NASSP Masters 5003F - Computational Astronomy Lecture 11 Single-dish spectra: baselines. Complex numbers refresher Fourier refresher Correlation in general.

NASSP Masters 5003F - Computational Astronomy Spectra - baselines The spectrum of interest sits on top of a high ‘mesa’ due to system and background ‘temperature’ (ie noise). We usually want to subtract the mesa and just leave the spectrum. We could do that by alternating between on- and off-source observations, and subtracting the two: –But this needs 4 times as much observing time to reach the same SNR!

NASSP Masters 5003F - Computational Astronomy Spectra - baselines More commonly, the mesa is slowly varying compared to the spectrum, so one can fit some fairly smooth function to the mesa, then subtract it. The examples (for which this has already been done) show it is not always so simple! These show the infamous ‘Parkes ripple.’

NASSP Masters 5003F - Computational Astronomy The ‘Parkes ripple’ A weak Fabry-Perot resonance occurs between the dish and the feed. D ~ 26 m 2D = nλ = (n+1)(λ-Δλ) => Δ ν ~ 5.5 MHz.

NASSP Masters 5003F - Computational Astronomy Complex numbers REAL IMAGINARY

NASSP Masters 5003F - Computational Astronomy Complex numbers REAL IMAGINARY NONSENSE! There IS no √-1.

NASSP Masters 5003F - Computational Astronomy Let’s ‘forget’ about complex numbers for a bit......and talk about 2-component vectors instead. x y v x y θ

NASSP Masters 5003F - Computational Astronomy What can we do if we have two of them? x y v1v1 v2v2 We could define something like addition: There are lots of operations one could define, but only a few of them turn out to be interesting. v sum

NASSP Masters 5003F - Computational Astronomy The following has interesting properties: x y v1v1 v2v2 But it isn’t very like scalar multiplication except when all ys are zero. It’s fairly easy to show that: v prod θ2θ2 θ1θ1 θ prod =θ 1 + θ 2

NASSP Masters 5003F - Computational Astronomy These are just complex numbers! Note that: This, plus the angle-summing properties of the product, leads to the following typographical shorthand: Instead of the mysterious we should just note the simple identity

NASSP Masters 5003F - Computational Astronomy The lessons to learn: Complex numbers are just vectors. The ‘imaginary’ part is just as real as the ‘real’ part. Don’t be fooled by the fact that the same symbols ‘+’ and ‘x’ are used both for scalar addition/multiplication and for what turn out to be vector operations. This is a historical typographical laziness. –Be aware however that the notation I have used here, although (IMO) more sensible, is not standard. –So better go with the flow until you get to be a big shot, and stick with the silly x+iy notation.

NASSP Masters 5003F - Computational Astronomy fringes  point (delta function). Fourier refresher

NASSP Masters 5003F - Computational Astronomy higher spatial frequency  further from the origin. Fourier refresher

NASSP Masters 5003F - Computational Astronomy multiplication  convolution. Fourier refresher

NASSP Masters 5003F - Computational Astronomy gaussian  gaussian. Fourier refresher

NASSP Masters 5003F - Computational Astronomy Fourier refresher Hermitian  real.

NASSP Masters 5003F - Computational Astronomy Correlation in general. The (normalized) correlation (or cross- correlation) R 1,2 ( τ ) between two signals y 1 (t) and y 2 (t) is Its Fourier transform F {R} is ie the FT of y 1 times the conjugate of the FT of y 2.

NASSP Masters 5003F - Computational Astronomy This has many uses. We can calculate at τ =0. (Note, this single number R 1,2 (0) is sometimes written ie the expectation value of the product.) 1.Suppose y 1 and y 2 come from the same source, but different antennas. Suppose for simplicity that the signals are narrow-band. There is in general a phase difference of φ radians between signals gathered from different antennas. (In the next lecture I’ll show why.) In other words, y 1 (t) = |y| cos(2 πν t) y 2 (t) = |y| cos(2 πν t + φ)

NASSP Masters 5003F - Computational Astronomy R 1,2 (0) or is given by If however we have some way to shift the phase of y 2 by 90° (and such circuits exist), then and These are just the two components (‘real’ and ‘imaginary’ parts if you will) of a complex number |y| 2 e i φ which encodes the phase φ. Cross-corr for different antennas continued. y 2 (t) = |y| cos(2 πν t + φ – π/2 ) = |y| sin(2 πν t + φ)

NASSP Masters 5003F - Computational Astronomy Cross-corr to get polarization. 2.Suppose now that y 1 and y 2 come from the same antenna, but from feeds sensitive to opposite polarisations – say one sensitive to left-hand polarized, the other to RH. Let’s rename the signals y L and y R for clarity in this case. Then the coherency matrix of slide 6 lecture 10 is just (I haven’t got this entirely figured out to my own satisfaction yet.)

NASSP Masters 5003F - Computational Astronomy Cross-correlation at a range of τ : We can calculate R over a range of values of τ. The Fourier transform of this is a cross-power spectrum. –If y 1 =y 2 =y: the correlation is an autocorrelation (already mentioned in slide 14, lecture 3). Its FT is the power spectrum of y. –If y 1 and y 2 come from the same source, but different antennas: the FT is complex valued, and contains both spatial and spectral information.

NASSP Masters 5003F - Computational Astronomy The first thing necessary is to sample each continuous y at a number of times kΔt. Then R 1,2 (kΔt) is approximated by But, how many bits to use to store each y k value? Digital correlation y ykyk t k

NASSP Masters 5003F - Computational Astronomy Digital correlation 1 Surprisingly, 1 bit works pretty well! Multiplication becomes a boolean NOT(XOR). Allows us to use simple boolean logic circuits (cheap). SNR drops by about 2/ π though. 2 or 3 bits improves the SNR without too much increase in circuit cost y y k >0 k t