{
"cells": [
{
"cell_type": "markdown",
"id": "c247a198",
"metadata": {},
"source": [
"# FFT\n",
"This tutorial illustrates the Fast Fourier Transforms interface in ROOT.\n",
"FFT transform types provided in ROOT:\n",
"\n",
" - \"C2CFORWARD\" - a complex input/output discrete Fourier transform (DFT)\n",
"in one or more dimensions, -1 in the exponent\n",
" - \"C2CBACKWARD\"- a complex input/output discrete Fourier transform (DFT)\n",
" in one or more dimensions, +1 in the exponent\n",
" - \"R2C\" - a real-input/complex-output discrete Fourier transform (DFT)\n",
" in one or more dimensions,\n",
" - \"C2R\" - inverse transforms to \"R2C\", taking complex input\n",
" (storing the non-redundant half of a logically Hermitian array)\n",
" to real output\n",
" - \"R2HC\" - a real-input DFT with output in \"halfcomplex\" format,\n",
" i.e. real and imaginary parts for a transform of size n stored as\n",
" r0, r1, r2, ..., rn/2, i(n+1)/2-1, ..., i2, i1\n",
" - \"HC2R\" - computes the reverse of FFTW_R2HC, above\n",
" - \"DHT\" - computes a discrete Hartley transform\n",
"\n",
"\n",
"\n",
"Sine/cosine transforms:\n",
"\n",
" - DCT-I (REDFT00 in FFTW3 notation)\n",
" - DCT-II (REDFT10 in FFTW3 notation)\n",
" - DCT-III(REDFT01 in FFTW3 notation)\n",
" - DCT-IV (REDFT11 in FFTW3 notation)\n",
" - DST-I (RODFT00 in FFTW3 notation)\n",
" - DST-II (RODFT10 in FFTW3 notation)\n",
" - DST-III(RODFT01 in FFTW3 notation)\n",
" - DST-IV (RODFT11 in FFTW3 notation)\n",
"\n",
"First part of the tutorial shows how to transform the histograms\n",
"Second part shows how to transform the data arrays directly\n",
"\n",
"\n",
"\n",
"\n",
"**Author:** Anna Kreshuk, Jens Hoffmann \n",
"This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, May 19, 2026 at 08:24 PM."
]
},
{
"cell_type": "markdown",
"id": "2cec6e6d",
"metadata": {},
"source": [
"Histograms\n",
"=========\n",
"prepare the canvas for drawing"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "d6f2ad82",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:30.919115Z",
"iopub.status.busy": "2026-05-19T20:24:30.918988Z",
"iopub.status.idle": "2026-05-19T20:24:31.267983Z",
"shell.execute_reply": "2026-05-19T20:24:31.267303Z"
}
},
"outputs": [],
"source": [
"TCanvas *myc = new TCanvas(\"myc\", \"Fast Fourier Transform\", 800, 600);\n",
"myc->SetFillColor(45);\n",
"TPad *c1_1 = new TPad(\"c1_1\", \"c1_1\",0.01,0.67,0.49,0.99);\n",
"TPad *c1_2 = new TPad(\"c1_2\", \"c1_2\",0.51,0.67,0.99,0.99);\n",
"TPad *c1_3 = new TPad(\"c1_3\", \"c1_3\",0.01,0.34,0.49,0.65);\n",
"TPad *c1_4 = new TPad(\"c1_4\", \"c1_4\",0.51,0.34,0.99,0.65);\n",
"TPad *c1_5 = new TPad(\"c1_5\", \"c1_5\",0.01,0.01,0.49,0.32);\n",
"TPad *c1_6 = new TPad(\"c1_6\", \"c1_6\",0.51,0.01,0.99,0.32);\n",
"c1_1->Draw();\n",
"c1_2->Draw();\n",
"c1_3->Draw();\n",
"c1_4->Draw();\n",
"c1_5->Draw();\n",
"c1_6->Draw();\n",
"c1_1->SetFillColor(30);\n",
"c1_1->SetFrameFillColor(42);\n",
"c1_2->SetFillColor(30);\n",
"c1_2->SetFrameFillColor(42);\n",
"c1_3->SetFillColor(30);\n",
"c1_3->SetFrameFillColor(42);\n",
"c1_4->SetFillColor(30);\n",
"c1_4->SetFrameFillColor(42);\n",
"c1_5->SetFillColor(30);\n",
"c1_5->SetFrameFillColor(42);\n",
"c1_6->SetFillColor(30);\n",
"c1_6->SetFrameFillColor(42);\n",
"\n",
"c1_1->cd();\n",
"TDirectory::TContext ctx{nullptr}; // Don't register histograms to the current directory"
]
},
{
"cell_type": "markdown",
"id": "30363861",
"metadata": {},
"source": [
"A function to sample"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "1c957286",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:31.269814Z",
"iopub.status.busy": "2026-05-19T20:24:31.269662Z",
"iopub.status.idle": "2026-05-19T20:24:31.475947Z",
"shell.execute_reply": "2026-05-19T20:24:31.475258Z"
}
},
"outputs": [],
"source": [
"TF1 *fsin = new TF1(\"fsin\", \"sin(x)+sin(2*x)+sin(0.5*x)+1\", 0, 4*TMath::Pi());\n",
"fsin->Draw();\n",
"\n",
"Int_t n=25;\n",
"TH1D *hsin = new TH1D(\"hsin\", \"hsin\", n+1, 0, 4*TMath::Pi());\n",
"Double_t x;"
]
},
{
"cell_type": "markdown",
"id": "a1527b75",
"metadata": {},
"source": [
"Fill the histogram with function values"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "c00de51f",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:31.477724Z",
"iopub.status.busy": "2026-05-19T20:24:31.477571Z",
"iopub.status.idle": "2026-05-19T20:24:31.700837Z",
"shell.execute_reply": "2026-05-19T20:24:31.691425Z"
}
},
"outputs": [],
"source": [
"for (Int_t i=0; i<=n; i++){\n",
" x = (Double_t(i)/n)*(4*TMath::Pi());\n",
" hsin->SetBinContent(i+1, fsin->Eval(x));\n",
"}\n",
"hsin->Draw(\"same\");\n",
"fsin->GetXaxis()->SetLabelSize(0.05);\n",
"fsin->GetYaxis()->SetLabelSize(0.05);\n",
"\n",
"c1_2->cd();"
]
},
{
"cell_type": "markdown",
"id": "e253472f",
"metadata": {},
"source": [
"Compute the transform and look at the magnitude of the output"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "49aa6fd2",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:31.702537Z",
"iopub.status.busy": "2026-05-19T20:24:31.702386Z",
"iopub.status.idle": "2026-05-19T20:24:31.908887Z",
"shell.execute_reply": "2026-05-19T20:24:31.907999Z"
}
},
"outputs": [],
"source": [
"TH1 *hm =nullptr;\n",
"TVirtualFFT::SetTransform(nullptr);\n",
"hm = hsin->FFT(hm, \"MAG\");\n",
"hm->SetTitle(\"Magnitude of the 1st transform\");\n",
"hm->Draw();"
]
},
{
"cell_type": "markdown",
"id": "a34e2faf",
"metadata": {},
"source": [
"NOTE: for \"real\" frequencies you have to divide the x-axes range with the range of your function\n",
"(in this case 4*Pi); y-axes has to be rescaled by a factor of 1/SQRT(n) to be right: this is not done automatically!"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "e7271c46",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:31.910436Z",
"iopub.status.busy": "2026-05-19T20:24:31.910303Z",
"iopub.status.idle": "2026-05-19T20:24:32.146918Z",
"shell.execute_reply": "2026-05-19T20:24:32.130393Z"
}
},
"outputs": [],
"source": [
"hm->SetStats(kFALSE);\n",
"hm->GetXaxis()->SetLabelSize(0.05);\n",
"hm->GetYaxis()->SetLabelSize(0.05);\n",
"c1_3->cd();"
]
},
{
"cell_type": "markdown",
"id": "3f3ddbf2",
"metadata": {},
"source": [
"Look at the phase of the output"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "ee93baa5",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:32.152319Z",
"iopub.status.busy": "2026-05-19T20:24:32.152184Z",
"iopub.status.idle": "2026-05-19T20:24:32.369289Z",
"shell.execute_reply": "2026-05-19T20:24:32.361480Z"
}
},
"outputs": [],
"source": [
"TH1 *hp = nullptr;\n",
"hp = hsin->FFT(hp, \"PH\");\n",
"hp->SetTitle(\"Phase of the 1st transform\");\n",
"hp->Draw();\n",
"hp->SetStats(kFALSE);\n",
"hp->GetXaxis()->SetLabelSize(0.05);\n",
"hp->GetYaxis()->SetLabelSize(0.05);"
]
},
{
"cell_type": "markdown",
"id": "4a01d8b5",
"metadata": {},
"source": [
"Look at the DC component and the Nyquist harmonic:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "0ad82ecb",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:32.371055Z",
"iopub.status.busy": "2026-05-19T20:24:32.370886Z",
"iopub.status.idle": "2026-05-19T20:24:32.577417Z",
"shell.execute_reply": "2026-05-19T20:24:32.576700Z"
}
},
"outputs": [],
"source": [
"Double_t re, im;"
]
},
{
"cell_type": "markdown",
"id": "323d41d6",
"metadata": {},
"source": [
"That's the way to get the current transform object:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "bcca01aa",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:32.579396Z",
"iopub.status.busy": "2026-05-19T20:24:32.579255Z",
"iopub.status.idle": "2026-05-19T20:24:32.785663Z",
"shell.execute_reply": "2026-05-19T20:24:32.785034Z"
}
},
"outputs": [],
"source": [
"TVirtualFFT *fft = TVirtualFFT::GetCurrentTransform();\n",
"c1_4->cd();"
]
},
{
"cell_type": "markdown",
"id": "a0f32f90",
"metadata": {},
"source": [
"Use the following method to get just one point of the output"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "c3147209",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:32.788130Z",
"iopub.status.busy": "2026-05-19T20:24:32.787984Z",
"iopub.status.idle": "2026-05-19T20:24:33.017625Z",
"shell.execute_reply": "2026-05-19T20:24:33.017224Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1st transform: DC component: 26.000000\n",
"1st transform: Nyquist harmonic: -0.932840\n"
]
}
],
"source": [
"fft->GetPointComplex(0, re, im);\n",
"printf(\"1st transform: DC component: %f\\n\", re);\n",
"fft->GetPointComplex(n/2+1, re, im);\n",
"printf(\"1st transform: Nyquist harmonic: %f\\n\", re);"
]
},
{
"cell_type": "markdown",
"id": "f788077f",
"metadata": {},
"source": [
"Use the following method to get the full output:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "1a6ef4e8",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:33.042317Z",
"iopub.status.busy": "2026-05-19T20:24:33.042173Z",
"iopub.status.idle": "2026-05-19T20:24:33.250095Z",
"shell.execute_reply": "2026-05-19T20:24:33.248839Z"
}
},
"outputs": [],
"source": [
"Double_t *re_full = new Double_t[n];\n",
"Double_t *im_full = new Double_t[n];\n",
"fft->GetPointsComplex(re_full,im_full);"
]
},
{
"cell_type": "markdown",
"id": "8316c160",
"metadata": {},
"source": [
"Now let's make a backward transform:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "393d75fe",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:33.251804Z",
"iopub.status.busy": "2026-05-19T20:24:33.251657Z",
"iopub.status.idle": "2026-05-19T20:24:33.478003Z",
"shell.execute_reply": "2026-05-19T20:24:33.477280Z"
}
},
"outputs": [],
"source": [
"TVirtualFFT *fft_back = TVirtualFFT::FFT(1, &n, \"C2R M K\");\n",
"fft_back->SetPointsComplex(re_full,im_full);\n",
"fft_back->Transform();\n",
"TH1 *hb = nullptr;"
]
},
{
"cell_type": "markdown",
"id": "58d2ed2c",
"metadata": {},
"source": [
"Let's look at the output"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "d6fdc0e4",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:33.486158Z",
"iopub.status.busy": "2026-05-19T20:24:33.486035Z",
"iopub.status.idle": "2026-05-19T20:24:33.692274Z",
"shell.execute_reply": "2026-05-19T20:24:33.691687Z"
}
},
"outputs": [],
"source": [
"hb = TH1::TransformHisto(fft_back,hb,\"Re\");\n",
"hb->SetTitle(\"The backward transform result\");\n",
"hb->Draw();"
]
},
{
"cell_type": "markdown",
"id": "80f48c4e",
"metadata": {},
"source": [
"NOTE: here you get at the x-axes number of bins and not real values\n",
"(in this case 25 bins has to be rescaled to a range between 0 and 4*Pi;\n",
"also here the y-axes has to be rescaled (factor 1/bins)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "b1d4969f",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:33.693998Z",
"iopub.status.busy": "2026-05-19T20:24:33.693886Z",
"iopub.status.idle": "2026-05-19T20:24:33.929596Z",
"shell.execute_reply": "2026-05-19T20:24:33.929067Z"
}
},
"outputs": [],
"source": [
"hb->SetStats(kFALSE);\n",
"hb->GetXaxis()->SetLabelSize(0.05);\n",
"hb->GetYaxis()->SetLabelSize(0.05);\n",
"delete fft_back;\n",
"fft_back=nullptr;"
]
},
{
"cell_type": "markdown",
"id": "bbfbe053",
"metadata": {},
"source": [
"Data array - same transform\n",
"==========================="
]
},
{
"cell_type": "markdown",
"id": "28a690b8",
"metadata": {},
"source": [
"Allocate an array big enough to hold the transform output\n",
"Transform output in 1d contains, for a transform of size N,\n",
"N/2+1 complex numbers, i.e. 2*(N/2+1) real numbers\n",
"our transform is of size n+1, because the histogram has n+1 bins"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "96da6c56",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:33.959522Z",
"iopub.status.busy": "2026-05-19T20:24:33.959373Z",
"iopub.status.idle": "2026-05-19T20:24:34.161668Z",
"shell.execute_reply": "2026-05-19T20:24:34.160998Z"
}
},
"outputs": [],
"source": [
"Double_t *in = new Double_t[2*((n+1)/2+1)];\n",
"Double_t re_2,im_2;\n",
"for (Int_t i=0; i<=n; i++){\n",
" x = (Double_t(i)/n)*(4*TMath::Pi());\n",
" in[i] = fsin->Eval(x);\n",
"}"
]
},
{
"cell_type": "markdown",
"id": "c15713af",
"metadata": {},
"source": [
"Make our own TVirtualFFT object (using option \"K\")\n",
"Third parameter (option) consists of 3 parts:\n",
"- transform type:\n",
"real input/complex output in our case\n",
"- transform flag:\n",
"the amount of time spent in planning\n",
"the transform (see TVirtualFFT class description)\n",
"- to create a new TVirtualFFT object (option \"K\") or use the global (default)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "94fe9bb1",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:34.163647Z",
"iopub.status.busy": "2026-05-19T20:24:34.163506Z",
"iopub.status.idle": "2026-05-19T20:24:34.369904Z",
"shell.execute_reply": "2026-05-19T20:24:34.369288Z"
}
},
"outputs": [],
"source": [
"Int_t n_size = n+1;\n",
"TVirtualFFT *fft_own = TVirtualFFT::FFT(1, &n_size, \"R2C ES K\");\n",
"if (!fft_own) return;\n",
"fft_own->SetPoints(in);\n",
"fft_own->Transform();"
]
},
{
"cell_type": "markdown",
"id": "ed7d59b1",
"metadata": {},
"source": [
"Copy all the output points:"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "6b8e1f7a",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:34.371899Z",
"iopub.status.busy": "2026-05-19T20:24:34.371780Z",
"iopub.status.idle": "2026-05-19T20:24:34.591040Z",
"shell.execute_reply": "2026-05-19T20:24:34.590299Z"
}
},
"outputs": [],
"source": [
"fft_own->GetPoints(in);"
]
},
{
"cell_type": "markdown",
"id": "ed5dab48",
"metadata": {},
"source": [
"Draw the real part of the output"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "52fa8a71",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:34.592898Z",
"iopub.status.busy": "2026-05-19T20:24:34.592783Z",
"iopub.status.idle": "2026-05-19T20:24:34.799038Z",
"shell.execute_reply": "2026-05-19T20:24:34.798392Z"
}
},
"outputs": [],
"source": [
"c1_5->cd();\n",
"TH1 *hr = nullptr;\n",
"hr = TH1::TransformHisto(fft_own, hr, \"RE\");\n",
"hr->SetTitle(\"Real part of the 3rd (array) transform\");\n",
"hr->Draw();\n",
"hr->SetStats(kFALSE);\n",
"hr->GetXaxis()->SetLabelSize(0.05);\n",
"hr->GetYaxis()->SetLabelSize(0.05);\n",
"c1_6->cd();\n",
"TH1 *him = nullptr;\n",
"him = TH1::TransformHisto(fft_own, him, \"IM\");\n",
"him->SetTitle(\"Im. part of the 3rd (array) transform\");\n",
"him->Draw();\n",
"him->SetStats(kFALSE);\n",
"him->GetXaxis()->SetLabelSize(0.05);\n",
"him->GetYaxis()->SetLabelSize(0.05);\n",
"\n",
"myc->cd();"
]
},
{
"cell_type": "markdown",
"id": "ab034446",
"metadata": {},
"source": [
"Now let's make another transform of the same size\n",
"The same transform object can be used, as the size and the type of the transform\n",
"haven't changed"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "f4b0678c",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:34.800891Z",
"iopub.status.busy": "2026-05-19T20:24:34.800778Z",
"iopub.status.idle": "2026-05-19T20:24:35.007616Z",
"shell.execute_reply": "2026-05-19T20:24:35.007304Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2nd transform: DC component: 29.000000\n",
"2nd transform: Nyquist harmonic: -0.000000\n"
]
}
],
"source": [
"TF1 *fcos = new TF1(\"fcos\", \"cos(x)+cos(0.5*x)+cos(2*x)+1\", 0, 4*TMath::Pi());\n",
"for (Int_t i=0; i<=n; i++){\n",
" x = (Double_t(i)/n)*(4*TMath::Pi());\n",
" in[i] = fcos->Eval(x);\n",
"}\n",
"fft_own->SetPoints(in);\n",
"fft_own->Transform();\n",
"fft_own->GetPointComplex(0, re_2, im_2);\n",
"printf(\"2nd transform: DC component: %f\\n\", re_2);\n",
"fft_own->GetPointComplex(n/2+1, re_2, im_2);\n",
"printf(\"2nd transform: Nyquist harmonic: %f\\n\", re_2);\n",
"delete fft_own;\n",
"delete [] in;\n",
"delete [] re_full;\n",
"delete [] im_full;"
]
},
{
"cell_type": "markdown",
"id": "2023bbd6",
"metadata": {},
"source": [
"Draw all canvases "
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "e60dc06e",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:24:35.026431Z",
"iopub.status.busy": "2026-05-19T20:24:35.026302Z",
"iopub.status.idle": "2026-05-19T20:24:35.253802Z",
"shell.execute_reply": "2026-05-19T20:24:35.253433Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
"
\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%jsroot on\n",
"gROOT->GetListOfCanvases()->Draw()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "ROOT C++",
"language": "c++",
"name": "root"
},
"language_info": {
"codemirror_mode": "text/x-c++src",
"file_extension": ".C",
"mimetype": " text/x-c++src",
"name": "c++"
}
},
"nbformat": 4,
"nbformat_minor": 5
}