Biomedical Engineering Reference
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area is irradiated. The sum of the output of each irradiation gives the nal
spectrum.
The nal spectrum can thus be viewed as consisting of measurements re-
peated many times under nominally identical conditions to gradually build
up a portrait of the probability distribution of arrival times for each m=z.
Though to derive the exact arrival time distribution is rather complicated,
it is clear that, for the sake of reasonable mass resolution, the velocity dis-
tribution at the moment that ions enter the drift region should be sharply
peaked around some nominal value v
(m) with certain width (m). Then
from maximum entropy perspective, this would lead to the choice of a Gaus-
sian for the velocity distribution:
(vv
)
2
2
2
1
p
g (v) =
2
exp
:
(3)
The choice of the maximum-entropy distribution is founded upon the
principle that it maximizes the number of possible outcomes of repeated
observations that are consistent with this assignment of the probability
19
.
Hence, it is the least biased assignment of the probability that is consistent
with our limited knowledge of the initial distribution. Transforming from
3 to the temporal peak shape, i.e., the arrival time distribution, requires
solving a Fokker-Planck equation, which denes how the probability density
function g evolves along the ying path. The most simple case involves the
ions entering the drift region and ying directly to the detector, receiving
no extra action. The arrival time distribution would be:
!
2
D
t
D
v
p(t
k
jv
; ) =
p
2t
2
exp
;
(4)
2
2
where D is the ying distance.
The nal spectrum is a summation of many independent, repeated mea-
surements of the sample under identical conditions and each measurement
is a sampling from a population characterized by the arrival time distribu-
tion p(t
k
jv
; ), which is determined by instrumental function and sample
surface. The ion counts between time t
k
and t
k
+ t, s
k
, in the nal time
seriesfsg, is then independent of those ion counts that arrive at any other
time t
j
, even when t
k
and t
j
are associated with the same ion peak. This
independence will be a crucial assumption that underlies the entire analy-
sis we pursue. Any correlations in the signal are assumed to be due to the
electronics and should be taken into account as a part of the model used.
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