1. NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 16 Further with interferometry – Digital correlation Earth-rotation synthesis and non-planar arrays Resolution and the field of view; Binning in frequency and time, and its effects on the image; Noise in cross-correlation; Gridding and its pros and cons.

2. NASSP Masters 5003F - Computational Astronomy - 2009 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

3. NASSP Masters 5003F - Computational Astronomy - 2009 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. 1111 00000000 111111111 0 1 000 y y k >0 k t

4. NASSP Masters 5003F - Computational Astronomy - 2009 Earth-rotation synthesis Apply appropriate delays: like measuring V with ‘virtual antennas’ in a plane normal to the direction of the phase centre.

5. NASSP Masters 5003F - Computational Astronomy - 2009 Earth-rotation synthesis Apply appropriate delays: like measuring V with ‘virtual antennas’ in a plane normal to the direction of the phase centre.

6. NASSP Masters 5003F - Computational Astronomy - 2009 Earth-rotation synthesis Apply appropriate delays: like measuring V with ‘virtual antennas’ in a plane normal to the direction of the phase centre.

7. NASSP Masters 5003F - Computational Astronomy - 2009 Field of view and resolution. Single dish: FOV and resolution are the same. FOV ~ λ /d (d = dish diameter) Resolution ~ λ /d

8. NASSP Masters 5003F - Computational Astronomy - 2009 Field of view and resolution. Aperture synthesis array: FOV is much larger than resolution. FOV ~ λ /dResolution ~ λ /D (D = longest baseline) d D

9. NASSP Masters 5003F - Computational Astronomy - 2009 Field of view and resolution. Phased array: Signals delayed then added. FOV again = resolution. FOV ~ λ /DResolution ~ λ /D d D Good for spectroscopy, VLBI.

10. NASSP Masters 5003F - Computational Astronomy - 2009 LOFAR – can see the whole sky at once.

11. NASSP Masters 5003F - Computational Astronomy - 2009 Reconstructing the image. The basic relation of aperture synthesis: where all the (l,m) functions have been bundled into I´. We can easily recover the true brightness distribution from this. The inverse relationship is: But, we have seen, we don’t know V everywhere.

12. NASSP Masters 5003F - Computational Astronomy - 2009 Sampling function and dirty image Instead, we have samples of V. Ie V is multiplied by a sampling function S. Since the FT of a product is a convolution, where the ‘dirty beam’ B is the FT of the sampling function: I D is called the ‘dirty image’.

13. NASSP Masters 5003F - Computational Astronomy - 2009 Painting in V as the Earth rotates

14. NASSP Masters 5003F - Computational Astronomy - 2009 Painting in V as the Earth rotates

15. NASSP Masters 5003F - Computational Astronomy - 2009 But we must ‘bin up’ in ν and t. This smears out the finer ripples. Fourier theory says: finer ripples come from distant sources. Therefore want small Δ ν, Δt for wide-field imaging. But:  huge files.

16. NASSP Masters 5003F - Computational Astronomy - 2009 We further pretend that these samples are points.

17. NASSP Masters 5003F - Computational Astronomy - 2009 What’s the noise in these measurements? Theory of noise in a cross-correlation is a little involved... but if we assume the source flux S is weak compared to sky+system noise, then If antennas the same, Root 2 smaller SNR from single-dish of combined area (lecture 9). –Because autocorrelations not done  information lost.

18. NASSP Masters 5003F - Computational Astronomy - 2009 Resulting noise in the image: Spatially uniform – but not ‘white’. (Note: noise in real and imaginary parts of the visibility is uncorrelated.)

19. NASSP Masters 5003F - Computational Astronomy - 2009 Transforming to the image plane: Can calculate the FT directly, by summing sine and cosine terms. –Computationally expensive - particularly with lots of samples. MeerKAT: a day’s observing will generate about 80*79*17000*500=5.4e10 samples. FFT: –quicker, but requires data to be on a regular grid.

20. NASSP Masters 5003F - Computational Astronomy - 2009 How to regrid the samples? Could simply add samples in each box.

21. NASSP Masters 5003F - Computational Astronomy - 2009 But this can be expressed as a convolution. Samples convolved with a square box.

22. NASSP Masters 5003F - Computational Astronomy - 2009 Convolution  gridding. ‘Square box’ convolver is Gives But the benefit of this formulation is that we are not restricted to a ‘square box’ convolver. –Reasons for selecting the convolver carefully will be presented shortly.

23. NASSP Masters 5003F - Computational Astronomy - 2009 What does this do to the image? Fourier theory: –Convolution  Multiplication. –Sampling onto a grid  ‘aliasing’.

24. NASSP Masters 5003F - Computational Astronomy - 2009 A 1-dimensional example ‘dirty image’ I D : V  I via direct FT:

25. NASSP Masters 5003F - Computational Astronomy - 2009 A 1-dimensional example ‘dirty image’ I D : Multiplied by the FT of the convolver:

26. NASSP Masters 5003F - Computational Astronomy - 2009 A 1-dimensional example: The aliased result is in green: Image boundaries become cyclic.

27. NASSP Masters 5003F - Computational Astronomy - 2009 A 1-dimensional example: Finally, dividing by the FT of the convolver:

28. NASSP Masters 5003F - Computational Astronomy - 2009 Effect on image noise: Direct FT Gridded then FFT

29. NASSP Masters 5003F - Computational Astronomy - 2009 Aliasing of sources – none in DT This is a direct transform. The green box indicates the limits of a gridded image.

30. NASSP Masters 5003F - Computational Astronomy - 2009 Aliasing of sources – FFT suffers from this. The far 2 sources are now wrapped or ‘aliased’ into the field – and imperfectly suppressed by the gridding convolver.