FYSA21 Mathematical Tools in Science Signal analysis in astronomy Lennart Lindegren Department of Astronomy and Theoretical Physics, Lund University 1 Aim of the project This project will illustrate an application of the discrete Fourier transform (DFT), and in particular the relation between the time and frequency domains, by estimating the amplitude spectra of astronomical time series. 2 Intended learning outcomes Having completed this exercise, the student should have acquired the following skills and abilities: To apply the discrete Fourier transform to regular time series in order to estimate their amplitude spectra To understand the relation between the time and frequency domains, for example by identifying corresponding features in the time series and amplitude spectra To understand how sampling effects distort the amplitude spectra (resolution and aliasing) To understand how measurement noise shows up in the time series and amplitude spectra To interpret amplitude spectra in terms of different kinds of signal (periodic, quasi-periodic, multi-periodic, stochastic) and noise components To estimate Q values for periodic and quasi-periodic phenomena and understand their meaning in terms of coherence times To describe and discuss, in concise writing, different time series with respect to the various concepts mentioned above. 1
3 Signal, noise, and signal analysis In physics and engineering, a signal usually means a time-varying physical quantity that is used to transmit information from one point to another. Familiar examples are acoustic signals (sound waves), electromagnetic signals (e.g., radio waves and light) and electrical signals (e.g., from a microphone). Unintentional, or unwanted, variations of the same physical quantity make up what is called noise. Signal processing refers to a wide range of techniques for extracting information from a signal. Very often, this involves (among other things) the suppression of noise. In signal analysis we often do not know exactly what to look for, or even whether there is any signal at all, or just noise. Signal analysis uses a number of tools designed to detect specific certain kinds of signals, if they are present. Tools based on the Fourier transform are especially useful for detecting periodic signals. In astronomy, most of what we know about the universe outside the Earths atmosphere comes from studying electromagnetic radiation received from various objects. Depending on their temperatures and other factors, astronomical objects emit electromagnetic radiation at a broad range of frequencies (radio waves, infrared radiation, visible and ultraviolet light, X-ray and γ-ray radiation). Sometimes the radiation is highly time-variable, and signal analysis can help us to learn more about the objects and the physical processes that are causing the variations. 4 Time series A physical signal can usually be thought of as a continuous function of time, x(t), where x stands for the physical quantity such as the intensity of light or pressure of the sound wave at a certain point. The observation or measurement process converts this into a series of discrete values x 0, x 1, x 2,... called a time series. In the simplest case the time series just represents x(t) at discrete points in time: x 0 = x(t 0 ), x 1 = x(t 1 ), etc. The process of converting a continuous function to a sequence of discrete values is called sampling. The time series considered in this exercise are all regular, i.e., successive points in time are separated by a constant time interval t. Thus t k = t 0 + k t and x k = x(t k ) = x(t 0 + k t). t is called the sampling interval. The number of samples per unit time is given by f s = 1/ t which is called the sampling frequency. For example, if the intensity of a star is measured once every minute, we obtain a time series with sampling interval t = 1 min = 60 s, and sampling frequency f s = 1 min 1 = 0.016667 Hz = 16.667 mhz. In digital sound recording and transmission, the sound wave (or rather the electrical signal from microphone and amplifier) must be sampled several thousand times per second; for example, the sampling frequency used for audio CDs is f s = 44.1 khz. 2
Figures 1 3. Schematic illustration of the amplitude spectra for different kinds of signals. Figure 1 (top): a periodic signal with fundamental frequency f 0. Figure 2 (middle): quasi-periodic oscillations (the broad peaks) superposed on stochastic signals. Figure 3 (bottom): stochastic signals (the solid curve shows an almost white spectrum, the dashed curve a pink spectrum). 3
5 Spectra for different kinds of variations A very common task in signal analysis is to find and characterize periodic (or not so periodic) variations in the signal. This is usually done by calculating the spectrum of the signal. The spectrum shows the decomposition of the signal into its various frequencies. The amplitude spectrum shows the amplitude of the components versus frequency. In a power spectrum one usually plots a different quantity versus frequency, namely the power spectral density, which is proportional to the square of the amplitude. In this exercise we use amplitude spectra. With reference to Figs. 1 3, we shall distinguish between three basic types of variations in regard to their periodicity: periodic, quasi-periodic, and non-periodic signals. This does not by any means cover all possible kinds of signals, but it is a useful first-order classification of astronomical signals. A periodic signal repeats itself cyclically with a certain period P. Its amplitude spectrum (Fig. 1) consists of a single narrow peak at the fundamental frequency f 0 = 1/P, plus additional peaks at the harmonics (overtones) nf 0, where n > 1 are integers. Only a purely sinusoidal signal has no harmonics. In astronomy, a strictly periodic signal is usually associated with a rotating body, or a body orbiting around another body, in which case f 0 can be identified with the basic rotation or orbital frequency. In some cases there are multiple periods that do not form a harmonic series. A quasi-periodic signal has a peak at a certain frequency, but it is much wider than for a periodic signal (Fig. 2), meaning that the frequency is not very well defined. This is common in physical systems where the oscillations depend on local factors, for example ionized gas (plasma) surrounding a star may oscillate with a frequency that varies with the density of the plasma. A stochastic signal has no recognizable peaks at all, although the amplitude often increases towards lower frequencies (Fig. 3). Such a spectrum is called white (if it is flat) or pink (if it increases towards zero frequency). Noise is nearly always stochastic, but stochastic variations are not necessarily noise. There are many astronomical objects that emit radiation varying in a stochastic manner. Observational noise is often white (e.g., photon noise). In practice one often has a combination of stochastic and periodic or quasi-periodic components, including stochastic noise. It should also be remembered that real, computed spectra do not at all look as smooth and clean as Figs. 1 3. They are always very ragged, due to the fact that the calculated amplitude at each frequency has some uncertainty. Figure 4 shows an example of a real calculated amplitude spectrum for an astronomical X-ray source. A useful concept in connection with periodic and quasi-periodic signals is the Q factor, which is the ratio of the central frequency to the bandwidth: Q = f 0 /B. The bandwidth B is usually defined as the full width of the quasi-periodic peak at half maximum intensity. A periodic signal has a large or very large Q factor; a quasi- 4
Figure 4. The computed amplitude spectrum for an X-ray variable. It shows a pink noise component intrinsic to the object (best seen at f < 10 Hz), a flat photon noise component (best seen ay f > 50 Hz), and a quasi-periodic variation with a central frequency around 25 Hz. All other features seen in the spectrum are statistical fluctuations. periodic signal has moderate or small Q factor. The signal in Fig. 4 has Q 5. The Q factor is approximately equal to the number of oscillation periods during which the amplitude and phase of the oscillation remains roughly the same. 6 Resolution, aliasing, etc There are two important properties of the time series that determine how well the spectrum can be determined: the sampling interval t (i.e., the time between successive points) and the total length of the time series. The total length is given by N, the number of samples, or equivalently by the total length in time T = N t. (Note that the time interval from the first to the last sample is (N 1) t.) The total length T of the time series determines the frequency resolution f = 1 T = 1 (1) N t This tells us how close in frequency two signal components can be and still be (just) recognized as two components. The sampling interval determines the sampling frequency f s = 1/ t and the Nyquist frequency f Ny = 1 2 f s = 1 2 t 5 (2)
Figure 5. Aliasing: in the calculated amplitude spectrum of a sampled signal it is not possible to know which of the peaks A represents the true oscillation frequency (and similarly for the peaks at B). which is half the sampling frequency. The significance of the Nyquist frequency is shown by the sampling theorem: If a continuous signal does not contain any frequency components above f Ny, then no information is lost by the sampling. Conversely, if the signal does contain components at frequencies f > f Ny, then in the sampled time series it is not possible to distinguish such components from other components with f < f Ny. This phenomenon is called aliasing. In the amplitude spectrum it appears as if the frequency f > f Ny is mirrored in the Nyquist frequency and therefore is seen also at f s f (Fig. 5). The opposite is also true: any real frequency f < f Ny is mirrored at f s f above the Nyquist frequency. As a result, the computed spectrum is always symmetric around f Ny. 7 Calculating spectra using MATLAB Sampling the continuous signal x(t) at t = 0, t, 2 t,..., (N 1) t results in the time series x 0, x 1, x 2,..., x N 1. The amplitude spectrum is given by X k, where X k = 1 N N 1 j=0 x j exp( i2πjk/n) (3) is the discrete Fourier transform calculated for k = 0, 1,..., N 1. The index k can be interpreted as frequency expressed in the number of periods per N points, so that f = k f = k (4) N t In practice Eq. (3) is never used to compute the discrete Fourier transform. Instead, use the Fast Fourier Tranform (FFT), which gives the same result with much less 6
computation (especially if N is large). In MATLAB you can use the function fft (Fig. 6), which however differs from Eq. (3) in two respects: 1. Elements of arrays (vectors) in MATLAB are referenced by an index starting at 1, which the indices j och k in Eq. (3) starts at 0. The frequency components in MATLAB have indices 1, 2,..., N, corresponding to k = 0, 1,..., N 1 in our equations. 2. The factor 1/N in Eq. (3) is not included in the transform calculated by MATLAB. (It is instead included in the inverse FFT, ifft.) To calculate the amplitudes according to Eq. (3) you therefore need to apply the factor 1/N to the FFT returned by MATLAB. 8 Data to be used: Astronomical time series Astronomical objects may vary on very different time scales, from fractions of a second to hundreds of years. Some stars vary considerably in intensity; they are called variable stars. But even apparently stable stars such as our Sun show tiny variations in intensity when measured with high accuracy. Some very different examples of astronomical time series can be found on the web page www.astro.lu.se/ lennart/teaching: sunspot numbers over a few hundred years, the variable star V834 Cen observed in visible light, X-ray radiations for the active nucleus of the galaxy NGC5506, X-ray radiation from the pulsar Her X 1, and solar intensity variations measured with the SOHO satellite. For each object there is a short text file explaining the data. It contains, among other things, information about the number of data points (N) and the sampling interval ( t). 9 Tasks to be performed Examine the time series for the five different objects on the web page indicated above. Calculate their amplitude spectra. For some of the objects several segments of the time series are given. Use any one of them. (The segments are adjacent in time, so that they can be concatenated into a single, long time series. This increases the frequency resolution as discussed in Sect. 5. But it may also be instructive to compute separate amplitude spectra for the different segments and compare them.) For each time series, plot the data values versus time (or plot just a small part of it in order to show the details better). Then plot the amplitude spectrum, or part of it (depending on where the interesting bit is). Make sure that the time axis for the time series and the frequency axis for the spectrum are correctly plotted, using sensible and consistent units. For example, if the time series is shown with a time axis expressed in seconds, then the spectrum should have a frequency axis in Hz. Use convenient units (e.g, seconds, hours, years, and the corresponding frequency units) to avoid very large/small numerical values. 7
Figure 6. The MATLAB manual page for fft. Describe and compare the different time series. In particular, classify each series as periodic, quasi-periodic or stochastic, and motivate the choice. For periodic and quasi-periodic series, estimate the Q value and what it means e.g. in terms of how well the variations can be predicted. Discuss if there are several superposed signal components, and which part of the spectrum is likely to be noise. 10 The written report The written report should contain a brief description of the aims of the exercise and how it was done. For each time series, include at least two diagrams showing (part of) the time series and (part of) the amplitude spectrum, with correctly labelled axes and units. In each case, the time series and spectrum should be concisely described and interpreted in terms of the different signal and noise components that can be discerned in the spectrum. The report can be handed in on paper (Astronomihuset, reception desk on 1st floor), or sent by e-mail as a PDF file (no other format accepted) to lennart@astro.lu.se. 8