ipcv/julia_notebook/dft_fft.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "using FFTW"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dft (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function dft(f)\n",
    "    if ndims(f) == 1\n",
    "        M = length(f)\n",
    "        [sum([x * exp(-2im*pi*u * (idx - 1) / M ) for (idx, x) in enumerate(f)]) for u=0:M-1]\n",
    "    elseif ndims(f) == 2\n",
    "        M, N = size(f)\n",
    "        [sum([f[x+1, y+1] * exp(-2im*pi*(u*x/M + v*y/M)) for x=0:M-1, y=0:N-1]) for u=0:M-1, v=0:N-1]\n",
    "    else\n",
    "        error(\"Wrong dimension\")\n",
    "    end\n",
    "end        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fft2d (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fft1d(array)\n",
    "  #= https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm\n",
    "  #\n",
    "  # Xk = Ek + e^(-2pi*i/N * k) * Ok\n",
    "  #\n",
    "  # Ek = Event DFT\n",
    "  # Ok = Odd DFT\n",
    "  #\n",
    "  =#\n",
    "  N = size(array)[1]\n",
    "  if N == 1\n",
    "    return array\n",
    "  elseif N % 2 != 0\n",
    "    error(\"Wrong dimension\")\n",
    "  end\n",
    "\n",
    "  # people count from zero while julia counts from 1\n",
    "  # either way, it should be the same name\n",
    "  even = fft1d(array[1:2:end])\n",
    "  odd = fft1d(array[2:2:end])\n",
    "  w = exp.(-2im * pi / N * [i for i=0:N/2-1])\n",
    "  half1 = even .+ (odd .* w)\n",
    "  half2 = even .- (odd .* w)\n",
    "  vcat(half1,half2)\n",
    "end\n",
    "\n",
    "function fft2d(matrix)\n",
    "  if ndims(matrix) != 2\n",
    "    error(\"I dont handle other than 2 dimension matrix\")\n",
    "  end\n",
    "  matrix_fft1 = hcat(fft1d.(eachrow(matrix))...)\n",
    "  hcat(fft1d.(eachrow(matrix_fft1))...)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{Complex{Float64},1}:\n",
       " 36.0 + 0.0im               \n",
       " -4.0 + 9.65685424949238im  \n",
       " -4.0 + 4.0im               \n",
       " -4.0 + 1.6568542494923806im\n",
       " -4.0 + 0.0im               \n",
       " -4.0 - 1.6568542494923806im\n",
       " -4.0 - 4.0im               \n",
       " -4.0 - 9.65685424949238im  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "8-element Array{Complex{Float64},1}:\n",
       "                36.0 - 0.0im               \n",
       "                -4.0 + 9.65685424949238im  \n",
       "                -4.0 + 4.0im               \n",
       "                -4.0 + 1.6568542494923797im\n",
       "                -4.0 + 0.0im               \n",
       " -3.9999999999999996 - 1.6568542494923797im\n",
       " -3.9999999999999996 - 4.0im               \n",
       " -3.9999999999999987 - 9.65685424949238im  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "8-element Array{Complex{Float64},1}:\n",
       "                36.0 - 0.0im                 \n",
       "  -4.000000000000003 + 9.65685424949238im    \n",
       "  -4.000000000000002 + 3.9999999999999982im  \n",
       "                -4.0 + 1.656854249492386im   \n",
       "                -4.0 - 3.91886975727153e-15im\n",
       " -4.0000000000000115 - 1.6568542494923912im  \n",
       "  -4.000000000000018 - 4.000000000000005im   \n",
       " -3.9999999999999725 - 9.656854249492369im   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "2×8 Array{Complex{Float64},2}:\n",
       " 72.0+0.0im  -8.0+19.3137im  -8.0+8.0im  …  -8.0-8.0im  -8.0-19.3137im\n",
       "  0.0+0.0im   0.0+0.0im       0.0+0.0im      0.0+0.0im   0.0+0.0im    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "2×8 Array{Complex{Float64},2}:\n",
       " 72.0-0.0im  -8.0+19.3137im  -8.0+8.0im  …  -8.0-8.0im  -8.0-19.3137im\n",
       "  0.0+0.0im   0.0+0.0im       0.0+0.0im      0.0+0.0im   0.0+0.0im    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "2×8 Array{Complex{Float64},2}:\n",
       " 72.0-0.0im          -8.0-7.83774e-15im  …  -8.0-3.39081e-13im\n",
       "  0.0-4.40873e-15im   0.0+4.89859e-16im      0.0-1.87804e-13im"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(fft(1:8))\n",
    "display(fft1d(1:8))\n",
    "display(dft(1:8))\n",
    "\n",
    "\n",
    "display(fft([[1:8...] [1:8...]]'))\n",
    "display(fft2d([[1:8...] [1:8...]]'))\n",
    "display(dft([[1:8...] [1:8...]]'))\n"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Julia 1.2.0-DEV",
   "language": "julia",
   "name": "julia-1.2"
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  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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}