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
-
 ~ x

(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
|
-
 ~ x

(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
-
 ~ x

(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
|
-
 ~ x

(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:
1. the name of the channel, and
2. 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:
1. 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.
2. 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
 DCorrInit apr00.config
followed by
 DEnvCorr apr00.config
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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On 3 May 2006, 16:38.