Ideally, frequency-switching may be described as a process of convolution; a pattern c(x) disposed as at the top of Fig. 4 (click here) can be convolved with an array to reproduce the appearance of a frequency-switched spectrum. For some but not all choices of the frequency-switching interval, the original spectrum may be recovered by a naive linear deconvolution in the ideal case of noise-free data. That is, the convolution of c(x) and a spectrum f(x) is the observed spectrum g(x) = c(x)*f(x). If the Fourier transform of a function is denoted by F(), the deconvolution theorem states that and F(f(x)) = F(g(x))/F(c(x)). As long as no Fourier components of c(x) have 0-modulus, f(x) can be recovered exactly from noiseless data simply by doing an inverse Fourier transform. As shown by EKH and many choices for the frequency-switching interval yield zeros in the components of F(c(x)) when a simple FFT is performed, including but by no means limited to channel shifts of channels.
In practice the naive linear deconvolution is nearly useless with real data. Because the signal and reference phases are accumulated independently, neither their noise nor their baseline shapes, etc. are exact shifted copies of one another: the data only approximate a convolution. Even more importantly, a one-sided function like the c(x) shown at the top Fig. 4 (click here) is grossly insensitive to some Fourier components even when it has no exact zeros. This is manifested by the many small moduli in the periodogram power spectrum at the bottom of that figure and is not rectified by available alternatives such as switching symmetrically to either side. Unless noise in the frequency-switched spectrum g(x) is coincidentally missing or heavily filtered at these frequencies, a simple linear deconvolution will amplify it greatly.
Figure: As in Fig. 1 (click here), but for a pathological case. The resulting spectrum (bottom) does not reproduce the original signal when presented to the algorithm described in Fig. 2 (click here)
Some degree of selective attenuation of the Fourier components of the incoming signal is inevitable with frequency-switching schemes. One insight of EKH was to show that components of F(c(x)) may be selected such that a region of appreciable extent can be recovered without unduly increasing the noise through deconvolution. Note that the aliasing referred to at the end of Sect. 1 (see Fig. 3 (click here)) is equivalent to linear deconvolution with improper switching intervals etc.
Symbolically we represent a spectrum by f(x). The process of creating a
frequency-switched spectrum involves forming the difference
where is the frequency-switching
interval (most telescopes actually specify offsets for the signal and
reference phases individually; is their separation). The standard
algorithm for creating an approximation to the unknown f(x)
from the data is (see Fig. 2 (click here))
Substituting 1a into 1b, one finds
which shows that an arbitrary spectrum is recovered at position x only when is large enough that . When this the case, the error in is smaller than the noise in g(x) by a factor . Switching within the signal region, as in Fig. 3 (click here), creates insurmountable problems for this algorithm.
To ameliorate the difficulties caused by switching within the line, imagine applying a different algorithm to the spectrum at the bottom of Fig. 3 (click here), i.e. step to the right in the spectrum calculating
This recursive approach, which simply adds back to each channel a close-at-hand estimate of what should have been subtracted from it, identically recovers the signal phase of the spectrum from the g(x) at the bottom in Fig. 3 (click here). Similarly, a leftward progression calculating
recovers the reference phase, whence
Figure 4: Frequency switching viewed as a convolution. At top, a typical convolution function c(x) for frequency-switching (the interval is 27 channels). A periodogram power spectrum of this function appears at bottom showing the amplitudes (not their squares) of its Fourier components. Many of these are quite small, making it hard to do deconvolution on noisy data. Many choices for c(x), such as intervals of channels, have actual zeroes in their power spectra, making linear deconvolution impossible. Some Fourier components of the incoming signal are heavily attenuated by the switching process
Figure 5: Treatment of actual frequency-switched data. At top is a CO J=2-1 spectrum observed with a 122.88 channel frequency-switching interval (the strongest feature is telluric). A linear baseline and standing-wave have been removed to create the "baseline" spectrum and this baselined spectrum is shown as processed by naive linear deconvolution, the convential shift and subtract, and by the bootstrap algorithm described here
In fact this procedure is equivalent to the dual beam restoration algorithm of EKH, although such may not be entirely obvious from a casual reading of the earlier work. EKH show that it is possible to do a worthwhile selective recovery of the signal using only those Fourier components which are not attenuated by more than a factor of two in switching. They then show a) that this selection is equivalent to requiring six samples per cycle of F(c(x)); b) that a region of extent can be recovered; and c) that the deconvolution may be implemented without Fourier transforms simply by convolving g(x) with a simple comb (their Fig. 6iii; see our Fig. 6 and Sect. 3.4 here as well). In the caption for their Fig. 6 (click here), it is noted that using only half the comb is eqivalent to our Eqs. (3a) or (3b), from which it follows that use of all of it is equivalent to our Eq. (3c).
Figure: Unfolding functions. The convolution of these functions with frequency-switched spectra implements the bootstrap in the mannner discussed by Emerson et al. (1979) for shifts of 34 (top) and channels. There are 256 channels and the central spike is the 128th channel from the left in either case. The phase of the comb has been chosen so that the recovered spectrum appears in channels occupied by the so-called signal phase of the unfolded data
The non-recursive approach in Eq. (1) should of course be used whereever possible. However the recursive bootstrap is worthwhile even when it is not absolutely needed because it is an efficient means of testing whether the incoming signal was inadvertantly corrupted by improper frequency- switching. It produces not only a faithful representation of the more obvious signal-bearing channels, but a cleaner view of the passband as well (i.e. the presumed signal-free channels on either side of the line), which could in principle harbor signals which otherwise would be missed in the process of shifting and subtracting.
The recursive approach does not necessarily achieve a noise level corresponding to the full integration time because the signal and reference phases may not be fully present in the spectra upon which it works. There are more subtractions, some of which increase the noise (Eqs. (3a) and (3b)) because they are not compensated by averaging which only occurs in Eq. (3c). Unlike most conventional algorithms, however, this one produces a noise level which may be lower over its representation of the signal-bearing region than over the baseline regions. If the support of the signal and reference phases is apparent, the procedures implied by Eqs. (3a) and (3b) can be limited to that region.
If the switching interval is very narrow compared to the signal-bearing region, use of recursion in traversing a region of extent in steps of will degrade the noise by a factor (also see EKH). When , noise in the output spectrum of the recursive algorithm varies in broad contiguous patches across the band, which we explored by doing Monte Carlo simulations of frequency-switching a broad, flat-topped profile. For a profile with noise which would have been well-recovered by the shift and subtract technique (resulting in a uniform noise level ) the bootstrap had differing noise levels in four regimes; over the region of the line; between and in the adjacent baseline region; in that portion of the output passband where the reference phase was originally present; and elsewhere. For a profile switched within itself by half the width, the resultant noise level was over the region of the line and the adjacent passband, and elsewhere.
It is very important to recognize that the recursion can introduce serious artifacts from imperfections such as bad baselines. On the other hand, the use of narrower frequency switching intervals, made possible by the use of the recursive algorithm, may do much to eliminate the root cause of such problems.
Figure 5 (click here) illustrates the processes discussed here. At top is a 30-minute frequency-switched integration on the CO(J=2-1) line toward the continuum source B0355+508; a profile of the J=1-0 line of CO appears in Liszt & Wilson (1993). The channel interval of the switch was 122.88 channels or 7.8 km in units along the x-axis. The profile was baselined by removing a first-order binomial and a long-period standing-wave (shown superposed at the top) and the baselined profile was processed in three ways: by linear deconvolution using FFT's, by the usual shift and subtraction, and by the bootstrap recursive solution discussed above. Clearly only the bootstrap produces a reasonable profile without substantial further intervention although its noise is noticeably larger than that in the (rather useless) spectrum directly above it.