The Lag Spectrum, Gibbs Phenomenon and Hanning Smoothing

The problems connected with frequency response (artifacts in the spectra caused by the sampling and the Fourier transform) are due to the fact that the lag spectrum, as produced by the correlator, is a real function of finite length. The VLA correlator produces a real lag spectrum consisting of $n$ positive and $n$ negative lags. A frequency spectrum containing $n$ complex visibilities will be generated from these data. The measured lag spectrum, being a real function, will generate a frequency spectrum that is Hermitian. The result of this is that we will find the mirror image/complex conjugate of the ``real'' spectrum at negative frequencies. This causes a phase discontinuity at $\nu$ = 0 if the phase is non-zero -- which it usually is for uncalibrated data.

The VLA gets by with the minimum number of lags ($2n$) required to obtain a complex spectrum of length $n$. Since the true lag spectrum is much longer, this amounts to multiplying it with a box function. In terms of the frequency spectrum, this corresponds to a convolution with a sinc function ($\sin (x)/x$). The nulls of this function are spaced by the channel separation. If a monochromatic signal falls exactly on a channel, the other channels will be placed on its nulls, and all one will see is a good-looking spike -- a truthful representation of the real spectrum. However, signals are seldom monochromatic, nor do they usually fall on exact channel frequencies; hence, we have to cope with the 22% sidelobes of the sinc function. An unresolved spectral line in one channel will give a spurious response in an adjacent channel at the 22% level, diminishing in amplitude as one moves away from the line. These ripples are especially annoying in spectra with strong narrow features (e.g., masers, planetary radar experiments, or in the presence of interference) and the effect is referred to as ``ringing''.

The Gibbs phenomenon is the ringing at the edges of the frequency spectrum that results from the truncation of the temporal (lag) cross-correlation spectrum. This truncation corresponds to a convolution of the complex bandpass with a sinc function, generating a ripple at the low frequency side of the spectrum (high frequency for U-band; both sides for band width code 9). The ripple appears both in amplitude and phase, since the discontinuity is really in the imaginary part of the spectrum and will only be there in the presence of a continuum signal. Since the magnitude of the effect depends on the instrumental as well as the source phase for a particular baseline, the ripple is impossible to calibrate out. When observing with a narrow band and many spectral line channels, of order 128, Gibbs phenomenon is less destructive since the amplitude of the ripple decreases as one moves away from the band edge. If one is willing to tolerate some small ripple, the amplitude has decreased to about 2% around channel 20, leaving enough channels for the line of interest.

One can ``soften'' the edge at the end of the lag spectrum or, equivalently, the ``ringing'' around sharp spectral features by tapering. The tapering function used at the VLA is the Hanning taper, $h(\tau)$:

$$h(\tau ) = \frac{1}{2} \times (1 + \cos (\pi \tau /T))$$ (2.3)

where $\tau$ is the lag and $T$ the maximum lag. This is a smoothing function in the frequency domain which adds three adjacent channels together with weights $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$. Hanning smoothing works by taking every even channel, $i$, of the original dataset and creating a new channel, $i^{\prime}=i/2$, such that $i^{\prime}=(0.5{\times}i) +(0.25{\times}(i-1))+(0.25{\times}(i+1))$. The Hanning taper will effectively suppress ringing and Gibbs phenomenon, and the side lobes will be reduced to less than 3%. The frequency resolution (FWHM) of untapered spectra is $1.2 \times \Delta\nu$ (where $\Delta\nu$ is the channel spacing); the resolution of Hanning tapered spectra is $2.0 \times \Delta\nu$. The noise equivalent bandwidth for these cases is $1.0 \times \Delta\nu$ and $2.0 \times \Delta\nu$, respectively. Hanning smoothing may be applied on-line in real time, in which case half of the channels are dropped although the full number of channels is used in the lag-to-frequency transform. The channel spacing is changed, making it equal to the resolution. Alternatively, Hanning smoothing may be applied during the data reduction stage, in which case the channel data are smoothed, all channels are retained, and adjacent channels are no longer independent.

A comprehensive discussion of frequency response in an XF correlator can be found in Chapter 2, Appendix 2, of the Westerbork Synthesis Radio Telescope Users Manual written by Willis and Bregman. Gibbs phenomenon is described in Chapters 4 and 18 of SIRA as well.