Linear Integral Transforms
Math.NET Numerics currently supports two linear integral transforms: The discrete Fourier transform and the discrete Hartley transform. Both are strongly localized in the frequency spectrum, but while the Fourier transform operates on complex values, the Hartley
transform operates on real values only.
The transforms implement a separate forward and inverse transform method. How the forward and inverse methods are related to each other and what exact definition is to be used can be specified by an additional
Fourier Space: Discrete Fourier Transform and FFT
Wikipedia has an extensive
article on the discrete fourier transform (DFT)
. We provide implementations of the following algorithms:
- Naive Discrete Fourier Transform (DFT): Out-place transform for arbitrary vector lengths. Mainly intended for verifying faster algorithms:
- Radix-2 Fast Fourier Transform (FFT): In-place fast fourier transform for vectors with a power-of-two length (Radix-2):
Furthermore, the Transform
class provides a shortcut for the Bluestein FFT using static methods which are even easier to use:
Code Sample using the Transform class:
// create a complex sample vector of length 96
Complex samples = SignalGenerator.EquidistantInterval(
t => new Complex(1.0 / (t * t + 1.0), t / (t * t + 1.0)),
-16, 16, 96);
// inplace bluestein FFT with default options
- Default: Uses a negative exponent sign in forward transformations, and symmetric scaling (that is, sqrt(1/N) for both forward and inverse transformation). This is the convention used in Maple and is widely accepted in the educational sector (due
to the symmetry).
- AsymmetricScaling: Set this flag to suppress scaling on the forward transformation but scale the inverse transform with 1/N.
- NoScaling: Set this flag to suppress scaling for both forward and inverse transformation. Note that in this case if you apply first the forward and then inverse transformation you won't get back the original signal (by factor N/2).
- InverseExponent: Uses the positive instead of the negative sign in the forward exponent, and the negative (instead of positive) exponent in the inverse transformation.
- Matlab: Use this flag if you need Matlab compatibility. Equals to setting the
AsymmetricScaling flag. This matches the definition used in the
- NumericalRecipes: Use this flag if you need Numerical Recipes compatibility. Equal to setting both the
InverseExponent and the NoScaling flags.
Useful symmetries of the fourier transform:
- h(t) is real valued <=> real part of H(f) is even, imgainary part of H(f) is odd
- h(t) is imaginary valued <=> real part of H(f) is odd, imaginary part of H(f) is even
- h(t) is even <=> H(f) is even
- h(t) is odd <=> H(f) is odd
- h(t) is real-valued even <=> H(f) is real-valued even
- h(t) is real-valued odd <=> H(f) is imaginary-valued odd
- h(t) is imaginary-valued even <=> H(f) is imaginary-valued even
- h(t) is imaginary-valued odd <=> H(f) is real-valued odd
Hartley Space: Discrete Hartley Transform