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

**Author:** Anna Kreshuk, Jens Hoffmann

*This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, August 16, 2022 at 09:32 AM.*

prepare the canvas for drawing

In [1]:

```
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();
TH1::AddDirectory(kFALSE);
```

A function to sample

In [2]:

```
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());
Double_t x;
```

Fill the histogram with function values

In [3]:

```
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

In [4]:

```
TH1 *hm =0;
TVirtualFFT::SetTransform(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!

In [5]:

```
hm->SetStats(kFALSE);
hm->GetXaxis()->SetLabelSize(0.05);
hm->GetYaxis()->SetLabelSize(0.05);
c1_3->cd();
```

Look at the phase of the output

In [6]:

```
TH1 *hp = 0;
hp = hsin->FFT(hp, "PH");
hp->SetTitle("Phase of the 1st transform");
hp->Draw();
hp->SetStats(kFALSE);
hp->GetXaxis()->SetLabelSize(0.05);
hp->GetYaxis()->SetLabelSize(0.05);
```

Look at the DC component and the Nyquist harmonic:

In [7]:

```
Double_t re, im;
```

That's the way to get the current transform object:

In [8]:

```
TVirtualFFT *fft = TVirtualFFT::GetCurrentTransform();
c1_4->cd();
```

Use the following method to get just one point of the output

In [9]:

```
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:

In [10]:

```
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:

In [11]:

```
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

In [12]:

```
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)

In [13]:

```
hb->SetStats(kFALSE);
hb->GetXaxis()->SetLabelSize(0.05);
hb->GetYaxis()->SetLabelSize(0.05);
delete fft_back;
fft_back=0;
```

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

In [14]:

```
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)

In [15]:

```
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:

In [16]:

```
fft_own->GetPoints(in);
```

Draw the real part of the output

In [17]:

```
c1_5->cd();
TH1 *hr = 0;
hr = TH1::TransformHisto(fft_own, hr, "RE");
hr->SetTitle("Real part of the 3rd (array) transform");
hr->Draw();
hr->SetStats(kFALSE);
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->SetStats(kFALSE);
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

In [18]:

```
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;
```

Draw all canvases

In [19]:

```
%jsroot on
gROOT->GetListOfCanvases()->Draw()
```