Frequency-switching is a commonly used technique in spectral line observing. As illustrated in Fig. 1 (click here), a spectrum is accumulated in two interleaved phases, one of which is inverted and shifted with respect to the other. If the signal is in emission, the upward-pointing phase is commonly described as the "signal" while the other is called the "reference". The signal and reference phases are observed sequentially at moderate switching rates (typically 1 Hz) and combined to form an output spectrum (Fig. 1 (click here) at bottom) which has shifted and oppositely- directed representations of the incoming profile.

The advantages of frequency switching are that the 1 Hz rate is rapid with respect to variations in the hardware and environment, which are partly cancelled when the two phases are combined; that there is no need to find a signal-free reference position elsewhere on the sky; and that the source is observed without interruption. Its main defect is an often-poor output passband shape which prevents detection of broad lines. If a spectral shape is impressed on the signal and reference phases, shifting and differencing them will cause something resembling a crude numerical derivative to appear in the output. If the spectral shape is of zero or first order, differencing will remove it. If the spectral shape is discontinuous or complicated, the output frequency-switched spectrum may be hopelessly compromised.

**Figure 1:** Construction of a frequency-switched spectrum. A spectrum
(top) observed in the signal phase is shifted and inverted (middle) in
the reference phase. The signal and reference are added (bottom) to produce
the "observed" spectrum

Proper data-handling in extant reduction algorithms requires that broader lines be observed with wider frequency-switching intervals. Such wider switching further degrades the output passband, often rendering the frequency-switching technique inappropriate for all but the narrowest and simplest spectra. This is a significant hindrance when emission-free regions of the sky are either unknown, difficult to find, or very distant.

The standard algorithm for recovering the signal from a frequency-switched spectrum like that at the bottom of Fig. 1 (click here) is simply to shift by the known frequency-switching interval and subtract, dividing by a factor 2, as shown schematically in Fig. 2 (click here). When both the signal and reference phases contain a good representation of the complete line profile, with adequate signal-free regions between and to either side, the output of the shift and subtract algorithm is an error-free reconstruction of the line profile (Fig. 2 (click here) at bottom).

Occasionally, the frequency-switching interval is not chosen so felicitously. In Fig. 3 (click here), the interval accidentally coincides with part of the structure of the line in such a way that the output spectrum (at bottom) is corrupted. In this particular case, use of the shift and subtract algorithm to recover the profile produces a result which has the correct shape but only half the correct amplitude. This example also illustrates the fact that the frequency-switching interval cannot necessarily be deduced from the appearance of a frequency-switched spectrum. There is a common aliasing problem such that a given output frequency-switched spectrum can often be reproduced by various combinations of signal shape and switching interval.

The failure of the standard algorithm to recover the proper spectrum
from data like that in Fig. 3 (click here) is unfortunate because the output
frequency-switched spectrum at bottom clearly contains a faithful
copy of every signal-bearing channel; the component on the left appears
in the signal phase while that on the right appears in the reference.
In fact, noiseless spectra are identically recoverable for *any* frequency-
switching interval of at least one channel. No matter how complicated
or damaged they may appear, it should not be necessary to abandon even noisy,
imperfect data simply for lack of a competent algorithm. The essentials
of this matter were recognized some time ago by Emerson et al.
(1979: EKH) in considering how to map spatially extended objects
with a small telescope beam throw. The solution we propose for spectral
line work is functionally equivalent to the EKH algorithm, although this
may not be apparent at first glance.

**Figure 2:** Reconstruction of a frequency-switched spectrum by the standard
algorithm. The observed spectrum (top) is shifted (middle) and differenced
with itself, and divided by 2 to recreate the original signal

Section 2 discusses several methods of recovering frequency-switched spectra. The effect of each one, operating on real-world data, is shown in the next- to-last figure. Section 3 briefly discusses a generalization to the case when the frequency-switching interval is not a whole number of channels, using the formalism of EKH.

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