[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

J. Pety and N. Rodríguez-Fernández: Revisiting the theory of interferometric wide-field synthesistel-00726959, version 1 - 31 Aug 2012A.3. GriddingThe gridding kernel can be defined as the product of two functions,each one operating in its own dimension. We use this tostudy separately the effect of gridding in the uv and sky planes.We then use the intermediate results to get the effect of griddingsimultaneously in both planes.A.4. Gridding in the uv planeWe define the sampled visibility function gridded in the uvplane asSV ( ) g u p ,α s ≡{g⋆SVα s} ( )u p∫= g(u p − u ′ p ) SVα s(u ′ p )du′ p .(A.13)u ′ pUsing that the gridding is here applied on the u p dimension,while the Fourier transform is applied on the α s dimension, it iseasy to show that the gridding and Fourier-transform operationscommute:SV g u p(u s ) =∫∫g ( ( ) ( )u p − u p) ′ S u′p ,α s V u′p ,α s e−i2πα s u sdα s du ′ p (A.14)α s u ′ p∫= g ( u p − u p) ′ SVu ′ p(u s ) du ′ p. (A.15)u ′ pDefining the Fourier transform of the uv gridded dirty image, wederiveI g dirty (u) ≡ 〈 SV g〉 (u)(A.16)∫∫= W ( ) ( ( )u p , u−u p g up −u p) ′ SVu ′ p u−up dup du ′ p. (A.17)u p u ′ pUsing Eq. (31) to replace SV u ′ p(u − u p ), we can write the Fouriertransform of the uv gridded dirty image asI g dirty (u) = ∫u ′ D g ( u ′ , u − u ′) I ( u ′) du ′ , (A.18)withD g ( u ′ , u−u ′) ∫≡ W ( ) gu p , u−u ( p Σ u p , u−u p , u ′) du pu pand(A.19)Σ g ( u p , u s , u ′) ≡∫g ( ( (u p − u p) ′ S u′p , u s − u ′ + u p) ′ B u′p − u ′) du ′ p . (A.20)Usingu ′ pS ( u ′ p , u s − u ′ + u ′ p)=[∫α sS u ′ p(α s ) e −i2πα su sdα s]e −i2π(u′ p−u ′ )α s,and(A.21)Σ g ( u p ,α s , u ′) α s⊃usΣ g ( u p , u s , u ′) , (A.22)we deriveΣ g ( u p ,α s , u ′) =∫g(u p − u ′ p) S ( ) (u ′ p,α s B u′p − u ′) e −i2π(u′ p −u′ )α sdu ′ p,u ′ porΣ g ( u p ,α s , u ′) ∫=u ′ p(A.23)g ( u p − u ′ p)Σ(u′p ,α s , u ′ p − u′) du ′ p . (A.24)Thus, Σ g is the uv gridded version of the generalized samplingfunction Σ.A.4.1. Gridding in the sky planeWe define the sampled visibility function gridded in the skyplane asSV ( ) { } γ u p ,α s ≡ γ⋆SVup (αs )∫= γ(α s − α ′ s) SV up (α ′ s)dα ′ s.α ′ s(A.25)(A.26)Applying the convolution theorem on the Fourier transformalong the α s dimension, we deriveSV γ u p(u s ) = γ (u s ) SV up (u s ) .(A.27)Defining the Fourier transform of the sky-gridded dirty image,we deriveI γ dirty (u) ≡ 〈 SV γ〉 (u)(A.28)∫= W ( ) ( ) ( )u p , u−u p γ u−up SVup u−up dup . (A.29)u pUsing Eq. (31) to replace SV up (u − u p ), we can write the Fouriertransform of the sky-gridded dirty image asI γ dirty (u) = ∫u ′ D γ ( u ′ , u − u ′) I ( u ′) du ′ (A.30)withD γ ( u ′ , u − u ′) ≡∫W ( ) γu p , u − u ( p Σ u p , u − u p , u p − u ′) du pu pandΣ γ ( )u p , u s , u ′′s ≡ γ (us ) S ( )u p , u s + u ′′ (s B u′′) s ,(A.31)(A.32)or, with the definition of Σ (i.e., Eq. (45)),Σ γ ( )up , u s , u ′′s ≡ γ (us ) Σ ( )u p , u s , u ′′s . (A.33)UsingΣ γ ( )up ,α s , u ′′ α ss ⊃usΣ γ ( )up , u s , u ′′s , (A.34)and the convolution theorem when taking the inverse Fouriertransform of Σ γ ,wederiveΣ γ ( )∫u p ,α s , u ′′s = γ ( α s − α ′ ) ( )s Σ up ,α ′ s, u ′′s dα′s . (A.35)α ′ sThus, Σ γ is the sky gridded version of the generalized samplingfunction Σ.Page 17 of 21