7 Time scale spectrum of an ampligram

The ampligram may be used for calculation of average wavelet spectra, one for each coefficient magnitude. It is equivalent to performing, once again, the forward wavelet transform on the filtered, inverse transformed data, that constitute the ampligram.

The procedure will generate a 3-D graph showing the time scale of the signal on the x-axis, the wavelet coefficient magnitude of the original signal, in percent of its max value, on the y-axis and the wavelet coefficient magnitude (corresponding to the power spectral density) of the decomposed component as the color scale. A graph of that kind will show the average properties of the different modes, if such exist, during the entire sample period.

The algorithm is as follows. Each row of the ampligram matrix, Y, is wavelet transformed, resulting in L matrices. We then time-average these matrices (average along rows) leading to L arrays, $\bar{w}_l$, with J elements. Construct an $L\times J$ matrix, $\bar{Y}$, with $\bar{w}_l$ as rows. This matrix $\bar{Y}$ is the time scale spectrum of the ampligram.

Examples of time scale spectra of total ampligrams (left) and low-20 ampligrams (right) of the data ROR 701246 are shown in Fig. 11.

Figure 11: Time scale spectra of the total ampligram (left) and of the low-20 ampligram (right) of the X-ray data from ROSAT, ROR 701246

The interesting property of this graph is that deterministic periodic or semi-periodic structures in the data are mapped on the graph as vertically elongated features, while purely stochastic structures are mapped as horizontally elongated features. That property of the time scale spectrum may be illustrated as follows: A pure and stationary sine-like signal will be mapped on the time scale spectrum as a single dot. Introducing random phase variations, but without changing the signal amplitude will broaden the dot in the horizontal direction. On the other hand, introducing random amplitude fluctuations, without scrambling the phase, will broaden the dot in the vertical direction.

For the X-ray pulsar ampligram of Fig. 6 the time scale spectrum may be useful to resolve different frequency components. Since the pulsar 1E2259+586 has a secondary maximum, located asymmetrically with respect to the middle of the period, it may be expected that the scalogram of the photon count series will be quite complex. According to Fig. 3 of Parmar et al. Parmar98 the secondary maximum is approximately 4.17 s (T1) from the preceding main maximum and 2.81 s (T2) from the following main maximum. With the pulse period T of approximately 6.98 s, time scales of Table 1 may be identified in time scale spectrum of ampligram of Fig. 6 shown in Fig. 12.

  

Table 1: Time scales present in time scale spectrum of 1E2259+586

Positions of above components in the time scale spectrum are indicated by white lines, numbered as in Table 1. At least a part of information contained in the occurrence of those time scale components, together with their relative intensities may reflect the nature of the process responsible for the pulsar's emission.