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The measurement equation of an instrument is the relationship between the
sky intensity and the measured quantities. The measurement equation for a
millimeter interferometer is to a good approximation (after calibration)

(4.1) 
where
is the bidimensional Fourier transform of
the function taken at the spatial frequency ,
the sky intensity distribution,
the primary beam of the
interferometer (i.e. a Gaussian whose FWHM is the natural resolution of the
singledish antenna composing the interferometer), some thermal noise
and the calibrated visibility at the spatial frequency .
This measurement equation implies different kinds of problems.
 The presence of noise leads to sensitivity problems.
 The multiplication of the sky intensity by the primary beam implies a
distortion of the information about the intensity distribution of the
source.
 The presence of the Fourier transform implies that visibilities
belongs to the Fourier space while most (radio)astronomers are used to
work in the image space. A step of imaging is thus required to go
from the plane to the image plane.
 Finally, the main problem implied by this measurement equation is
certainly the irregular, limited sampling of the plane because it
implies that the information about the source intensity distribution is
incomplete.
To show that deconvolution techniques are needed to overcome the incomplete
sampling of the plane, we need additional definitions
 Let's call
the
continuous visibility function.
 The sampling function is defined as

at spatial frequencies where
visibilities are measured by the interferometer. is the rms
noise predicted from the system temperature, antenna efficiency,
integration time and bandwidth. The sampling function thus contains
information on the relative weights of each visibility.
 elsewhere.
 We finally call
the dirty
beam.
If we forget about the noise, we can thus rewrite the measurement equation
as

(4.2) 
Using the property #1 of the Fourier transform (see Appendix), we obtain

(4.3) 
where is the convolution symbol. Thus, the incompleteness of the
sampling translate in the image plane as a convolution by the dirty
beam, implying the need of deconvolution. From the last equation, it is
easy to show that the dirty beam is point spread function of the
interferometer, i.e. its response at a point source. Indeed, for a point
source at the phase center,
at the phase center and 0 elsewhere and the convolution with
a point source is equal to the simple product:
for a point source of
intensity
Jy.
Next: Imaging
Up: Singlefield imaging and deconvolution
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20190219