window at t
0
, we have N isolated data points, s =
(s
1
; s
2
; s
3
;; s
N
), from the spectrum, and we have an N-component vec-
tor that describes the peak lineshape:
x = (x
1
; : : : x
N
)(f(t
1
t
0
); : : : f(t
N
t
0
)) :
(5)
For convenience, let x be normalized to have unit area:
X
N
x
k
= 1:
(6)
k=1
This will only introduce a constant correction for the peak intensity
computed later.
The rst thing we want to determine is whether or not there is a peak
in the window. This is a comparison between two hypotheses:
H
1
= There is a single peak in the window around t
0
with the peak
lineshape described by x but of unknown intensity, embedded in
noise of assumed type. Deviations from this shape in the data are
due to noise. Let us call the associated peak-plus-noise model M
1
;
H
0
= There is no peak in the widow t
0
. The data are noise of the
assumed type. Let us call the associated pure-noise model M
0
;
We want to emphasize that for each model M
0
and M
1
, we mean a
particular choice of peak lineshape x and noise type. If the noise is additive,
for example, the white Gaussian noise, then, hypothesis H
1
is equivalent to
assume that the observed signal s within the window is given by:
s
k
= ax
k
+
k
;
k = 1; 2; : : : N;
(7)
where a is an unknown intensity and = (
1
;
2
N
) is a random process
of the assumed type. Similarly, for hypothesis H
0
, we have:
s
k
=
k
k = 1; 2N:
(8)
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