Dispersion and Techniques for its Correction

As mentioned earlier, propagation of pulsar signals through the tenuous plasma of the ISM produces dispersion of the pulses. This is because the speed of propagation through a plasma varies with the frequency of the wave (see chapter 16). Low frequency waves travel progressively slowly, with a cut-off in propagation at the plasma frequency. At high frequencies, the velocity reaches the velocity of light asymptotically. The difference in travel time between two radio frequencies $f_{1}$ and $f_{2}$ is given by

$$t_{d} ~~=~~ K\,DM\,\left(\frac {1}{f_{1}^{2}} ~-~ \frac {1}{f_{2}^{2}} \right) ~~~,$$ (17.4.1)

where $DM ~=~ \int n_{e}\,dl$ is the dispersion measure of the pulsar, usually measured in the somewhat unusual units of $pc\,cm^{-3}$, and $K \,=\, 4.149 \times 10^{6}$ is a constant. In this equation, $t_{d}$ is in units of millisec and $f_{1}$, $f_{2}$ are in units of MHz. For the typical ISM, a path length of 1 kiloparsec amounts to a $DM$ of about $30\, pc\,cm^{-3}$. Equation (1) can be used to derive the following approximate relationship for the dispersion smear time for a bandwidth $B$ centred at a frequency of observation $f_{0}$, for the case when $B \ll f_{0}$

$$\tau_{disp} ~~\simeq~~ {\left(\frac{202}{f_{0}}\right)}^{3} \, DM \, B ~~~,$$ (17.4.2)

where $\tau_{disp}$ is in millisec, $f_{0}$ and $B$ are in MHz and DM is in the units given earlier. Interstellar dispersion degrades the effective time resolution of pulsar data due to smearing, and this effect becomes worse with decreasing frequency of observation. For example, the dispersion smear time is about 0.25 millisec per MHz of bandwidth per unit DM at an observing frequency of 325 MHz. This means that a pulse of 25 millisec width would be broadened to twice its true width when observed with a bandwidth of 10 MHz, for a DM of $10\,pc\,cm^{-3}$. Even worse, signal from a pulsar of period 25 millisec would be completely smeared out and not be visible as individual pulses. Thus it is important to reduce the effect of interstellar dispersion in pulsar data. This is called dedispersion.

There are two main methods used for dedispersion - incoherent dedispersion and coherent dedispersion. In incoherent dedisperion, which is a post-detection technique, the total observing band (of bandwidth B) is split into $N_{ch}$ channels and the pulsar signal is acquired and detected in each of these. The dispersion smearing in each channel is less than the total smearing across the whole band, by a factor of $N_{ch}$. The detected signal from each channel is delayed by the appropriate amount so that the dispersion delay between the centers of the channels is compensated. These differentially delayed data trains from the $N_{ch}$ channels are added to obtain a final signal that has the dispersion smearing time commensurate with a bandwidth of $B/N_{ch}$, thereby reducing the effect of dispersion. In practical realisations of this scheme, the splitting of the band into narrow channels is usually carried out on-line in dedicated hardware (as described in section 17.3) while the process of delaying and adding the detected signals from the channels can be done on-line using special purpose hardware or can be carried out off-line on the recorded, multi-channel data. In this scheme, the final time resolution obtained for a given pulsar observation is limited by the number of frequency channels that the band is split into.

In coherent dedispersion, one attempts to correct for interstellar dispersion in a pulsar signal of bandwidth B before the signal goes through a detector, i.e. when it is still a voltage signal. It is based on the fact that the effect of interstellar scintillation on the electromagnetic signal from the pulsar can be modelled as a linear filtering operation. This means that, if the response of the filter is known, the original signal can be deconvolved from the received voltage signal by an inverse filtering operation. The time resolution achievable in this scheme is $1/B$ - the maximum possible for a signal of bandwidth B. Thus coherent dedispersion gives a better time resolution than incoherent dedispersion, for the same bandwidth of observation. It is the preferred scheme when very high time resolution studies are required - as in studies of profiles of millisecond pulsars and microstructure studies of slow pulsars. The main drawback of coherent dedispersion is that practical realisations of this scheme are not easy as it is a highly compute intensive operation. This is because the duration of the impulse response of the dedispersion filter (equal to the dispersion smear time across the bandwidth) can be quite long. To reduce the computational load, the deconvolution operation of the filtering is carried out in the Fourier domain, rather than in the time domain. Nevertheless, real time realisations of this scheme are limited in their bandwidth handling capability. Most coherent dedispersion schemes are implemented as off-line schemes where the final baseband signal from the telescope is recorded on high speed recorders and analysed using fast computers.