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Register a new userImaging on a Sphere with Interferometers: the Spherical Wave Harmonic Transform
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T. D. Carozzi
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I present an exact and explicit solution to the scalar (Stokes flux intensity) radio interferometer imaging equation on a spherical surface which is valid also for non-coplanar interferometer configurations. This imaging equation is comparable to $w$-term imaging algorithms, but by using a spherical rather than a Cartesian formulation this term has no special significance. The solution presented also allows direct identification of the scalar (spin 0 weighted) spherical harmonics on the sky. The method should be of interest for future multi-spacecraft interferometers, wide-field imaging with non-coplanar arrays, and CMB spherical harmonic measurements using interferometers.
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The millihertz gravitational-wave frequency band is expected to contain a rich symphony of signals with sources ranging from galactic white dwarf binaries to extreme mass ratio inspirals. Many of these gravitational-wave signals will not be individually resolvable. Instead, they will incoherently add to produce stochastic gravitational-wave confusion noise whose frequency content will be governed by the dynamics of the sources. The angular structure of the power of the confusion noise will be modulated by the distribution of the sources across the sky. Measurement of this structure can yield important information about the distribution of sources on galactic and extra-galactic scales, their astrophysics and their evolution over cosmic timescales. Moreover, since the confusion noise is part of the noise budget of LISA, mapping it will also be essential for studying resolvable signals. In this paper, we present a Bayesian algorithm to probe the angular distribution of the stochastic gravitational-wave confusion noise with LISA using a spherical harmonic basis. We develop a technique based on Clebsch-Gordan coefficients to mathematically constrain the spherical harmonics to yield a non-negative distribution, making them optimal for expanding the gravitational-wave power and amenable to Bayesian inference. We demonstrate these techniques using a series of simulations and analyses, including recovery of simulated distributed and localized sources of gravitational-wave power. We also apply this method to map the gravitational-wave foreground from galactic white-dwarfs using a simplified model of the galactic white dwarf distribution.
We present the generalized iterative residual fitting (IRF) for the computation of the spherical harmonic transform (SHT) of band-limited signals on the sphere. The proposed method is based on the partitioning of the subspace of band-limited signals into orthogonal subspaces. There exist sampling schemes on the sphere which support accurate computation of SHT. However, there are applications where samples~(or measurements) are not taken over the predefined grid due to nature of the signal and/or acquisition set-up. To support such applications, the proposed IRF method enables accurate computation of SHTs of signals with randomly distributed sufficient number of samples. In order to improve the accuracy of the computation of the SHT, we also present the so-called multi-pass IRF which adds multiple iterative passes to the IRF. We analyse the multi-pass IRF for different sampling schemes and for different size partitions. Furthermore, we conduct numerical experiments to illustrate that the multi-pass IRF allows sufficiently accurate computation of SHTs.
We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.
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