**11.6.3. Data Reduction**

*a) Frequency Fourier Transform*

If a correlation receiver has been used, then the first stage is a Fourier transform to recover the visibility spectrum of the signal. It should be noted that the shape of the frequency, filters is under our control as we can apply a weighting function to the correlation function before computing the Fourier transform (this is equivalent to convolving the spectrum after the Fourier transform). Thus we can reduce the 22% frequency sidelobes associated with a uniform weighting function, at the expense of an increased half-width. The frequency Fourier transform is often made in an on-line computer, which receives data directly from the correlator.

*b) Calibrations*

There are two basic calibrations in addition to the usual interferometer baseline and system gain-and-phase calibrations of a broad-band interferometer (see Chapter 10). These are

to determine the relative sensitivities of the frequency channels

to determine the sensitivity and system phase of the telescope as a function of frequency.

*c) Spatial Fourier Transform*

Each frequency channel may be separately Fourier inverted as for a broad-band interferometer, as discussed in Chapter 10. The chief problem is the large quantity of data to be processed, and the computation may take several hours - even on the largest computers using fast Fourier transform techniques.

*d) Presentation of the Data*

The end-product of the above processing is a series of maps of the
hydrogen distribution in the different frequency channels. These
maps form the basic data, which can be considered as a three-dimensional
array with *x*, *y*, and frequency coordinates. The maps of
frequency channels containing line emission
may be combined to produce a map of the
integrated line emission and a map of frequency spectra at a grid of
points on the area of sky studied. A velocity (or profile width)
can then be fitted to the frequency spectra
and a map of isovelocity contours drawn
across the source. There is obviously a considerable data-handling
problem associated with mapping a 1° square of sky, say, with an
angular resolution of 1 minute arc at 60 different velocities, and the
basic problem is presenting the information in a digestible
form. It is to be expected that fairly exotic techniques will eventually
be used in this final and most important stage of data reduction,
namely, in the interface between the data and the astronomer.