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Reference Guide

Detailed Description

View in nbviewer Open in SWAN This tutorial illustrates the Fast Fourier Transforms interface in ROOT.

FFT transform types provided in ROOT:

  • "C2CFORWARD" - a complex input/output discrete Fourier transform (DFT) in one or more dimensions, -1 in the exponent
  • "C2CBACKWARD"- a complex input/output discrete Fourier transform (DFT) in one or more dimensions, +1 in the exponent
  • "R2C" - a real-input/complex-output discrete Fourier transform (DFT) in one or more dimensions,
  • "C2R" - inverse transforms to "R2C", taking complex input (storing the non-redundant half of a logically Hermitian array) to real output
  • "R2HC" - a real-input DFT with output in "halfcomplex" format, i.e. real and imaginary parts for a transform of size n stored as r0, r1, r2, ..., rn/2, i(n+1)/2-1, ..., i2, i1
  • "HC2R" - computes the reverse of FFTW_R2HC, above
  • "DHT" - computes a discrete Hartley transform

Sine/cosine transforms:

  • DCT-I (REDFT00 in FFTW3 notation)
  • DCT-II (REDFT10 in FFTW3 notation)
  • DCT-III(REDFT01 in FFTW3 notation)
  • DCT-IV (REDFT11 in FFTW3 notation)
  • DST-I (RODFT00 in FFTW3 notation)
  • DST-II (RODFT10 in FFTW3 notation)
  • DST-III(RODFT01 in FFTW3 notation)
  • DST-IV (RODFT11 in FFTW3 notation)

First part of the tutorial shows how to transform the histograms Second part shows how to transform the data arrays directly

1st transform: DC component: 26.000000
1st transform: Nyquist harmonic: -0.932840
2nd transform: DC component: 29.000000
2nd transform: Nyquist harmonic: -0.000000
#include "TH1D.h"
#include "TVirtualFFT.h"
#include "TF1.h"
#include "TCanvas.h"
#include "TMath.h"
void FFT()
{
// Histograms
// =========
//prepare the canvas for drawing
TCanvas *myc = new TCanvas("myc", "Fast Fourier Transform", 800, 600);
myc->SetFillColor(45);
TPad *c1_1 = new TPad("c1_1", "c1_1",0.01,0.67,0.49,0.99);
TPad *c1_2 = new TPad("c1_2", "c1_2",0.51,0.67,0.99,0.99);
TPad *c1_3 = new TPad("c1_3", "c1_3",0.01,0.34,0.49,0.65);
TPad *c1_4 = new TPad("c1_4", "c1_4",0.51,0.34,0.99,0.65);
TPad *c1_5 = new TPad("c1_5", "c1_5",0.01,0.01,0.49,0.32);
TPad *c1_6 = new TPad("c1_6", "c1_6",0.51,0.01,0.99,0.32);
c1_1->Draw();
c1_2->Draw();
c1_3->Draw();
c1_4->Draw();
c1_5->Draw();
c1_6->Draw();
c1_1->SetFillColor(30);
c1_1->SetFrameFillColor(42);
c1_2->SetFillColor(30);
c1_2->SetFrameFillColor(42);
c1_3->SetFillColor(30);
c1_3->SetFrameFillColor(42);
c1_4->SetFillColor(30);
c1_4->SetFrameFillColor(42);
c1_5->SetFillColor(30);
c1_5->SetFrameFillColor(42);
c1_6->SetFillColor(30);
c1_6->SetFrameFillColor(42);
c1_1->cd();
//A function to sample
TF1 *fsin = new TF1("fsin", "sin(x)+sin(2*x)+sin(0.5*x)+1", 0, 4*TMath::Pi());
fsin->Draw();
Int_t n=25;
TH1D *hsin = new TH1D("hsin", "hsin", n+1, 0, 4*TMath::Pi());
//Fill the histogram with function values
for (Int_t i=0; i<=n; i++){
x = (Double_t(i)/n)*(4*TMath::Pi());
hsin->SetBinContent(i+1, fsin->Eval(x));
}
hsin->Draw("same");
fsin->GetXaxis()->SetLabelSize(0.05);
fsin->GetYaxis()->SetLabelSize(0.05);
c1_2->cd();
//Compute the transform and look at the magnitude of the output
TH1 *hm =0;
hm = hsin->FFT(hm, "MAG");
hm->SetTitle("Magnitude of the 1st transform");
hm->Draw();
//NOTE: for "real" frequencies you have to divide the x-axes range with the range of your function
//(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!
hm->GetXaxis()->SetLabelSize(0.05);
hm->GetYaxis()->SetLabelSize(0.05);
c1_3->cd();
//Look at the phase of the output
TH1 *hp = 0;
hp = hsin->FFT(hp, "PH");
hp->SetTitle("Phase of the 1st transform");
hp->Draw();
hp->GetXaxis()->SetLabelSize(0.05);
hp->GetYaxis()->SetLabelSize(0.05);
//Look at the DC component and the Nyquist harmonic:
Double_t re, im;
//That's the way to get the current transform object:
c1_4->cd();
//Use the following method to get just one point of the output
fft->GetPointComplex(0, re, im);
printf("1st transform: DC component: %f\n", re);
fft->GetPointComplex(n/2+1, re, im);
printf("1st transform: Nyquist harmonic: %f\n", re);
//Use the following method to get the full output:
Double_t *re_full = new Double_t[n];
Double_t *im_full = new Double_t[n];
fft->GetPointsComplex(re_full,im_full);
//Now let's make a backward transform:
TVirtualFFT *fft_back = TVirtualFFT::FFT(1, &n, "C2R M K");
fft_back->SetPointsComplex(re_full,im_full);
fft_back->Transform();
TH1 *hb = 0;
//Let's look at the output
hb = TH1::TransformHisto(fft_back,hb,"Re");
hb->SetTitle("The backward transform result");
hb->Draw();
//NOTE: here you get at the x-axes number of bins and not real values
//(in this case 25 bins has to be rescaled to a range between 0 and 4*Pi;
//also here the y-axes has to be rescaled (factor 1/bins)
hb->GetXaxis()->SetLabelSize(0.05);
hb->GetYaxis()->SetLabelSize(0.05);
delete fft_back;
fft_back=0;
// Data array - same transform
// ===========================
//Allocate an array big enough to hold the transform output
//Transform output in 1d contains, for a transform of size N,
//N/2+1 complex numbers, i.e. 2*(N/2+1) real numbers
//our transform is of size n+1, because the histogram has n+1 bins
Double_t *in = new Double_t[2*((n+1)/2+1)];
Double_t re_2,im_2;
for (Int_t i=0; i<=n; i++){
x = (Double_t(i)/n)*(4*TMath::Pi());
in[i] = fsin->Eval(x);
}
//Make our own TVirtualFFT object (using option "K")
//Third parameter (option) consists of 3 parts:
//- transform type:
// real input/complex output in our case
//- transform flag:
// the amount of time spent in planning
// the transform (see TVirtualFFT class description)
//- to create a new TVirtualFFT object (option "K") or use the global (default)
Int_t n_size = n+1;
TVirtualFFT *fft_own = TVirtualFFT::FFT(1, &n_size, "R2C ES K");
if (!fft_own) return;
fft_own->SetPoints(in);
fft_own->Transform();
//Copy all the output points:
fft_own->GetPoints(in);
//Draw the real part of the output
c1_5->cd();
TH1 *hr = 0;
hr = TH1::TransformHisto(fft_own, hr, "RE");
hr->SetTitle("Real part of the 3rd (array) tranfsorm");
hr->Draw();
hr->GetXaxis()->SetLabelSize(0.05);
hr->GetYaxis()->SetLabelSize(0.05);
c1_6->cd();
TH1 *him = 0;
him = TH1::TransformHisto(fft_own, him, "IM");
him->SetTitle("Im. part of the 3rd (array) transform");
him->Draw();
him->GetXaxis()->SetLabelSize(0.05);
him->GetYaxis()->SetLabelSize(0.05);
myc->cd();
//Now let's make another transform of the same size
//The same transform object can be used, as the size and the type of the transform
//haven't changed
TF1 *fcos = new TF1("fcos", "cos(x)+cos(0.5*x)+cos(2*x)+1", 0, 4*TMath::Pi());
for (Int_t i=0; i<=n; i++){
x = (Double_t(i)/n)*(4*TMath::Pi());
in[i] = fcos->Eval(x);
}
fft_own->SetPoints(in);
fft_own->Transform();
fft_own->GetPointComplex(0, re_2, im_2);
printf("2nd transform: DC component: %f\n", re_2);
fft_own->GetPointComplex(n/2+1, re_2, im_2);
printf("2nd transform: Nyquist harmonic: %f\n", re_2);
delete fft_own;
delete [] in;
delete [] re_full;
delete [] im_full;
}
int Int_t
Definition: RtypesCore.h:45
const Bool_t kFALSE
Definition: RtypesCore.h:101
double Double_t
Definition: RtypesCore.h:59
virtual void SetLabelSize(Float_t size=0.04)
Set size of axis labels.
Definition: TAttAxis.cxx:205
virtual void SetFillColor(Color_t fcolor)
Set the fill area color.
Definition: TAttFill.h:37
void SetFrameFillColor(Color_t color=1)
Definition: TAttPad.h:73
The Canvas class.
Definition: TCanvas.h:23
TVirtualPad * cd(Int_t subpadnumber=0) override
Set current canvas & pad.
Definition: TCanvas.cxx:706
1-Dim function class
Definition: TF1.h:213
TAxis * GetYaxis() const
Get y axis of the function.
Definition: TF1.cxx:2405
virtual void Draw(Option_t *option="")
Draw this function with its current attributes.
Definition: TF1.cxx:1328
virtual Double_t Eval(Double_t x, Double_t y=0, Double_t z=0, Double_t t=0) const
Evaluate this function.
Definition: TF1.cxx:1440
TAxis * GetXaxis() const
Get x axis of the function.
Definition: TF1.cxx:2394
1-D histogram with a double per channel (see TH1 documentation)}
Definition: TH1.h:618
TH1 is the base class of all histogram classes in ROOT.
Definition: TH1.h:58
virtual void SetTitle(const char *title)
See GetStatOverflows for more information.
Definition: TH1.cxx:6680
static TH1 * TransformHisto(TVirtualFFT *fft, TH1 *h_output, Option_t *option)
For a given transform (first parameter), fills the histogram (second parameter) with the transform ou...
Definition: TH1.cxx:9163
static void AddDirectory(Bool_t add=kTRUE)
Sets the flag controlling the automatic add of histograms in memory.
Definition: TH1.cxx:1283
TAxis * GetXaxis()
Get the behaviour adopted by the object about the statoverflows. See EStatOverflows for more informat...
Definition: TH1.h:320
virtual TH1 * FFT(TH1 *h_output, Option_t *option)
This function allows to do discrete Fourier transforms of TH1 and TH2.
Definition: TH1.cxx:3291
TAxis * GetYaxis()
Definition: TH1.h:321
virtual void SetBinContent(Int_t bin, Double_t content)
Set bin content see convention for numbering bins in TH1::GetBin In case the bin number is greater th...
Definition: TH1.cxx:9065
virtual void Draw(Option_t *option="")
Draw this histogram with options.
Definition: TH1.cxx:3074
virtual void SetStats(Bool_t stats=kTRUE)
Set statistics option on/off.
Definition: TH1.cxx:8833
The most important graphics class in the ROOT system.
Definition: TPad.h:26
TVirtualPad * cd(Int_t subpadnumber=0) override
Set Current pad.
Definition: TPad.cxx:604
void Draw(Option_t *option="") override
Draw Pad in Current pad (re-parent pad if necessary).
Definition: TPad.cxx:1300
TVirtualFFT is an interface class for Fast Fourier Transforms.
Definition: TVirtualFFT.h:88
static void SetTransform(TVirtualFFT *fft)
static: set the current transfrom to parameter
virtual void GetPoints(Double_t *data, Bool_t fromInput=kFALSE) const =0
static TVirtualFFT * FFT(Int_t ndim, Int_t *n, Option_t *option)
Returns a pointer to the FFT of requested size and type.
static TVirtualFFT * GetCurrentTransform()
static: return current fgFFT
virtual void SetPoints(const Double_t *data)=0
virtual void SetPointsComplex(const Double_t *re, const Double_t *im)=0
virtual void Transform()=0
virtual void GetPointComplex(Int_t ipoint, Double_t &re, Double_t &im, Bool_t fromInput=kFALSE) const =0
virtual void GetPointsComplex(Double_t *re, Double_t *im, Bool_t fromInput=kFALSE) const =0
Double_t x[n]
Definition: legend1.C:17
const Int_t n
Definition: legend1.C:16
constexpr Double_t Pi()
Definition: TMath.h:37
Authors
Anna Kreshuk, Jens Hoffmann

Definition in file FFT.C.