# Processing System for Coherent Dedispersion of Pulsar Radio Emission

I. A. Girin  S. F. Likhachev  A. S. Andrianov  M. S. Burgin  M. V. Popov  A.
G. Rudnitskiy  V. A. Soglasnov  V. A. Zuga

###### Abstract

The work describes a system for converting VLBI observation data using the
algorithms of coherent dedispersion and compensation of two-bit signal
sampling. Coherent dedispersion is important for processing pulsar
observations to obtain the best temporal resolution, while correction for
signal sampling makes it possible to get rid of a number of parasitic effects
that interfere with the analysis of the diffraction pattern of pulsars. A
pipeline has been established that uses the developed converter and the ASC
Software Correlator, which will allow reprocessing all archived data of
Radioastron pulsar observations and to conduct a search for giant pulses,
which requires the best temporal resolution.

[aff]organization=Astro Space Center, Lebedev Physical Institute, Russian
Academy of Sciences,addressline=Profsoyuznaya str. 84/32, city=Moscow,
postcode=117997, country=Russian Federation

##  1 Introduction

The Radioastron project comprised the 10-meter space radio telescope (SRT)
that together with ground VLBI antennas formed a space-ground interferometer
with a maximum baseline projection of up to 380000 km and a record angular
resolution of about 8 $\mu$as (Baan et al., 2022). The SRT was launched on the
18th of July 2011, and successfully operated till January 2019.

The strategy of the Radioastron mission was to archive all the original raw
baseband data. Such an approach provides an ability to re-correlate the
original data if new scientific problems arise or improved methods of data
reduction and interpretation are developed. At the end of the mission
operation, the total volume of the raw data was approximately 3500 TB. More
details about the archive and observations database can be found in Shatskaya
et al. (2020a, b).

Pulsar observations were an important part of the Radioastron scientific
program (Kardashev et al., 2013, 2017). They were conducted at 324 MHz
(P-band) or 1668 MHz (L-band) and in some cases at both frequencies
simultaneously. The single intermediate frequency (IF) bandwidth of the
Radioastron was 16 MHz. The P-band receiver supported one sub-band, while 1668
MHz had two 16 MHz sub-bands. In total, the data of 25 pulsars were
accumulated during the Radioastron operation, which included 98 observations
having total duration of 250 hours. Usually, at least two large ground-based
radio telescopes participated in the space-ground VLBI sessions, such as GBT,
Arecibo, or the WSRT aperture synthesis system.

As compared to single dish observations of pulsars that are usually performed
using temporal resolution, $\Delta t$, of the order of $1\,\mu{\mathrm{s}}$,
interferometric observations are conducted with lower values of $\Delta t$. In
particular, for the Radioastron pulsar observations $\Delta t=62\,$ns. High
temporal resolution of the recorded signal combined with high sensitivity of
the ground-based telescopes participating in the observations and their
ability to simultaneously measure the flux density in two polarization
channels permits, in principle, to study the phenomena that involve rapid
variations of intensity and/or polarization. An example of such a phenomenon
is the longitudinal dependence of the polarization of giant pulses of the Crab
pulsar. An analysis of that dependence was carried out by Main et al. (2021),
who detected the difference between locations of emitting regions where pulses
and interpulses originate.

In studying the physics of pulsars, to take full advantage of the high
temporal resolution of Radioastron observations it is necessary to
“dedisperse” the received signal, that is to correct it for smearing caused by
the frequency dependence of the propagation speed of radio waves in the
intervening interstellar plasma. The standard Radioastron data processing
includes the procedure of the so called “incoherent” dedispersion, but precise
measurements of rapid intrapulse variations in the data strongly influenced by
the interstellar plasma dispersion may require the application of more
accurate and much more computer-intensive “coherent” dedispersion (see Hankins
and Rickett (1975), van Straten and Bailes (2011)).

High temporal resolution of the interferometric data is achieved partly
through the low number of bits in each individual readout of the observed
signal. The ground- and space-based observations in the Radioastron project
were performed using one- and two-bit digitizing, respectively. Precision of
observations with a low number of digitizing levels depends critically on
proper choice of the quantization thresholds. The standard Radioastron data
processing routines are based on the assumption that the thresholds are set to
values close to optimal as described by Thompson et al. (2017).

The assumption usually holds for objects with slowly varying flux density,
where necessary adjustments are performed by an automatic gain control (AGC)
system. However, in observing pulsars the AGC is not used as it cannot
function properly when the received signal is rapidly variable. Consequently,
the digitizing may be performed with quantization thresholds that deviate
significantly from optimal level, and reduction of the data using the standard
approach introduces additional error in the final results. To minimize that
error, the algorithms of processing of the digitized signal should be
generalized to the case of arbitrary values of quantization thresholds. This
problem was considered by Jenet and Anderson (1998).

An example of how using a standard procedure for processing pulsar
observations leads to erroneous conclusions was presented by Popov et al.
(2023). In that paper it was shown that the dynamic spectra of the
interstellar scintillations of the PSR B1237+25 obtained using the standard
data processing algorithms exhibit frequency shifts that depend on the
longitude and, if real, could be interpreted as a signature of the
“interstellar interferometer” effect. However, the shift almost disappears
when the coherent dedispersion and proper correction for the actual values of
quantization thresholds is applied.

In this paper we describe the software for processing pulsar observations
performed by the Radioastron. The newly developed preprocessor of the raw
observational data and the modified ASC software correlator allow one to
perform the coherent dedispersion and correctly account for digitization with
non-optimal quantization thresholds.

The paper is organized as follows: in Section 2 we briefly describe the
methods of dedispersion, and the numerical method used for correction for
digitizing is outlined in Section 3. Technical details of the implementation
are described in Section 4. We present the results of application of the
described methods to Radioastron observations of the PSR B1237+25 in Section 5
and conclude in Section 6.

##  2 Correction of distortions caused by the dispersion

Due to the dispersion of radio waves, emission propagating through the ionized
interstellar plasma arrives at the observer with a delay depending on the
signal frequency. A short quasi-monochromatic impulse emitted at the low-
frequency edge of the observing band will be received later than a
synchronously emitted impulse at the high-frequency edge. The relative delay,
$\tau_{\mathrm{d}}$, is given by

$\tau_{\mathrm{d}}=D\cdot\left(\frac{1}{f_{0}^{2}}-\frac{1}{(f_{0}+B)^{2}}\right),$
(1)

where $f_{0}$ and $B$ are the lowest observing frequency and the bandwidth,
respectively. The coefficient $D$ (s$\cdot$MHz2) is related to the dispersion
measure $DM$ (pc$\cdot$cm-3) by

$DM=2.410\times 10^{-4}\cdot D,$ (2)

and the dispersion measure is defined as the integral of the electron column
density along the line of sight to the pulsar. As a result of relative delay,
the variability in integrated flux on time scales shorter than
$\tau_{\mathrm{d}}$ is temporally smeared.

There exist two methods to reduce the effect of smearing: post-detection or
incoherent dedispersion and pre-detection or phase-coherent dedispersion.
Incoherent dedispersion is performed by dividing the frequency band into many
smaller bins. The signal received in each bin is shifted in time to compensate
for the difference in arrival time, then the shifted signals are summed. Such
a method compensates for the delay between the bins, but the dispersion within
individual bins remains uncompensated.

The coherent dedispersion is free from such a drawback. The method is based on
the fact that the effect of dispersion on the signal received from a pulsar
can be modeled as a linear filtering operation and the original signal can be
recovered from the received signal by performing the inverse filtering. The
effect is best described in the frequency domain. The dispersion causes the
phase shift,

$\phi(f)=\frac{2\pi Df^{2}}{f_{0}^{2}\cdot(f_{0}+f)},$ (3)

of a Fourier component of the original signal corresponding to frequency $f$.
Consequently, the Fourier transform of the dedispersed signal may be computed
by shifting the phase of each Fourier component of the observed signal by
$-\phi(f)$.

In the observations of pulsars, this procedure is complicated by the fact that
the function sought to be measured is not the Fourier spectrum of the signal
in the strict mathematical sense, but the so called dynamic spectrum.
Computation of the dynamic spectrum consists in dividing the observing session
in many non-intersecting time intervals, usually of equal duration and
covering the whole session, and performing the Fourier analysis of the signal
on each interval separately. Because of the dispersion, the interval at which
the signal is received depends not only on the moment of emission, but also on
the frequency. For this reason, coherent dedispersion can not be performed
independently on each separate interval. An approach to overcome this
difficulty was proposed by Hankins and Rickett (1975), who presented an
elaborate description of the coherent dedispersion technique that we follow in
our study.

##  3 Correction for 2-bit Sampling

Usually, ground telescopes record the signal using 2-bit (four-level)
digitizing. Four levels prescribed for such digitizing are -3, -1, +1, and +3.

The transition threshold between the levels $1$ and $3$ is supposed to be
close to the value of the original analog signal RMS ($\sigma$). The optimal
value for the threshold is $t=0.9674\sigma$ in the case of 2-bit sampling
(Thompson et al., 2017).

Besides, it is important to switch off the telescope’s automatic gain control
(AGC) system in pulsar observations because the system inertia would not
operate properly at the ON-pulse and OFF-pulse stages of pulsar observation.
Hence a pulsar signal is an example of a non-stationary noise process. Thus,
records for such signals must be corrected for 2-bit digitizing. The problem
was considered by Jenet and Anderson (1998). They have demonstrated that
digitizing the signal before removing the dispersive effects generates
unwanted systematic artifacts in the data. Namely, the “negative” dips appear
around the pulse in the average pulse profile.

We follow the method described in Jenet and Anderson (1998). First, it is
necessary to estimate the undigitized power level $\sigma_{0}$ using a
fraction of samples $\Phi$ with values of $\pm$1 in the given portion of the
signal record:

$\Phi(t)=\frac{1}{\sqrt{2\pi}\sigma}\int\limits_{0}^{t}\exp{\left({-{\frac{x^{2}}{2\sigma^{2}}}}\right)}dx={\rm
erf}\left(\frac{t}{\sqrt{2}\sigma}\right)$ (4)

With the value of $\sigma_{0}$ known it is possible to calculate the corrected
values of signal $y1$ and $y3$ instead of $\pm$1 and $\pm$3 using Eq. (40) and
(41) from Jenet and Anderson (1998). In practice, we calculated all values
using $\sigma_{0}$ as an independent variable, and then we consider the
results with independent variable $\Phi$. These results are shown in Fig. 1.
To make corrected values closer to VLBI standard, the obtained $y1$ and $y3$
were multiplied by a factor of 2. Let us define the calculated threshold value
from Eq. (4) as $\Lambda$. For example, $y1(0.67)=1.085$ and $y3(0.67)=3.23$
and $\Phi=0.67$ for $t=0.9674$ is an optimal threshold.

Figure 1: The corrected values $y1$ and $y3$ versus observed fraction of
samples equal $\pm 1$.

Approximation functions were introduced to calculate $y1$ and $y3$ from the
measured fraction of samples $\Phi$ with digitized values as $\pm 1$:

$y1(\Lambda)=a1+b1\cdot\Lambda+c1\cdot\Lambda^{2}+d1\cdot\Lambda^{3}+e1\cdot\Lambda^{4}$
(5)
$y3(\Lambda)=a3\cdot\exp{\left(-\frac{\Lambda^{b3}}{d3}\right)}+c3+e3\cdot\Lambda$
(6)

The polynomial approximation is sufficient for $y1$, while an approximation by
a combination of an exponent and linear term is more suitable for $y3$. The
coefficients used in approximating functions are the following:

$a1=1.1438(5)$, $b1=0.169(7)$, $c1=-0.96(3)$, $d1=1.62(4)$, $e1=-1.13(2)$;

$a3=1920(200)$, $b3=0.228(5)$, $c3=3.24(4)$, $d3=0.119(1)$, $e3=-1.39(3)$.

RMS uncertainties of the coefficients are given in brackets. The difference
between the calculated and approximated values can’t be distinguished well due
to the coarse scale in the figures, . In fact, RMS residuals are 0.0012 for
$y1$ approximation and 0.011 for $y3$.

##  4 Implementation of Coherent Dedispersion

###  4.1 ASC Software Correlator

The VLBI data are processed using a correlator. There are two types of
correlators: software and hardware, which can be XF and FX correlators.
Software correlators perform data processing on conventional computers or on
computing clusters running operating systems such as Microsoft Windows or
Linux. In turn, hardware correlators are implemented using programmable logic
integrated circuits (FPGAs). Correlators of the XF and FX types differ in the
sequence of operations performed, where X stands for multiplication and F for
Fourier transform. Accordingly, in XF correlators, the signals are first
multiplied, and then the Fourier transform is performed. In FX correlators,
the Fourier transform performed and the result is multiplied.

A dedicated software correlator (the ASC correlator) was developed in Astro
Space Center to process the data of the Radioastron mission. It is a software
FX correlator that calculates auto and cross spectra for all possible
combinations of baselines and polarizations. The detailed description is
provided in Likhachev et al. (2017).

The ASC software correlator operates on a high-performance computing (HPC)
cluster using the message-passing interface (MPI) for parallel calculations.
Its total computing power is 1 Tflops. This software correlator supports
several modes of data processing: continuous spectrum, spectral lines, and
several types of pulsar data processing, including the search and correlation
of giant pulses. It supports widespread nomenclature of VLBI baseband data
formats such as MarkIV/A, Mark5B, VDIF, RDF, LBA, etc. The ASC Correlator
processed about 95% of Radioastron data, including the pulsar one. The
complexity of the pulsar VLBI data processing primarily lies in the need to
compensate for the dispersion of the signal.

###  4.2 Compensation of Dispersion

####  4.2.1 Incoherent Dedispersion

Incoherent dedispersion implemented in the ASC software correlator is
performed in several steps. The pulsar period is divided into bins. The
correlator calculates the spectra for each bin. The delay of signal in each
bin then is compensated using a polynomial model provided by the TEMPO2
software package (Edwards et al., 2006). Further, each bin is averaged over
the observation time. As a result, each bin will contain the signal with
removed dispersion. The dispersion effect will decrease linearly with the
increasing number of bins. Accordingly, it is necessary to sum only the data
that contain fringes of pulses and discard the rest. It is important to
compensate the pulse shape with a dispersive delay (Likhachev et al., 2017).

####  4.2.2 Coherent Dedispersion

Up to now, the ASC correlator has not been able to perform data processing
using coherent dedispersion. For these purposes we have developed a data
converter that coherently removes the dispersion effects. This converter
creates the data in a new complex format, which is supported by the updated
version of the correlator. The converted data then is processed in the
standard mode but with the dedispersion switched off. Such an approach
minimized the changes in the ASC software correlator structure. Based on the
converter, a pipeline was developed for processing pulsar observations. The
operation steps of this pipeline are the following:

At first, one has to calculate the discrete Fourier transform of the recorded
signal. Next, the result of the Fourier transform is multiplied by the complex
frequency response function given by Eq. (3). Inverse Fourier transform will
yield the recovered signal. It is equivalent to a convolution of impulse
response function with the recorded voltage signal in a time domain. The
discrete Fourier transform is represented with the periodical assumption.
Thus, the first $n$ points corresponding to the time interval of
$\tau_{\mathrm{d}}$ will be erroneously multiplied with the data wrapped
around from the end of the segment and it must be discarded. So in this
approach the first portion of the signal will be lost in the interval
$\tau_{d}$.

Figure 2: The scheme of sequential sampling of recording sections with a
duration $T$ from a continuous record.

Fig. 2 illustrates the scheme of sequential sampling of signal sections of
duration $T$ from a continuous recording. Sequential values of starting time
$t_{i}$ are determined by the relation $t_{i}=t_{0}+i(T-\tau_{\mathrm{d}})$.
It is clear that $T$ must be larger than $\tau_{\mathrm{d}}$ and the
efficiency of data reduction is $1-\tau_{\mathrm{d}}/T$. More details on data
reduction approach used in our processing system are given below in Section 4.

  * 1.

Read raw baseband data sample of $M$ size. Supported data formats are the same
as for ASC Correlator: RDF (Radioastron Data Format), MarkIV/A, Mark5B, K5A,
LBA, and VDIF.

  * 2.

Split the sample $M$ into $P$ pieces, each piece of $N$ size ($P=M/N$).

  * 3.

Determine a parameter $\Lambda$ by counting portion of samples equal to $\pm$1
and calculate the values $y1$ and $y3$ for each data sample in the $P$th piece
using equations 5,6;

  * 4.

Replace every sample $s_{i}$ by the values $y1$ or $y3$, thus performing 2-bit
sampling correction;

  * 5.

Calculate the matrix $\phi_{i}$ of the phase shifts depending on the frequency
$f_{i}$:

$\phi_{i}=\frac{2\pi Df_{i}^{2}}{f_{0}^{2}\cdot(f_{0}+f_{i})}$ (7)

  * 6.

Perform a complex $M$-point Fourier transform for each data chunk numbered
$k=1,...,P$;

  * 7.

Multiply Fourier spectra by a correction complex function
$R_{i}=e^{-\phi_{i}}$;

  * 8.

Add $M$ more zeros to the end of the array to get an array of $2M$ values;

  * 9.

Perform inverse FFT to obtain complex $M$ samples;

  * 10.

Write the converted data to the output file in half-precision float (2 bytes)
format.

The converter is capable to adjust the parameters of the FFT size, and the
data sample size. At the output a new data file is generated in the special
format, which can be used in the ASC correlator.

Bit statistics (the number of $\pm 1$ and $\pm 3$) are estimated with a
floating average method for $P$ data piece size ($N$). The statistics are
calculated for each data point in the interval [current position - N/2,
current position + N/2]. Further, the 2-bit sampling correction coefficient is
considered by a function of estimated statistics (5 and 6).

The correlated data will be in a UVX format where cross and auto spectra are
represented as a float (4 bytes number format). The UVX files can be converted
straight to the widespread IDI-FITS format for further analysis. Finally,
bandpass correction and noise cleaning can be applied to the correlated data
using the corresponding calibration tools of ASL software package Likhachev et
al. (2020).

####  4.2.3 Data Format

The developed converter uses its special format as an output (No Packet Data
or NPD), which is supported by the ASC software correlator. The first 512
bytes are dedicated to the header, which is a string that contains the
information about the observation parameters separated by commas and includes:
version, date (DD/MM/YYYY), time (HH:MM:SS.sss), bandwidth ($1\cdot
10^{-6}\times 1/dt$ – full bandwidth including all sub-bands), number of
channels ($NCH$, polarization channels and sub-band channels), number of bits
per sample.

The header is followed by half-precision float FP16$\times$NCH data. The
single-time data sample is (2$\times$NCH) bytes. The order of channels is the
same as for the initial input raw baseband data. At the same time, the lower
sub-band is converted to the upper one, which in turn requires a shift in the
reference frequency for it when correlating the converted data.

##  5 Results

We have performed several data processing experiments with the Radioastron
observations of the pulsar B1237+25 to test the effectiveness of the developed
coherent dedispersion data converter. The observations were conducted at 324
MHz with a 16 MHz IF bandwidth. Arecibo and Green Bank radio telescopes took
part in the session as ground support. The dispersion measure of the pulsar is
$9.3$ pc$\cdot$cm-3. The estimated time smearing is 36.35 ms in the band of
316-332 MHz. For this test, we used the following parameters of the number of
samples and pieces: $M=10^{6}$ and $P=10^{3}$.

Fig. 3 (left panel) shows the reconstructed signal smoothed by 1 ms for one
individual pulse recorded at the Green Bank radio telescope. The lower curve
shows the signal after coherent dedispersion, but without 2-bit sampling
correction. The upper curve shows the signal with sampling correction applied.
It can be seen that the applied 2-bit correction eliminates parasitic signal
dips around the pulse and improves the SNR by about 20%.

Figure 3: Comparison of pulsar signals for B1237+25 restored without
correction for 2-bit sampling (lower curves), an with such correction applied
(upper curves). The left panel shows a single pulse, while the right panel
presents a pulse averaged over 10 seconds (7 pulses). Time resolution is one
millisecond.

Fig. 3 (right panel) gives a similar illustration for a pulse averaged over 10
s. in time (7 individual pulses). The distortions caused by two-bit
digitization are complex and need to be corrected.

Two-bit sampling of the signal causes additional distortion in the radio
spectra of pulsar pulses. These radio spectra contain important information
about the scattering and scintillation parameters of the interstellar plasma
inhomogeneities, namely, the so-called diffraction pattern. From an
astrophysical point of view, it is of interest to compare the diffraction
pattern for different parts of the pulsar averaged profile. The detection of a
shift in the diffraction pattern with longitude would indicate that the radio
emission at different longitudes occurs in spatially different parts of the
pulsar’s magnetosphere, i.e. it would be possible to measure such
localization.

Figure 4: Frequency shift between dynamic spectra obtained at different
longitudes of the mean profile of B1237+25. The solid curve passing through
the squares corresponds to the analysis of the spectra obtained without two-
bit digitization correction, and the dotted line passing through the circles
corresponds to the spectra obtained for the corrected signal.

Fig. 4 shows measurements of the frequency shift of the diffraction pattern in
the spectrum of the B1237+25 pulsar as a function of the longitude of the
averaged profile. The observations of 20 minutes were carried out with the
Arecibo radio telescope. The solid curve with squares corresponds to the
measurements obtained without two-bit digitization correction and the dotted
line with circles corresponds to the measurements for the corrected signal.
The notable effect of the diffraction pattern drift with longitude disappears
after the correction. This case is considered a separate study.

The program supports multi-threading and we have estimated the approximate
performance on the VDIF data of the Green Bank telescope of B1237+25 which had
two polarization channels (LCP and RCP) and one sub-band with a bandwidth of
16 MHz. The measured time ratio was 1:14, thus it takes 14 seconds of real-
time to convert one second of such data.

##  6 Conclusions

The developed converter allows not only perform of coherent dedispersion but
also correct the digitized signal for two-bit sampling. This tool, together
with the ASC software correlator and the ASL software package, forms a new
pipeline for processing the pulsar data of the Radioastron project. This
pipeline was tested on the observational data of the pulsar B1237+25 observed
by the Radioastron space-ground interferometer. The corrected data then were
used in the studies in Popov et al. (2023).

Processing of pulsar data with compensation for both dispersion and 2-bit
sampling is an important point and allows the removal of the parasitic effects
in the data that interfere with their subsequent correct interpretation. It
was shown, that the distortions caused by the 2-bit sampling can introduce
interference leading to false pulsar diffraction pattern drift in longitude
disappearing after bit statistic compensation.

Further, this pipeline will be used to reprocess the Radioastron pulsar raw
baseband data with the maximum available temporal resolution, not smeared by
dispersion, and also to process any pulsar VLBI data of modern known and
supported formats.

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