De-dispersion

Pulsar searching 37 probability of having a 3-σ or greater detection in all beams is therefore 10⁻¹⁸%. Miti-gating against impulsive RFI therefore becomes a case of selecting a threshold number of beams in which a signal must be detected above a given significance in a single time bin⁶. Signals which satisfy these criteria may be replaced by Gaussian noise based on the statistics of the surrounding data.

A similar methodology may be applied to mitigate against multi-beam RFI in the Fourier domain. In this case, the data from each beam may be examined in the Fourier domain, with Fourier bins which have power above some threshold in multiple beams, being removed. For a discussion on thresholds for signals in the Fourier and time domain see Sections 2.2.4.6 and 2.2.5.2.

As part of this work, we have developed a system to perform multi-beam RFI excision using all of the methods outlined above. In particular, this is the first system to im-plement multi-beam RFI filtering using the full time and frequency resolution of the incoming data. This system has been used to good effect in the processing of data from the High Time Resolution Universe North pulsar survey (see Chapter 4).

Other, arguably more elegant, forms of multi-beam RFI excision exist, such as the method proposed by Kocz et al. (2012), which considers the covariance matrix for each time sample of the observation. Unfortunately, this method, however powerful, is cur-rently limited by the amount of computation it requires.

Pulsar searching 38

1510 1450 1390 1330 1270

O bs er vi ng fr eq ue nc y (M H z)

1210

Pre

₋

dedispersion

0 1/2 1

Pulse phase

0 5 10 15 20

S / N

Post

₋

dedispersion

0 1/2 1

Pulse phase

Figure 2.4: An example single pulse from PSR B0355+54. Left-hand panels: Subband plot and pulse profile without taking into account the pulsar’s DM.Right-hand panels:

Subband plot and pulse profile after de-dispersion at the correct DM.

whereF is a stream of filterbank data containingN_chans channels and > j+ ∆bin_Nchans time samples. This process is known as standard incoherent de-dispersion and is the most common form of de-dispersion used for pulsar search observations. Figure 2.4 shows a frequency-resolved pulse before and after de-dispersion.

The method described above is the simplest realisation of the de-dispersion algorithm. A different realisation of this algorithm can be found in tree de-dispersion (Taylor, 1974).

Although tree de-dispersion is computationally faster than the method described above, it requires that the dispersive delay across the band be treated as a linear function. Such an effect can be achieved by the simple addition of extra channels to the filterbank data, as was shown to good effect in the first PMPS processing (Manchester et al., 2001).

Despite this, the power of modern computer processors has rendered tree de-dispersion somewhat obsolete, with speed improvements being generally outweighed by the added complexity the tree algorithm brings to any processing pipeline.

Standard incoherent de-dispersion is often split into two steps through the process of subbanding. Here the data are initially de-dispersed in groups of channels to form a set of de-dispersed subbands. The subbanded data may simply be thought of as a reduced-frequency-resolution filterbank file in which each channel is a de-dispersed time series.

By adding subbands together with appropriate time delays (see Equation 2.1), further de-dispersion to DMs similar to the DM of the subbands can be quickly performed.

Pulsar searching 39 When performing a search for unknown pulsars, we must de-disperse the data at many trial DMs to retain sensitivity to pulsars at all DMs. The choice of DMs to de-disperse to is an important factor in the processing pipeline, as using too many trial DMs will result in unnecessary computation, whereas using too few trial DMs will result in a lack of sensitivity to fast pulsars, especially at high DMs. To optimise the number and separation of the DM trials used in a pulsar survey, we must consider the deformation of a standard pulse profile due to instrumental and ISM effects.

The effective width, W_eff, of a pulse can be given by the quadrature sum of a series of geometrical factors, ∆t, which act to widen that pulse (see e.g. Manchester et al., 2001).

For a pulsar of intrinsic width W_int, W_eff =q

W_int² + ∆t²_scatter+ ∆t²_dispersion+ ∆t²_sampling, (2.3) where ∆t_scatter is the scattering timescale (see Section 1.1.7), ∆t_dispersion is the disper-sive timescale and ∆t_sampling is the timescale of the intrinsic smearing associated with sampled data.

Although ∆t_sampling ultimately determines the resolution of the pulse at 0 DM, its effect at higher DMs is negligible and thus the term may be safely ignored. The scattering timescale may be given by the empirical relation of Bhat et al. (2004) (see Equation 1.18). By this relation, scattering should be the dominant delay at high DMs. However, we note that the intrinsic scatter of delay estimates about this relation is several orders of magnitude. For this reason, scattering is normally not taken into account when de-termining optimal trial DM values. Therefore, the only remaining term is the dispersive smearing across the full band, given by (see e.g. Lorimer and Kramer, 2005)

∆tdispersion= 4.15×10³ MHz

f_bottom⁻² −MHz f_top²

! DM pc cm⁻³

sec, (2.4)

where f_bottom and f_top are the frequencies at the bottom and top of the band, respec-tively.

A logical choice for a DM step therefore becomes the DM for which the degree of smearing across the band is equal to the sampling time of the observation, i.e.

∆DM = 2.41×10⁻⁴

tsamp sec

MHz

f_bottom⁻² −MHz f_top²

!

pc cm⁻³ (2.5)

In this case thei^th DM trial is given byi×∆DM.

At higher DMs the dispersive smearing in individual channels becomes the dominant contribution to the effective pulse width. Considering that the channel bandwidth is

Pulsar searching 40

−60 −40 −20 0 20 40 60

∆DM pc cm⁻³

0.0 0.2 0.4 0.6 0.8 1.0

RelativeS/N

Period : 1 ms Period : 10 ms Period : 100 ms Period : 1000 ms

Figure 2.5: Relative S/N as a func-tion of DM for a selecfunc-tion of pulsar pe-riods. ∆DM = 0 at the true DM of the pulsar. All pulsars shown have a 5%

duty-cycle.

−10 −5 0 5 10

∆DM pc cm⁻³

0.0 0.2 0.4 0.6 0.8 1.0

RelativeS/N

Duty−cycle : 1%

Duty−cycle : 10%

Duty−cycle : 50%

Figure 2.6: Relative S/N as a func-tion of DM for a selecfunc-tion of pulsar duty-cycles. ∆DM = 0 at the true DM of the pulsar. All pulsars shown have a

10 ms period.

much smaller than the total bandwidth, we may represent the smearing in a single channel, ∆t_channel, of width ∆f, by the derivative of Equation 1.15,

∆t_channel = 8.3×10³

∆f MHz

MHz f³

DM pc cm⁻³

sec. (2.6)

In the case where ∆t_channel ≥nt_samp, the so-calleddiagonal DM, the data are oversam-pled and may be reduced in time resolution by a factor of n, where n∈ Z and n ≥2.

To do this, the data in each frequency channel are averaged over n adjacent samples.

As the sampling time is increased through the re-sampling, the DM step size may be recalculated through Equation 2.5.

To examine how the choice of DM step will affect the signal-to-noise ratio, S/N, of a pulsar with periodP, we may use Equation 1 of Dewey et al. (1985),

S/N ∝

rP −W_eff

W_eff . (2.7)

Combining the above equation with Equations 2.3 and 2.4 we can calculate the S/N behaviour of a pulsar at ∆DM away from its optimal DM. Figure 2.5 shows the DM evolution of the relative S/N of 1-, 10-, 100- and 1000-ms pulsars with duty-cycles of 5%. Figure 2.6 shows the DM evolution of the relative S/N of three, 10-ms pulsars with 1%, 10% and 50% duty-cycles, respectively.

For each trial DM, a de-dispersed time series is created which is then searched for periodic and accelerated signals.

Pulsar searching 41

Im Dokument Searching for Pulsars with the Effelsberg Telescope (Seite 59-63)