Studies based on limited data usually compute the autocorrelation function (ACF) and define
T
charin terms of the ACF, for exampleT
charT0.5orT
charT1/e, while Ceduna datatypically capture several scintles in each observing period, and thus enable
T
charTperiodtobe estimated with confidence. In addition, as shown later in this chapter, the ACF fails to properly compute T0.5and T1/ein certain situations. The ACF does a better job of identifying
Tperiodin these situations, but its value is sometimes clearer in the frequency domain.
This motivates estimation of the Power Spectral Densityfunction (PSD), PI (= 2f). This
could be estimated directly from a flux density time series, I(t), using a discrete Fourier
transform, not a fast Fourier transform, since I(t) has non-uniform data (Deeming, 1974;
Madisetti & Williams, 1998). This approach is not fully appropriate for a random process time series, which cannot be truly represented by a discrete Fourier series (Newland, 1975), but is expected to be acceptable for a flux density time series whose variability is due to the quasi-periodic scintillation phenomenon: the power spectra of several snapshots of the scintillation pattern should be similar within limits set by sampling errors.
However, any problems of using a discrete Fourier transform can be avoided by computing the PSD from the ACF of the time series. For a random process, the ACF meets the Fourier transform condition, and it contains information about the frequency content of I(t), since it
is a maximum when I(t) and I(t+) are in phase, and a minimum when they are in antiphase
(Newland, 1975).
By Wold’s theorem, the PSD, PI(), is the Fourier transform of the ACF, RI():
d e P R d e I R P I I i i x I
2 1 2 1The PSD estimation approach used herein is to compute a periodogram, and this time a Fast Fourier Transform does work:
2 1 0 exp 1 0 j N j j I I R t i t N P
The terms PSD and periodogram are often used interchangeably in the literature, although strictly the periodogram is only an estimate of the PSD, since I(t) only samples the random
process (Scargle, 1982). The periodogram is a non-parametricapproach to estimating the
PSD. Both the basic method and variations thereon are described by Madisetti & Williams (1998), and are available in the Matlabsignal processing toolbox (Mathworks, 2002).
Considering the main options, the Bartlett Method subdivides I(t) into segments, and the
estimated PSD is the average of each segment’s PSD. The Welch Method is similar, but with overlapping segments. Neither method is appropriate for a Ceduna time series, which has only a limited number of scintles in a 10-15 day time series, thereby leading to a set of very different segment PSDs, while using a longer time series sample would risk violating the stationarity condition. The Blackman-Tukey Method weights the ACF in favour of shorter lag times, which have greater contribution to the calculation, and this method could be applied. However, the binning strategy of the Edelson & Krolik (1988) method has a similar effect, and this method is preferred given its prior usage by other researchers. Alternative PSD estimation approaches that do not rely on manipulation of the ACF are discussed by Mathworks (2002). Subspace methodsare effective in detecting sinusoids in
signals with low signal-to-noise ratios, but Ceduna data have reasonably good signal-to- noise ratios, as shown in the next section, even before application of the data filtering procedures described later in this chapter. Parametric methodsmodel the process and then
estimate the PSD in terms of the model parameters. In this case the process is scintillation, and the standard ISS model described in Chapter 2 might be developed to describe the scintillation characteristics of a given source. In this case, a parametric approach might be of value in estimating the PSD from data recorded by unusually short observing periods, which occur from time to time for various reasons.
Table 4.1 gives Matlabcode for determining the PSD from the ACF, written by the author
based on Mathworks (2002). It is convenient to measure lag times in days, so the frequency scale has units of days-1, and the PSD has units of power per unit frequency. This exercise is
of key importance to the spectral analysis, and thus is explained in detail below.
1. ACF = ACF – mean(ACF); % Compute zero mean ACF
2. NFFT=2048; % Require NFFT-point FFT
3. NFFTnext=2^( nextpow2(length(ACF)) ); % Find next highest power of 2 4. If NNFTnext>NFFT; disp(‘NFFT Warning’); end; % We want to pad ACF, not truncate it 5. PSDtwo = fft( ACF, NFFT ); % FFT of ACF, zero padded to NFFT 6. PSDtwo_real = PSDtwo.* conj(PSDtwo) / NFFT; % Compute real part of PSD
7. PSDsingle = 2*PSDtwo(1:1+NFFT/2); % Single sided PSD
8. Freq = (1/Laginc) * (0:NFFT/2) / NFFT; % Frequencies out to NFFT/2 9. plot( Freq, (NFFT / NFFTnext)*PSDsingle); % PSD plot (NFFT equivalent)
Table 4.1 Matlab code for computing the Power Spectral Density function (PSD)
In Table 4.1, the MatlabFast Fourier Transform (FFT) routine fftcomputes the complex
two-sided (–∞ to ∞) PSD, PSDtwo,from the ACF. Code lines 6 and 7 then produce a real
single-sided (0 to ∞) PSD, PSDsingle,in terms of frequency instead of angular frequency. PSDsinglecontains the zero frequency component as its first element, PSsingle(1), which is
zero since line 1 recalculates the ACF to have zero mean.
The routine ffteither pads or truncates the ACF to NFFTpoints before computing its Fourier
transform, where NFFTis a power of two to facilitate the computation. The ACF of a
Ceduna time series is not very long, so padding is preferable to truncation, and larger values of NFFTenable the PSD peaks to be more accurately identified. The computing strategy in
Table 4.1 (lines 2, 3 and 4) is to specify an NFFT= 211= 2048 point FFT, and check that
this is larger than NFFTnext, the next highest power of 2 greater than the ACF’s length.
The Nyquist frequency component isPSDsingle(1+NFFT/2). If the ACF is computed at lag
intervals of = 1/6 days (4 hours), the Nyquist frequency is (2)-1days-1= 3 days-1, which
corresponds to a minimum resolvable period of 8 hours. Higher frequencies can be studied by recomputing RI() at smaller lags: if future telescope operation manages to avoid the
problem of systematic intensity fluctuations, discussed later in this Chapter and in Chapters 3 and 5, Ceduna should be able to resolve flux density variations on the order of an hour. Line 7 of the Table 4.1 code applies a factor of two multiplier in extracting the single-sided PSD from the two-sided PSD. In addition, line 9 of the code converts PSD plots to their NFFT-point FFT equivalent, since a PSD function computed using an Npoint FFT should
be divided by two in order to compare it to a PSD function computed using a 2Npoint FFT.
Figure 4.8 shows the PSD for the PKS B1144-379 data of observing period 4, computed from the ACF shown in Figure 4.7. The spectral peak has a central frequency of 0.46 days-1
which corresponds to
T
period2.2 days, and agrees with previous estimates. Secondaryspectral features are discussed in the next section.
As a Ceduna time series analysis tool, the main purpose of the PSD function is to identify the variability period, Tperiod= 1 / fo. Actual spectral power values are not overly important.
The practice followed herein is to remove the ACF’s normalisation ahead of computing the PSD, such that the total spectral power depends on the ACF lag increments, and on the maximum lag of the ACF, including its zero padded part.
Conclusion of the PKS B1144-379 case study analysis
The variability time scale, Tperiod, for PKS B1144-379 during observing period 4 has been
unambiguously estimated to be just over 2 days by all three analysis tools: spectral analysis, supported by scintle counting and data folding cross-checks.