[daip] Gridding convolution function implementation

[Eric Greisen]

Nithyanandan Thyagarajan wrote:
> Dear Eric,
> 
> I am Nithya, a post-doc at RRI, Bangalore, India. I am writing to you 
> upon Dwarkanath's suggestion.
> 
> I plan to work on the MWA array. In trying to simulate the array and its 
> response, I came across the need to perform gridding convolution to 
> reduce aliasing. I found the book from the synthesis imaging workshop 
> mentioning that a zeroth order prolate spheroidal function is found to 
> be the most optimal for such a requirement.  This function is denoted by 
> \psi(\alpha, 0, \pi*m/2, \eta(u)). m=6 and alpha = 1 are typically used 
> in AIPS as default.
> 
> Generally reading about spheroidal functions, they seem to be of two 
> kinds, namely, the first and the second kind (these are the two 
> solutions to the second order differential equation for which the 
> spheroidal functions are solutions). They are further decomposable into 
> products of radial and angular functions. I am planning to use Python 
> and SciPy has different routines to compute all these different 
> functions (first and second kinds, radial and angular functions, etc.) 
> here -- http://docs.scipy.org/doc/scipy/reference/special.html under the 
> heading "Spheroidal Wave Functions". You can see some routines named 
> pro_rad1, pro_rad2, pro_ang1, etc.
> 
> I have checked many references such as the synthesis imaging book, AIPS 
> explain section under IMAGR, adverbs such as XTYPE, YTYPE, UV5TYPE, AIPS 
> cookbook glossary. None of these references give me the specific detail 
> I am looking for. My question is, is \psi(\alpha, 0, \pi*m/2, \eta(u)) a 
> zeroth order prolate spheroidal function of the first or the second 
> kind? Is this function composed of both the radial and angular parts? In 
> other words, can I obtain this function by multiplying the radial and 
> angular routines found in SciPy.
> 
> I have tried reading the VLA memos 123, 124 and 129 but they all seem to 
> precede the reference Schwab (1984) and unfortunately I could not find 
> the memos that talk specifics on the spheroidal functions and their 
> implementation.
> 
> It would be very helpful to know specifically what kind of zeroth order 
> prolate spheroidal function is being used.  I kindly request you to 
> point me in the right direction.
> 
> Appreciate your help.
> Thanks a lot,
> Nithya

> 

I do not know this detail.  The function involved is implemented with
subroutine $APLNOT/SPHFN.FOR which has the following precursor comments.

C   SPHFN is a subroutine to evaluate rational approximations to
C   selected zero-order spheroidal functions, psi(c,eta), which are, in
C   a sense defined in VLA Scientific Memorandum No. 132, optimal for
C   gridding interferometer data.  The approximations are taken from
C   VLA Computer Memorandum No. 156.  The parameter c is related to the
C   support width, m, of the convoluting function according to
C   c=pi*m/2.  The parameter alpha determines a weight function in the
C   definition of the criterion by which the function is optimal.
C   SPHFN incorporates approximations to 25 of the spheroidal func-
C   tions, corresponding to 5 choices of m (4, 5, 6, 7, or 8 cells)
C   and 5 choices of the weighting exponent (0, 1/2, 1, 3/2, or 2).

Perhaps Fred Schwab can answer this for you.  I will CC him on this reply.

Eric Greisen