Smoothing

The cross power spectrum is obtained by measuring the correlation of signals from different antennas as a function of time offset between them. A spectrum with a bandwidth $\delta\nu$ and N channels is produced by cross correlating signals sampled at interval of $\tau$ with relative time offset in the range -N$\tau$ to (N-1)$\tau$, where $\tau$ = 1/(2$\delta\nu$). Because of this truncation in the offset time range amounting to a rectangular window, the resulting spectrum is equivalent to convolving the true spectrum by a Sinc function. Thus, a delta function in frequency (a narrow spectral line, for e.g.) will result in an appropriately shifted sin( $N\pi\nu/\delta\nu)/(N\pi\nu/\delta\nu$) pattern, where $\delta\nu$/N is the channel separation. The full width at half maximum of the Sinc function is 1.2$\delta\nu$/N. This is the effective resolution. Any sharp edge in the spectrum will result in an oscillating function of this form. This is called the Gibbs' phenomenon. There are different smoothing functions that bring down this unwanted ringing, but at the cost of spectral resolution. One of the commonly used smoothing functions in radio astronomy is that due to Hanning weighting of the correlation function. This smoothing reduces the first sidelobe from 22% (for the Sinc function) to 2.7%. The effective resolution will be 2$\delta\nu$/N. After such a smoothing, one retains only the alternate channels. For Nyquist sampled data, the Hanning smoothing is achieved by replacing every sample by the sum of one half of its original value and one quarter the original values at the two adjacent positions.

Apart from Hanning smoothing which is required to reduce the ringing, additional smoothing of the spectra might be desirable. The basic point being that a spectral line of given width will have the best signal-to-noise ratio when observed with a spectral resolution that matches its width. This is the concept of $'$matched-filtering$'$ and is particularly important in detection experiments.