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Chapter6
Temporal Analysis in X–ray Astronomy
Now we will apply all the mathematical tools developed in Part I to real data. In particular we
will explore the techniques that are commonly used in timing studies of X–ray sources. The
regime we will be referring to is that of equidistantly binned timing data, the background noise
of which is dominated by counting statistics. If there are gaps in the data, they are far apart
and the data are not “sparse” in the sense that nearly all time bins are empty. This kind of data
are eminently suited to analysis with FFT techniques, and the discussed methods will be based
on these techniques.
6.1 Power Spectra in X–ray Astronomy
If we indicate with x k , k = 0,1,2,..., N − 1, the number of photons detected in bin k by our
instrument, then the discrete Fourier transform a j , with j = −N /2,..., N /2−1, decomposes this
signal into N sine waves. The following expressions describe the signal transform pair:
Definition 6.1 (Discrete Fourier transform in X–ray astronomy).
N−1 ∑
a j = x k e 2πi j k/N j = − N 2 ,..., N 2 − 1 (6.1a)
k−0
x k = 1 N
∑
N /2−1
j =−N /2
a j e −2πi j k/N k = 0,1,..., N − 1 (6.1b)
Important Note: please note the difference between this definition and the definition (3.11). The
signs in the exponents in (6.1) are reversed with respect to the ones in (3.11). In X–ray astronomy
it is customary the use of the convention as in Press et al. Accordingly, the prefactor 1/N is
present in the inverse discrete Fourier transform and not in the direct one. The consequence is
that a 0 will not be anymore the average, but the total number of counts N ph = ∑ k x k . As we
said before, it is only a question of convention.
If the signal is an equidistant time series of length T , so that x k refers to a time t k = k(T /N ),
then the transform is an equidistant “frequency series”, and a j refers to a frequency ω j =
2πν j = 2πj /T . The time step is δt = T /N; the frequency step is δν = 1/T .
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