4 The ampligram

There is a straightforward generalization of the above technique, which may be used to separate independent components of the signal, assuming that the different components are characterized by different wavelet coefficient magnitudes (spectral densities). The experience from studies of oscillations in complex mechanical systems indicate that a given oscillation mode usually occurs with a certain amplitude/spectral density. The amplitude ratios between possible modes are usually constant in such a system. That observation may be used to generalize the above non-linear filtering technique. For a discrete-time signal y(t_i) the following operations are performed

1.

Wavelet transform y(t_i). This results in a

complex $J\times N$ matrix $W=\left\{w_{j,k}\right\}$.

2.

Instead of using the low-pass or high-pass filtering of wavelet coefficient magnitudes, as described in Sect. 3, a kind of band-pass filtering of wavelet coefficient magnitudes is used.

Find the maximum magnitude among the wavelet coefficients,

$$\vert W\vert = \max_{j,k} \vert w_{j,k}\vert.$$

Define L magnitude intervals

$$I_l = \left[ (l-1)\Delta w, l\Delta w \right], \quad l = 1,2,\ldots,L$$ (7)

with $\Delta w = \vert W\vert/L$.

Construct L matrices W_l, $l = 1,2,\ldots,L$, such that

$$\left[ W_l \right]_{j,k} = \left\{ \begin{array}{cl} w_{j,k}... ...\vert\in I_l, \\ 0 & \mbox{otherwise.} \end{array} \right.$$

3.

Inverse wavelet transform W_l to get new time-signals y_l(t_i), $l = 1,2,\ldots,L$. Each y_l(t_i) is what the signal would look like if only a narrow range of wavelet coefficient amplitude would be present in the signal.

4.

Construct an $L\times N$ matrix Y with y_l(t_i) as rows. This matrix Y is the ampligram of the original time-signal y(t_i).

Note that if we just want to examine a subset of wavelet coefficient magnitudes, the construction of the intervals I_l in the above algorithm can be changed, e.g., choosing $\Delta w = 0.2\,\vert W\vert/L$ results in a "low-20'' ampligram.

An interesting application of the method is to study variability of a X-ray pulsar, even using data with low counting rate. Here, an ampligram of a 1024 seconds sample from 1E2259+586 taken by ROSAT PSPC instrument (file number ROR 400314), is shown in Fig. 6. That pulsar has been extensively studied [Parmar et al.1998] by means of the BeppoSAX satellite. According to the above reference, the 0.5-10 keV pulse shape is characterized by a double peaked profile, with the amplitude of the second peak about 50% of that of the main peak. At the time when the analyzed ROSAT file was taken, the pulse period was about 6.97885 seconds (cf. Fig. 5 of Parmar et al. 1998). In order to construct the ampligram, photons were counted in 0.24 s bins. The bandwidth of 4% in the wavelet coefficient magnitude domain was used. The 4% band was moved in 1% steps over the range 0-20% of the maximum wavelet coefficient magnitude. It may be seen that 7 s pulses cover the range of wavelet coefficient magnitude between 5 and 14%. The ampligram reveals a semi-regular pattern of variations of the coefficient magnitude, corresponding to 7 s pulses, with a period of about 100 s. That period is clearly seen in the bottom of ampligram as an independent component with coefficient magnitudes less than 4%.

Figure 5: A total ampligram of a 1024-point sample of photon counts with a frequency spectrum shown in Fig. 4. Sampling bin width is 1 s. The color scale is expressed in counts/second - zero corresponds to the average counts. Columns of the ampligram matrix are plotted in horizontal direction

Figure 6: A low-20 ampligram of the X-ray pulsar 1E2259+586, ROR 400314. Sampling bin width is 0.25 s

The summation along colums of the matrix Y should result in the original sample $y\left(t_{i}\right)$, if there would be no energy leakage from outside the filter band (7). Figure 7 shows results of summation (thick line) of seconds 700-800 of the ampligram in Fig. 5 together with the measured data (thin line). It may be seen that the differences between the observed and reconstructed data are of the order of 10%, which means that the energy leakage from outside the pass-band is not very important.

Figure 7: Results of summation (thick line) of the first 5.5 s of ampligram in Fig. 5 together with the measured data (thin line)

In order to investigate the presence of weak components in the signal the low-20 ampligram is a useful tool. The low-20 ampligram for the data shown in Fig. 5 is shown in Fig. 8. Another presentation method is used here. Since all horizontal crossection of the ampligram are bipolar, the graphical presentation may be simpler if only positive (or negative) portion of the signal is plotted. Here only positive portion of the ampligram is plotted for clarity. The color scale shows the natural logarithm of the amplitude $y_l\left(t\right)$. The use of logarithmic z-scale enhances the lowest amplitudes.

Figure 8: The low-20 ampligram of the same data as shown in Fig. 5. Only positive portion of the ampligram is shown for clarity, logarithmic z-scale

The ampligram demonstrate the amplitude and phase of components of the signal corresponding to different spectral densities. The ampligram is an useful method of presentation of the physical properties of the signal. The signal in Fig. 5 with highest coefficient magnitudes (40-100%) is burst-like and random. Regular structures seem to be dominant below 10% of highest coefficient magnitudes (see Fig. 8). It may be determined whether there is one or more semi-regular components in the signal. A decomposition technique for an ampligram is discussed in the following section.