Convolutions Corelations & FFT.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import signal as sg\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = b = np.arange(25).reshape((5, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8,  9],\n",
       "       [10, 11, 12, 13, 14],\n",
       "       [15, 16, 17, 18, 19],\n",
       "       [20, 21, 22, 23, 24]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8,  9],\n",
       "       [10, 11, 12, 13, 14],\n",
       "       [15, 16, 17, 18, 19],\n",
       "       [20, 21, 22, 23, 24]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2D Convolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   0,    0,    1,    4,   10,   20,   25,   24,   16],\n",
       "       [   0,   10,   32,   68,  120,  140,  140,  118,   72],\n",
       "       [  25,   80,  168,  292,  455,  460,  420,  332,  193],\n",
       "       [ 100,  260,  484,  776, 1140, 1080,  940,  716,  404],\n",
       "       [ 250,  600, 1055, 1620, 2300, 2100, 1775, 1320,  730],\n",
       "       [ 500, 1100, 1804, 2616, 3540, 3080, 2500, 1796,  964],\n",
       "       [ 625, 1340, 2148, 3052, 4055, 3460, 2760, 1952, 1033],\n",
       "       [ 600, 1270, 2012, 2828, 3720, 3140, 2480, 1738,  912],\n",
       "       [ 400,  840, 1321, 1844, 2410, 2020, 1585, 1104,  576]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.convolve2d(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2D Corelation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   0,   24,   71,  140,  230,  220,  191,  144,   80],\n",
       "       [ 120,  278,  472,  700,  960,  820,  652,  458,  240],\n",
       "       [ 335,  712, 1128, 1580, 2065, 1700, 1308,  892,  455],\n",
       "       [ 620, 1276, 1964, 2680, 3420, 2760, 2084, 1396,  700],\n",
       "       [ 950, 1920, 2905, 3900, 4900, 3900, 2905, 1920,  950],\n",
       "       [ 700, 1396, 2084, 2760, 3420, 2680, 1964, 1276,  620],\n",
       "       [ 455,  892, 1308, 1700, 2065, 1580, 1128,  712,  335],\n",
       "       [ 240,  458,  652,  820,  960,  700,  472,  278,  120],\n",
       "       [  80,  144,  191,  220,  230,  140,   71,   24,    0]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.correlate2d(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### For 3D Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = b = np.arange(125).reshape((5, 5, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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       "             16],\n",
       "        [     0,     10,     32,     68,    120,    140,    140,    118,\n",
       "             72],\n",
       "        [    25,     80,    168,    292,    455,    460,    420,    332,\n",
       "            193],\n",
       "        [   100,    260,    484,    776,   1140,   1080,    940,    716,\n",
       "            404],\n",
       "        [   250,    600,   1055,   1620,   2300,   2100,   1775,   1320,\n",
       "            730],\n",
       "        [   500,   1100,   1804,   2616,   3540,   3080,   2500,   1796,\n",
       "            964],\n",
       "        [   625,   1340,   2148,   3052,   4055,   3460,   2760,   1952,\n",
       "           1033],\n",
       "        [   600,   1270,   2012,   2828,   3720,   3140,   2480,   1738,\n",
       "            912],\n",
       "        [   400,    840,   1321,   1844,   2410,   2020,   1585,   1104,\n",
       "            576]],\n",
       "\n",
       "       [[     0,     50,    152,    308,    520,    540,    500,    398,\n",
       "            232],\n",
       "        [   250,    620,   1114,   1736,   2490,   2280,   1930,   1436,\n",
       "            794],\n",
       "        [   800,   1810,   3036,   4484,   6160,   5420,   4440,   3214,\n",
       "           1736],\n",
       "        [  1700,   3720,   6068,   8752,  11780,  10160,   8180,   5832,\n",
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       "        [  2950,   6140,   9574,  13256,  17190,  14280,  11110,   7676,\n",
       "           3974],\n",
       "        [  1800,   3730,   5792,   7988,  10320,   8540,   6620,   4558,\n",
       "           2352]],\n",
       "\n",
       "       [[   625,   1400,   2328,   3412,   4655,   4060,   3300,   2372,\n",
       "           1273],\n",
       "        [  2000,   4330,   6996,  10004,  13360,  11420,   9120,   6454,\n",
       "           3416],\n",
       "        [  4200,   8940,  14229,  20076,  26490,  22380,  17685,  12396,\n",
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       "        [  4825,   9920,  15288,  20932,  26855,  22060,  16980,  11612,\n",
       "           5953]],\n",
       "\n",
       "       [[  2500,   5300,   8404,  11816,  15540,  13080,  10300,   7196,\n",
       "           3764],\n",
       "        [  6500,  13640,  21428,  29872,  38980,  32560,  25460,  17672,\n",
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       "        [ 19400,  40240,  62536,  86304, 111560,  92320,  71560,  49264,\n",
       "          25416],\n",
       "        [ 28500,  58900,  91220, 125480, 161700, 133400, 103100,  70780,\n",
       "          36420],\n",
       "        [ 27000,  55600,  85816, 117664, 151160, 124320,  95800,  65584,\n",
       "          33656],\n",
       "        [ 23500,  48260,  74292, 101608, 130220, 106840,  82140,  56108,\n",
       "          28732],\n",
       "        [ 17900,  36680,  56348,  76912,  98380,  80560,  61820,  42152,\n",
       "          21548],\n",
       "        [ 10100,  20660,  31684,  43176,  55140,  45080,  34540,  23516,\n",
       "          12004]],\n",
       "\n",
       "       [[  6250,  13000,  20255,  28020,  36300,  30100,  23375,  16120,\n",
       "           8330],\n",
       "        [ 15000,  31050,  48160,  66340,  85600,  70700,  54700,  37590,\n",
       "          19360],\n",
       "        [ 26375,  54400,  84090, 115460, 148525, 122300,  94350,  64660,\n",
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       "        [ 40500,  83300, 128420, 175880, 225700, 185400, 142700,  97580,\n",
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       "        [ 57500, 118000, 181525, 248100, 317750, 260500, 200125, 136600,\n",
       "          69900],\n",
       "        [ 52500, 107500, 165020, 225080, 287700, 235400, 180500, 122980,\n",
       "          62820],\n",
       "        [ 44375,  90700, 138990, 189260, 241525, 197300, 151050, 102760,\n",
       "          52415],\n",
       "        [ 33000,  67350, 103060, 140140, 178600, 145700, 111400,  75690,\n",
       "          38560],\n",
       "        [ 18250,  37200,  56855,  77220,  98300,  80100,  61175,  41520,\n",
       "          21130]],\n",
       "\n",
       "       [[ 12500,  25500,  39004,  53016,  67540,  55080,  42100,  28596,\n",
       "          14564],\n",
       "        [ 27500,  56040,  85628, 116272, 147980, 120560,  92060,  62472,\n",
       "          31788],\n",
       "        [ 45100,  91820, 140172, 190168, 241820, 196840, 150180, 101828,\n",
       "          51772],\n",
       "        [ 65400, 133040, 202936, 275104, 349560, 284320, 216760, 146864,\n",
       "          74616],\n",
       "        [ 88500, 179900, 274220, 371480, 471700, 383400, 292100, 197780,\n",
       "         100420],\n",
       "        [ 77000, 156400, 238216, 322464, 409160, 332320, 253000, 171184,\n",
       "          86856],\n",
       "        [ 62500, 126860, 193092, 261208, 331220, 268840, 204540, 138308,\n",
       "          70132],\n",
       "        [ 44900,  91080, 138548, 187312, 237380, 192560, 146420,  98952,\n",
       "          50148],\n",
       "        [ 24100,  48860,  74284, 100376, 127140, 103080,  78340,  52916,\n",
       "          26804]],\n",
       "\n",
       "       [[ 15625,  31700,  48228,  65212,  82655,  67060,  51000,  34472,\n",
       "          17473],\n",
       "        [ 33500,  67930, 103296, 139604, 176860, 143420, 109020,  73654,\n",
       "          37316],\n",
       "        [ 53700, 108840, 165429, 223476, 282990, 229380, 174285, 117696,\n",
       "          59604],\n",
       "        [ 76300, 154580, 234852, 317128, 401420, 325240, 247020, 166748,\n",
       "          84412],\n",
       "        [101375, 205300, 311790, 420860, 532525, 431300, 327450, 220960,\n",
       "         111815],\n",
       "        [ 86500, 175100, 265812, 358648, 453620, 367240, 278700, 187988,\n",
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       "        [ 69000, 139620, 211869, 285756, 361290, 292380, 221805, 149556,\n",
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       "        [ 48800,  98710, 149736, 201884, 255160, 206420, 156540, 105514,\n",
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       "\n",
       "       [[ 15000,  30350,  46052,  62108,  78520,  63540,  48200,  32498,\n",
       "          16432],\n",
       "        [ 31750,  64220,  97414, 131336, 165990, 134280, 101830,  68636,\n",
       "          34694],\n",
       "        [ 50300, 101710, 154236, 207884, 262660, 212420, 161040, 108514,\n",
       "          54836],\n",
       "        [ 70700, 142920, 216668, 291952, 368780, 298160, 225980, 152232,\n",
       "          76908],\n",
       "        [ 93000, 187950, 284860, 383740, 484600, 391700, 296800, 199890,\n",
       "         100960],\n",
       "        [ 78500, 158600, 240308, 323632, 408580, 330160, 250100, 168392,\n",
       "          85028],\n",
       "        [ 62000, 125230, 189696, 255404, 322360, 260420, 197220, 132754,\n",
       "          67016],\n",
       "        [ 43450,  87740, 132874, 178856, 225690, 182280, 138010,  92876,\n",
       "          46874],\n",
       "        [ 22800,  46030,  69692,  93788, 118320,  95540,  72320,  48658,\n",
       "          24552]],\n",
       "\n",
       "       [[ 10000,  20200,  30601,  41204,  52010,  42020,  31825,  21424,\n",
       "          10816],\n",
       "        [ 21000,  42410,  64232,  86468, 109120,  88140,  66740,  44918,\n",
       "          22672],\n",
       "        [ 33025,  66680, 100968, 135892, 171455, 138460, 104820,  70532,\n",
       "          35593],\n",
       "        [ 46100,  93060, 140884, 189576, 239140, 193080, 146140,  98316,\n",
       "          49604],\n",
       "        [ 60250, 121600, 184055, 247620, 312300, 252100, 190775, 128320,\n",
       "          64730],\n",
       "        [ 50500, 101900, 154204, 207416, 261540, 211080, 159700, 107396,\n",
       "          54164],\n",
       "        [ 39625,  79940, 120948, 162652, 205055, 165460, 125160,  84152,\n",
       "          42433],\n",
       "        [ 27600,  55670,  84212, 113228, 142720, 115140,  87080,  58538,\n",
       "          29512],\n",
       "        [ 14400,  29040,  43921,  59044,  74410,  60020,  45385,  30504,\n",
       "          15376]]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.convolve(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[     0,    124,    371,    740,   1230,   1220,   1091,    844,\n",
       "            480],\n",
       "        [   620,   1478,   2572,   3900,   5460,   4820,   3952,   2858,\n",
       "           1540],\n",
       "        [  1835,   4012,   6528,   9380,  12565,  10700,   8508,   5992,\n",
       "           3155],\n",
       "        [  3620,   7676,  12164,  17080,  22420,  18760,  14684,  10196,\n",
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       "        [  4955,  10192,  15708,  21500,  27565,  22580,  17328,  11812,\n",
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       "        [  2080,   4244,   6491,   8820,  11230,   9140,   6971,   4724,\n",
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       "\n",
       "       [[  3100,   6398,   9892,  13580,  17460,  14340,  11032,   7538,\n",
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       "\n",
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       "\n",
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       "\n",
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       "        [ 55340, 110548, 165612, 220520, 275260, 219800, 164532, 109468,\n",
       "          54620],\n",
       "        [ 73480, 146704, 219656, 292320, 364680, 291040, 217736, 144784,\n",
       "          72200],\n",
       "        [ 91300, 182180, 272620, 362600, 452100, 360600, 269620, 179180,\n",
       "          89300],\n",
       "        [ 71800, 143184, 214136, 284640, 354680, 282720, 211256, 140304,\n",
       "          69880],\n",
       "        [ 52820, 105268, 157332, 209000, 260260, 207320, 154812, 102748,\n",
       "          51140],\n",
       "        [ 34460,  68632, 102508, 136080, 169340, 134800, 100588,  66712,\n",
       "          33180],\n",
       "        [ 16820,  33476,  49964,  66280,  82420,  65560,  48884,  32396,\n",
       "          16100]],\n",
       "\n",
       "       [[ 12275,  24472,  36588,  48620,  60565,  48260,  36048,  23932,\n",
       "          11915],\n",
       "        [ 24160,  48134,  71916,  95500, 118880,  94660,  70656,  46874,\n",
       "          23320],\n",
       "        [ 35580,  70836, 105759, 140340, 174570, 138900, 103599,  68676,\n",
       "          34140],\n",
       "        [ 46460,  92428, 137892, 182840, 227260, 180680, 134652,  89188,\n",
       "          44300],\n",
       "        [ 56725, 112760, 168090, 222700, 276575, 219700, 163590, 108260,\n",
       "          53725],\n",
       "        [ 43700,  86788, 129252, 171080, 212260, 168440, 125292,  82828,\n",
       "          41060],\n",
       "        [ 31440,  62376,  92799, 122700, 152070, 120540,  89559,  59136,\n",
       "          29280],\n",
       "        [ 20020,  39674,  58956,  77860,  96380,  76300,  56616,  37334,\n",
       "          18460],\n",
       "        [  9515,  18832,  27948,  36860,  45565,  36020,  26688,  17572,\n",
       "           8675]],\n",
       "\n",
       "       [[  6700,  13298,  19792,  26180,  32460,  25740,  19132,  12638,\n",
       "           6260],\n",
       "        [ 12890,  25556,  37994,  50200,  62170,  49240,  36554,  24116,\n",
       "          11930],\n",
       "        [ 18520,  36674,  54456,  71860,  88880,  70300,  52116,  34334,\n",
       "          16960],\n",
       "        [ 23540,  46552,  69028,  90960, 112340,  88720,  65668,  43192,\n",
       "          21300],\n",
       "        [ 27900,  55090,  81560, 107300, 132300, 104300,  77060,  50590,\n",
       "          24900],\n",
       "        [ 20700,  40792,  60268,  79120,  97340,  76560,  56428,  36952,\n",
       "          18140],\n",
       "        [ 14260,  28034,  41316,  54100,  66380,  52060,  38256,  24974,\n",
       "          12220],\n",
       "        [  8630,  16916,  24854,  32440,  39670,  31000,  22694,  14756,\n",
       "           7190],\n",
       "        [  3860,   7538,  11032,  14340,  17460,  13580,   9892,   6398,\n",
       "           3100]],\n",
       "\n",
       "       [[  2400,   4724,   6971,   9140,  11230,   8820,   6491,   4244,\n",
       "           2080],\n",
       "        [  4420,   8678,  12772,  16700,  20460,  16020,  11752,   7658,\n",
       "           3740],\n",
       "        [  6035,  11812,  17328,  22580,  27565,  21500,  15708,  10192,\n",
       "           4955],\n",
       "        [  7220,  14076,  20564,  26680,  32420,  25160,  18284,  11796,\n",
       "           5700],\n",
       "        [  7950,  15420,  22405,  28900,  34900,  26900,  19405,  12420,\n",
       "           5950],\n",
       "        [  5300,  10196,  14684,  18760,  22420,  17080,  12164,   7676,\n",
       "           3620],\n",
       "        [  3155,   5992,   8508,  10700,  12565,   9380,   6528,   4012,\n",
       "           1835],\n",
       "        [  1540,   2858,   3952,   4820,   5460,   3900,   2572,   1478,\n",
       "            620],\n",
       "        [   480,    844,   1091,   1220,   1230,    740,    371,    124,\n",
       "              0]]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.correlate(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fast Fourier Transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[ 1.50319263e-12,  1.95793998e-13,  1.00000000e+00,\n",
       "          4.00000000e+00,  1.00000000e+01,  2.00000000e+01,\n",
       "          2.50000000e+01,  2.40000000e+01,  1.60000000e+01],\n",
       "        [-5.62118253e-12,  1.00000000e+01,  3.20000000e+01,\n",
       "          6.80000000e+01,  1.20000000e+02,  1.40000000e+02,\n",
       "          1.40000000e+02,  1.18000000e+02,  7.20000000e+01],\n",
       "        [ 2.50000000e+01,  8.00000000e+01,  1.68000000e+02,\n",
       "          2.92000000e+02,  4.55000000e+02,  4.60000000e+02,\n",
       "          4.20000000e+02,  3.32000000e+02,  1.93000000e+02],\n",
       "        [ 1.00000000e+02,  2.60000000e+02,  4.84000000e+02,\n",
       "          7.76000000e+02,  1.14000000e+03,  1.08000000e+03,\n",
       "          9.40000000e+02,  7.16000000e+02,  4.04000000e+02],\n",
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       "          1.58500000e+03,  1.10400000e+03,  5.76000000e+02]],\n",
       "\n",
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       "          1.36200000e+04,  9.45400000e+03,  4.91600000e+03],\n",
       "        [ 2.95000000e+03,  6.14000000e+03,  9.57400000e+03,\n",
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       "          6.62000000e+03,  4.55800000e+03,  2.35200000e+03]],\n",
       "\n",
       "       [[ 6.25000000e+02,  1.40000000e+03,  2.32800000e+03,\n",
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       "          3.30000000e+03,  2.37200000e+03,  1.27300000e+03],\n",
       "        [ 2.00000000e+03,  4.33000000e+03,  6.99600000e+03,\n",
       "          1.00040000e+04,  1.33600000e+04,  1.14200000e+04,\n",
       "          9.12000000e+03,  6.45400000e+03,  3.41600000e+03],\n",
       "        [ 4.20000000e+03,  8.94000000e+03,  1.42290000e+04,\n",
       "          2.00760000e+04,  2.64900000e+04,  2.23800000e+04,\n",
       "          1.76850000e+04,  1.23960000e+04,  6.50400000e+03],\n",
       "        [ 7.30000000e+03,  1.53800000e+04,  2.42520000e+04,\n",
       "          3.39280000e+04,  4.44200000e+04,  3.72400000e+04,\n",
       "          2.92200000e+04,  2.03480000e+04,  1.06120000e+04],\n",
       "        [ 1.13750000e+04,  2.38000000e+04,  3.72900000e+04,\n",
       "          5.18600000e+04,  6.75250000e+04,  5.63000000e+04,\n",
       "          4.39500000e+04,  3.04600000e+04,  1.58150000e+04],\n",
       "        [ 1.15000000e+04,  2.39000000e+04,  3.72120000e+04,\n",
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       "          4.29000000e+04,  2.95880000e+04,  1.52920000e+04],\n",
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       "          3.82050000e+04,  2.62560000e+04,  1.35240000e+04],\n",
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       "          2.09320000e+04,  2.68550000e+04,  2.20600000e+04,\n",
       "          1.69800000e+04,  1.16120000e+04,  5.95300000e+03]],\n",
       "\n",
       "       [[ 2.50000000e+03,  5.30000000e+03,  8.40400000e+03,\n",
       "          1.18160000e+04,  1.55400000e+04,  1.30800000e+04,\n",
       "          1.03000000e+04,  7.19600000e+03,  3.76400000e+03],\n",
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       "          2.54600000e+04,  1.76720000e+04,  9.18800000e+03],\n",
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       "          4.57800000e+04,  3.16280000e+04,  1.63720000e+04],\n",
       "        [ 1.94000000e+04,  4.02400000e+04,  6.25360000e+04,\n",
       "          8.63040000e+04,  1.11560000e+05,  9.23200000e+04,\n",
       "          7.15600000e+04,  4.92640000e+04,  2.54160000e+04],\n",
       "        [ 2.85000000e+04,  5.89000000e+04,  9.12200000e+04,\n",
       "          1.25480000e+05,  1.61700000e+05,  1.33400000e+05,\n",
       "          1.03100000e+05,  7.07800000e+04,  3.64200000e+04],\n",
       "        [ 2.70000000e+04,  5.56000000e+04,  8.58160000e+04,\n",
       "          1.17664000e+05,  1.51160000e+05,  1.24320000e+05,\n",
       "          9.58000000e+04,  6.55840000e+04,  3.36560000e+04],\n",
       "        [ 2.35000000e+04,  4.82600000e+04,  7.42920000e+04,\n",
       "          1.01608000e+05,  1.30220000e+05,  1.06840000e+05,\n",
       "          8.21400000e+04,  5.61080000e+04,  2.87320000e+04],\n",
       "        [ 1.79000000e+04,  3.66800000e+04,  5.63480000e+04,\n",
       "          7.69120000e+04,  9.83800000e+04,  8.05600000e+04,\n",
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       "          4.31760000e+04,  5.51400000e+04,  4.50800000e+04,\n",
       "          3.45400000e+04,  2.35160000e+04,  1.20040000e+04]],\n",
       "\n",
       "       [[ 6.25000000e+03,  1.30000000e+04,  2.02550000e+04,\n",
       "          2.80200000e+04,  3.63000000e+04,  3.01000000e+04,\n",
       "          2.33750000e+04,  1.61200000e+04,  8.33000000e+03],\n",
       "        [ 1.50000000e+04,  3.10500000e+04,  4.81600000e+04,\n",
       "          6.63400000e+04,  8.56000000e+04,  7.07000000e+04,\n",
       "          5.47000000e+04,  3.75900000e+04,  1.93600000e+04],\n",
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       "          1.42700000e+05,  9.75800000e+04,  5.00200000e+04],\n",
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       "          2.48100000e+05,  3.17750000e+05,  2.60500000e+05,\n",
       "          2.00125000e+05,  1.36600000e+05,  6.99000000e+04],\n",
       "        [ 5.25000000e+04,  1.07500000e+05,  1.65020000e+05,\n",
       "          2.25080000e+05,  2.87700000e+05,  2.35400000e+05,\n",
       "          1.80500000e+05,  1.22980000e+05,  6.28200000e+04],\n",
       "        [ 4.43750000e+04,  9.07000000e+04,  1.38990000e+05,\n",
       "          1.89260000e+05,  2.41525000e+05,  1.97300000e+05,\n",
       "          1.51050000e+05,  1.02760000e+05,  5.24150000e+04],\n",
       "        [ 3.30000000e+04,  6.73500000e+04,  1.03060000e+05,\n",
       "          1.40140000e+05,  1.78600000e+05,  1.45700000e+05,\n",
       "          1.11400000e+05,  7.56900000e+04,  3.85600000e+04],\n",
       "        [ 1.82500000e+04,  3.72000000e+04,  5.68550000e+04,\n",
       "          7.72200000e+04,  9.83000000e+04,  8.01000000e+04,\n",
       "          6.11750000e+04,  4.15200000e+04,  2.11300000e+04]],\n",
       "\n",
       "       [[ 1.25000000e+04,  2.55000000e+04,  3.90040000e+04,\n",
       "          5.30160000e+04,  6.75400000e+04,  5.50800000e+04,\n",
       "          4.21000000e+04,  2.85960000e+04,  1.45640000e+04],\n",
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       "          9.20600000e+04,  6.24720000e+04,  3.17880000e+04],\n",
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       "          7.83400000e+04,  5.29160000e+04,  2.68040000e+04]],\n",
       "\n",
       "       [[ 1.56250000e+04,  3.17000000e+04,  4.82280000e+04,\n",
       "          6.52120000e+04,  8.26550000e+04,  6.70600000e+04,\n",
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       "          1.74285000e+05,  1.17696000e+05,  5.96040000e+04],\n",
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       "          3.17128000e+05,  4.01420000e+05,  3.25240000e+05,\n",
       "          2.47020000e+05,  1.66748000e+05,  8.44120000e+04],\n",
       "        [ 1.01375000e+05,  2.05300000e+05,  3.11790000e+05,\n",
       "          4.20860000e+05,  5.32525000e+05,  4.31300000e+05,\n",
       "          3.27450000e+05,  2.20960000e+05,  1.11815000e+05],\n",
       "        [ 8.65000000e+04,  1.75100000e+05,  2.65812000e+05,\n",
       "          3.58648000e+05,  4.53620000e+05,  3.67240000e+05,\n",
       "          2.78700000e+05,  1.87988000e+05,  9.50920000e+04],\n",
       "        [ 6.90000000e+04,  1.39620000e+05,  2.11869000e+05,\n",
       "          2.85756000e+05,  3.61290000e+05,  2.92380000e+05,\n",
       "          2.21805000e+05,  1.49556000e+05,  7.56240000e+04],\n",
       "        [ 4.88000000e+04,  9.87100000e+04,  1.49736000e+05,\n",
       "          2.01884000e+05,  2.55160000e+05,  2.06420000e+05,\n",
       "          1.56540000e+05,  1.05514000e+05,  5.33360000e+04],\n",
       "        [ 2.58250000e+04,  5.22200000e+04,  7.91880000e+04,\n",
       "          1.06732000e+05,  1.34855000e+05,  1.09060000e+05,\n",
       "          8.26800000e+04,  5.57120000e+04,  2.81530000e+04]],\n",
       "\n",
       "       [[ 1.50000000e+04,  3.03500000e+04,  4.60520000e+04,\n",
       "          6.21080000e+04,  7.85200000e+04,  6.35400000e+04,\n",
       "          4.82000000e+04,  3.24980000e+04,  1.64320000e+04],\n",
       "        [ 3.17500000e+04,  6.42200000e+04,  9.74140000e+04,\n",
       "          1.31336000e+05,  1.65990000e+05,  1.34280000e+05,\n",
       "          1.01830000e+05,  6.86360000e+04,  3.46940000e+04],\n",
       "        [ 5.03000000e+04,  1.01710000e+05,  1.54236000e+05,\n",
       "          2.07884000e+05,  2.62660000e+05,  2.12420000e+05,\n",
       "          1.61040000e+05,  1.08514000e+05,  5.48360000e+04],\n",
       "        [ 7.07000000e+04,  1.42920000e+05,  2.16668000e+05,\n",
       "          2.91952000e+05,  3.68780000e+05,  2.98160000e+05,\n",
       "          2.25980000e+05,  1.52232000e+05,  7.69080000e+04],\n",
       "        [ 9.30000000e+04,  1.87950000e+05,  2.84860000e+05,\n",
       "          3.83740000e+05,  4.84600000e+05,  3.91700000e+05,\n",
       "          2.96800000e+05,  1.99890000e+05,  1.00960000e+05],\n",
       "        [ 7.85000000e+04,  1.58600000e+05,  2.40308000e+05,\n",
       "          3.23632000e+05,  4.08580000e+05,  3.30160000e+05,\n",
       "          2.50100000e+05,  1.68392000e+05,  8.50280000e+04],\n",
       "        [ 6.20000000e+04,  1.25230000e+05,  1.89696000e+05,\n",
       "          2.55404000e+05,  3.22360000e+05,  2.60420000e+05,\n",
       "          1.97220000e+05,  1.32754000e+05,  6.70160000e+04],\n",
       "        [ 4.34500000e+04,  8.77400000e+04,  1.32874000e+05,\n",
       "          1.78856000e+05,  2.25690000e+05,  1.82280000e+05,\n",
       "          1.38010000e+05,  9.28760000e+04,  4.68740000e+04],\n",
       "        [ 2.28000000e+04,  4.60300000e+04,  6.96920000e+04,\n",
       "          9.37880000e+04,  1.18320000e+05,  9.55400000e+04,\n",
       "          7.23200000e+04,  4.86580000e+04,  2.45520000e+04]],\n",
       "\n",
       "       [[ 1.00000000e+04,  2.02000000e+04,  3.06010000e+04,\n",
       "          4.12040000e+04,  5.20100000e+04,  4.20200000e+04,\n",
       "          3.18250000e+04,  2.14240000e+04,  1.08160000e+04],\n",
       "        [ 2.10000000e+04,  4.24100000e+04,  6.42320000e+04,\n",
       "          8.64680000e+04,  1.09120000e+05,  8.81400000e+04,\n",
       "          6.67400000e+04,  4.49180000e+04,  2.26720000e+04],\n",
       "        [ 3.30250000e+04,  6.66800000e+04,  1.00968000e+05,\n",
       "          1.35892000e+05,  1.71455000e+05,  1.38460000e+05,\n",
       "          1.04820000e+05,  7.05320000e+04,  3.55930000e+04],\n",
       "        [ 4.61000000e+04,  9.30600000e+04,  1.40884000e+05,\n",
       "          1.89576000e+05,  2.39140000e+05,  1.93080000e+05,\n",
       "          1.46140000e+05,  9.83160000e+04,  4.96040000e+04],\n",
       "        [ 6.02500000e+04,  1.21600000e+05,  1.84055000e+05,\n",
       "          2.47620000e+05,  3.12300000e+05,  2.52100000e+05,\n",
       "          1.90775000e+05,  1.28320000e+05,  6.47300000e+04],\n",
       "        [ 5.05000000e+04,  1.01900000e+05,  1.54204000e+05,\n",
       "          2.07416000e+05,  2.61540000e+05,  2.11080000e+05,\n",
       "          1.59700000e+05,  1.07396000e+05,  5.41640000e+04],\n",
       "        [ 3.96250000e+04,  7.99400000e+04,  1.20948000e+05,\n",
       "          1.62652000e+05,  2.05055000e+05,  1.65460000e+05,\n",
       "          1.25160000e+05,  8.41520000e+04,  4.24330000e+04],\n",
       "        [ 2.76000000e+04,  5.56700000e+04,  8.42120000e+04,\n",
       "          1.13228000e+05,  1.42720000e+05,  1.15140000e+05,\n",
       "          8.70800000e+04,  5.85380000e+04,  2.95120000e+04],\n",
       "        [ 1.44000000e+04,  2.90400000e+04,  4.39210000e+04,\n",
       "          5.90440000e+04,  7.44100000e+04,  6.00200000e+04,\n",
       "          4.53850000e+04,  3.05040000e+04,  1.53760000e+04]]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.fftconvolve(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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