1.(a) Let f~(w) be the Fourer transform of f(t). What is the Fourier transform of (i) f(t - t0) ; (ii) eipt f(t); (iii) f(t/s) ?
[Just use the definition and changes of variable]
(b) By straightforward integration, what are the Fourier transforms of (i) f(t)=1, |t|<T, f(t)=0, |t|>T; (ii) e-a|t| ? (iii) What (by (ii) or otherwise) is the Fourier transform of 1/( t2+τ2) ?
(c) The Fourer transform of f(t) = exp(- 1/2 t2/a2) is (2π)1/2 a exp(- 1/2 w2 a2). Confirm this numerically using numerical Fourier Transform (eg. MatLab fft) imposing period T>10a on f(t) and choosing discrete time interval Δt<a/10.
[You will need to think carefully how you make f(t) periodic such that negative values of t are adequately represented; exploiting part (a) results is one way but not the simplest. I hope to see a labelled comparison between expected and computed Fourier transforms. I would prefer to see axes labelled with scales expressed in terms of a , but if you canot rise to that then declare a particular value of a. ]
2. (a) The file data1.txt contains a clean time series of some 100000 points. You can presume it is statistically stationary and (for simplicity!) that periodic boundary conditions are appropriate. (i) Produce an estimated graph of its power spectrum, with statistical errors assuming this to be a smooth function. (ii) Produce a graph of !!the significant part of!! its autocrrelation function.
(b) [not for credit] Let f~(w) be the Fourier transform of f(t). Show from the Convolution Theorem that: the Fourier transform of the Autocorrelation function is the Power Spectrum |f~(w)|2, finding in the process the correct form for the autocorrelation function when f(t) is complex valued. [Hint: What function has Fourier transform f~(w)* ? ]
3. The following link will give you access to the time series of X-ray emission by a particular source in the sky. Your task is to estimate the autocovariance of the emission of this X-ray source. [Autocovariance = autocorrelation of (signal with average subtracted).]
Link: http://xte.mit.edu/asmlc/srcs/x1705-440.html In its Data window please leave the days blank and select: "One-day Average Light Curve" and for data columns just "Sum Band Intensity" then click "Retrieve...". You should end up with a text file in which the SECOND number on each line is the X-ray intensity for each successive day (averaged over that day), for each of 4238 successive days.
(a) Produce a simple graph of the series with time average removed.
FOR PARTS (b) and (c) plot your graphs over a limited range of time chosen to bring out the meaningful results and comparison. Error bars not required.
(b) Compute and compare graphically the autocovariance obtained by: (i) directly imposing periodic conditions on the time dependence; (ii) padding the data with zeros first; (iii) as (ii) but dividing by the autocorrelation of the window; (iv) [not for credit] as (iii) but using a smoother window function of your choice.
(c) For ONE of the above choices, compare graphically the autocovariance functions of different subintervals of the data (for example in percent, 0-50, 10-60, 20-70 .....50-100).
(d) Based on the above, discuss briefly how much you are confident of about the true autocovaiance of the source. Alternatively, produce a graph of the autocovariance with error bars.