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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

**Naive Discrete Fourier Transform (DFT):**Out-place transform for arbitrary vector lengths. Mainly intended for verifying faster algorithms:*NaiveForward*,*NaiveInverse*

**Radix-2 Fast Fourier Transform (FFT):**In-place fast fourier transform for vectors with a power-of-two length (Radix-2):*Radix2Forward*,*Radix2Inverse*

**Bluestein Fast Fourier Transform (FFT):**In-place fast fourier transform for arbitrary vector lengths:*BluesteinForward*,*BluesteinInverse*

Furthermore, the

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 Transform.FourierForward(samples);

Fourier 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 wikipedia article.**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

Last edited Jul 20, 2012 at 2:28 PM by cdrnet, version 1