What is Cyclic Spectroscopy?#
Cyclic spectroscopy (CS) is an advanced signal processing technique that takes advantage of the period nature of pulsars to simultaneously achieve high radio frequency and pulse phase resolution. The application of CS to pulsars is described in Demorest [2011].
Traditional signal processing techniques measure radio-frequency power spectra via a Fourier transform. There are different approaches for doing so (e.g. autocorrelation spectroscopy and polyphase filterbanks) but they are all subject to the Fourier transform uncertainty principle. Specifically, the frequency resolution one can achieve is inversely proportional to the temporal resolution. In other words, to resolve narrowband frequency structure, one has to sacrifice time resolution.
Pulsars, in particular millisecond pulsars (MSPs), frequently exhibit temporal structure on the order of 0.1 - 0.01 milliseconds, and resolving this temporal structure is critical for high-precision pulsar timing. As such, pulsar observers typically prioritize high time resolution at the expense of frequency resolution. A common observing setup obtains frequency resolution of about 1 MHz and time resolution of about 1 ms.
The Fourier transform uncertainty principle specifically applies to wide-sense stationary processes. However, the periodic nature of pulsars makes them cyclostationary, i.e. their statistical properties are periodic in nature. This property allows one to use advanced signal processing techniques to break the degeneracy between time and frequency resolution and simultaneously achieve very high radio frequency and pulse phase resolution. The resulting cyclic spectrum measures amplitude and phase as a function of radio frequency (\(\nu\)) and cycle frequency (\(\alpha\)), with a frequency resolution limited only by \(1/P\), where \(P\) is the pulsar period.
When to Use Cyclic Spectroscopy#
CS is useful when observing pulsars that are highly scattered by the ionized interstellar medium (IISM). Density variations in the IISM diffract pulsar signals at radio frequencies, scattering rays into an observer’s line of sight that would not otherwise intersect with the Earth. The differing path lengths upon which these signals travel causes them to arrive at Earth with different phases, leading to constructive and destructive interference known as scintillation. The resulting interference pattern varies with time and frequency leading to increases in flux density that are localized over a characteristic scintillation bandwidth, \(\Delta \nu_{\rm scint}\), and timescale, \(t_{\rm scint}\). Multi-path propagation also causes pulsed emission to arrive at the Earth with varying time delays that are functionally described as a one-sided exponential that decays by \(1/e\) over a scattering timescale, \(\tau_{\rm s}\). The relationship between the scattering timescale and scintillation bandwidth is given by
where C1 is a constant of proportionality related to the geometry of density variations in the IISM. For a uniform electron density model with a Kolmogorov turbulence spectrum \(C_1\) = 1.16, while for a thin-screen density model and Kolmogorov spectrum \(C_1\) = 0.957.
Within pulsar astronomy a measurement of pulse flux density as a function of time and frequency (i.e. an observation of the diffractive interference pattern) is known as a dynamic spectrum, and such a measurement encodes a wealth of information about the IISM. Specifically, measurement of the scintillation bandwidth can be used to estimate the scattering delay up to the value of \(C_1\). Furthermore, a 2-D Fourier transform of the dynamic spectrum, known as a secondary spectrum, often reveals parabolic arcs and inverted parabolic arclets which are related to the distance to the scattering screens. The secondary spectrum is thus a valuable probe of the geometry of the IISM.
While scattering is useful for studying the IISM, it serves as a nuisance term for high-precision pulsar timing. Scattering delays bias measurements of pulse times of arrival (TOAs), and epoch-to-epoch changes in ts are a source of stochastic red noise in pulsar timing models. The influence of nanohertz-frequency gravitational waves (GWs) also manifests as a stochastic red noise process, so unmodeled scattering delays decrease pulsar timing array sensitivity to GWs.
Scattering delays can be estimated by measuring \(\Delta \nu_{\rm scint}\) from dynamic spectra (which, as previously mentioned, are also useful for studying the IISM). However, very highly scatted pulsars may have \(\Delta \nu_{\rm scint}\) that are too narrow to resolve with traditional techniques. Thus CS may be useful for resolving scintles in highly scattered pulsars.
This has several benefits. First, it may be possible to measure \(\Delta \nu_{\rm scint}\), and thus estimate \(\tau_{\rm s}\), while maintaining adequate pulse phase resolution for high-precision pulsar timing. Second, measuring the average phase slope of the cyclic spectrum provides another measure of the total time delay that a pulsar signal experiences. Finally, under certain conditions it may be possible to measure the impulse response function (IRF) of the IISM, and to thus measure ts directly.
The GBT Cyclic Spectroscopy Backend#
CS is computationally demanding and its use was historically limited to special instruments. GBO now offers an observatory-supported CS backend that operates in close-to-real time and that is controlled using the standard GBT observing interface (Astrid). It operates in parallel with VEGAS, producing traditional data products and periodic spectra (which are related to the cylclic spectrum via a Fourier transform).
The Cyclic Spectroscopy reference section contains detailed technical information on the CS backend.
The How-to guide on Cyclic Spectroscopy Observations of a Pulsar contains detailed instructions on how to obtain the cyclic spectrum of known pulsars.