In astronomy as in many physical sciences there is frequent need to measure signal from a data series. In measuring a specific attribute of this signal such as redshift, the power of Fourier analysis has long been recognized (e.g. Sargent et al. 1977; Tonry & Davis 1979). It is the purpose here to draw attention to the power of Fourier analysis in the less well-defined case of attempting general signal measurements from a data series.
This problem of accurate measurement of signal parameters inevitably comes down to assessment of ``zero-level" or continuum height. Many techniques are invoked -- from eye-ball sketching, through least-squares fits of polynomials of low orders, heavy-smoothing, and spline-fitting. The difficulty is inevitably the signal. Those parts of the scan with signal must be removed from consideration in order to place the continuum; and with irregular and a priori unknown spacing of the signal, development of a formal technique becomes prejudicial or perhaps impossible. Moreover, smoothing techniques and polynomial fits make initial assumptions which the data may not justify. For some types of signal such as emission or absorption lines with extreme breadth of wings, the behaviour of the continuum in the regions masked by signal is critical in measurement of that signal.
There are formal tools to apply. For example Bayesian spectral analysis (e.g. Sivia & Carlile 1992) is appropriate when some specific prior knowledge such as line-width is available. However the analysis must be repeated for each different prior-knowledge set and for each different question posed of the data. The situation frequently arising in spectral analysis is one in which the prior knowledge is the somewhat unquantifiable recognition of which parts of the spectrum are signal-free, while very general parameter sets (e.g. line-shape, line-width, line-flux, equivalent width, centroid position) may be required from the measurements.
A simple harmonic-analysis technique is described here which provides a good basis for such analysis, and which can be optimized in cases with some a priori knowledge of signal form. It gives results which are stable from observer to observer.
There will always remain instances in which
baseline assessment is
impossible, even additional resolution being unable to
rescue a continuum such as Ly
forests in QSO
spectra, some regions of the solar spectrum, and the
spectra of stars of types much later than solar. These represent
uni-directional confusion of signal, absorbed or negative
signal in these cases, with no possibility of true continuum
appearing on both sides of a signal-dominated area to give the analyst
a chance. Such cases must be dealt with by techniques beyond
the scope of this note, techniques involving a priori knowledge
or models of
instrument performance and line patterns.
However, cases of pure confusion of signal
in which the probabilities of positive
and negative deflections are equal, do lend themselves
to baseline assessment of the type described here (Wall et al. 1982).
The paper is concerned with data sets of the type represented by the optical spectra in the upper panels of Fig. 1 (click here) and Fig. 2 (click here). In the first case, the spectral appearance is dominated by substantial emission lines covering more than 30% of the length of the spectrum. In the second case, the continuum slope is severe and the spectrum is dominated by broad and deep absorption lines.
Figure 1:
A spectrum of 3C 47 obtained by Laing et al. (1994) with the
Faint Object Spectrograph of the William Herschel Telescope, La Palma.
The redshift is 0.345; broad lines of the hydrogen Balmer series can
be seen, together with narrow lines of [OIII]. The upper panel shows the
flux-calibrated spectrum, while the middle panel shows the ``patched"
spectrum, the scan
with portions of apparent signal patched out,
together with a straight-line least-squares fit. The lower panel shows the
continuum obtained from using the low-frequency Fourier
components of this patched spectrum, superposed on the original data
Figure 2:
A spectrum of RZ Cas (Maxted et al. 1994). The panels
are as described in the caption for Fig. 1