6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
281
curve of the source as a function of frequency or, equivalently, the product of the
Fourier transform of the signal by its complex conjugate. If, instead, one multiplies
the Fourier transform of a signal by the complex conjugate of another signal, both
functions of frequency, the result is the cross-spectrum. In the same way that the
power spectrum measures the variance of the signal per unit frequency (through
Parseval’s theorem), the modulus and the argument of the cross-spectrum measure,
respectively, the covariance per unit frequency and the phase difference, also called
phase lag, Δφ, between the two signals as a function of Fourier frequency. In the
same way that the power spectrum gives the degree of correlation of a light curve
with itself, the autocorrelation of the light curve, the cross-spectrum gives the degree
of correlation of one light curve with the other, the cross-correlation between the two
light curves. If the signals are uncorrelated, at each Fourier frequency the covariance
and the phase lag will be on average 0. (Notice that uncorrelated signals give a 0
phase lag, but a 0 phase lag does not imply that the signals are uncorrelated.) At any
given frequency, ν, the phase lag can be converted into a time lag, Δt (ν) =
Δφ(ν)
2πν
.
The phase lags are defined 1 from −π to π, while the time lags run between −1/(2ν)
and 1/(2ν). Since both quantities are related, depending on the context, we will
either use the term phase or time lags to refer to the delay between the two light
curves in the Fourier space.
If the two light curves used to compute the cross-spectrum come from two
different energy bands, the time lag at each Fourier frequency represents the time
delay between the light curves in those two energy bands at each Fourier frequency.
For a QPO (and any other somewhat broad component) with a centroid frequency
ν 0 and a FWHM Δ, we call the phase (or time) lag of the QPO to the average of
the phase (or time) lags over a frequency range around the centroid frequency of
the QPO, e.g. from ν 0 − Δ to ν 0 + Δ. It is customary to take the light curve at
the lowest energy band as the reference band and to measure the phase lag, with
respect to reference band, of the light curve in the bands, called subject bands, at
energies above the energy of the reference band. Under this convention, a positive
phase/time lag, also called hard lag, indicates that the hard light curve lags (follows
after) the soft one, whereas a negative phase/time lag, when the soft light curve leads
(comes before) the hard light curve, is called soft lag. Alternatively, one can use the
full band as the reference band, and narrow bands within the full band to measure
the lags, provided that one corrects for the correlation introduced by the part of the
signal that is both in the subject and the reference bands. In the end one obtains
the energy dependent phase lags, of the subject bands with respect to the reference
band, over the frequency range in which the QPOs dominate the variability of the
source.
Because the lower kHz QPO is usually narrower and, therefore usually more
significantly detected, than the upper, the first measurements of lags where obtained
1 Phase lags equal to Δφ ± 2nπ, with n any integer number, cannot be distinguished from a phase
lag Δφ.
281
curve of the source as a function of frequency or, equivalently, the product of the
Fourier transform of the signal by its complex conjugate. If, instead, one multiplies
the Fourier transform of a signal by the complex conjugate of another signal, both
functions of frequency, the result is the cross-spectrum. In the same way that the
power spectrum measures the variance of the signal per unit frequency (through
Parseval’s theorem), the modulus and the argument of the cross-spectrum measure,
respectively, the covariance per unit frequency and the phase difference, also called
phase lag, Δφ, between the two signals as a function of Fourier frequency. In the
same way that the power spectrum gives the degree of correlation of a light curve
with itself, the autocorrelation of the light curve, the cross-spectrum gives the degree
of correlation of one light curve with the other, the cross-correlation between the two
light curves. If the signals are uncorrelated, at each Fourier frequency the covariance
and the phase lag will be on average 0. (Notice that uncorrelated signals give a 0
phase lag, but a 0 phase lag does not imply that the signals are uncorrelated.) At any
given frequency, ν, the phase lag can be converted into a time lag, Δt (ν) =
Δφ(ν)
2πν
.
The phase lags are defined 1 from −π to π, while the time lags run between −1/(2ν)
and 1/(2ν). Since both quantities are related, depending on the context, we will
either use the term phase or time lags to refer to the delay between the two light
curves in the Fourier space.
If the two light curves used to compute the cross-spectrum come from two
different energy bands, the time lag at each Fourier frequency represents the time
delay between the light curves in those two energy bands at each Fourier frequency.
For a QPO (and any other somewhat broad component) with a centroid frequency
ν 0 and a FWHM Δ, we call the phase (or time) lag of the QPO to the average of
the phase (or time) lags over a frequency range around the centroid frequency of
the QPO, e.g. from ν 0 − Δ to ν 0 + Δ. It is customary to take the light curve at
the lowest energy band as the reference band and to measure the phase lag, with
respect to reference band, of the light curve in the bands, called subject bands, at
energies above the energy of the reference band. Under this convention, a positive
phase/time lag, also called hard lag, indicates that the hard light curve lags (follows
after) the soft one, whereas a negative phase/time lag, when the soft light curve leads
(comes before) the hard light curve, is called soft lag. Alternatively, one can use the
full band as the reference band, and narrow bands within the full band to measure
the lags, provided that one corrects for the correlation introduced by the part of the
signal that is both in the subject and the reference bands. In the end one obtains
the energy dependent phase lags, of the subject bands with respect to the reference
band, over the frequency range in which the QPOs dominate the variability of the
source.
Because the lower kHz QPO is usually narrower and, therefore usually more
significantly detected, than the upper, the first measurements of lags where obtained
1 Phase lags equal to Δφ ± 2nπ, with n any integer number, cannot be distinguished from a phase
lag Δφ.