Signal Processing - January 2017 - 108

C1,1

C2,1

C3,1

C1,2

C2,2

C3,2

CNC,2

C1,m1

C2,m2

C3,m3

CNC,mN

C1,MC

C2,MC

C,1

C3,MC

CNC,MC

Figure 2. A trellis diagram representing all the possible combinations of components and their local
neighborhoods [18].

where N C = " 1, .., N , . For each value
j
j
j
of N C, let A j = " C 1, C 2, ..., C N C , be
a set of triplets for one of the possible
combinations, where j is from 1 to
j
N!/N C ! (N - N C) ! and C i corresponds
to the ith component in the jth combination. The corresponding set of neighborj
j
j
hoods is V A j = " V 1, V 2, ..., V N C ,, where
j
j
j
j
V i = (~ i , a i , z i ) denotes the neighborj
hood of candidate components C i . The
j
t , for a specific N C
optimal set of Aj, A
value, is the one associated with the
lowest MMSE and corresponds to (5),
shown in the box at the bottom of the

500
0

Lomb−Scargle
Periodogram

−500

100

MMSE
Periodogram

Radial
Velocity (m/s)

where d defines half of the neighborhood band around each component and
set to some value that incorporates all
the significant components around the
selected peaks of S P . Hence, for each
component C i (~ i, a ~ i, z ~ i) there are
M C i candidates, where M C i is the cardinality of Vi .
Subsequently, a trellis analysis for
all of the possible combinations of components and their local neighborhoods
is performed, as schematically shown
in Figure 2. Now, all possible combinations of candidates NC is evaluated,

1,000

2,000
3,000
Time (Days)
(a)

4,000

5,000

50
0

0.2

0.4

0.6
0.8
Frequency (Hz)

1.2
× 10−5

(b)

20
0

0.2

0.4

0.6
0.8
Frequency (Hz)
(c)

Figure 3. Radial velocity data of (a) star GJ876, (b) the LS and MMSE periodograms, and (c) the
planets initially detected shown by asterisks.

108

IEEE Signal Processing Magazine

January 2017

j
j
j
previous page [18], where ^~t i , at i , zt i h
j
! V i , 1 # i # N C, providing an optimal
set of triplets for each NC. Finally, the
optimal set of triplets having the global
minimum MMSE is selected at which
its length, defines the number of the
most important sinusoidal components
in the nonuniformly sampled signal,
while its elements are their frequencies,
amplitudes, and phases, respectively. It
is worth mentioning that the problem of
order selection has also been addressed
by using statistical significance analysis
[20] and extreme value theory [21].
Figure 3 illustrates, as an example,
the Keck and the High Accuracy Radial
Velocity Planetary Searcher radial velocity data of the M-dwarf planet host star
GJ876 and the corresponding LS and
MMSE periodograms. GJ876 is known
to host a system of planets that contains
at least two short-period gas giants [18].
Signals can be searched for using the
gradient-based approach that starts by
searching for one signal only, and when
one is detected it is then subtracted out of
the data and a new search is made using
the residuals all over again by treating
them as an independent time series from
the original observed data. This process is
then repeated until the noise floor of the
data is reached. By applying this method,
the following signals [with periods in days
(d)] were detected with the MMSE method [18]: 61.03 d, 30.23 d, 15.04 d, 1.94 d,
10.01 d, and 124.69 d. This system was
chosen because the two large-amplitude
signals could be detected in both halves
of the time series separately. The MMSE
and trellis technique allows studying the
phase of the detected signals as a function
of time, showing that the phase difference between both planets is stable over
the length of the time series and therefore
adding weight to the reality of these signals. This analysis shows the power of this
method over previous periodogram techniques, such as the LS method, that gives
no information on the signal parameters
other than the frequency. However, phase
variations with time for the 1.94 d, 10.01 d,
and 15.04 d signals were found, which
could cast doubt on the origin of these signals as being from orbiting planets. This
was consistent with previous Newtonian
integrational methods. This highlights that

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