Collaboration is encouraged but please acknowledge it. With the numerics it is important you end up knowing how to do it yourself on your computer. For symbolic questions I expect to see SOME indication of how results are obtained, but complete worked answers not required. These questions are designed to help you understand the material as it is taught - don't just wait for the deadlines to get close!
Please attempt questions 1 and 2 as soon as possible after the relevant lectures (probably Oct 12, 14 ). Skip the not-for-credit items first time through. Deadline for submission to Complexity Office: Wed 20 October 12 noon.
1.(a) Let f~(w) be the Fourer transform of f(t). What is the Fourier transform of (i) f(t - t0) ; [NOT FOR CREDIT: (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; [NOT FOR CREDIT: (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) (i) 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 200000 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)* ? ]
Please attempt questions 3 and 4 as soon as possible in week 3. Deadline for submission to Complexity Office: Wed 27 October 12 noon.
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.
4. 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 denoise this time series by wavelet methods. [Note: in contrast to the example discussed by Dubok de Wit, the noise here is at shorter scales (higher frequncies) and the useful signal at longer scales (lower frequencies).]
Link: http://xte.mit.edu/asmlc/srcs/x0512-401.html In its Data window please leave the days blank and select: "One-day Average Light Curve" and for data columns "Sum Band Intensity" and "Sum Band Uncertainties" 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 4717 successive days, and the THIRD number is an uncertainty in that measurement.
(a) Produce a simple graph of the "Sum Band Intensity" series with its time average removed.
(b) Satisfy yourself that the rms (Root Mean Square) value of the uncertainty is quite close to 0.5. Generate a model error series comprising independent gaussian dsitributed random numbers with mean zero and standard deviation 0.5, and graph this for comparison with (a).
(c) Compare scalograms [intensity maps of the power of the wavelet transform of] your series from (a) and (b) and comment as to which parts of the scalogram of (a) appear significant on this basis. Please show results for the Haar wavelet. Open-endedly: do you find any other wavelet more informative?
(d) Produce a graph of the significant denoised time dependence. Please show result for the Haar wavelet, plus any you find does better.
[Note: I have deliberately made the question answerable using the Haar wavelet alone: it is so simple you could programme its calculation directly (I just did so). However it would be much more productive to master the Wavelet Toolkit in MATLAB and thereby access other wavelet choices.]