SIGNAL-TO-NOISE ENHANCEMENT TECHNIQUES
Signal-to-noise enhancement by environmental cross-correlation
There are many situations of interest in which data are contaminated
by the environment. Often this contamination is understood, and by
monitoring the environment it is possible to "clean up" or "reduce"
the data, by subtracting the effects of the environment from the
signal or signals of interest. In the case of the data stream from an
interferometric gravitational radiation detector, the signal of
interest is the differential displacement of suspended
test masses. This displacement arises from gravitational waves but
also has contributions arising from other contaminating sources, such
as the shaking of the optical tables (seismic noise) and forces due to
ambient environmental magnetic fields. The key point is that the
gravitational waves are not correlated with any of these environmental
artifacts.
The method implemented here works by estimating the linear transfer
function between the principal channel and specified environmental
channels on the basis of the correlations over a certain bandwidth in
Fourier space. The method is explained in detail in the paper
`Automatic cross-talk removal from multi-channel data'
(WISC-MILW-99-TH-04)1
Here we will just give a very brief overview to introduce the quantities calculated.
We denote the channel of interest, for example, the InterFerOmeter Differential
Mode Read-out, by X or Y1. The other sampled channels consist
of environmental and instrumental monitors which we denote
Y2,...,YN. The channels are decimated
so that all channels are sampled at the slowest rate of any channel
Y1,...,YN.
We assume that the contribution of channel i to channel 1 is
described by an (unknown!) linear transfer function Ri(t-t¢). The
basic idea of the method is to use the data to estimate the transfer
functions Ri. For the reasons discussed in the paper, we work
with the data in Fourier space. The transfer function is estimated by
averaging over a frequency band, that is a given number of frequency
bins. The number of bins in any band is denoted by F in
the cited paper and correlation_width in
the associated programs. The method assumes that [R\tilde]i can be well
approximated by a complex constant within each frequency band,
in other words that the transfer function
does not vary rapidly over the frequency bandwidth Df = F/T
where T is total time of the data section under consideration.
The choices 32, 64 and 128 appear most appropriate for F.
Within a given band, b, the Fourier components of the field may be thought
of as the components of an F-dimensional vector, Yi(b).
Correlation between two channels (or the auto-correlation of
a channel with itself) may be expressed by the standard inner product
(Yi(b),Yj(b)) = Yi(b)· Yj(b)* (no summation over b). Our assumption that [R\tilde]i
is constant over each band means that the `true' channel of interest
(the principal channel with environmental influences subtracted) can
be written
|
|
-
|
(b)
|
= |
~
X
|
(b)
|
- |
N
å
j=2
|
r(b)j |
~
Y
|
(b)
j
|
. |
| (1) |
where r(b)j, j=2,...,N are constants.
The fundamental assumption is that the best estimate of the transfer
function in the frequency band b is given by the complex vector
(r(b)2,...,r(b)N) that minimises
|[`([(x)\tilde])](b)|2.
To measure the `improvement' in the signal we define |r|2 by
|
| |
-
|
(b)
|
|2 = | |
~
X
|
(b)
|
|2 ( 1 - |r|2 ) . |
| (2) |
denoted by rho2 in the programs below.
By definition 0 £ |r|2 £ 1.
If any of the environmental channels are strongly correlated
with the channel of interest, a significant reduction in
|[`([(x)\tilde])](b)|2 is
obtained, that is, |r|2 will be close to 1.
To understand the origin of the `improvement' it is also convenient to
study the best estimate that can be obtained using any given single
environmental channel. Thus we define
|
|
-
|
(b)
i
|
= |
~
X
|
(b)
|
- r¢(b)i |
~
Y
|
(b)
i
|
|
| (3) |
and choose the complex number r¢(b)i to minimise
|[`([(x)\tilde])](b)i|2. Of course, in general this
will not correspond to the ith component of the vector
used in the multi-channel case.
The corresponding improvement |ri|2 given by
|
| |
-
|
(b)
i
|
|2 = | |
~
X
|
(b)
|
|2 ( 1 - |ri|2 ) |
| (4) |
is denoted by rho2_pairwise in the programs below.
By definition 0 £ |ri|2 £ 1.
If the ith environmental channel is strongly correlated
with the channel of interest, a significant reduction in
|[`([(x)\tilde])](b)i|2 is
obtained, that is, |ri|2 will be close to 1.
Outline
Calculation of environmental correlations using the routines presented
in this chapter proceeds through the
establishment of a configuration file, called here apr00.config
with the following structure:
# DMT test configuration file using April 2000
# engineering run data
EngMC
7
64.0
H2:IOO-MC_I 16384
H2:IOO-MC_L 256
H2:IOO-MC_F 16384
H2:SUS-MC1_SENSOR_UL 256
H2:SUS-MC1_SENSOR_UR 256
H2:SUS-MC2_SENSOR_UL 256
H2:SUS-MC2_SENSOR_UR 256
1
The file may begin with any number of comment lines beginning
with an initial #. The next line is a character string
describing the set of channels being examined, this is just used for
naming intermediate files and the ROOT canvas.
The following line gives the total number of channels (signal plus
environmental).
The next line gives the time over which we look for correlations.
There follow lines each containing
two columns, the first of these lines pertains to the signal
the remainder to environmental channels. The two columns are:
- the name of the channel,
and
- the sample rate of the channel.
Finally, there is a line containing a single number. This should be
set to 1 if the user wants to obtain `cleaned' output and 0 if the
user just wants to see correlation data. (Note: This line is not used by
DCorrInit described below, but only by DEnvCorr. Thus it
is possible to change one's mind about whether to find the cleaned
signal without having to rerun DCorrInit.)
The configuration file is used by the two basic programs:
- DCorrInit which calculates the Fourier transforms and writes
binary data files in a data directory named `descriptor'_fft, so
EngMC_fft in the above example. Only those frequencies
appropriate to the slowest channel are saved.
- DEnvCorr which calculates the correlations between each
environmental channel and the signal channel and pops up a graph
of these correlations.
The data for this graph is stored in the same data directory as the
FFT data. DEnvCorr also produces a data file
rho2_`signal_name'.dat in `descriptor'_fft
which enables this graph to be reproduced later without running
DEnvCorr again. If the configuration file asks for the
signal to be cleaned DEnvCorr will also produce an ASCII data file
giving the (FFT of the) `cleaned' signal and also the total fractional reduction
in noise obtained by the method.
Again this file is stored in the same data directory, its first line
gives the frequency spacing and the following lines the real and
imaginary parts of the FFT of the cleaned signal. (To avoid plotting
difficulties with the DC component is arbitrarily set equal
to that of the first bin.)
Thus, having created the appropriate configuration file one would
type DCorrInit apr00.config and then DEnvCorr apr00.config
(or CorrInit(äpr00.config") and EnvCorr(äpr00.config")
from within root).
(Of course, the environment variable DMTINPUT must first
be set to the directory containing the appropriate frames.)
- Note: These programs perform linear algebra by calls to
clapack routines included in the shared library constructed
from EnvUtility.cc. These routines use f2c and, in
particular, use the corresponding structure to deal with complex
numbers.
Example: Correlations in data from the April 2000
Engineering Run
The output below was produced starting with 64 seconds of data
from the April 2000 Engineering Run (starting at frame H-638834111.GDS).
The fast channels are decimated
so that all channels are effectively sampled at the slowest channel rate
of 256 Hz. This yields a (real) time series with 16384
samples and correspondingly a (complex) FFT of length 8192.
Averaging is carried out over 32 frequency bins but this may
be varied by changing the variable correlationWidth.
(It is not necessary to rerun DCorrInit after changing
correlationWidth.)
Figure
Figure 1:
The graphical display of the contents of the
output file EngMC_fft/rho2_H2:IOO-MC_I_32.dat produced by DEnvCorr
illustrating environmental cross-correlation.
The output was obtained from the commands
followed by
where apr00.config is the configuration file printed above.
The data below show sections of the output file EngMC_fft/rho2_H2:IOO-MC_I_32.dat
illustrating strong environmental cross-correlation
at around 15 Hz with H2:IOO-MC_L and H2:IOO-MC_F
and at around 60 Hz with all other channels
...
13.250 0.196 0.000 0.000 0.196 0.000 0.000 0.000
13.750 0.000 0.000 0.000 0.000 0.000 0.000 0.000
14.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
14.750 0.791 0.791 0.791 0.000 0.000 0.000 0.000
15.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
15.750 0.000 0.000 0.000 0.000 0.000 0.000 0.000
16.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
...
58.750 0.000 0.000 0.000 0.000 0.000 0.000 0.000
59.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
59.750 1.075 0.000 0.147 0.000 0.177 0.000 0.183
60.250 0.980 0.733 0.966 0.499 0.762 0.766 0.713
60.750 0.158 0.000 0.135 0.000 0.000 0.000 0.000
61.250 0.175 0.000 0.104 0.000 0.000 0.000 0.000
...
Figure
Figure 2:
The spectrum of the H2:IOO-MC_I_32 channel
before (blue) and after `cleaning' based on the three environmental
channels discussed in the text
using a correlation width of 32 bins (red).
The output file EngMC_fft/fftclean_H2:IOO-MC_I_32.dat contains the Fourier
transform of the corresponding signal `cleaned'
by estimating the transfer functions over a correlation width of
32 bins.
Figure 2 shows the spectrum of the H2:IOO-MC_I_32 channel
before and after `cleaning' based on the 6 `environmental' channels.
Footnotes:
1available from
http://www.lsc-group.phys.uwm.edu/ ~ www/docs/pub_table/gravpub.html.
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