Fourier Transforms

Forward Fourier Transform

We apply the Discrete Fourier Transform to decompose the signal and represent it as a sum of periodic components. On decomposition we get a real component and an imaginary component. For DFT on real signals, the frequencies are mirrored around the centre.

In this library, we are able to get all 4 variations:

• Only Magnitude
• Only Phase (in Radians/Degrees)
• Magnitude and Phase (in Radians/Degrees)
• Complex Signal

The examples provided here use this signal:

$\sin(40\pi t) + \frac{1}{2}\sin(400\pi t)$

CODE

_Fourier dft = new DiscreteFourier(signal); //Works well for short signals (<200 points)
OR
_Fourier dft = new FastFourier(signal); //Works well for longer signals (>200 points)
dft.transform();

CODE

boolean onlyPositive = true;
double[] out = dft.getMagnitude(onlyPositive); //Positive Absolute

boolean onlyPositive = true;
double[][] out = dft.getComplex2D(onlyPositive); //Positive Complex

boolean onlyPositive = false;
double[] out = dft.getMagnitude(onlyPositive); //Full Absolute

boolean onlyPositive = false;
double[][] out = dft.getComplex2D(onlyPositive); //Full Complex

Inverse Fourier Transform

We apply the Inverse Discrete Fourier Transform to a complex sequence to reconstruct the original signal. This process also generate a signal with a real and an imaginary component.

In this library, the IDFT works in 2 modes:

• For a real sequence
• For a complex sequence

In the examples provided here the source signal is:

$\sin(40\pi t) + \frac{1}{2}\sin(400\pi t)$

CODE

// Automatically detects real and complex sequence and performs IDFT accordingly
_InverseFourier idft = new InverseDiscreteFourier(complexSequence); //When DFT is performed using DiscreteFourier
OR
_InverseFourier idft = new InverseFastFourier(complexSequence); //When DFT is performed using FasrFourier
idft.transform();

CODE

double[] outReal = idft.getReal(); //To get only the real component
double[] outAbsolute = idft.getMagnitude(); //To get the absolute value of the complex signal
double[][] out = idft.getComplex2D(); // To get the complex signal

The image shows the plot of the complex output. As you can see, the real component represents the original signal: $\sin(40\pi t) + \frac{1}{2}\sin(400\pi t)$