NA62 beam studies (23/3/23) (main plots from beam/html) -------------------------- Time series analysis. -------------------- The objective of a time series analysis is to determine the trend, periodic, irregular and random components of the time series. Here the time series is the rate of NA62 triggers taken to be characteristic of the SPS spill. Introduction. ------------ To illustrate some of the analysis techniques, a simulation of the spill and and its analysis using Fourier transforms , periodograms and difference plots is shown here Fig 1 shows a simulation of the spill. A histogram of the number of events (triggers) per ms is plotted over a one second period. 10**5 events are plotted. The trend is ~100 events per ms and the random component is consequently ~10 events per ms. In the simulation, sinusoidal components of 10, 50, 100 and 150 Hz have been added to the trend with the 100 Hz present at the 10% level. The 10 Hz signal has been added to illustrate the effect of a low frequency component (see the autocorrelation plot). In addition, as frequently found in the data, there are gaps in the spill; this has been simulated by the random addition of 25 1 ms gaps in the spill. In reality, the length of these gaps varies. These gaps constitute the irregular component of the time series. Figure 2 shows two analyses that measure the periodic components of the histogram of the spill. Figure 2a shows the sample autocorrelation. The dominant 100 Hz component of the time series, modulated by the 10 Hz low frequency component, is clear. Carefull inspection also indicates the influence of the 50 and 150 Hz signals. A FFT analysis is shown in Fig. 2b. With the exception of the 150 Hz signal, the Fourier transform separates the periodic parts of the spill fronm the background. The same periodic analysis is repeated in Fig 3 using a periodogram and a Fourier transform with a log axis that shows the zero frequency term measuring the total number of events in Figure 1. The periodogram confirms the results of the Fourier transform. Figure 3 has two histogram showing the gross characteristics of the spill. Fig. 3a plots a histogram of the size of the difference between two successive bins of Fig. 1. This is, in effect, the first derivative of Fig 1 and should therefore be a gaussian centered at zero with a width of sqrt(2*trend) in the absence of periodic and irregular terms. Fig 3a shows that the periodic terms have only a small effect on the gaussian peak. The irregular part of the spill forms tails about the peak; consequently, the ratio RMS/Sigma is a good measure of the significance of the irregular component of the spill. Fig. 3b shows a histogram of the number of entries/ms of the spill. As would be expected, the histogram is closely gaussian with a peak at trend and a width larger than sqrt(trend) due to the periodic components. Any deviation from a uniform trend will also increase the width of this distribution. The irregular component appears as a separate peak at low counts for the chosen form of simulation. Data Analysis ------------- The techniques discussed in the introduction have been applied to nine bursts from the 2022 run. The sections below follow approximately a sequence from low frequency millisecond general parameterisation of the spill to spill characterisation in the microsecond range. 1) Frequency and first difference analysis. The plots here show, for each burst, in Fig.1 the spill histogrammed in 1 ms bins, together with histogram of the first difference, again in 1 ms intervals. In the absence of irregular noise this histogram would lie within the 2 sd cyan lines drawn on this plot. Fig. 2 shows a frequency analysis of the spill in four one second time intervals. Difference plots for 1.5 - 5.5 secs of the spill are shown in Fig. 3. In these plots RMS refers to the root mean square(rms) of the histogram and sigma to the rms of the gaussian fit to the central peak. The ratio, RMS/sigma, is thus a measure of the significance of tails in the difference plots.. The legend 'expected sigma' is, as explained in the Introduction, sqrt(2*mean). The plot in Fig. 4 is a histogram of the number of entries per ms for the 1.5 to 5.5 second region of the spill and shows clearly any irregular component of the spill as a peak at ~ 20 counts. The first burst analysed, 12567/336 , is typical of the majority of the bursts in the sequence presented here: the periodic frequency 100 Hz is dominant, 50 Hz is present over part of the spill, and an irregular term is present in the middle of the spill. There is also low frequency 'noise' present which is largest in the center of the spill and consequently may result from irregular gaps in the spill. The final burst in this list, 12066/1129, has no irregular component. Consequently, there are no tails to the first difference distrubution and RMS/sigma ~ 1.0 . The spill distribution, however, does not have the expected sigma, sqrt(mean), since the trend of the spill is not flat. 2) High frequency spill characteristics. The plots to be discussed in this Section show the high frequency characterists of the spill that result, in part, from the 5 x 2 structure of the SPS fill. Fig. 1 shows the time structure of the spill in 5 ms bins, with Fig. 2 showing the same distrution in 0.2 bins to illustrate the fine structure. The large-scale characteristics are tabulated in Fig. 3 together with a histogram of the number of triggers per ms. Typically, Fig 3 shows that the spill has negative skewness due to the gaps in the spill, kurtosis greater than the 3 expected for a gaussian distribution, and a mean number of triggers/ms ~ 100. Bursts such as 12066/1129, that have no irregular term, tend to have a positive skewness due to the periodic component. Fig. 4 demonstrates the spill structure that results from the SPS. Here is plotted the spill trigger time (ms) vs the trigger time in 25 ns bins modulo 923.99xx (the 'folded time'). The 923.99xx term ( the fold time) is found by minimizing the signal in the gaps in the projection of the spill time on the folded time axis after correcting the folded time. The correction to the folded time used is 2.4*(spill_time -1.5)**3 in folded time units with spill time in secs. This correcion partially removes the effect of the reduction in momentum of the beam as it circulates the SPS. The corrected spill vs folded time is shown in Fig 5. Three projections on the folded time axis are shown in Fig. 6 to illustrate the influence of the correction and the fine structure of the beam. These two Figures suggest that the beam is rarely fully debunched and that, especially at the start of the spill, there can be substantial variations in intensity in a bunch. Figure 7 has autocorrelation plots for folded time (7a) and spill time (7b). Fig 7a has a peak at zero time lag showing a strong correlation over short time intervals. Other peaks correspond to the bunch structure. Fig. 7b shows the dominant 50 Hz or 100 Hz periodic term in the time series of triggers that is often modulated by a low freqency component of the spill. Figures 8a and b illustrate a section of the spill and its associated Fourier analysis indicating the dominant low frequency periodic components of the trigger times. 3) Time difference analysis Histograms of the interval in time between succesive triggers are shown here These plots illustrate the higher frequency components of the spill originating from the SPS.. The first three Figures show the general spill characteristics: firstly, a histogram af the spill in 1 ms bins; secondly, a smoothed histograms of the spill to indicate the dominant periodicity and, thirdly, a histogram of the number of entries per bin in the first histogram to give the mean and rms of the trend of the spill. Figure 4 is a log plot of a histogram of the time between triggers in microsecond bins. As expected for a Poisson distribution this is a straight line with slope ~0.1 events/musec. Deviations from the straight line are evident at half the circulation period of the SPS. Figure 5 repeats this plot over a wider range of time intervals. The majority of bursts have peaks at ~ 0.2 and 0.33 ms. that are probably the irregular terms in the spill. Finally, Figure 6 repeats Figure 5 with a plot of the difference between the histogram of time differences and the ftted line. The plot again shows evidence for the 5 X 2 bunch structure of the SPS beam. Summary and conclusions. ----------------------- The time series of the triggers of 2022 NA62 detector have been examined with the object of determining the trend, periodic, irregular and random components of this series. This study has been motivated by the fact that any departure from statistical uniformity of the time series may reduce the efficiency of data taking. The periodic components have been identified using periodograms and Fourier analysis together with sample autocorrelation plots. The random (shot noise) and irregular components in the spill have been measured using first and second difference plots. It has been found that the periodic frequencies 50 Hz and 100 Hz are preent at the levels of typically 5% and 10% ,respectively. Low frequency noise and an irregular component are found at a level about twice that expected from the trend of the data ie at ~ 2*sqrt(N), where N is the trend level of the triggers, nominally ~ 100 events/ms. A 5X2 structure of the triggers, charactersitic of the SPS bunch filling and extraction, is evident at the microsecond level giving rise to a distortion of the time structure that is otherwise Poissonian. The SPS spill is rarely completely debunched, the bunches may have diferent intensities, and there are frequently intensity variations across a bunch. The effect on the efficiency of data taking of the above described time characteristics of the triggers has yet to be determined, but is likely to be in the few percent range. plots in ~/public_html/newplots
