A real-time cyclic spectroscopy backend for the Green Bank Telescope

Scintillometry 2024 • University of Central Florida • October 31, 2024

Ross Jennings & Jacob Turner

What is cyclic spectroscopy?

A specialized signal processing technique for characterizing cyclostationary signals.

• Cyclostationary = “statistically periodic” (stationary, for fixed pulse phase). Pulsar signals are cyclostationary!
• Cyclic spectrum = power spectrum, plus more information, indexed by cycle frequency (harmonics of pulse frequency).
• Periodic spectrum = Fourier transform of cyclic spectrum wrt cycle frequency. Roughly: intensity vs. frequency and pulse phase.

Why cyclic spectroscopy?

For one thing: much higher frequency resolution for pulsar observations! (Turner et al., 2024).

But – can’t a regular filterbank do that too?

CS allows you to go to high frequency resolution without sacrificing pulse phase resolution!

This “should” be impossible (uncertainty principle), and in fact is impossible for non-cyclostationary signals!

The Gabor limit, and how to beat it

A general filterbank setup (STFT):

• Can choose w(t) arbitrarily, but subject to uncertainty principle:

A narrower w(t) means a wider ŵ(ν).

• At a point in time, the signal is just one number – to find its frequency content, you need to use nearby points as well. This reduces the time resolution.
• Workaround: Instead of using nearby points, use points a whole number of periods away! This only works because of the cyclostationary property.

Uncertainty principle (Kennard bound):

A cyclostationary covariance matrix

• For a continuous signal, Σ_ij = ⟨x_i x_j*⟩ is replaced by C(t₁, t₂) = ⟨x(t₁) x*(t₂)⟩.
• For a cyclostationary signal, shifting both t₁ and t₂ by P is a symmetry:

C(t₁ + P, t₂ + P) = C(t₁, t₂).

Keep in mind:

A normal distribution is completely described by Σ (and μ).

From covariance matrix to cyclic spectrum

Periodic spectrum

Cyclic spectrum

represents frequency domain correlations

Covariance matrix in symmetric coordinates:

defined only at cycle frequencies (α_k = k/P)

The cyclic spectrum of a pulsar

Before scintillation:

Dolch et al. (2021)

Demorest (2011)

Periodic spectrum

Folded filterbank

Cyclic spectrum intensity

Cyclic spectrum phase

Scintillation can be represented by convolution with an impulse response function (IRF) h(t), with Fourier transform (filter function) H(ν):

Cyclic spectrum:

The phase of H(ν) is (partially) preserved!

pulse profile

Deconvolving the ISM impulse response

• Because the phase H(ν) is preserved, it can be reconstructed from the observed cyclic spectrum.
• This allows the IRF, h(t), to be fully deconvolved from the intrinsic pulsar emission.
• This is a major advantage compared to traditional methods, which worked with either |H(ν)|² or |h(t)|². (Turner et al. 2023).

Walker et al. (2013)

Simulating CS data

Code is available on GitHub!

How can cyclic spectroscopy benefit scintillometry efforts?

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Scintillation Arcs (and arclets) Now Resolvable!

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Credit: Turner et al. (2024)

Cyclic Spectroscopy Recovers Amplitude AND Phase!

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Transfer Function of ISM can be Recovered from Cyclic Spectrum

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Multiple Ways to Approach Recovery

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Transfer Function of ISM can be Recovered from Cyclic Spectrum

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Multiple Ways to Approach Recovery

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• Iterative Deconvolution (see Pycyc)

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Transfer Function of ISM can be Recovered from Cyclic Spectrum

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Multiple Ways to Approach Recovery

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• Iterative Deconvolution (see Pycyc)

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• Directly from cyclic spectrum (Assuming )

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Transfer Function of ISM can be Recovered from Cyclic Spectrum

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Multiple Ways to Approach Recovery

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• Iterative Deconvolution (see Pycyc)

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• Directly from cyclic spectrum (Assuming )

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• Eigendecomposition? (Work in progress…)

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Stack Recovered Transfer Functions over many Subintegrations to get Dynamic Wavefield

Dynamic Wavefield Power

Images from Walker, Demorest, & van Straten (2013)

Stack Recovered Transfer Functions over many Subintegrations to get Dynamic Wavefield

Take 2D Fourier Transform of Dynamic Wavefield to get Conjugate Wavefield

Secondary Wavefield

Dynamic Wavefield Power

Images from Walker, Demorest, & van Straten (2013)

Introducing The Green Bank Observatory Cyclic Spectroscopy Backend

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Keywords

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• Roach - FPGA Board

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• VEGAS - Primary GBT Pulsar Backend

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• Cyclid - Cycspec Processing Pipeline

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• Guppi daq - Backend software for GUPPI (modified for VEGAS)

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Results from Testing

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Approaching Real-Time Processing Capabilities for Preferred Configurations

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User Interface

Cyclops: A way to monitor Cycspec Observations

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User Interface

Additional Astrid Commands/Proposal Info

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Available for proposals beginning in the 26A semester! Be on the lookout for an announcement in June/July 2025!

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