Loading...
 

First, one needs to know the sampling interval and duration of the time series. The sampling interval is the minimum spacing between measurements, and will be denoted \Delta t. The duration of the time series is the difference in time between the first and last data point, and will be denoted T.

Second, one needs to define the frequency domain. We generally work with frequencies, because that is the Fourier conjugate of time. However sometimes it is useful to talk about periods, where the period is P=1/f. The frequency is The maximum frequency of a signal that can be unambiguously identified in a time series is that which will be sampled at least twice during each period. This maximum frequency is called the Nyquist frequency, and is given by f_{\rm Nyq} = 1/(2 \Delta t).

The frequency resolution for independent measurements, and the minimum frequency, is determined by the duration of the time series. Two signals will be independent (that is, they will not interfere with each other), if the first undergoes n cycles during a time T, and the other undergoes at least one more (or one fewer) cycles, i.e., n+1. The periods of these two signals are T/n and T/(n+1), and the frequencies are n/T and (n+1)/T. The difference in the frequencies is (n+1)/T - n/T = 1/T. This gives the frequency resolution. It is also the minimum frequency, or, equivalently, the longest period that goes through one cycle during the time series.

Third, one needs to be aware that noise will produce power in the Fourier domain. This will inevitably limit one's ability to identify weak, periodic signals. In the worst-case scenario, one might be led to believe that noise signals are real, and lead you down a time-wasting path toward numerology. The statistics of signals in white noise is discussed in Groth (1975) and Vaughan et al (1994). However, one should be aware that random, low-frequency noise (often called red noise) could be present in data, either caused by changes in observing conditions or by correlated noise in the astronomical source. If there is a lot of power in your Fourier data, you should be very, very cautious about calling any of it "periodic", or else you are likely to see your work ridiculed on the Internet.

Fourth, one needs to be aware of the window function for the data. The Fourier transform is defined on the domain (-\infty, +\infty), but data will never be taken for that long. The function representing the times when the data are taken (and those when it is not) is referred to as a window function. Mathematically, the window function, w(t), can be thought of as multiplying an infinite time series to obtain the data set, d(t) = w(t)f(t). The Fourier transform of the data is then the Fourier transform of the infinite time series convolved with the Fourier transform of the window function,

{EQUATION()}D(\nu) = \int_{-\infty}^{+\infty} w(t)f(t) e^{-i\pi\nu\t} = W(\nu) * F(\nu).{EQUATION}


(See Wikipedia for properties of the Fourier transform). If the data selected is contiguous, the Fourier transform of a square window function is just a sinc function with a width equal to the frequency resolution. However, if the data is sampled irregularly, as will happen if your telescope can only observe at night, then any periodic signal in the data will also produce spurious signals elsewhere in the frequency domain. In this sense, window function can be nasty. However, engineers often use window functions to improve the response of signal processing devices, so they aren't all bad. For more details on window functions, see Numerical Recipes in C.

Fifth, one needs to be aware that sampling a continuous time series will impact ones ability to detect some signals. Imagine, for instance, that you have a signal with a frequency equal to the Nyquist value, so that you only sample it twice per period. Give it the form \sin(\pi t/\Delta t). If you are unlucky and sample it at times t = k\Delta t, the sampled signal will be d(t_k) = 0. Likewise, one won't be able to sample the full power of signals that aren't at an integer multiple of the lowest frequency (i.e., that don't fall at the center of a Fourier bin). Therefore, sampling the data lowers the sensitvity to some frequencies. See Leahy et al. (1983) and Ransom et al. (2002).

Sixth, in carrying out a Fourier analysis, one is implicitly assuming that there are no signals with periods longer than the time series. If there are long-term variations, one usually tries to remove them by fitting a low-order polynomial to the data, and dividing the data by that polynomial. This is sometimes called de-trending the data.

Finally, if the frequency of the signal you are looking for changes during an observation, it will appear in multiple Fourier bins, and may be hard to identify. If you expect regular variations, as one would from a pulsar in a binary orbit, there are techniques to attempt to correct for these variations. These fall under the rubric of acceleration Searches, such as Vaughan et al. (1994) and Ransom et al. (2005).


Page last modified on Saturday 30 of January, 2010 15:22:48 EST