# Discrete Fourier Transform¶

## Description¶

This treatment performs a Discrete Fourier Transform (DFT) on all the given variables of a 3-D finite time-marching result.

The Fourier transform of a continuous-time signal $$x(t)$$ may be defined as:

$$X(f) = \int_{-\infty}^{+\infty} x(t) e^{-j2 \pi ft}dt$$

The Discrete Fourier Transform implemented replaces the infinite integral with a finite sum:

$$X(k) = \frac{1}{N}\sum_{n=0}^{N-1} x(n) e^{-j2 \pi \frac{nk}{N}}dt$$

where $$x(n)$$ is the $$N$$ sampling terms of an analogic signal $$x(t) = x(n\Delta t)$$ and the $$N$$ terms $$X(k)$$ are an approximation of the Fourier transform of this signal at the mode frequency defined as $$f_k = k\Delta f/N = k/T$$.

with:
• the sampling frequency: $$\Delta f = \frac{1}{\Delta t}$$

• the sampling interval: $$T = N \Delta t$$

• the mode $$k = f_k \times T$$

## Construction¶

import antares
myt = antares.Treatment('dft')


## Parameters¶

• base: antares.Base

The base on which the Fourier modes will be computed. It can contain several zones and several instants. DFT is performed on all variables except coordinates.

• type: str, default= ‘mod/phi’

The DFT type of the output data: ‘mod/phi’ for modulus/phase decomposition or ‘im/re’ for imaginery/real part decomposition. The phase is expressed in degrees.

• coordinates: list(str), default= antares.Base.coordinate_names

The variable names that define the set of coordinates. The coordinates will not be computed by the DFT treatment.

• mode: lists(int), default= None

Give one mode or a list of mode ([1, 2, 4] for example). If empty, this returns all the mode including the mean part.

## Preconditions¶

All the zones must have the same instant.

## Postconditions¶

If dtype_in = ‘mod/phi’, the phase is expressed in degrees.

## Example¶

import antares
myt = antares.Treatment('dft')
myt['base'] = base
myt['type'] = 'mod/phi'
myt['mode'] = [4, 18, 36]
dft_modes = myt.execute()


Warning

A mode is defined as $$k=f_k \times T$$, with $$T$$ the sampling interval.

## Main functions¶

class antares.treatment.TreatmentDft.TreatmentDft
execute()

Execute the treatment.

Returns

a base that contains many zones. Each zone contains one instant. Each instant contains two arrays (the FFT parts depending on the type of decomposition).

Return type

antares.Base

## Example¶

"""
This example illustrates the Discrete Fourier Transform
treatment of Antares.
"""
import os
if not os.path.isdir('OUTPUT'):
os.makedirs('OUTPUT')

from antares import Reader, Treatment, Writer

# ------------------
# Reading the files
# ------------------
reader = Reader('bin_tp')
reader['filename'] = os.path.join('..', 'data', 'ROTOR37', 'GENERIC', 'flow_<zone>_<instant>.dat')
base = reader.read()

# ----
# DFT
# ----
treatment = Treatment('dft')
treatment['base'] = base
treatment['type'] = 'mod/phi'
treatment['mode'] = list(range(0, 2))
result = treatment.execute()

# -------------------
# Writing the result
# -------------------
writer = Writer('bin_tp')
writer['filename'] = os.path.join('OUTPUT', 'ex_dft_<instant>.plt')
writer['base'] = result
writer.dump()