How to find the Fourier Transform of a list?

Question

I have a list (list) in the form of {x,y} as

list = {{0., 16}, {0.00392157, 19}, {0.00784314, 6}, {0.0117647, 
  5}, {0.0156863, 8}, {0.0196078, 5}, {0.0235294, 6}, {0.027451, 
  7}, {0.0313725, 4}, {0.0352941, 10}, {0.0392157, 7}, {0.0431373, 
  9}, {0.0470588, 8}, {0.0509804, 11}, {0.054902, 9}, {0.0588235, 
  15}, {0.0627451, 10}, {0.0666667, 13}, {0.0705882, 16}, {0.0745098, 
  21}, {0.0784314, 24}, {0.0823529, 20}, {0.0862745, 25}, {0.0901961, 
  19}, {0.0941176, 24}, {0.0980392, 27}, {0.101961, 22}, {0.105882, 
  23}, {0.109804, 29}, {0.113725, 28}, {0.117647, 28}, {0.121569, 
  37}, {0.12549, 41}, {0.129412, 44}, {0.133333, 41}, {0.137255, 
  44}, {0.141176, 43}, {0.145098, 22}, {0.14902, 50}, {0.152941, 
  39}, {0.156863, 59}, {0.160784, 48}, {0.164706, 58}, {0.168627, 
  45}, {0.172549, 56}, {0.176471, 52}, {0.180392, 70}, {0.184314, 
  53}, {0.188235, 61}, {0.192157, 61}, {0.196078, 58}, {0.2, 
  41}, {0.203922, 47}, {0.207843, 53}, {0.211765, 59}, {0.215686, 
  54}, {0.219608, 58}, {0.223529, 63}, {0.227451, 55}, {0.231373, 59},
  {0.235294, 77}, {0.239216, 72}, {0.243137, 66}, {0.247059, 
  70}, {0.25098, 53}, {0.254902, 61}, {0.258824, 67}, {0.262745, 
  58}, {0.266667, 54}, {0.270588, 51}, {0.27451, 57}, {0.278431, 
  62}, {0.282353, 57}, {0.286275, 63}, {0.290196, 59}, {0.294118, 
  64}, {0.298039, 50}, {0.301961, 46}, {0.305882, 52}, {0.309804, 
  58}, {0.313725, 51}, {0.317647, 39}, {0.321569, 63}, {0.32549, 
  54}, {0.329412, 41}, {0.333333, 48}, {0.337255, 48}, {0.341176, 
  47}, {0.345098, 56}, {0.34902, 42}, {0.352941, 65}, {0.356863, 
  46}, {0.360784, 36}, {0.364706, 45}, {0.368627, 41}, {0.372549, 
  43}, {0.376471, 34}, {0.380392, 43}, {0.384314, 37}, {0.388235, 
  40}, {0.392157, 34}, {0.396078, 32}, {0.4, 32}, {0.403922, 
  50}, {0.407843, 40}, {0.411765, 31}, {0.415686, 34}, {0.419608, 
  45}, {0.423529, 37}, {0.427451, 28}, {0.431373, 44}, {0.435294, 
  28}, {0.439216, 32}, {0.443137, 24}, {0.447059, 24}, {0.45098, 
  28}, {0.454902, 34}, {0.458824, 38}, {0.462745, 36}, {0.466667, 
  25}, {0.470588, 31}, {0.47451, 15}, {0.478431, 24}, {0.482353, 
  27}, {0.486275, 20}, {0.490196, 24}, {0.494118, 31}, {0.498039, 
  19}, {0.501961, 11}, {0.505882, 25}, {0.509804, 25}, {0.513725, 
  32}, {0.517647, 17}, {0.521569, 17}, {0.52549, 21}, {0.529412, 
  25}, {0.533333, 16}, {0.537255, 16}, {0.541176, 20}, {0.545098, 
  17}, {0.54902, 17}, {0.552941, 16}, {0.556863, 19}, {0.560784, 
  22}, {0.564706, 14}, {0.568627, 18}, {0.572549, 16}, {0.576471, 
  20}, {0.580392, 12}, {0.584314, 15}, {0.588235, 17}, {0.592157, 
  5}, {0.596078, 17}, {0.6, 11}, {0.603922, 19}, {0.607843, 
  6}, {0.611765, 8}, {0.615686, 9}, {0.619608, 10}, {0.623529, 
  10}, {0.627451, 9}, {0.631373, 9}, {0.635294, 5}, {0.639216, 
  5}, {0.643137, 7}, {0.647059, 9}, {0.65098, 7}, {0.654902, 
  7}, {0.658824, 10}, {0.662745, 5}, {0.666667, 6}, {0.670588, 
  6}, {0.67451, 6}, {0.678431, 4}, {0.682353, 8}, {0.686275, 
  6}, {0.690196, 3}, {0.694118, 2}, {0.698039, 8}, {0.701961, 
  3}, {0.705882, 6}, {0.709804, 5}, {0.713725, 3}, {0.717647, 
  1}, {0.721569, 3}, {0.72549, 1}, {0.729412, 5}, {0.733333, 
  1}, {0.737255, 1}, {0.741176, 2}, {0.745098, 2}, {0.74902, 
  3}, {0.752941, 2}, {0.756863, 1}, {0.760784, 1}, {0.764706, 
  2}, {0.768627, 1}, {0.776471, 1}, {0.784314, 1}, {0.788235, 
  1}, {0.792157, 1}, {0.8, 1}, {0.807843, 1}, {0.827451, 
  1}, {0.835294, 1}, {0.870588, 1}};

Since the x values are not equally spaced, I am not sure whether I can extract the frequency components using Fourier.

How can I find the frequency components present in the list?

Try to interpolate your data in an equally spaced grid, now you can calculate the FourierSeries. Or you can approximate(least square) your data by a trigonometric polynom... — Ulrich Neumann, May 25 '18 at 13:50
@UlrichNeumann Yes, I thought about interpolation. Are there any other alternatives? — user36426, May 25 '18 at 13:55
I think interpolation is the fastest way(see Hugh's answer). What's your purpose using FourierSeries for your data which doesn't look "periodic"? — Ulrich Neumann, May 25 '18 at 14:02
@UlrichNeumann I have a set of such lists and I want to compare them by looking at their frequency contents. — user36426, May 25 '18 at 14:06
See also: https://en.wikipedia.org/wiki/Non-uniform_discrete_Fourier_transform — Daniel Lichtblau, May 26 '18 at 16:20

score 2 · Answer 1 · answered May 25 '18 at 13:54

Here is a way using Interpolation.

f = Interpolation[list];
{x1, x2} = list[[All, 1]][[{1, -1}]];
inc = (x2 - x1)/(Length[list] - 1);

Then your list with equal x increments is

newList = Table[{x, f[x]}, {x, x1, x2, inc}];
ListLinePlot[{list, newList}]

Mathematica graphics

Is the newList a good enough approximation for you? It looks very noisy. Also it does not have a definite periodic content so I don't know the purpose of using Fourier. Is this for smoothing? Anyway

ft = Fourier[newList[[All, 2]], FourierParameters -> {-1, -1}];
ListLinePlot[Abs[ft]]

Mathematica graphics

See here for notes on using Fourier and getting axes etc.

Hope that helps.

How to find the Fourier Transform of a list?

1 Answers1