One-dimensional Fourier Transform

Solve the following problems.

(1) DFT

  Compute DFT for the following three data.
  After that, compare the results.
  Note that the results of DFT are complex valued functions.

• Data 1: cosine function(32 points, 4 periods, the end points are not duplicated)
• Data 2: sine function(32 points, 4 periods,the end points are not duplicated)
• Data 3: cosine function(33 points, 4 periods, the end points are duplicated)

[File format]
  Each line has three data (x_n, Re{f_n}, Im{f_n}) that are divided by space characters.
  The line number corresponds to the sequential data number, n.
  Lines with begining hash marks (#) are comment lines.
  (You can remove these comment lines.)

(2) Inverse DFT

  Apply an inverse DFT for the result of DFT of Data 2 obtained by the problem (1).
   (The number of points is same to the original one, i.e. 32 points.)

(3) Interpolation using DFT

  Interpolate f(x) for the result of DFT of Data 2.
  The number of points after the interpolation is 128 that is 4 times of the original one.

[Note]

   (1) You must write the codes yourself; i.e. you can not use packages or libraries for Fourier transform.
   (2) In your report, describe the procedure using equations as well as the results of figures.
       Furthermore, show the discussions.
   (3) Append the program code you write.
       (If the structure of codes can be understand easily,
        you can use any kind of computer language, e.g. Fortran, C, Mathematica, etc.)

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