Editing Array processing (section)

== Correlation spectrometer ==
The problem of computing pairwise correlation as a function of frequency can be solved by two mathematically equivalent but distinct ways. By using [[Discrete Fourier Transform]] (DFT) it is possible to analyze signals in the time domain as well as in the spectral domain. The first approach is "XF" correlation because it first cross-correlates antennas (the "X" operation) using a time-domain "lag" convolution, and then computes the spectrum (the "F" operation) for each resulting baseline. The second approach "FX" takes advantage of the fact that convolution is equivalent to multiplication in Fourier domain. It first computes the spectrum for each individual antenna (the F operation), and then multiplies pairwise all antennas for each spectral channel (the X operation). A FX correlator has an advantage over a XF correlators in that the computational complexity is [[Big O notation|O]](N<sup>2</sup>). Therefore, FX correlators are more efficient for larger arrays.<ref>{{cite journal
 | last1=Parsons |first1=Aaron |last2=Backer |first2=Donald |last3=Siemion |first3=Andrew |authorlink3=Andrew Siemion
 | date = September 12, 2008
 | title = A Scalable Correlator Architecture Based on Modular FPGA Hardware, Reuseable Gateware, and Data Packetization
 |journal=Publications of the Astronomical Society of the Pacific |volume=120 |issue=873 |pages=1207–1221 | doi = 10.1086/593053
 |arxiv = 0809.2266 |bibcode=2008PASP..120.1207P|s2cid=14152210 }}</ref>

Correlation spectrometers like the [[Michelson interferometer]] vary the time lag between signals obtain the power spectrum of input signals. The power spectrum <math>S_{\text{XX}}(f)</math> of a signal is related to its autocorrelation function by a Fourier transform:<ref name="Harris">[http://www.sofia.usra.edu/det_workshop/papers/session4/4-04harris_edjw021022.pdf ''Spectrometers for Heterodyne Detection''] {{webarchive |url=https://web.archive.org/web/20160307051932/http://www.sofia.usra.edu/det_workshop/papers/session4/4-04harris_edjw021022.pdf |date=March 7, 2016 }} Andrew Harris</ref>

{{NumBlk|:|<math>S_{\text{XX}}(f) = \int_{-\infty}^{\infty} R_{\text{XX}}(\tau) \cos(2 \pi f \tau),\mathrm{d}\tau</math>|{{EquationRef|I}}}}

where the autocorrelation function <math>R_{\text{XX}}(\tau)</math> for signal X as a function of time delay <math>\tau</math> is

{{NumBlk|:|<math>R_{\text{XX}}(\tau) = \left( V_X(t) V_X(t + \tau)\right)</math>|{{EquationRef|II}}}}

Cross-correlation spectroscopy with spatial interferometry, is possible by simply substituting a signal with voltage <math>V_Y(t)</math> in equation {{EquationNote|Eq.II}} to produce the cross-correlation <math>R_{\text{XY}}(\tau)</math> and the cross-spectrum <math>S_{\text{XY}}(f)</math>.