Identification of gravitational-wave bursts in high noise using Padé filtering

We consider the case of highly noisy data coming from two different antennas, with each data set containing a damped signal with the same frequency and decay factor but different amplitude, phase, starting point, and noise. Formally, we treat the first data set as real numbers and the second one as purely imaginary and we add them together. This complex set of data is analyzed using Padé approximations applied to its $Z$ transform. Complex conjugate poles are representative of the signal; other poles represent the noise, and this property allows us to identify the signal even in strong noise. The product of the residues of the complex conjugate poles is related to the relative phase of the signal in the two channels and is purely imaginary when the signal amplitudes are equal. Examples are presented on the detection of a fabricated gravitational-wave burst received by two antennas in the presence of either white or highly colored noise.

Figure 1

The output of two antennas receiving the same damped sinusoidal signal (black in the first channel, red in the second one). Amplitudes are equal and phases are opposite. The signal starts after about 140 data points and lasts for about 500 data points.

Figure 2

Poles in the upper half of the complex plane (black crosses) and c.c.’s of the poles in the lower half of the complex plane (red crosses) for the signals in Fig. 1 in white Gaussian noise whose standard deviation in each channel is 0.1 times the maximum amplitude of the signal in the first channel. The traveling window is of length 100 data points, with step 2 data points.

Figure 3

Poles in the upper half of the complex plane (black crosses) and c.c.’s of the poles in the lower half of the complex plane (red crosses) for the signals in Fig. 1 in white Gaussian noise whose standard deviation in each channel is 0.1 times the maximum amplitude of the signal in the first channel. The traveling window is of length 100 data points, with step 2 data points. The noise realization is changed at each window.

Figure 4

Poles in the upper half of the complex plane (black crosses) and c.c.’s of the poles in the lower half of the complex plane (red crosses) for the signal in Fig. 1 in the Hanford and Livingston colored noise. Its standard deviation in each channel is 0.36 times the maximum amplitude of the signal in the first channel. The traveling window is of length 100 data points, with step 2 data points.

Figure 5

Coincidences of c.c. poles from Fig. 4; ${\delta }_1={\delta }_2=0.01$. The color of the dots represents the quantity in Eq. (17).

Figure 6

The output of two antennas receiving the same damped sinusoidal signal (black in the first channel, red in the second one). Both amplitudes and starting points are different; the signals are almost in quadrature.

Figure 7

Poles in the upper half of the complex plane (black crosses), c.c.’s of the poles in the lower half of the complex plane (red crosses), and coincidences of c.c. poles (colored circles) for the signal in Fig. 6 in the Hanford and Virgo colored noise. Its standard deviation in each channel is 0.36 times the maximum amplitude of the signal in the first channel; ${\delta }_1={\delta }_2=0.01$. The color of the circles represents the quantity in Eq. (17).

Figure 8

Coincidences for the signal in Fig. 1 in white Gaussian noise whose standard deviation in each channel is 1.0 times the maximum amplitude of the signal in the first channel; ${\delta }_1={\delta }_2=0.04$. The color of the dots represents the quantity in Eq. (17).

Figure 9

Coincidences for the signal in Fig. 1 in the Hanford and Livingston colored noise whose standard deviation in each channel is 3.6 times the maximum amplitude of the signal in the first channel; ${\delta }_1={\delta }_2=0.02$. The color of the dots represents the quantity in Eq. (17).

Figure 10

Same as in Fig. 9, but we also added white Gaussian noise whose standard deviation in each channel is 0.3 times the maximum amplitude of the signal in the first channel.

Figure 11

Same as in Fig. 10, but ${\delta }_1={\delta }_2=0.03$.

Figure 12

Noise (black) and signal (red) FFT amplitude spectra; the noise spectrum is averaged over 100 consecutive data segments.

Figure 13

Coincidences for the signal in Fig. 6 in the Hanford and Virgo colored noise whose standard deviation in each channel is 1.44 times the maximum amplitude of the signal in the first channel; ${\delta }_1={\delta }_2=0.02$. The color of the dots represents the quantity in Eq. (17).

Figure 14

Same as in Fig. 13, but we also added white Gaussian noise whose standard deviation in each channel is 0.12 times the maximum amplitude of the signal in the first channel; ${\delta }_1={\delta }_2=0.03$.