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5 X-ray Pulsar-Based Navigation: Theories and Experiments
between both, including the constant deviation, scale transformation factor, time
delay and random noise. In other words, for an observation epoch t, the measuring
pulse profile p(t) can be expressed by the standard pulse profile s(t-d) with a time
delay of d, i.e.,
p(t) = a + b · s(t − d ) + g(t),
(5.107)
where a is the constant deviation between the two pulse profiles; b is the scale
transformation factor of the pulse profile; d is the time delay or displacement of the
origin of time; g(t) is the background noises of X-ray pulse signals, such as space
background noise, high-energy particle noise, nebula noise and detector self-noise,
all of which are assumed as the additive random noise process.
The purpose of the cross-correlation processing between the measuring and standard pulse profiles is to determine the values of three parameters a, b and d. Within
the given observation time, the d will gradually be increased to the starting observation time of getting the first pulse arrival-time, so that the peak of the measuring pulse
profile is aligned with that of the standard pulse profile. In the process of aligning
the peak of pulse profile, the time delay from the SSB to spacecraft along the line
of sight of the pulsar is obtained at the same time, so as to get the pulse arrival-time
[28]. In this way, the measurement of the pulse arrival-time has been converted into
the problem of solving the time delay d of two pulse profiles, so the d is sometimes referred to as the pulse arrival-time. The Discrete Fourier Transform (DFT)
is a commonly used cross-correlation processing method for the pulse profiles. By
comparing the two pulse profiles in the frequency domain, the accuracy of parameter
estimation can be evaluated directly and has nothing to do with the photon sampling
interval. The time-domain cross-correlation method can also be used to estimate the
pulse arrival-time, but its accuracy depends on the photon sampling interval.
In the process of time transformation, the prior knowledge of positions and velocities of the spacecraft is required. However, for the absolute navigation, these parameters are often unknown or approximate estimates. If the proper time series of the
photons arriving at the onboard detector within the given observation time are folded
directly in the spacecraft-body coordinate system, the shape of the pulse profile will
distort, and its SNR is so low that it is difficult to discriminate the pulse peaks from
the photon flux intensity. In addition, the Doppler effects have a great influence on the
pulse profiles. When the spacecrafts relative to the pulsars move rapidly, it is required
to use the velocity parameters as accurate as possible in the time transformation to
reduce the influence of the Doppler effects on the extracted pulse profiles.
5.6.4 Integer Ambiguity of Pulse Phase
The pulse periods of X-ray pulsars are extremely stable, which is conducive to
extracting the pulse profile, and through the cross-correlation processing of the pulse
profiles to obtain the time of arrival of the pulses. Meanwhile, this kind of stable
5 X-ray Pulsar-Based Navigation: Theories and Experiments
between both, including the constant deviation, scale transformation factor, time
delay and random noise. In other words, for an observation epoch t, the measuring
pulse profile p(t) can be expressed by the standard pulse profile s(t-d) with a time
delay of d, i.e.,
p(t) = a + b · s(t − d ) + g(t),
(5.107)
where a is the constant deviation between the two pulse profiles; b is the scale
transformation factor of the pulse profile; d is the time delay or displacement of the
origin of time; g(t) is the background noises of X-ray pulse signals, such as space
background noise, high-energy particle noise, nebula noise and detector self-noise,
all of which are assumed as the additive random noise process.
The purpose of the cross-correlation processing between the measuring and standard pulse profiles is to determine the values of three parameters a, b and d. Within
the given observation time, the d will gradually be increased to the starting observation time of getting the first pulse arrival-time, so that the peak of the measuring pulse
profile is aligned with that of the standard pulse profile. In the process of aligning
the peak of pulse profile, the time delay from the SSB to spacecraft along the line
of sight of the pulsar is obtained at the same time, so as to get the pulse arrival-time
[28]. In this way, the measurement of the pulse arrival-time has been converted into
the problem of solving the time delay d of two pulse profiles, so the d is sometimes referred to as the pulse arrival-time. The Discrete Fourier Transform (DFT)
is a commonly used cross-correlation processing method for the pulse profiles. By
comparing the two pulse profiles in the frequency domain, the accuracy of parameter
estimation can be evaluated directly and has nothing to do with the photon sampling
interval. The time-domain cross-correlation method can also be used to estimate the
pulse arrival-time, but its accuracy depends on the photon sampling interval.
In the process of time transformation, the prior knowledge of positions and velocities of the spacecraft is required. However, for the absolute navigation, these parameters are often unknown or approximate estimates. If the proper time series of the
photons arriving at the onboard detector within the given observation time are folded
directly in the spacecraft-body coordinate system, the shape of the pulse profile will
distort, and its SNR is so low that it is difficult to discriminate the pulse peaks from
the photon flux intensity. In addition, the Doppler effects have a great influence on the
pulse profiles. When the spacecrafts relative to the pulsars move rapidly, it is required
to use the velocity parameters as accurate as possible in the time transformation to
reduce the influence of the Doppler effects on the extracted pulse profiles.
5.6.4 Integer Ambiguity of Pulse Phase
The pulse periods of X-ray pulsars are extremely stable, which is conducive to
extracting the pulse profile, and through the cross-correlation processing of the pulse
profiles to obtain the time of arrival of the pulses. Meanwhile, this kind of stable