Fourier Transforms in Radio Astronomy
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1 Fourier Transforms in Radio Astronomy Kavilan Moodley, UKZN Slides taken from N Gupta s lectures: SKA School 2013
2 van-cittert Zernike theorem Extended, quasi-monochromatic, incoherent source X (l,m) Y Field at P 1 and P 2 due to an element at (l,m): to (l,m) P 1 P 2 y x (Thompson, Moran & Swenson)
3 van-cittert Zernike theorem The complex correlation at P 1 and P 2 for zero time offset: * I(l,m) small wrt receiver BW -1 u=(x 2 -x 1 )!/c v=(y 2 -y 1 )!/c (R 2 -R 1 )=(ul+vm)c/! R 2 R 1 = R V(u,v) I(l,m) (Thompson, Moran & Swenson)
4 I(l,m) is real; V(u,v) is Hermitian. Hermitian function Re f(x) Im f(x) f(x) = f*(-x)! f(x) = E(x) + i O(x) Real part is even;! Imag part is odd. Since V(u,v) is hermitian we measure only half of the (u,v)- plane and fill the other half with the complex conjugates. V(u,v) = V*(-u,-v)
5 V(u,v) FT I(l,m) I(l,m) is real; V(u,v) is Hermitian.
6 (u,v) -plane (u,v) tracks as ellipses (Thompson, Moran & Swenson)
7 (u,v) -plane (u,v) tracks as ellipses Holes correspond to missing information. (Thompson, Moran & Swenson)
8 Image reconstruction: phase vs amplitude
9 Digital images: Quantization Lim (1990)
10 Digital images: Pixelization Lim (1990)
11 Phase/amplitude-only synthesis? Lim (1990)
12 Phase/amplitude-only synthesis Lim (1990)
13 Phase/amplitude-only synthesis (Taylor, C. A. & Lipson, H., Optical Transforms, Bell, London 1964)
14 Phase/amplitude-only synthesis Duck (Amplitude) + Cat (Phase) Cat (Amplitude) + Duck (Phase) (Taylor, C. A. & Lipson, H., Optical Transforms, Bell, London 1964)
15 Amplitude: magnitude of the spatial frequency.! Phase: it s location.
16 Image analysis Lim (1990)
17 Image reconstruction: a few components required 12% 5% Lim (1990)
18 Sampling V(u,v)
19 Nyquist rate Any continuous band-limited signal can be reconstructed if sampled at the Nyquist rate. Sampling rate = 1/2f Higher frequency components will be aliased to the lower frequencies in the sampled band.
20 Sampling V(u,v) V(u) III(u/!u)= (u-i!u) o!u u o o o o o o 1/!u l If adequately sampled convolution with sinc provides exact interpolation of the original function from the samples. (Thompson, Moran & Swenson)
21 Under-Sampling: Aliasing V(u) III(u/!u)= (u-i!u) u l If aliasing is avoided convolution with sinc provides exact interpolation of the original function from the samples. (Thompson, Moran & Swenson)
22 Fast Fourier Transform: V(u, v) - I(l,m) Faster.! Requires data on uniform grid.! Gridding to resample V(u,v).
23 Fast Fourier Transform: Image domain Holes correspond to missing information.! Longest baseline: limit on resolution! Inner hole: no information on large scales (Thompson, Moran & Swenson) Pixel size: 1/(2u max ), 1/(2v max ) i.e. satisfy sampling theorem.
24 Fast Fourier Transform: Image domain Image size: whole primary beam; sources in the side lobe will be aliased back. Solution: make larger image! (Thompson, Moran & Swenson)
25 Errors in V(u,v)
26 Effect of Amplitude error (Thomson, Moran & Swenson)
27 Taylor et al. lecture (NRAO Synthesis Imaging School 2012)
28 Taylor et al. lecture (NRAO Synthesis Imaging School 2012)
29 V (obs) = G ij V(true) -Observing set-up! -bad data.! poor calibration/! baseline-based errors
30 Diffuse extended emission 1) Weighting: surface brightness sensitivity! 2) Masking: deconvolution & flux density! (Flux calculated correctly for cleaned map.).! (Briggs et al. 1999)
31 Emission at various scales.! (Konar et al. 2006)
32 Emission at various scales.! 80x63 mas Heavier Gaussian taper 36x33 mas Momjian et al. 2003
33 Summary V(u,v) I(l,m)! Radio interferometer samples V(u, v): fourier transform to get image.! Fourier transforms also useful in identifying problems.! Use Flagging, Gridding and Weighting of the visibility to get appropriate image. References and further reading Bracewell: The Fourier Transform and its applications.! Thompson, Moran & Swenson: Interferometry and.! Synthesis in Radio Astronomy.! Synthesis Imaging in Radio Astronomy II: the NRAO lecture series.
34 END - PART I1
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