A simple and efficient line detection algorithm applied to Virgo data

We propose a new method for the detection of spectral lines in random noise. It mimics the processing scheme of matching filtering i.e., a whitening procedure combined with the measurement of the correlation between the data and a template. Thanks to the original noise spectrum estimate used in the whitening procedure, the algorithm can easily be tuned to the various types of noise. It can thus be applied to the data taken from a wide class of sensors. This versatility and its small computational cost make this method particularly well suited for real-time monitoring in gravitational wave experiments. We show the results of its application to Virgo C4 commissioning data. Persistent and narrow spectral peaks, known as lines are a typical feature of the data from gravitational wave interferometric detectors (ITF). Lines can originate from the ITF functioning (e.g., mirror or suspension resonant modes) and operation (e.g., calibration lines) or from environmental perturbations (e.g., vacuum pumps). Detecting the presence of such lines in ITF readout and control channels is very important for the studying and monitoring of the detector performance. In this paper we present a simple and versatile processing tool to address this issue. Optimal strategies can be designed for the detection of sinusoidal signals in random noise [1]. However, they are computationally expensive and thus not very well-suited for monitoring purposes. In addition, the amplitude of the lines we are dealing with, are usually large as compared to the noise level. A statistic which is suboptimal, but computationally acceptable, suffices in this context. Contrarily to [2, 3], we are not interested here in the removal of the line once detected. Though estimates of the line characteristics (such as the central frequency, amplitude and width) can be obtained from the proposed tool, we don’t discuss this point here and concentrate on the detection issue. In Sect. 1, we give a general description of the line detection problem and point out its main difficulties. In Sect. 2, we detail the proposed line search method. In Sect. 3, we give several rules of thumb for adequately setting the free parameters of the line search. In Sect. 4, we present the results of its application to Virgo C4 commissioning data. 1. A tricky detection problem ITFs are sophisticated apparatus controlled by a complex network of feedback loops. One way of locating the origin of a line in such a system is to check for the coincident occurrence of the line in channels including the ITF readouts, the ITF controls and the environmental sensors. The line search has thus to be applied to many different channels. Generally, the spectral lines are superimposed to a broadband random noise, hereafter referred to as “background” noise. The power spectral density (PSD) of the background noise can have very different shapes. In most cases, we only have a rough idea of the PSD shape A simple and efficient line detection algorithm applied to Virgo data 3 which may also change with time. Therefore, the problem to address is the statistical detection of a signal (lines) in a random noise of unknown PSD which thus needs to be estimated from the data. This turns out to be difficult because lines and background noise mix at all times i.e., there is no ”background noise only” data. The proposed method tackles with this difficulty. 2. A simple line detection algorithm The basic ingredient of the proposed method is the spectrogram. Let us assume that the data x(t+ tsn), n ≥ 0 are sampled at Nyquist rate fs = 1/ts and collected during the time period from t to t+ T . We divide this time period into N non-overlapping intervals of equal duration T /N = tsN (which thus contain the same number of samples N). The spectrogram S(t, f) is defined for f ∈ [0, fs/2] by [4] S(t, f) ≡ 1 N N−1 ∑ k=0 ∣∣∣∣∣ ts (k+1)N−1 ∑ j=kN x(t+ tsj) h(tk − tsj) e−2πi tsjf ∣∣∣∣∣ 2 , where tk ≡ ts(kN + (N − 1)/2) is the center of the kth interval and h(t) is a window function (e.g., Hanning type) centered around t = 0 and scaled to unit L norm. Matched filtering [5] is an efficient method for detecting deterministic signals in random noise. It can be viewed as a two-step process: a whitening of the data followed by a scalar product with a template. The line search algorithm mimics this structure. We assume that S(t, f) has been computed for some given time t and we detail now the detection procedure. step 1. background PSD estimate We use S(t, f) to get a robust estimate of the background PSD: (1) the frequency axis f ∈ [0, fs/2] is tiled into intervals. Their size is chosen sufficiently small that each of them contains only a few lines ( 5) and that the background PSD can be considered almost linear within the interval. (2) In each interval, we remove the Nq points with the largest amplitude (these “outliers” are essentially corresponding to the few frequency peaks belonging to the interval). (3) We make a least mean square linear fit of the remaining spectrogram data points. The collection of the fits performed in all frequency intervals yields the estimate of the background noise PSD, Ŝ(t, f). Clearly, step (2) prevents that the lines (of large amplitude) bias this estimate. step 2. whitening and scalar product with template Roughly speaking, the lines can be described by the general model a(t) cosφ(t) where the envelope a(t) is slowly varying and the phase φ(t) is well approximated by a linear function for time periods of duration not smaller than T . The template matching to a line of duration T with a constant envelope a(t) = C and a constant frequency f0 = (2π)−1dφ/dt is a “simple” cosine function of the same duration. The FFT (of time base T ) does exactly the scalar product with this template. The above definition of the spectrogram is not identical to this (because of the division of T into intervals and the modulus averaging) but it is similar. This suggests to use S(t, f) in place of the exact template match. We consider the following statistic W (t, f) ≡ S(t, f)/Ŝ(t, f) which includes both whitening and template match. A simple and efficient line detection algorithm applied to Virgo data 4 step 3. detection and post-processing The line detection is then made by comparing the statistic to a threshold: W (t, f) > η. Once the three steps are completed, we restart the procedure from step 1 to process the spectrogram computed over the next data chunk. 3. Choosing the parameters Some tuning of the free parameters is required to ensure that the procedure works properly. There is a total of six free parameters. The computation of S(t, f) uses two of them: the total observation time T (typically T ≈ 300s) and the FFT time base N ≡ T fs/N (set to a power of 2 for efficiency) which also sets the frequency resolution (the orders of magnitude are: fs ≈ 10 kHz and we choose N ∼ 10 data points corresponding to a frequency resolution of 100 mHz). The PSD fit requires a value for Nq. A good value is given by the mean number of lines per frequency interval times the peak average width (at half height, expressed in bin). The threshold η determines the minimum detectable signal-to-noise ratio (or, more specifically, the ratio of the line amplitude to the neighboring background noise level) and at the same time, the rate of false alarms. Typically, we choose η 3 to 4. The two remaining parameters are related to the tiling of the frequency axis mentioned in Sect. 2. We discuss the question of choosing their value in the next two sections.